Model: krishmajumdar/arxiv-finetuned
Author: krishmajumdar
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, dense, generated_from_trainer, dataset_size:1000, loss:MultipleNegativesRankingLoss, dataset_size:75000, arxiv:1908.10084, arxiv:1705.00652, base_model:sentence-transformers/all-MiniLM-L6-v2, base_model:finetune:sentence-transformers/all-MiniLM-L6-v2, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: sentence-transformers/all-MiniLM-L6-v2
Card metadata: tags: sentence-transformers, sentence-similarity, feature-extraction, dense, generated_from_trainer, dataset_size:1000, loss:MultipleNegativesRankingLoss, dataset_size:75000
base_model: sentence-transformers/all-MiniLM-L6-v2
widget: source_sentence: <S> dimer vacancy ( dv ) defect complexes in the si(001)@xmath0 surface were investigated using high - resolution scanning tunneling microscopy and first principles calculations . </S> <S> we find that under low bias filled - state tunneling conditions , isolated ` split - off ' dimers in these defect complexes are imaged as pairs of protrusions while the surrounding si surface dimers appear as the usual `` bean - shaped '' protrusions . </S> <S> we attribute this to the formation of @xmath1-bonds between the two atoms of the split - off dimer and second layer atoms , and present charge density plots to support this assignment . </S> <S> we observe a local brightness enhancement due to strain for different dv complexes and provide the first experimental confirmation of an earlier prediction that the 1 + 2-dv induces less surface strain than other dv complexes </S> <S> . finally , we present a previously unreported triangular shaped split - off dimer defect complex that exists at s@xmath2-type step edges , and propose a structure for this defect involving a bound si monomer . </S>
sentences: any _ in vivo _ quantum system is in a contact with its environment . although typically weak , this interaction becomes relevant when studying the evolution of a system over long time scales . in particular , the asymptotic state of such an _ open _ system depends both on the unitary action induced by the system hamiltonian , and the action of the environment , conventionally called ` dissipation ' . a recent idea of `` engineering by dissipation '' @xcite , the creation of designated pure and highly entangled states of many - body quantum systems by using specially designed dissipative operators , has promoted dissipation to the same level of importance as the underlying unitary dynamics . the use of time - periodic modulations constitutes another channel to impact states of a quantum system . in the coherent limit , when the system is isolated from the environment , modulations imply an explicit time - periodicity of the system hamiltonian , @xmath3 . the dynamics of the system are determined by the basis of time - periodic _ floquet eigenstates _ the properties of the floquet states depend on various modulation parameters . modulations being resonant with intrinsic system frequencies can create a set of non - equilibrium eigenstates with properties drastically different from those with time - independent hamiltonians . modulations enrich the physics occurring in fields such as quantum optics , optomechanics , solid state and ultra - cold atom physics @xcite and disclose a spectrum of new phenomena @xcite .      as an object of mathematical physics , ( [ lind ] ) has a specific structure and possesses a variety of important properties @xcite . in the case of a time - independent , stationary hamiltonian @xmath15 , the generator @xmath5 induces a continuous set of completely positive quantum maps @xmath16 @xcite . under some conditions ( ` quantum ergodicity ' ) , the system evolves from an initial state @xmath17 to a unique and time - independent asymptotic state @xmath18 , @xmath19 @xcite . when time - periodic modulations are present , eq .  ( [ lind ] ) preserves the complete positivity of the evolution if all coupling rates are non - negative at any instance of time , @xmath20 @xcite . under some suitable , experimentally relevant assumptions , a set - up `` time - dependent hamiltonian and time - independent dissipation '' provides a valid approximation @xcite .    here , we address the particular case of quench - like , time - periodic dependence of the hamiltonian , @xmath21 , corresponding to periodic switches between several constant hamiltonians @xcite . a popular choice is the set - up composed of two hamiltonians , @xmath22 where @xmath23 , @xmath24 $ ] . this minimal form has recently been used to investigate the connection between integrability and thermalization @xcite or , alike , for disorder - induced localization @xcite in _ isolated _ periodically modulated many - body systems .    from a mathematical point of view , ( [ lind],[pc ] ) define a linear equation with a time - periodic generator @xmath25 . therefore , floquet theory applies and asymptotic solutions of the equation are all time - periodic with temporal period @xmath26 @xcite . @xmath25 is a dissipative operator and , in the absence of relevant symmetries @xcite , the system evolution in the asymptotic limit @xmath27 is determined by a unique ` quantum attractor ' , i.e. , by an asymptotic , time - periodic density operator , @xmath28 , @xmath24 $ ] and @xmath29 . the challenge here consists in explicit numerical evaluation of the matrix form of this operator .    to use spectral methods ( complete / partial diagonalization and different kinds of iterative algorithms @xcite ) to calculate @xmath30 as eigen - element of a superoperator would mean to deal with @xmath31 scaling of computationally expensive operations . in the case of periodically modulated systems it restricts the use of spectral methods to @xmath32 , @xmath33 and the corresponding lindblad operators @xmath34,@xmath35 are sparse , the floquet map @xmath36 is a dense matrix . therefore , the numerical evaluation can not benefit from sparse - matrix methods . ] . a direct propagation of eq .  ( [ lind ] ) for a time span long enough for @xmath37 to approach the attractor is not feasible for @xmath1 for two reasons : first , direct propagation requires to numerically propagate @xmath38 complex differential equations with time - dependent coefficients , and second , accuracy might become a problem for large times . although the accuracy may be improved by implementing high - order integration schemes @xcite or faber and newton polynomial integrators @xcite , this approach is hardly parallelizable ) by using the so - called time evolving block decimation ( tebd ) technique @xcite . the numerical effort scales as @xmath39 . however , this algorithm can only be used for lattice systems , i.e. , systems that can be partitioned into @xmath40 ` pieces ' coupled by next - neighbor interactions , both unitary and dissipative . in this case it gives a correct answer when the asymptotic state is characterized by a low entanglement . ] . systems with @xmath41 states may still be too small , for example , to explore mbl effects in open periodically - modulated systems . is it possible to go beyond this limit ? and if so how far ? we attempt to answer these questions by unraveling of the quantum master equation ( [ lind ] ) into a set of stochastic realizations , called `` quantum trajectories '' @xcite . this method allows us to transform the problem of the numerical solution of eqs . ( [ lind],[pc ] ) into a task of statistical sampling over quantum trajectories which form vectors of the size @xmath4 . the price to pay for the reduction from @xmath31 to @xmath4 is that we now have to sample over many realizations . this problem is very well suited for parallelization @xcite and we can definitely benefit from the use of a computation cluster . if the number of realizations @xmath42 becomes large , the sampling of the density operator @xmath37 with initial condition @xmath43 converges to the solution of eq . ( [ lind ] ) @xcite . this method is popular in the field of quantum optics where it adequately describes the physics of experiments @xcite .    [ cols="^,^ " , ]     finally , we estimated the computation time to propagate a single trajectory on a single - core as a function of system size @xmath4 , see table [ table2 ] . for the model specified by eqs . ( [ eq : hamiltonian ] , [ eq : jump ] ) this time scales as @xmath44 ; this is due to the multiplication of the quadratic scaling of a dense matrix - vector multiplication and a linear scaling of the jump frequency . the latter scaling is , however , model specific and may differ for other physical systems so that the overall computation time may vary substantially with the type of hamiltonian and/or dissipators under study . on top , the numbers we present in table [ table1 ] depend on the values of the coupling constant @xmath45 and the period of modulations @xmath26 . this is so because these parameters control the rate of the jumps . therefore , the obtained estimates are specific and can not be used as universal quantifiers . we now report the results of our simulations obtained for the physical setup with eqs . ( [ eq : hamiltonian ] , [ eq : jump ] ) . we start with the performance of the algorithm , table  [ table1 ] . the idea of the algorithm mimics a float : the algorithm constantly attempts to ` float to the surface ' , i.e. , to increase the time step of integration towards its maximal value @xmath46 while every next jump pulls it downwards to @xmath47 , see fig .  [ fig : husimi ] . the average time between two consequent jumps is the mean of the local maxima in the depicted saw - like time sequence of @xmath48 . there is no problem in overestimating @xmath46 , simply because the time step will rarely reach its maximum . the shortest time step , @xmath47 , or , equivalently , the depth @xmath49 , is tuned to the values needed to reach the desired accuracy . next we turn to the averages @xmath50 over the realizations and the corresponding statistical variances @xmath51 $ ] of the matrix elements , as discussed in detail in section  [ sampling ] . both quantities converge to `` limit cycles '' if the propagation time @xmath52 , @xmath53 , is much larger than all relaxation times . this means that for @xmath54 the density matrix converges to a time - periodic quantum attractor , i.e. , @xmath55 , and the variances also become time - periodic functions , @xmath56 \backsimeq \operatorname{var}[\varrho^\mathrm{att}_{kl}(\tau)]$ ] , see fig . [ fig:3]b . the crumpled caustic - like shapes of the limit cycles is a result of the projection on a plane of a global limit - cycle living in @xmath57-dimensional space . this limit - cycle is not a topological product of @xmath31 two - dimensional limit cycles ; elements of the asymptotic density matrix do not evolve independently , they do interact so that their means and variances are coupled .    in the asymptotic regime sampling can be performed stroboscopically , i.e. , after every period @xmath26 . in our simulations we used as the transient time @xmath58 and then performed the stroboscopic sampling of @xmath59 . the attractor density matrix at any other instant of time @xmath60 can be sampled by shifting the starting time of the sampling , @xmath61 , or simply by performing an extra - sampling at all needed intermediate points .    with @xmath62 samples per trajectory ( that amounts to an additional propagation for the time @xmath63 ) it became possible to collect @xmath64 samples for the model of dimension @xmath65 ( i.e. , with @xmath66 indistinguishable bosons ) by running the program on @xmath67 cores during three days . for some choice of parameters the sample density matrix displays a standard diagonal - dominated structure , see fig . [ fig:3]a . the husimi distribution of the sample density matrix is depicted in fig . [ fig:4 ] . there is an intriguing similarity between the distribution of the quantum attractor and the phase - space structure of the classical attractor ( its stroboscopic section , to be more precise ) produced by the mean - field equations , eqs . ( [ eq : thetadot ] ) . this allows us to conjecture that the attractor density matrix is resolved with good accuracy . the @xmath68 cores allowed us to sample the same number of realizations for the model of the dimension @xmath69 during approximately one week , [ eq : jump ] ) involve summation over series of binomial coefficients of the order @xmath4 . it was not possible , however , to go beyond @xmath70 when evaluating the corresponding husimi distributions . ] . the primary aim of this study was to estimate the numerical horizon of a high - accuracy sampling of non - equilibrium dissipative quantum attractors , i.e. , the asymptotic solutions of periodically driven open systems , by using the quantum trajectory method . our main result is that it is possible to resolve asymptotic density matrices of systems of several thousand of states by using even a small cluster ( with @xmath71 cores ) on a time scale of a few days . the benefit of having access to the whole density matrix is the possibility to extract more detailed information about non - equilibrium regimes encoded in the matrix structure such as purity and many - body entanglement @xcite . naturally , the next step must be a systematic analysis of the errors , statistical and numerical ones . we reserve this objective for further studies . here , we would like to surmise on further optimization of the sampling procedure . an immediate idea is to use a reduced adaptive basis , constructed by using coherent states @xcite . this however demands a priori knowledge of the attractor s structure . this insight can be obtained by using models of small sizes , @xmath72 . another possibility is the choice of more optimal initial conditions which in turn will substantially reduce the transient time @xmath73 ( for example , one can use an extrapolation of the largest - eigenvalue eigenstate of the density matrix for a small system ) . altogether , we expect that such improvements could increase @xmath4 further by a factor of @xmath74 . research areas where quantum attractors are of potential interest were already mentioned in section [ introduction ] . here , we briefly recall them . first , this is many - body localization @xcite where the action of temporal modulations @xcite and dissipation @xcite so far have been considered separately . a combined action of both factors presents an intriguing challenge . a complementary task is to extend the idea of `` dissipative engineering '' @xcite to periodically - modulated quantum systems . finally , a survival of floquet topological insulators @xcite in the presence of dissipation is a timely question . the authors acknowledge support by the russian science foundation ( grant no . 15 - 12 - 20029 ) . we thank i. vakulchik for the help with preparing the figures . f. pastawski , l. clemente , and j. i. cirac , phys . a * 83 * ( 2011 ) 012304 ; m. j. kastoryano , m. m. wolf , and j. eisert , phys . rev * 110 * ( 2013 ) 110501 ; e. kapit , phys * 116 * ( 2016)150501 .                                                                                          k. j. strm and b. bernhardsson , _ comparison of periodic and event based sampling for first - order stochastic systems_. in preprints 14th world congress of ifac , beijing , p.r . china , july 1999 . ; k. j. strm and b. bernhardsson , lecture notes in control and information sciences * 286 * ( 2002 ) 1 . d. poletti , j .- s . bernier , a. georges , c. kollath , phys . * 109 * ( 2012 ) 045302 . c. gross , t. zibold , e. nicklas , j. esteve , m.k . oberthaler , nature * 464 * ( 2010 ) 1165 . j. tomkovic , w. muessel , h. strobel , s. lock , p. schlagheck , r. ketzmerick , m. k. oberthaler , _ observing the emergence of chaos in a many - particle quantum system _ ,, there are currently several exciting proposals to use the ( 001 ) surface of silicon for the construction of atomic - scale electronic devices , including single electron transistors  @xcite , ultra - dense memories  @xcite and quantum computers  @xcite . however , since any random charge or spin defects in the vicinity of these devices could potentially destroy their operation , a thorough understanding of the nature of crystalline defects on this surface is essential . the si(001 ) surface was first observed in real space at atomic resolution using scanning tunneling microscopy ( stm ) by tromp _ _ et . al.__@xcite in 1985 . in this study they observed the surface consisted of rows of `` bean - shaped '' protrusions which were interpreted as tunneling from the @xmath1-bonds of surface si dimers , thereby establishing the dimer model as the correct model for this surface . since then , stm has been instrumental in further elucidating the characteristics of this surface , and in particular atomic - scale defects present on the surface@xcite . the simplest defect of the si(001 ) surface is the single dimer vacancy defect ( 1-dv ) , shown schematically in figs .  [ def1](a ) and [ def1](b ) . this defect consists of the absence of a single dimer from the surface and can either expose four second - layer atoms ( fig . [ def1](a ) ) or form a more stable structure where rebonding of the second - layer atoms occurs  @xcite as shown in fig . [ def1](b ) . while the rebonded 1-dv strains the bonds of its neighboring dimers it also results in a lowering of the number of surface dangling bonds and has been found to be more stable than the nonbonded structure . @xcite single dimer vacancy defects can also cluster to form larger defects such as the double dimer vacancy defect ( 2-dv ) and the triple dimer vacancy defect ( 3-dv ) . more complex clusters also form , the most commonly observed@xcite example is the 1 + 2-dv consisting of a 1-dv and a 2-dv separated by a single surface dimer , the so - called `` split - off dimer '' . the accepted structure of the 1 + 2-dv , as proposed by wang _ et . based on total energy calculations,@xcite is shown in fig . [ def1](c ) and consists of a rebonded 1-dv ( left ) , a split - off dimer , and a 2-dv with a rebonding atom ( right ) . recently we have observed another dv complex that contains a split - off dimer , called the 1 + 1-dv , which consists of a rebonded 1-dv and a nonbonded 1-dv separated by a split - off dimer , as shown in fig . [ def1](d ) .        here we present a detailed investigation of dv defect complexes that contain split - off dimers . using high - resolution , low - bias stm we observe that split - off dimers appear as well - resolved pairs of protrusions under imaging conditions where normal si dimers appear as single `` bean - shaped '' protrusions . we show that this difference arises from an absence of the expected @xmath1-bonding between the two atoms of the split - off dimer but instead the formation of @xmath1-bonds between the split - off dimer atoms and second layer atoms . electron charge density plots obtained using first principles calculations support this interpretation . we observe an intensity enhancement surrounding some split - off dimer defect complexes in our stm images and thereby discuss the local strain induced in the formation of these defects . finally , we present a model for a previously unreported triangular - shaped split - off dimer defect complex that exists at s@xmath2-type step edges . experiments were performed in two separate but identical variable temperature stm systems ( omicron vt - stm ) . the base pressure of the ultra - high vacuum ( uhv ) chamber was @xmath3 mbar . phosphorus doped @xmath4 and @xmath5 @xmath6 wafers , orientated towards the [ 001 ] direction were used . these wafers were cleaved into @xmath7 mm@xmath8 sized samples , mounted in sample holders , and then transferred into the uhv chamber . wafers and samples were handled using ceramic tweezers and mounted in tantalum / molybdenum / ceramic sample holders to avoid contamination from metals such as ni and w. sample preparation@xcite was performed in vacuum without prior _ ex - situ _ treatment by outgassing overnight at 850  k using a resistive heater element , followed by flashing to 1400  k by passing a direct current through the sample . after flashing , the samples were cooled slowly ( @xmath9  k / s ) from 1150  k to room temperature . the sample preparation procedure outlined above routinely produced samples with very low surface defect densities . however , the density of defects , including split - off dimer defects , was found to increase over time with repeated sample preparation and stm imaging , as reported previously.@xcite it is known that split - off dimer defects are induced on the si(001 ) surface by the presence of metal contamination such as ni ,  @xcite and w  @xcite . the appearance of these defects in our samples therefore points to a build up of metal contamination , either ni from in - vacuum stainless steel parts , or more likely w contamination from the stm tip . after using an old w stm tip to scratch a @xmath10  1  mm line on a si(001 ) sample in vacuum and then reflashing , the concentration of split - off dimer defects on the surface was found to have dramatically increased , confirming the stm tip as the source of the metal contamination . figure  [ sods ] shows an stm image of a si(001 ) surface containing a @xmath10  10% coverage of split - off dimer defects . the majority of the defects in this image can be identified as 1 + 2-dvs , however , two 1 + 1-dvs are also present , as indicated . the most striking feature of this image is the difference in appearance of the split - off dimers in contrast to the surrounding normal surface dimers . each split - off dimer in this image appears as a double - lobed protrusion , while the surrounding normal si dimers each appear as a single `` bean - shaped '' protrusion , as expected at this tunneling bias . @xcite line profiles taken across a 1 + 2-dv both parallel and perpendicular to the dimer row direction are shown in fig . [ sods](b ) . the line profile parallel to the dimer row direction agrees with previously reported profiles over 1 + 2-dvs and fits well with the accepted structure ,  @xcite as shown by the overlayed ball and stick model . the line profile taken perpendicular to the dimer row direction , however , clearly shows that the split - off dimer of this defect is separated into two protrusions while the neighboring si dimers are single protrusions . this is the first recognition and explanation of split - off dimers appearing as double - lobed protrusions . 1 surface with split - off dimer defects is shown in ( a ) . tunneling conditions for this image were @xmath11  v sample bias and 0.8  na tunnel current . line profiles are taken across a single 1 + 2-dv both parallel , x  x@xmath12 ( b ) , and perpendicular , y  y@xmath12 ( c ) , to the dimer row direction , as indicated in ( a ) . the schematic ( d ) is a top view ball and stick model of a 1 + 2-dv with the approximate positions of @xmath1-bonds indicated by shaded ellipses . ]    to understand why split - off dimers appear as double - lobed protrusions we must consider the structure of these defects shown in figs . [ def1](c ) and [ def1](d ) . normally si(001 ) surface dimers appear as `` bean - shaped '' protrusions in stm images because the dangling bonds of each si dimer atom mix to form a @xmath1-bond between the two dimer atoms . however , if we examine the split - off dimer structure closely ( figs . [ def1](c ) and [ def1](d ) ) we see that unlike normal surface dimers , the split - off dimer has two nearest neighbor second layer atoms that each have a dangling bond . the separation distance between the split - off dimer atoms and these second layer atoms is sufficiently close to allow the formation of @xmath1-bonds . the resulting four - atom structure can therefore be referred to as a _ tetramer_. we propose that the four dangling bonds of the split - off dimer tetramer interact primarily along the backbonds between the split - off dimer atoms and the second layer atoms to form @xmath1-bonds down the backbonds , as drawn schematically in fig . [ sods](c ) . these two spatially separated @xmath1-bonds therefore lead to the double - lobed appearance of the split - off dimers under low bias filled - state tunneling conditions , which we confirm in section  [ theory1 ] with charge density calculations .    in an attempt to fully characterize the appearance of these split - off dimers in stm images , we have performed a series of experiments observing split - off dimers with changing stm sample bias . figure  [ sodv ] summarizes our results , showing images where a 1 + 2-dv and a 1 + 1-dv located next to each other are observed at four different sample biases  two filled - state images and two empty - state images . in the filled - state image of fig . [ sodv](a ) we see that at @xmath13  v the split - off dimer of both the 1 + 2-dv and the 1 + 1-dv appear as double - lobed protrusions similar to those in fig . [ sods](a ) . however , when the filled - state bias is increased in magnitude to @xmath14  v , fig . [ sodv](b ) , the split - off dimers become single - protrusions and appear very similar to the surrounding normal si surface dimers . this is because as the bias magnitude is increased towards @xmath14  v , the dimer @xmath15-bond and bulk states contribute increasingly to the tunneling current  @xcite and the image of the split - off dimer reverts to the bean - shaped protrusion in the same manner as normal surface si dimers . in both of the empty - state images , figs . [ sodv](c ) and [ sodv](d ) , acquired at + 0.8  v and + 2  v , respectively , the appearance of the split - off dimers is very similar to that of the surrounding normal surface dimers . this is because under empty - state tunneling conditions electrons tunnel into the @xmath16-antibonding orbitals of the dimers , resulting in the normal si dimers appearing as double - lobed protrusions . @xcite it is therefore only under low bias magnitude filled - state tunneling conditions that split - off dimers appear significantly different to the surrounding normal si surface dimers . v , ( b ) @xmath14  v , ( c ) @xmath17  v , ( d ) @xmath18  v. ]      another noticeable feature of figs .  [ sods](a ) and [ sodv](a ) is the enhanced brightness of the 1 + 1-dv compared to the 1 + 2-dv . this is a reproducible effect that we attribute to an increased amount of surface strain induced by the 1 + 1-dv . figure  [ strain ] shows a series of adjacent defects forming a short vacancy line channel in the surface . this channel is composed of individual 1-dv , 3-dv , 1 + 2-dv , and 1 + 1-dv defects ( see figure caption ) . in the filled - state image , fig . [ strain](a ) , there is a clear brightening of the dimers on one end of the 1 + 1-dvs and the dimers on both ends of the 1-dv , which is not present for the 1 + 2-dvs . in the empty - state image of the line of defect complexes , fig . [ strain](b ) , we notice that there is a darkening of the same dimers that are enhanced in the filled - state image . v , @xmath19  v , 0.15  na ) of a short chain of dvs in a si(001 ) surface . the individual defects are ( from top left to bottom right ) : 1 + 1-dv , 1 + 1-dv , 1 + 2-dv , 3-dv , 1 + 2-dv , 1 + 2-dv , 1 + 1-dv , 1-dv , and 1 + 2-dv . note the strain - induced brightening of the 1-dv and 1 + 1-dvs in the filled - state ( a ) and the corresponding darkening in the empty - state ( b ) ]    owen _ et . _ ,  @xcite have shown using low bias stm and first principles calculations , that the dimers neighboring a rebonded 1-dv are enhanced in low bias filled - state stm images due to the strain induced by the defect shifting the surface states upwards in energy toward the fermi energy . this effect can be seen for the 1-dv in fig . [ strain](a ) , where the neighboring dimers in the same row as the 1-dv are enhanced in intensity , with the magnitude of the enhancement decaying with distance from the 1-dv . a very similar enhancement can be seen around the 1 + 1-dv sites in this image , with the split - off dimer in particular appearing much brighter than the surrounding normal surface dimers . however , for the 1 + 1-dv only the dimers on one end of the defect are enhanced in intensity while the dimers on the other end of the defect are not . this observation can be readily explained since the 1 + 1-dv is composed of a rebonded 1-dv adjacent to a nonbonded 1-dv ( fig .  [ def1](d ) ) and owen  _ et . @xcite have shown that while the rebonded 1-dv results in strain - induced image enhancement , the nonbonded 1-dv does not . the observation of an asymmetric strain - induced enhancement of the 1 + 1-dv in fig . [ strain](a ) can therefore be taken as an experimental confirmation of the structure of this defect ( fig . [ def1](d ) ) and the first application of the method of owen  _ et . @xcite for identifying strain in more complex surface defect structures . the fact that the 1 + 2-dv causes no enhancement of its neighboring dimers over the surrounding normal surface dimers suggests that the 1 + 2-dv , unlike the 1-dv and 1 + 1-dvs , does not increase the strain of the surface . this at first seems strange , since the 1 + 2-dv involves a rebonded 1-dv similar to the 1 + 1-dv structure . however , wang _ et . _  @xcite have shown , using total energy calculations , that the junction formed between the 1-dv and the 2-dv to create the 1 + 2-dv releases the surface strain that is present when these two defects exist separately from one another . the stm data that we have presented here is therefore the first experimental verification of this calculation . the fact that both the 1-dv and the 1 + 1-dv show local enhancement due to strain , while the 1 + 2-dv does not , indicates that the 1 + 2-dv structure induces less local strain than the 1-dv . in their paper , owen _ et . do not present empty state stm images , nor do they consider empty states in their tight binding calculations . in fig . [ strain](b ) , we show an empty state image of the same line of defects shown in fig . [ strain](a ) . interestingly , in this empty state image the dimers that were enhanced in brightness surrounding the 1-dv and 1 + 1-dvs in the filled - state image are less bright than the surrounding si dimers in the empty - state image . this suggests that the strain associated with these defects causes the lowest unoccupied molecular orbital ( lumo ) of the adjacent dimers to also shift higher in energy , away from the fermi energy . to confirm the interpretation of our stm images , we have performed first principles electronic structure calculations of both the 1 + 2-dv and 1 + 1-dv complexes using the car - parrinello molecular dynamics program . @xcite valence electrons were described using goedecker pseudopotentials  @xcite expanded in a basis set of plane waves with an energy cutoff of 18 rydbergs and the exchange - correlation functional was of the blyp form . @xcite slab calculations contained between 124 and 128 si atoms in a @xmath20  @xmath21 supercell , corresponding to six layers of vacuum in the @xmath22-direction , and all calculations were performed with gamma point sampling of the brillouin zone only . a reference calculation was performed with no surface vacancies and assuming the @xmath23 structure in which the dimers buckled alternately along the row . a single 256 atom calculation with a duplication along the y - axis confirmed that the effect of dispersion across the rows is minor as has been noted elsewhere . @xcite    both zero temperature geometry optimization and high temperature molecular dynamics calculations were used to explore a variety of surface and second - layer bonding configurations for the 1 + 2-dv and 1 + 1-dv . the results confirm the configurations in figs . [ def1](c ) and [ def1](d ) are the lowest energy geometries of both defect complexes . the dimers are drawn symmetric in these schematics , however , the true minimum energy structure at zero temperature involved charge - transfer buckling of the si dimers . it is well known that at room temperature the barrier is sufficiently small for the dimers to flip - flop between the two equivalent configurations . @xcite our calculations show that the split - off dimer tetramer also has two symmetrically equivalent buckling configurations , with charge transfer between the atoms of the tetramer buckling adjacent atoms in alternate directions . by analogy with the normal dimers we can expect room - temperature stm measurements of the tetramer to image the average of the two configurations . the chemical potential was determined from a 512-atom bulk calculation , which yielded a formation energy of 0.85  ev for the 1 + 2-dv , similar to the value of 0.65  ev computed by wang _ at . al_.  @xcite the 1 + 1-dv formation energy has not been previously reported , and we found it to be 1.13  ev . we note that this value is high , but this is consistent with the rarity of observation of the 1 + 1-dv in stm experiments .    in fig . [ 1 + 2-dv ] we present a series of calculated electron density slices through various regions of the 1 + 2-dv marked by ( a ) , ( b ) , ( c ) , and ( d ) in the ball and stick schematic . the charge density shown in the figure is the sum of the occupied kohn - sham orbitals within 0.25  ev of the highest occupied molecular orbital ( homo ) . taking into account the @xmath24  ev surface band gap of si(001 ) and the n - type doping of the experimental samples , these states correspond approximately to the accessible states for a @xmath25  v sample bias and can therefore be directly compared to the experimental data in fig . [ sodv](a ) , which was acquired with a @xmath13  v sample bias . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) 2-dv edge dimer . ]    the four charge density slices in fig . [ 1 + 2-dv ] show : fig . [ 1 + 2-dv](a ) the 1-dv edge dimer , fig . [ 1 + 2-dv](b ) the split - off dimer , fig . [ 1 + 2-dv](c ) the backbond of the split - off dimer , and fig . [ 1 + 2-dv](d ) the 2-dv edge dimer , as indicated schematically in fig . [ 1 + 2-dv](e ) . the charge densities of both buckling configurations of the dimers and backbond atoms are averaged , and the positions of the dimer and tetramer atoms are shown superimposed in both buckling configurations . in the case of the backbonds , the two configurations are not coincident , and so the atoms and bonds are shown in projection onto the plane in fig . [ 1 + 2-dv](c ) . the 1-dv edge dimer in fig . [ 1 + 2-dv](a ) shows a clear three - lobed character with significant overlap between the up - atom charge density of the two buckling orientations , and a single lobe beneath the plane of the surface at the mid - point of the dimer . density functional calculations by hata  _ et . al . _  @xcite and tight - binding green s function calculations by pollman  _ et @xcite have separately identified this three - lobed feature as being characteristic of @xmath1-bonding in flip - flop dimers on the silicon surface , and we can therefore take this three - lobed feature as a signature of @xmath1-bonding in this work . the backbond of the split - off dimer in fig . [ 1 + 2-dv](c ) connects a first - layer atom to a second - layer atom and also shows a three - lobed structure . by analogy with the surface dimer in fig . [ 1 + 2-dv](a ) we characterize this bond as having @xmath1-character and have indicated this by the shaded ellipse ( c ) shown in fig . [ 1 + 2-dv](e ) . the split - off dimer itself in fig . [ 1 + 2-dv](b ) , however , does not exhibit three - lobed character . instead , the split - off dimer has four lobes ; two located above the up - atoms of the dimer in each buckling configuration , and a second pair of spatially separated lobes beneath the bond . the calculations thus show that @xmath1-bonding occurs down the backbonds of the split - off dimer , but not across the dimer itself . the absence of the @xmath1-bond across the split - off dimer correlates with the double - protrusions observed in the stm images . finally , we also consider the charge density of the 2-dv edge dimer , fig . [ 1 + 2-dv](d ) , and note that it also exhibits three - lobed character , indicative of @xmath1-bonding . this gives the 2-dv edge dimer a bean - shaped appearance in the stm image , as for the 1-dv dimer in fig . [ 1 + 2-dv](a ) .    a similar situation exists for the 1 + 1-dv charge density slices shown in fig .  [ 1 + 1-dv ] the first three charge density slices , figs . [ 1 + 1-dv](a )  [ 1 + 1-dv](c ) , are analogous to the slices for the 1 + 2-dv as was the case for the 1 + 2-dv , the rebonded 1-dv edge dimer , fig . [ 1 + 1-dv](a ) and the split - off dimer backbonds , fig . [ 1 + 1-dv](c ) exhibit three - lobed @xmath1-like character , while the split - off dimer , fig . [ 1 + 1-dv](b ) exhibits four - lobed character , consistent with an end - on view of @xmath1-bonding down the backbonds . finally , another slice is presented in fig . [ 1 + 1-dv](d ) , which is through the nonbonded 1-dv edge dimer as indicated schematically in fig . [ 1 + 1-dv](e ) . it can be seen that the nonbonded 1-dv edge dimer appears quite different to the charge density slices discussed so far . in particular , we notice that the nonbonded 1-dv edge dimer has a much reduced charge density compared to the other slices , fig . [ 1 + 1-dv](a )  [ 1 + 1-dv](c ) . examination of the structure identifies strain as the characteristic that differentiates the dimer in fig . [ 1 + 1-dv](d ) from the other dimers . since the dimer in fig . [ 1 + 1-dv](d ) is part of a tetramer , one might expect its appearance to resemble the split - off dimer which is also part of the tetramer shown figs .  [ 1 + 1-dv](b ) and [ 1 + 1-dv](c ) . however , a detailed examination of the simulated structure reveals that the nonbonded 1-dv tetramer is relaxed , since there is one adjacent dimer present , while the split - off tetramer is highly strained because of the rebonding in the second - layer . since the nonbonded 1-dv tetramer is much less strained , its occupied states lie further from the fermi level , explaining the charge reduction observed in calculations in fig .  [ 1 + 1-dv](d ) . as discussed in ref . , the minimum energy arrangement of the electrons in a tetramer is one where the @xmath1-states are delocalized across the four atoms , to form three bonding segments , as indicated by the ellipses in fig . [ 1 + 1-dv](e ) . the charge density slice of fig . [ 1 + 1-dv](d ) is consistent with such an arrangement where the charge density is shared between @xmath1-like bonds on both backbonds and across the dimer atoms . we conclude that this charge density arrangement forms for the nonbonded 1-dv tetramer because it is allowed to relax . in the case of the split - off dimer , the tetramer is constrained by the rebonding and instead forms a higher energy configuration in which the @xmath1-bonds conjugate to form two @xmath1-bonds down its backbonds . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) nonbonded 1-dv edge dimer . ] having presented a detailed understanding of the electronic structure of previously observed split - off dimer defects in the si(001 ) surface using both stm and first - principles calculations , we now turn our attention to elucidating the structure of a previously unreported split - off dimer defect . in figs . [ triangular](a ) and [ triangular](b ) we show filled- and empty - state stm images of dv defects at a single - layer s@xmath2-type step edge . at the top of these images white arrows indicate are three defects known as s@xmath2-dvs , which are rebonded 1-dvs at the step edge , which leave a single split - off dimer as the last dimer before the lower terrace begins . @xcite as was the case for the 1 + 1-dv and 1 + 2-dv , the split - off dimers in s@xmath2-dvs appear as double - lobed protrusions under low - bias filled - state imaging conditions , fig . [ triangular](a ) . at the bottom of fig . [ triangular](a ) two similar dv complexes can be observed , as indicated by black arrows , however these defects have a third protrusion giving them a triangular appearance . in empty - state imaging , fig . [ triangular](b ) , however , the additional third feature is not present . these triangular - shaped defects have not been reported on the si(001 ) surface before and most likely arise due to the presence of w contamination . 1.2  v ) of dv defects at an s@xmath2-type step edge . white arrows indicate s@xmath2-dvs ,  @xcite while black arrows point to a previously unreported defect that exhibits a third protrusion in the filled - state giving it a triangular appearance . we propose the structure ( c ) as a model for this defect . calculated charge density slices at a constant @xmath22-height for the dashed region of ( c ) are shown in ( d ) and ( e ) ( for kohn - sham orbitals summed over 0.45  ev below the homo and 0.45  ev above the lumo , respectively ) . these contour slices are in good agreement with the stm images in ( a ) and ( b ) , in particular predicting the correct spacing of 6.4    between the split - off dimer and third protrusion and also the disappearance of the third protrusion in the empty - state . the horizontal tic - marks in ( d ) and ( e ) indicate the dimer positions on the defect - free surface . ] our proposed structural model of the triangular - shaped defects in fig . [ triangular](a ) is shown in fig . [ triangular](c ) . this model consists of a nonbonded 1-dv defect at an s@xmath2-type step edge , followed by a rebonded split - off dimer and a bound si monomer . swartzentruber has previously observed si monomers on the si(001 ) surface using high - resolution stm after depositing a few percent of a monolayer of si atoms to the surface . @xcite these monomers were bound at rebonded s@xmath2-type step edges , confirming the minimum energy binding position predicted by first principles calculations . the binding position of the monomer in our proposed structure , fig . [ triangular](c ) , is essentially the same position observed by swartzentruber , with the difference being the presence of the dv defect adjacent to the step edge . swartzentruber also observed that the si monomers bound at s@xmath2-type step edges were visible in one bias polarity ( empty - state ) but invisible in the other ( filled - state ) . our images reveal a similar effect , however the feature we observe appears in filled - state images while being invisible in empty - state images . we have performed first - principles calculations to produce charge density contours for our proposed structure . figure 7(d ) shows a constant @xmath22-height contour slice taken 1.2    above the monomer for occupied kohn - sham orbitals within 0.45  ev of the homo . we see in this charge density contour slice the two lobes expected for the split - off dimer as well as a third lobe due to the bound monomer . moreover , the distance between the split - off dimer lobes and the monomer lobe is 6.4   in agreement with the separation seen in the stm image . in fig .  [ triangular](e ) we show an empty - state slice taken at the same @xmath22-height and summed over kohn - sham orbitals up to 0.45  ev above the lumo . in this contour the double lobe of the split - off dimer is still present but the monomer lobe is significantly lessened in intensity . the results of our first - principles calculations therefore give good agreement between our proposed structure and the observed defect . the presence of the split - off dimer must therefore be responsible for the reversal of the filled- and empty - state monomer characteristics when compared to those observed for monomers bound to rebonded s@xmath2-type step edges . we have investigated split - off dimers on the si(001)2@xmath261 surface using high resolution stm and first principles calculations . we find that split - off dimers form @xmath1-bonds with second layer atoms which gives them a double - lobed appearance in low bias filled - state stm images . we apply the method of owen  _ et . @xcite for identifying local areas of increased surface strain to dimer vacancy defect complexes and thereby present the first experimental confirmation of the predicted strain relief offered by the 1 + 2-dv . finally , we have presented a previously unreported triangular - shaped defect on the si(001 ) surface and a proposed model for this structure involving a bound si monomer ., recently , the study of the dynamics of @xmath3 on the @xmath2-character variety of @xmath5 for a compact hyperbolizable @xmath6-manifold @xmath0 with nonempty boundary and no toroidal boundary components , has led to the somewhat surprising discovery that often the dynamical and geometric decompositions of the @xmath2-character variety do not coincide . the @xmath2-character variety of @xmath5 denoted @xmath7 is the geometric quotient of @xmath8 by inner automorphisms of @xmath9 sitting inside @xmath10 is @xmath11 the set of discrete and faithful representations , which can equivalently be thought of as the deformation space of hyperbolic @xmath6-manifolds homotopy equivalent to @xmath12 the action of @xmath3 on the interior of @xmath13 is properly discontinuous ( see canary @xcite ) , but if @xmath0 has a primitive essential annulus , the action can not be properly discontinuous on all of @xmath13 ( canary - storm @xcite ) . minsky in @xcite was the first to observe that there exists a domain of discontinuity , called the set of primitive - stable representations , for @xmath3 containing the interior of @xmath13 as well as points on @xmath14 when @xmath0 is a handlebody . in particular , the set of primitive - stable representations contains both discrete and faithful representations and dense representations . jeon - kim - ohshika - lecuire @xcite gave a complete criterion for a discrete and faithful representation to be primitive - stable in terms of disc - busting properties of the ending lamination and parabolic loci of the associated hyperbolic @xmath6-manifold . in her thesis @xcite , the second author gave a generalization of the primitive - stable condition to representations of the fundamental group of a compression body into @xmath2 , called the separable - stable condition , and she showed that the set of separable - stable representations is a domain of discontinuity for the action of @xmath3 that contains the interior of @xmath13 as well as point on @xmath15 in this paper , we give a characterization of which discrete and faithful representations of @xmath1 into @xmath2 are separable - stable analogous to the jeon - kim - ohshika - lecuire criterion . a compression body @xmath0 is the boundary connect sum of a @xmath6-ball , a collection of trivial @xmath16-bundles over surfaces , and a handlebody , where the other components are connected to the @xmath6-ball along disjoint discs . its fundamental group is a free product of closed surface groups and a free group . if @xmath0 is not the boundary connect sum of two trivial @xmath16-bundles over closed surfaces , then an element in @xmath1 is separable if it lies in a proper factor of a free decomposition of @xmath4 if @xmath0 is the boundary connect sum of @xmath17 and @xmath18 where @xmath19 and @xmath20 are closed surfaces , then an element is separable if it lies in a factor of a decomposition @xmath21 where @xmath22 is homotopic to a simple closed curve on @xmath19 or @xmath20 . a homomorphism @xmath23 is separable - stable if every geodesic defined by a separable element in the cayley graph of @xmath24 is mapped to a uniform quasi - geodesic in @xmath25 . see section [ sep - stab ] for a precise definition . for any discrete and faithful representation @xmath26 we let @xmath27 denote the corresponding hyperbolic @xmath6-manifold obtained from taking the quotient @xmath28    the main theorem of this article is :    [ thm : main ] let @xmath0 be a nontrivial hyperbolizable compression body without toroidal boundary components that is not the boundary connect sum of @xmath17 and @xmath18 where @xmath19 and @xmath20 are closed surfaces with genus at least two . let @xmath29 be a discrete and faithful representation of @xmath1 into @xmath2 . then , @xmath29 is separable - stable if and only if in @xmath30 each component of the parabolic loci and each ending lamination is disc - busting . a measured lamination @xmath31 is _ disc - busting _ if there exists @xmath32 such that @xmath33 for all unweighted meridians @xmath34 . otherwise it is _ disc - dodging_. we say that an ending lamination is disc - busting if it is the support of a measured lamination that is disc - busting . such a lamination is used by jeon - kim - ohshika - lecuire in @xcite for the characterization of primitive stable representations of free groups , and has a root in doubly incompressible laminations ( see kim - lecuire - oshika @xcite or lecuire @xcite ) . in @xcite , the second author showed that representations obtained by pinching masur domain curves or masur domain laminations on @xmath35 are separable - stable .    for the case where @xmath0 is the boundary connect sum of @xmath36 and @xmath37 where @xmath19 and @xmath20 are closed surfaces , we establish the following analogous characterization of the set of discrete and faithful representations that are separable - stable .    [ thm : special case ] let @xmath0 be the boundary connect sum of @xmath17 and @xmath18 where @xmath19 and @xmath20 are closed surfaces with genus at least two . let @xmath26 be a discrete and faithful representation . then , @xmath29 is separable - stable if and only if in @xmath30 each component of the parabolic loci and each ending lamination is annulus - busting . a measured lamination @xmath31 is _ annulus busting _ if there exists @xmath38 such that @xmath39 for any essential annulus @xmath40 a _ compression body _ is a compact , orientable , irreducible @xmath6-manifold @xmath0 with a boundary component , @xmath41 called the _ exterior boundary _ , such that the inclusion @xmath42 induces a surjection on the level of fundamental groups @xmath43 . the other boundary components are called _ interior boundary components_. equivalently , a compression body is the boundary connect sum of a @xmath6-ball , a collection of trivial @xmath16-bundles over surfaces , and a handlebody , where the other components are connected to the @xmath6-ball along disjoint discs . the fundamental group of a compression body can be decomposed as a free product , @xmath44 where @xmath45 is isomorphic to a closed surface group for @xmath46 and @xmath47 is infinite cyclic @xmath48 . by grushko s theorem ( @xcite ) and kurosh s subgroup theorem ( @xcite ) , any other decomposition of the fundamental group into a free product , @xmath49 where each factor is freely indecomposable , satisfies @xmath50 and @xmath51 , up to re - ordering . a compression body is _ trivial _ if @xmath52 is an isomorphism , i.e. if @xmath0 is a trivial @xmath16-bundle . we say that @xmath0 is _ uniquely freely decomposable _ if @xmath0 is the boundary connect sum of two trivial @xmath16-bundles over closed surfaces since , in this case , the decomposition of @xmath1 is essentially unique . an _ essential disc _ in @xmath0 is a properly embedded disc whose boundary is nontrivial in @xmath53 a compression body @xmath0 has the property that any splitting of @xmath1 as a free product or as an hnn - extension over the trivial group can be realized by an essential disc in @xmath0 in the following sense . suppose that @xmath54 is a separating essential disc and @xmath55 if @xmath56 for @xmath57 then @xmath58 where @xmath59 is inclusion and the base point is chosen to lie in @xmath60 in this case , we say that @xmath54 realizes the splitting of the fundamental group . as we can move the basepoint around , this is only well - defined up to conjugation . if @xmath54 is a non - separating essential disc and @xmath61 then @xmath62 we say that @xmath54 realizes this hnn - extension .    [ lem : freeprod ] let @xmath0 be a compression body and let @xmath63 be a nontrivial splitting of @xmath1 into a free product . then , the splitting is realizable by an essential disc . [ lem : hnn ] let @xmath0 be a compression body . any splitting of the form @xmath64 is realizable by an essential disc . recall that @xmath65 is isomorphic to @xmath66 by lemma [ lem : freeprod ] , we can realize the latter splitting by an essential disc @xmath67 . then , @xmath67 separates @xmath68 into two components @xmath69 and @xmath70 where @xmath71 is conjugate to @xmath72 and @xmath70 is a compact , oriented , irreducible @xmath6-manifold with infinite cyclic fundamental group , i.e. a solid torus . then , the essential disc @xmath54 in the solid torus , realizes the original splitting .      in a free group , an element is called _ primitive _ if it can be completed to a free generating set and an element is _ separable _ if it lies in a free factor . the notion of separability is originally due to stallings ( @xcite ) in his study of separable sets in free groups that generalized whitehead s ( @xcite ) study of primitive elements of free groups . although the notion of a primitive element does not generalize to other freely decomposable groups , the notion of a separable element does .    if @xmath0 is a compression body that is not uniquely freely decomposable , then an element in @xmath1 is _ separable _ if it lies in a proper factor of a decomposition of @xmath1 into a nontrivial free product . if @xmath0 is the boundary connect sum of two trivial @xmath16-bundles over closed surfaces @xmath19 and @xmath20 , then an element in @xmath1 is _ separable _ if it lies in a factor of a decomposition @xmath73 where @xmath22 is homotopic to a simple closed curve on @xmath19 or @xmath20    one can think of separability in the following geometric way . if @xmath0 is not uniquely freely decomposable , by lemma [ lem : freeprod ] , an element @xmath74 of @xmath1 lies in a proper factor of a free decomposition if and only if its associated curve in @xmath0 is homotopic to one that misses an essential disc . similarly , lying in a factor of a decomposition @xmath73 where @xmath22 is homotopic to a simple closed curve on @xmath19 or @xmath20 is equivalent to missing the corresponding essential annulus @xmath75 up to homotopy . essential annulus _ in @xmath0 is a properly embedded incompressible annulus that is not homotopic , relative to its boundary , into the boundary of @xmath12    let @xmath24 denote the fundamental group of @xmath0 and let @xmath76 be a finite symmetric generating set for @xmath77 the _ cayley graph of @xmath24 with respect to @xmath76 _ , denoted @xmath78 , is a graph where the vertices are in one - to - one correspondence with the elements of @xmath24 and there is an edge between @xmath74 and @xmath79 if there exists an element @xmath80 in the generating set @xmath76 such that @xmath81 . the group @xmath24 acts on @xmath78 by left multiplication on the vertices . since @xmath24 is torsion - free and gromov hyperbolic each element @xmath74 of @xmath24 has two fixed points on @xmath82 . for each @xmath74 in @xmath24 , let @xmath83 and @xmath84 denote the repelling and attracting fixed points of @xmath74 acting on @xmath82 . let @xmath85 denote the set of geodesics connecting @xmath83 and @xmath84 and @xmath86 denote the set of geodesics @xmath87 in @xmath78 such that @xmath87 is contained in @xmath85 for some separable element @xmath74 .    given a representation @xmath88 and a basepoint @xmath89 in @xmath25 , there exists a unique @xmath29-equivariant map @xmath90 taking the identity to @xmath89 and edges to geodesic segments . a representation @xmath88 is called _ @xmath91-separable - stable _ if there exists a basepoint @xmath89 in @xmath25 such that @xmath92 is a @xmath91-quasi - geodesic for any @xmath87 in @xmath93    we will call @xmath29 , a representation , _ separable - stable _ if there exists @xmath91 such that @xmath29 is @xmath91-separable - stable . separable - stability is independent of the choice of basepoint in @xmath25 and the choice of generators @xmath76 for @xmath24 , and separable - stability is invariant under conjugation ( see lemma 16 in lee @xcite ) . the set of separable - stable representations forms a domain of discontinuity for the action of @xmath3 strictly larger than the interior of @xmath13 . namely the following is true . let @xmath0 be a nontrivial compression body without toroidal boundary components . the outer automorphism group @xmath3 acts properly discontinuously on the set of separable - stable representations , which contains the interior of @xmath13 as well as points on the boundary of @xmath94      in this section , we describe otal s ( @xcite ) generalization of the whitehead graph for a free group to the whitehead graph for a compression body , which we will use in proving the sufficient condition . let @xmath0 be a nontrivial compression body without toroidal boundary components . a _ system of meridians _ @xmath95 on @xmath0 is a collection of pairwise nonisotopic and pairwise disjoint meridians @xmath96 bounding discs @xmath97 such that @xmath98 is a disjoint union of trivial @xmath16-bundles over closed surfaces , where @xmath99 is a regular neighborhood of @xmath97 .    by thurston s hyperbolization theorem ( see kapovich @xcite ) and marden ( @xcite ) , we can fix @xmath100 a convex cocompact representation of @xmath1 such that @xmath101 is homeomorphic to @xmath0 , where @xmath102 is the conformal boundary of @xmath103 let @xmath104 be the limit set of @xmath105 let @xmath106 be a closed subset of @xmath107 that is @xmath108-invariant and also invariant under switching the two factors . in this paper , @xmath106 will be one of the following two examples .    * if @xmath109 is a closed geodesic in @xmath110 , then @xmath111 is the set of endpoints of all the lifts of @xmath112 * if @xmath31 is a lamination on @xmath35 that is in tight position with respect to @xmath95 , then @xmath113 is the set of endpoints of all lifts of all leaves of @xmath114    see the discussion before proposition [ discbusting ] for the definition of a lamination in tight position and lemma [ endpoints ] to see why @xmath113 is well - defined . identify @xmath35 with @xmath102 . let @xmath95 be a system of meridians bounding the discs @xmath115 and suppose @xmath116 is the disjoint union of @xmath117 where each @xmath118 is a closed surface of genus at least two . the _ whitehead graph for @xmath0 of @xmath106 with respect to @xmath95 _ , denoted @xmath119 is a collection of not necessarily connected graphs , @xmath120 where the elements in the collection are in one - to - one correspondence with the components of @xmath116 . in @xmath116 , there are two copies @xmath121 and @xmath122 of each @xmath97 in @xmath54 . given a component @xmath123 of @xmath116 , the vertices in the corresponding graph @xmath124 are in one - to - one correspondence with the components of @xmath125 in the frontier of @xmath123 . abusing notation , relabel the vertices @xmath126 fix a jordan curve @xmath127 in @xmath128 that is invariant under a conjugate of @xmath129 , which we will continue to denote @xmath129 . let @xmath130 denote the boundary component of @xmath123 coming from @xmath131 . fix a lift @xmath132 of each @xmath133 in @xmath134 such that @xmath135 lies in the component of the preimage of @xmath130 in @xmath134 containing @xmath127 on its boundary . let @xmath136 be the open set in @xmath134 , bounded by @xmath132 not containing @xmath127 . the edges from @xmath137 to @xmath138 will be in one - to - one correspondence with elements @xmath74 in @xmath129 such that @xmath139 is nonempty . we will denote such an edge @xmath140 . although these edges are directed , for each edge from @xmath136 to @xmath141 labeled @xmath74 , there is an edge from @xmath141 to @xmath136 labeled @xmath142 . [ otal ] a connected component of @xmath143 is _ strongly connected _ if there exists a cycle that represents a nontrivial element of @xmath129 . a connected component of @xmath143 has a _ strong cutpoint _ if we can express the graph as the union of two graphs @xmath144 and @xmath145 that intersect in a single vertex such that either @xmath144 or @xmath145 is not strongly connected . we take the convention that a cycle @xmath146 corresponds to the group element @xmath147 . although we made several choices when defining @xmath148 , the two properties defined above , strong connectedness and the presence of a strong cutpoint are independent of these choices ( see section 3.1.2 in lee @xcite ) . otal in @xcite , also , gave a dichotomy between the whitehead graph of a separable element and the whitehead graph of a disc - busting lamination in tight position with respect to a system of meridians . in this section , we define and collect the relevant properties of a lamination in tight position . the results in this section are essentially all due to otal in @xcite . since @xcite is difficult to obtain for each result we either give an alternate source containing a proof or we provide a proof . a lamination @xmath31 is in _ tight position with respect to a system of meridians @xmath95 _ if there are no waves disjoint from @xmath31 where a _ wave _ is an arc @xmath149 satisfying the following two conditions :    1 . @xmath149 has endpoints on @xmath95 , but its interior is disjoint from @xmath95 and , 2 . @xmath149 is homotopic , relative to its endpoints , in @xmath0 but not in @xmath35 into @xmath150    for any disc - busting lamination @xmath31 we can find a system of meridians @xmath95 such that @xmath31 is in tight position with respect to @xmath150 otal proved this for masur domain laminations but his proof applies in this more general case . [ discbusting ] if @xmath31 is a disc - busting lamination , there exists a system of meridians @xmath95 with respect to which @xmath31 is in tight position . suppose that @xmath31 is not in tight position with respect to a system of meridians @xmath95 . let @xmath149 be a wave and suppose its endpoints lie on @xmath151 then the endpoints of @xmath149 split @xmath96 into two arcs @xmath152 and @xmath153 such that @xmath154 and @xmath155 are the endpoints of @xmath156 since @xmath149 is homotopic into @xmath96 through @xmath0 but not in @xmath157 we have that @xmath158 and @xmath159 are both meridians . since @xmath31 is disc - busting , @xmath160 and @xmath161 by construction both @xmath162 and @xmath163 are disjoint from @xmath164 for any @xmath165 replacing @xmath96 with either @xmath162 or @xmath163 will produce a new collection of meridians @xmath166 and @xmath167 respectively , whose intersection number with @xmath31 is at least @xmath168 less than @xmath169    we claim that at least one of these new collections of meridians @xmath166 or @xmath167 is a system of meridians . by construction @xmath170 and @xmath163 bound a pair of pants on @xmath53 since they are all meridians and @xmath0 is irreducible , the discs they bound , @xmath171 and @xmath172 bound a ball in @xmath0 . so @xmath173 contains a component that is a ball @xmath174 . we will use a @xmath175 superscript to denote the copy of @xmath97 ( or @xmath176 ) that lies on the boundary of the ball , i.e. @xmath177 lie on the boundary of @xmath178 since @xmath179 is a collection of trivial @xmath16-bundles over closed surfaces , @xmath180 must consist of a collection of trivial @xmath16-bundles over closed surfaces and two balls , one of which is the ball @xmath174 described above . the other ball must contain either @xmath181 or @xmath182 on its boundary ( possibly both ) . if it contains @xmath183 then @xmath167 will be a system of meridians , and vice versa . so , replacing @xmath96 with one of @xmath162 and @xmath163 results in a system of meridians whose intersection number with @xmath31 is at least @xmath168 less than @xmath169 if @xmath31 is not in tight position with respect to this new system of meridians , repeat this process . since @xmath184 for any system of meridians @xmath185 , eventually , this process must terminate . an important property of disc - busting , minimal laminations in tight position with respect to a system of meridians @xmath95 is that any lift of a leaf has well - defined endpoints . [ endpoints ] let @xmath0 be a nontrivial compression body . let @xmath186 be a convex cocompact representation uniformizing @xmath0 . let @xmath31 be a lamination on @xmath102 in tight position with respect to @xmath150 let @xmath87 be a leaf @xmath31 such that any half leaf of @xmath87 intersects @xmath150 then , any lift @xmath187 of @xmath87 to @xmath188 has two well - defined endpoints in @xmath189    propositions [ discbusting ] and lemma [ endpoints ] imply that if @xmath31 is a disc - busting ending lamination , then @xmath113 , the set of endpoints of lifts of leaves of @xmath190 is well - defined . in particular , we will refer to the whitehead graph of @xmath31 as the whitehead graph of @xmath191    otal proved the following two results , which provide a dichotomy between separable elements and disc - busting laminations . [ separableelement ] let @xmath0 be a nontrivial compression body that is not the boundary connect sum of two trivial @xmath16-bundles over closed surfaces . if @xmath74 is separable , then for any system of meridians @xmath95 some connected component of @xmath192 is either not strongly connected , or has a strong cut - point . [ otal ] if @xmath31 is a disc - busting lamination in tight position with respect to a system of meridians @xmath193 then the whitehead graph of @xmath31 with respect to @xmath95 is strongly connected and without strong cutpoints .    in @xcite , otal states proposition [ otal ] only for masur domain laminations , but in fact his proof applies in this more general case . in general , for @xmath194 a discrete and faithful representation , the associated hyperbolic @xmath6-manifold @xmath27 will be homotopy equivalent but not necessarily homeomorphic to @xmath12 since otal s results only apply in the case when @xmath0 is a compression body , in this section we associate a whitehead graph to any hyperbolizable @xmath6-manifold @xmath195 homotopy equivalent , but not necessarily homeomorphic to @xmath0 . we can do this by using the characteristic compression body introduced by bonahon in @xcite . we also describe how to compare whitehead graphs for @xmath0 and @xmath195 . let @xmath196 denote the set of compact , orientable , irreducible , atoroidal ( unmarked ) @xmath6-manifolds homotopy equivalent to @xmath0 , up to homeomorphism . let @xmath197 , \ldots , [ m_n]$ ] be the elements in @xmath198 where @xmath199 is homeomorphic to @xmath12 for each @xmath200 fix a convex cocompact representation @xmath201 such that @xmath202 is homeomorphic to the interior of @xmath203 let @xmath204 be an orbit map for @xmath201 as in the definition of separable - stable representations where @xmath205 . then , @xmath206 is a quasi - isometric embedding and there is a continuous extension , @xmath207 which is a homeomorphism . we suppressed the basepoint in @xmath206 since @xmath208 is independent of the choice of basepoint . then , let @xmath209 be the homeomorphism from @xmath210 to @xmath211    in order to compare the whitehead graph of @xmath199 and @xmath200 we need a way to compare systems of meridians on the two manifolds . define a map @xmath212 from the set @xmath213 of unweighted meridians on @xmath200 to the set @xmath214 of unweighted meridians on @xmath199 in the following way ( see figure [ pic ] ) .     on the characteristic compression body @xmath215 of @xmath200 is mapped to the system of meridians @xmath216 on @xmath217,width=491 ]    let @xmath218 denote the essential disc bounded by @xmath150 first suppose that @xmath95 is a meridian in @xmath200 such that @xmath218 separates @xmath203 then , @xmath218 realizes a splitting @xmath219 up to conjugation . in @xmath199 we can realize the splitting by an essential disc @xmath67 . let @xmath220 be the boundary of @xmath221 this is well - defined up to isotopy . indeed if we consider @xmath220 on the conformal boundary of @xmath222 then some lift @xmath223 separates @xmath224 into two components one of which contains all the attracting fixed points of any element whose reduced word representation starts with an element in @xmath72 and the other contains all the attracting fixed points of any element whose reduced word representation starts with an element in @xmath178 since such points are dense in @xmath225 this determines @xmath226 up to isotopy on @xmath227    now suppose that @xmath218 is non - separating . then , @xmath218 realizes an hnn extension , @xmath228 by lemma [ lem : hnn ] , we can realize such a splitting in @xmath199 by an essential disc @xmath67 . as before , let @xmath220 be the boundary of @xmath221 again , to see that @xmath220 is well - defined notice that some lift @xmath229 separates @xmath134 into two components one of which contains all the attracting fixed points of any element whose reduced word representation starts with an element of @xmath72 or @xmath230 and the other contains all the attracting fixed points of any element whose reduced word representation starts with @xmath231 where @xmath230 is the new generator associated to the hnn - extension . any manifold @xmath0 with compressible boundary has a _ characteristic compression body _ @xmath232 , unique up to isotopy that has the following properties ( see bonahon @xcite or mccullough - miller @xcite ) . the exterior boundary of @xmath232 coincides with @xmath35 . the closure of @xmath233 contains no essential compression discs and no component of @xmath234 is a 3-ball . let @xmath215 denote the characteristic compression body of @xmath235 with any trivial compression body components removed . [ lem : mermap ]    let @xmath236 be a system of meridians on @xmath237 then , the collection of meridians @xmath238 is a system of meridians on @xmath217    first we will show that the meridians @xmath238 can be realized such that they are pairwise disjoint and pairwise nonisotopic . to see that we can realize @xmath239 and @xmath240 disjointly , observe that since @xmath164 and @xmath241 are disjoint , any two lifts @xmath242 and @xmath243 , have the property that there exist @xmath141 a component of @xmath244 and @xmath245 a component of @xmath246 such that @xmath247 and @xmath248 are each non - empty and disjoint from each other . there exists lifts @xmath249 and @xmath250 and components @xmath251 and @xmath252 of @xmath253 and @xmath254 respectively , such that @xmath255 and @xmath256 then , we can homotope , @xmath239 and @xmath240 to be disjoint .    to see that @xmath239 and @xmath240 are not isotopic , notice that if they were isotopic , then there exists lifts @xmath249 and @xmath250 that bound a region in @xmath134 containing no limit points . then , the same would be true about the corresponding lifts @xmath242 and @xmath257 which contradicts that they are not isotopic . let @xmath258 be the disc bounded by @xmath239 in @xmath217 finally , we want to see that @xmath259 is a collection of trivial @xmath16-bundles over closed surfaces . we will first show that @xmath260 is a collection of trivial @xmath16-bundles over closed surfaces . recall that @xmath261 where @xmath215 is the characteristic compression body and @xmath262 has incompressible boundary . by definition , @xmath263 is a collection of trivial @xmath16-bundles over closed surfaces . we claim that the components of @xmath262 are all trivial @xmath16-bundles over closed surfaces . let @xmath127 be a component of @xmath264 and let @xmath265 be a component of @xmath266 since @xmath127 has incompressible boundary and can not be a @xmath6-ball , we have that @xmath267 lies in a surface group factor of some maximal free decomposition of @xmath1 i.e. @xmath267 is a subgroup of some @xmath268 . the image of @xmath269 must be finite index in @xmath268 since they are both closed surface groups . this implies that @xmath269 is finite index in @xmath270 which in turn implies that @xmath127 is an @xmath16-bundle ( theorem 10.5 in hempel @xcite ) . so @xmath127 is either a trivial @xmath16-bundle or twisted @xmath16-bundle . it remains to check that @xmath127 can not be a twisted @xmath16-bundle . if @xmath127 were a twisted @xmath16-bundle , then the fundamental group of @xmath127 would be the fundamental group of a non - orientable surface , but this is impossible as @xmath267 is finite index in @xmath271 so , we have shown that the components of @xmath262 are all trivial @xmath16-bundles over surfaces . now @xmath272 is formed by gluing @xmath262 and @xmath263 along their boundaries , so the result will be a collection of trivial @xmath16-bundles over closed surfaces . now , consider @xmath273 a component of @xmath274 . we want to see that @xmath273 is a trivial @xmath16-bundle over a closed surface . if not , then either @xmath273 is a @xmath6-ball , or @xmath273 has compressible boundary . in the former case , let @xmath238 be the components of @xmath220 that lie on the boundary of @xmath273 ( notice that we can have two copies of @xmath164 on the boundary of @xmath273 , relabel accordingly ) . fix a lift @xmath275 of @xmath276 then , the corresponding lifts @xmath277 bound a connected region in @xmath134 containing no limit points . the corresponding lifts of @xmath278 of @xmath95 would have the same property . this implies that there is a @xmath6-ball component of @xmath279 a contradiction . if @xmath273 has compressible boundary , then there exists a closed curve @xmath109 in @xmath273 such that @xmath109 does not lie in any surface group factor of a free decomposition of @xmath4 moreover , any lift @xmath280 of @xmath109 has the property that both endpoints lie in the same component of @xmath253 for any lift of any @xmath281 since @xmath109 misses any @xmath282 then , we would also have that @xmath280 has both endpoints in the same component of @xmath283 so , @xmath109 misses all the meridians @xmath281 a contradiction , since @xmath284 consists of trivial @xmath16-bundles over closed surfaces .    using lemma [ lem : mermap ] we can compare the whitehead graphs of @xmath215 and @xmath217 recall from section [ white ] the definition of the whitehead graph of a compression body . for @xmath285 we will abuse notation , and let @xmath286 denote the image of @xmath106 under @xmath287    [ whotherhomeo ] let @xmath200 lie in @xmath196 and let @xmath215 be the characteristic compression body of @xmath203 if @xmath288 is strongly connected and without strong cutpoint , then @xmath289 is strongly connected and without strong cutpoint . recall that we have a @xmath24-equivariant homeomorphism @xmath290 although @xmath291 is only defined on the limit set of @xmath292 it induces a natural map on lifts of @xmath293 to lifts of @xmath294 for example , given a lift @xmath242 of @xmath164 , there exists a corresponding lift @xmath249 of @xmath295 that partitions @xmath296 in the same way as @xmath242 via @xmath297 we ll abuse notation to denote such a lift @xmath298 as @xmath299 more precisely , suppose that @xmath300 then , @xmath301 is the lift @xmath249 of @xmath239 such that if @xmath302 then @xmath303 where @xmath304 up to switching @xmath305 and @xmath306    suppose that @xmath123 is a component of @xmath307 recall that when defining the whitehead graph , we fixed @xmath127 a jordan curve in @xmath134 invariant under @xmath129 . we let @xmath130 denote the boundary component of @xmath123 intersecting @xmath95 and we fixed the lift @xmath308 of @xmath130 whose boundary is @xmath309 if @xmath242 corresponds to a vertex of @xmath310 then map it to @xmath299    we want to show that this is a well - defined map on the vertices . first notice that since @xmath311 is a system of meridians , @xmath312 consists of trivial @xmath16-bundles over surfaces . one of these components @xmath313 must have fundamental group conjugate to @xmath314 by the uniqueness of the free decomposition of @xmath1 ( see section [ sec : compbody ] ) . let @xmath315 be the jordan curve invariant under @xmath129 . note that @xmath316 let @xmath317 be the component of @xmath123 intersecting @xmath220 and let @xmath318 denote the component of the preimage of @xmath317 with @xmath315 on its boundary . then , we claim that @xmath301 lies on @xmath319 indeed , this is clear from the fact that @xmath301 partitions @xmath296 in the same way as @xmath242 via @xmath320 as described above . so we see that the map on the vertices is well - defined . now suppose that @xmath321 is an edge in @xmath322 this implies that the intersection of @xmath106 and @xmath323 is nonempty . by construction , this is true if and only if the same is true for @xmath286 and @xmath324 so , we have a bijection between the whitehead graph of @xmath288 and that of @xmath325 where an edge in @xmath288 labeled @xmath74 is mapped to an edge in @xmath325 with the same label . in particular , we see that @xmath288 is strongly connected and without cutpoint if and only if the same is true for @xmath326 recently , mj ( @xcite ) has proved the existence of cannon - thurston maps for general kleinian groups . in particular , given a kleinian group @xmath24 and a discrete and faithful representation @xmath327 , he showed that there exists a continuous extension @xmath328 of @xmath329 moreover , he gives a characterization , which we describe below , of which points are mapped non - injectively by @xmath330 .    before stating mj s result we recall some facts from hyperbolic geometry that we will need . we will restrict to the case that @xmath24 has no non - cyclic abelian subgroups since this is the case in this paper . there exists a constant @xmath331 called the margulis constant , such that for any hyperbolic @xmath6-manfiold , and any @xmath332 each component of the @xmath333-thin part of @xmath334 is a metric neighborhood of a closed geodesic or a parabolic cusp homeomorphic to @xmath335 given @xmath336 the _ non - cuspidal part of @xmath27 _ , denoted @xmath337 is the submanifold of @xmath27 obtained by removing the noncompact components of the @xmath333-thin part of @xmath338    when @xmath339 has parabolics , @xmath340 has a _ relative compact core , _ @xmath341 that is a compact submanifold whose inclusion is a homotopy equivalence and that intersects each cusp neighborhood along an annulus whose core curve we call a _ parabolic curve _ ( mccullough @xcite ) . let @xmath342 denote the union of these annuli . moreover , by the tameness ( agol @xcite or calegari - gabai @xcite ) and the uniqueness of cores ( mccullough - miller - swarup @xcite ) @xmath27 is homeomorphic to the interior of @xmath127 . each frontier component @xmath343 of @xmath344 in @xmath345 faces an end @xmath346 with a well - defined ending lamination @xmath347 obtained from taking the support of a projective limit of any sequence of simple closed geodesics exiting the end @xmath348 the ending lamination @xmath349 is a filling minimal lamination on @xmath343 ( canary @xcite ) .    let @xmath350 be the union of @xmath349 and the parabolic curves , and assume that each is disc - busting . if we fix a convex cocompact structure @xmath110 on @xmath127 , then we can identify @xmath351 with a geodesic lamination on @xmath102 . by lemma [ endpoints ] any lift @xmath187 of a leaf @xmath87 of @xmath349 for any @xmath59 has two well defined endpoints at infinity on @xmath128 . since @xmath100 is convex cocompact , we can identify @xmath82 with @xmath128 . we introduce a relation @xmath352 for points in @xmath128 ( and so also on @xmath82 ) such that @xmath353 if and only if either @xmath354 and @xmath355 are the endpoints of a leaf of @xmath356 or ideal vertices of a complementary region of @xmath356 . in contrast to the case of groups without parabolics , there are complementary regions of @xmath357 which are ideal polygons with infinitely many sides . these are exactly regions touching lifts of parabolic curves , which are isolated leaves . the relation @xmath358 may not be transitive . we define @xmath359 to be the transitive closure of @xmath352 , which is an equivalence relation . mj s theorem about the identified points of a cannon - thurston map can be adapted to our case as follows . [ mj2 ] let @xmath23 be a discrete and faithful representation . the identification of @xmath78 with its image under @xmath360 extends continuously to a map @xmath361 . moreover , for @xmath362 , we have @xmath363 if and only if @xmath364 . in this section , we use the dichotomy between the whitehead graph of a separable element and a disc - busting lamination to show the sufficient condition of theorem [ thm : main ] , namely that when @xmath1 is not uniquely freely decomposable , a representation is separable - stable if each end invariant is disc - busting . for the sufficient condition , we ll use mj s result described in the previous section . the following lemma , whose proof can be found in lee @xcite lemma 21 , sets the stage for using mj s result .    [ notss ] let @xmath26 be discrete and faithful . if @xmath29 is not separable - stable , then there exists a sequence of separable elements @xmath365 such that if @xmath366 and @xmath367 are the endpoints of @xmath368 then @xmath369 and @xmath370 converge , up to subsequence , to the same point in @xmath371 but @xmath366 and @xmath367 converge to @xmath372 and @xmath373 in @xmath374 where @xmath375    [ sufficient ] let @xmath0 be a nontrivial compression body , without toroidal boundary components , that is not the boundary connect sum of two trivial @xmath16-bundles over closed surfaces . let @xmath26 be a discrete and faithful representation . if each parabolic locus and ending lamination of @xmath27 is disc - busting , then @xmath29 is separable - stable . notice that the statement of the above proposition does not make any assumption about the homeomorphism type of @xmath338    suppose that @xmath27 is homeomorphic to the interior of @xmath376 where @xmath377 \in a(m).$ ] let @xmath378 be the union of the parabolic loci and ending laminations on @xmath379 if @xmath29 is not separable - stable , then by lemma [ notss ] there exists a sequence of separable elements @xmath365 such that the fixed points of @xmath365 in @xmath82 converge to @xmath372 and @xmath380 but the endpoints of @xmath381 converge to the same point in @xmath382 by theorem [ mj2 ] we have that @xmath383 and @xmath384 are each an endpoint of a leaf of one of the end invariants , although not necessarily the same end - invariant . suppose that @xmath383 contains an endpoint of a leaf of @xmath385    let @xmath386 be the set of limit points of @xmath387 then @xmath388 is @xmath389-invariant and also invariant under switching the two factors . moreover , @xmath390 lies in @xmath391 this implies that @xmath392 contains @xmath393 for any system of meridians @xmath95 ( see the proof of proposition 22 in lee @xcite ) . fix a system of meridians @xmath95 such that @xmath31 is in tight position with respect to @xmath150 then , @xmath393 is strongly connected and without any strong cutpoints by proposition [ otal ] .    by lemma [ whotherhomeo ] , the whitehead graph @xmath394 is strongly connected and without strong cutpoint . observe that @xmath393 is a finite graph since edges correspond to homotopy classes of arcs in @xmath190 so @xmath394 is also finite . since the graph @xmath392 contains @xmath393 , for @xmath59 large enough @xmath395 contains the whitehead graph @xmath396 this implies @xmath397 contains @xmath398 for @xmath59 large enough . but then the whitehead graph @xmath397 is strongly connected and without strong cutpoint . notice , that @xmath399 is the same as @xmath400 this contradicts proposition [ separableelement ] . in this section , we show that when @xmath1 is not uniquely freely decomposable , if @xmath26 is discrete and faithful , then the condition that each end invariant of @xmath27 is disc - busting is also necessary . notice that if each end invariant is disc - busting , then , in particular , @xmath27 contains only one compressible boundary component . we will make use of the following characterization of which discrete and faithful representations are separable - stable . [ compact ] let @xmath26 be discrete and faithful . then , @xmath29 is separable - stable if and only if every separable curve is homotopic to a closed geodesic and the set of separable geodesics is contained in a compact set of @xmath338    if @xmath31 is a minimal lamination that is not a simple closed curve , then @xmath401 the _ supporting surface of _ @xmath190 is the unique minimal compact subsurface with geodesic boundary containing @xmath114 let @xmath402 be the unique , up to isotopy , maximum , simple , multi - curve of @xmath403 disjoint from @xmath114 the multi - curve @xmath402 contains @xmath404 . [ necessary ] let @xmath0 be a nontrivial compression body , without toroidal boundary components , that is not the boundary connect sum of two trivial @xmath16-bundles over closed surfaces . suppose @xmath405 is a discrete and faithful representation . if @xmath29 is separable - stable , then each end invariant of @xmath27 is disc - busting . the proof of the necessary condition follows almost exactly the proof in the case when @xmath0 is a handlebody from jeon - kim - ohshika - lecuire @xcite ( see section 5 ) . we include it here for completeness .    by lemma [ compact ] , if a representation maps a separable element of @xmath24 to a parabolic element in @xmath406 then the representation is not separable - stable . since any disc - dodging , closed curve is separable , if there is a disc - dodging parabolic curve in @xmath27 , then @xmath29 is not separable - stable . so we may assume that each parabolic curve is disc - busting . similarly , if there exists an ending lamination @xmath31 that misses an essential disc @xmath54 , then @xmath31 is the limit of separable elements . in particular , there exists a sequence of separable elements exiting every compact set of @xmath338 by lemma [ compact ] , @xmath29 is not separable - stable . so , we may also assume that @xmath407 for any essential disc @xmath54 and any ending lamination @xmath31 of @xmath338    then , proposition [ necessary ] follows from the following two lemmas whose proofs we include below . we first explain how the necessary condition follows from the following two lemmas , and we include the proofs of the lemmas , afterwards . [ esan ] let @xmath0 have compressible boundary . for any essential annulus @xmath72 in @xmath0 , there is a meridian which is disjoint from @xmath72 . [ limitannuli ] let @xmath26 be discrete and faithful . suppose that @xmath31 is a disc - dodging ending lamination of @xmath27 . if some component @xmath22 of @xmath402 is disc - busting , then @xmath31 is the unique minimal component in the hausdorff limit of a sequence @xmath408 where @xmath409 is an essential annulus . indeed , by lemma [ limitannuli ] , if @xmath27 has an ending lamination that is disc - dodging , then there exists a sequence of simple closed curves @xmath410 , that are homotopic to core curves of essential annuli , such that @xmath411 exits every compact set . by lemma [ esan ] these curves are separable . hence , there exists a sequence of separable curves exiting every compact set . by lemma [ compact ] , the representation is not separable - stable . let @xmath72 be an essential annulus in @xmath12 since @xmath0 has compressible boundary there exists an essential disc @xmath60 we can isotope @xmath54 so that there are no inessential intersections between @xmath54 and @xmath72 . if @xmath412 , then the boundary of @xmath54 is a meridian disjoint from @xmath40 if not , consider an outermost arc @xmath149 in @xmath413 then , @xmath149 together with an arc in @xmath414 bounds a disc @xmath415 . if both of the endpoints of @xmath149 lie on the same component of @xmath416 , then together with an arc in @xmath417 the arc @xmath149 cuts off a disc @xmath418 from @xmath419 and @xmath420 is an essential disc which can be isotoped off @xmath72 . if @xmath149 connects two components of @xmath416 , then we can boundary - compress @xmath72 along @xmath415 , and to get an essential disc disjoint from @xmath40    the proof of lemma [ limitannuli ] makes use of a result of lecuire in @xcite that states the following . if @xmath0 is a compact hyperbolizable @xmath6-manifold , and there exists a lamination @xmath421 such that    * each closed leaf has weight at most @xmath422 and * for any compressing disc @xmath54 , we have @xmath423    then , the following two conditions are equivalent .    1 .   there exists @xmath168 such that @xmath424 for any essential annulus @xmath40 2 .   if @xmath425 and @xmath426 are disjoint geodesic rays on @xmath427 such that two lifts @xmath428 and @xmath429 share the same endpoint in @xmath371 then they are asymptotic . the first step is finding a lamination containing @xmath31 that satisfies the first two conditions ( a ) and ( b ) . let @xmath430 be the union of @xmath31 and @xmath431 where we place a weight of @xmath422 on each closed curve . if there exists an essential annulus @xmath72 that is disjoint from @xmath432 then cut @xmath0 along @xmath72 and let @xmath433 be the component containing @xmath114 we claim that @xmath31 is still disc - dodging on @xmath434 indeed , suppose that @xmath435 is a sequence of meridians on @xmath0 such that @xmath436 then , as in the proof of lemma [ esan ] we can find a sequence of meridians @xmath437 disjoint from @xmath419 such that @xmath438 hence , @xmath31 is still disc dodging on @xmath434 let @xmath439 be the union of @xmath430 and the core curve or curves of @xmath419 depending on whether @xmath72 separates @xmath0 . put a weight of @xmath422 on each core curve that is added . notice that this new sequence of meridians also has the property that @xmath440    if there exists an essential annulus on @xmath433 disjoint from @xmath441 we repeat the procedure above . since there exists at most finitely many disjoint non - isotopic essential annuli on @xmath68 this procedure must terminate . let @xmath442 be the manifold obtained at the end of this procedure , and let @xmath443 be the corresponding lamination . then , any closed leaf has weight at most @xmath444 by construction . to see that @xmath445 for any essential disc @xmath446 it suffices to show that @xmath414 intersects an element in @xmath402 twice . indeed , this implies that @xmath447 and since @xmath448 we have that @xmath449 since there is a disc - busting component @xmath22 of @xmath431 we know that @xmath22 intersects @xmath54 at least once . if @xmath22 intersects @xmath54 exactly once , then a regular neighborhood of @xmath450 is a solid torus , @xmath451 we can form an essential disc missing @xmath22 by taking the closure of @xmath452 , a contradiction since @xmath22 is disc - busting . so @xmath443 satisfies conditions @xmath453 and @xmath454 above . secondly , we want to find two geodesic rays satisfying the hypothesis of condition @xmath455 above . namely , we want to find two disjoint , geodesic rays on @xmath456 that have lifts with a common endpoint in @xmath382 recall that there exists a sequence of meridians @xmath457 such that @xmath458 and @xmath459 up to subsequence , @xmath457 converges to a geodesic lamination @xmath460 in the hausdorff topology . by casson s criterion ( see casson - long @xcite or theoreme b1 in lecuire @xcite ) , @xmath106 contains a homoclinic leaf @xmath461 we want to show that @xmath79 has two half - leaves @xmath462 and @xmath463 that are disjoint from @xmath464 then , if @xmath22 is the disc - busting component in @xmath431 we have that @xmath462 and @xmath463 will lie on @xmath465 since @xmath466 is an incompressible subsurface , any lift @xmath467 has a well - defined endpoint in @xmath468 where @xmath469 is the subgroup of @xmath470 corresponding to the subsurface @xmath471 since @xmath79 is homoclinic , @xmath462 and @xmath463 must have the same endpoint .    to see why two such half - leaves exist , first note that if @xmath79 intersects @xmath31 transversely , then it contains an arc @xmath149 such that @xmath472 this implies that for @xmath59 large enough , @xmath473 which is a contradiction since @xmath474 so @xmath79 does not intersect @xmath31 transversely . it follows that @xmath79 has two half - leaves @xmath475 that are disjoint from @xmath476 similarly , if @xmath79 intersects a leaf @xmath109 of @xmath443 outside of @xmath401 then @xmath477 for @xmath59 large enough , a contradiction since @xmath459 so @xmath79 has two half - leaves @xmath475 that are disjoint from @xmath464    next , we want to see that @xmath443 does not satisfy condition ( c ) above . indeed we want to show that if we take lifts @xmath478 and @xmath479 with the same endpoint , then they are not asymptotic . suppose they are asymptotic and take a sequence of geodesic arcs @xmath480 joining @xmath478 and @xmath479 such that the length of @xmath480 goes to zero . let @xmath481 be the projection of @xmath480 to @xmath482 . then @xmath483 since @xmath31 is minimal and not a simple closed curve . since @xmath478 and @xmath479 are asymptotic and do not transversely intersect @xmath31 we can assume that @xmath481 are homotopic relative to their endpoints and that the homotopy is through arcs transverse to @xmath114 then , we must have @xmath484 for all @xmath485 . in particular , the arc @xmath486 is disjoint from @xmath114 then , adjusting @xmath487 so that it begins at @xmath488 , if we take @xmath489 this projects to a nontrivial curve @xmath34 on @xmath482 disjoint from @xmath31 that is trivial in @xmath442 . it follows from the proof of dehn s lemma that there is a compressing disc disjoint from @xmath190 a contradiction . since condition ( c ) is not true , we have that condition ( c ) is also not true . in particular , there exists a sequence @xmath409 of essential annuli such that @xmath490 since each component of @xmath491 is a closed curve with weight @xmath422 , we have that @xmath492 for @xmath59 large enough . up to subsequence , @xmath493 converges , in the hausdorff topology , to a geodesic lamination @xmath494 then , @xmath495 and @xmath31 can not intersect transversely . since @xmath496 we can not have that @xmath495 and @xmath31 are disjoint . since @xmath492 for @xmath59 large enough , we have that @xmath497 in particular , @xmath495 lies on @xmath498 since @xmath31 is filling on @xmath499 , we have that @xmath31 and @xmath495 coincide as geodesic laminations . since @xmath493 is disjoint from all the essential annuli used to construct @xmath442 , we can view @xmath495 as a lamination on @xmath35 and we are done . this section deals with the missing case when @xmath1 is uniquely freely decomposable , i.e. when @xmath0 is the boundary connect sum of @xmath17 and @xmath18 where @xmath343 is a closed surfaces of genus at least two for @xmath500 here an element of @xmath1 is separable if and only if it misses an essential annulus in one of the two surface group factors .    by lemma [ esan ] for any essential annulus @xmath419 there exists an essential disc disjoint from @xmath40 since there is only one essential disc , @xmath446 up to isotopy , in fact , @xmath72 lies in one of the two surface group factors . if @xmath72 is contained in @xmath501 then @xmath72 is either an essential annulus in @xmath502 or homotopic ( relative to its boundary ) into the boundary of @xmath503 if the latter is true , since @xmath72 is essential in @xmath68 if @xmath504 and @xmath72 lies in @xmath505 then @xmath416 bounds an annulus in @xmath506 containing @xmath60 in either case , if a curve or lamination misses @xmath419 it misses an essential annulus in @xmath17 of @xmath507    we will consider annulus - busting laminations on @xmath35 . recall that a measured lamination @xmath31 is annulus - busting if there exists an @xmath168 such that @xmath508 for any essential annulus @xmath40    let @xmath0 be the boundary connect sum of @xmath17 and @xmath18 where @xmath19 and @xmath20 are closed surfaces of genus at least two . let @xmath26 be discrete and faithful . then , @xmath29 is separable - stable if and only if each end invariant of @xmath27 is annulus - busting . we will start with the sufficient condition , and we will follow the general outline of proposition [ sufficient ] . in this case , we have the following simplifications . firstly , @xmath196 has only one element , i.e. any manifold homotopy equivalent to @xmath0 is homeomorphic to @xmath0 . secondly , we have only one meridian @xmath95 . if @xmath31 is annulus - busting , then it must intersect @xmath193 and so it is automatically in tight position with respect to @xmath150    now suppose that @xmath26 is a discrete and faithful representation such that each end invariant and parabolic locus of @xmath27 is annulus - busting . if @xmath29 is not separable - stable , by lemma [ notss ] , there exists a sequence of separable elements @xmath365 with endpoints @xmath509 such that @xmath510 and @xmath511 then , by theorem [ mj2 ] , the image @xmath512 is an endpoint of one of the end invariants @xmath31 . here @xmath31 can either be an ending lamination or a simple closed curve that is mapped to a parabolic element under @xmath513 as before , this implies that for @xmath59 large enough @xmath514 contains @xmath515 if @xmath31 is annulus - busting , then @xmath516 intersects each essential annulus in the following sense . suppose that @xmath72 is an annulus in @xmath503 let @xmath517 . in defining the whitehead graph , we fixed lifts @xmath518 of @xmath343 . if we take a lift @xmath519 of @xmath520 and the lift @xmath521 of @xmath522 with the same endpoints as @xmath519 , then @xmath523 forms a loop in @xmath134 . we will say that an edge @xmath524 intersects @xmath72 if there exists lifts @xmath519 and @xmath521 in @xmath518 as above such that @xmath136 and @xmath525 lie in different components of @xmath526 . since @xmath514 contains @xmath516 for @xmath59 large enough , we have that @xmath514 `` intersects '' any essential annulus . this implies that the geodesic representative @xmath410 of @xmath365 intersects any essential annulus , a contradiction . for the necessary condition , suppose that @xmath527 is a discrete , faithful and separable - stable representation and some end invariant of @xmath27 is annulus - dodging . if a parabolic curve @xmath22 is annulus - dodging , then @xmath22 is separable , a contradiction to lemma [ compact ] . so suppose that all parabolic curves are annulus - busting , but there is an ending lamination @xmath31 that is annulus - dodging . if @xmath31 is disjoint from an essential annulus @xmath419 then we can find a sequence of separable elements approaching @xmath190 a contradiction to lemma [ compact ] . so we can assume that @xmath528 for any essential annulus . since @xmath31 is annulus - dodging , there exists a sequence @xmath409 of essential annuli such that @xmath529 since each essential annulus misses the essential disc , @xmath446 up to a subsequence , we can assume that @xmath409 is contained in one of the two surface group factors for all @xmath530 without loss of generality , assume that @xmath409 is contained in @xmath531 for all @xmath530 let @xmath351 be the limit , up to subsequence , of @xmath493 in @xmath532 the space of projective measured laminations . since @xmath493 is disjoint from the meridian for all @xmath59 , so is @xmath351 . in particular , @xmath351 lies on @xmath533 since @xmath534 either @xmath31 and @xmath351 are disjoint or @xmath535 since @xmath31 is minimal . if @xmath31 and @xmath351 are disjoint , then we can find a simple closed curve @xmath22 on @xmath19 that is disjoint from @xmath114 then , @xmath22 is the boundary of an essential annulus @xmath536 that misses @xmath190 a contradiction . if @xmath537 then @xmath31 lies on @xmath19 and misses any essential annulus in @xmath18 a contradiction we thank the organizers of the workshop on  ergodic decompositions of representation varieties " held in princeton in october 2011 , where the layout of the paper was conceived . the second author thanks dick canary for several useful conversations . 99 i. agol , _ tameness of hyperbolic 3-manifolds _ , preprint available at arxiv : math\0405568v1 f. bonahon , _ cobordism of automorphisms of surfaces _ , ann . sci . cole norm . 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sentences: primordial gravity waves are expected to be produced during cosmic inflation in addition to scalar perturbations . if indeed present , they would leave a characteristic footprint on the polarized anisotropies of the cosmic microwave background ( cmb ) , as they are considered to be essentially the sole source of the so - called primordial @xmath0-mode residing at the super - horizon scales at the time of the last scattering . a detection of the @xmath0-mode angular power spectrum at large angular scales would be then treated as a smoking gun of inflation , while a precise measurement of its amplitude would constrain the energy scale of inflation , or , geometrically speaking , the expansion rate of the universe during inflation @xcite . this amplitude is expressed by the tensor - to - scalar ratio , @xmath2 , defined as the relative power of primordial gravity waves with respect to that of the scalar perturbations at some pivot scale @xmath8 , chosen here to be equal to @xmath9 mpc@xmath10 . currently , the most stringent upper bound on @xmath2 using temperature anisotropies has been derived by the planck  collaboration : @xmath11 at 95% cl @xcite , while a recent joint analysis of the planck  and  polarized data set an upper limit @xmath12 at 95% cl @xcite . the measurement of the tensor - to - scalar ratio @xmath2 could allow to discriminate between different inflationary models . in particular , if this upper bound @xmath13 is indeed realized in nature , this would imply a rather high amount of primordial gravity waves thus favoring large - fields inflationary models @xcite .    at smaller angular scales , the @xmath0-mode is dominated by the lensing induced signal . this signal is generated by the gravitational lensing of the cmb photons due to the large scale structure @xcite . the lensing contribution is well - understood from a theoretical point of view and can be uniquely predicted given the primary @xmath1-modes anisotropies and the lensing deflection field @xcite . such predictions have been been recently confirmed by the sptpol  @xcite and polarbear  experiments  @xcite , with also constraints on the cmb lensing @xmath0-mode power spectrum  @xcite . the lensing @xmath0-mode signal does not depend on @xmath2 . it therefore acts as a source of an additional noise masking the primordial , @xmath2-dependent @xmath0-mode , and making its detection more difficult . striving for a detection of @xmath2 , one has to either try to remove this lensing signal  @xcite or rely solely on the large angular scales . in this latter case two features of the primordial @xmath0-mode spectrum are of particular interest as they are anticipated to be particularly prominent . these are so - called reionization and recombination bumps peaking at @xmath14 and at @xmath15 , respectively . + measuring @xmath0-mode is made even more difficult by the fact that measurements as performed by the majority of current experiments , which scan the sky in order to produce its maps , are straightforwardly expressed only in terms of the stokes parameters , @xmath16 and @xmath17 . the @xmath1- and @xmath0-mode are mathematically related to the stokes parameters  @xcite and can be therefore recovered from the observational data . this however is only simple , if full - sky data were available . in contrast , realistic cmb experiments provide maps of polarized anisotropies , which only cover a reduced fraction of the celestial sphere , ranging from @xmath18 for balloon - borne and ground - based experiments to @xmath19 for satellite missions . in the context of pseudospectrum estimation of the angular power spectra on an incomplete sky part of the @xmath1-mode signal is unavoidably mislabelled as @xmath0-modes and vice verse . though such leakages can be corrected on average , the leaked signal inevitably contributes to the sampling variance of the other reconstructed spectrum . this dramatically increases the uncertainties of the estimated @xmath0-mode spectrum since the cosmological @xmath1-mode is expected to be at least two orders of magnitude higher than the @xmath0-mode in terms of their power spectrum  @xcite . the nature of the leakages and approaches to their removal were investigated in ref . @xcite and a relevant pseudospectrum estimator , referred to as the pure pseudospectrum estimator , was proposed subsequently in ref . this estimator has been thoroughly investigated and extended to include an optimization of the sky apodization  @xcite , cross - spectrum approaches  @xcite , and , @xmath20 and @xmath21 cross - correlations  @xcite . alternative constructions of pseudospectrum estimators correcting for @xmath1-to-@xmath0 leakages have been also proposed  @xcite . nevertheless , the pure pseudospectrum method has been found the most mature and efficient one , particularly due to its ability of optimizing the sky apodizations  @xcite , making it a method of choice for many practical applications . it is worth pointing out that the leakages are indeed ubiquities and correcting for them is as mandatory for small - scale experiments , covering @xmath18 of the sky , as for satellite - like missions , with access to as much as @xmath19 of the sky  @xcite . though the impact of the @xmath1-to-@xmath0 leakage on the variance of the @xmath0-mode power spectrum is generally acknowledged , it is rarely included in projecting performance of planned cmb experiments or instrumental concepts from the point of view of their setting constraints on the tensor - to - scalar ratio , @xmath2 . instead , the major body of work ( see  @xcite for some recent examples ) in this area is based on simplified mode - counting arguments ( see , however e.g. ,  @xcite for some exceptions ) . this stemmed mostly from the practical reasons , as the impact of the leakage is neither calculable analytically nor analysis method independent . + the objective of this work is to fill this gap and present a more systematic study of the impact of the presence of the leakage on the performance forecasts of cmb b - mode experiments . the paper consists of two parts . in the first part , sec . [ sec : cap ] , we consider idealized observations of azimuthally symmetric sky areas with homogenous noise and study differences between performance forecasts derived applying three different approaches for different assumed sky area sizes . subsequently , from these three different perspectives we revisit the issue of the optimal sky area , which would permit setting the most stringent constraints on the scalar - to - tensor ratio , @xmath2 , given a fixed length and sensitivity of the experiment .    in the second part , [ sec : realistic ] , we complete those considerations by discussing more realistic sky areas defined for three types of experiments : small - scale observations covering @xmath22% of the sky , an array of ground - based telescope covering @xmath23% and a satellite - like mission capable of delivering up to @xmath24% of the foreground clean sky . our conclusions are drawn out in sec . [ sec : conclu ] , where we also briefly sketch the implications for constraining inflationary models .    throughout this work we neglect complications such as polarized diffuse foregrounds , e.g. ,  @xcite and account for resolved points sources only by appropriately tailoring the adopted mask . we also assume that no subtraction of the lensing @xmath0-mode has been attempted  @xcite . we consider first the case of small - scale experiments in an idealized way . the observed part of the celestial sphere is assumed to be azimuthally symmetric , given by a spherical cap . we however let vary the sky coverage from 0.5% to 10% . the noise is an homogeneous , white noise , and its level is fixed at @xmath25k - arcminute for @xmath26 ( a typical level for ongoing small - scale experiments ) . for a fixed sensitivity and a fixed time of observation , the noise level ( in @xmath27k - arcminute ) scales as : @xmath28}{1\%}}\times n_p(1\%).\ ] ] the instrumental noise reprojected on the sky thus varies from @xmath29k - arcminute to @xmath30k - arcminute for an observed fraction of the sky of 0.5% and 10% , respectively . finally , the angular resolution is given by an azimuthally symmetric , gaussian beam with a width of 8 arminutes . we subsequently investigate the signal - to - noise ratio , @xmath31 , as a function of the sky coverage . this will be done considering four values of the tensor - to - scalar ratio : @xmath32 and @xmath33 . we note that the last two values are disfavored by the current data , nevertheless we include them in our considerations as they are useful in demonstrating some of the effects we describe hereafter .      translating the uncertainties on the @xmath0-mode angular power spectrum reconstruction into error bars on the measured tensor - to - scalar ratio , @xmath34 , can be done using a fisher matrix formalism .    for the rather small observed fractions of the celestial sphere here - considered , the @xmath0-mode angular power spectrum is reconstructed within bandpowers , labelled @xmath35 hereafter , with bandwidths @xmath36 . the binned power spectrum is given by @xmath37 , where the binning operator is defined as : @xmath38 ( our specific choice of the binning will be given in sec . [ ssec : powspec ] . ) the error bars on @xmath2 are then derived from the fisher matrix via : @xmath39 with @xmath40 , which stands for the covariance matrix of the reconstructed , binned angular power spectrum of the @xmath0-mode . ( note that @xmath41 denotes the _ estimator _ of the angular power spectrum , @xmath42 . ) the @xmath0-mode angular power spectrum as a function of @xmath2 is modeled as : @xmath43 with @xmath44 and @xmath45 two fiducial angular power spectra , which do not depend on @xmath2 . the former is just obtained as the contribution of primordial gravity waves for @xmath46 ( taking into account that the primordial @xmath0-mode is itself lensed ) . the latter corresponds to the contribution of primary @xmath1-mode transferred into @xmath0-mode because of the gravitational lensing of large scale structures . we do not consider here a potential _ delensing _ of the @xmath0-mode anisotropies , and such a contribution will be assumed to act as an additional gaussian noise for the measurement of @xmath2 . this is a simplifying assumption since the lensing - induced @xmath0-mode is non - gaussian leading to an additional , non - gaussian contribution to the covariance @xcite . gaussianity remains however a good approximation for bandpowers which are narrow enough ( @xmath47 ) @xcite , which is the case in our study . the covariance matrix @xmath48 is estimated using three different approaches , as described here . first , we rely on a nave mode - counting expression ( or so - called @xmath49-formula ) . in this case , the covariance on @xmath50 is approximated by : @xmath51 with @xmath52 the noise power spectrum , @xmath53 the beam of the telescope , and , @xmath49 the portion of the celestial sphere , which is observed ( or kept in the analysis ) . the noise power spectrum scales linearly with the sky coverage . the covariance matrix for the binned power spectrum is thus given by : @xmath54\delta_{b , b'}.       \label{eq : mcountcov}\ ] ] this is essentially used as a benchmark as such an evaluation of the statistical error bars on the @xmath0-mode reconstruction underestimates the error bars coming from any numerical methods to be applied to the data . second we consider the error bars that could be incurred by using a minimum variance quadratic estimator @xcite . the estimator is defined as follows : @xmath55-\tilde{n}_{\ell'}\right\}.      \label{eq : copt}\ ] ] in the above , @xmath56 is the covariance matrix of the maps of the stokes parameter , and @xmath57 is the column vector composed of @xmath58 ( the trace operation is across pixels ) . the quantity @xmath59 stands for the noise debias . finally , @xmath60 is the fisher information matrix given by : @xmath61 . \label{eq : fishopt}\ ] ] it is then shown that the covariance of the above estimator is given by the inverse of the fisher matrix , i.e. @xmath62 . we remind that this estimator is precisely built to be the quadratic estimator with the lowest variance . if the @xmath0-mode power spectrum is indeed estimated for each multipole , @xmath63 ( that is chosing @xmath64 ) , this directly gives the following expression for the error bars expected on @xmath2 : @xmath65,\ ] ] with @xmath66 the same covariance matrix _ but _ assuming that only @xmath67 does depend on @xmath2 , in line with our approach consisting in constraining the tensor - to - scalar ratio from the @xmath0-mode s measurements only given by : @xmath68,\ ] ] with @xmath69 the covariance matrix assuming that all the angular power spectra do depend on @xmath2 , the equation ( [ eq : fisher ] ) is therefore replaced by : @xmath70 in the above , the indices @xmath71 runs over @xmath72 and @xmath21 . the equation ( [ eq : fisher ] ) is finally obtained assuming that only @xmath67 in @xmath69 does depend on the tensor - to - scalar ratio . ] . in the case of azimuthally symmetric patches , the numerical computation of such fisher matrices ( either @xmath73 or @xmath74 ) , can be performed in a reasonable time using the expression found in the appendix f of ref . @xcite , and by using the s@xmath75hat package to perform spherical harmonic transforms @xcite . ( the use of this massively parallel package allows for a rapid computation of the covariance matrix for large sky coverages . ) . in the standard case , for brute force calculation the fisher matrix requires @xmath76 operations to be evaluated , but in this calculation , evaluating the spin harmonics by recursion in @xmath63 makes the computational cost as @xmath77 , where @xmath78 is the number of rings actually used .    practically speaking , one should nonetheless include the impact of binning , done as follows . first one defines the so - called _ optimal pseudospectrum _ : @xmath79-\tilde{n}_{\ell}.\ ] ] one easily checks that @xmath80 ( where we also include the impact of an azimuthally symmetric beam ) . one then introduces the matrix : @xmath81 with the interpolation operator , @xmath82 : @xmath83 the binned estimator , @xmath84 , is finally defined as : @xmath85 with @xmath86 the binned , optimal pseudospectrum . from that last definition , and making use of eqs . ( [ eq : copt ] ) and ( [ eq : fishopt ] ) , it is straightforward to show that : @xmath87_{bb_1}\left[p_{b_1\ell_1}~\mathbf{f}_{\ell_1\ell'_1}~p_{b'_1\ell'_1}\right][(\tilde{\mathbf{f}}^{-1})^\dag]_{b'_1b ' } ,       \label{eq : qmlcov}\ ] ] where summations over repeated indices ( i.e. @xmath88 and @xmath89 ) is implicitly assumed , and @xmath90 means the transpose operation . + we note that this way of estimating the uncertainties on the power spectrum reconstruction is also relevant for maximum - likelihood approaches , see e.g. ref . @xcite . third , we make use of the x@xmath75pure code and monte - carlo simulations to estimate the covariance matrix expected for the pure pseudospectrum approach . details on the pure pseudospectrum estimator can be found in refs . @xcite . in practice , the power spectrum is estimated within bandpower and the covariance matrix reconstructed from the mc simulations is directly @xmath91 . the numerical cost of this scales as @xmath92 , allowing for rapid mc simulations .    for each sky coverage and for each value of @xmath2 here - considered , we compute optimized sky apodizations to apply to the maps of @xmath16 and @xmath17 . those optimized sky apodizations are described in refs . @xcite and they allow for having the smallest error bars on the @xmath0-mode power spectrum reconstruction within the context of pure pseudospectrum techniques . those sky apodizations are a set of spin-0 , spin-1 and spin-2 window functions to be applied to the maps of the stokes parameter . they can be interpreted as the window functions , which make the pure pseudospectrum estimator as close as possible to the minimum - variance quadratic estimator @xcite . numerically speaking , computing those optimized sky apodizations may be long , especially for intricate shape of the observed region and/or low level of noise . using an iterative method , the numerical cost is @xmath93 , with @xmath94 a number of iterations ranging from few tens to few hundreds for simple patch geometry and noise level considered here ( see sec . iii in @xcite ) . we stress that for a given sky patch , those sky apodizations are to be optimized for each value of the tensor - to - scalar ratio and bin - by - bin . taking into account the number of bins ( see sec . [ ssec : powspec ] ) , the number of sky fractions and the number of values of @xmath2 , which are sampled in this study , this means that 1664 of such sky apodizations have to be computed . fortunately , in the case of homogeneous noise and patches with relatively simple contours ( which is obviously the case for a spherical cap ) , it was demonstrated in ref . @xcite that an approximated but numerically fast computation of those sky apodizations in the harmonic domain is possible , and indeed leads to error bars equal to those obtained thanks to a direct , pixel - based computation of the optimized sky apodizations . the numerical cost of this technique is reduced to @xmath92 which allows us to derive optimized sky apodizations for each value of the sky coverage , and for each value of the tensor - to - scalar ratio . our chosen bandpowers for reconstructing @xmath67 are the following . the first bin starts at @xmath95 and we use a constant bandwidth , @xmath96 . our last bin extends up to @xmath97 . the value of the maximum multipole is chosen in order to include all the relevant contributions from the primordial @xmath0-mode , that is until the lensing @xmath0-mode be the dominant contribution to the total @xmath0-mode power . in addition , for the experimental cases under consideration in this paper , we use beamwidths up to 8 arcminutes , corresponding to a cut - off of @xmath98 . the choice of the bandwidth is mainly motivated by the use of the pure pseudospectrum estimator . especially , it is mandatory for the numerical inversion of the mode - mixing matrices to be possible . we also note that the bandwidth is wide enough so that the correlations between different bins are nearly uncorrelated in the covariance of pure pseudospectrum estimator . we discuss the role of the bin width later on . we stress that the multipoles ranging from @xmath99 to @xmath95 ( corresponding to the reionization peak , and gathered in one bandpower ) are actually used in the pure pseudo-@xmath100 estimation of @xmath67 . however , given the limited sky coverages considered here such low multipoles are difficult to estimate and hardly constrained by the data . this bin is therefore not included in our analysis of the signal - to - noise ratio .    the uncertainties on the estimated power spectrum of the @xmath0-mode as functions of the sky fraction , are shown in fig . [ fig : powunc1 ] where four selected values of the sky coverage are depicted : 1% , 3.5% , 7% and 10% . the tensor - to - scalar ratio chosen for this figure is @xmath101 . each panel corresponds to a different approach to derive the covariance matrix , @xmath91 : mode - counting , minimum variance quadratic estimator , and , pure pseudospectrum estimator ( from left to right ) . as expected , we do observe that the lowest error bars are the ones from the mode - counting estimation of the uncertainties , while the highest error bars are obtained from the pure pseudo-@xmath100 estimator . the error bars from the minimum variance , quadratic estimator lie between those two . at the largest accessible scales , @xmath102 , the error bars from the pure pseudospectrum estimator are @xmath4 greater than the optimistic mode - counting estimation . similarly , the error bars from the pure pseudospectrum estimators are at most @xmath103 higher than the ones derived from the minimum variance , quadratic estimators . at the smaller angular scales where lensing dominates , the three approaches lead to almost the same uncertainties . + the behaviour of the uncertainties as a function of the sky fraction is common to the three approaches . at the smaller angular scales first ( for multipoles greater than @xmath104 ) , the behavior is monotonic since the uncertainties systematically increase with the value of @xmath49 . this is because at these scales , the variance is dominated by the noise , which increases with the sky fraction . at larger scales however ( for multipoles smaller than @xmath104 ) , the uncertainties have a more intricate behaviour . first one notes that the uncertainties _ decrease _ from @xmath26 to @xmath105 . this is because the variance is dominated by sampling variance , which decreases for higher values of @xmath49 . second , one notes that uncertainties at @xmath106 then _ increases _ for a sky coverage ranging from @xmath107 to 10% . this means that for @xmath108 , the noise is now dominating the variance . once this transition value of @xmath109 is crossed , the noise contribution dominates the variance for all our considered angular scales , @xmath110 $ ] . therefore , the variance will monotonically increase with @xmath49 at all the relevant angular scales once @xmath108 . ( note that an identical behaviour is observed for the other values of @xmath2 , though the specific value of @xmath49 at which the transition occurs depends on the specific value of @xmath2 . ) the signal - to - noise ratio on @xmath2 is computing using eq . ( [ eq : fisher ] ) considering the three above - described methods to estimate the uncertainties on the @xmath0-mode reconstruction , @xmath91 . we remind that the summation in ( [ eq : fisher ] ) is performed over bandpowers with a bandwidth of @xmath96 and considering a range of multipoles from @xmath95 to @xmath97 . our numerical results on the signal - to - noise ratio for @xmath2 are gathered in fig . [ fig : snrfsky ] , showing @xmath111 as a function of the sky coverage . each panel corresponds to a given value of the tensor - to - scalar ratio , @xmath32 , and , @xmath33 ( from top to bottom ) . for each panel , the black , red , and blue crosses correspond to the signal - to - noise ratio derived by using the mode - counting , the minimum variance quadratic estimator , and , the pure pseudo-@xmath100 estimator , respectively . the horizontal , dashed line marks a @xmath112 detection . the sky fraction varies from 0.1% to 10% , what is wide enough to sample the maximal values of the signal - to - noise ratio . we note that the signal - to - noise ratio keeps decreasing for @xmath113 . this is because for the level of noise and values of @xmath2 here considered , the uncertainties on the reconstructed @xmath0-mode are noise dominated at all scales for @xmath113 . similarly , the ( s / n)@xmath114 keeps decreasing for @xmath115 , because the uncertainties on angular scales greater than a degree are dominated by the sampling variance for such low values of the sky fraction . +     from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm   from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm   from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm   from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig : " ]    for the case of the mode - counting first , the signal - to - noise ratio is systematically greater than 3 for all the considered values of the sky coverage , and for all the considered values of the tensor - to - scalar ratio . considering then the case of the minimum variance , quadratic estimator , the signal - to - noise ratio on @xmath2 is systematically greater than 3 for @xmath116 and @xmath33 . for a tensor - to - scalr ratio of @xmath117 , the ( s / n)@xmath114 is greater or equal to three for @xmath118 . assuming finally a pure pseudospectrum reconstruction of the @xmath0-mode , the signal - to - noise ratio is systematically greater than 3 for @xmath119 and @xmath120 only . for a tensor - to - scalar ratio of @xmath101 , a measurement of it with a ( s / n)@xmath114 of at least 3 , is possible for @xmath121 . for a smaller value of @xmath117 , its measurement with ( s / n)@xmath122 is possible assuming @xmath123 . we note however here that for @xmath117 and @xmath101 , the signal - to - noise ratios remains greater than 2 assuming a pure pseudospectrum reconstruction of @xmath42 . + as expected from the error bars on the reconstructed @xmath0-mode , the highest and lowest ( s / n)@xmath114 s are respectively obtained from the mode - counting estimation , and the pure pseudospectrum estimator , while the ( s / n)@xmath114 from the minimum variance quadratic estimator lies between those two . this is the case for all the values of the tensor - to - scalar ratio we consider . at the peak , the signal - to - noise ratio from the pure pseudospectrum estimation of the @xmath42 is @xmath12415% ( @xmath120 ) to @xmath12420% ( @xmath117 ) smaller than the signal - to - noise ratio derived from the optimistic mode - counting . this means that the statistical significance on the measurement of @xmath2 by using the optimistic mode - counting is overestimated by a factor @xmath4 as compared to the more realistic case of the pure pseudoreconstruction of the @xmath0-mode .    similarly , the ( s / n)@xmath114 from the pure pseudospectrum estimator is @xmath1241.5% ( @xmath120 ) to @xmath1248% ( @xmath117 ) smaller than the signal - to - noise ratio derived from the minimum variance , quadratic estimator . using the minimum variance , quadratic estimator to estimate the @xmath0-mode , as compared to the use of the pure pseudospectrum , thus translates into a gain in the statistical significance on the measurement of @xmath2 , of a factor 1.01 to 1.08 . this gain appears rather small but is larger for smaller values of the tensor - to - scalar ratio .      as clearly shown in figs . [ fig : snrfsky ] , there exists a value of the sky coverage , which maximizes the signal - to - noise ratio on @xmath2 . this _ optimal _ value of @xmath49 was already observed in ref . @xcite , using only the mode - counting expression for the statistical error bars on the @xmath0-mode estimation though . we found that such an optimal value also exists using the minimum variance quadratic estimator or the pure pseudo-@xmath100 estimator . this is intuitively understood as follows . the _ statistical _ uncertainties on the angular power spectrum estimation have two sources , the sampling variance , which is dominant at the largest angular scales , and the noise variance dominating at the smallest angular scales . reducing the sampling variance is obtained by covering a large fraction of the sky . however , for a given sensitivity and a given time of observation , covering a large fraction of the sky inevitably translates into a higher level of noise per pixel . one should therefore find the good balance between sampling and noise variance so as to minimize the total error on given targetted parameters , which is @xmath2 here . the salient features of those results are summarized in the table [ tab : fsky ] . for each value of the tensor - to - scalar ratio and for each techniques used to compute uncertainties on the @xmath0-mode , we provide the values of the sky fraction maximizing the signal - to - noise ratio , @xmath125 . its associated ( thus maximal ) value of the signal - to - noise ratio , ( s / n)@xmath126 is also reported in this table . we stress that the position of the peak of ( s / n)@xmath114 is well defined for the mode - counting and the minimum variance , quadratic estimator . such a position of the peaking signal - to - noise ratio is however less pronounced for the case of the pure pseudo-@xmath100 estimation of the @xmath0-mode ( see e.g. the case @xmath127 for which a range of @xmath128 leads roughly to the same ( s / n)@xmath114 ) . this means that the values of @xmath125 reported in tab . [ tab : fsky ] for the case of the pure pseudospectrum approach are more indicative than a sharply defined value . +    lm0.5cmm0.75cmm0.75cmm0.75cmm0.75 cm @xmath2 & & 0.07 & 0.1 & 0.15 & 0.2 + @xmath129 $ ] : & & & & & + mode - counting & & 2.0 & 3.0 & 4.0 & 5.0 + minimum - variance @xmath100 & & 2.5 & 2.5 & 3.5 & 5.0 + pure pseudo-@xmath100 & & 3.5 & 3.5 & 5.0 & 4.0 + ( s / n)@xmath126 : & & & & & + mode - counting & & 4.4 & 5.5 & 7.0 & 8.2 + minimum - variance @xmath100 & & 3.7 & 4.7 & 5.9 & 7.0 + pure pseudo-@xmath100 & & 3.4 & 4.4 & 5.8 & 6.9 +    for all the approaches used to estimate the uncertainties on the @xmath0-mode , we observe that the optimal sky fraction increases with the value of the tensor - to - scalar ratio . this is because for higher values of @xmath2 , the signal in the @xmath0-mode is higher . one should therefore minimize first the sampling variance by increasing the observed part of the sky . + except for the case of @xmath120 , we note that the optimal sky coverage assuming a minimum variance , quadratic estimator slightly differs by 0.5% ( either higher or lower ) than the value of @xmath130 ( in % ) as inferred from the mode - counting . we also note that the optimal sky fraction obtained for the pure pseudo-@xmath100 reconstruction of the @xmath0-mode differs by 1% to 1.5% ( depending on the value of @xmath2 ) from the one inferred from the mode - counting estimation of the uncertainties on the @xmath0-mode . nevertheless , the values of the sky fraction for which the detection of the tensor - to - scalar ratio is peaking in the case of the mode - counting expression and the minimum - variance quadratic estimator fall in the range of optimized @xmath49 as derived from the pure pseudo-@xmath100 estimator . those numerical results therefore show that ( at least ) for the range of values of @xmath2 here - considered , the value of the sky coverage , which maximizes the measurement of the tensor - to - scalar ratio is rather independent on the adopted method for evaluating the statistical uncertainties on the @xmath0-mode reconstruction . this means that using the mode - counting expression , though underestimating the error bars , allows for a rapid and reliable search of the range of values of the optimized sky fraction . ( obviously , such an optimization of @xmath49 based on the mode - counting expression is reliable providing the final data set to be analyzed using either the minimum - variance quadratic estimator or the pure pseudospectrum estimator . )      for the two specific cases of the mode - counting uncertainties and the minimum variance , quadratic estimator , we note that an explicit reconstruction of the power spectrum is not mandatory to derive the ( s / n)@xmath114 in the fisher formalism . one can indeed directly plugged in eq . ( [ eq : fisher ] ) the formulas ( [ eq : mcountcov ] ) or ( [ eq : qmlcov ] ) . this allows for a study of the impact of binning on the signal - to - noise ratio , letting the bandwidth to vary from @xmath64 ( i.e. no binning ) to @xmath96 ( i.e. the binning imposed by the use of the pseudospectrum estimator in this analysis ) .     as a function of the observed sky fraction , derived for three methods used to estimate the uncertainties on the @xmath0-mode reconstruction : mode - counting ( black area ) , minimum variance quadratic estimator ( red area ) , and , pure pseudospectrum estimator ( blue crosses ) . for the two first methods , we let the bandwidth of the bins to vary from @xmath64 ( highest ( s / n)@xmath114 ) to @xmath96 ( lowest ( s / n)@xmath114 ) . for the specific case of the pure pseudospectrum estimator , the reconstruction of the @xmath42 requires to use the bandwidth @xmath96 . ( we remind that the range of multipoles used to compute the signal - to - noise ratio is @xmath131 . ) ]    the impact of binning is illustrated in fig . [ fig : snrbin ] showing the signal - to - noise ratio on @xmath101 as a function of the sky coverage . the grey ( red ) area corresponds to the ( s / n)@xmath114 using the mode - counting ( minimum variance quadratic estimator ) to estimate the uncertainties the angular power spectrum of the @xmath0-mode . for each shaded area , the highest signal - to - noise ratio is obtained for @xmath64 and the lowest for @xmath96 . as a reference , we also show the ( s / n)@xmath114 obtained with pure pseudospectrum reconstruction ( thus using a bandwidth of @xmath96 ) depicted by the blue crosses . the overall effect of increasing the width of the bandpower is to lower the signal - to - noise ratio . the decrease is however more pronounced for the case of the minimum variance , quadratic estimator than for the mode - counting estimation of the error bars on the reconstructed @xmath0-mode . this is due to the fact that correlations between multipoles ( or bandpowers ) are accounted for in the minimum variance , quadratic estimator , while those are supposed to be systematically vanishing for the mode - counting estimation of the covariance matrix . this additional piece of information contained in the correlations is therefore partially lost by averaging over bandpowers . we also checked that artificially imposing those off - diagonal correlations to be zero lowered the signal - to - noise ratio in the minimum variance method , although we note that once the bins are sufficiently wide the effect of the binwidth on the ( s / n)@xmath114 should be weak . the maximum values of the ( s / n)@xmath114 obtained using a bandwidth of @xmath64 , and a bandwidth of @xmath96 , are reported in tab . [ tab : bin ] , for each values of @xmath2 and for the mode - counting and the minimum variance quadratic estimator . for each cases , we also report the value of the sky coverage corresponding to that maximum . for the mode - counting approach , increasing the bandwidth from @xmath64 to @xmath96 , degrades the maximum ( s / n)@xmath114 by a factor @xmath132 for @xmath117 and @xmath133 , and , by a factor @xmath134 for @xmath119 and @xmath33 . this however only mildly affects the values of the sky fraction at which the maximum is achieved . the impact of binning is more marked for the minimum variance quadratic estimator however . increasing the bandwidth from @xmath64 to @xmath96 , here degrades the maximum ( s / n)@xmath114 by a factor @xmath135 for all the values of the tensor - to - scalar ratio considered in this study . similarly , the values of the sky fraction ( in % ) at which this maximum is achieved is systematically lowered ( except for the case @xmath127 ) , by 1% for @xmath101 and by 2% for @xmath120 . we note that despite these changes in the value of @xmath130 with the bandwidth , the optimized values of the sky fraction still fall in the range of optimized @xmath49 as derived from the pure pseudo-@xmath100 estimator . @xmath2 & & 0.07 & 0.1 & 0.15 & 0.2 + ( s / n)@xmath126 and @xmath125 : & & & & & + mode - counting : & & & & & + @xmath64 & & 4.6  ( 2.5% ) & 5.7  ( 3% ) & 7.2  ( 4% ) & 8.4  ( 5.5% ) + @xmath96 & & 4.4  ( 2.0% ) & 5.5  ( 3% ) & 7.0  ( 4% ) & 8.2  ( 5% ) + minimum - variance @xmath100 : & & & & & + @xmath64 & & 4.2  ( 2.5% ) & 5.3  ( 3.5% ) & 6.8  ( 5% ) & 8.0  ( 7% ) + @xmath96 & & 3.7  ( 2.5% ) & 4.7  ( 2.5% ) & 5.9  ( 3.5% ) & 7.0  ( 5% ) + we turn to the question of the detection of @xmath2 in more realistic cases . clearly , a spherical cap is ideal . the issue of leakages is strongly related to the detailed shape of the contours of the observed ( or kept - in - the - analysis ) portion of the sky ( see e.g. the figure 20 of ref . @xcite for the impact of the shape of the mask on the statistical error bars ) . a spherical cap then leads to the smallest amount of leakages for a given sky fraction since its contour has the smallest perimeter for that given sky fraction . to this end , we consider three archetypal cases , which capture the main characteristics of ongoing , or being - deployed , small - scale experiments ( ground - based or balloon - borne ) , a possible upgrade of those ground - based experiments to an array covering a rather large fraction of the sky ( @xmath136 ) , and , a possible satellite mission covering the entire celestial sphere . +    [ cols="^,^,^ " , ]     a couple of comments about the numerical computation of the pixel - based , minimum - variance sky apodizations are in order here . they are theoretically built to give the smallest uncertainties in the context of the pure pseudospectrum estimators . however , they are pratically computed from a preconditionned conjugate gradient ( pcg ) algorithm , which efficiency strongly depends on the experimental configurations , especially with respect to the noise level and its distribution over the patch , as well as with respect to the complexity of the contours of the mask . first , the number of iterations in our implemented pcg rapidly increases for lower levels of noise : at the largest angular scales ( @xmath137 ) the number of iterations ranges from @xmath104 for a noise level of 5.75@xmath27k - arcminute to @xmath138 for a noise level of 1@xmath27k - arminute ( the number of iterations being one order of magnitude smaller for smaller angular scales , @xmath139 ) . the @xmath0-mode angular power spectrum is estimated using the same binning as in the previous section , leading to @xmath140 , and we selected 6 values of @xmath2 . considering 3 experimental setups , this would translate into @xmath124500 optimized sky apodization to compute , which is numerically too costly . we therefore compute the optimized sky apodization for @xmath141 only but use them for all the here - considered values of @xmath2 , meaning that the signal - to - noise ratios obtained for @xmath142 may be suboptimal within the context of the pure pseudospectrum estimator - modes leaking into the @xmath0-mode , and poorly affected by the amplitude of the primordial @xmath0-mode . this means that the resulting sky apodizations may be mildly dependant on the assumed value of @xmath2 and that the derived signal - to - noise ratios are only slightly suboptimal for the case of small - scale experiments . ] . second , it is not guaranteed that the algorithm converges towards the optimal solution , especially for inhomogeneous noise ( see sec . iv c of ref . @xcite where it was shown that trimming out the external , noisiest pixels is required ) , or a very low level of noise ( see ref . @xcite mentionning that convergence is not reached for a noise level of @xmath143k - arminute , corresponding to the level of the array of telescopes case ) . this means that the performances of those sky apodizations have to be assessed using numerical simulations at the level of power spectrum reconstruction , comparing the resulting error bars on the estimated power spectra to the error bars obtained by using the other types of sky apodizations . the relative performances of the different sky apodizations are appraised at the level of power spectrum uncertainties . for each case we performed a series of 500 monte - carlo simulations to compute the statistical uncertainties on the reconstructed @xmath0-mode angular power spectra , assuming different kinds of sky apodizations . such performances have been exhaustively studied for the small - scale experiment case and the satellite mission case ( see refs . @xcite and ref . @xcite , resp . ) . on the contrary , the applicability of the pure pseudospectrum estimator for the case of an array of telescopes was hitherto not studied . we then performed numerical simulations using the different classes of sky apodizations to assess the efficiency of the pure pseudospectrum reconstruction of the @xmath0-mode , and subsequently select those sky apodizations , which lead to the smallest uncertainties .    in this section , we only briefly review the major conclusions concerning the cases of a small - scale experiment and a satellite mission . then , we present the results of our numerical investigations for the case of an array of telescopes .      for the small - scale experiment case , it was shown that the lowest uncertainties in the range @xmath144 $ ] were obtained using either the pixel - based optimized sky apodizations or analytic sky apodizations appropriately chosen to minimize the variance per each bin . however , the harmonic - based computation of the sky apodization fails in providing error bars comparable to the previous ones in the entire range of multipoles considered here , simply because the noise is inhomogeneous . ( we refer to fig . 24 of ref . @xcite and discussions therein . )    for the satellite mission case , the pixel - based computation of the minimum - variance sky apodization yields the smallest uncertainties for the range @xmath144 $ ] . similar performance is obtained by using the harmonic - based computation of these sky apodizations for @xmath145 . the error bars however drastically increase for larger angular scales , the reason for that being the intricate contours of the galactic mask and the point - sources mask , which require to relax the neuman and dirichlet boundary conditions to keep ( part of ) the informations about the @xmath0-mode contained in the ambiguous modes . for such a case , analytic sky apodizations fails in providing comparable error bars at the largest angular scales . ( we refer to fig . @xcite and discussions therein . )      for the case of an array of telescopes , we systematically search for the type of sky apodizations , which lead to the smallest uncertainties bin per bin and for each values of @xmath2 considered in this study . we first found that the harmonic - based , optimized sky apodizations yield to error bars higher than the analytic sky apodizations or the pixel - based , optimized sky apodizations . this is similar to what was observed for the case of a satellite mission and the inefficiency of the harmonic - based , optimized sky apodizations is due to the complexity of the contours of the mask . an example of the uncertainties for @xmath101 and using the pixel - based , optimized sky apodizations ( the dashed - red curve ) or the analytic ones for different values of the apodization length ( the dashed - blue curves ) is shown in fig . [ fig : clhalfsky ] . the solid - black curve stands for the input angular power spectrum and the dashed - black curve stands for the binned , mode - counting computation of the error bars . this first shows that the pixel - based , optimized sky apodizations perform the best at the largest angular scales . this is systematically so for the two first bins . for the third bin , the pixel - based , sky apodizations and the analytic ones perform the same for @xmath146 , while for @xmath147 , the analytic sky apodizations with an apodization length of 4 degrees lead to a smaller error bar than the pixel - based , optimized sky apodization . however at smaller scales , @xmath148 , the smallest error bars are systematically obtained by using an analytic sky apodizations with an apodization length of 2 degrees for intermediate scales , @xmath149 , and an apodization length of 1 degree for small scales , @xmath150 . we found this to be independant of the value of @xmath2 ( at least for the grid of values considered here ) . + we note that the apparent failure of the pixel - based , optimized sky apodizations here may be rather due to practical difficulties in computing such apodizations sufficiently accurately , rather than an indication of some fundamental problems . indeed , we have found that for the noise levels the iterative solver used to compute the apodizations converges extremely slowly ( as also observed in ref . @xcite ) potentially preventing us in practice from achieving sufficient precision . -mode for the case of an array of telescopes . the solid - black curve stands for the input angular power spectrum with a tensor - to - scalar ratio equal to @xmath101 . the dashed - black curve is for the binned , mode - counting uncertainties used as a benchmark . the error bars obtained by using the pixel - based , optimized sky apodizations ( called pcg in the figure legend ) is represented by the dashed - red curve . the different dashed - blue curves correspond to the error bars obtained by using analytic sky apodizations with an apodization length ranging from 1 degree to 6 degrees . ]      as a summary , the smallest statistical uncertainties obtained for @xmath101 are shown in fig . [ fig : cellallexp ] in which the orange , red and burgundy curves stand for the small - scale experiment , an array of telescopes and a satellite mission respectively . for each experimental setup , we show the smallest error bars , which are attained for each bandpower . for the cases of a small - scale experiment and a satellite mission , this is obtained by using the pixel - based , optimized sky apodizations throughout the entire range of angular scales . for the case of an array of telescopes , the pixel - based , optimized sky apodizations are used for multipoles lower than 100 , while analytic sky apodizations with an apodization of 2 degrees and 1 degree are used in the range @xmath151 and in the range @xmath152 , respectively . +    -mode s angular power spectra with the pure pseudospectrum estimator . the orange , red and burgundy lines stand for the small - scale experiment , an array of telescopes and a satellite mission respectively . the black line corresponds to the input angular power spectrum for @xmath101 . ]    as expected , the higher uncertainties are the ones from a small - scale experiment due to the tiny fraction of the sky it covers , and the relatively high level of instrumental noise . we provide the uncertainties for the first bandpower , @xmath153 , for completeness . these scales are nonetheless unaccessible starting from a map covering 1% of the sky due to the high uncertainties , as already stated in sec . [ sec : cap ] .    for angular scales going from @xmath99 to @xmath154 , the smallest error bars corresponds to a satellite mission . this is because at these angular scales , the uncertainties are dominated by sampling variance and a satellite mission , as compared to an array of telescopes , benefits from its larger sky fraction . for multipoles smaller than @xmath104 , the error bars from a satellite mission are roughly 1.5 to 2 times smaller than the error bars obtained from an array of telescopes , which is in line with the fact that the sky fraction for a satellite mission is 2 times higher than the sky fraction observed by an array of telescopes , thus reducing the error bars by a factor of @xmath155 as compared to the error bars from an array of telescope . nevertheless , at small scales , @xmath156 , smaller error bars are obtained from an observation by an array of telescopes . this is because in that regime , the uncertainties for the case of a satellite mission are dominated by the noise term , @xmath157 . since the noise for a satellite mission is four times higher ( in power spectrum ) than the noise for an array of telescopes , and the beam is more than two times higher , this increase of the variance largely overcome the decrease due to a larger sky coverage . this quantitatively explains why at those small angular scales , the lowest error bars on the @xmath0-mode reconstruction are obtained from an array of telescope . ( one can even notice that for the range of angular scales considered here , the uncertainties obtained for an array of telescopes in sampling variance dominated . ) the computation of the signal - to - noise ratio on the tensor - to - scalar ratio is done by using the same fisher matrix formalism as employed in the previous section , eq . ( [ eq : fisher ] ) . for each experimental configurations and each value of @xmath2 , we select the smallest error bars we obtained _ bin per bin_. this means that for the specific case of an array of telescope , the estimation of the @xmath0-mode angular power spectra is done by mixing different kind of sky apodizations . we use the same bandpowers as in sec . [ sec : cap ] and now add the largest angular scales , from @xmath99 to @xmath95 gathered in one single bandpower , in the summation in eq . ( [ eq : fisher ] ) . adding these scales is relevant for the case of an array of telescopes , and the case of a satellite mission . we will first add this bandpower at the largest scales for the three experimental setups , sec . [ sssec : numres ] . we will subsequently study its impact on the measurement of @xmath2 , sec . [ sec : firstbin ] . ( note that we use the binned covariance for both the modecounting and the pure pseudospectrum reconstruction of @xmath42 . ) our results on the signal - to - noise ratio for @xmath2 ranging from @xmath158 to @xmath33 are shown in fig . [ fig : snrallexp ] ( note that for the specific case of a satellite mission the value @xmath159 has been added in order to fall below the @xmath112 limit ) . the red and black crosses correspond to a covariance matrix computed using the mode - counting expression for error bars on @xmath67 , and the pure pseudospectrum error bars , respectively . the solid red line is the @xmath112 limit . the left panel corresponds to a small - scale experiment covering @xmath1241% of the celestial sphere with a highly inhomogeneous noise distribution . the middle panel corresponds to an array of telescopes covering @xmath12436% of the sky with a low level of ( homogeneous ) noise . finally , the right panel is for a satellite mission covering @xmath12471% of the sky with a low level of homogeneous noise . +         for the case of a small - scale experiment , the signal - to - noise ratio on @xmath2 ranges from 0.06 for @xmath160 to 4 for @xmath120 assuming a pure pseudo-@xmath100 reconstruction of the @xmath0-mode power spectrum ( meaning that a `` measurement '' of @xmath160 would be consistent with @xmath161 ) . this has to be compared to what would be inferred from the idealized mode - counting evaluation of the uncertainties , for which the signal - to - noise ratio varies from 0.25 for @xmath160 to 6.7 for @xmath120 . for @xmath162 and @xmath133 , the ( s / n)@xmath114 derived from a mode - counting estimation of the uncertainties on the @xmath0-mode is overestimated by a factor @xmath163 , @xmath164 and @xmath165 , resp . , as compared to the ( s / n)@xmath114 obtained from a pure pseudo-@xmath100 reconstruction of the angular power spectrum .    for the case of an array of telescopes and assuming the pure pseudosepctrum estimation of the @xmath0-mode , the signal - to - noise ratio varies from 0.67 to 41 with @xmath2 varying from 0.001 to 0.2 . values of @xmath166 and @xmath101 would be measured with a statistical significance of 5.75@xmath167 and @xmath168 , respectively . using instead the mode - counting estimation of the uncertainties on @xmath67 , the ( s / n)@xmath114 varies from 3 to 54 for @xmath2 ranging from 0.001 to 0.2 . for the three selected values of @xmath162 and 0.1 , the signal - to - noise ratio obtained from the mode - counting approach is respectively overestimated by a factor 4.5 , 2.5 and 1.3 , as compared to the realistic ( s / n)@xmath114 derived from the pure pseudo-@xmath100 estimation of @xmath67 .    for the case of a satellite mission , the signal - to - noise ratio varies from 0.66 for @xmath169 to 59 for @xmath120 , and assuming the pure pseudospectrum estimation of @xmath67 . the values of @xmath160 , @xmath166 and @xmath101 would be detected with a statistical significance of 1.34 , 10.84 and 46.19 , respectively . if one instead makes use of the mode - counting estimation of the error bars on the @xmath0-mode reconstruction , the ( s / n)@xmath114 varies from 2 to 72 for values of the tensor - to - scalar ratio ranging from 0.0005 to 0.2 . for @xmath162 and @xmath133 , the mode - counting evaluation overestimates the signal - to - noise ratio , as compared to the pure pseudo-@xmath100 reconstruction , by a factor 2.2 , 1.59 and 1.22 . + from a qualitative viewpoint , the signal - to - noise ratios computed in the framework of the mode - counting expression are always higher compared to the signal - to - noise ratios assuming the pure pseudospectrum reconstruction of @xmath0-mode . ( this is obviously expected from the fact that the mode - counting approach is an idealized and _ underestimated _ computation of the uncertainties . ) we observe that the overestimation using the mode - counting expression ( as compared to the more realistic pure pseudospectrum reconstruction of @xmath67 ) is less marked for higher values of @xmath2 . this behavior is common to the three experimental configurations here - considered , though there are differences from a quantitative viewpoint . the reason is that for low values of @xmath2 , most of the information comes from the largest scales , which is precisely at those large scales that the underestimation of the @xmath0-mode reconstruction using the mode - counting formul  is more marked . we also stress that in the case of mode - counting approach , the leakages are ignored . on the contrary , the pure pseudospectrum approach consistently includes them but correct them in the analysis . this explains why the mode counting approach overestimate the signal - to - noise ratio on @xmath2 . cmb observations covering a large fraction of the sky are automatically contaminated by various astrophysical foregrounds with complex physics involved among which the emission from our galaxy is the strongest . masks are used to remove from the analysis the portion of sky with the highest foreground level , but the foreground emission is present on the entire celestial sphere . usually techniques - such as parametric component separation @xcite used to determinate the spectral parameters or template fitting method , which deprojects the template of the foreground from the map @xcite - are used to minimize the impact of the foreground . the residual level of foreground contaminants depends on the technique actually chosen . however , the power spectrum of the galactic dust , polarized emission ( which is the major contaminant of cmb measurements at frequencies above @xmath104ghz ) behaves as @xmath170 , to be compared to @xmath171 for the cmb @xmath0-mode angular power spectrum at scales above a degree @xcite . the impact of such a galactic foreground is therefore expected to be more pronounced at the largest angular scales .    here , we considered the worst case scenario where the foreground contamination could not be removed at all on the largest scale , meaning that the information from the reionization peak is no more taken into account in the computation of the signal - to - noise ratio . in practice , we discard the first bin ( @xmath172 ) from the analysis , which necessarily lowers the signal - to - noise ratio on @xmath2 . we define this relative decrease as : @xmath173 with ( s / n)@xmath114 the signal - to - noise on @xmath2 accounting for _ all _ the angular scales , and ( s / n)@xmath174 the signal - to - noise ratio obtained by _ discarding _ the first bandpower . this relative decrease can alternatively be interpreted as the relative contribution from the first bin to the signal - to - noise on @xmath2 since : @xmath175 with ( s / n)@xmath176 the signal - to - noise ratio on @xmath2 that would be obtained by using the first bandpower _ only_. + this relative decrease of ( s / n)@xmath114 is shown in fig . [ fig : degradesnrallexp ] . the red crosses correspond to the mode - counting estimation of the error bars on the reconstruction of @xmath67 while the black crosses correspond to the error bars from a pure pseudospectrum estimation of @xmath67 . the left , middle and right panels respectively stand for the case of a small - scale experiment , an array of telescopes and a satellite mission . the case of a small - scale experiment is poorly affected by the removal of the first bin using the pure peudospectrum reconstruction of @xmath67 , the relative decrease being systematically smaller than 0.1% . this is because in such a case the signal - to - noise ratio for the first bandpower , @xmath177 , is much smaller than unity for all the values of @xmath2 considered here . this bandpower therefore does not bring any significant amount of informations on @xmath2 . this drastically differs if one uses the mode - counting evaluation for which @xmath178 varies from 0.7% for @xmath120 to 32% for @xmath160 . this is because in this case , the signal - to - noise ratio in the first bandpower , @xmath179 with @xmath153 , becomes greater than unity though the sky coverage is only of @xmath18 . the fact that the relative decrease is more pronounced for small values of @xmath2 is understood as follows . for lower values of @xmath2 , the recombination bump at the degree scale , falls below the lensing part of the @xmath0-mode while the reionization bump in the first bandpower remains above the lensing signal . as a consequence , the reionization peak carries more information , relative to the informations carried by the recombination peak , for lower values of the tensor - to - scalar ratio .    for the case of an array of telescopes , the relative decrease ranges from 0.4% for @xmath120 to roughly 3% for @xmath160 . we note that this relative decrease is now roughly constant from @xmath160 to @xmath166 and then decreases for higher values of the tensor - to - scalar ratio . this behavior of @xmath178 is explained by the very same reason explaining why @xmath178 decreases for higher values of @xmath2 if one makes use of the mode - counting estimation of the uncertainties on the estimated @xmath67 , and because for an array of telescopes , the angular power spectrum in the first bandpower can now be measured with a signal - to - noise ratio greater than unity . we note that the relative decrease using the mode - counting behaves the same as in the case of a small - scale experiment ( with minor quantitative differences at high values of @xmath2 ) .    for the case of a satellite mission , the relative decrease varies from 0.35% for @xmath120 to 9% for @xmath169 , with @xmath180 for @xmath160 . this relative decrease now monotonically increases with lower values of @xmath2 ( though our results suggest that a plateau is reached for @xmath181 ) . this behavior is explained by the same reason explaining the decrease of @xmath178 for higher values of @xmath2 in the case of an array of telescopes . we also note that @xmath178 obtained from the mode - counting expression behaves the same as in the case of a small - scale experiment and an array of telescopes . + as is clear from fig . [ fig : degradesnrallexp ] , the shape of @xmath178 as derived using the mode - counting expression , is qualitatively the same for the three experimental configurations , though sky fractions and shapes of the masks drastically change . this is because the impact of the limited sky fraction is simply modelled as an overall renormalization of the error bars , equally applied at all angular scales ( see eq . ( [ eq : mcount ] ) ) . neglecting the noise contribution to the error bars on the @xmath0-mode reconstruction ( which is a relatively fair assumption here ) , it is easy to figure out that this overall @xmath182 does not enter in the final expression of @xmath178 . ( we note that minor differences are however expected because of the different noise level and beamwidth . ) at low values of @xmath2 , the relative decrease is much less marked in the context of the pure pseudospectrum reconstruction of @xmath67 , with respect to the mode - counting expression . this is because the different leakages have stronger impacts at large scales ( in term of increase of the error bars on the estimated @xmath67 ) , thus reducing the relative contribution of the first bandpower to the constraint that can be set on @xmath2 . the impact of leakages in terms of error bars on @xmath67 at large angular scales increases with smaller @xmath49 , which therefore reduces the relative contribution of the first bandpower to the constraints on @xmath2 . this is clearly seen in fig . [ fig : degradesnrallexp ] where @xmath178 is more important from the case of a small - experiment , to the case of an array of telescopes , to the case of a satellite mission .      as a result , at a given @xmath2 and considering all the angular scales from 2 to 1020 , the value of the signal - to - noise ratio is the highest in the case of a satellite mission . as an example , a tensor - to - scalar ratio @xmath183 would be detected at a statistical significance of about @xmath184 . in the case of an array of telescope , the value of the signal - to - noise ratio remains high for a large range of values of the tensor - to - scalar ratio , showing a detection of 28@xmath167 at @xmath185 . finally , a small scale experiment would set mild constraints on low values of the tensor - to - scalar ratio , reaching 3@xmath167 at @xmath185 . + in the frame of the primordial @xmath0-mode detection prospects , the minimal value of the tensor - to - scalar ratio @xmath2 that could be detected regarding the experimental setups is a relevant result . the table [ tab : snr ] summarizes in this perspective the aforementioned results , considering a measurement of @xmath2 with at least a 3@xmath167 statistical significance as a threshold . the minimal accessible value of @xmath2 is shown with respect to the experimental setups and the estimation of the @xmath0-mode variance over all the range of multipoles ( referred to as _ case a _ in the table ) . as explained above , the mode counting estimation of the variance overestimates the forecasts made on the minimal accessible @xmath2 as compared to the realistic @xmath0-mode estimation . in the case of a potential satellite mission for instance , the lowest accessible @xmath2 we could realistically expect is 2.88 greater than the one estimated using the mode - counting estimation . these results therefore highlight the inaccuracy that an approximative estimation of the @xmath0-mode induces on the performed forecasts of the detectable @xmath2 values . thus considering the realistic forecasts performed thanks to the pure estimation of the @xmath0-mode power spectrum , we conclude that a satellite mission would give access to the largest range of @xmath2 , with a minimal @xmath2 value of @xmath186 . a typical small scale experiment is indeed expected to reach only @xmath2 higher than @xmath133 at 3@xmath167 ( note that @xmath141 is detectable at @xmath187 ) . between these two cases lies the one of an array of telescopes , which warrants a detection of the tensor - to - scalar ratio if it is higher than @xmath188 . as a result , each studied experiments widens the accessible range of the tensor - to - scalar ratio @xmath2 . in terms of minimal detectable vlaue of @xmath2 , one gains about a factor 20 from small - scale experiments to an array of telescopes , and about a factor 2 between the latter and a satellite mission . +    p2.4cmm1.9cmm1.9cmm1.9 cm & small - scale exp . & telescopes array & satellite mission + mode - counting : & & & +  case a & @xmath189 & @xmath190 & @xmath191 +  case b & @xmath192 & @xmath193 & @xmath194 + pure pseudo-@xmath100 : & & & +  case a & @xmath195 & @xmath196 & @xmath197 +  case b & @xmath195 & @xmath198 & @xmath199 +    furthermore , the table [ tab : snr ] also displays the minimum accessible @xmath2 obtained without the information from the first bin ( @xmath200 ) of the @xmath0-mode power spectrum ( referred to as the _ case b _ ) . this lack of information obviously leads to a smallest range of accessible @xmath2 than in the _ case a _ for a naive estimation of the @xmath0-mode variances , as explained in the previous subsection . in particular , while the accessible @xmath2 range is little affected by removing the first bin in the case of a small scale experiment , the minimal accessible @xmath2 is @xmath201 ( @xmath202 resp . ) greater for a satellite mission ( an array of telescopes resp . ) as the largest angular scales are relevant for these experimental setups . we note here that contrary to what one might expect , a large scale experiment would still succeed in detecting @xmath2 of at least @xmath203 . for @xmath63 between 20 and 90 , the amplitude of the primordial signal is roughly 10% of the lensing signal while the total ( mode - counting estimated ) error budget varies from few percents to 10% of the lensing signal . summing over the multipoles range thus enables a detection of @xmath204 with a @xmath112 statistical significance .    nonetheless , in this _ case b _ , the orders of magnitude of the realistic forecasts remain unchanged if the @xmath0-mode power spectrum is reconstructed from the pure pseudo-@xmath100 approach . the values of @xmath2 that could be detected at @xmath112 increase by a factor of less than one percent for a small - scale experiment , a factor of few percents for an array of telescopes , and , a factor of ten percents for a satellite mission . ( this obvisouly reflects the values of @xmath178 found in the previous section , sec . [ sec : firstbin ] . ) this means that the pure pseudo-@xmath100 estimation of the reionization peak of the @xmath0-mode mildly constraints the tensor - to - scalar ratio . to take full advantages of the range @xmath205 ( so as to lower the minimal detectable value of @xmath2 and to enlarge the lever arm to constraint e.g. the spectral index ) , one should probably rely on more optimal techniques for reconstructing @xmath67 at those largest angular scales at large scales is also plagued by other sources of uncertainties such as the level of residual foregrounds and/or the impact of filtering of the maps . ] . we have investigated the detection of the tensor - to - scalar ratio , @xmath2 , from forthcoming and potential future measurements of the cmb polarized anisotropies . we considered the @xmath0-mode as the main source of information on @xmath2 and assumed the pure pseudospectrum reconstruction of its angular power spectra from the maps of stokes @xmath16 and stokes @xmath17 , previously shown to be a method of choice for analyzing coming data sets . we focused on realistic statistical uncertainties ( i.e. sampling and noise variance ) as incurred by such a numerical method , and we purposefully did not consider the potential gain thanks to delensing , nor the loss due to polarized foreground contamination and instrumental systematics . we emphasize that in this paper we consider only the @xmath1-to-@xmath0 leakage due to a cut sky . in cmb practice there are numerous other potential sources of such leakages . for instance , they can arise from instrument limitations , such as beam mismatch  @xcite or polarimeter orientation uncertainty  @xcite , or be generated by data processing , say , via time - domain filtering  @xcite . such leakages would also have an effect on estimated @xmath0-mode power spectrum . the effect will in general depend on a specific method used for the estimation but also on the detailed nature of the leakage itself , and thus would have to be studied cased by case . in many situations , such leakages could be corrected for already on the map - making stage , leaving therefore the cut - sky as the only fundamental source of the leakage to contend with on the power spectrum estimation level as assumed in this work . in contrast , we include the effects of the gravitational lensing , i.e. , of the `` cosmological @xmath1-to-@xmath0 leakage '' , in the total uncertainty budget , in spite of the fact that map - making - level , delensing procedures , which could correct for part of this effect have been proposed  @xcite . the improvements on the detection of @xmath2 those methods could give depend on the noise level and the resolution of the experiment , and , on the potential use of external datasets ( if delensing can not be done internally ) . by including the lensing - induced @xmath0-mode , we adopt a more conservative viewpoint as far as forecasts on the tensor - to - scalar ratio are concerned . + in this framework , we first consider the case of small - scale ( either ground - based or balloon - borne ) experiments in an idealized way , assuming the observed sky patch is azimuthally symmetric . we consider four values of @xmath32 and @xmath33 , and let the sky fraction to vary from 0.5% to 10% ( with a noise level of @xmath206k - arcminute at @xmath26 . we compare the signal - to - noise on @xmath2 as obtained from the pure pseudospectrum reconstruction of the @xmath0-mode to the signal - to - noise ratio that would obtained assuming either the mode - counting estimation of the uncertainties on the @xmath0-mode , or a minimum variance , quadratic estimator . we show that the statistical significance on the detection of @xmath2 using the mode - counting is overestimated by a factor @xmath4 as compared to the more realistic case of the pure pseudospectrum estimation . ( the mode - counting also overestimate this significance by a factor @xmath3 as compared to the minimum variance , quadratic estimator . ) similarly , the ( s / n)@xmath114 obtained from the pure pseudospectrum estimator is reduced by 1.5% ( at @xmath120 ) to 8% ( at @xmath117 ) as compared to the lossless minimum variance , quadratic estimator .    for the case of small - scale experiment for which the reionization bump is not accessible , and in the limitation of azimuthally symmetric patches , the pure pseudospectrum approach for @xmath0-modes reconstruction is thus almost as accurate as the more computationally costly minimum variance , quadratic estimator ( the former scaling as @xmath92 and the latter as @xmath207 if the observed sky patch is not azimuthally symmetric ) . as shown in fig . 20 of @xcite , non - azimuthal symmetry basically does not change the uncertainties on the @xmath0-mode reconstruction with the pure pseudospectrum estimator ( except for unrealistic , highly squeezed shapes ) . we can thus except this conclusion to holds for more intricate shapes of the observed sky . our results ( summarized in tab . [ tab : fsky ] ) show that for a given sensitivity typical of forthcoming small - scale experiments , the value of the sky fraction maximizing the signal - to - noise ratio on @xmath2 is rather insensitive to the method adopted to compute the uncertainties on the reconstructed @xmath67 ( either the mode - counting expression , a minimum - variance quadratic estimator or the pure pseudo-@xmath100 approach ) . we also show that the choice of the bandwidth only mildly affect this optimized sky fraction in the case of the mode - counting approach to estimate the uncertainties on the @xmath0-mode reconstruction ( see tab . [ tab : bin ] ) . this means that using the mode - counting expression provide a rather reliable estimate of the optimized sky fraction from the viewpoint of statistical uncertainties though being underestimated . + second , we consider the detection of the tensor - to - scalar ratio for three selected examples , each of them mimicking three archetypal experimental configurations . realistic sky coverage ( with intricate contours ) and realistic noise distribution for the small - scale experimental setup are considered and the statistical uncertainties on the @xmath0-mode reconstruction are derived from the mode - counting expression first ( used as a benchmark ) , and second , from the pure pseudospectrum estimators using optimized sky apodizations . our results are summarized in the table [ tab : snr ] . for each experimental setups , it shows the minimal values of @xmath2 that could be measured with at least a statistical significance of @xmath112 . one gains more than one order of magnitude for the minimal detectable value of @xmath2 from the small - scale experiment to an array of telescopes , and another factor 2 from an array of telescopes to a satellite mission . this conclusion stands even if the largest angular scales ( @xmath208 ) can not be used in the analysis . let us briefly discuss the impact of those results in the context of single - field , slow - roll inflation . our purpose here is to give a rough translation of the potential measurement of the tensor - to - scalar ratio with the pure pseudo-@xmath100 estimation of the @xmath0-mode , into a potential discrimination between small fields and large fields models of inflation . ( a more detailed study of inflationary models can be found in ref . @xcite , though it is restricted to satellite missions and assumes a different evaluation of the error budget for the @xmath0-mode reconstruction . ) the tensor - to - scalar ratio is an valuable source of information for the physics of the primordial universe . first , it is a direct measure of the energy scale of inflation , @xmath209 with @xmath210 the value of the inflaton potential during inflation : @xmath211 this means that a measured value of @xmath212 corresponds to test a physical regime in the playground of grand unified theories . second , the tensor - to - scalar ratio is directly related to the number of e - folds , @xmath213 , and the excursion of the scalar field , @xmath214 , from the instant when cosmological fluctuations observed in the cmb are created during inflation , to the end of inflation @xcite : @xmath215 with @xmath216 the reduced planck mass . ( we note that @xmath213 can be determined from the knowledge of the inflaton potential . we however let it free in order not to assume a too specific shape of this potential . ) single field inflationary models can be roughly classified between large fields models and small fields models , whether the excursion of the scalar field is transplanckian or subplanckian , respectively . though the value @xmath217 should not be considered as a sharp and univoquely defined frontier between small fields and large fields models , a precise measure of @xmath2 then allows for discriminating between this two classes of models . for @xmath218 and considering zero runnings of the spectral index ( see @xcite for extensions of the lyth bound with runnings ) , values of @xmath2 greater than @xmath219 would correspond to large fields models of inflation ( see also ref . @xcite and references therein for examples of small fields models evading the lyth bound ) . , that could be observed with , at least , a @xmath112 significance , as functions of the number of e - folds during inflation . darker blue to lighter blue respectively stands for small - scale experiment , an array of telescopes and a satellite mission . the minimal , detectable value of @xmath2 at @xmath112 allowing for such a measurement is the one derived from the pure pseudo-@xmath100 estimation of the @xmath0-mode angular power spectra . ] the figure [ fig : lyth ] shows the ranges of @xmath220 as a function of @xmath213 accessible assuming that the tensor - to - scalar ratio has been measured with at least a @xmath112 statistical significance . blue areas correspond to the accessible range for each experimental configurations ( notice that the higher @xmath214 , the higher @xmath2 ) . the dark blue region is for the case of small - scale experiments , while the somewhat lighter blue and light blue regions corresponds to the case of the array of telescopes and of the satellite mission respectively . the minimal detectable value of @xmath2 with at least @xmath112 is the one derived from a pure pseudospectrum reconstruction of the angular power spectra of the @xmath0-mode and using the entire set of angular scales ( the _ case a _ of tab . [ tab : snr ] ) . this shows that a measurement of @xmath2 from the pure pseudo-@xmath100 reconstruction of the @xmath0-mode thanks to datasets coming from a small - scale experiment , is impossible if small fields models appear to be realized in the early universe . though a detection is possible in the large field models , there is still a range of such models for which the level of primordial gravity waves is still undetectable by a small - scale experiment . small fields models are only marginally accessible from the pure pseudo-@xmath100 estimation of the @xmath0-mode using datasets from an array of telescopes , as @xmath221 is accessible for @xmath213 smaller than @xmath222 . a detection of @xmath2 consistent with zero with a @xmath112 confidence level implies an excursion of the scalar field ( in planck units ) smaller than 0.8 to 1.8 for @xmath213 varying from 30 to 70 . finally , datasets coming from a satellite mission allows for a detection of primordial gravity waves in the 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time that owing to planar property and mutual focussing , colliding plane waves ( cpw ) result in spacelike singularities [ 1 ] . these singularities are somewhat weakened when the waves are endowed with a relative cross polarization prior to the collision . a solution given by chandrasekhar and xanthopoulos ( cx ) [ 2 ] , however constitutes an example contrasting this category , namely , it possesses a cauchy horizon ( ch ) instead of a spacelike singularity . naturally , this solution initiated a literature devoted entirely on the quest of stability of horizons formed hitherto . ch formed in spacetimes of cpw was shown by yurtsever to be unstable against plane - symmetric perturbations [ 3 ] . a linear perturbation analysis by cx reveals also an analogues result [ 4 ] . any such perturbation applied to a cpw spacetime will turn the ch into an essential singularity . a second factor that proved effective in weakening the strength of a singularity in cpw is the electromagnetic ( em ) field itself . in other words , the degree of divergence in the curvature scalars of colliding pure gravitational waves turn out to be stronger than the case when em field is coupled to gravity . in particular , collision of pure em waves must have a special significance as far as singularity formation is concerned . such an interesting solution was given by bell and szekeres ( bs ) which describes the collision of two linearly polarized step em waves [ 5 ] . the singularity ( in fact a ch ) formed in the interaction region of the bs solution was shown to be removable by a coordinate transformation . on the null boundaries , however it possesses esential curvature singularities which can not be removed by any means . since cross polarization and em field both play roles in the nature of resulting singularity it is worthwhile to purse these features together . this invokes a cross polarized version of the bs ( cpbs ) solution which was found long time ago [ 6,7 ] . this metric had the nice feature that the weyl scalars are all regular in the interaction region . cross polarization , however , does not remove the singularities formed on the null boundaries . in this paper we choose cpbs solution as a test ground , instead of bs , with various added test fields to justify the validity of a ch stability conjecture proposed previously by helliwell and konkowski ( hk ) [ 8,9 ] . unlike the tedious perturbation analysis of both cx and yurtsever the conjecture seems to be much economical in reaching a direct conclusion about the stability of a ch . this is our main motivation for considering the problem anew , for the case of untested solutions in cpw . in this paper we look at the spacetimes : a ) single plane wave with added colliding test fields and b ) colliding plane waves having non - singular interaction regions with test field added , fig.1 illustrates these cases .    the terminology of singularities should be follwed from the classification presented by ellis and schmidt [ 10 ] . singularities in maximal four dimensional spacetimes can be divided into three basic types : quasiregular ( qr ) , scalar curvature ( sc ) and non - scalar curvature ( nsc ) . the ch stability conjecture due to hk is defined as follows .    for all maximally extended spacetimes with ch , the backreaction due to a field ( whose test stress - energy tensor is @xmath0 ) will affect the horizon in one of the following manners . @xmath1 , @xmath2 and any null dust density @xmath3 are finite , and if the stress energy tensor @xmath4 in all parallel propagated orthonormal ( ppon ) frames is finite , then the ch remains non - singular . b)if @xmath1 , @xmath2 and any null dust density @xmath3 are finite , but @xmath4 diverges in some ppon frames , then an nsc singularity will be formed at the ch . c)if @xmath1 , @xmath2 and any null dust density @xmath3 diverges , then an sc singularity will be formed at the ch .    expressed otherwise , the conjecture suggests to put a test field into the background geometry and study the reaction it will experience . if certain scalars diverge then in an exact back - reaction solution the field will respond with an infinite strength to the geometry ( i.e action versus reaction ) . such an infinite back - reaction will render a ch unstable and convert it into a scalar singularity . + the paper is organized as follows . in section ii , we review the cpbs solution and the correct nature of the singularity structure is presented in appendix a. section iii , deals with geodesics and test em and scalar field analyses . in section iv , we present an exact back reaction calculation for the collision of cross polarized em field coupled with scalar field . the derivation of weyl and maxwell scalars are given in appendix b. the insertion of test null dusts to the background cpbs spacetime and its exact back reaction solution is studied in section v. appendix c is devoted for the properties of this solution . the paper is concluded with a discussion in section vi . the metric that describes collision of em waves with the cross polarization was found to be [ 7 ] + @xmath5    in this representation of the metric our notations are + @xmath6 in which @xmath7 is a constant measuring the second polarization , @xmath8 are constant of energy and @xmath9 stand for the usual null coordinates . it can be seen easily that for @xmath10 the metric reduces to bs . unlike the bs metric , however , this is conformally non - flat for @xmath11 , where the conformal curvature is generated by the cross polarization . as a matter of fact this solution is a minimal extension of the bs metric . a completely different generalization of the bs solution with second polarization was given by cx [ 11 ] . their solution , however , employs an ehlers transformation and involves two essential parameters which is therefore different from ours . both solutions form ch in the interaction region . our result drown out in this paper , namely , that the horizon is unstable against added sources can also be shown to apply to the cx metric as well . as it was shown before the interaction region @xmath12 of the above metric is of type - d without scalar curvature singularities . we wish to check now the possible singularities of metric ( 1 ) . the single component of the weyl scalar suffices to serve our purpose . we find that the real part of the weyl scalar @xmath13 is given by @xmath14\end{aligned}\ ] ] where we have used the abbreviations @xmath15 as @xmath16 we obtain @xmath17 which reduces to the singularity form of the bs spacetime given by @xmath18 . we see that the same singularity remains unaffected by the introduction of the cross polarization . a similar calculation for @xmath19 gives the symmetrical singular hypersurface sitting on @xmath20 . now in order to explore the true nature of the singularity we concentrate our account on the incoming region ii @xmath21 . the metric in this region is expressed in the form @xmath22\ ] ] where @xmath23 we observe that for @xmath24 , @xmath25 is a bounded positive definite function which suggests that no additional singularities arise except the one occuring already in the bs case , namely at @xmath26 . to justify this we have calculated all riemann components in local and ppon frames ( see appendix a ) . it is found that all riemann tensor components vanish as @xmath27 . in the ppon frame , however , they are all finite and according to the classification scheme of ellis and schmidt such a singularity is called a quasiregular ( qr ) singularity . this is said to be the mildest type among all types of singularities . to check whether the qr is stable or not we consider generic test fields added to such a background geometry and study the effects . this we will do in the follwing sections . we are interested in the stability of qr singularities that are developed at @xmath28 in region ii and @xmath29 in region iii . to investigate their stability we will express geodesics and behaviour of test em and scalar fields by calculating stress - energy tensor in local and ppon frames . + our discussion on geodesics will be restricted in region ii only . we shall consider the geodesics that originate at the wave front and move toward the quasiregular singularity . solution of geodesics equations in region ii can be obtained by geodesics lagrangian method and using @xmath30 as a parameter . the results are @xmath31}{a}\tan(au ) +   \frac{3p_{x_{0}}\left[5q^{2}+2 - 2\sqrt{1+q^{2}}\right]u}{4 } \nonumber \\ & & - \frac{p_{x_{0}}\left[\sqrt{1+q^{2}}-1\right]^{2}}{8a } \sin(2au ) -   \frac{2p_{y_{0}}q}{a\cos(au ) } \nonumber \\ y - y_{0}&=&-\frac{2qp_{x_{0}}}{a\cos(au)}-\frac{2p_{y_{0}}\tan(au)}{a }   \nonumber \\ v - v_{0}&=&\frac{\tan(au)}{a}\left[p^{2}_{x_0}(1 + 2q^2 ) + p^{2}_{y_{0}}\right ]   + u\left[\frac{\epsilon}{4}(1 + 3\sqrt{1+q^{2}})\right . \nonumber \\ & & \left.-\frac{3p^2_{x_0}}{8}(5q^{2}+2 - 2\sqrt{1+q^{2}})\right ] +   \frac{2p_{x_{0}}p_{y_{0}}q}{a\cos(au ) }    \nonumber \\ & & + \frac { ( \sqrt{1+q^{2}}-1 ) \sin(2au)}{8a}\left[\frac{p^{2}_{x_{0}}}{2 } (   \sqrt{1+q^{2}}-1)-\epsilon\right]\end{aligned}\ ] ] where @xmath32 for null and @xmath33 for time like geodesics and @xmath34 and @xmath35 are constants of integration . it can be checked easily that for @xmath10 our geodesics agree with those of the region ii of the bs metric [ 8 ] . it is clear to see that if either @xmath36 or @xmath35 is nonzero then @xmath37 becomes positive for @xmath38 , and particles can pass from region ii to the region iv . geodesics that remain in region ii are @xmath39 where @xmath40 . the effect of cross polarization is that more geodesics remains in region ii relative to the parallel polarization case . on physical grounds this result could be anticipated because cross polarization behaves like rotation which creates a pushing out effect in the non - inertial frames .      to test the stability of quasiregular singularity , let us consider a test em field whose vector potential is choosen appropriately as in [ 9 ] to be @xmath41 with arbitrary functions @xmath42 and @xmath43 . the only nonzero energy - momentum for this test em field is @xmath44\ ] ] in which a prime denotes derivative with respect to @xmath37 . both of scalars @xmath45 and @xmath2 vanish , predicting that qr singularities are not transformed into a scs . in the ppon frame . @xmath46 we find that @xmath47 are given in terms of @xmath48 by @xmath49 for@xmath50 and @xmath51 , otherwise . the divergence of this quantity predicts the occurence of nscs and therefore qr singularity must be unstable . + the stability conjecture therefore correctly finds that these qr singularities are unstable . however , the same stability conjecture does not find correctly the nature of the singularity . as we have discussed in section ii , the interior of the interaction region has no scs . the only scs is on the null boundaries . clarke and hayward have analysed these singular points for a collinear bs spacetime and found that the singularity nature of surfaces @xmath52 and @xmath53 are qr . this observation can also be used in the cross polarized version of bs spacetime , because the order of diverging terms in @xmath54 and @xmath13 are the same . the qr singularity structure formed in the incoming region of bs problem remains unchanged in the case of cross polarized version of the same problem . let us now consider the stability of these qr singularities by imposing a test scalar field in region ii which is the one of the incoming region bounded by the qr singularity . the massless scalar field equation is given by @xmath55 where we consider @xmath56 independent scalar waves so that a particular solution to this equation is obtained as in the ref ( ) @xmath57 where @xmath58 and @xmath59 are arbitrary functions . the stress energy tensor is given by @xmath60 the corresponding non - zero stress - energy tensors for the test scalar wave is obtained by taking @xmath61 as , @xmath62f(v)f'(v ) } {   8\pi f^{2 } } \nonumber \\ t_{xy}&=&t_{yx}=\frac{aq\sin(au ) \tan(au ) f(v)f'(v)}{8 \pi f^{2}}\end{aligned}\ ] ] it is observed that each component diverges as the qr singularity @xmath63 is approached . + next we consider the stress energy tensor in a ppon frame . such frame vectors are given in equation ( 11 ) . the stress - energy tensor is @xmath64 the nonzero components are ; @xmath65 \nonumber   \\ t_{01}&=&t_{10}=\left(\frac{\sec^{2}(au)}{16 \pi}\right)\left[\frac{a^{2 }   \tan^{2}(au)f^2}{f^{2}}-f'^{2}(v)\right ] \nonumber   \\ t_{22}&=&t_{33}=\left(\frac{a\sec^{2}(au)\tan(au)f(v)f'(v)}{8\pi   f^{3}}\right)\left[f^{2 } + 2q^{2}\sin^{2}(au)\right ] \nonumber \\ t_{32}&=&t_{23}=\frac{aq\sec^{3}(au)\sin^{2}(au)f'(v)f(v)\sqrt{f^{2 }   + q^{2}\sin^{2}(au)}}{4\pi f^{3}}\end{aligned}\ ] ] these components also diverge as the singularity @xmath66 is approached .    by the conjecture , this indicates that the qr singularity will be transformed into a curvature singularity . finally we calculate the scalar @xmath67 . @xmath68 \right\}\end{aligned}\ ] ] which also diverges as @xmath69 . from these analyses we conclude that the curvature singularity formed will be an scs . + hence , the conjecture predicts that the qr singularities of cross polarized version of bs spacetime are unstable . it is predicted that the qr singularities will be converted to scalar curvature singularities if generic waves are added . the similar results have also been obtained by hk for the bs spacetime . hk was unable to compare the validity of the conjecture by an exact back - reaction solution . in the next section we present an explicit example that represents cross - polarized em field coupled with scalar field . in the former sections , we applied hk stability conjecture to test the stability of qr singularities in the incoming region of cpbs spacetime . according to the conjecture these mild singularities are unstable . in order to see the validity of the conjecture we introduce this new solution . + let the metric be ; @xmath70\ ] ] the new solution is obtained from the electrovacuum solution . the ems solution is generated in the following manner . the lagrangian density of the system is defined by @xmath71\end{aligned}\ ] ] which correctly generate all ems field equations from a variational principle . here @xmath72 is the scalar field and we define the em potential one - form ( with coupling constant @xmath73 ) by @xmath74 where @xmath75 and @xmath76 are the components in the killing directions . variation with respect to @xmath77 and @xmath72 yields the following ems equations . @xmath78   \\ 2b_{uv}&=&-v_{v}b_{u}-v_{u}b_{v}-tanhw\left(w_{v}b_{u}+w_{u}b_{v}\right ) \nonumber \\ & & -e^{v}\left[2b_{uv}tanhw + w_{u}b_{v}+w_{v}b_{u}\right]\end{aligned}\ ] ] where @xmath79 and @xmath80 are the newmann - penrose spinors for em plane waves given as follows @xmath81 \nonumber \\ \phi_{0}&= & \frac{e^{u/2}}{\sqrt{2}}\left[e^{-v/2}\left(isinh\frac{w}{2 }   + cosh\frac{w}{2}\right)a_{v}\right . \nonumber \\ & & \left.+e^{v/2}\left(icosh\frac{w}{2}+sinh\frac{w}{2}\right)b_{v}\right ] \end{aligned}\ ] ] the remaining two equations which do not follow from the variations , namely @xmath82 are automatically satisfied by virtue of integrability equations . + the metric function @xmath83 can be shifted now in accordance with + @xmath84 where @xmath85 and @xmath86 satisfy the electrovacuum em equations . integrability condition for the equation ( 31 ) imposes the constraint , @xmath87=0\ ] ] from which we can generate a large class of ems solution . as an example , for any @xmath88 satisfying the massless scalar field equation the corresponding @xmath89 is obtained from @xmath90 the only effect of coupling a scalar field to the cross polarized em wave is to alter the metric into the form , @xmath91 here @xmath92 and @xmath93 represents the solution of electrovacuum equations and @xmath89 is the function that derives from the presence of the scalar field . + it can be easily seen that for @xmath10 our solution represents pure em bs solution coupled with scalar field . it constitutes therefore an exact back reaction solution to the test scalar field solution in the bs spacetime considered by hk ( ) . it is clear to see that the weyl scalars are nonzero and scs is forming on the surface @xmath94 . this is in aggrement with the requirement of stability conjecture introduced by hk . for @xmath24 the obtained solution forms the exact back reaction solution of the test scalar field solution in the cpbs spacetime . in appendix b , we present the weyl and maxwell scalars explicitly . + from the explicit solutions we note that , the coulomb component @xmath95 remains regular but @xmath13 and @xmath54 are singular when @xmath96 or @xmath97 . this indicates that the singularity structure of the new solution is a typical scs . this result is in complete agreement with the stability conjecture . * a ) * let us assume first null test dusts moving in the cpbs background . for simplicity we consider two different cases the @xmath98 and @xmath99 projections of the spacetime . we have in the first case @xmath100 where we have used the coordinates @xmath101 according to @xmath102 the energy - momentum tensor for two oppositely moving null dusts can be chosen as @xmath103 where @xmath104 and @xmath105 are the finite energy densities of the dusts . the null propagation directions @xmath106 and @xmath107 are @xmath108 with @xmath109 @xmath110 we observe from ( 1 ) that @xmath111 which is finite for @xmath112 . the components of energy - momentum tensors in ppon frames are proportional to the expression ( 38 ) . this proportionality makes all the components of energy - momentum tensors finite . in the limit as @xmath113 which reduces our line element to the bs this expression diverges on the horizon @xmath114 . trace of the energy - momentum is obviously zero therefore we must extract information from the scalar @xmath115 . one obtains , @xmath116 the projection on @xmath99 , however is not as promising as the @xmath98 case . consider the reduced metric @xmath117 a similar analysis with the null vectors @xmath118 with @xmath119 @xmath120 yields the scalar @xmath121 from the metric ( 1 ) we see that @xmath122 which diverges on the horizon @xmath123 . the scalar @xmath124 constructed from the test dusts therefore diverges . the ppon components of the energy momentum tensors are also calculated and it is found that all of the components are proportional to the expression ( 42 ) . this indicates that the components of energy - momentum tensor diverges as the singularity is approached . when we consider the hk stability conjecture an exact back reaction solution must give a singular solution . we present now an exact back reaction solution of two colliding null shells in the interaction region of the cpbs spacetime . + * b ) * the metric given by @xmath125 where @xmath126 with @xmath127 positive constants was considered by wang [ 12 ] to represent collision of two null shells ( or impulsive dusts ) . the interaction region is transformable to the de sitter cosmological spacetime . in other words the tail of two crossing null shells is energetically equivalent to the de sitter space . it can be shown also that the choice of the conformal factor @xmath128 , with @xmath129 positive constants becomes isomorphic to the anti - de sitter space . + the combined metric of cpbs and colliding shells can be represented by @xmath130 this amounts to the substitutions @xmath131 where @xmath132 correspond to the metric functions of the cpbs solution . under this substitutions the scale invariant weyl scalars remain invariant ( or at most multiplied by a conformal factor ) because @xmath133 is the combination that arise in those scalars . the scalar curvature , however , which was zero in the case of cpbs now arises as nonzero and becomes divergent as we approach the horizon . appendix c gives the scalar quantities @xmath134 and @xmath135 . thus the exact back reaction solution is a singular one as predicted by the conjecture . it is further seen that choosing @xmath136 , which removes one of the shells leaves us with a single shell propagating in the @xmath37- direction . from the scalars we see that even a single shell gives rise to a divergent back reaction by the spacetime . the horizon , in effect , is unstable and transforms into a singularity in the presence of two colliding , or even a single propagating null shell . let us note as an alternative interpretation that the metric ( 43 ) may be considered as a colliding em waves in a de sitter background . collision of em waves creates an unstable horizon in the de sitter space which is otherwise regular for @xmath137 and @xmath138 . in this paper we have analysed the stability of qr singularities in the cpbs spacetime . three types of test fields are used to probe the stability . first we have applied test em field to the background cpbs spacetime . from the analyses we observed that the qr singularity in the incoming region becomes unstable according to the conjecture , and it is transformed into nsc singularity . this is the prediction of the conjecture . however , the exact back - reaction solution shoes that beside the true singularities on the null boundaries @xmath139 and @xmath140 . there are quasiregular singularities in the incoming regions . the interior of interaction region is singularity free and the hypersurface @xmath141 is a killing - cauchy horizon . as it was pointed out by hk in the case of colliding em step waves , conjecture fails to predict the correct nature of the singularity in the interaction region . we have also discovered the same behaviour for the cross polarized version of colliding em step waves . the addition of cross polarization does not alter the existing property . + secondly we have applied test scalar field to the background cpbs spacetime . the effect of scalar field on the qr singularity is stronger than the effect of em test field case . we have obtained that the qr singularity is unstable and transformes into a scs . to check the validity of the conjecture , we have constructed a new solution constituting an exact back reaction solution to the test scalar field in the cpbs spacetime . the solution represents the collision of cross polarized em field coupled to a scalar field . an examination of weyl and maxwell scalars shows that @xmath142 and @xmath143 diverge as the singularity is approached and unlike the test em field case the conjecture predicts the nature of singularity formed correctly . + finally , we have introduced test null dusts into the interaction region of cpbs spacetime . the conjecture predicts that the ch are unstable and transforms into scs . this result is compared with the exact back - reaction solution and observed that the conjecture works . to determine the type of singularity in the incoming region of cpbs spacetime , the riemann tensor in local and in ppon frame must be evaluated . non - zero riemann tensors in local coordinates are found as follows . @xmath144 \nonumber \\ -r_{uyuy}&= & e^{-u - v } \left [ \phi_{22 } coshw + ( i m \psi_4 ) sinhw - re \psi_4    \right ] \nonumber \\ r_{uxuy}&= & e^{-u}\left [ \phi_{22 } sinhw + ( im\psi_4)coshw\right]\end{aligned}\ ] ] where @xmath145 \nonumber \\ i m \psi_4&=&-\frac{i}{2}\left(w_{uu } -u_uw_u + m_uw_u -v^2_u coshw sinhw   \right ) \nonumber \\ \phi_{22}&=&\frac{1}{4}\left(2u_{uu}-u^2_u - w^2_u - v^2_u cosh^2w \right)\end{aligned}\ ] ] note that in region ii the weyl scalar @xmath146 , therefore only @xmath143 exists . it is clear that @xmath147 in the limit @xmath27 , so that all of the components vanish @xmath148 to find the riemann tensor in a ppon frame , we define the following ppon frame vectors ; @xmath149 in this frame the non - zero components of the riemann tensors are : @xmath150 in the limit of @xmath27 , we have the following results @xmath151 which are all finite . this indicates that the apparent singularity in region ii ( one of the incoming region ) is a quasiregular singularity . in order to calculate the weyl and maxwell scalars we make use of the cx line element @xmath152 where @xmath153 @xmath154 is given in equation ( 4 ) and we have chosen @xmath155 , such that the new coordinates @xmath156 are related to @xmath157 by @xmath158 the weyl and maxwell scalars are found as @xmath159 \\ & & \nonumber \\ \psi_0&=&-e^{\gamma - i\lambda } \left [ 3r   + \frac{1}{4f\sigma \sin \theta \sin \psi } \left(\sigma \sin(\psi -\theta)-\sigma_{\theta } \sin \psi \sin\theta \right . \nonumber \\ & & \nonumber \\ & & \left.\left.+ i\sin \alpha \sin^2 \theta \sin \psi \right ) \left(\gamma_{\psi}+\gamma_{\theta}\right)\right ] \\ & & \nonumber \\ 2\phi_{00}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac {   \sin(\psi+\theta)(\gamma_{\theta}+\gamma_{\psi})}{2f\sin \psi \sin \theta }    \right ] \\ & & \nonumber \\ 2\phi_{22}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac{\sin(\theta-   \psi)(\gamma_{\theta}-\gamma_{\psi})}{2f\sin \psi \sin \theta } \right ] \\ & & \nonumber \\ -2\phi_{02}&=&e^{\gamma + i\lambda}\frac{\cos \alpha}{\sigma^2}\end{aligned}\ ] ] where @xmath160 \\ & & \\ e^{i\lambda}&=&\frac{\sin \theta + \sigma \sin \psi + i \sin \psi \sin \theta   \cos \psi } { \sin \theta + \sigma \sin \psi - i \sin \psi \sin \theta \cos \psi}\end{aligned}\ ] ] the non - zero weyl and maxwell scalars for the collision of null shells in the background of cpbs spacetime are found as follows . @xmath161 \theta(u ) \theta(v ) \\ & & \nonumber \\ 4\phi e^{-m}\lambda&= & \left [ ( a\beta + \alpha b)\tan(au+bv)+(a \beta -\alpha b)\tan(au - bv)\right . \nonumber \\ & & \nonumber \\ & & \left.+\frac{4\alpha \beta}{\phi } \right ] \theta(u ) \theta(v ) \\ & & \nonumber \\ \phi_{22}&=&(\phi_{22})_{cpbs } + \left(\frac{\alpha e^m } { \phi}\right)\left [   \delta(u ) \right.\nonumber \\ & & \nonumber \\ & & \left . - \theta(u)\left(a \pi + \frac{u}{(1-u^2)(1-v^2)}\right ) \right]\\ & & \nonumber   \\ \phi_{00}&=&(\phi_{00})_{cpbs } + \left(\frac{\beta e^m } { \phi}\right)\left [   \delta(v ) \right . \nonumber   \\ & & \nonumber \\ & & \left.+ \theta(v)\left(b \pi - \frac{v}{(1-u^2)(1-v^2)}\right)\right ] \\ & & \nonumber \\ \phi_{02}&= & ( \phi_{02})_{cpbs}+\left ( \frac{e^m}{4fy\phi}\right ) \left[\frac{1}{f}\left(\alpha q \theta(u ) + \beta p \theta(v)\right ) \right . \nonumber \\ & & \nonumber \\ & & \left . + iq\left(\alpha l \theta(u ) + \beta k \theta(v)\right)\right]\end{aligned}\ ] ] where @xmath162 \\ & & \\ p&=&a\left[2q^2\sin(au+bv)\cos(au - bv)+f^2\left(\tan(au - bv)-\tan(au+bv)\right )   \right . \\ & & \\ & & \left . + 2f\cos(au - bv)\sin(au - bv)\left(\sqrt{1+q^2}-1\right)\right ] \\ & & \\ y&=&\left(1+\frac{q^2}{f^2}\tan(au+bv)\sin(au+bv)\cos(au - bv)\right)^{1/2}\\ & & \\ k&=&\frac{a}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } {   \cos(au+bv)}+\sin2au \right . \\ & & \\ & & \left . -\frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv )   \tan(au - bv)}{f}\right ] \\ & & \\ l&=&\frac{b}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } {   \cos(au+bv)}+\sin2bv \right . \\ & & \\ & & \left . + \frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv )   \tan(au - bv)}{f}\right ] \\ & & \\ \pi&=&\frac{\left(\sqrt{1+q^2}-1\right)\sin(2au-2bv)}{\sqrt{1+q^2}+1 +   \left(\sqrt{1+q^2}-1\right)\sin^2(au - bv)}\end{aligned}\ ] ]    99 j.b . griffiths , colliding plane waves in general relativity , oxford , clarendon press ( 1991 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 408 * , 175 ( 1986 ) . u. yurtsever , phys . rev . * d 36 * , 1662 ( 1987 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 415 * , 329 ( 1988 ) . p. bell and p. szekeres , gen . * 5 * , 275 ( 1974 ) . m. halilsoy , phys . rev . * d,37 * , 2121 , ( 1988 ) . m. halilsoy , j. math . phys . * 31 * , 2694 , ( 1990 ) . d. a. konkowski . and t. m. helliwell , phys . rev . * d 43 * , 609 ( 1991 ) . d. a. konkowski . and t. m. helliwell , class . quantum grav . * 16 * , 2709 ( 1999 ) . g. f. r. ellis and b. g. schmidt , gen . * 8 * , 915 ( 1977 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 410 * , 311 ( 1987 ) . a. z. wang , j. math . phys . * 33 * , 1065 , ( 1992 ) . fig.1(a ) : single plane waves with added colliding test fields indicated by arrows . ch exists on the surface @xmath163 . + ( b ) : colliding plane wave spacetime with ch s in the incoming regions at @xmath164 and @xmath165 . test fields are added to test the stability of ch existing in region iv ., in recent years considerable progress has been achieved in the partonic interpretations of diffractive processes in @xmath1 collisions ( see e.g.  @xcite ) , most of diffractive studies performed up to now at hera have been based on the characteristic presence of a rapidity gap in the diffractive final state . the precision of this method is limited by the uncertainty related to the presence of dissociated proton background events . the only precise and unambiguous way of studying diffraction is by tagging the diffracted proton and measuring its four momentum by means of a proton spectrometer . such devices have been used by the h1 and zeus collaborations and have delivered interesting results , but their acceptances are small , with the result that the collected statistics are limited and large systematic errors affect the measurements . to fully profit from the hera luminosity upgrade in the study of diffraction after the year 2003 , a very forward proton spectrometer ( vfps ) which identifies and measures the momentum of the diffracted proton with a high acceptance has therefore been installed by h1 . this contribution reports the vfps tagging performance using diffractive events collected during hera running at high energy ( 27.5 gev for the electron / positron beam and 920 gev for the proton ) in 2006 and 2007 . the vfps @xcite is a set of two `` roman pots '' located at 118 m and 222 m downstream of the h1 interaction point . each pot consists of an insert into the beam pipe , allowing two tracking detectors equipped with scintillating fibres to be moved very close to the proton beam . + many aspects of the design of the roman pots , including the stainless plunger vessel and the scintillating fiber detectors , are adaptations of the fps proton spectrometer  @xcite , installed and operational in h1 since 1994 . both detectors of each roman pot consists of two planes of scintillating fibres oriented at @xmath2 w.r.t .  the horizontal plane and moving perpendicularly to the beam line direction . each detector allows to reconstruct of the position of one impact point of the scattered proton trajectory with a precision of about 100 @xmath3 m . for triggering purposes each detector is sandwiched between 2 scintillating planes which are connected to different pm s . a trigger signal , corresponding to a activity in at least 3 planes out of four , is delivered separately for each station at the first trigger level . the vfps have been installed at the very end of 2003 . radiation damage of the optical readout fiber prohibited data taking during 2004 . hence data available for physics    r0.5 analysis started in 2005 . the bulk of data were taken in 2006 and 2007 , they correspond to and integrated luminosity of 140 @xmath4 . from an operational point of view , the vfps was into data taking position for 70% of the luminosity collected by h1 . + the @xmath5 beam orbit has been modified in april 2006 to increase the vfps acceptance . a large fraction of protons with a energy loss above 2% w.r.t . the beam energy are hitting the beam pipe around 200 m when the nominal orbit is used . the orbit has been changed moving the @xmath5 outwards hera by 6 mm at about 200 m from the interaction point . this procedure leads to an increase of the rate of events tagged by the vfps at @xmath6 ( see fig .  [ fig : kick ] ) . to study the description of the beam optics and of the vfps system by our simulation , first , vfps tagged events , i.e. with a fired trigger , are compared to the full sample of diffractive events selected using the information from the main detector using the rapidity gap method ( see e.g.  @xcite ) . the full event sample is selected asking for an electromagnetic cluster in the backward ( lepton beam direction ) calorimeter spacal of more than 10 gev ( corresponding to the scattered electron candidate ) , a reconstructed vertex and that the most forward particle in the main detector has pseudo - rapidity of less than 2.5 ( this latest condition is equivalent to asking for a rapidity gap ) . additionally the forward muon detector should not have recorded a signal above the noise level . one can then look to what fraction of this sample largely dominated by diffractive events is tagged by vfps . this selection was applied to produce the fig . [ fig : kick ] discussed in the previous section . over the 140 @xmath4 collected , 880,000 events are tagged by the vfps . if a kinematic cut of @xmath7 gev@xmath8 is applied , 215,000 events remain . + this sample is compared to the sum of diffractive and background contributions as estimated by monte carlo . in fig . [ fig : dis ] data corresponding to about 1 month of running in @xmath9 mode with a 6 mm bump applied ( 24 @xmath4 ) are compared to monte carlo predictions ( see figure caption for details ) . +    ( 100,120 ) ( -10,-10 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ] ( 206,-10 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ] ( 120,69 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ]    a good agreement is found between the full data sample and the monte carlo ( normalized to the data luminosity ) . the trend in @xmath0 of vfps tagged sample is described by the monte carlo . a more precise understanding of the vfps acceptance is needed and will lead to a better description in @xmath0 . the analysis of diffractive dijets in dis regime is based on 42.6@xmath10 and 54.7@xmath11 @xmath4 taken in 2006 . additionally to the selection applied in the previous section , a requirement of at least two jets ( using the @xmath12 algorithm ) is asked , with a minimal transverse momentum in the photon - proton frame of @xmath13 and @xmath14 respectively for the first and the second jet . the jets are asked to be well contained in the main detector , by requirering @xmath15 $ ] . + the @xmath0 distribution is shown in fig . [ fig : jets]a comparing the full dijet sample and vfps tagged dijet sample . this plot illustrates the well suited acceptance of the vfps for the dijets production in diffraction . in fig . [ fig : jets]b the transverse momentum of the first jet in the laboratory frame is shown . here again the full dijet sample is compared to the vfps tagged dijet sample . ( 100,80 ) ( 0,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 140,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 280,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 22,58)a ) ( 162,58)b ) ( 300,65)c )      to record with a high efficiency diffractive dijet events in photoproduction tagged by the vfps , a special trigger has been developed . it allows to lower the threshold in jet transverse momentum down to 5 gev . a luminosity of 23.7 @xmath4 has been collected in 2006 and 2007 with that trigger corresponding to a selected sample of 6000 events . the selection criteria are the same as in the dijet dis case except that the scattered electron escapes undetected , at small angle , in the beam pipe . figure [ fig : jets]c shows the transverse momentum of the first jet in the laboratory frame ( equivalent to the photon - proton frame for the transverse direction in the present photoproduction case ) . the distribution of vfps tagged events can not be compared to a full dijet sample as no trigger allowed to keep efficiently those events down to a transverse momentum of 5 gev . the vfps has run successfully collecting a luminosity of 140 @xmath4 . the observed acceptance is high ( above 60% ) in a region of @xmath0 around @xmath16 . the trend in @xmath0 of diffractive events tagged by the vfps is described by the monte carlo simulation . important statistics have been collected ( 880,000 diffractive dis events , 800 dijets diffractive dis events and 6000 dijets diffractive events in photoproduction ) for diffractive structure function measurement and qcd factorisation tests . the proton momentum reconstruction based on vfps fiber information is still in progress . 99 l.  favart , _ experimental review of diffractive phenomena _ , in proceedings of the 10@xmath17 international baryons conference ( baryons 2004 ) , [ hep - ex/0501052 ] . , proposal submitted to the physics research committee , * prc-01/00 * , h1 note * h1 - 05/00 - 582 * , http://www-h1.desy.de/h1det/tracker/vfps/, source_sentence: <S> the universal curve @xmath0 of nuclear photoabsorption is investigated within a fermi gas model of nuclear matter . an energy range from pion threshold up to 400 mev </S> <S> is considered . </S> <S> the interactions between nucleon , pion , @xmath1-isobar and photon are considered in the non - relativistic approximation with corrections of the order @xmath2 taken into account with respect to proton mass . </S> <S> analytical expressions are obtained , in which the influence of nuclear correlations , two - nucleon contributions and relativistic corrections is studied explicitely . </S> <S> an extension of the model calculation to nucleon knock - out reactions is discussed .    </S> <S> contribution of real and virtual pions to nuclear photoabsorption at intermediate energies   + m .- th.htt@xmath3 , a.i.milstein@xmath4 and m.schumacher@xmath3 + ( a ) ii . </S> <S> institut der universitt gttingen , gttingen , germany    \(b ) budker institute of nuclear physics , 630090 novosibirsk , russia    0em 1.5ex plus 0.5ex minus 0.5ex    _ pacs code : _ </S> <S> 25.20.-x    _ keywords : _ photoabsorption , mesonic exchange currents , nuclear correlation functions </S>
sentences: one of the most basic questions that arises in trying to understand the nonperturbative structure of string theory concerns the classification of vector bundles over real and complex manifolds . in the presence of d - branes one encounters gauge theories in spacetime dimensionalities up to ten . already more than 20 years ago , bps - type equations in higher dimensions were proposed  @xcite as a generalization of the self - duality equations in four dimensions . for nonabelian gauge theory on a khler manifold the most natural bps  condition lies in the donaldson - uhlenbeck - yau equations  @xcite , which arise , for instance , in compactifications down to four - dimensional minkowski spacetime as the condition for at least one unbroken supersymmetry .    while the criteria for solvability of these bps equations are by now very well understood , in practice it is usually quite difficult to write down explicit solutions of them . one recent line of attack has been to consider noncommutative deformations of these field theories  @xcite@xcite . in certain instances , d - branes can be realized as noncommutative solitons  @xcite , which is a consequence  @xcite of the relationship between d - branes and k - theory  @xcite@xcite . all celebrated bps configurations in field theories , such as instantons  @xcite , monopoles  @xcite and vortices  @xcite , have been generalized to the noncommutative case , originally in  @xcite , in  @xcite and in  @xcite , respectively ( see  @xcite for reviews and further references ) . solution generating techniques such as the adhm construction  @xcite , splitting  @xcite and dressing  @xcite methods have also been generalized to the noncommutative setting in  @xcite and in  @xcite . solutions of the generalized self - duality equations  @xcite were investigated in  @xcite , for example . noncommutative instantons in higher dimensions and their interpretations as d - branes in string theory have been considered in  @xcite@xcite . in all of these constructions the usual worldvolume description of d - branes emerges from the equivalence between analytic and topological formulations of k - homology .    in this paper we will complete the construction initiated in  @xcite of multi - instanton solutions of the yang - mills equations on the manifold which is the product of noncommutative euclidean space @xmath3 with an ordinary two - sphere @xmath4 . we consider both bps and non - bps solutions , and extend previous solutions to those which are explicitly @xmath1-equivariant for any value of the dirac monopole charge characterizing the gauge field components along the @xmath4 directions . dimensional reduction techniques are used to establish an equivalence between multi - instantons on @xmath0 and nonabelian vortices on @xmath2 . the configurations can be interpreted in type  iia superstring theory as _ chains _ of branes and antibranes with higgs - like open string excitations between neighbouring sets of d - branes . the equivalence between instantons and vortices may then be attributed to the decay of an unstable configuration of d@xmath5-branes into a state of d0-branes ( there are no higher brane charges induced because @xmath6 is equivariantly contractible ) . the d0-brane charges are classified by @xmath1-equivariant k - theory and the low - energy dynamics may be succinctly encoded into a simple quiver gauge theory . unlike the standard brane - antibrane systems , the effective action can not be recast using the formalism of superconnections  @xcite but requires a more general formulation in terms of new geometrical entities that we call `` graded connections '' . this formalism makes manifest the interplay between the assignment of k - theory classes to the explicit instanton solutions and their realization in terms of a quiver gauge theory . the organisation of this paper is as follows . the material is naturally divided into two parts . sections  25 deal with _ ordinary _ gauge theory on a generic khler manifold of the form @xmath7 in order to highlight the geometric structures that arise due to dimensional reduction and which play a prominent role throughout the paper . sections 610 are then concerned with the noncommutative deformation @xmath8 and they construct explicit solutions of the dimensionally reduced yang - mills equations , emphasizing their interpretations in the context of equivariant k - theory , quiver gauge theory , and ultimately as states of d - branes . in section 2 we introduce basic definitions and set some of our notation , and present the field equations that are to be solved . in section 3 we write down an explicit ansatz for the gauge field which is used in the @xmath1-equivariant dimensional reduction . in section  4 we describe three different interpretations of the ansatz as configurations of d - branes , as charges in equivariant k - theory , and as field configurations in a quiver gauge theory ( later on these three descriptions are shown to be equivalent ) . in section  5 the dimensional reduction mechanism is explained in detail in the new language of graded connections and the resulting nonabelian vortex equations , arising from reduction of the donaldson - uhlenbeck - yau equations , are written down . in section  6 we introduce the noncommutative deformations of all these structures . in section  7 we find explicit bps and non - bps solutions of the noncommutative yang - mills equations and show how they naturally realize representations of the pertinent quiver . in section  8 we develop an @xmath1-equivariant generalization of the ( noncommutative ) atiyah - bott - shapiro construction , which provides an explicit and convenient representation of our solution in terms of k - homology classes . in section  9 we compute the topological charge of our instanton solutions directly in the noncommutative gauge theory , and show that it coincides with the corresponding k - theory charge , which then allows us to assign d0-brane charges to the solutions from a worldvolume perspective . finally , in section  10 we construct some novel bps solutions in the vacuum sectors of the noncommutative field theory and describe their relation to stable states of brane - antibrane systems . in this section we will introduce the basic definitions and notation that will be used throughout this paper , as well as the pertinent field equations that we will solve . * the manifold @xmath9 .  * let @xmath10 be a real @xmath11-dimensional lorentzian manifold with nondegenerate metric of signature @xmath12 , and @xmath13 the standard two - sphere of constant radius @xmath14 . we shall consider the manifold @xmath15 with local real coordinates @xmath16 on @xmath10 and coordinates @xmath17 $ ] , @xmath18 $ ] on @xmath4 . in these coordinates the metric on @xmath15 reads @xmath19 where hatted indices @xmath20 run over @xmath21 while primed indices @xmath22 run through @xmath23 . we use the einstein summation convention for repeated spacetime indices .    * the khler manifold @xmath24 .  * as a special instance of the manifold @xmath10 we shall consider the product @xmath25 of dimension @xmath26 with metric @xmath27 here @xmath28 is a khler manifold of real dimension @xmath29 with local real coordinates @xmath30 , where the indices @xmath31 run through @xmath32 . the cartesian product @xmath33 is also a khler manifold with local complex coordinates @xmath34 and their complex conjugates , where @xmath35 while @xmath36 are stereographic coordinates on the northern hemisphere of @xmath4 . in these coordinates the metric on @xmath37 takes the form @xmath38 while the khler two - form @xmath39 is given by @xmath40    * yang - mills equations .  * consider a rank @xmath41 hermitean vector bundle @xmath42 with gauge connection @xmath43 of curvature @xmath44 . in local coordinates , wherein @xmath45 , the two - form @xmath46 has components @xmath47 $ ] , where @xmath48 . both @xmath49 and @xmath50 take values in the lie algebra @xmath51 . for the usual yang - mills lagrangian can be introduced via the redefinition @xmath52 . ] @xmath53 the equations of motion are @xmath54=0 \ , \ ] ] where @xmath55 . the curvature two - form can be written in local coordinates on @xmath56 as @xmath57 and the yang - mills lagrangian becomes l^  _ ym&=&-  tr^  _ kk  . [ lagrprod ]    * donaldson - uhlenbeck - yau equations .  * for static field configurations in the temporal gauge @xmath58 , the yang - mills equations ( [ ym ] ) on @xmath59 reduce to equations on @xmath37 . their stable solutions are provided by solutions of the donaldson - uhlenbeck - yau ( duy ) equations which can be formulated on any khler manifold  @xcite . the importance of these equations derives from the fact that they yield the bps solutions of the full yang - mills equations . the duy equations on @xmath7 are @xmath60 where @xmath61 is the hodge duality operator and @xmath62 is the khler decomposition of the gauge field strength . in the local complex coordinates @xmath63 these equations take the form @xmath64\label{duy2 } { { { \cal{f}}}}_{{{\bar{z}}}^{{{\bar{a}}}}{{\bar{z}}}^{{{\bar{b}}}}}&=&0~=~{{\cal{f}}}_{z^az^b } \ , \\[4pt]\label{duy3 } { { { \cal{f}}}}_{{{\bar{z}}}^{{{\bar{a}}}}{{\bar{y}}}}&=&0~=~{{\cal{f}}}_{z^ay } \ , \end{aligned}\ ] ] where the indices @xmath65 run through @xmath66 . ( [ duy1 ] ) is a hermitean condition on the gauge field strength tensor , while eqs . ( [ duy2 ] ) and ( [ duy3 ] ) are integrability conditions implying that the bundle @xmath67 endowed with a connection @xmath43 is holomorphic . it is easy to show that any solution of these @xmath68 equations also satisfies the full yang - mills equations . in this section we shall write down the fundamental form of the gauge potential @xmath43 on @xmath56 that will be used later on to dimensionally reduce the yang - mills equations for @xmath43 to equations on @xmath10 . this will be achieved by prescribing a specific @xmath69 dependence for @xmath43 , which we proceed to describe first .    * monopole bundles . * consider the hermitean line bundle @xmath70 over the sphere with @xmath71 and unique @xmath1-invariant unitary connection @xmath72 having , in the local complex coordinate @xmath73 on @xmath69 , the form @xmath74 where @xmath75 is an integer . the curvature of this connection is @xmath76 the topological charge of this gauge field configuration is given by the first chern number ( equivalently the degree ) of the associated complex line bundle as @xmath77 in terms of the spherical coordinates @xmath78 the configuration ( [ f1],[f2 ] ) has the form @xmath79 it describes @xmath80 dirac monopoles or antimonopoles sitting on top of each other . the @xmath75-monopole bundle is classified by the hopf fibration @xmath81 . for each @xmath82 there is a one - dimensional representation @xmath83 of the circle group @xmath84 defined by _ m  : v  v=^mv  s^1 v  . [ u1irreps]we denote this irreducible @xmath85-module by @xmath86 . regarding the sphere as the homogeneous space @xmath87 , the @xmath1-equivariant line bundle @xmath88 corresponds to the representation @xmath89 in the sense that it can be expressed as l^m = su(2)_u(1)_m  , [ monmexpl]where the quotient on @xmath90 is by the @xmath91 action @xmath92 for @xmath93 , @xmath94 and @xmath95 . the action of @xmath1 on @xmath90 given by @xmath96 descends to an action on ( [ monmexpl ] ) . any @xmath1-equivariant hermitean vector bundle over the sphere is a whitney sum of bundles ( [ monmexpl ] ) . there is an alternative description in terms of the holomorphic line bundle @xmath97 defined as the @xmath75-th power of the tautological bundle over the complex projective line . the universal complexification of the lie group @xmath1 is @xmath98 , and we may regard the sphere as a projective variety through the natural diffeomorphism @xmath99 , where @xmath100 is the parabolic subgroup of lower triangular matrices in @xmath98 . the @xmath1 action on ( [ monmexpl ] ) lifts to a smooth @xmath98 action , and the complexification of ( [ monmexpl ] ) is realized as the @xmath98-equivariant line bundle o(m)=sl(2,)_p_m [ monmholexpl]over @xmath69 . only the cartan subgroup @xmath101 of non - zero complex numbers acts non - trivially on the modules @xmath102 , with the @xmath103 action defined analogously to ( [ u1irreps ] ) . the two descriptions are equivalent after the introduction of a hermitean metric on the fibres of @xmath104 . this holomorphic line bundle has transition function @xmath105 transforming sections from the northern hemisphere to the southern hemisphere of @xmath4 . however , the monopole connection ( [ f1 ] ) is transformed on the intersection of the two patches covering @xmath69 via the transition function @xmath106 , which is the unitary reduction of the holomorphic transition function @xmath105 . thus the bundle @xmath104 regarded as a hermitean line bundle has transition function @xmath106 and can be substituted for the monopole bundle @xmath107 . * su(2)-invariant gauge potential .  * the form of our ansatz for the gauge connection on @xmath56 is fixed by imposing invariance under the @xmath1 isometry group of @xmath69 acting through rigid rotations of the sphere . let @xmath108 be an @xmath1-equivariant @xmath109-bundle , with the group @xmath1 acting trivially on @xmath10 and in the standard way on @xmath110 . let @xmath43 be a connection on @xmath111 . imposing the condition of @xmath1-equivariance means that we should look for representations of the group @xmath1 inside the @xmath109 structure group , i.e. for homomorphisms @xmath112 . the ansatz for @xmath43 is thus given by @xmath41-dimensional representations of @xmath1 . up to isomorphism , for each positive integer @xmath113 there is a unique irreducible @xmath1-module @xmath114 of dimension @xmath113 . therefore , for each positive integer @xmath75 , the module = _ i=0^m _ k_i _ i=0^mk_ik [ genrepsu2uk]gives a representation @xmath115 of @xmath1 inside @xmath116 . the total number of such homomorphisms is the number of partitions of the positive integer @xmath117 into @xmath118 components . the original @xmath109 gauge symmetry is then broken down to the centralizer subgroup of @xmath119 in @xmath120 as ( k )  _ i=0^mu(k_i )  . [ gaugebroken ]    it is natural to allow for gauge transformations to accompany the @xmath1 action  @xcite , and so some `` twisting '' can occur in the reduction of the connection @xmath43 on @xmath56 . the @xmath69 dependence in this case is uniquely determined by the above @xmath1-invariant dirac monopole configurations  @xcite . the @xmath51-valued gauge potential @xmath43 thus splits into @xmath121 blocks @xmath122 , @xmath123 where the indices @xmath124 run over @xmath125 , @xmath126 and [ f5 ] ^ii&=&a^i(x)1 + _ k_ia_m-2i(y )  , + [ f6 ] ^ii+1  = :  ^  _ i+1&= & ^  _ i+1(x)|(y )  , + [ f7 ] ^i+1i  =  -(^ii+1)^  =  -(^  _ i+1 ) ^&=&-_i+1^ ( x)(y )  , + [ f9 ] ^ii+l&=&0  =  ^i+li l  2  . here @xmath127 are the unique covariantly constant , @xmath128-invariant forms of type @xmath129 and @xmath130 such that the khler @xmath131-form on @xmath69 is @xmath132 . they respectively take values in the bundles @xmath133 and @xmath134 . it is easy to see that the gauge potential @xmath43 given by ( [ f5])([f9 ] ) is anti - hermitean and @xmath1-invariant . note that we do not use the einstein summation convention for the repeated indices @xmath135 labelling the components of the irreducible representation @xmath136 of the group @xmath1 . thus the gauge potential @xmath137 decomposes into gauge potentials @xmath138 with @xmath139 and a multiplet of scalar fields @xmath140 with @xmath141 transforming in the bi - fundamental representations @xmath142 of the subgroup @xmath143 of the original @xmath109 gauge group . all fields @xmath144 depend only on the coordinates @xmath145 . every @xmath1-invariant unitary connection @xmath43 on @xmath56 is of the form given in ( [ f4])([f9 ] )  @xcite , which follow from the fact that the complexified cotangent bundle of @xmath69 is @xmath146 . this ansatz amounts to an equivariant decomposition of the original rank  @xmath41 @xmath1-equivariant bundle @xmath147 in the form = _ i_k_il^m-2i  , [ caleansatz]where @xmath148 is a hermitean vector bundle of rank @xmath149 with typical fibre the module @xmath150 , and @xmath151 is the bundle with fibres @xmath152 . by regarding @xmath153 for @xmath154 and defining @xmath155 , we can summarize our ansatz through the following chain of bundles : @xmath156    * field strength tensor .  * the calculation of the curvature ( [ curvprod ] ) for @xmath43 of the form ( [ f4])([f9 ] ) yields @xmath157 where [ f11 ] ^ii&=&f^i+ f_m-2i+ ( ^  _ i+1^_i+1 - _ i^^  _ i)|  , + [ f12 ] ^ii+1&=&d _ i+1|  , + [ f13 ] ^i+1i  =  - ( ^ii+1)^&=&- ( d _ i+1)^  , + [ f14 ] ^ii+l&=&0  =  ^i+li l  2  . here we have defined @xmath158 and introduced the bi - fundamental covariant derivatives @xmath159 from ( [ f11])([f14 ] ) we find the non - vanishing field strength components [ f15 ] ^ii_&=&f_^i  , + [ f16 ] ^ii+1_|y&= & d_ _ i+1  =  -(^i+1i_y)^  , + [ fyyb ] ^ii_y|y&= & - ( m-2i+_i^^  _ i - ^  _ i+1^_i+1 )  . in this section we shall clarify some features of the ansatz constructed in the previous section from three different points of view . to set the stage for the string theory interpretations of the solutions that we will construct later on , we begin by indicating how the ansatz can be interpreted in terms of configurations of d - branes in type  ii superstring theory . this leads into a discussion of how the ansatz is realized in topological k - theory , which classifies the ramond - ramond charges of these brane systems , and we will derive the decomposition ( [ caleansatz ] ) directly within the framework of @xmath1-equivariant k - theory . we will then explain how seeking explicit realizations of our ansatz is equivalent to finding representations of the @xmath160 quiver . one of the goals of the subsequent sections will be to establish the precise link between these three descriptions , showing that they are all equivalent .    * physical interpretation .  * before entering into the formal mathematical characterizations of the ansatz of the previous section , let us first explain the physical situation which they will describe . our ansatz implies an equivalence between brane - antibrane systems on @xmath10 and wrapped branes on @xmath56 . in the standard d - brane interpretation , our initial rank @xmath41 hermitean vector bundle @xmath161 corresponds to @xmath41 coincident d(@xmath11 + 1)-branes wrapping the worldvolume manifold @xmath162 . the condition of @xmath1-equivariance imposed on this bundle fixes the dependence on the coordinates of @xmath69 and breaks the gauge group u(@xmath41 ) as in ( [ gaugebroken ] ) . the rank @xmath149 sub - bundle @xmath148 of this bundle is twisted by the dirac multi - monopole bundle @xmath163 . the system of @xmath41 coincident d(@xmath11 + 1)-branes thereby splits into blocks of @xmath164 coincident d(@xmath11 + 1)-branes , associated to irreducible representations of @xmath1 and wrapping a common sphere @xmath69 with the monopole fields . this system is equivalent to a system of @xmath165   d(@xmath1661)-branes carrying different magnetic fluxes on their common worldvolume @xmath10 . the d(@xmath1661)-branes which carry negative magnetic flux have opposite orientation with respect to the d(@xmath1661)-branes with positive magnetic flux , i.e. they are antibranes . this will become evident from the k - theory formalism , which will eventually lead to an explicit worldvolume construction , and also from the explicit calculation of the topological charges of the instanton solutions . in addition to the usual chan - paton gauge field degrees of freedom @xmath167 living on each block of branes , the field content on the brane configuration contains bi - fundamental scalar fields @xmath168 corresponding to massless open string excitations between neighbouring blocks of @xmath149 and @xmath169 d(@xmath1661)-branes . other excitations are suppressed by the condition of @xmath1-equivariance . however , as we shall see explicitly in the following , the fields @xmath170 should not be regarded as tachyon fields , but rather only as ( holomorphic ) higgs fields responsible for the symmetry breaking ( [ gaugebroken ] ) . only the brane - antibrane pairs whose constituents carry equal and opposite monopole charges are neutral and can thus annihilate to the vacuum , which carries no monopole charge ( although it can carry a k - theory charge from the virtual chan - paton bundles over @xmath10 ) . other brane pairs are stable because their overall non - vanishing chern number over @xmath69 is an obstruction to decay , and the monopole bundles thereby act as a source of flux stabilization for such brane pairs by giving them a conserved topological charge . in particular , neighbouring blocks of d@xmath171-branes are marginally bound by the massless open strings stretching between them . in this sense , the @xmath1-invariant reduction of d - branes on @xmath56 induces brane - antibrane systems on @xmath10 . note that while the system on @xmath10 is generically unstable , the original brane configuration on @xmath56 can be nonetheless stable .    * k - theory charges .  * given that the charges of configurations of d - branes in string theory are classified topologically by k - theory  @xcite , let us now seek the k - theory representation of the above physical situation . the one - monopole bundle @xmath172 is a crucial object in establishing the bott periodicity isomorphism k(m_qp^1)=k(m_q ) [ bottper]in topological k - theory . the isomorphism is generated by taking the k - theory product of the tachyon field @xmath173 of a virtual bundle @xmath174\in{{\rm k}}({{\cal m}}_q)$ ] with that of the class of the line bundle @xmath172 which represents the bott generator of @xmath175  @xcite . the topological equivalence ( [ bottper ] ) then implies the equivalence of brane - antibrane systems on @xmath56 and @xmath10 , with the brane and antibrane systems each carrying a single unit of monopole charge . when they carry @xmath176 units of charge , the isomorphism breaks down , and it is necessary to introduce the notion of `` d - operations '' to establish the relationship  @xcite . while these operations are natural , they are not isomorphisms and they reflect the fact that the explicit solutions in this setting are not @xmath128-invariant , so that the equivalence breaks down due to spurious moduli dependences of the system of branes on the @xmath69 factor . in what follows we will derive a modification of the relation ( [ bottper ] ) in _ equivariant _ k - theory which will naturally give the desired isomorphism , reflecting the equivalence of the brane - antibrane systems for arbitrary monopole charge , and bypass the need for introducing d - operations this is only possible by augmenting the basic brane - antibrane system to a _ chain _ of @xmath177 branes and antibranes with varying units of monopole charge as described above , and we will thereby arrive at an independent purely k - theoretic derivation of our ansatz . the representation ring @xmath178 of a group @xmath179  @xcite is the grothendieck ring of the category of finite dimensional representations of @xmath179 , with addition induced by direct sum of vector spaces , @xmath180+[\,\underline{v'}\,]:= [ \,\underline{v}\oplus\underline{v'}\,]$ ] , and multiplication induced by tensor product of modules , @xmath180\cdot[\,\underline{v'}\,]:= [ \,\underline{v}\otimes\underline{v'}\,]$ ] . as an abelian group it is generated by the irreducible representations of @xmath179 alternatively , since the isomorphism class of a @xmath179-module @xmath181 is completely determined by its character @xmath182 , the map @xmath183 identifies @xmath178 as a subring of the ring of @xmath179-invariant functions on @xmath179 . if @xmath10 is a @xmath179-space , then the grothendieck group of @xmath179-equivariant bundles over @xmath10 is called the @xmath179-equivariant k - theory group @xmath184 . this group unifies ordinary k - theory with group representation theory , in the sense that for the trivial space @xmath185 is the representation ring of @xmath179 , while for the trivial group @xmath186 is the ordinary k - theory of @xmath10 . the former property implies that @xmath184 is an @xmath178-module and the coefficient ring in equivariant k - theory is @xmath178 , rather than just @xmath187 as in the ordinary case . if the @xmath179-action on @xmath10 is trivial , then any @xmath179-equivariant bundle @xmath188 may be decomposed as a finite whitney sum e= _ ( g)hom^  _ g(11^  _ , e ) 11^  _ [ trivialedecomp]where @xmath189 is the trivial bundle over @xmath10 with fibre the irreducible @xmath179-module @xmath181 . it follows that for trivial @xmath179-actions the equivariant k - theory takes the simple form k_g(m_q)=k(m_q)r_g  . [ kgxqtrivialg]the @xmath190-functor behaves analogously to the ordinary @xmath191-functor , and in addition @xmath190 is functorial with respect to group homomorphisms . a useful computational tool is the equivariant excision theorem . if @xmath192 is a closed subgroup of @xmath179 and @xmath10 is an @xmath192-space , then the inclusion @xmath193 induces an isomorphism  @xcite ^*:k_g(g_fm_q )   k_f(m_q )  , [ excision]where the quotient on @xmath194 is by the @xmath192-action @xmath195 for @xmath196 , @xmath145 and @xmath197 . the @xmath179-action on @xmath198 descends from that on @xmath194 given by @xmath199 . let us specialize to our case of interest by taking @xmath200 , @xmath201 and the trivial action of @xmath1 on the space @xmath10 . using ( [ kgxqtrivialg ] ) and ( [ excision ] ) we may then compute k_su(2)(m_qp^1)&=&k_su(2)(su(2)_u(1)m_q ) + & = & k_u(1)(m_q )  =  k(m_q)r_u(1 )  . [ excisionslc]this k - theoretic equality asserts a one - to - one correspondence between classes of @xmath1-equivariant bundles over @xmath56 and classes of @xmath91-equivariant bundles over @xmath10 with @xmath91 acting trivially on @xmath10 . the isomorphism ( [ excisionslc ] ) of equivariant k - theory groups is constructed explicitly as follows  @xcite . given an @xmath1-equivariant bundle @xmath147 , we can induce a @xmath91-equivariant bundle @xmath202 by restriction to the slice @xmath203 . conversely , if @xmath188 is a @xmath91-equivariant bundle , then @xmath204 is an @xmath1-equivariant bundle , where the quotient on @xmath205 is by the action of @xmath91 on both factors , @xmath206 for @xmath207 , and the action of @xmath208 on @xmath209 descends from that on @xmath205 given by @xmath210 . this construction defines equivalence functors between the categories of @xmath1-equivariant vector bundles over @xmath56 and @xmath91-equivariant vector bundles over @xmath10 , and hence the corresponding grothendieck groups coincide , as in ( [ excisionslc ] ) . the role of the representation ring @xmath211 is unveiled by setting @xmath212 in ( [ excisionslc ] ) to get k_su(2)(p^1)=r_u(1 )  , [ kslccp1rp]which establishes a one - to - one correspondence between classes of homogeneous vector bundles over the sphere @xmath69 and classes of finite - dimensional representations of @xmath91 . since the corresponding irreducible representations are the @xmath89 given by ( [ u1irreps ] ) , the representation ring of @xmath91 is the ring of formal laurent polynomials in the variable @xmath213 , @xmath214 $ ] . using ( [ monmexpl ] ) we can associate the monopole bundle @xmath172 to the generator @xmath213 , and thereby identify ( [ kslccp1rp ] ) as the laurent polynomial ring k_su(2)(p^1)=  . [ kslccp1mon]in particular , the relationship ( [ excisionslc ] ) can be expressed as k_su(2)(m_qp^1)=k(m_q )  . [ botteq]this is the appropriate modification of the bott periodicity isomorphism ( [ bottper ] ) to the present setting . the crucial difference now is that virtual bundles over @xmath10 are multiplied by arbitrary powers of the one - monopole bundle , allowing us to extend the equivalence to arbitrary monopole charges @xmath82 . in the equivariant setting , there is no need to use external twists of the monopole bundle , nor the ensuing k - theory product as done in  @xcite . the monopole fluxes are now naturally incorporated by the coefficient ring @xmath211 of the @xmath91-equivariant k - theory , superseding the need for introducing d - operations . it is instructive to see precisely how the correspondence ( [ botteq ] ) works . for this , it is convenient to work instead in the category of holomorphic @xmath98-equivariant bundles  @xcite . if @xmath215 is an @xmath1-equivariant vector bundle over @xmath56 , then the action of @xmath1 can be extended to an @xmath98 action . everything we have said above carries through by replacing the group @xmath1 with its complexification @xmath98 and the cartan torus @xmath216 with the subgroup @xmath217 of lower triangular matrices . we are then interested in @xmath100-equivariant bundles over @xmath10 with @xmath100 acting trivially on @xmath10 . the lie algebra @xmath218 is generated by the three pauli matrices _ 3=    1&0 + 0&-1    , _ + =    0&1 + 0&0    _ -=    0&0 + 1&0    [ sl2cmatrices]with the commutation relations = 2_=_3  . [ sl2clie]the lie algebra of the subgroup @xmath100 is generated by the elements @xmath219 and @xmath220 , while the cartan subgroup @xmath101 is generated by the element @xmath219 with the corresponding irreducible representations being the @xmath89 given by ( [ u1irreps ] ) . since the manifold @xmath10 carries a trivial action of the subgroup @xmath103 , any @xmath103-equivariant bundle @xmath221 can be written using ( [ trivialedecomp ] ) as a finite whitney sum e=_l(e)e_l^  _ l  , [ ewhitneysl]where @xmath222 is the set of eigenvalues for the @xmath103-action on @xmath223 and @xmath224 are bundles carrying the trivial @xmath103-action . the rest of the @xmath100-equivariant structure is determined by the generator @xmath220 . since @xmath225=-2\,\sigma_-$ ] , the action of @xmath220 on @xmath226 corresponds to holomorphic bundle morphisms @xmath227 and the trivial @xmath220-action on the irreducible @xmath103-modules @xmath228 . thus every indecomposable @xmath100-equivariant bundle @xmath221 has weight set of the form @xmath229 for some @xmath230 with @xmath231 . after an appropriate twist by a @xmath103-module and a relabelling , the @xmath219-action is given by the @xmath103-equivariant decomposition e=_i=0^me_k_i_m-2i [ hactione]while the @xmath220-action is determined by a _ chain _ 0  e_k_m  e_k_m-1    e_k_1   e_k_0  0 [ holchain]of holomorphic bundle maps between consecutive @xmath232 s . we can now consider the underlying @xmath91-equivariant hermitean vector bundle defined by the unitary @xmath109 reduction of the @xmath233 structure group of the holomorphic bundle ( [ hactione ] ) , after introducing a hermitean metric on its fibres . then the corresponding bundle @xmath234 is given by = su(2)_u(1)e  . [ caleinduction]using ( [ monmexpl ] ) one finds that ( [ caleinduction ] ) coincides with the original equivariant decomposition ( [ caleansatz ] ) . conversely , given an @xmath1-equivariant bundle @xmath234 , its restriction @xmath235 defines a @xmath91-equivariant bundle over @xmath10 which thereby admits an isotopical decomposition of the form ( [ hactione ] ) and @xmath67 may be recovered from ( [ caleinduction ] ) . * quiver gauge theory . * the ansatz for the gauge potential on @xmath56 , represented symbolically by the bundle chain ( [ bundlechain ] ) , corresponds to the disjoint union of two copies of the quiver    [ aquiver]with the second copy obtained from ( [ aquiver ] ) by reversing the directions of the arrows and replacing @xmath236 with @xmath237 for each @xmath154 . the vertices of the quiver are labelled by the degrees of the monopole bundles @xmath238 , while the arrows correspond to module morphisms @xmath239 ( locally at each point @xmath240 ) . equivalently , the vertices may be labelled by irreducible chiral representations of the group @xmath100 . thus our ansatz determines a representation of the quiver @xmath241 in the category of complex vector bundles over the manifold @xmath10 such a representation is called an @xmath241-bundle . many properties of the explicit solutions that we construct later on find their most natural explanation in the context of such a quiver gauge theory , which provides a more refined description of the brane configurations than just their k - theory charges . this framework encompasses the algebraic and representation theoretic aspects of the problem  @xcite . the quiver graph ( [ aquiver ] ) is identical to the dynkin diagram of the lie algebra @xmath160 . the adjacency matrix of the quiver has matrix elements specifying the number of links between each pair of vertices @xmath242 , and in the case ( [ aquiver ] ) it is given by @xmath243 . the matrix elements @xmath244 are then identical to those of the cartan matrix @xmath245 , where @xmath246 , @xmath247 are the simple roots of @xmath160 . corresponding to the gauge symmetry breaking ( [ gaugebroken ] ) , the dimension vector @xmath248 can be regarded as a positive root of @xmath160 associated with the cartan matrix @xmath249 by writing it as k^  _ = _ i=0^mk_i  e_i    one - to - one correspondence between the isomorphism classes of indecomposable representations of the quiver @xmath241 and the set of positive roots of the lie algebra @xmath160 . this property is a consequence of the @xmath1-invariance of our ansatz . let us focus for a while on the case @xmath212 . in this case eq . ( [ caleansatz ] ) , with the @xmath75-monopole bundles @xmath107 substituted everywhere by the holomorphic line bundles ( [ monmholexpl ] ) , gives a relation between the categories of homogeneous holomorphic vector bundles over @xmath250 and of finite - dimensional chiral representations of @xmath100 , while the quiver representation further gives a relation with the abelian category of finite - dimensional representations of @xmath241  @xcite . to describe this latter category , it is convenient to introduce the notion of a path @xmath251 in @xmath241 , which is generally defined as a sequence of arrows of the quiver which compose . in the present case any path is of the form    [ path]with @xmath252 . we will denote it by the formal vector @xmath253 . the non - negative integer @xmath254 is the length of the path  ( [ path ] ) . the trivial path of length  @xmath255 based at a single vertex @xmath256 is denoted @xmath257 . the path algebra @xmath258 of the quiver ( [ aquiver ] ) is then defined as the algebra generated by all paths @xmath251 of @xmath241 , i.e. as the vector space a_m+1=_^m|m_0,  ,m_1 ) [ pathalg]together with the @xmath259-linear multiplication induced by ( left ) concatenation of paths where possible , |m_0,  ,m_1)|n_0,  ,n_1)= _ m_1n_0  |m_0,  [ pathalgmult]this makes @xmath258 into a finite - dimensional quasi - free algebra . the path algebra has a natural @xmath260-grading by path length , a_m+1=_i=0^m(a_m+1)_i ( a_m+1)_i=_m_0=-m^m-2i |m_0,  ,m_0 + 2i )  , [ pathgrading]and can thereby be alternatively described as the tensor algebra over the ring c_0=_i=0^m|m-2i )  ^m+1 [ c0ring]of the @xmath261-bimodule c_1=_i=0^m|m-2i , m-2i+2 )  . [ c1bimodule ]    the importance of the path algebra stems from the fact that the category of representations of the quiver @xmath241 is equivalent to the category of ( left ) @xmath258-modules  @xcite . given a representation @xmath262 , @xmath154 , of @xmath241 , the associated @xmath258-module @xmath263 is = _ i=0^m_m-2i [ calwdef]with multiplication extended @xmath259-linearly from the definitions |m-2i)w_j=_ij w_j |m-2i , m-2i+2)w_j=_i , j+1  _ j(w_j ) [ quivalgmod]for @xmath264 . conversely , given a left @xmath258-module @xmath263 , we can set @xmath265 for @xmath247 and define @xmath266 for @xmath154 by _ i(w_i)=|m-2i , m-2i+2)w_i  . [ etaiwi]one can further show that morphisms of representations of @xmath241 correspond to @xmath258-module homomorphisms  @xcite . thus , the problem of determining finite - dimensional representations of the quiver @xmath241 , or equivalently homogeneous vector bundles over @xmath69 , is equivalent to finding representations of its path algebra .    as an example , consider the @xmath267 quiver    [ a2quiver]it represents the standard brane - antibrane system , and as expected @xmath1-equivariance implies that it can only carry @xmath268 unit of monopole charge  @xcite . the corresponding path algebra is a_2=|-1 )  |+1 )  |-1,+1)=    & + 0 &    . [ a2pathalg]representations of this algebra yield the standard superconnections characterizing the low - energy field content on the worldvolume of a brane - antibrane system  @xcite . in the next section we will show how to generalize the superconnection formalism to account for representations of generic path algebras ( [ pathalg ] ) . later on we shall write down explicit solutions with generic monopole charge @xmath82 that also correspond to the basic brane - antibrane system . our technique for generating d - branes from a quiver gauge theory on @xmath10 arises via a quotient with respect to a generalized @xmath1-action on chan - paton bundles over @xmath56 . this new construction is rather different from the well - known quiver gauge theories that arise from orbifolds with respect to the action of a _ discrete _ group @xmath179  @xcite . in the latter case the nodes of a quiver represent the irreducible representation fractional branes into which a regular representation d - brane decays into when it is taken to an orbifold point of @xmath269 , and they can be thought of in terms of a projection of branes sitting on the leaves of the covering space @xmath10 . while our quiver gauge theory is fundamentally different , it shares many of the physical features of orbifold theories of d - branes . for instance , the blowing up of vortices on @xmath10 into instantons on @xmath56 is reminescent of the blowing up of fractional d@xmath171-branes into d@xmath270-branes wrapping a non - contractible @xmath69 that is used to resolve the orbifold singularity in @xmath269 . our solutions provide explicit realizations of this blowing up phenomenon , but in a completely smooth setting . the condition of @xmath1-equivariance uniquely prescribes a specific @xmath69 dependence for the gauge potential @xmath43 and reduces the yang - mills equations ( [ ym ] ) on @xmath162 to equations on @xmath10 . in this section we will formulate this reduction in detail and relate it to representations of the path algebra ( [ pathalg ] ) . this will be done by developing a new formalism of @xmath260-graded connections which describes the field content corresponding to the bundle chains ( [ bundlechain ] ) and ( [ holchain ] ) , and which generalizes the standard superconnection field theories on the worldvolumes of brane - antibrane systems  @xcite . this formalism will be the crux to merging together the three interpretations of the previous section . * reduction of the yang - mills functional . * the dimensional reduction of the yang - mills equations can be seen at the level of the yang - mills lagrangian ( [ lagr ] ) . substituting ( [ f15])([fyyb ] ) into ( [ lagrprod ] ) and performing the integral over @xmath69 we arrive at the action , @xmath247 can be introduced via the redefinitions @xmath271 . ] s^  _ ym&:=&_m_q ^q+2x  l_ym^   + & = & r^2_m_q^qx   _ i=0^mtr^  _ k_ik_i  , [ symred]where @xmath272 . in the remainder of this paper we shall only consider static field configurations on @xmath25 in the temporal gauge @xmath58 . in this case one can introduce the corresponding energy functional [ ef ] e^  _ _ m_2n^2nx  _ i=0^m tr^ _ k_ik_i  , where @xmath273 . the functional ( [ ef ] ) is non - negative .    * graded connections .  * the energy functional ( [ ef ] ) is analysed most efficiently by introducing a framework specific to connections on the rank @xmath274 @xmath260-graded vector bundle e:=_i=0^me_k_i [ gradedbundle]over @xmath28 whose typical fibre is the module ( [ genrepsu2uk ] ) . the endomorphism algebra bundle corresponding to ( [ gradedbundle ] ) is given by the direct sum decomposition ( e)=_i=0^mend(e_k_i )   _ ^m hom(e_k_i , e_k_j )  . [ endgraded]we may naturally associate to ( [ endgraded ] ) a distinguished representation of the @xmath160 quiver . for this , we note that the path algebra @xmath258 is itself a @xmath258-module , and that the elements @xmath275 define a complete set of orthogonal projectors of the path algebra , i.e. @xmath276 for @xmath277 with @xmath278 . analogously to the construction of ( [ calwdef])([etaiwi ] ) , we may thereby define a _ projective _ @xmath258-module @xmath279 for each @xmath247  @xcite , which is the subspace of @xmath280 generated by all paths which start at the @xmath135-th vertex of the quiver @xmath241 . then @xmath281 is the vector space generated by the path from the @xmath135-th vertex to the @xmath282-th vertex , and the corresponding dimension vector is k^  _ _ i=_j = i^me_j  . [ pidimvec]the modules @xmath283 , @xmath247 are exactly the set of all indecomposable projective representations of the @xmath284 quiver  @xcite , with a_m+1=_i=0^m_i  . [ pathalgmoddecomp ]    the importance of this path algebra representation stems from the fact that , for any quiver representation ( [ genrepsu2uk ] ) , there is a natural isomorphism  @xcite ( _ i , ) _ [ hompathnatural]we may thereby identify @xmath285 in terms of appropriate combinations of the spaces ( _ j,_i )   |m-2j)a_m+1|m-2i )  . [ hompij]this is the vector space generated by the path from the @xmath135-th vertex to the @xmath282-th vertex of @xmath241 . a natural representation of this path is by a matrix of dimension @xmath286 with @xmath287 in its @xmath288-th entry and @xmath255 s everywhere else . the path algebra ( [ pathalgmoddecomp ] ) is thereby identified with the algebra of upper triangular @xmath286 complex matrices  @xcite . for a given quiver representation ( [ genrepsu2uk ] ) , this algebra may be represented by assembling the chiral higgs fields @xmath289 into the @xmath290 matrix ^  _ ( m):=    0&_1&0&  &0 + 0&0&_2&  &0 + & & & & + 0&0&0&  &_m + 0&0&0&  &0    [ mgradedphidef]with respect to the decomposition ( [ gradedbundle ] ) . this object generates a representation of the path algebra in the category of complex vector bundles over @xmath28 , corresponding to the off - diagonal @xmath291 components of the decomposition ( [ endgraded ] ) . the finite dimensionality of @xmath280 is reflected in the property that generically ^  _ ( m ) , ( ^  _ ( m))^2 ,  , ( ^  _ ( m))^m  0 ( ^  _ ( m))^m+10  . [ mphi0s]the field configuration ( [ mgradedphidef ] ) generates the basic zero - form component of a geometric object that we shall refer to as a `` @xmath260-graded connection '' on @xmath28 . for @xmath268 it corresponds to a standard superconnection  @xcite , while for @xmath176 it is the appropriate entity that constructs representations corresponding to the enlargement of the path algebra @xmath280 . its matrix form is similar to ( [ f4])([f9 ] ) , but without the one - forms on @xmath69 .    to formulate the definition precisely , we note that the algebra @xmath292 of differential forms on @xmath28 with values in the bundle ( [ gradedbundle ] ) has a natural @xmath293 grading , where the @xmath187-grading is by form degree . we can thereby induce a total @xmath260-grading by the decomposition _ ( m_2n , e)=_p=0^m_(p)(m_2n , e ) _ ( p)(m_2n , e)= _ i+j^  _ m+1p^i(m_2n , e_k_j )  , [ omegagrading]where @xmath294 denotes congruence modulo @xmath177 . by using ( [ endgraded ] ) and the usual tensor product grading , this induces a @xmath260-grading on the corresponding endomorphism algebra as _ ( m_2n , end  e)=_p=0^m_(p)(m_2n , end  e ) [ omegaenddecomp]with _ ( p)(m_2n , end  e)= _ i_a^  _ m+1(p - a )  ^i_a(m_2n)(e_k_i , e_k_i+a )  . [ omegaendgrading]a _ graded connection _ on ( [ omegagrading ] ) is defined to be a linear operator @xmath295 which shifts the total @xmath260-grading by @xmath287 modulo @xmath177 , i.e. an element of _ ( 1)(m_2n , end  e)&=&_i=0^m  ( _ i_1^  _ m+11^i_1(m_2n)(e_k_i ) . + & &  .  _ m+10 ^i_0(m_2n)(e_k_i , e_k_i+1 ) )  , [ omega1conn]and which satisfies the usual leibniz rule on @xmath296 . as in the standard cases , the @xmath260-graded connections form an affine space modelled on a set of local operators .    in our case we retain only the @xmath297 and @xmath298 components of ( [ omega1conn ] ) corresponding to the lowest lying massless degrees of freedom on the given configuration of d - branes . from the leibniz rule it follows that the pertinent graded connections are then of the form @xmath299 , where ^(m):=_i=0^ma^i_i [ madef]and @xmath300 are the canonical orthogonal projections of rank  @xmath287 , _ i_j=_ij  _ i  , [ piortho]which may be represented , with respect to the decomposition ( [ gradedbundle ] ) , by diagonal matrices @xmath301 of unit trace . in this geometric framework all @xmath302 are assumed to anticommute with a given local basis @xmath303 of the cotangent bundle of the khler manifold @xmath28 , as if they were @xmath75 basic odd complex elements of a superalgebra . this requisite property may be explicitly realized by extending the graded connection formalism to @xmath7 . for this , we rewrite the ansatz ( [ f4])([f8 ] ) in terms of the above field configurations as _ & = & ( ^(m))_1  , [ calagradedmu ] + _ y&=&_k ( ^(m))_y- ( ^  _ ( m))^_y  , [ calagradedy ] + _ |y&=&_k ( ^(m))_|y+ ( ^  _ ( m))|_|y  , [ calagraded]where ^(m):=_i=0^ma_m-2i_i [ madef]and @xmath304 are the canonical projections on ( [ caleansatz ] ) . the coupling of @xmath305 to @xmath306 in ( [ calagraded ] ) yields the desired anticommutativity with @xmath303 . alternatively , we may use the canonical isomorphism @xmath307 to map the cotangent basis @xmath308 onto the generators of the clifford algebra ^^+^^= -2g^  _ 2^n+1 , 1,  [ 2n2cliffalg]the gamma - matrices in ( [ 2n2cliffalg ] ) may be decomposed as \{^}=\{^,^y , ^|y } ^=^_2  ,   ^y=^y ^|y=^|y  , [ gamma2n2decomp]where the @xmath309 matrices @xmath310 act on the spinor module @xmath311 over the clifford algebra @xmath312 , ^^+^^=-2g^  _ 2^n , 1,  ,2n  , [ 2ncliffalg]while =  _ _ 1_2n ^_1^_2n ( ) ^2=_2^n ^=-^[chiralityop]is the corresponding chirality operator . here @xmath313 is the levi - civita symbol with @xmath314 . the action of the clifford algebra @xmath315 on the spinor module @xmath316 is generated by ^y=1r^2(r^2+y|y)^y ^|y=1r^2(r^2+y|y)^|y[cp1cliffalg]with constant @xmath317 pauli matrices @xmath318 and @xmath319 obeying @xmath320=-\sigma_3 $ ] . the gauge potential ( [ f4])([f9 ] ) may then be written in an algebraic form as & : = & ^ _ + & = & ^ ( ^(m))__2 + ( ^  _ ( m ) ) ^|y|_|y- ( ^  _ ( m))^ ^y_y + & & + ( ^y ( ^(m))_y+^|y ( ^(m))_|y )  , [ calagammas]and the coupling of ( [ mgradedphidef ] ) with the chirality operator ( [ chiralityop ] ) realizes the desired anticommutativity with the one - form representatives @xmath321 . note that the products ( ^  _ ( m))^|y|_|y= 1r ( ^  _ ( m))^|y ( ^  _ ( m))^^y_y= 1r ( ^  _ ( m))^^y [ cp1indepprods]are independent of the coordinates @xmath322 . the curvature @xmath323 of the graded connection is also most elegantly expressed through dimensional reduction from @xmath7 . from ( [ f10])([fyyb ] ) it is given by & : = & _ + & = & ( ^(m))__2 - 1r(^ d _ ^  _ ( m))^^y-1r(^ d _ ^  _ ( m))^|y + & & + 12r^2 ( ^  _ ( m)+ ( ^  _ ( m))^ ( ^  _ ( m))- ( ^  _ ( m ) ) ( ^  _ ( m))^ ) _ 2^n_3 [ gradedcurv]where @xmath324 and ^  _ ( m):=_i=0^m(m-2i )  _ i  . [ mupdef]the contribution ( [ mupdef ] ) is generated by the monopole connection on @xmath69 in ( [ calagammas ] ) , while the higgs potentials in ( [ gradedcurv ] ) are produced by ( [ cp1indepprods ] ) . the graded curvature is independent of @xmath325 , and the standard gamma - matrix trace formulas tr^  _ ^2^n+1 ( ^^_2)&=&-2^n+1 g^  , [ trgammaid1 ] + tr^  _ ^2^n+1 ( ^^ ^^_2)&=&2^n+1 ( g^g^+g^g^- g^g^ )  , [ trgammaid2 ] + tr^  _ ^2^n+1 ( _ 2)&=&2^n+3(g^g^-g^ g^ )  , [ trgammaid3 ] + tr^  _ ^2^n+1 ( ^ ^^|y^y ) & = & -2^ng^  =  tr^  _ ^2^n+1(^ ^^y^|y ) [ trgammaid4]imply that the energy functional ( [ ef ] ) can be compactly written in terms of ( [ gradedcurv ] ) as e^  _ ym=_m_2n^2nx   tr_kk^  tr^  _ ^2^n+1  ^2  . [ efgraded ]    * nonabelian coupled vortex equations . * let us now examine the reduction of the duy equations on @xmath37 for a gauge potential of the form proposed in section 3 ( with static configurations in the gauge @xmath326 ) . substituting ( [ f11])([f14 ] ) into ( [ duy1])([duy3 ] ) , we obtain [ f24 ] g^a|bf^i_a|b&=&(m-2i+_i^ ^  _ i -^  _ i+1^_i+1 )  , + [ f240 ] f_|a|b^i&=&0  =  f_ab^i  , + [ f25 ] |^  _ |a^  _ i+1 + a^i_|a^  _ i+1 - ^  _ i+1a^i+1_|a&=&0 for each @xmath247 , where @xmath327 . recall that there is no summation over @xmath135 in these equations . we have abbreviated @xmath328 etc . , and defined the derivatives @xmath329 and @xmath330 with @xmath331 . we shall call ( [ f24])([f25 ] ) the nonabelian coupled vortex equations . ( [ f240 ] ) implies that the vector bundles @xmath332 are holomorphic , while eq . ( [ f25 ] ) implies that the higgs fields @xmath333 are holomorphic maps . by using a bogomolny - type transformation  @xcite one can show that solutions to these equations realize absolute minima of the energy functional ( [ ef ] ) . these field configurations describe supersymmetric bps states of d - branes .    * seiberg - witten monopole equations .  * for @xmath334 , @xmath268 and @xmath335 ( so that @xmath336 ) , the equations ( [ f24])([f25 ] ) coincide with the perturbed abelian seiberg - witten monopole equations on a khler four - manifold @xmath337  @xcite . in this case we have @xmath338 and the equations ( [ f24])([f25 ] ) reduce to [ f26 ] g^a|bf_a|b&=&(1-| )  , + f_|a|b&=&0  =  f_ab  , + [ f27 ] |_|a+ 2a_|a&=&0  . the perturbation , i.e. the term @xmath339 in ( [ f26 ] ) , is needed whenever @xmath337 has non - negative scalar curvature in order to produce a non - trivial and non - singular moduli space of finite energy @xmath340-solutions . it is usually introduced into the seiberg - witten equations by hand . in the present context , it arises automatically from the extra space @xmath69 and the reduction from @xmath341 to @xmath337 . to build further on the interpretation of our ansatz in terms of configurations of d - branes as described in section  [ ansatzdescr ] , we should now proceed to construct explicit solutions of the reduced yang - mills equations on @xmath28 . unfortunately , even solutions of the vortex equations ( [ f24])([f25 ] ) are difficult to come by and there is no known general method for explicitly constructing the appropriate field configurations . as we will demonstrate in the following , explicit realizations of these d - brane states are possible in the context of _ noncommutative _ gauge theory , which can be mapped afterwards onto commutative worldvolume configurations . for this , we will now specialize the khler manifold @xmath37 to be @xmath342 with metric tensor @xmath343 on @xmath6 and pass to a noncommutative deformation of the flat part of the space , i.e. @xmath344 . note that the @xmath69 factor remains a commutative space throughout this paper . then we will deform the yang - mills , duy and nonabelian coupled vortex equations , and in the subsequent sections construct various solutions of them . * noncommutative deformation . * field theory on @xmath345 may be realized in an operator formalism which turns schwartz functions  @xmath346 on @xmath6 into compact operators  @xmath347 acting on the @xmath348-harmonic oscillator fock space  @xmath349  @xcite . the noncommutative space  @xmath345 is then defined by declaring its coordinate functions @xmath350 to obey the heisenberg algebra relations @xmath351 = { \,\mathrm{i}\,}\th^{\mu\nu}\ ] ] with a constant real antisymmetric tensor  @xmath352 . via an orthogonal transformation of the coordinates , the matrix @xmath353 can be rotated into its canonical block - diagonal form with non - vanishing components @xmath354 for @xmath355 . we will assume for definiteness that all @xmath356 . the noncommutative version of the complex coordinates ( [ zz ] ) has the non - vanishing commutators @xmath357 \= -2\,\de^{a{{\bar{b}}}}\,\th^a \ = : \ \th^{a{{\bar{b } } } } \= -\th^{{{\bar{b}}}a}\ \le\ 0 \ .\ ] ] taking the product of @xmath345 with the commutative sphere @xmath69 means extending the noncommutativity matrix @xmath358 by vanishing entries along the two new directions . the fock space  @xmath349 may be realized as the linear span @xmath359 are connected by the action of creation and annihilation operators subject to the commutation relations @xmath360 = \de^{a{{\bar{b } } } } \ .\ ] ] in the weyl operator realization @xmath361 , coordinate derivatives are given by inner derivations of the noncommutative algebra according to @xmath362 \ = : \ \pa_{{\hat{z}}^a } { \hat{f}}\qquad\textrm{and}\qquad \widehat{\pa_{{{\bar{z}}}^{{{\bar{a } } } } } f}\=\th_{{{\bar{a}}}b}\,\big[{\hat{z}}^b \,,\ , { \hat{f}}\,\big ] \ \pa_{{\hat{\bar{z}}}^{\,{{\bar{a } } } } } { \hat{f}}\ , \ ] ] where @xmath363 is defined via @xmath364 so that @xmath365 . on the other hand , integrals are given by traces over the fock space @xmath349 as @xmath366    the transition to the noncommutative yang - mills and duy equations is trivially achieved by going over to operator - valued objects everywhere . in particular , vector bundles @xmath367 whose typical fibres are complex vector spaces @xmath181 are replaced by the corresponding ( trivial ) projective modules @xmath368 over @xmath3 . the field strength components along @xmath2 in  ( [ ym ] ) and ( [ duy1])([duy3 ] ) read @xmath369 $ ] , where @xmath370 are simultaneously @xmath51 and operator valued . to avoid a cluttered notation , we drop the hats over operators from now on . thus all our equations have the same form as previously but are considered now as operator equations .    * noncommutative coupled vortex equations .  * by reducing the noncommutative version of the duy equations on @xmath371 to @xmath345 we obtain the noncommutative nonabelian coupled vortex equations . instead of working with the gauge potentials @xmath372 we shall use the operators @xmath373 defined by @xmath374 in terms of these operators the field strength tensor reads @xmath375 + \th_{a{{\bar{b } } } } \ , ~~ f_{{{{\bar{a}}}}{{{\bar{b}}}}}^{i}\ = \ \big[x_{{{{\bar{a}}}}}^{i}\,,\ , x_{{{\bar{b}}}}^{i}\,\big ] \quad\textrm{and}\quad f_{{a}{b}}^{i}\ = \ \big[x_{{a}}^{i}\,,\ , x_{b}^{i}\,\big]\ , \label{fieldstrengthx}\ ] ] while the bi - fundamental covariant derivatives become d_|a^  ^  _ i+1x_|a^i^  _ i+1 x_|a^i+1 d_a^  ^  _ i+1x_a^i ^  _ i+1x_a^i+1  . [ bifundx ]    the nonabelian vortex equations ( [ f24])([f25 ] ) can then be rewritten as [ ddd1 ] ^a|b(+ _ a|b ) & = & ( m-2i+_i^^  _ i - ^  _ i+1^_i+1 )  , + [ ddd ] & = & 0  =   , + [ ddd2 ] x_|a^i^  _ i+1 - ^  _ i+1x_|a^i+1&=&0 for @xmath247 . note that for @xmath268 we obtain the equations [ f28 ] ^a|bf^0_a|b&=&(1-^  _ 1 _ 1^ )  f_|a|b^0  =  0  =  f_ab^0  , + [ f29 ] ^a|bf_a|b^1&=&-(1-_1^ ^  _ 1 ) f_|a|b^1  =  0  =  f_ab^1  , + [ f30 ] |^  _ |a^  _ 1 + a^0_|a^  _ 1 - ^  _ 1a^1_|a&=&0 which are considered in  @xcite . in particular , for @xmath334 and @xmath335 the equations ( [ f28])([f30 ] ) coincide with the perturbed seiberg - witten @xmath376 monopole equations on @xmath377 as considered in  @xcite . we are now ready to construct solutions to the yang - mills equations on @xmath378 . we shall first present the generic non - bps solutions of the full yang - mills equations , and then proceed to solve the nonabelian coupled vortex equations ( [ ddd1])([ddd2 ] ) , and thus the duy equations on  @xmath378 , which describe the stable bps states . our technique will make use of appropriate partial isometry operators @xmath379 in the noncommutative space .    * ansatz for explicit solutions . * let us fix a monopole charge @xmath380 and an arbitrary integer @xmath381 . consider the ansatz [ ansatz3 ] x_a^i & = & _ a|bt^  _ n_i|z^|bt_n_i^ x_|a^i  =  ^  _ |ab t^  _ n_i z^bt_n_i^  , + [ ansatz3p ] ^  _ i+1 & = & ^ _ i+1t^  _ n_it_n_i+1^ ^_i+1  =  |^  _ n_i+1t_n_i^ for @xmath247 , where @xmath382 are some constants with @xmath383 . denoting by @xmath349 the @xmath348-oscillator fock space , the toeplitz operators @xmath384 are partial isometries described by _ rectangular _ @xmath385 matrices ( with operator entries acting on @xmath349 ) possessing the properties @xmath386 where @xmath387 is a hermitean projector of finite rank @xmath388 on the fock space @xmath389 so that p_n_i^2p^  _ n_ip^_n_i tr^  _ _ k_ih  p^  _ n_in_i  . [ pnitrace]from ( [ ansatz4 ] ) it follows that the operator @xmath379 has a trivial kernel , while the kernel of @xmath390 is the @xmath388-dimensional subspace of @xmath389 corresponding to the range of @xmath391 . thus t^  _ n_i0 t^_n_in_i  . [ dimkertni ]    substituted into ( [ fieldstrengthx ] ) this ansatz yields the gauge field strength @xmath392 while from ( [ bifundx ] ) one finds the covariant derivatives d_|a_i+10d_a_i+1  . [ ansatzcovconst]thus our ansatz describes _ holomorphic _ fields , and the projector @xmath391 defines a noncommutative gauge field configuration of rank @xmath388 and constant curvature in the subspace @xmath393 . in particular , the higgs fields @xmath170 are covariantly constant with ^_i^  _ i=|_i|^2 ( ^  _ k_i- p^  _ n_i ) ^  _ i+1^_i+1=|_i+1|^2 ( ^  _ k_i - p^  _ n_i )  . [ phiphidags ]    the ranks @xmath388 are generically non - negative integers . if some @xmath394 , then we should formally set @xmath395 , @xmath396 and @xmath397 in the @xmath135-th component of the ansatz . then x_a^i=_a|b|z^|bx^i_|a=^  _ |abz^b [ xni0]which leads to the vacuum gauge field configuration a^i0 f^i0  . [ afni0]these matter fields correspond to open strings with one end on a d - brane and the other end on the closed string vacuum . our ansatz has a natural interpretation in quiver gauge theory . consider the module  : =  _ i=0^mt_n_i^k^  _ = _ i=0^mn_i  e_i [ vnam]over the quiver @xmath241 , which is a finite - dimensional submodule of the infinite - dimensional representation @xmath398 of @xmath241 given by the noncommutative quiver bundle . let us fix an integer @xmath399 , and take @xmath400 for all @xmath401 and @xmath394 for all @xmath402 . the quiver representation ( [ vnam ] ) is a combination of the indecomposable projective representations @xmath283 of @xmath241 that we encountered in section  [ dimred ] . the @xmath283 s form a complete set of projective representations in the sense that any quiver representation has a projective resolution in terms of sums of them  @xcite . in particular , the canonical ringel resolution of ( [ vnam ] ) is given by the exact sequence 0  _ i=1^s_i-1t_n_i^  _ i=0^s _ it_n_i^   0  . [ ringelres ]    * solving the yang - mills equations .  * we shall now demonstrate that the field configurations ( [ ansatz3])([ansatz4 ] ) yield solutions of the full yang - mills equations on @xmath403 for any values of @xmath75 , @xmath404 and @xmath405 . for this , we write the ansatz in the form [ chia ] _ a- _ a|b|z^|b & = & _ i=0^m x^i_a^  _ i  =   _ a|b_i=0^m t^  _ n_i|z^|bt_n_i^ ^  _ i  , + [ chiab ] _ |ab z^b & = & _ i=0^m x^i_|a^  _ i  =   _ |a b_i=0^m t^ _ n_iz^bt_n_i^ ^  _ i  . we also have _ y^ii&= &  _ k_i  , [ ayii ] + _ |y^ii&=&-  _ k_i  , [ aybii ] + [ avt ] _ |y^i i+1 & = & ^  _ i+1  =  t^  _ n_it_n_i+1^  , + [ avp ] _ y^i+1 i & = & - ^_i+1  =   -t^  _ n_i+1t_n_i^  , with _ j j  i , i+1  . [ av0]thus for the ansatz ( [ ansatz3])([ansatz4 ] ) the field strength tensor is given by [ cfabb ] _ a|b & = & _ a|b _ i=0^m p^  _ n_i^  _ i  , + [ cfvtvp ] _ y|y & = & - _ i=0^m(m-2i+(|^  _ i|^2 -|^  _ i+1|^2)(_k_i - p^  _ n_i ) ) ^  _ i  , with all other components of @xmath406 vanishing . let us now insert these expressions into the yang - mills equations ( [ ym ] ) ( for static configurations with @xmath407 ) . it is enough to consider the cases @xmath408 and @xmath409 , since the cases @xmath410 and @xmath411 can be obtained by hermitean conjugation of ( [ ym ] ) due to the anti - hermiticity of @xmath412 and @xmath413 . for @xmath408 , eq . ( [ ym ] ) becomes @xmath414\bigr)=0\ ] ] which is equivalent to @xmath415=0\ .\ ] ] substituting ( [ chiab ] ) and ( [ cfabb ] ) , we see that ( [ rym ] ) is satisfied due to the identities ( [ piortho ] ) and @xmath416 in the case @xmath417 , eq . ( [ ym ] ) simplifies to @xmath418=0\ ] ] with @xmath419 . substituting ( [ ayii ] ) , ( [ avp ] ) , ( [ av0 ] ) and ( [ cfvtvp ] ) , we find that ( [ rrym ] ) is also satisfied due to the identities ( [ piortho ] ) and ( [ iden ] ) . hence , the yang - mills equations on @xmath378 are solved by our choice of ansatz .    * finite - energy solutions . * the arbitrary coefficients @xmath420 can be fixed ( up to a phase ) by demanding that the solution ( [ ansatz3])([ansatz4 ] ) yield finite - energy field configurations . for this , we evaluate the energy functional ( [ ef ] ) using ( [ intnc ] ) . from ( [ ansatzfieldstrength ] ) we may compute ( f_^i)^(f^i)= 8^a|c^d|bf_a|b^if_d|c^i2 ( _ a=1^n1(^a)^2 )  p^  _ n_i  , [ efncfieldstrength]and combining this with ( [ ansatzcovconst ] ) and ( [ phiphidags ] ) we find the noncommutative yang - mills energy e^  _ ym&=&2r^2(_a=1^n2^a ) _ i=0^mtr^  _ _ [ efncgen]because of the trace over the infinite - dimensional fock space @xmath349 , the constant terms in ( [ efncgen ] ) which are not proportional to the projectors @xmath391 must all vanish in order for the energy to be finite . this leads to the finite - energy conditions @xmath421 for each @xmath247 .    with @xmath383 , the constraints ( [ cnstr1 ] ) are solved by @xmath422 ) can thereby be written as e^  _ ym=2r^2(_a=1^n2^a ) _ i=0^m2(n_i+n_m - i )  . [ efncfinite]we have naturally split the sum over nodes @xmath135 into contributions from dirac monopoles and antimonopoles , which for each @xmath247 have the same yang - mills energy on the sphere @xmath69 . later on we will see that this splitting corresponds to a @xmath423-grading of the chain of d - branes into brane - antibrane pairs . the monopole independent terms in ( [ efncfinite ] ) can be interpreted as the tension of @xmath424 d0-branes inside a d@xmath5-brane  @xcite in the seiberg - witten decoupling limit  @xcite .    * bps solutions .  * the solutions we have described generically yield non - bps solutions of the full yang - mills equations on @xmath403 . on the other hand , the duy equations on @xmath403 are bps conditions for the yang - mills equations . inserting ( [ ansatz3])([ansatz4 ] ) and ( [ ansatzfieldstrength])([phiphidags ] ) into our nonabelian vortex equations ( [ ddd1])([ddd2 ] ) , we find that ( [ ddd ] ) and ( [ ddd2 ] ) are automatically satisfied . the vanishing of the constant term ( not proportional to @xmath391 ) in ( [ ddd1 ] ) is precisely the finite - energy constraint ( [ cnstr1 ] ) , whose solution is given in ( [ 7.8 ] ) . equating the coefficients of @xmath391 in ( [ ddd1 ] ) for each @xmath400 leads to the additional constraints @xmath425    for @xmath426 the conditions ( [ cnstr2 ] ) are incompatible with one another , implying that the ansatz ( [ ansatz3])([ansatz4 ] ) with @xmath426 does not allow for bps configurations . for @xmath427 , the equation ( [ cnstr2 ] ) relates the radius @xmath14 of the sphere to the noncommutativity parameters @xmath428 of @xmath3 . in this case we obtain the explicit solutions of the noncommutative vortex and duy equations parametrized by the partial isometry operators @xmath429 as [ explsl0 ] x^0_a&=&_a|bt^  _ n_0|z^|b t_n_0^ ^  _ 1=^  _ 1t^  _ n_0  , + x^i_a&=&_a|b|z^|b  ^  _ i=^  _ i  _ k_1 0  <  i  m  . [ explsl1]the bps conditions ( [ ddd1])([ddd2 ] ) force us to take @xmath430 corresponding to the gauge symmetry breaking @xmath431 , so that @xmath432 , @xmath433 with @xmath434 and @xmath435 . the configurations with @xmath436 correspond to the vacuum gauge fields ( [ afni0 ] ) with trivial bundle maps @xmath302 given as multiplication by the complex numbers @xmath437 satisfying ( [ 7.8 ] ) . using ( [ cnstr2 ] ) and ( [ efncfinite ] ) we find that the energies of these bps states are given by e^  _ bps=2(2)^n+1r^2(_^n _ ^n^a )  n_0  . [ ebps ]    these solutions have a natural physical interpretation along the lines described in section  [ ansatzdescr ] . the original noncommutative duy equations are fixed by the positive integers @xmath348 and @xmath41 . our ansatz ( [ f4])([f9 ] ) and ( [ ansatz3])([ansatz4 ] ) is labelled by the collection of positive integers @xmath438 with @xmath439 . according to the standard identification of d - branes as noncommutative solitons  @xcite , the configuration ( [ explsl0],[explsl1 ] ) with @xmath427 describes a collection of @xmath440 bps d0-branes as a stable bound state ( i.e. a vortex - like solution on @xmath3 ) in a system of @xmath441 d(2@xmath348 ) branes and antibranes . but from the point of view of the initial branes wrapped on @xmath403 , they are spherical @xmath442 d2-branes . this means that instantons on @xmath403 are the spherical extensions of vortices which are points in @xmath2 . for @xmath426 the configuration ( [ ansatz3])([ansatz4 ] ) describes an unstable system of @xmath443 d0-branes ( vortices ) in a d(2@xmath348 ) brane - antibrane system , because @xmath444 for each @xmath445 . again they form a system of spherical d2-branes ( i.e. an su(2)-symmetric multi - instanton ) in the initial brane - antibrane system on @xmath403 . their orientation depends on the sign of the magnetic charge @xmath446 for each @xmath439 , which determines whether we have d2-branes or d2-antibranes . if more than one @xmath447 then the ansatz either describes pairs of d0-branes with overall non - vanishing monopole charges , or both d0-branes and anti - d0-branes . such systems can not be stable , i.e. the corresponding configuration ( [ ansatz3])([ansatz4 ] ) can not satisfy the noncommutative vortex and duy equations .    the distinction between bps versus non - bps solutions is very natural in quiver gauge theory . the bps configurations are described by the simple schur representations @xmath448 , @xmath247 of the @xmath160 quiver given by a one - dimensional vector space at vertex @xmath135 with all maps equal to  @xmath255 , i.e. the @xmath241-module with @xmath449 and dimension vector @xmath450 . the bps states constructed above then correspond to the quiver representations @xmath451 . together with the projective modules @xmath283 , the schur modules @xmath448 admit the projective resolutions 0  _ 0  _ 0 & & 0  , [ plres0 ] + 0   _ i-1  _ i   _ ,s [ plresi]and satisfy the relations  @xcite ( _ i,_j)=_ij  hom(_i,_j )  . [ plhomrels ]    the resolutions ( [ ringelres ] ) and ( [ plres0],[plresi ] ) exhibit a sharp homological distinction between bps and non - bps solutions . the constituent d - branes at the vertices of the quiver @xmath241 are associated with the basic representations @xmath448 . sums @xmath452 for fixed @xmath135 correspond to bps states , associated generally with the symmetry breaking @xmath453 , which are constructed analogously to ( [ explsl0],[explsl1 ] ) but with the vacuum higgs configurations @xmath454 for @xmath455 and @xmath456 for @xmath457 . a generic non - bps state , associated to the quiver representation ( [ vnam ] ) , corresponds to the decay of the original @xmath1-symmetric branes wrapped on @xmath403 into the collection of constituent branes @xmath458 in @xmath3 . for @xmath426 this collection is unstable . in the quiver gauge theory , we have thereby arrived at a natural construction of the unstable d - brane configurations in terms of stable bps states of d - branes , which may be succinctly summarized through the sequence of distinguished triangles of quiver representations    ( _ 0)^n_0  = & _ 0 & & _ 1 &   & _ m-1 & & _ m&=   + &  & & & &  & & & + & & ( _ 1)^n_1 & & & & ( _ m)^n_m & &    [ bpstriangles]where @xmath459 and the horizontal maps are the canonical inclusions of submodules . this exact sequence expresses the fact that , for each @xmath460 , the non - bps module @xmath461 is an extension of the bps module @xmath462 by the non - bps module @xmath463 . in this section we shall construct an explicit realization of the basic partial isometry operators @xmath464 which will be particularly useful for putting the d - brane interpretation of our noncommutative multi - instanton solutions on firmer ground . it is based on an @xmath1-equivariant generalization of the ( noncommutative ) atiyah - bott - shapiro ( abs ) construction of tachyon field configurations  @xcite@xcite .    * equivariant abs construction .  * if @xmath179 is a group and @xmath465 , we denote by @xmath466 the grothendieck group of isomorphism classes of finite - dimensional @xmath423-graded @xmath467 modules , i.e. clifford modules possessing an even ( @xmath423-degree preserving ) @xmath179-action which commutes with the @xmath468-action . more precisely , we consider representations of @xmath469\otimes{{{\rm c}\ell}}_{2n}$ ] with @xmath469 $ ] the group ring of @xmath179 . the inclusion @xmath470 of clifford algebras induces a restriction map _ g(2n)^*:r_spin_g(2n+1 )  r_spin_g(2n ) [ restrmapg]on equivariant clifford modules . following the standard abs construction  @xcite , we may then obtain the @xmath179-equivariant k - theory @xmath471 ( with compact support ) through the descendent isomorphism k_g(^2n )  =  coker_g(2n)^ *  =  r_spin_g(2n)/ _ g(2n)^*r_spin_g(2n+1 )  . [ descisog]the image of @xmath472 in @xmath466 contains classes of clifford modules @xmath180 $ ] which admit a @xmath467-equivariant involution @xmath473 , where @xmath474 is the clifford module @xmath181 with its @xmath423-parity reversed .    in our case , we take @xmath475 acting trivially on @xmath6 , and thereby consider @xmath476-modules with the @xmath91-action commuting with the clifford action . any such module is a direct sum of tensor products of a @xmath91-module and a spinor module , and hence r_spin_u(1)(2n)r_spin(2n)r_u(1)_u(1)(2n)^*=(2n)^ *  . [ spinortrivdecomp]since from the standard abs construction one has the isomorphism  @xcite k(^2n )  =  coker(2n)^ *  =  r_spin(2n)/ ( 2n)^*r_spin(2n+1 ) [ usualabsiso]of abelian groups , we can reduce ( [ descisog ] ) for @xmath477 to the isomorphism k_u(1)(^2n)=k(^2n)r_u(1)[eqabsiso]of @xmath211-modules , where @xmath478 ( note that the isomorphism @xmath479 also follows from the fact that @xmath6 is equivariantly contractible to a point ) . we may describe the isomorphism ( [ eqabsiso ] ) along the lines explained in section  [ ansatzdescr ] . in particular , the spinor module @xmath480 admits the isotopical decomposition _ i hom^  _ u(1)(_m-2i,_2n ) [ spinmoddecomp]obtained by restricting @xmath481 to representations of @xmath482 . the @xmath483 s in ( [ spinmoddecomp ] ) are the corresponding multiplicity spaces .    the most instructive and useful way to explicitly realize the decomposition ( [ spinmoddecomp ] ) is to use the equivariant excision theorem ( [ excisionslc ] ) directly and consider the @xmath1-invariant dimensional reduction of spinors from @xmath484 to @xmath6 . for this , we introduce the twisted dirac operator on @xmath484 using the graded connection formalism of section  [ dimred ] to write the @xmath260-graded clifford connection  : =  ^d _  =  ^d__2 + ( ^  _ ( m))^|y_|y- ( ^  _ ( m))^^y _ y+_p^1  , [ diracgradeddef]where _ p^1  : =  ^yd_y+^|yd_|y  =   ^y(_y+_y+ ( ^(m))_y)+ ^|y(_|y+_|y+ ( ^(m))_|y ) [ diracs2def]and @xmath485 are the components of the levi - civita spin connection on the tangent bundle of @xmath69 . from ( [ diracgradeddef ] ) we see that the monopole charges @xmath446 in the yang - mills energy functional ( [ ef ] ) can be understood as originating from the dirac operator ( [ diracs2def ] ) on @xmath69 . the operator ( [ diracgradeddef ] ) acts on spinors @xmath486 which are sections of the bundle =    ^+ + ^-    _ i=0^m(e_k_i_2n )    l^m-2i+1 + l^m-2i-1    [ spinortotgen]over @xmath484 , where @xmath487 are the twisted spinor bundles of rank  @xmath488 over the sphere @xmath69 . we are therefore interested in the twisted spinor module @xmath489 over the clifford algebra @xmath490 which is the product of the spinor module @xmath491 with the fundamental representation ( [ genrepsu2uk ] ) of the gauge group @xmath109 broken as in ( [ gaugebroken ] ) . the symmetric fermions on @xmath6 that we are interested in correspond to @xmath1-invariant spinors on @xmath484 . they belong to the kernel of the dirac operator ( [ diracs2def ] ) on @xmath69 and will be massless on @xmath6 . one can write _ p^1=_i=0^m_m-2i= _ i=0^m    0&^-_m-2i + ^+_m-2i&0    , [ diracs2decomp]where ^+_m-2i&=&1r^2  , [ diracs2p ] + ^-_m-2i&=&-1r^2  . [ diracs2m]the operator ( [ diracs2decomp ] ) acts on sections of the bundle ( [ spinortotgen ] ) which we write with respect to this decomposition as = _ ( m-2i)^+ + _ ( m-2i)^-    , [ psidecomp]where @xmath492 are sections of @xmath493 taking values in @xmath494 with coefficients depending on  @xmath495 .    to describe the kernel of the dirac operator ( [ diracs2decomp ] ) , we need to solve the differential equations ^+_m-2i^+_(m-2i)0 ^-_m-2i^-_(m-2i)0 [ diracpkernel]for the positive and negative chirality spinors @xmath496 and @xmath497 in @xmath498 and @xmath499 . by recalling the form of the transition functions for the monopole bundles from section  [ invgauge ] , one easily sees that the only solutions of these equations which are regular on both the northern and southern hemispheres of @xmath4 are of the form _ ( m-2i)^+=1(r^2+y|y)^t_i/2 _ = 0^t_i^+_(m-2i)(x )  y^_(m-2i)^-0 m-2i  <  0 [ solschargeneg]and _ ( m-2i)^-=1(r^2+y|y)^t_i/2 _ = 0^t_i^-_(m-2i)(x )  |y^_(m-2i)^+0 m-2i  >  0  . [ solschargepos]here @xmath500 and the component functions @xmath501 on @xmath6 with @xmath502 form the irreducible representation @xmath503 of the group @xmath1 . thus the chirality grading is by the sign of the magnetic charges . this analysis is valid when the monopole charge @xmath75 is an even or odd integer . however , when @xmath75 is even there is precisely one term in ( [ spinortotgen ] ) with @xmath504 for which the sub - bundle @xmath505 is twisted by the ordinary spinor bundle @xmath506 of vanishing magnetic charge . this bundle admits an infinite - dimensional vector space of symmetric @xmath507-sections comprised of spinor harmonics @xmath508 with @xmath509 , @xmath510 and @xmath511  @xcite . the spectrum of the ( untwisted ) dirac operator @xmath512 consists of the eigenvalues @xmath513 , each of even multiplicity @xmath514 . after dimensional reduction , this produces an infinite tower of massive spinors on @xmath6 , and such fermions of zero magnetic charge have no immediate interpretation in the present context . however , one has @xmath515 , and this will be enough for our purposes . we will therefore fix one of these vector spaces , such that after integration over @xmath69 it corresponds to the space _ p  ^2^p+1 p1,3,5 ,   . [ spinorharm]all of our subsequent results will be independent of the particular choice of eigenspace ( [ spinorharm ] ) . we have thereby shown that the @xmath1-equivariant reduction of the twisted spinor representation of @xmath490 decomposes as a @xmath423-graded bundle giving _ ( ^2np^1 ) ^su(2)=_2n(_^+  _ ^- )  m   , [ twistedspingrad]where _ ^+=_i = m_+^m _ k_i_|m-2i| _ ^-= _ i=0^m_- _ k_i_m-2i [ twistedspinpm]with @xmath516 and @xmath517 . when @xmath75 is an even integer , one should also couple the eigenspace ( [ spinorharm ] ) giving _ ( ^2np^1 ) ^su(2)=_2n(_^+  ( _ k_m2_p )  _ ^- )  m   [ twistedspingradeven]with @xmath518 and @xmath517 . it remains to work out the corresponding action of clifford multiplication ^  _ : _ ^-   _ ^+  . [ cliffmultv]for this , we recall from section  [ ansatzdescr ] that the action of the generators of the parabolic subgroup @xmath217 on the equivariant decomposition ( [ twistedspingrad],[twistedspinpm ] ) is given by @xmath519 and @xmath520 . since the clifford action is required to commute with this action , the map ( [ cliffmultv ] ) is thereby uniquely fixed on the isotopical components in the form ^  _ ^  _ i:_k_i_|m-2i| _ k_i+m2 + 1 _ |m-2i| i0,1,  [ cliffmultisot]furthermore , since @xmath521 for all @xmath522 , the space of spinor harmonics must lie in the kernel of the clifford map and one has ^  _ _ m20  m   . [ cliffordmulteven ]    it is also illuminating to formulate this equivariant dimensional reduction from a dynamical point of view , as we did for the gauge fields in section  [ dimred ] . using the gauged dirac operator ( [ diracgradeddef ] ) we may define a fermionic energy functional on the space of sections of the bundle ( [ spinortotgen ] ) by e^  _ d:=_^2np^1 ^ 2n+2x  g   ^  . [ eddef]one has ^ ( ( ^  _ ( m))^|y- ( ^  _ ( m))^^y)=    ( ^+)^&(^-)^    ( ^  _ ( m))^(^- ) + ( ^  _ ( m))(^+ )    . [ psiphiid]substituting ( [ diracs2decomp])([solschargepos ] ) , we see that ( [ psiphiid ] ) vanishes on symmetric spinors and after integration over @xmath69 the energy functional ( [ eddef ] ) for @xmath75 odd becomes e^  _ d&=&4r^2_^2n^2nx  . [ efred]the symmetric fermion energy functional for @xmath75 even also contains mass terms for fermions of vanishing magnetic charge which are proportional to the multiplicity @xmath523 of the spinor harmonics . * explicit form of the operators @xmath524 .  * the operators @xmath525 parametrizing the solutions of the previous section may be realized explicitly by appealing to a noncommutative version of the above construction . for this , we first note that the ( trivial ) action of @xmath216 on @xmath6 induces an action on functions @xmath346 on @xmath6 by @xmath526 for @xmath95 . this in turn defines an action of @xmath91 on the noncommutative space @xmath527 through automorphisms @xmath528 of the weyl operator algebra , i.e. a representation of @xmath91 in the automorphism group of the algebra . we will assume that the fock space ( [ fockspace ] ) carries a unitary representation of @xmath91 . we can then decompose it into its isotopical components in the usual way as h=_i=0^mh_i_m-2i  . [ fockisotop]for @xmath95 we denote the corresponding unitary operator on @xmath349 by @xmath529 . if we demand that the representations of @xmath91 above are covariant with respect to each other  @xcite , f^-1=  , [ covariantreps]then they define a representation of the crossed - product of the algebra of weyl operators with the group @xmath91 . this defines the ( trivial ) noncommutative @xmath91-space @xmath530 , and equivariant field configurations are operators belonging to the commutant of @xmath91 in the crossed - product algebra . in quiver gauge theory , the pertinent representation of @xmath241 thus labels isotopical components of the hilbert space of the noncommutative gauge theory . since the @xmath91-action is trivial here , the isotopical components of the fock space ( [ fockisotop ] ) are given by @xmath531 for each @xmath247 . note that one has an isomorphism @xmath532 by the usual hilbert hotel argument . we will now construct a representation on ( [ fockisotop ] ) of the partial isometry operators @xmath379 in @xmath530 . for this , let us put @xmath533 and consider the operators  @xcite @xmath534 where @xmath535 , @xmath536 and the @xmath537 matrices @xmath538 are subject to the anticommutation relations @xmath539 eq .  ( [ sigmaanti ] ) implies that the matrices @xmath540 generate the clifford algebra @xmath468 . note that for @xmath541 we have @xmath542 , @xmath543 and @xmath544 , which yields @xmath545 and we obtain the standard shift operator @xmath546 on the fock space @xmath349 in this case . generally , the operators ( [ tt ] ) obey @xmath547 where @xmath548 is a projector of rank @xmath388 on the vector space @xmath549 , and @xmath550 are the irreducible chiral spinor modules of dimension @xmath551 ( with @xmath552 ) on which the matrices @xmath553 act . the partial isometry operators @xmath554 in @xmath527 do not act on the isotopical decomposition ( [ fockisotop ] ) and thus do not properly incorporate the @xmath1-equivariant reduction of the original system of d - branes . the desired operators @xmath464 in @xmath555 are obtained by first projecting these partial isometries onto constituent brane subspaces . with @xmath556 the rank  @xmath287 projector onto the @xmath135-th isotopical component in ( [ fockisotop ] ) , we thereby define the @xmath537 matrices @xmath557 the operator @xmath558 acts as the shift operator @xmath554 on @xmath559 and as the identity operator @xmath560 on @xmath561 for all @xmath562 . it is easy to see that these matrices satisfy the equations ( t^(0)_n_i)^(t^(0)_n_i)=_r ( t^(0)_n_i)(t^(0)_n_i ) ^=_r - p_n_i^(0 ) [ tni0eqs]with @xmath563 a projector of rank @xmath388 on the fock space @xmath549 . they also satisfy the algebra ( t^(0)_n_i)^nt^(0)_n_in t^(0)_n_it_n_j^(0)t^(0)_n_i+t_n_j^(0)-^  _ r= t^(0)_n_jt_n_i^(0 ) i  j  . [ tnitnj]the operator ( [ 8 ] ) may be regarded as a linear map @xmath564 in particular , the map @xmath565 has a trivial kernel , while @xmath566 has a one - dimensional kernel which is spanned by the vector @xmath567 where @xmath568 denotes the lowest - weight spinor of so(@xmath29 ) . finally , the desired _ rectangular _ @xmath385 toeplitz operators @xmath379 may be realized in terms of the partial isometries ( [ 8 ] ) by appealing to the hilbert hotel argument . for this , we introduce a lexicographic ordering @xmath569 on the fock space @xmath349 so that @xmath570 with @xmath571 , and fix an orthonormal basis @xmath572 of the chiral spinor representation space @xmath573 . then @xmath574 , @xmath575 is an orthonormal basis for @xmath576 and there is a one - to - one correspondence @xmath577 of basis states . similarly , by fixing an orthonormal basis @xmath578 of the @xmath1 representation space @xmath579 , there is a one - to - one correspondence latexmath:[$\vec\lambda_{a_i}^{\,i}\otimes|q_i\rangle\leftrightarrow    corresponding orthonormal basis of @xmath389 . let us now introduce unitary isomorphisms @xmath581 and @xmath582 by the formulas [ uipdef ] u^  _ i&=&_a=0^r-1  _ a_i=0^k_i-1  _ ^|k_iq_i+a_irq+a| + & = & _ a=0^r-1 _ a_i=0^k_i-1  _ ^^i_a_i ^_a^|q_iq|  , + u^_i&=&_a=0^r-1 _ a_i=0^k_i-1  _ ^|rq+ak_iq_i+a_i| + & = & _ a=0^r-1   _ a_i=0^k_i-1  _ ^_a^ ^i_a_i^|qq_i|  . [ uimdef]by using the shift operators ( [ 8 ] ) , we then define the operators t^  _ n_i^  _ i(t_n_i^(0 ) ) t_n_i^=(t_n_i^(0))^u^_i [ tniuidef]on @xmath583 and @xmath584 . they satisfy the requisite equations ( [ ansatz4 ] ) , with the @xmath585 matrix p^  _ n_i = u_i^  ( p^  _ n_i^  _ i ) u_i^[pnifinal]a projector of rank @xmath388 on the fock space @xmath389 . notice that the rank @xmath551 used in this construction is an even integer for @xmath586 . to work with odd ranks @xmath587 one may introduce the @xmath588 matrices @xmath589 where @xmath590 is defined as above and @xmath591 is a shift operator on the fock space @xmath349 . then the operators ( [ 10 ] ) satisfy the equations ( [ tni0eqs ] ) with @xmath592 a projector of rank @xmath388 on the fock space @xmath593 , where @xmath594 . in this case the toeplitz operators @xmath464 are obtained by substituting ( [ 10 ] ) into ( [ tniuidef ] ) with the replacement @xmath595 . note also that the partial isometry operator t^(0)&:=&_i=0^mt^(0)_n_i + & = & _ r^  + _ i=0^m ( t^(0)_n_i-_r^  )  =  _ r^  + _ i=0^m(()^n_i-_r^  )  , [ totaltn]together with the above representations of the @xmath91 group on the weyl operator algebra of @xmath3 and on the fock space @xmath349 , defines a cycle in the @xmath91-equivariant analytic k - homology @xmath596 . after a twisting appropriate to the inclusion of the pertinent magnetic monopole bundles , it describes the @xmath1-invariant configurations of d - branes as branes on the ( trivial ) quotient space @xmath597 . the charge of this class is the same as that of the cocycle @xmath598 $ ] built earlier in the topological k - theory ( [ eqabsiso ] ) from the standard abs brane - antibrane class @xmath599 $ ] which is the generator of ( [ usualabsiso ] )  @xcite@xcite,@xcite . the computation of the topological charge , as well as the equivalence between the commutative ( topological ) and noncommutative ( analytic ) k - homology descriptions of the d - brane configurations , will be presented in the next section .    * moduli space of solutions .  * the realization ( [ 8 ] ) can be generalized in order to introduce @xmath600 real moduli into the solution which specify the locations of the various noncommutative solitons in  @xmath6  @xcite . for this , one first has to introduce `` shifted ground states '' centered at @xmath601 , @xmath602 for each @xmath247 . the operators ( [ 8 ] ) are rewritten as t^(0)_n_i=^  _ i)+(^i_1^i_2 ^i_n_i)^  _ i  , [ tnirewrite]where each @xmath603 , @xmath604 , @xmath247 is of the form of the shift operator @xmath605 in ( [ tt ] ) but with the coordinates  @xmath606 shifted to @xmath607 . they behave just like @xmath605 except that now the kernel of @xmath608 is spanned by the vector @xmath609 , where @xmath610 is the fermionic ground state and the shifted ground state @xmath611 is a coherent state in the @xmath348-oscillator fock space @xmath349 , i.e. @xmath612 . the states @xmath613 and @xmath614 for @xmath615 span the kernel of the operator @xmath616 given by ( [ tnirewrite ] ) , and we find that the equations ( [ tni0eqs ] ) are obeyed with @xmath617 the orthogonal projection onto @xmath618 .    the space of partial isometries ( [ totaltn ] ) may thereby be described as the complex manifold @xmath619 . after a quotient by the appropriate discrete symmetry group , the moduli space for the full solution consisting of the rectangular toeplitz operators ( [ tniuidef ] ) is given by ( n;k^  _ , k^  _ ) = q(k^  _ ) _ i=0^mhilb^n_i ( ^n )  , [ partisospace]where @xmath620 is the moduli space of @xmath388 noncommutative solitons on @xmath2  @xcite which is given as the ( singular ) hilbert scheme of @xmath388 points in @xmath621 , i.e. the set of ideals @xmath622 of codimension @xmath388 in the polynomial ring @xmath623 $ ] . the factor @xmath624 is the moduli space of isomorphism classes of quiver representations ( [ genrepsu2uk ] ) of dimension  @xcite ( k^  _ ) = 1-k^  _ c k^  _ 1+_i=0^mk_i(k_i+1-k_i )  . [ dimcalq]note that real roots ( having @xmath625 ) correspond to rigid representations of the quiver @xmath241 with no moduli , while imaginary roots ( having @xmath626 ) carry moduli associated to the gauge symmetry breaking ( [ gaugebroken ] ) . the points of the moduli space ( [ partisospace ] ) label the positions of well - separated d - branes , and it coincides in the low - energy limit with the moduli space of the commutative brane description  @xcite . in this section we will compute the topological charge of our multi - instanton solutions in essentially two distinct ways . the first one is a direct field theoretic calculation of the @xmath627-th chern number of our gauge field configurations on @xmath0 , which can also be computed using the @xmath260-graded connection formalism of section  [ dimred ] . the second one is a homological calculation of the index class of our solutions in k - theory , which is also equivalent to the euler - ringel character of the pertinent representations of the quiver @xmath241 . the equivalence of these two calculations will then lead us directly into a worldvolume description whereby we can interpret the topological charge in terms of cycles in topological equivariant k - homology , yielding the claimed d - brane interpretation of our solutions . the results of this section bridge together the descriptions presented in section  [ ansatzdescr ] and justify the brane interpretations that have been given throughout this paper thus far . * field theory calculation . * we will first compute the topological charge of the configurations ( [ ansatz3])([ansatz4 ] ) . for this , it is convenient to parametrize the two - sphere by the angular coordinates @xmath628 and @xmath629 defined in ( [ zn1 ] ) . in these coordinates @xmath630 and we have @xmath631 { { { \cal{f}}}}^{~}_{\vt\vp } \= & -{\,\mathrm{i}\,}\,\frac{\sin\vt}{2}\ , \sum^m_{i=0}\,(m-2i)~p^{~}_{n_i}\otimes\pi^{~}_i\end{aligned}\ ] ] giving @xmath632 \=&\,(-{\,\mathrm{i}\,})^{n+1}\,\frac{\sin\vt}{2\,\prod\limits^n_{a=1}\th^a}\ , \sum^m_{i=0}\,(m-2i)~p^{~}_{n_i}\otimes\pi^{~}_i \ , \end{aligned}\ ] ] where we have used the definitions ( [ pnitrace ] ) and ( [ piortho ] ) of the projectors @xmath391 and @xmath633 . the instanton charge is then given by the @xmath627-th chern number @xmath634 \= & \,\bigl(\frac{{\,\mathrm{i}\,}}{2\pi}\bigr)^{n+1}\ , \frac{(-{\,\mathrm{i}\,})^{n+1}}{2\,\prod\limits^n_{b=1}\th^b}\ , \bigl(\,\prod_{a=1}^n{2\pi\,\th^a}\bigr)\ , \sum^m_{i=0}\,(m-2i)\,{n_i}\ , \int_{s^2 } \sin\vt~{\mathrm{d}}\vt\wedge{\mathrm{d}}\vp \ . \label{chernnum}\end{aligned}\ ] ] after splitting the sum over @xmath135 into contributions from monopoles and antimonopoles analogously to ( [ efncfinite ] ) , this becomes q=^m2_i=0(m-2i ) ( n_i - n_m - i )  , [ topchargeym]where we recall that @xmath635 for @xmath247 . the formula ( [ topchargeym ] ) clarifies the d - brane interpretation of the configuration ( [ ansatz3])([ansatz4 ] ) . it describes a collection of @xmath636 d0-branes for @xmath637 and @xmath638 anti - d0-branes for @xmath639 as a bound state ( i.e. a vortex - like configuration on @xmath2 ) in a system of @xmath640 d(2@xmath348 ) branes and antibranes . however , from the point of view of the initial brane - antibrane system on @xmath0 , they are spherical @xmath641  d2-branes or d2-antibranes depending on the sign of the monopole charge @xmath446 . note that the vortices with @xmath642 , which always exist for even @xmath75 , have vanishing instanton charge since they couple with the trivial line bundle @xmath643 . thus they are not extended to instantons on @xmath0 , but are rather unstable and simply decay into the vacuum . the topological charge can be alternatively computed within the graded connection formalism of section  [ dimred ] . recalling the equivariant abs construction ( [ twistedspingrad])([twistedspingradeven ] ) , we note that the @xmath260-graded vector space ( [ genrepsu2uk ] ) ( the fibre of the @xmath260-graded bundle ( [ gradedbundle ] ) ) also has a _ natural _ @xmath423-grading by the sign of the magnetic charge , i.e. by the involution @xmath644 defined by @xmath645 for @xmath646 , where throughout we use the convention @xmath647 . the corresponding supertrace is given by str^  _ kk  x  : =  tr_kk^  ( x )  =  _ i=0^msgn(m-2i )  tr_k_ik_i^  x_i [ supertracedef]for any linear operator @xmath648 with block - diagonal components @xmath649 . this extends to a supertrace @xmath650 which we may use to express the chern number in terms of the graded curvature ( [ gradedcurv ] ) as q=  ( ) ^n+1 ( _ a=1^n2^a )  str^  _ tr^  _ ^2^n+1  ( ^n+1 ) _ asym  , [ topchargegraded]where @xmath651 and the antisymmetrized product of gamma - matrices ( ^_1^_q)_asym : = 1q!_s_q sgn ( )  ^_(1)^_(q ) [ asymgamma]mimicks the algebraic structure of the exterior product of differential forms . the formula ( [ topchargeym ] ) follows from ( [ topchargegraded ] ) upon repeated application of the clifford algebra and the trace identities ( [ trgammaid1])([trgammaid4 ] ) , with the supertrace ( [ supertracedef ] ) giving the appropriate sign alternations . * k - theory calculation . * the origin of the topological charge lies in the _ graded chern character _ @xmath652 . standard transgression arguments can be used to show that the cohomology class defined by this closed differential form is independent of the choice of graded connection  @xcite . in particular , we may either compute it by setting the off - diagonal higgs fields @xmath653 or by setting the diagonal gauge fields @xmath654 . it is instructive to recall how this works in the case @xmath268 corresponding to the basic brane - antibrane system represented by the chain ( [ a2quiver ] )  @xcite . in the former case we would obtain the difference @xmath655 of topological charges on the branes and antibranes . in the latter case we would compute the index of the tachyon field @xmath656 , or equivalently the euler characteristic of the two - term complex @xmath657 . the virtual euler class generated by the cohomology of this complex is the analytic k - homology class @xmath658\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n})$ ] of the brane configuration . the equivalence of the two computations is asserted by the index theorem . the situation for @xmath176 is more subtle . the action of the graded connection zero - form ( [ mgradedphidef ] ) on the bundle ( [ gradedbundle ] ) produces the holomorphic chain ( [ holchain ] ) . in general this is _ not _ a complex because , according to ( [ mphi0s ] ) , @xmath659 for @xmath176 , i.e. @xmath660 . the only physical instance in which such a chain generates a complex is when it corresponds to an alternating sequence of branes and antibranes  @xcite . but if one has a tachyon field which is a holomorphic map from an antibrane to a brane , then the adjoint map is antiholomorphic . recalling ( [ f25 ] ) , we see that in our chain ( [ holchain ] ) all maps @xmath302 are _ holomorphic _ and thus do not represent tachyon fields between pairs of branes and antibranes . furthermore , the maps @xmath302 obtained as solutions of the vortex equations , which can be associated with the @xmath160 quiver and are obtained by @xmath1-invariant reduction , can never satisfy the constraints @xmath661  @xcite . the solution to this problem is to fold the given holomorphic chain into maps between branes and antibranes . let us first carry out the calculation in the case that the monopole chern number @xmath75 is an odd integer . by using the @xmath423-grading @xmath662 introduced above , we explicitly decompose ( [ genrepsu2uk ] ) as a @xmath423-graded module into the @xmath663 eigenspaces of the involution @xmath664 giving = _ + _ - _ + = _ i=0^m_- _ k_i _ -= _ i = m_+^m_k_i  . [ calvz2module]using ( [ mgradedphidef ] ) and ( [ mphi0s ] ) we now introduce the operator t^  _ ( m):= ( ^  _ ( m))^m2 + 1  . [ mtdef]with respect to the @xmath423-grading ( [ calvz2module ] ) , it is an odd map ^  _ ( m):_-h   _ + h ( ^  _ ( m))^20  . [ mtmap]thus the triple @xmath665 $ ] defines a two - term complex and represents a brane - antibrane system with tachyon field given in terms of the graded connection by ( [ mtdef ] ) . the corresponding index class @xmath666\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n})$ ] is thus the analytic k - homology class of our configuration of d - branes . in particular , on isotopical components one has ^  _ ( m)^  _ i+1+m2 = ^  _ i+1^ _ i+1+m2=(^  _ i+1^  _ i+1+m2 )  t^  _ n_it^_n_i+1+m2 [ mtisotopexpl]while @xmath667 , where @xmath668 . the tachyon field is thus a holomorphic map between branes of equal and opposite magnetic charge , ^  _ ( m)^  _ i+m2 + 1 : _ k_i+m2 + 1h  _ k_i h  , [ mtisotopmap]and from ( [ dimkertni ] ) it follows that it has a finite dimensional kernel and cokernel with ( ^  _ ( m)^  _ i+m2 + 1 ) n_i+m2 + 1 ( ^  _ ( m)^  _ i+m2 + 1 ) ^n_i  . [ dimkermt ]    to incorporate the twistings by the magnetic monopole bundles , we use the abs construction ( [ twistedspingrad])([cliffordmulteven ] ) to define the tachyon field ^  _ ( m )  : =   ^  _ ( m):_^+h  _ ^-h  . [ mcaltdef]it behaves like a noncommutative version of clifford multiplication @xmath669 in ( [ cliffmultv],[cliffmultisot ] ) . since @xmath670 , from ( [ dimkermt ] ) it follows that the index of the tachyon field ( [ mcaltdef ] ) is given by   ^  _ ( m)&= & ( ^  _ ( m))- ( ^  _ ( m))^ + & = & _ i = m_+^m|m-2i|n_i-_i=0^m_- |m-2i|n_i =- q  . [ indexmcalt]thus the k - theory charge of the noncommutative soliton configuration ( [ ansatz3])([ansatz4 ] ) coincides with the yang - mills instanton charge ( [ chernnum],[topchargeym ] ) on @xmath0 .    when the monopole charge @xmath75 is even , we introduce the tachyon field @xmath671 by the same formula ( [ mtdef ] ) . the only difference now is that the subspace @xmath672 is annihilated by both operators @xmath673 and @xmath674 so that _ k_m2h ( ^  _ ( m ) ) ( ^  _ ( m))^  . [ mtvm20]according to ( [ twistedspingradeven ] ) , this subspace should be coupled to the eigenspace ( [ spinorharm ] ) of spinor harmonics on @xmath69 when defining the extended tachyon field ( [ mcaltdef ] ) . analogously to ( [ cliffordmulteven ] ) , one then has ( ^  _ ( m)^  _ m2)= ( ^  _ ( m)^  _ m2)^= _ k_m2_ph . [ kermcaltm2]with a suitable regularization of the infinite dimensions of the kernel and cokernel of the operator @xmath675 , these subspaces will make no contribution to the index ( [ indexmcalt ] ) . this statement will be justified below by the fact that @xmath676 and that the index class of the noncommutative tachyon field coincides with that of the twisted @xmath1-invariant dirac operator on @xmath484 .    we can give a more detailed picture of how the topological charge of the system of d - branes arises by relating the index to a homological computation in the corresponding quiver gauge theory , which shows precisely how the original brane configuration folds itself into branes and antibranes . consider the @xmath241-module ( [ vnam ] ) defined by a generic ( non - bps ) solution of the yang - mills equations on @xmath403 , and let = _ i=0^m_i k^  _ = _ i=0^mw_i  e_i [ anywrep]be any quiver representation . applying the functor @xmath677 to the projective resolution ( [ ringelres ] ) gives a complex whose cohomology in the @xmath522-th position defines the extension groups @xmath678 , with @xmath679 and @xmath680 . we may then define the relative euler character between these two representations through the corresponding euler form ( , ) : = _ p0(-1)^p  ^p ( , )  . [ eulerformgen]since the @xmath160 quiver has no relations , one has @xmath681 for all @xmath682 in the present case  @xcite .    by using ( [ hompathnatural ] ) , the resolution ( [ ringelres ] ) induces an exact sequence of extension groups given by 0  ( , ) & & _ i=0^s ( t_n_i^,_i )   + & & _ i=0^s-1 ( t_n_i+1^,_i )   ( , )  0 [ extexactseq]from which we may compute the euler form ( [ eulerformgen ] ) explicitly to get ( , ) & = & ( , ) - ( , ) + & = & _ i=0^s(t_n_i^ , _ i)-_i=0^s-1 ( t_n_i+1^,_i ) + & = & _ i=0^mn_iw_i-_i=0^m-1n_i+1w_i  . [ eulerwgen]thus the relative euler character depends only on the dimension vectors of the corresponding representations and coincides with the ringel form @xmath683 on the representation ring @xmath684 of the @xmath160 quiver  @xcite . the map @xmath685\mapsto\vec k^{~}_{\underline{\cal w}}$ ] gives a linear map @xmath686 which is an isomorphism of abelian groups since @xmath684 is generated by the schur modules @xmath448 , @xmath247 . by using ( [ ansatz3p ] ) and ( [ dimkertni ] ) we can write this bilinear pairing in the suggestive form ( , ) = -_i=0^mw_i  index(_i+1 )  . [ eulerindexphi ]    the appropriate representation @xmath263 to couple with in the present case is dictated by the correct incorporation of magnetic charges . as before , the fact that the higgs fields @xmath170 in ( [ eulerindexphi ] ) themselves are not tachyonic , i.e. do not generate a complex , means that we have to fold the @xmath1 representations @xmath687 appearing in the abs construction ( [ twistedspinpm ] ) appropriately . the correct folding is expressed by the collection of distinguished triangles ( [ bpstriangles ] ) which shows that we should couple an increasing sequence @xmath688 of representations as we move along the chain of constituent d - branes of the quiver , so that the @xmath1-module @xmath689 gives an extension of the monopole field carried by the elementary brane state at vertex @xmath135 by the @xmath1-module @xmath690 . thus we take @xmath691 and embed its class into the representation ring @xmath692 using the @xmath423-grading above as the element = _ j=0^isgn(m-2j )   sgn(m-2i )   + [ widef]of virtual dimension w_i=_j=0^i(m-2j)(i+1)(m - i ) [ wiwidef]for each @xmath247 . in this case the euler - ringel form ( [ eulerwgen ] ) becomes ( , ) = _ i=0^m(i+1)(m - i)(n_i - n_i+1)=_i=0^m(m-2i ) n_iq [ eulertopcharge]and it also coincides with the instanton charge of the gauge field configurations on @xmath0 . the equivalence of the relative euler character with the index of the tachyon field above is a consequence of the grothendieck - riemann - roch theorem .    * worldvolume construction . * we can now present a very explicit geometric description of the equivalence between the brane configurations on @xmath484 and on @xmath6 . the crux of the formulation is the well - known map in k - theory between analytic ( noncommutative ) and topological ( commutative ) descriptions  @xcite . if @xmath693 is the usual dirac operator on @xmath6 , then its index coincides with that of the noncommutative abs configuration  ( [ tt ] ) giving  = index  . [ indexsigmadirac]this coincides with the k - theory charge of the bott class @xmath599 \in{{\rm k}}({{\mathbb{r}}}^{2n})$ ] given by the ordinary abs construction  @xcite , where @xmath694 is clifford multiplication by @xmath495 . in particular , the dirac operator itself can be used to represent the analytic k - homology class @xmath695=[\dirac]$ ] described by the noncommutative abs field . let us represent a system of @xmath41 type  iia d - branes wrapped on @xmath484 with virtual chan - paton bundle @xmath696 by the k - cycle @xmath697 $ ] in the topological k - homology @xmath698 . its equivalence class is invariant under the usual relations of bordism , direct sum and vector bundle modification  @xcite . there is an isomorphism @xmath699 of abelian groups which sends this k - cycle to the analytic k - homology class @xmath700 $ ] defined by the corresponding twisted dirac operator on @xmath484 . similarly , if @xmath701 and @xmath702 is the slice induced by the inclusion @xmath703 of groups , then the topological k - cycle @xmath704\in{{\rm k}}^{\rm    t}({{\mathbb{r}}}^{2n}\times{{\mathbb{c}}}p^1)$ ] corresponds to the analytic k - homology class @xmath705\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n}\times{{\mathbb{c}}}p^1)$ ] , where @xmath706 is the twisted dirac operator on @xmath6 .    now consider the @xmath1-equivariant reduction of these cycles . from the construction of the previous section with @xmath707 and the equivariant excision theorem of section  [ ansatzdescr ] we have the equality ^su(2)=_*^u(1)[diracsu2eq]in @xmath708 which leads to = su(2)_u(1)[kcyclesu2eq]in the left - hand side of ( [ kcyclesu2eq ] ) corresponds to the class of d@xmath5 brane - antibrane pairs wrapping @xmath6 , while the right - hand side corresponds to d@xmath710 brane - antibrane pairs wrapping @xmath484 . this is just the equivalence between instantons on @xmath484 and vortices on @xmath6 . we note that in the case @xmath268 , the monopole field is automatically spherically symmetric on @xmath69 and one can formulate the equivalence ( [ kcyclesu2eq ] ) using only the requirement of vector bundle modification in _ ordinary _ topological k - homology  @xcite , which is equivalent to bott periodicity ( [ bottper ] ) . in contrast , for @xmath176 one must appeal to an @xmath1-equivariant framework and the identification ( [ kcyclesu2eq ] ) of k - cycles is far more intricate . in this case it is a result of the equivariant excision theorem , and _ not _ of bott periodicity in equivariant k - theory . it is this intricacy that leads to a more complicated brane - antibrane system when @xmath176 .    using the equivariant abs construction of the previous section , the k - homology class of the multi - instanton solution ( [ ansatz3])([ansatz4 ] ) is given by the left - hand side of ( [ kcyclesu2eq ] ) with =  , [ ximultiinst]where _ e^+=_i = m_+^m e_k_i_|m-2i| _ e^-= _ i=0^m_- e_k_i_m-2i [ twistedebundles]while ^  _ n_0,n_1,  ,n_m=_i=0^m_- ( ^  _ e^  _ i)^n_i_j = m_+^m ( _ e^_j^  ) ^n_j [ muns]with @xmath711 acting fibrewise as clifford multiplication ( [ cliffmultv],[cliffmultisot ] ) . the class ( [ ximultiinst ] ) is the k - theory class of the noncommutative soliton field ( [ totaltn ] ) . the relation ( [ kcyclesu2eq ] ) equates the resulting k - homology class with that defined by =  , [ ximultiinst]where the projection @xmath712 is a left inverse to the inclusion @xmath713 , i.e. @xmath714 . through the standard process of tachyon condensation on the system of d@xmath710 branes and antibranes wrapping @xmath6 , the right - hand side of ( [ kcyclesu2eq ] ) then describes @xmath715 d2-branes and @xmath716 d2-antibranes . on the left - hand side of ( [ kcyclesu2eq ] ) , these are instead d0-branes corresponding to vortices left over from condensation in the transverse space @xmath6 . one can also compute the topological charge in this worldvolume picture and explicitly demonstrate that the k - theory charges on both sides of ( [ kcyclesu2eq ] ) are the same . the natural charge of branes defined by elements of equivariant k - theory is given by the equivariant index @xmath717 , which may be computed by using the @xmath1-index theorem  @xcite ^  _ su(2 )  _ = -_^2np^1 ^  _ su(2)()(^2np^1 ) , [ equiindexthm]where @xmath718 is the equivariant chern character taking values in @xmath1-equivariant rational cohomology . since this index depends only on the equivariant k - homology class of the dirac operator on @xmath484 , we may explicitly use ( [ diracsu2eq ] ) and perform the dimensional reduction to write the index ( [ equiindexthm ] ) as ^  _ su(2 )  _ = -_^2n^  _ su(2 ) ( )  . [ equiindexred ] since the chern character in ( [ equiindexred ] ) is a ring homomorphism between @xmath719 and @xmath720 , upon substitution of ( [ ximultiinst],[twistedebundles ] ) we can use its additivity and multiplicativity to compute ^  _ su(2)()=^  _ su(2 ) ( _ e^+_e^-)=_i = m_+^m ( e_k_i)^  _ _ |m-2i|-_i=0^m_- ( e_k_i)^  _ _ |m-2i|  , [ chernmultadd]where @xmath721 are the characters of the @xmath1 representations @xmath722 . this enables us to write the equivariant index on @xmath484 in terms of ordinary indices on @xmath6 to get ^  _ su(2 )  _ = _ i=0^m_- index(_e_k_i ) ^  _ _ |m-2i| -_i = m_+^mindex(_e_k_i)^  _ _ |m-2i|  . [ indexord]we can turn ( [ indexord ] ) into a linear map @xmath723 by composing it with the projection @xmath724 onto the trivial representation . acting on the character ring this gives _ 0(^  _ _ |m-2i|)= ^  _ _ |m-2i|(id ) = _ |m-2i||m-2i| [ pi0compose]and one finally arrives at _ 0(index^  _ su(2 )  _ ) = _ i=0^m_- |m-2i|  index(_e_k_i ) -_i = m_+^m|m-2i|  index(_e_k_i )  . [ pi0indexfinal ]    alternatively , one may arrive at the same formula by directly computing the _ ordinary _ index of the dirac operator ( [ diracgradeddef ] ) with @xmath707 using ( [ diracs2decomp ] ) and ( [ diracpkernel])([solschargepos ] ) . since  _ m-2i=^+_m-2i- ^-_m-2i=-(m-2i )  , [ indexcp1]the index of ( [ diracgradeddef ] ) acting on sections of the bundle ( [ spinortotgen ] ) coincides with ( [ pi0indexfinal ] ) . for a gauge field configuration appropriate to the k - theory class defined by the tachyon field ( [ muns ] ) , these topological charges coincide with ( [ topchargeym ] ) . the extremal cases for which the higgs fields have the configurations @xmath725 and @xmath726 , fall outside of the general scope of the previous analysis and are worth special consideration . they correspond to vacuum sectors of the noncommutative gauge theory and are associated with indecomposable representations of the quiver @xmath241 that have no arrows . nevertheless , these vacuum sectors admit non - trivial bps solutions which signal the presence of stable d - branes attached to the closed string vacuum after condensation on the brane - antibrane system . we shall now study them in some detail . * monopole vacuum . * let us first look at the case @xmath727 . the nonabelian coupled vortex equations ( [ ddd1])([ddd2 ] ) then imply @xmath728 which is only possible in the equal rank case @xmath729 corresponding to the gauge symmetry breaking pattern @xmath730 with @xmath731 . thus we take @xmath732 and @xmath733 with @xmath734 , where @xmath735 are given in ( [ 7.8 ] ) . in quiver gauge theory , the bps conditions in this sector thus correspond to the representation of @xmath241 which is @xmath587 copies of the indecomposable quiver representation @xmath736 . they also require @xmath737 which are simply the duy equations on @xmath2 . note that ( [ fyyb ] ) implies @xmath738 in this case , giving the trivial dimensional reduction to @xmath2 . after switching to matrix form via ( [ x ] ) , we obtain @xmath739 + \de^{a{{\bar{b}}}}\,\th_{a{{\bar{b}}}}\=0 \qquad\textrm{and}\qquad \big[x_{{{\bar{a}}}}\,,\,x_{{{\bar{b}}}}\big]\=0\= \big[x_a\,,\,x_b\big ] \ .\ ] ]    the obvious solution to ( [ mf ] ) is the trivial one with @xmath740 , giving @xmath741 . this sector can be understood physically as the endpoint of tachyon condensation , wherein the higgs fields @xmath140 have rolled to their minima at @xmath742 and the fluxes have been radiated away to infinity . here the d0-branes have been completely dissolved into the d(2@xmath348)-branes . however , non - trivial solutions of the equations ( [ mf ] ) also exist . for this , let us restrict ourselves to the abelian case @xmath542 and simplify matters by taking @xmath743 for all @xmath744 . we fix an integer @xmath745 and consider the ansatz  @xcite @xmath746 where @xmath346 is a real function of the `` total number operator '' @xmath747 with the property that @xmath748 for @xmath749 . the shift operator @xmath750 in ( [ ans ] ) is defined to obey @xmath751 with @xmath752 where @xmath753 with @xmath754 . note that @xmath755 and @xmath756 projects all states with @xmath757 out of the fock space @xmath758 . one easily sees that ( [ ans ] ) fulfills the homogeneous equations in ( [ mf ] ) . remembering that @xmath759 , we also obtain @xmath760\ & = \ \th_{a{{\bar{c}}}}\,\th_{{{\bar{b}}}d}\,\sigma_{l}^\+\,\left\ { f({{\cal n}}\,)\,{{{\bar{z}}}}^{{{\bar{c}}}}\,(1{-}p^{~}_{l})\,z^d\,f ( { { \cal n}}\,)-z^d\,f({{\cal n}}\,)\,(1{-}p^{~}_{l})\,f({{\cal n}}\ , ) \,{{\bar{z}}}^{{{\bar{c}}}}\right\}\,\sigma^{~}_{l } \\[4pt ] \label{mf4 } & = \ -\frac{1}{4\,\th^2}\,\de_{a{{{\bar{c}}}}}\,\de_{d{{{\bar{b}}}}}\,\sigma_{l}^\+\,\left\ { f^2({{\cal n}}\,)\,{{{\bar{z}}}}^{{{\bar{c } } } } z^d - f^2({{\cal n}}{-}1)\,z^d { { { \bar{z}}}}^{{{\bar{c}}}}\right\}\,\sigma^{~}_{l}\end{aligned}\ ] ] with the help of the identities @xmath761 where @xmath762 . we have also used @xmath763 substituting ( [ mf4 ] ) into ( [ mf ] ) , we employ @xmath764 to find the conditions @xmath765 + \de^{a{{\bar{b}}}}\,\th_{a{{\bar{b } } } } \\[4pt ] \nonumber & \=-\frac{1}{2\,\th}\ , \sigma_{l}^\+\,\biggl\ { f^2({{\cal n}}\,)\,({{\cal n}}{+}n)- f^2({{\cal n}}{-}1)\,{{\cal n}}\biggr\}\,\sigma^{~}_{l}+ \frac{n}{2\,\th } \\[4pt ] \label{mf5 } & \=\frac{1}{2\,\th}\,\sigma_{l}^\+\,\biggl\ { { { \cal n}}\,f^2({{\cal n}}{-}1)- ( { { \cal n}}{+}n)\,f^2({{\cal n}}\,)+ n\biggr\}\,\sigma^{~}_{l}\end{aligned}\ ] ] on the operator @xmath346 . with the initial conditions @xmath766 and the finite - energy condition @xmath767 as @xmath768 , these recursion relations are solved by @xmath769 where @xmath770 is the number of states in @xmath349 with @xmath771 , i.e. the number of states removed by the operator @xmath772 . we arrive finally at the non - trivial gauge field configuration given by @xmath773 the field strength @xmath192 on @xmath774 obtained from ( [ xsol ] ) has finite @xmath348-th chern number @xmath775  @xcite . the topological charge @xmath775 given by ( [ qpm ] ) is calculated here via an integral over @xmath2 . however , the @xmath776-th chern number for this configuration considered as a gauge field on @xmath378 with @xmath777 vanishes . moreover , this configuration has finite energy ( [ ef ] ) proportional to the topological charge  @xcite , e^  _ bps=(2)^n+1r^2n(n-1)q  , [ ebpsm0]as usual for a bps instanton solution .    * higgs vacuum .  * the choice @xmath778 for all @xmath734 is somewhat more interesting since from ( [ fyyb ] ) and ( [ mupdef ] ) we then have @xmath779 with @xmath780 this configuration gives the local maximum of the higgs potential corresponding to the open string vacuum containing d - branes . in this case the vortex equations ( [ ddd1])([ddd2 ] ) reduce to @xmath781 after switching to matrix form via ( [ x ] ) we obtain @xmath782 + \de^{a{{\bar{b}}}}\,\bigl(1- \frac{(m-2i)\,\th}{2n\,r^2}\,\bigr)\,\th_{a{{\bar{b}}}}\=0 \qquad\textrm{and}\qquad \big[x^i_{{{\bar{a}}}}\,,\,x^i_{{{\bar{b}}}}\big]\=0 \=\big[x^i_a\,,\,x^i_b\big]\ , \ ] ] where we have used the formula @xmath783 . recall that there is no summation over the index @xmath247 in the equations ( [ mf1 ] ) .    by comparing ( [ mf1 ] ) and ( [ mf ] ) , we conclude that ( [ mf1 ] ) can be solved for each @xmath135 by the same ansatz as for ( [ mf ] ) . for this , let us restrict ourselves again to the abelian case for all nodes @xmath247 ( so that @xmath784 ) , and fix @xmath785 positive integers @xmath786 . we take @xmath787 analogously to ( [ ans])([projl ] ) . producing then the same calculations as before , we obtain the gauge field configuration @xmath788 where @xmath789 and @xmath790 we have chosen the radius @xmath14 of the sphere so that @xmath791 . the solutions ( [ xso ] ) coincide with those given by ( [ xsol ] ) if one assigns different noncommutativity parameters @xmath792 to the worldvolumes of d(@xmath29)-branes carrying different magnetic fluxes proportional to @xmath446 . then the field strength @xmath793 on @xmath794 obtained from ( [ xso ] ) will have finite topological charge @xmath795 given by ( [ qi ] ) and corresponding finite bps energy analogous to ( [ ebpsm0 ] ) , and the configuration thus described extends to instantons on @xmath403 . the interesting idea of introducing distinct noncommutativity parameters on multiple coincident d - 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nucleus reactions can be described within one consistent framework @xcite . in the case of photonuclear reactions , however , serious descrepancies remain , which partially have been accounted for by including non - resonant background terms @xcite . nevertheless , such a procedure , in particular for nuclear photoabsorption , either leads to contradictions with previous @xmath1-hole results or lacks the complete agreement with experimental data @xcite . the question arises , whether theoretical ingredients other than in - medium @xmath1-hole propagations can lead to a similar degree of accuracy . therefore it is natural to address the situation from a different point of view asking to what extend the absorption process can be described , when only a very simple @xmath1-nucleon interaction is used and all additional effects are accounted for in a purely diagrammatic approach . when combined with a simple form of nucleon momentum distribution inside the nucleus , namely a fermi gas model , this leads to analytical expressions , in which different corrections to this lowest - order approximation can be studied . this is the aim of the present article . in our formalism we follow closely wakamatsu and matsumoto @xcite . however , we do not introduce a phenomenological potential to account for the deviation of the nucleon wave functions from plane waves , but study the influence of such corrections in a perturbative way , similar to our previous work @xcite . in addition , our focus is on the energy - dependence of the total photoabsorption , rather than on the differential cross section as a function of the momentum of the outgoing proton . a characteristic feature of wakamatsu s and matsumoto s approach is the technically equal treatment of the ( @xmath8,pn ) and the ( @xmath8,pp ) knock - out process , which allowed them to obtain a natural explanation for the supression of the two - proton knock - out . this characteristic property is also present in our calculation , where it is related to a vanishing trace in spin space . in the last years the ratio ( @xmath8,pp)/(@xmath8,pn ) has been investigated in depth within different formalisms . in an extension of wakamatsu s and matsumoto s work by boato and giannini @xcite , finite - size effects have been calculated and , more recently , a combination of pion exchange and shell - model wave functions was used to investigate this quantity @xcite . currently , two complementary approaches for the description of nuclear knock - out reactions exist . carrasco and oset @xcite used a diagram - oriented many - body expansion in a fermi gas . the evaluation of self - energy diagrams leads to an accuracy for medium effects high enough to study knock - out reactions in great detail . with their primary goal being thus different from ours , their formalism does not yield isolated expressions for the resonant and non - resonant parts of the mechanisms of nuclear photoabsorption a different approach is used by the gent group @xcite , where the main emphasis lies in the construction of realistic shell - model wave functions , rather than on a microscopic description fully based on the evaluation of feynman diagrams . as the nuclear photoabsorption is almost insensitive to structural differences between nuclei , the quality of their approach becomes obvious in the investigation of differential cross sections for nucleon knock - out , rather than of photoabsorption . nuclear photoabsorption provides an interesting tool to study the interplay between one - nucleon and two - nucleon contributions . we obtained analytical expressions for these contributions , as well as for their resonant and non - resonant parts . this set of results can be used as a starting point to include ( and test ) further nuclear or nucleonic effects . in section 2 the basic notations are listed , as well as the interaction terms and the most important model assumptions and approximations , which are present in this calculation . the main results and their most prominent properties , e.g. the effects of relativistic corrections and nuclear structure , are discussed in section 3 , where also the following possible extension of such an approach is considered : as the angular dependence of the two - nucleon process is not very strong ( cf . @xcite ) , the different mechanisms contributing to the photoabsorption curve can also be used to understand qualitative features of experimental data for nucleon knock - out processes as a function of the photon energy . in section 4 some concluding remarks are made , with an emphasis on the applicability of the partial cross sections , whose analytical forms are given in the appendix . the starting point of our investigation is the static hamiltonian for the pion - nucleon interaction , @xmath9 together with a minimal coupling to the photon field . in eq.([s1eq1 ] ) @xmath10 is the pion mass . for all coupling constants we use the same notation and values as given in @xcite , in particular @xmath11=0.08 . in eq.([s1eq1 ] ) underlined symbols denote vectors in ( cartesian ) isospin space , while an arrow indicates a vector in coordinate space . in the static limit the interactions with the @xmath1-isobar excitation of the nucleon are determined by the following hamiltonians ( see e.g. @xcite ) : @xmath12 and @xmath13 with the hermitian conjugate to be added in both cases . here @xmath14 and @xmath15 are the 1/2-to-3/2 transition operators in spin space and isospin space , respectively ; @xmath16 is the proton charge , @xmath17 . for the coupling constants we have @xmath18=2 and @xmath19=0.35 . in all cases , where high momentum transfers occur at the pion - nucleon vertices we regularize the vertex functions by introducing dipole form factors @xmath20 as was also done e.g. in @xcite . the value for the cut - off parameter @xmath21 has been taken to be 800 mev . this value gives the best agreement of our predictions with the experimental data . in addition , a similar value has been used in @xcite . the general expression for the total cross section @xmath22 of the photoabsorption with one nucleon outside the fermi sphere and one pion in the final state ( one - nucleon process ) is of the form @xmath23\ ; . \label{s1eq5}\end{aligned}\ ] ] for the total cross section @xmath24 of the process with two free nucleons in the final state it is given by @xmath25\left [ { 1-n(\vec p_4)\ ; }      \right ]          \label{s1eq6}\ ; .\end{aligned}\ ] ] the notation for the external momenta is shown in fig.([figa ] ) . the function @xmath26 is the occupation number , @xmath27 is the step function . furthermore , @xmath28 is the nuclear volume , @xmath29 is the fermi momentum , @xmath30 is the mass of the proton , @xmath28 is the nuclear volume , @xmath31 , and @xmath32 is the energy of the outgoing pion .    both amplitudes @xmath33 and @xmath34 consist of non - resonant and resonant parts , @xmath35 . diagrammatically this decomposition is shown in fig.([figb ] ) . the second ( crossed ) contribution to the resonant part is small due to the big energy denominator and will be skipped in the following . in the energy @xmath36-function in eq.([s1eq5 ] ) and eq.([s1eq6 ] ) we will usually not consider the smallest term connected with the kinetic energy of the incoming nucleon , as it is smaller than @xmath37mev . indeed , as we expect its contribution to introduce only a small modification of the actual @xmath38-dependence in the integrand in ( [ s1eq5 ] ) , we substitute it by its average value @xmath39 , which results in an overall shift of the absorption cross section . for the one - nucleon process we find first - order relativistic corrections to be essential for a successful treatment of the absorption process . in the case of the resonant contribution , such corrections are accounted for by making the following substitutions in the vertices @xcite : @xmath40 where @xmath41 is the @xmath42-isobar mass and @xmath43 . for the non - resonant part , the corrections give amplitude @xmath33 of the following form : @xmath44-{{2i(\vec \sigma \cdot ( \vec q-\vec k))(\vec \varepsilon \cdot \vec q)}\over { ( \vec q-\vec k)^2+m^2-(\varepsilon _ q-\omega ) ^2 } } \nonumber \\ & -&{{2i(\vec \sigma \cdot \vec q)(\vec \varepsilon \cdot ( \vec k+\vec p-\vec q))}\over { \,\left [ { ( \vec k+\vec p-\vec q)^2 - 2m\omega -(\vec p-\vec q)^2}\right]}}\left . { \matrix{{}\cr { } \cr } } \right\ }   ,          \label{s1eq9}\end{aligned}\ ] ] which corresponds to the production of a @xmath45 . in comparison with @xcite we neglected those terms in ( [ s1eq9 ] ) , which modify the result by less than 2 per cent . it should be noted that no free parameters are present in our approach . a certain model dependence exists , however , in the selection of diagrams . we neglect all those diagrams , which are suppressed by some mechanism . in fig.([figaa ] ) two examples for suppression mechanisms are given . for fig.([figaa]a ) the contribution is small because in the case of infinite nuclear matter due to momentum conservation the four - momentum of the photon should be equal to that of the outgoing pion , which is impossible . for fig.([figaa]b ) let us consider the case , where the upper two nucleons ( incoming and outgoing ) are identified . furthermore , let us select a @xmath1-isobar as an intermediate state . then , one has a vanishing trace in spin space : @xmath46\,=\ , 0\ ] ] at @xmath47 . therefore , the contribution in this case vanishes in the non - relativistic limit considered here . as a result of applying such methods to the various diagrams , we have obtained that only those displayed in fig.([figa ] ) should be taken into acount . explicit evaluation of the diagrams shown in fig.([figa ] ) leads to amplitudes for the one- and two - nucleon contribution to nuclear photoabsorption . squaring these amplitudes and summing over spin and isospin states of the nucleons by using trace methods as previously @xcite one finds the expressions @xmath48 and @xmath49 , which enter into eqs.([s1eq5 ] ) and ( [ s1eq6 ] ) . it has turned out to be convenient to investigate the resonant and non - resonant parts of each of these contributions separately . this can be done by neglecting the interference terms , which are highly suppressed ( cf . fig.([figh ] ) ) . shown here exemplary for the resonant parts , one obtains the following expressions , which serve as a starting point for the integrations with respect to nucleon momenta : @xmath50-q^2-m^2 } \right)\times          \nonumber \\          & & n(\vec p-\vec k)\;\left [ { 1-n(\vec p-\vec q ) } \right]\,{{3\left [ { ( \vec q\times          \vec k)\cdot \vec \varepsilon } \right]^2+q^2(\vec k\times \vec \varepsilon ) ^2 }          \over { \left ( { \omega -\delta - p^2/2 m } \right)^2 +          \gamma^2 / 4 } }   ,          \label{s2eq1}\end{aligned}\ ] ] where @xmath51 and @xmath52\left [ { 1-n(\vec p_4)\ ; } \right]\times          \nonumber \\          & & \delta \left ( { \omega -{{p_3 ^ 2+p_4 ^ 2 }          \over { 2 m } } } \right)\,\delta ( \vec p_1+\vec p_2+\vec k-\vec p_3-\vec p_4)\times          \nonumber \\          & & \left\ { { { { 2a^2\left [ { a^2 + 3(\vec \varepsilon \cdot \vec a)^2 } \right ] }          \over { ( a^2+m^2)^2}}\,g_\pi ^4(a)+{{2\omega ^2(\vec \varepsilon \cdot          \vec a)(\vec \varepsilon \cdot \vec b ) }          \over { ( a^2+m^2)(b^2+m^2)}}\,g_\pi ^2(a)g_\pi ^2(b ) } \right\ }          \label{s2eq2}\end{aligned}\ ] ] with @xmath53 the analytical expressions for the resulting partial cross sections are given in the appendix . for the total absorption cross section @xmath54 a comparison with experimental data is shown in fig.([figc ] ) . the result of this model calculation compares favorably with the data . the one - nucleon and two - nucleon parts of the cross section are equally important at energies around 250 mev . as can be seen in this plot , significant features of the data , e.g. the position of the peak , are only obtained due to the interplay between the two mechanisms . as mentioned before , each of the contributions to the full curve in fig.([figc ] ) has a resonant and a non - resonant part . in fig.([figd ] ) and ( [ fige ] ) this decomposition is shown for the one - nucleon and the two - nucleon mechanism , respectively . here it is clearly seen that the non - resonant parts give an important contribution at lower energies . in the two - nucleon case the non - resonant part decreases with energy , while in the one - nucleon process it remains almost constant . it is interesting to see , in what way the one - nucleon partial cross sections are affected by the use of only the static limit of the interaction . neglecting the first - order relativistic corrections in the current one has @xmath55}}\,\int\limits_{q_{min}}^{q_{max } } { q\,dq}\,\left ( { 1+s_0(q ) }   \right)\times   & &          \nonumber \\          \left\ { { \left [ { \omega ^2-m^2-{{\omega q^2 }          \over m } } \right]+{3 \over 2 } \left ( { q^2-\left [ { q^2\left ( { 1+{\omega \over m } } \right)+m^2 } \right]^2{1 \over { 4\omega ^2 } } } \right ) }          \right\ }   & &          \label{s2eq3}\end{aligned}\ ] ] and @xmath56 ^ 2 } \right\}\left . { \matrix{{}\cr { } \cr } } \right ]   ,          \label{s2eq4}\end{aligned}\ ] ] where the integration limits in both cases are given by @xmath57 and the integrand contains the function @xmath58 in fig.([figf ] ) the corresponding cross sections are compared with those resulting from eq.([s2eq1 ] ) and its non - resonant counterpart . the relativistic corrections lead mainly to a shift of the one - nucleon curve . this effect is essential for obtaining a good agreement with the experimental data . although no relativistic corrections in the current are included , note that in ( [ s2eq3 ] ) and ( [ s2eq4 ] ) terms of that order have been kept in the kinematical contributions to the integrand . as the two - nucleon mechanism gives a comparatively small contribution at higher energies , we neglect relativistic corrections in @xmath59 . this has also been done in @xcite . a more difficult problem is the influence , which nucleon correlations inside the nucleus can have on the nuclear photoabsorption process . we investigate this aspect for the non - relativistic forms ( [ s2eq3 ] ) and ( [ s2eq4 ] ) of the one - nucleon case . the main reason for doing so is the fact that due to the square of the amplitudes we can express part of the integrand via the standard lowest - order central correlation function of a fermi gas ( see e.g. @xcite ) . the object , which is obtained by diagrammatically squaring the one - nucleon contribution of fig.([figa]a ) , can be coupled to an additional nucleon . in the incoherent case of the diagrammatical square also a further pion exchange can be allowed . the modifications of the correlation function , which occur due to such effects , have been investigated analytically in @xcite , where a corrected central correlator @xmath60 has been constructed . the function @xmath61 is given in fig.([figj ] ) . by making the substitution @xmath62 these further correlations can be incorporated effectively . in fig.([figg ] ) the one - nucleon contribution resulting from @xmath61 is compared with the original form , in which @xmath63 has been used . it can be seen that such medium effects modify the result by about 15 per cent . naturally , the use of this substitution technique can only be used to obtain an estimate for such a medium - induced modification . a full examination should involve the inclusion of the additional two - and three - nucleon diagrams in eq.([s1eq5 ] ) . the limit of our approach is certainly reached , when a comparison with differential cross sections is attempted . an extreme case is the comparison with @xmath64he , for which a recent measurement of both , the one - nucleon and the two - nucleon channel exists @xcite . we compare the calculated average cross section with the data for the differential cross section in c.m . frame , as we expect the angular dependence not to be strong in that frame . the corresponding plot is shown in fig.([figi ] ) . although no full agreement is obtained , it is interesting to note that the general features of the two cross sections are well reproduced , such as the peak positions and the relative size of the two processes . by these means it is possible to unambiguously identify the physical mechanisms behind the data points . a similar degree of agreement is obtained for other data @xcite . in the present paper we have developed a diagrammatical description of the nuclear photoabsorption process . the main result of our investigation is that the total photoabsorption cross section can be fully understood in terms of a simple physical picture , where point - like nucleons and @xmath65-isobars interacting via pion exchange are the relevant degrees of freedom . due to the diagram - oriented formalism and the fermi gas model as an approximate description of the nucleons in momentum space , we could obtain analytical expressions for all the relevant contributions to the photoabsorption curve . in this way a flexible and efficient description has been obtained , which can be used as a starting point for the investigation of additional effects . especially in the low - energy part of our calculated curve , the agreement with experiment comes about as a non - trivial interplay between the one - nucleon and two - nucleon contributions . it is worth noting that , as long as a comparatively low cut - off parameter in the vertex form factor is used , there seems to be no need for an explicit diagrammatical inclusion of the @xmath66-meson as an additional mechanism of the nucleon - nucleon interaction . we found relativistic corrections in the case of the one - nucleon process to be crucial for obtaining a good agreement . the aspect of additional nucleon correlations , which can be accounted for as a deviation of the nucleon wave functions from plane waves , deserves some further attention in future investigations . we could estimate the overall effect to be of the order of 15 per cent . we are most grateful to a.i . lvov for useful comments and discussions . one of us ( m.t.h . ) wishes to thank the budker institute , novosibirsk , for the kind hospitality accorded him during his stay , when part of this work was done . here we present the explicit expressions for the four contributions to the photoabsorption curve , which have been used to obtain the figures shown in section 3 . as was mentioned earlier , the interference terms between the resonant and the non - resonant contributions are small ( cf . fig.([figh ] ) ) . therefore , we can write the absorption cross section as a sum of four parts , @xmath67 . first , we deal with the one - nucleon case . in the case of the non - resonant contribution the absorption cross section can be represented in the following form : @xmath68\times \nonumber\\ & \displaystyle g(p , q,\omega)\,\theta(2q\omega - m^2-q^2-\frac{\omega}{m}p^2 ) ,          \label{app1}\end{aligned}\ ] ] where @xmath69          \label{app4}\ ] ]    and @xmath70 $ ] , @xmath71 , @xmath72 . in eq.([app1 ] ) the form factor @xmath73 is the same as in ( [ s1eq4 ] ) . note that the integration with respect to the variable @xmath38 can easily be performed , but the result is too lengthy to be given here explicitely . the resonant part of the one - nucleon process has the following form : @xmath74 ^ 2 + \gamma^2/4}\ ; h(p , y,\omega ) \label{app6}\end{aligned}\ ] ] where @xmath75 and the integrand is given by @xmath76 + \nonumber\\ & \displaystyle [ ( x+1)a_1-{1 \over 8}(x - x^3)a_2]\theta(f-|p - p_f|)\,\theta(p+p_f - f )       \ , , \end{aligned}\ ] ] with @xmath77^{1/2}\ ; , \ ; x=\frac{p^2+f^2-p_f^2}{2pf}\ , ,   \nonumber \\ & \displaystyle a_1=\omega^2 - 2\omega ay+a^2\ ; , \ ; a_2=\omega^2(1 - 3y^2)+4\omega ay-2a^2\ ; , \nonumber \\ & \displaystyle a=\frac{p\delta}{m}\left(\frac{1+\omega / m}{1+\delta / m}\right)\ , .\end{aligned}\ ] ] the mass difference @xmath1 between the proton and the @xmath78-excitation is @xmath79 mev , the width @xmath80 of the @xmath42-isobar has been taken to be 115 mev . again , the integration with respect to @xmath81 in eq.([app6 ] ) can be performed analytically , but due to its length the result is not presented here . for the partial cross sections of the two - nucleon case , we obtained the following result : @xmath82+\nonumber \\ & \displaystyle \theta(\omega -5\varepsilon ) \,\theta ( 9\varepsilon - \omega ) \,\int\limits_{l_1(q)}^{l_3(q)}\phi(\beta_1,\ , q ) \biggr\}g_\pi ^4(p)\left[g^{(nr)}(p,\omega ) + g^{(r)}(p,\omega ) \right]dp , \nonumber          \label{app10}\end{aligned}\ ] ] where @xmath83 , @xmath84 , @xmath85 the elementary function @xmath86 is @xmath87\ , +   \nonumber \\ & \displaystyle \sqrt 8pq\,[\frac{1}{3}(2q^2+p^2-p_f^2)\cos ^3x       -\frac{2}{5}q^2\cos ^5x\,]\biggr\}\ ,    \biggl.\biggr|_{x=\beta}^{x=\pi /2 -\beta}\ , .          \label{app13}\end{aligned}\ ] ] the functions in the integrand of eq.([app10 ] ) , which characterize the resonant and non - resonant part , are given by @xmath88 and @xmath89 \frac{p^2}{(p^2+m^2)^2 }          \label{app15}\ ] ] respectively . in eq.([app10 ] ) the integration limits are @xmath90 and @xmath91    99    m. hirata , j. koch , f. lenz and e. moniz , ann.phys.120 ( 1979 ) 205    e. oset and w. weise , nucl.phys.a329 ( 1979 ) 365    c. garcia - recio , l. salcedo , e. oset , d. strottman and m. lopez - santodomingo , nucl.phys.a526 ( 1991 ) 685    l. salcedo , e. oset , m. vicente - vacas and c. garcia - recio , nucl.phys.a484 ( 1988 ) 557    j. koch , e.j . moniz and n. ohtsuka , ann.phys.154 ( 1984 ) 99    e. oset and w. weise , nucl.phys.a368 ( 1981 ) 375    m. wakamatsu and k. matsumoto , nucl.phys.a392 ( 1983 ) 323    m .- th . htt and a.i . milstein , nucl.phys.a in press    l. boato and m. giannini , j.phys.g15 ( 1989 ) 1605    j. ryckebusch , m. vanderhaeghen , l. machenil , m. waroquier , nucl.phys.a568 ( 1994 ) 828    r.c . carrasco and e. oset , nucl.phys.a536 ( 1992 ) 445    m. vanderhaeghen , k. heyde , j. ryckebusch , m. waroquier , nucl.phys.a595 ( 1995 ) 219    j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.rev.c49 ( 1994 ) 2704    j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.lett.b291 ( 1992 ) 213    s. homma et al . , phys.rev.c36 ( 1987 ) 1623    t. ericson , w. weise , `` pions and nuclei '' , oxford univ . press 1988    d.o . riska , phys.rep.181 ( 1989 ) 207    t. sasakawa , s. ishikawa , y. wu and t. saito , phys.rev.lett.68 ( 1992 ) 3503    a. molinari , phys.rep.64 ( 1980 ) 283    r. wichmann et al . , submitted to phys.lett.b    s. homma et al . , phys.rev.lett.53 ( 1984 ) 2538    j. ahrens , nucl.phys.a446 ( 1985 ) 229c [ figa ] notation for three - momenta of the external particles a ) for the one - nucleon process and b ) for the two - nucleon reaction . the wavy lines denote photons and dashed lines denote pions . a circle indicates a bound , an arrow a free nucleon . [ figb ] diagrammatical forms of the resonant part @xmath92 ( first two terms ) and the non - resonant part @xmath93 ( last term ) of the amplitude @xmath94 . these contributions to the @xmath95-interaction enter into the diagrams shown in fig.([figa ] ) . [ figc ] comparison of the calculated curve @xmath96 for nuclear photoabsorption with the experimental data . the dotted curve is the one - nucleon contribution @xmath97 , while the dashed curve represents the two - nucleon mechanism @xmath97 . the data are taken from ref . the empty ( full ) circles correspond to @xmath98pb ( @xmath99c ) data , while the squares represent data on @xmath100o . [ figd ] resonant ( dashed ) and non - resonant ( dotted ) contribution to the one - nucleon reaction ( full line ) . the corresponding analytical expressions @xmath101 , @xmath102 and @xmath97 , respectively , can be found in the appendix . [ figh ] contribution of interference terms . the full ( dashed ) curve corresponds to the one- ( two- ) nucleon case . as can be seen from the overall scale , both for @xmath106 and @xmath107 this contribution is highly suppressed . [ figg ] effect of two- and three - nucleon correlation in the case of the non - relativistic one - nucleon part of the photoabsorption cross section . the dashed curve contains only @xmath63 , while in the full curve @xmath61 ( cf . fig.([figj ] ) ) has been included . [ figi ] approximate description of differential cross sections for helium . the differential cross section with respect to the direction of the outgoing proton is shown as a function of the energy of the incoming photon . data points are taken from @xcite . the full curve and filled circles correspond to the one - nucleon process , while the dashed curve and empty circles represent the two - nucleon case ., simple models for the interaction between fermions and bosons continue to be fascinating , as often very non - trivial results can be obtained from even the most primitive hamiltonians . exactly solvable models for the interaction of photons or phonons with electrons in quantum dots @xcite or quasi - one - dimensional systems @xcite provide the best examples , as they often provide a deep insight into rich and complex physics .    in this contribution , we re - consider a simple model for a single electron of mass @xmath0 in one dimension that interacts with a delta - barrier through a coupling parameter that itself is a dynamical quantity . the hamiltonian is written as @xmath1\right\}+\omega a^{\dagger}a.\end{aligned}\ ] ] here , @xmath2 creates a boson of frequency @xmath3 and @xmath4 $ ] is a dynamical contribution on top of the static coupling constant @xmath5 . the constant zero point energy is omitted since it merely shifts the energy scale by @xmath6 . the lattice version of this model was originally introduced by gelfand , schmitt - rink and levi @xcite years ago in the study of tunneling in presence of phonons , and was shortly afterwards compared to a corresponding time - dependent classical hamiltonian @xcite , the continuous version of which reads @xmath7 @xmath8 is obtained as the interaction picture hamiltonian of eq.([hamiltonian ] ) with respect to @xmath9 , after replacing the boson operators by @xmath10 .    in its time - dependent version , eq.([hamiltonianc ] ) has subsequently been used as a model for scattering in quasi - one - dimensional quantum wires by bagwell and co - workers @xcite , who found fano - type resonances in the transmission coefficient as a function of the energy of an incident electron . it soon turned out that the scattering properties of this hamiltonian are quite intriguing as they very much depend on the relative sign and strength of the two coupling parameters @xmath5 and @xmath11 . the interplay between evanescent modes , quasibound states @xcite , and the behaviour of the transmission amplitude in the complex energy plane @xcite have been studied recently . our focus here is on the quantum version eq . ( [ hamiltonian ] ) of the model and its peculiarities in comparison with @xmath12 . it turns out that beside transmission zeroes , there are points of perfect transparency in the fano resonance that only appear in the model @xmath13 but not in @xmath12 . perfect transmission and fano resonances have been found recently in the transmission of phonons through non - linear chains without delta impurities @xcite . although not discussed in detail here , these results indicate that there still is rich and non - trivial behavior to be discovered from models like eq.([hamiltonian ] ) . the total wave function @xmath14 of the coupled electron - boson system can be expanded in the oscillator basis @xmath15 as @xmath16 with wave function coefficients @xmath17 depending on the position @xmath18 of the electron . we solve the stationary schrdinger equation at total energy @xmath19 , implying a scattering condition for the electron part of the wave function in demanding that there is no electron incident from the right . for @xmath20 , the @xmath17 are superpositions of plane waves if @xmath21 is above the threshold for the @xmath22 boson energy , @xmath23 whereas normalizabale evanescent modes occur if @xmath21 is below the threshold , @xmath24 here and in the following we set @xmath25 . we impose the condition that the boson is in its ground state for an electron incoming from the left , @xmath26 where we set the corresponding amplitude @xmath27 to unity . continuity of @xmath17 at @xmath28 yields @xmath29 for all @xmath30 , whereas the jump in derivative of @xmath17 across the delta barrier leads to a recursion relation for the transmission amplitudes @xmath31 , @xmath32 where the @xmath33 are real ( imaginary ) above ( below ) the boson energy @xmath34 , @xmath35 the total transmission coefficient @xmath36 is obtained from the sum over all _ propagating _ modes , @xmath37}\frac{k_n(e)}{k_0(e)}|t_n(e)|^2,\end{aligned}\ ] ] where the sum runs up to the largest @xmath30 such that @xmath38 remains real . although eq.([transmission ] ) is a finite sum , its evaluation requires the solution of the _ infinite _ recursion relation eq.([recursion ] ) due to the fact that the propagating modes are coupled to all evanescent modes . the transmission amplitudes can be determined from the linear equation @xmath39 numercally , this can easily been solved by truncation of the matrix @xmath40 . alternatively , one can solve eq.([matrix ] ) recursively which actually is numerically more efficient . for example , the result for the zero - channel transmission amplitude @xmath41 can be written in a very intuitive form : defining the ` greens function ' @xmath42 by @xmath43^{-1},\end{aligned}\ ] ] one can write @xmath41 with the help of a recursively defined ` self energy ' @xmath44 , @xmath45 in fact , using @xmath46 , the self energy @xmath47 can be represented as a continued fraction @xmath48 this demonstrates that @xmath41 depends on @xmath11 only through @xmath49 . truncating the matrix @xmath40 to a @xmath50 matrix corresponds to the approximation that sets @xmath51 and recursively solves eq . ( [ selfenergy ] ) for @xmath52 down to @xmath47 . for example , truncating at @xmath53 we obtain the approximation @xmath54 an important observation can be made with respect to the stability of our theory for large coupling constants @xmath11 . in fact , the truncation at @xmath55 is only consistent if the truncated self energy @xmath44 is a small correction to the inverse ` free propagator ' , @xmath56 which by use of eq . ( [ g0def ] ) at large @xmath57 implies @xmath58 or @xmath59 . the tridiagonal form of the matrix , eq . ( [ matrix ] ) , actually implies that the method based on the recursion eq . ( [ recursion ] ) is perturbative in the coupling @xmath11 to the boson . we conjecture that for @xmath11 above the critical value , the perturbation based on the oscillator basis @xmath15 used here breaks down . a similar breakdown of numerical approaches that start from a weak coupling regime in single boson hamiltonians is known from the rabi hamiltonian @xcite , i.e. the coupling of a single boson mode to a spin @xmath60 . we mention that the lattice version of the present model would be a natural starting point for a more detailed analysis of the strong coupling ( small polaron ) limit . transmission coefficient through a dynamical one - dimensional delta barrier with repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) static part , cf . ( [ hamiltonian ] ) and ( [ hamiltonianc ] ) . @xmath21 is the energy of the incident particle.,title="fig:",scaledwidth=50.0% ] transmission coefficient through a dynamical one - dimensional delta barrier with repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) static part , cf . ( [ hamiltonian ] ) and ( [ hamiltonianc ] ) . @xmath21 is the energy of the incident particle.,title="fig:",scaledwidth=50.0% ]    the recursion relation corresponding to eq . ( [ recursion ] ) for the classical time - dependent hamiltonian , eq . ( [ hamiltonianc ] ) , was derived and discussed by bagwell and lake @xcite , @xmath63 here , @xmath31 is the coefficient of the time - dependent electron wave function in photon side - band @xmath30 , where @xmath30 runs through positive _ and negative _ integers @xmath30 . in further contrast to the recursion relation eq . ( [ recursion ] ) , there are no factors @xmath64 and @xmath65 multiplying the coupling constant @xmath11 . this latter fact is an important difference to the quantum case where these terms lead to the factors @xmath57 that multiply @xmath49 in the self energies @xmath44 , eq . ( [ selfenergy ] ) . this difference is eventually responsible for the breakdown of the perturbative approach for large @xmath11 in the quantum case . a continued fraction representation of @xmath41 for the classical case has been derived recently by martinez and reichl @xcite . the corresponding matrix defining the transmission amplitudes @xmath66 in the classical case is the infinite tridiagonal matrix @xmath67 with @xmath68 on the diagonal and @xmath11 on the lower and upper diagonals , @xmath69 fig . ( [ delta1.eps ] ) shows a comparison between the transmission coefficient @xmath36 , eq . ( [ transmission ] ) , for the quantum and the classical barrier . in the repulsive case with @xmath70 , the dynamical part of the barrier is essentially a weak perturbation to the unperturbed @xmath71 case . additional structures ( cusps ) appear at the boson ( photo side - band ) energies @xmath34 although the overall @xmath36-curve resembles the @xmath71 case . the more interesting case occurs for barriers with an attractive static part , @xmath62 ( fig . ( [ delta1.eps ] ) , right ) . a fano type resonance appears below the first threshold @xmath72 where the transmission coefficient has a zero in both the classical and the quantum case . in the classical case , this is a well - known phenomenon @xcite : the transmission zero for weak coupling ( small @xmath11 ) shows up when the fano resonance condition @xmath73 is fulfilled.there , the energy of the electron in the first side channel ( @xmath74 ) coincides with the bound state of the attractive delta barrier potential , @xmath75 . in the quantum case , the self energy in eq.([selfenergy ] ) diverges at the zeros of @xmath36 , @xmath76^{-1}=0.\end{aligned}\ ] ] for @xmath77 , @xmath78 , cf . eq.([t0approx ] ) , and the two conditions eq.([fano1 ] ) and eq.([fano1a ] ) coincide . the most interesting feature in the scattering properties of the dynamical quantum barrier is the appearance of an energy close to the first channel @xmath79 threshold where perfect transmission @xmath80 occurs . this is clearly visible in the vanishing of the reflection coefficient , @xmath81 , in the logarithmic plot fig . ( [ delta3.eps ] ) . for a repulsive static part , @xmath82 , this occurs at an energy below the energy where the reflection coefficient comes close to unity , and above that energy if the static part is attractive ( @xmath83 ) . in contrast , in the classical case the reflection coefficient never reaches zero in neither the repulsive nor the attractive case . this contrast becomes even more obvious in a two - dimensional plot where the zeros in @xmath84 correspond to ` ridges ' in the @xmath5-@xmath21 plane , cf . ( [ delta4.eps ] ) . logarithmic plot of reflection coefficient @xmath85 for dynamical delta barrier with static repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) core . , title="fig:",scaledwidth=45.0% ] logarithmic plot of reflection coefficient @xmath85 for dynamical delta barrier with static repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) core . , title="fig:",scaledwidth=45.0% ]    density plot of @xmath86 ( reflection coefficient ) for the quantum delta barrier at @xmath87 . exact solution from eq . ( [ selfenergy ] ) ( left ) , from the @xmath53 truncation eq . ( [ t0approx ] ) ( center ) , and from the classical model eq.([hamiltonianc ] ) ( right ) . the light ` ridges ' correspond to curves of perfect transmission @xmath88 , cf . ( [ perfecta ] ) and eqs . ( [ perfectb]).,scaledwidth=95.0% ]    perfect transparency ( @xmath89 ) can be understood by considering the transmission amplitude @xmath41 which determines the total transmission below the first sideband threshold . recalling that @xmath90 , in the quantum case the transmission coefficient becomes unity when @xmath91 our exact continued fraction expression for the self energy , eq.([sigmacontinued ] ) , implies that for @xmath92 , @xmath47 is real because @xmath93 is real for @xmath94 . the condition eq.([perfect ] ) then means that the self energy exactly renormalizes the static part @xmath5 of the scattering potential to zero .    for small @xmath11 , we can use our perturbative expression corresponding to truncating the matrix @xmath40 , eq.([matrix ] ) , to a two - by - two matrix . the perfect transparency condition eq.([perfect ] ) then becomes @xmath95 which determines the position of the perfect transmission energy . the solution of eq.([perfecta ] ) defines two curves in the @xmath21@xmath5-plane with perfect transmission for @xmath92 , @xmath96 these two curves can be clearly identified in the logarithmic density plots of the reflection coefficient @xmath97 , cf . ( [ delta4.eps ] ) . the @xmath53 approximation to the transmission amplitude , eq.([t0approx ] ) , turns out to reproduce these features quite well even at moderate coupling constants @xmath11 . the above analysis of the two models eq . ( [ hamiltonian ] ) and eq . ( [ hamiltonianc ] ) has revealed some interesting differences between scattering properties of simple electron - boson models and their classical counter - part . the strong coupling limit of the quantum model and its extension to more complicated situations like multi - channel scattering remain to be explored ., source_sentence: <S> we study an appell hypergeometric system @xmath0 of rank four which is reducible and show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a @xmath1-dimensional family of curves of genus 2 . </S>
sentences: schwarz maps for hypergeometric systems in single and several variables are studied by several authors ( cf . @xcite ) for more than hundred years . these systems treated were irreducible , maybe because specialists believed that reducible systems would not give interesting schwarz maps . we study in this paper appell s hypergeometric system @xmath0 of rank four when its parameters satisfy @xmath2 or @xmath3 . in this case , the system @xmath0 is reducible , and has a @xmath4-dimensional subsystem isomorphic to appell s @xmath5 ( proposition [ prop : s2 ] ) . if @xmath6 then @xmath0 has two such subsystems . by proposition [ prop : s2 g ] , the intersection of these subsystems is equal to the gauss hypergeometric equation . as a consequence , we have inclusions on @xmath0 , two @xmath5 s and @xmath7 ( theorem [ matome ] ) . we give the monodromy representation of the system @xmath0 which can be specialized to the case @xmath6 in theorem [ th : monod - rep ] . as for explicit circuit matrices with respect to a basis @xmath8 , see corollary [ cor : monod - matrix ] . we further specialize the parameters of the system @xmath0 as @xmath9 in  [ schmap ] . in this case , the restriction of its monodromy group to the invariant subspace is arithmetic and isomorphic to the triangle group of type @xmath10 $ ] . we show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a 1-dimensional family of curves of genus 2 in theorem [ th : gen - schwarz ] . the system @xmath0 is equivalent to the restriction of a hypergeometric system @xmath11 to a two dimensional stratum in the configuration space @xmath12 of six lines in the projective plane . in appendix [ 3-dim - s ] , we study a system of hypergeometric differential equations in three variables , which is obtained by restricting @xmath11 to the three dimensional strata corresponding to configurations only with one triple point . the methods to prove proposition [ prop : s2 ] are also applicable to this system under a reducibility condition . in appendix [ genus2 ] , we classify families of genus @xmath13 branched coverings of the projective line , whose period maps yield triangle groups .    in a forthcoming paper @xcite , we study this schwarz map using period domains for mixed hodge structures . moreover , we explicitly give its inverse in terms of theta functions . gauss hypergeometric series @xmath14 where @xmath15 , admits an integral representation : @xmath16 the function @xmath17 is a solution of the hypergeometric equation @xmath18 where @xmath19 the collection of solutions is denoted by @xmath20 . appell s hypergeometric series @xmath21 admits an integral representation : @xmath22 the function @xmath23 is a solution of the hypergeometric system @xmath24   ( d'(c-1+d+d')-y(a+d+d')(b'+d'))z=0 ,   \end{array } \right.\ ] ] where @xmath25 , which can be written as @xmath26 where @xmath27 & q_1(a , b , b',c;x , y)=y(1-y)\partial_{yy}+x(1-y)\partial_{yx }      + ( c-(a+b'+1)y)\partial_y - b'x\partial_x - ab ' , & \\[2 mm ] & r_1(a , b , b',c;x , y)=(x - y)\partial_{xy}-b'\partial_x+b\partial_y , \end{aligned}\ ] ] and @xmath28 , etc . the last equation @xmath29 is derived from the integrability condition of the first two equations . the collection of solutions is denoted by @xmath30 . appell s hypergeometric series @xmath31 admits an integral representation : @xmath32@xmath33 the function @xmath34 satisfies the system @xmath35 where @xmath36 & & q_2(a , b , b',c , c';x , y)=d'(c'-1+d')-y(a+d+d')(b'+d ) . \end{aligned}\ ] ] the collection of solutions is denoted by @xmath37 .      as for the reducibility of the systems @xmath0 and @xmath5 , the following is known :    [ redf2]@xmath38@xcite@xmath39 appell s system @xmath40 is reducible if and only if at least one of @xmath41 is an integer . [ redf1]@xmath38@xcite@xmath39 appell s system @xmath42 is reducible if and only if at least one of @xmath43 is an integer . the system @xmath40 is reducible when @xmath44 , fact [ redf2 ] . in fact , we see that the system @xmath45 is a subsystem of @xmath46 ; precisely , we have    [ prop : s2 ] @xmath47    we give three `` proof ' 's : one using power series , subsection [ subsec : power ] , one using integral representations , subsection [ subsec : integ ] , and one manipulating differential equations , subsection [ subsec : equat ] . the former two are valid only under some non - integral conditions on parameters , which we do not give explicitly . though the last one is valid for any parameters , it would be not easy to get a geometric meaning . the following fact explains the inclusion in proposition [ prop : s2 ] . [ bailey1 ] @xmath48      we consider the integral @xmath49 which is a solution of the system @xmath50 . we change the coordinate @xmath51 into @xmath52 as @xmath53 which sends @xmath54 the inverse map is @xmath55 since @xmath56 we have @xmath57 this implies , if @xmath58 , then the double integral above becomes the product of the beta integral @xmath59 and the integral @xmath60 which is an element of the space @xmath61 . this shows @xmath62 which is equivalent to @xmath63    the bi - rational coordinate change @xmath64 is so made that the lines defining the integrand of the integral @xmath65 may become the union of vertical lines and horizontal lines in the @xmath66-space . actual blow - up and down process is as follows ( see figure [ st ] ) . name the six lines in the @xmath67-projective plane as : @xmath68 blow up at the 4 points ( shown by circles ) @xmath69 and blow - down along the proper transforms of the line @xmath70 and two lines : @xmath71 these three lines are dotted . this takes the @xmath67-projective plane to @xmath72 . in the figure , lines labeled @xmath73 stand for @xmath74 , and the lines labeled @xmath75 on the right are the blow - ups of the intersection points @xmath76 , respectively . the line obtained by blowing up the point @xmath77 is the line defined by @xmath78 , which should be labeled by @xmath79 .    , width=491 ]      a proof of the inclusion in proposition [ prop : s2 ] that is valid for any parameters is done as follows . let @xmath80 be a solution of the system @xmath81 . then , the system @xmath82 yields @xmath83-linear expressions of @xmath84 and @xmath85 in terms of @xmath86 and @xmath80 . substitute these expressions into the system @xmath87 . then , we get two linear forms in @xmath86 and @xmath80 . we now have only to see their coefficients vanish for the given parameters after a change of coordinates and a change of the unknown by multiplying a simple factor . we do not here present the actual computation , because if we put @xmath88 in the proof of proposition [ x3fd ] in subsection [ secondproof ] , manipulating differential equations , it gives essentially a proof of proposition [ prop : s2 ] . when @xmath90 , applying proposition 1.3 also for @xmath91 , we see that the system @xmath92 has two subsystems isomorphic to @xmath5 . the intersection of the two @xmath5 s would be the gauss hypergeometric equation . in fact , we have the following proposition .    [ prop : s2 g ] @xmath93    similar to the argument of the previous subsection , we can give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . we give a sketch of them in the following . the following identity explains the inclusion above . [ bailey2 ] @xmath94 we continue the argument in  [ subsec : integ ] . in the integral @xmath95 in  [ subsec : integ ] above , change the coordinate from @xmath96 to @xmath97 as @xmath98 sending @xmath99 since @xmath100 we have @xmath101 this implies , if @xmath91 , then @xmath102 this shows @xmath103 which of course implies the inclusion relation in proposition [ prop : s2 g ] by combination with that in proposition [ prop : s2 ] .      put @xmath104 we have @xmath105 and so on . assume that @xmath106 . the equation @xmath107 gives a linear expression of @xmath108 in @xmath109 and @xmath110 . substitute these expressions in @xmath111 and we get the product of @xmath112 and a @xmath83-linear combination of @xmath110 and @xmath109 . the coefficients vanish if @xmath91 . if we do the same for @xmath113 , then we find that it vanishes when @xmath58 . we show that some indefinite integrals solve the system @xmath92 . we begin with some well - known facts . [ wellknown1 ] @xmath114 where @xmath115    note that @xmath116 this implies @xmath117 since @xmath118 we have @xmath119    & = \dsp b x(d_s+1)\frac{1-s}{x - s}\phi \\[4 mm ]    & = \dsp b x\frac{\partial}{\partial s}\left(\frac{s(1-s)}{x - s}\phi\right ) . \end{array}\ ] ]    [ wellknown2 ] the indefinite integral @xmath120 solves @xmath121 . in particular , @xmath122 .    since @xmath123 , we have @xmath124 lemma [ wellknown1 ] leads to @xmath125 let @xmath126 and @xmath127 be the operators generating the system @xmath121 : @xmath128 q_1(a,0,b , c;s , t ) &   = p(a , b , c;t)/t+s(1-t)\partial_{st}-bs\partial_s,\\[2 mm ] r_1(a,0,b , c;s , t)&=(s - t)\partial_{st}-b\partial_s;\end{array}\ ] ] refer to section 1 . note that @xmath129 . by using the above identities , we have @xmath130 and @xmath131 @xmath132 furthermore , for @xmath133 , the forms of operators above imply that @xmath80 lies in @xmath134 . we now use the following fact :    @xcite . [ bailey3 ] @xmath135    from this fact we get , when @xmath136 , @xmath137 if we put @xmath138 then @xmath139 thus we have @xmath140 & = ( 1-y)^{-b'}(1-x)^{-b}s_1\left(b,0,b',a;\dsp\frac{-x}{1-x},\frac{xy}{(1-x)(1-y)}\right)\\[3 mm ] & \supset ( 1-y)^{-b'}(1-x)^{-b } s\left(b , b',a;\dsp\frac{xy}{(1-x)(1-y)}\right ) . \end{array}\ ] ] this agrees with the inclusion in proposition [ prop : s2 g ] . in particular , by the inclusion @xmath141 and lemma [ wellknown2 ] we get solution of @xmath142 represented by the indefinite integral : @xmath143 starting point of the path of integration can be any point @xmath144 , so we choose @xmath145 , just for simplicity . by exchanging the role of @xmath146 and @xmath147 , we get an inclusion @xmath148 and another solution of @xmath142 represented by the indefinite integral : @xmath149 after the change @xmath150 , it can be also expressed as @xmath151 thus , we have :    [ matome ] we have the following inclusions of the spaces of solutions : @xmath152    moreover , the collection of solutions @xmath153 is spanned by @xmath154 and @xmath155    this will play a key role to understand the schwarz map of a system @xmath0 with specific parameters , which will be introduced in  [ e36 ] . in this section , we study the monodromy representation of @xmath156 . though it is assumed in @xcite that the parameters satisfy the irreducibility condition in fact 1.1 : @xmath157 in this section , we only assume the weaker condition @xmath158 we modify theorem 7.1 in @xcite so that the statements remain valid for these parameters . we will apply the result of this section in  [ special ] . let @xmath159 be the complement of the singular locus of @xmath0 . for each @xmath160 , we consider a multi - valued function @xmath161 on @xmath162    as in  1.6 and  1.7 of chapter 2 in @xcite , we define the twisted homology group @xmath163 associated with @xmath164 and locally finite one @xmath165 . under some genericity condition , the integral of @xmath164 over a twisted cycle gives a solution of @xmath0 . if @xmath166 then the natural map @xmath167 is bijective , and the inverse map @xmath168 is called the regularization . in general , the map @xmath169 is neither injective nor surjective , however we still have the isomorphism @xmath170    under condition ( 3.1 ) , thanks to the vanishing theorem of cohomology groups in @xcite , rank of @xmath165 and @xmath163 are equal to the euler number @xmath171 of @xmath172 , and the bilinear form  _ the intersection form _  @xmath173 is non - degenerate . let @xmath174 be a small simply connected domain in @xmath175 . we can identify the local solution space to @xmath0 on @xmath174 with the trivial vector bundle @xmath176 via the euler type integral representation of solutions to @xmath0 . the monodromy representation of @xmath0 is equivalent to that of the local system @xmath177 over @xmath175 . we also consider a local system @xmath178 over @xmath175 . we fix a small positive real number @xmath179 , and let @xmath180 a base point in @xmath175 . denote the germs at this point of the local systems @xmath181 and @xmath182 by @xmath183 respectively . let @xmath184 be the monodromy representations of @xmath181 and @xmath182 with respect to @xmath180 . @xmath185 and @xmath186 are called the circuit transformations along @xmath187 . [ prop : inv - subspace ]    * the image @xmath188 of the natural map @xmath189 is invariant under the monodromy representation @xmath190 . * the kernel @xmath191 of the natural map @xmath192 is invariant under the monodromy representation @xmath193 . it is clear that @xmath194 and @xmath195 are invariant under @xmath190 and @xmath193 , respectively . we have only to note that the natural maps @xmath196 and @xmath197 commute with @xmath190 and @xmath193 . we will see that if @xmath2 or @xmath3 , then both of @xmath188 and @xmath191 are proper subspaces . thus monodromy representations @xmath190 and @xmath193 are reducible in this case . [ lem : duality ]    1 . let @xmath198 and @xmath199 be elements of @xmath200 and @xmath195 , respectively . then we have @xmath201 2 . suppose that @xmath202 is decomposed into the direct sum of the eigenspaces @xmath203 of @xmath204 of eigenvalues @xmath205 . then @xmath206 is decomposed into the direct sum of the eigenspaces @xmath207 of @xmath208 of eigenvalues @xmath209 . the eigenspace @xmath210 is characterized as @xmath211    \(1 ) the intersection number @xmath212 is stable under small deformations of @xmath198 and @xmath199 . \(2 ) since @xmath204 belongs to the general linear group , we have @xmath213 . let @xmath214 be any element of @xmath215 @xmath216 and let @xmath217 be any element of @xmath218 . since we have @xmath219 for @xmath220 , @xmath221 belongs to the space @xmath222 . thus @xmath208 induces a linear transformation of @xmath222 . let @xmath223 be an eigenvalue of the restriction of @xmath208 to @xmath222 , and let @xmath224 be an eigenvector of @xmath223 . since the intersection from @xmath225 is non degenerate , there exists @xmath226 such that @xmath227 . note that @xmath228 hence we have @xmath229 , i.e. , @xmath230 and @xmath231 . similarly , we have @xmath232 for @xmath233 . to show @xmath234 , we consider the restriction of @xmath208 to @xmath235 , where @xmath236 take its eigenvector @xmath237 and repeat the above argument . in this way , we have @xmath238 since @xmath239 we have @xmath240 for @xmath241 . let @xmath242 @xmath243 be locally finite chains shown in figure [ fig : basis ] . we specify a branch of @xmath164 on each chain by the assignment of @xmath244 on it as in table [ tab : arg ] , where @xmath245 and load @xmath164 to get the locally finite twisted cycles @xmath246 . -cycles , width=377 ]        .list of @xmath244 on @xmath247 [ cols="^,^,^,^,^,^",options="header " , ]     it will be shown that @xmath248 form a basis of @xmath200 in corollary [ cor : basis ] . we choose elements in @xmath195 as @xmath249 @xmath250 where @xmath251 and @xmath252 note that in terms of @xmath253 s the irreducible condition in fact 1.1 is @xmath254 the condition ( 3.1 ) is @xmath255 and ( see figure [ p2fig ] ) @xmath256 and that @xmath257 satisfies ( 3.1 ) and @xmath258 for any @xmath259 . explicitly , @xmath260 and @xmath261 can be written as : @xmath262- \frac{\circlearrowleft_{0}^-}{\mu_1^{-1}-1}\big)\times   \big(\frac{\circlearrowleft_0^+}{\mu_3^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] @xmath263- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times   \big(\frac{\circlearrowleft_\infty^+}{\mu_{345}-1 } + [ \frac{-1}{\delta},-\delta]- \frac{\circlearrowleft_{0}^-}{\mu_3^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] where @xmath264 is a small positive real number , @xmath265 $ ] and @xmath266 $ ] are closed intervals , @xmath267 is the negatively oriented circle of which radius , center and terminal are @xmath268 , @xmath269 and @xmath270 , @xmath271 @xmath272 is the positively oriented circle of which radius , center and terminal are @xmath264 , @xmath273 and @xmath274 . notice that the definition of the twisted cycles @xmath275 @xmath276 make sense even in the case @xmath2 or @xmath3 . indeed , this specialization gives no harm to @xmath277 @xmath278 , and thanks to the above expression , when @xmath279 and @xmath280 , we have    @xmath281- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1 } } , \\ { { \widetilde{\delta}}_4}&= & \big [ \big(\frac{\circlearrowleft_0^+}{\mu_1^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times   \circlearrowleft_\infty^+\big]^{\psi^{-1}},\end{aligned}\ ] ]    respectively . [ rem : kernel ] suppose that @xmath282 . then the twisted cycles @xmath283 and @xmath284 are homologous to @xmath269 in @xmath285 , since they are the boundary of locally finite @xmath4-chains given by the replacement @xmath286 in their expressions , where @xmath287 is the annulus @xmath288 . they belong to @xmath191 . by proposition [ prop : int - mat ] , it turns out that @xmath188 is spanned by @xmath289 and @xmath290 . [ prop : int - mat ] the intersection matrix @xmath291 for @xmath292,@xmath293,@xmath294 and @xmath295,@xmath293,@xmath296 is given by @xmath297 its determinant is @xmath298 which does not vanish under the assumption @xmath299 . follow  3 of chapter viii in @xcite for the computation of the intersection numbers . by a straightforward calculation , we have its determinant .    proposition [ prop : int - mat ] yields the following corollary . [ cor : basis ] the twisted cycles @xmath292,@xmath293,@xmath300 and @xmath295,@xmath293,@xmath301 form a basis of @xmath200 and that of @xmath195 , respectively . we express the twisted cycles @xmath302 and @xmath303 as linear combinations of @xmath304 . [ lem : squares ] we have @xmath305    set @xmath306 and compute the intersection numbers @xmath307 . then we have @xmath308 which yields the expression of @xmath302 .    since @xmath309 we have the expression of @xmath303 .    in @xcite , we take a basis @xmath292 , @xmath310 , @xmath302 and @xmath303 of @xmath200 . if @xmath6 then each of @xmath302 and @xmath303 is a scalar multiple of @xmath292 by lemma [ lem : squares ] . similar to lemma [ lem : squares ] , we have the following .    [ lem : dual - squares ] we have @xmath311        we give generators of the fundamental group @xmath312 . let @xmath313 ( and @xmath314 , @xmath315 ) be a loop in @xmath316 starting from @xmath317 , approaching to the point @xmath318,(and @xmath319 , @xmath320 ) with @xmath321 , turning once around the point positively , and tracing back to @xmath180 . let @xmath322 ( and @xmath323 ) be a loop in @xmath324 starting from @xmath317 , approaching to the point @xmath325 , ( and @xmath326 ) with @xmath327 , turning once around the point positively , and tracing back to @xmath180 . it is known that the loops @xmath328 generate @xmath312 . the circuit matrices of @xmath190 and @xmath329 along @xmath330 will be denoted as @xmath331      [ th : monod - rep ] under the assumption ( 3.1 ) , the circuit transformations @xmath332 @xmath333 are given as @xmath334 @xmath335 where @xmath336 is any element of @xmath200 , @xmath337 @xmath338 is the submatrix of @xmath339 consisting of the @xmath340 , @xmath341 , @xmath342 and @xmath343 entries of @xmath339 , and @xmath344 @xmath345    under the condition @xmath346 , the linear transformation @xmath347 satisfies the assumption of lemma [ lem : duality ] ( 2 ) . in fact , a fundamental system of solutions can be given by the hypergeometric series @xmath348 multiplied by the power functions @xmath349 thus the eigenvalues of @xmath347 are @xmath1 and @xmath350 , and each of the eigenspaces is two dimensional . it is easy to see that the locally finite chains @xmath351 and @xmath352 are invariant under the deformation along @xmath313 . hence the twisted cycles @xmath289 and @xmath353 span the eigenspace of @xmath347 of eigenvalue @xmath1 . similarly we can show that the twisted cycles @xmath354 and @xmath355 span the eigenspace of @xmath356 of eigenvalue @xmath1 . by lemma [ lem : duality ] ( 2 ) , the eigenspace of @xmath347 of eigenvalue @xmath357 is @xmath358 it is easy to see that the right hand side @xmath359 of ( [ eq : monod1 ] ) satisfies @xmath360 by proposition [ prop : int - mat ] , @xmath8 form a basis even in the case @xmath361 or @xmath362 . thus the representation matrix @xmath363 of @xmath347 with respect to this basis is continuous on the parameters @xmath364 . on the other hand , the expression @xmath365 is also continuous since the factor @xmath366 in the denominator of @xmath367 \dfrac{\mu_4 - 1}{\mu_{34}(\mu_5 - 1 ) } & \dfrac{\mu_{34}-1}{\mu_{34}(\mu_5 - 1 ) } \end{pmatrix}.\ ] ] cancels by @xmath368 . similarly we have the expression of @xmath369 . to study @xmath370 , we work temporarily under the condition @xmath371 . we decompose @xmath315 into @xmath372 , where @xmath373 is the approach to @xmath374 and @xmath375 is the turning path . we trace the deformation of the triangle @xmath376 made by @xmath377 and @xmath378 along @xmath373 . after the deformation , this becomes a small triangle near the point @xmath379 . we see the argument of @xmath380 on this triangle . since @xmath381 and @xmath382 in @xmath373 , @xmath383 in this triangle , and @xmath384 on the line @xmath385 , @xmath380 varies negative to positive via the upper - half space , i.e , @xmath386 decreases by @xmath387 along @xmath373 . note that this change is compatible with our assignment of @xmath386 on @xmath377 and @xmath378 in table [ tab : arg ] . thus the twisted cycle @xmath388 plays the role of a vanishing cycle as the line @xmath389 approaches the point @xmath390 . since @xmath375 corresponds to the move of @xmath391 turning around the point @xmath392 the cycle @xmath393 is an eigenvector of @xmath370 of eigenvalue @xmath394 . we can similarly show that @xmath395 is an eigenvector of @xmath396 of eigenvalue @xmath397 . on the other hand , we can find three chambers not affected by the move of the line @xmath389 along @xmath375 . for example , @xmath398 , @xmath399 and @xmath400 . hence @xmath370 has three dimensional eigenspace of eigenvalue @xmath1 . lemma [ lem : duality ] ( 2 ) yields that this eigenspace is expressed as @xmath401 so @xmath370 has desired eigenvalues and eigenspaces . since the factor @xmath402 is continuous on @xmath394 at @xmath1 , the expression of @xmath370 is valid even in the case @xmath403 . similarly we have the expressions of @xmath404 and @xmath405 . [ cor : monod - rep - dual ] under the assumption ( 3.1 ) , the circuit transformations @xmath406 @xmath333 are given as @xmath407 @xmath408 where @xmath199 is any element of @xmath195 . let @xmath409 and @xmath410 @xmath333 be the circuit matrices along the loop @xmath330 with respect to the basis @xmath411 of @xmath200 , and to @xmath412 of @xmath195 , respectively . that is , we have transformations @xmath413 by the continuation along @xmath330 . [ cor : monod - matrix ] the circuit matrices are expressed as @xmath414 @xmath415 where @xmath416 @xmath417 @xmath418 @xmath419 their explicit forms are @xmath420 @xmath421    @xmath422    @xmath423    they satisfy @xmath424    we identify @xmath425 with the row and column vectors @xmath426 respectively . note that @xmath427 theorem [ th : monod - rep ] yields these expressions . we have @xmath428 by lemma [ lem : duality ] ( 1 ) .    as a result , @xmath409 @xmath333 coincides with the circuit matrix with respect to the basis @xmath8 by theorem 7.1 in @xcite . in this section we introduce a system @xmath430 , and describe its schwarz map , which is the main result of this paper . we introduce in this subsection a system @xmath430 , which is a system @xmath0 with specific parameters , and mention a reason why this system is of special interest . let @xmath12 be the configuration space of six lines @xmath432 in general position in the projective plane @xmath433 . we identify the space @xmath12 with @xmath434 where @xmath70 is the line at infinity in the @xmath435-plane given by @xmath436 . the system @xmath437 is generated by the linear differential equations which annihilate functions on @xmath12 defined by the integral @xmath438 the schwarz map of the system @xmath439 is studied ( cf . @xcite , @xcite ) in two cases @xmath440 and @xmath441 . we have been interested in the case @xmath442 . on the other hand , let @xmath443 be the 2-dimensional stratum defined by @xmath444 , which is the space of six lines such that the three lines @xmath445 meet at a point , the three lines @xmath446 meet at another point , and nothing further special occurs . it is known ( @xcite ) that the restriction of @xmath439 onto @xmath443 is the appell s hypergeometric system @xmath447 , which is projectively equivalent ( multiplying a function to the unknown ) to @xmath448 setting @xmath449 , we define @xmath450 we believe that the first step of understanding @xmath451 is the study of the system @xmath429 .    the schwarz map of a system is defined by the ratio of linearly independent solutions . the main objective of this paper is the schwarz map of the hypergeometric system @xmath452 the system @xmath429 admits solutions stated in proposition [ matome ] . the next subsection gives a geometric background of understanding these solutions . consider a family of curves of genus 2 given as triple covers of @xmath453 : @xmath454 branching at four points @xmath455 . we choose two linearly independent holomorphic 1-forms : @xmath456 and put @xmath457    for a fixed @xmath51 , the _ abel - jacobi map _ for the curve @xmath458 is a multi - valued map @xmath459 it is a single - valued map to its jacobian @xmath460 , where @xmath461 is a lattice generated by its periods : integrals over possible loops with base @xmath462 : @xmath463      proposition [ matome ] for @xmath464 implies that after the coordinate change @xmath465 and the change of unknown : @xmath466 , two linearly independent solutions of @xmath467 and the two indefinite integrals @xmath468 form a set of fundamental solutions of @xmath429 .    on the other hand , the integral representation of the gauss hypergeometric equation given in section 1 asserts that the integral above along any _ closed _ path gives a solution of @xmath469 . thus we find that the schwarz map of @xmath429 is the totality of the abel - jacobi map of the family @xmath470 after a slight modification ( multiplying @xmath471 to the second coordinate ) . thus we get    [ th : gen - schwarz ] if we change the coordinates @xmath472 of @xmath473 as @xmath474 the schwarz map of the system @xmath429 is equivalent to the projectivization of the family of the abel - jacobi map of the family @xmath470 of curves of genus 2 , explicitly given as @xmath475 the latter two @xmath476 and @xmath477 are @xmath478 and @xmath479 in section 2 . the map by means of the former two @xmath480 is the schwarz map of the hypergeometric equation @xmath481 . its image is a disc , and the inverse map of @xmath482 is single - valued automorphic function on the disc with respect to the triangle group of type @xmath10 $ ] ; in other words , the disc is tessellated by schwarz triangles of type @xmath10 $ ] . the image surface under @xmath483 can be regarded as lying in a fiber bundle with the @xmath482-image disc as its base and the jacobian variety of @xmath458 as the fiber on the image point @xmath484 . a triangle of type @xmath485 $ ] is a hyperbolic triangle with angles @xmath486 and @xmath487 ; the above triangle has angles @xmath488 and @xmath269 . the triangle group of type @xmath485 $ ] is the group consisting of the even products of the reflections with the sides of the triangle of type @xmath485 $ ] as axes . it is known that the triangle group of type @xmath10 $ ] is conjugate to the congruence subgroup @xmath489 for arithmetic triangle groups , see @xcite . other than this family of curves , there are two families of curves of genus 2 branching at four points in @xmath453 ; see appendix 2 .      from monodromy side , theorem [ th : gen - schwarz ] can be understood as follows . define @xmath490 and @xmath491 @xmath333 by substituting @xmath492 into @xmath409 and @xmath410 defined in  [ circuit ] , respectively . they are the circuit matrices for @xmath429 with respect to @xmath493 and those for @xmath494 with respect to @xmath495 where @xmath496{t_1(1-t_1)t_2(1-t_2)(1-t_1x - t_2y ) } } .\ ] ]    [ cor : monodromy ] we have @xmath497 @xmath498 they satisfy @xmath499    by proposition [ prop : inv - subspace ] and remark [ rem : kernel ] , the subspace spanned by solutions @xmath500 and @xmath501 is invariant under the monodromy representation . in fact , the top - left @xmath502 block matrices @xmath503 of @xmath490 @xmath333 act on this space . note that @xmath504 , @xmath505 . let @xmath506 be the group generated by @xmath507 , @xmath508 and @xmath509 . the group @xmath506 is isomorphic to the triangle group @xmath10 $ ] , and is contained in the unitary group @xmath510)\mid g h ' \;^t \overline { g}=h'= \begin{pmatrix } -1 & -\omega \\ \omega+1 & 0 \end{pmatrix } \big\}.\ ] ] by a matrix @xmath511 the hermite matrix @xmath512 and circuit matrices @xmath513 @xmath514 are transformed as @xmath515 @xmath516 hence the projectivization of @xmath506 is isomorphic to the congruence subgroup @xmath517 of @xmath518 and the ratio @xmath519 can be regarded as the map @xmath482 in theorem [ th : gen - schwarz ] and as an element of the upper - half space . let @xmath520 denote the stratum defined by @xmath521 and let us restrict the system @xmath439 to this stratum , which is the space of six lines such that the three lines @xmath445 meet at a point , and nothing further special occurs . this system is denoted by @xmath522 or @xmath523 . little is known about this system .    before stating the proposition in this section , we briefly recall the appell - lauricella s system @xmath524 @xmath525 where @xmath526 are variables , and @xmath527 and @xmath528 . this is a 3-variable version of the appell s @xmath5 . it admits solutions given by a power series @xmath529 where @xmath530 , and by an integral @xmath531the collection of solutions is denoted by @xmath532    in this section , we prove the following proposition .    [ x3fd ] if @xmath533 , then the system @xmath522 is reducible and has a subsystem isomorphic to the appell - lauricella s system @xmath534 in 3 variables with 4 free parameters . more precisely , the collection of the solutions of @xmath522 includes @xmath535 note @xmath536 . if we apply the proposition under the further restriction @xmath88 , we find a subsystem isomorphic to @xmath5 in @xmath0 , which is equivalent to proposition 1.1 . we give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . it is known that the system @xmath537 has a solution given by the series @xmath538 where @xmath539 ; refer to @xcite . a computation shows that the identity @xmath540 holds if and only if @xmath541      we manipulate the integral @xmath542 the three lines @xmath543 meet at @xmath544 . introduce new coordinate @xmath545 @xmath546 which send @xmath547 to @xmath548 . since @xmath549 @xmath550 we have @xmath551 @xmath552 where @xmath553 . if @xmath554 then the double integral above becomes the product of the beta integral @xmath555 and the integral @xmath556 which can be written as @xmath557 on the other hand , the integral @xmath558 solves @xmath559 . by solving the system @xmath560 we complete the proof of proposition [ x3fd ]      we manipulate the system @xmath523 ( given in [ msy , p.24 ] ) : @xmath561 where @xmath562 , @xmath563 , @xmath564 , and @xmath565 , and the appell - lauricella system @xmath566 . we show that the system @xmath567 is a subsystem of @xmath523 when @xmath533 . change the unknown @xmath568 of @xmath523 into @xmath110 by @xmath569 and the variables @xmath570 into @xmath571 as @xmath572 then @xmath523 can be written as @xmath573 where @xmath574 where @xmath575 write the system @xmath567 as @xmath576 , where @xmath577 eliminating the second derivatives in @xmath578 by using @xmath579 , we see that @xmath578 are linear combination , over @xmath580 , of the @xmath579 s if and only if @xmath581 and @xmath582 this completes the proof of proposition [ x3fd ] .    * remark : * actually we have , under the condition @xmath583 , @xmath584@xmath585 we encountered a family of curves @xmath458 of genus 2 given as triple covers of @xmath453 . this is the case 3 in the following proposition . indeed , since the @xmath587 fold cyclic cover @xmath588 of @xmath453 branching at four points with indices @xmath589 has euler characteristic @xmath590 if we assume the genus of @xmath588 is two ( euler characteristic of @xmath588 is @xmath591 ) , we have @xmath592 it is easy to see that only three cases above are possible .    the three cases can be realized by the following families of curves : @xmath593 { \rm case\ 6:}&c_t^{(6 ) } : & s^6=s^2(1-s)^4(t - s)^3,&\quad t:{\rm \ parameter},\\[2 mm ] { \rm case\ 4:}&c_t^{(4 ) } : & s^4=s^2(1-s)^2(s - t),&\quad t:{\rm \ parameter}. \end{array}\ ] ] note that the double cover of the base space of case 6 branching at the two points of index 2 is equivalent to case 3 . aomoto k. and kita m. , translated by k. iohara , _ theory of hypergeometric functions _ , springer verlag , now york , 2011 . w. n. bailey , _ generalized hypergeometric series _ , cambridge , 1935 e. bod , algebraicity of the appell - lauricella and horn hypergeometric functions , j. diff . equations * 252 * ( 2012 ) , 541566 . k. cho , a generalization of kita and noumi s vanishing theorems of cohomology groups of local system , nagoya math . j. , * 147 * ( 1997 ) , 6369 . k. mimachi and t. sasaki , irreducibility and reducibility of lauricella s system of differential equations @xmath594 and the jordan - pochhammer differential equation @xmath595 , kyushu j. math . * 66*(2012 ) , 6187 . k. matsumoto , t. sasaki and m. yoshida , the monodromy of the period map of a @xmath596-parameter family of k3 surfaces and the hypergeometric function of type @xmath597 , intern . j. math . * 3*(1992 ) , 1164 . k. matsumoto , t. sasaki , n. takayama and m. yoshida , monodromy of the hypergeometric differential equation of type @xmath597 , ii the unitary reflection group of order @xmath598 , annali della scuola normale superiore di pisa ( 4 ) * 20 * ( 1993 ) , 617631 . k. matsumoto and t. terasoma , period maps of reducible hypergeometric equations and mixed hodge structures , in preparation . k. matsumoto and m. yoshida , monodromy of lauricella s hypergeometric @xmath599-system , ann . ( 5 ) , * 13 * ( 2014 ) , 551577 . takeuchi , k , commensurability classes of arithmetic triangle groups , j. fac . tokyo sect . ia math . 24(1977 ) , 201212 . m. yoshida , _ hypergeometric functions , my love _ , vieweg , 1997 ., in this paper we consider the numerical integration of autonomous stochastic differential delay equations ( sddes ) in the it s sense @xmath5 with initial data @xmath6 $ ] . here @xmath7 is a delay term satisfying @xmath8 and @xmath9 , @xmath10 . we assume that the initial data is independent of the wiener measure driving the equations and @xmath11 is an @xmath12-dimensional wiener process defined on the complete probability space @xmath13 with a filtration @xmath14 satisfying the usual conditions ( that is , it is increasing and right continuous while @xmath15 contains all @xmath16-null sets ) .    for a given constant stepsize @xmath17 , we propose a split - step backward euler ( ssbe ) method for sddes ( [ sddes1 ] ) as follows    @xmath18    @xmath19    where @xmath20 and for @xmath21 @xmath22 for arbitrary stepsize @xmath23 , @xmath24 denotes the approximation of @xmath25 at time @xmath26 . we remark that @xmath27 in ( [ y*n ] ) depends on how memory values are handled on non - grid points . generally there are two ways , the first is to use piecewise constant interpolation , corresponding to @xmath28 , and the second to use piecewise linear interpolation . in later development , we prefer to assume @xmath29 to cover both cases . also , we mention that the scheme ( [ ssbe1])-([ssbe2 ] ) here is quite different from the ssbe method in @xcite , which will be explained at the end of this section .    in ( [ ssbe1])-([ssbe2 ] ) , @xmath30 serves as an intermediate stage value , and in order to continue the process , we have to solve the implicit equation ( [ ssbe1 ] ) at every step to acquire @xmath30 . existence and uniqueness of solutions to the implicit equations ( [ ssbe1 ] ) will be discussed in section 4 . here , we always assume that numerical solution of ( [ ssbe1 ] ) exists uniquely . and one can easily check that @xmath31 is @xmath32-measurable . the key aim in this work is to propose a new ssbe method for sddes with variable delay and its convergence and stability in mean - square sense are investigated under a non - globally lipschitz condition . this situation has been investigated in @xcite for stochastic differential equations ( sdes ) without delay . for sdes with delay , most of previous work has been based on the more restrictive assumption that the coefficients @xmath33 satisfies global lipschitz and linear growth conditions , see , for example , @xcite . in @xcite , the authors showed that the numerical solution produced by euler - maruyama ( em ) method will converge to the true solution of the sddes under the local lipschitz condition . note that the proof of the convergence result in this paper is based on techniques used in @xcite . in @xcite , by interpreting the implicit method ssbe as the em applied to a modified sde the authors were able to get a strong convergence result . this paper , however , provides an alternative way to get the convergence result for ssbe . that is , by giving a direct continuous - time extension we accomplished the convergence proof for ssbe without considering the modified sddes . also , in deriving moment bounds of numerical solution , due to the delay term of our ssbe , i.e. , @xmath34 in ( [ ssbe1 ] ) , @xmath35 can not be explicitly dominated by @xmath24 as ( 3.25 ) in @xcite . starting with a recurrence of @xmath35 given by substituting ( [ ssbe2 ] ) into ( [ ssbe1 ] ) , we overcome this difficulty and obtained the desired moment bounds . note that a similar approach is adopted in the stability analysis . of course , the most important contribution of this work is to propose an improved ssbe method for sddes and to verify its excellent stability property . in @xcite , the authors proposed a ssbe method for a linear scalar sdde with constant lag and its convergence and stability are studied there . it is worth emphasizing that our proposed method is a modified version of ssbe in @xcite . the changes are in two aspects : firstly , we drop the stepsize restriction @xmath36 and allow for arbitrary stepsize @xmath17 ; secondly and most importantly , the scheme has been modified to a new one . to see this , the two methods are applied to a linear scalar sdde in section [ linear_ms ] . one can observe that the second terms of @xmath33 in the scheme in @xcite is the numerical solution @xmath37 ( see ( [ ssbez ] ) below ) . while the corresponding terms in our scheme is the intermediate stage value @xmath38 ( see ( [ ssbew ] ) below ) . note that the modifications of the method do not raise the strong order of the numerical solution , but they indeed improve the stability of the method greatly . in fact , it is shown below that our method can well replicate exponential mean - square stability of nonlinear test problem , including the linear test equation as a special case , without any restrictions on stepsize @xmath4 . the convergence and stability results of ssbe can be regarded as an extension of those in @xcite for sdes without delay to variable delay case . this unconditional stability property of ( [ ssbe1])-([ssbe2 ] ) demonstrates that the proposed method is promising and will definitely be effective in solving systems with stiffness in the drift term , where stability investigations are particularly important . this article is organized as follows . in next section , a general convergence result ( theorem [ ssbemain ] ) is established . in section 3 , a convergence result is derived under a one - sided lipschitz condition ( assumption [ olc ] ) . section 4 and 5 are devoted to exponential mean - square stability property of the method . numerical experiments are included in section 6 . throughout the paper , let @xmath39 denote both the euclidean norm in @xmath40 and the trace norm(f - norm ) in @xmath41 . as the standing hypotheses , we make the following assumption . [ lcmc ] the system ( [ sddes1 ] ) has a unique solution @xmath25 on @xmath42 $ ] . and the functions @xmath3 and @xmath0 are both locally lipschitz continuous in @xmath1 and @xmath2 , i.e. , there exists a constant @xmath43 such that @xmath44 for all @xmath45 with @xmath46 . moreover , we assume that @xcite    [ init ] @xmath47 is hlder continuous in mean - square with exponent 1/2 , that is @xmath48 and @xmath7 is a continuous function satisfying @xmath49    in the following convergence analysis , we find it convenient to use continuous - time approximation solution . hence we define continuous version @xmath50 as follows @xmath51 where @xmath52 . for @xmath53 we can write it in integral form as follows @xmath54 where @xmath55 it is not hard to verify that @xmath56 , that is , @xmath50 coincides with the discrete solutions at the grid - points .    in additional to the above two assumptions , we will need another one . [ anmb ] the exact solution @xmath25 and its continuous - time approximation solution @xmath50 have p - th moment bounds , that is , there exist constants @xmath57 such that @xmath58 \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\bar{y}(t)|^{p}\right ]   \leq a \label{mb0}.\ ] ]    now we state our convergence theorem here and give a sequence of lemmas that lead to a proof . [ ssbemain]under assumptions [ lcmc],[init],[anmb ] , if the implicit equation ( [ ssbe1 ] ) admits a unique solution , then the continuous - time approximate solution @xmath50 ( [ ce1 ] ) will converge to the true solution of ( [ sddes1 ] ) in the mean - square sense , i.e. , @xmath59    we need several lemmas to complete the proof of theorem [ ssbemain ] . first , we will define three stopping times @xmath60 where as usual @xmath61 is set as @xmath62 ( @xmath63 denotes the empty set ) . [ lem1 ] under assumption [ lcmc ] , [ init ] , there exist constants @xmath64 , @xmath65 such that for @xmath66 and @xmath67 @xmath68    _ proof . _ for @xmath66 , by definition of @xmath69 and @xmath70 , @xmath71 noticing that for @xmath72 @xmath73 with @xmath74 . using linear growth condition of @xmath75 and moment bounds in ( [ mb ] ) , we have appropriate constant @xmath64 so that @xmath76 as for estimate ( [ yd2 ] ) , there are four cases as to the location of @xmath77 and @xmath78 :    @xmath79 1 ) @xmath80 ,    @xmath79 2 ) @xmath81 ,    @xmath79 3 ) @xmath82 ,    @xmath79 4 ) @xmath83 . + noticing that the delay @xmath84 satisfies lipschitz condition ( [ initial2 ] ) , one sees that @xmath85 in the case 1 ) , combining hlder continuity of initial data ( [ initial1 ] ) and ( [ tau ] ) gives the desired assertion . in the case 2 ) , without loss of generality , we assume @xmath86 , @xmath87 . thus we have from ( [ ssbe1 ] ) and ( [ y*n2 ] ) that @xmath88 \nonumber \\ & & + \mu \sum_{k = j+1}^{i-1 } \left[h f(y^*_{k+1},\tilde{y}^*_{k+1})+ g(y^*_{k } , \tilde{y}^*_{k})\delta w_{k } \right ] , \label{yd5}\end{aligned}\ ] ] where as usual we define the second summation equals zero when @xmath89 . noticing from ( [ tau ] ) that @xmath90 , and combining local linear growth bound ( [ lf ] ) for @xmath91 , global linear growth condition for @xmath75 and moment bounds ( [ mb ] ) , we can derive from ( [ yd5 ] ) that @xmath92 in the case 3 ) and 4 ) , using an elementary inequality gives @xmath93 then combining this with results obtained in case 1 ) and 2 ) gives the required result , with @xmath65 a universal constant independent of @xmath4 . [ lem2 ] under assumption [ lcmc ] , [ init ] , for stepsize @xmath94 , there exists a constant @xmath95 such that @xmath96 \leq c_r h,\ ] ] with @xmath95 dependent on @xmath97 , but independent of @xmath4 . _ for simplicity , denote @xmath98 from ( [ sddes1 ] ) and ( [ ce2 ] ) , we have @xmath99 = \mathbb{e } \left[\sup_{0 \leq s \leq t}\left|\bar{y}(s\wedge \sigma_r)-x(s\wedge \sigma_r)\right|^2 \right ] \nonumber \\ & = & \mathbb{e } \left[\sup_{0 \leq s \leq t}\left|\int_0^{s\wedge \sigma_r}f(y^*(r),\tilde{y}^*(r))-f(x(r),x(r-\tau(r)))\mbox{d}r \right.\right.\nonumber \\ & & \left.\left.+ \int_0^{s\wedge \sigma_r}g(y^*(r),\tilde{y}^*(r))-g(x(r),x(r-\tau(r)))\mbox{d}w(r)\right|^2\right ] \nonumber \\ & \leq &   2 t \mathbb{e } \int_0^{t\wedge \sigma_r } \left|f(y^*(s),\tilde{y}^*(s))-f(x(s),x(s-\tau(s)))\right|^2 \mbox{d}s \allowdisplaybreaks \nonumber \\ & & + 2\mathbb{e}\left[\sup_{0 \leq s \leq t } \left|\int_0^{s\wedge \sigma_r}g(y^*(r),\tilde{y}^*(r))-g(x(r),x(r-\tau(r)))\mbox{d}w(r)\right|^2\right ] \allowdisplaybreaks \nonumber \\ & \leq & 2(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r }    \label{3.14}\end{aligned}\ ] ] where hlder s inequality and the burkholder - davis - gundy inequality were used again . using the elementary inequality @xmath100 , one computes from ( [ 3.14 ] ) that @xmath101 \nonumber \\ & \leq & 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r }    & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } |\tilde{y}^*(s)- \bar{y}(s-\tau(s))|^2 + |\bar{y}(s-\tau(s))-x(s-\tau(s))|^2 \mbox{d}s \allowdisplaybreaks \nonumber \\ & \leq & 8(t+4)l_r \int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|\bar{y}(r\wedge \sigma_r)-x(r\wedge \sigma_r)|^2 ] \mbox{d}s \nonumber \\ & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r }    \bar{y}(s)|^2 \mbox{d}s \nonumber \\ & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } |\tilde{y}^*(s)- \bar{y}(s-\tau(s))|^2 \mbox{d}s , \label{3.15}\end{aligned}\ ] ] where the fact was used that @xmath102 . by taking lemma [ lem1 ] into account , we derive from ( [ 3.15 ] ) that , with suitable constants @xmath103 @xmath104 & \leq & 8(t+4)l_r \int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|\bar{y}(r\wedge \sigma_r)-x(r\wedge \sigma_r)|^2 ] \mbox{d}s \nonumber \\ & & + 4(t+4)tl_rc_1(r)h + 4(t+4)tl_rc_2(r)h \nonumber \\ & = & \tilde{c}_r\int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|e(r \wedge \sigma_r)|^2 ] \mbox{d}s + \bar{c}_rh.\end{aligned}\ ] ] hence continuous gronwall inequality gives the assertion . _ proof of theorem [ ssbemain ] . _ armed with lemma [ lem2 ] and assumption [ anmb ] , the result may be proved using a similar approach to that in ( * ? ? ? * theorem 2.2 ) and ( * ? ? ? * theorem 2.1 ) , where under the local lipschitz condition they showed the strong convergence of the em method for the sodes and sddes , respectively .    under the global lipschitz condition and linear growth condition ( cf @xcite ) , we can choose uniform constants @xmath64 , @xmath105 in previous lemma [ lem1],[lem2 ] to be independent of @xmath97 . accordingly we can recover the strong order of 1/2 by deriving @xmath106 \leq c h,\ ] ] where @xmath107 is independent of @xmath97 and @xmath4 . in this section , we will give some sufficient conditions on equations ( [ sddes1 ] ) to promise a unique global solution of sddes and a well - defined solution of the ssbe method . we make the following assumptions on the sddes . [ olc ] the functions @xmath3 are continuously differentiable in both @xmath1 and @xmath2 , and there exist constants @xmath108 , such that @xmath109 @xmath110 @xmath111    the inequalities ( [ olc1]),([olc2 ] ) indicate that the first argument @xmath1 of @xmath91 satisfies one - sided lipschitz condition and the second satisfies global lipschitz condition . it is worth noticing that conditions of the same type as ( [ olc1 ] ) and ( [ olc2 ] ) have been exploited successfully in the analysis of numerical methods for deterministic delay differential equations ( ddes)(see @xcite and references therein ) . as for sdes without delay , the conditions ( [ olc1 ] ) and ( [ olc3 ] ) has been used in @xcite . we compute from ( [ olc1])-([olc3 ] ) that @xmath112 @xmath113 on choosing the constant @xmath114 as @xmath115 the following condition holds @xmath116    in what follows we always assume that for @xmath117 the initial data satisfies @xmath118    [ eu ] assume that assumption [ olc ] is fulfilled . then there exists a unique global solution @xmath25 to system ( [ sddes1 ] ) . morever , for any @xmath119 , there exists constant @xmath120 @xmath58 \leq c(1+\mathbb{e}\|\psi\|^p).\ ] ]    _ proof . _ see the appendix . [ mblem ] assume that @xmath33 satisfy the condition ( [ mc ] ) and @xmath67 is sufficiently small , then for @xmath121 the following moment bounds hold @xmath122 \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\tilde{y}^*(t)|^{2p}\right ] \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\bar{y}(t)|^{2p}\right ]   \leq a \label{mb}.\ ] ]    _ proof . _ inserting ( [ ssbe2 ] ) into ( [ ssbe1 ] ) gives @xmath123 hence @xmath124 expanding it and employing ( [ mc ] ) yields @xmath125 by definition of @xmath126 , one obtains @xmath127 . taking this inequality into consideration and letting @xmath128 , we have from ( [ 3.1 ] ) that @xmath129 denoting @xmath130 , one computes that @xmath131 by recursive calculation , we obtain @xmath132 raising both sides to the power @xmath133 gives @xmath134^p + \alpha^p n^{p-1 } \sum_{j=0}^{n-1 }    thus @xmath135^p + \alpha^p m^{p-1}\mathbb{e}\sum_{j=0}^{m-1}|g(y^*_j,\tilde{y}^*_j)\delta w_j|^{2p}\right \}.\label{y*n3}\end{aligned}\ ] ] here @xmath136 , where @xmath137 is the largest integer number such that @xmath138 . now , using the burkholder - davis - gundy inequality ( theorem 1.7.3 in @xcite ) gives @xmath139^p \leq c_p\mathbb{e}\left [ \sum_{j=0}^{m-1}|y^*_j|^2|g(y^*_j,\tilde{y}^*_j)|^2h\right]^{p/2 } \allowdisplaybreaks\nonumber\\ & & \leq c_p ( kh)^{p/2}m^{p/2 - 1}\mathbb{e}\left[\sum_{j=0}^{m-1 }    & & \leq \frac{1}{2}c_pk^{p/2}t^{p/2 - 1}h \mathbb{e}\left[\sum_{j=0}^{m-1 } \left(|y^*_j|^{2p } + 3^{p-1}(1+|y^*_j|^{2p}+|\tilde{y}^*_j|^{2p})\right)\right ] . \label{estimate1}\end{aligned}\ ] ] noticing that @xmath140 inserting it into ( [ estimate1 ] ) , we can find out appropriate constants @xmath141 such that @xmath142^p \nonumber \\ & \leq & \bar{c}h \sum_{j=0}^{m-1}\mathbb{e}\max_{0 \leq i \leq j } |y_i^*|^{2p } + \bar{c}(\mathbb{e}\|\psi\|^{2p}+1 ) . \label{estimate2}\end{aligned}\ ] ] at the same time , noting the fact @xmath143 and @xmath144 is independent of @xmath145 , one can compute that , with @xmath146 a constant that may change line by line @xmath147 \allowdisplaybreaks \nonumber \\ & \leq & \hat{c}h^{p-1}(\mathbb{e}\|\psi\|^{2p}+1 ) + \hat{c}h^p \sum_{j=0}^{m-1}\mathbb{e}\max_{0 \leq i \leq j } |y^*_i|^{2p}. \label{estimate3}\end{aligned}\ ] ] by definition ( [ ssbe1 ] ) , one sees that @xmath148 then using a similar approach used before , we can find out a constant @xmath149 to ensure that @xmath150 inserting ( [ estimate2]),([estimate3 ] ) into ( [ y*n3 ] ) and considering ( [ y*0 ] ) and @xmath67 , we have , with suitable constants @xmath151 @xmath152 thus using the discrete - type gronwall inequality , we derive from ( [ mbend ] ) that @xmath153 $ ] is bounded by a constant independent of @xmath137 . then by considering the elementary inequality @xmath154 , boundedness of @xmath155 $ ] is immediate .    to bound @xmath156 $ ] , we shall first bound @xmath157 $ ] . from ( [ ssbe2 ] ) , we have @xmath158 & \leq & 2^{2p-1}\left\ { \mathbb{e}\left[\sup_{0 \leq nh \leq t}|y^*_n|^{2p}\right ] + \mathbb{e}\left[\sup_{0 \leq nh \leq t}|g(y^*_n,\tilde{y}_n^*)\delta w_n|^{2p}\right]\right\ } \nonumber \\ & \leq & 2^{2p-1}\left\ { \mathbb{e}\left[\sup_{0 \leq nh \leq t}|y^*_n|^{2p}\right ] + \mathbb{e}\sum_{j=0}^{n}|g(y^*_j,\tilde{y}^*_j)\delta w_j|^{2p}\right\}. \nonumber\end{aligned}\ ] ] now ( [ estimate3 ] ) and bound of @xmath159 $ ] gives the bound of @xmath157 $ ] .    to bound @xmath156 $ ] , we denote by @xmath160 the integer for which @xmath161 . by definitions of ( [ ssbe1 ] ) and ( [ ce1 ] ) , for @xmath162 , @xmath163 where @xmath164 . thus @xmath165 & \leq & 2^{2p-1}\left\ { \gamma \mathbb{e}\left[\sup\limits_{0 \leq nh \leq t}|y^*_n|^{2p}\right]+(1-\gamma ) \mathbb{e}\left[\sup\limits_{0 \leq nh \leq t}|y_n|^{2p}\right]\right . \nonumber \\ & & \left.+\mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|g(y^*_{n_t},\tilde{y}^*_{n_t})\delta w_{n_t}(t)|^{2p}\right ] \right\}.\label{mb1}\end{aligned}\ ] ] using doob s martingale inequality ( * ? ? ? * theorem 1.3.8 ) , we derive that @xmath166 \leq \sum_{n=0}^{n}\mathbb{e}\left[\sup\limits_{0 \leq s \leq h}|g(y^*_{n},\tilde{y}^*_{n})\delta w_{n}(s)|^{2p}\right ] \nonumber \\ \leq \left(\frac{2p}{2p-1}\right)^{2p}\sum_{n=0}^{n}\mathbb{e } \left[|g(y^*_{n},\tilde{y}^*_{n})\delta w_{n}(h)|^{2p}\right].\label{ge}\end{aligned}\ ] ] thus the last term in ( [ mb1 ] ) is bounded by considering ( [ estimate3 ] ) and bounds of @xmath167 $ ] , @xmath168 $ ] . now boundedness of @xmath169 $ ] follows immediately . [ lem6 ] under assumption [ olc ] , if @xmath170 , the implicit equation in ( [ ssbe1 ] ) admits a unique solution . _ let @xmath171 , then the implicit equation ( [ ssbe1 ] ) takes the form as @xmath172 where at each step , @xmath173 are known . observing that @xmath174 the assertion follows immediately from theorem 14.2 of @xcite .    under assumption [ init],[olc ] , if @xmath175 , then the numerical solution produced by ( [ ssbe1])-([ssbe2 ] ) is well - defined and will converge to the true solution in the mean - square sense , i.e. , @xmath59    _ proof . _ noticing that assumption [ olc ] implies assumptions [ lcmc],[anmb ] by theorem [ eu ] and lemma [ mblem ] , and taking lemma [ lem6 ] into consideration , the result follows directly from theorem [ ssbemain ] . we remark that the problem class satisfying condition ( [ initial2 ] ) includes plenty of important models . in particular , stochastic pantograph differential equations ( see , e.g. , @xcite ) with @xmath176 and sddes with constant lag fall into this class and therefore corresponding convergence results follow immediately . in this section , we will investigate how ssbe shares exponential mean - square stability of general nonlinear systems . in deterministic case , nonlinear stability analysis of numerical methods are carried on under a one - sided lipschitz condition . this phenomenon has been well studied in the deterministic case ( @xcite and references therein ) and stochastic case without delay @xcite . in what follows , we choose the test problem satisfying conditions ( [ olc1])-([olc3 ] ) . moreover , we assume that variable delay is bounded , that is , there exists @xmath177 , for @xmath178 @xmath179 we remark that this assumption does not impose additional restrictions on the stepsize @xmath4 and admits arbitrary large @xmath4 on choosing @xmath180 and @xmath181 close to 1 . to begin with , we shall first give a sufficient condition for exponential mean - square stability of analytical solution to underlying problem . [ ems1 ] under the conditions ( [ olc1]),([olc2]),([olc3 ] ) and ( [ bd ] ) , and with @xmath182 obeying @xmath183 any two solutions @xmath184 and @xmath185 with @xmath186 and @xmath187 satisfy @xmath188 where @xmath189 $ ] is the zero of @xmath190 , with @xmath191 . _ by it formula , we have @xmath192 } \mathbb{e}|x(r)-y(r)|^2 \mbox{d}s.\end{aligned}\ ] ] letting @xmath193 and noticing that @xmath194 exists for @xmath195 and is continuous , we derive from ( [ 5.12 ] ) that @xmath196}u(s),\ ] ] where the upper dini derivative @xmath197 is defined as @xmath198 using theorem 7 in @xcite leads to the desired result . based on this stability result , we are going to investigate stability of the numerical method . [ ems3 ] under the conditions ( [ olc1]),([olc2]),([olc3 ] ) and ( [ bd ] ) , if @xmath199 , then for all @xmath23 , any two solutions @xmath200 produced by ssbe ( [ ssbe1])-([ssbe2 ] ) with @xmath186 and @xmath187 satisfy @xmath201 where @xmath202 is defined as @xmath203    _ proof . _ under @xmath199 , the first part is an immediate result from lemma [ lem6 ] . for the second part , in order to state conveniently , we introduce some notations @xmath204 from ( [ y*n2 ] ) , we have @xmath205 thus @xmath206 taking expectation and using ( [ olc3 ] ) yields @xmath207 now using the cauchy - schwarz inequality and conditions ( [ olc1])-([olc2 ] ) , we have @xmath208 inserting it into ( [ 5.3 ] ) gives @xmath209 here we have to consider which approach is chosen to treat memory values on non - grid points , piecewise constant interpolation ( @xmath210 ) or piecewise linear interpolation . in the latter case , let us consider two possible cases :    @xmath79 if @xmath211 , then @xmath212 inserting ( [ 5.10 ] ) , we derive from ( [ 5.4 ] ) that @xmath213\mathbb{e}|x_n^*-y_n^*|^2 \\ & \leq(1 + h\gamma_3+\tilde{\mu } h \gamma_2)\mathbb{e}|x_{n-1}^*-y_{n-1}^*|^2 + h\gamma_4 \mathbb{e}|\tilde{x}_{n-1}^*-\tilde{y}_{n-1}^*|^2 . \end{split}\ ] ] hence using the fact @xmath199 in ( [ beta ] ) gives @xmath214    @xmath79 if @xmath215 , it follows from ( [ 5.4 ] ) and @xmath216 that @xmath217 therefore , it is always true that inequality ( [ 5.6 ] ) holds for piecewise linear interpolation case . obviously ( [ 5.6 ] ) also stands in piecewise constant interpolation case . further , from ( [ ssbe1 ] ) one sees @xmath218 using a similar approach as before , one can derive @xmath219 denote @xmath220 noticing that @xmath199 , one can readily derive @xmath221 , we can deduce from ( [ 5.6 ] ) and ( [ 5.11 ] ) that @xmath222 here @xmath223 denotes the greatest integer less than or equal to @xmath1 . + finally from ( [ ssbe2 ] ) , we have for large @xmath224 such that @xmath225 @xmath226 where @xmath227 is defined as in ( [ nuh2 ] ) . the stability result indicates that the method ( [ ssbe1])-([ssbe2 ] ) can well reproduce long - time stability of the continuous system satisfying conditions stated in theorem [ ems1 ] . note that the exponential mean - square stability under non - global lipschitz conditions has been studied in @xcite in the case of nonlinear sdes without delay . the preceding results can be regarded as an extension of those in @xcite to delay case . although the main focus of this work is on nonlinear sddes , in this section we show that the ssbe ( [ ssbe1])-([ssbe2 ] ) has a very desirable linear stability property . hence , we consider the scalar , linear test equation @xcite given by @xmath228 note that ( [ lineartest ] ) is a special case of ( [ sddes1 ] ) with @xmath229 , and satisfies conditions ( [ olc1])-([olc3 ] ) with @xmath230 by theorem [ ems1 ] , ( [ lineartest ] ) is mean - square stable if @xmath231 for constraint stepsize @xmath232 , i.e. , @xmath233 in ( [ bd ] ) , the ssbe proposed in our work applied to ( [ lineartest ] ) produces @xmath234 , \\       y_{n+1 } & =   y_n^ * + [ cy_n^ * + d y_{n-\kappa}^*]\delta w_n . \end{array } \right.\end{aligned}\ ] ] in @xcite , the authors constructed a different ssbe for the linear test equation ( [ lineartest ] ) and their method applied to ( [ lineartest ] ) reads @xmath235 , \\ z_{n+1 } & =   z_n^ * + [ cz_n^ * + d z_{n-\kappa+1}]\delta w_n . \end{array } \right.\end{aligned}\ ] ] the stability results there ( * ? ? ? * theorem 4.1 ) indicate that under ( [ linearms ] ) the method ( [ ssbez ] ) can only preserve mean - square stability of ( [ lineartest ] ) with stepsize restrictions , but the new scheme ( [ ssbew ] ) exhibits a better stability property .    for the linear equation ( [ lineartest ] ) , if ( [ linearms ] ) holds , then the ssbe ( [ ssbew ] ) is mean - square stable for any stepsize @xmath236 . _ the assertion readily follows from theorem [ ems3 ] . apparently , the ssbe ( [ ssbew ] ) achieves an advantage over ( [ ssbez ] ) in stability property that the ssbe ( [ ssbew ] ) is able to inherit stability of ( [ lineartest ] ) for any stepsize @xmath236 . if one drops the stepsize restriction @xmath36 and allow for arbitrary stepsize @xmath17 , one can arrive at a sharper stability result from theorem [ ems3 ] .    for the linear equation ( [ lineartest ] ) , if ( [ linearms ] ) holds , then the ssbe([ssbe1])-([ssbe2 ] ) is mean - square stable for any stepsize @xmath237 . in this section we give several numerical examples to illustrate intuitively the strong convergence and the mean - square stability obtained in previous sections . the first test equation is a linear it sdde @xmath238 . \end{array } \right.\end{aligned}\ ] ] denoting @xmath239 as the numerical approximation to @xmath240 at end point @xmath241 in the @xmath242-th simulation of all @xmath243 simulations , we approximate means of absolute errors @xmath244 as @xmath245 in our experiments , we use the ssbe ( [ ssbew ] ) to compute an `` exact solution '' with small stepsize @xmath246 and @xmath247 . we choose two sets of parameters as follows    @xmath79 example i : @xmath248    @xmath79 example ii : @xmath249    @xmath79 example iii : @xmath250     with @xmath251 versus @xmath252 for example i ( left ) and example ii ( right).,title="fig:",width=240,height=172 ]   with @xmath251 versus @xmath252 for example i ( left ) and example ii ( right).,title="fig:",width=240,height=172 ]    16cm@ccccccc + & & example ii & & & example iii + ( r)2 - 4 ( l)5 - 7 @xmath4 & em & ssbe ( [ ssbez ] ) & ssbe ( [ ssbew ] ) & em & ssbe ( [ ssbez ] ) & ssbe ( [ ssbew ] ) +   + @xmath253 & 0.0008 & 0.0011 & 0.0008 & 0.0014 & 0.0020 & 0.0014 + @xmath254 & 0.0013 & 0.0016 & 0.0013&0.0025 & 0.0036 & 0.0023 + @xmath255 & 0.0021 & 0.0029 & 0.0019 & 0.0058 & 0.0070 & 0.0035 + @xmath256 & 0.0034 & 0.0058 & 0.0027 & 0.2744 & 0.0157 & 0.0053 + @xmath257 & 0.0086 & 0.0148 & 0.0038 & 6.1598e+010 & 0.0628 & 0.0078 +    in figure [ 1 ] , computational errors @xmath244 versus stepsize @xmath4 on a log - log scale are plotted and dashed lines of slope one half are added . one can clearly see that ssbe ( [ ssbew ] ) for linear test equation ( [ lssde ] ) is convergent and has strong order of 1/2 . in table [ table1 ] , computational errors @xmath244 with @xmath258 are presented for the well - known euler - maruyama method @xcite , the ssbe method ( [ ssbez ] ) and the improved ssbe method ( [ ssbew ] ) in this paper . there one can find that the improved ssbe method ( [ ssbew ] ) has the best accuracy among the three methods . in particular , for example iii with stiffness in drift term ( i.e. , @xmath259 ) , when the moderate stepsize @xmath260 was used , the euler - maruyama method becomes unstable and the two ssbe methods still remain stable , but with the improved ssbe ( [ ssbew ] ) producing better result . to compare stability property of the improved ssbe and ssbe in @xcite , simulations by ssbe ( [ ssbew ] ) and ( [ ssbez ] ) are both depicted in figure [ 2 ] , [ 3 ] . there solutions produced by ( [ ssbew ] ) and ( [ ssbez ] ) are plotted in solid line and dashed line , respectively . as is shown in the figures , methods ( [ ssbew ] ) and ( [ ssbez ] ) exhibit different stability behavior . one can observe from figure [ 2 ] that ( [ ssbew ] ) for example ii is mean - square stable for @xmath261 . but ( [ ssbez ] ) is unstable for @xmath262 . for example iii , the improved ssbe ( [ ssbew ] ) is always stable for @xmath263 , but ( [ ssbez ] ) becomes stable when the stepsize @xmath4 decreases to @xmath264 . the numerical results demonstrate that the scheme ( [ ssbew ] ) has a greater advantage in mean - square stability than ( [ ssbez ] ) .    ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ]    ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ]      consider a nonlinear sdde with a time - varying delay as follows @xmath273 dt + \left [ x(t)+x(t-\tau(t))\right ] dw(t ) , t > 0 , \\ x(t ) =   1 , \quad t \in [ -1,0 ] ,      \end{array } \right.\end{aligned}\ ] ] where @xmath274 . obviously , equation ( [ nlsdde ] ) satisfies conditions ( [ olc1])-([olc3 ] ) in assumption [ olc ] , with @xmath275 . thus @xmath276 and the problem is exponentially mean - square stable . as is shown in figure [ 4 ] , the ssbe ( [ ssbew ] ) can well reproduce stability for quite large stepsize @xmath277 . this is consistent with our result established in theorem [ ems3 ] .    ) by ssbe ( [ ssbew ] ) using various stepsizes.,width=288,height=240 ] _ proof of theorem [ eu ] . _ since both @xmath91 and @xmath75 are locally lipschitz continuous , theorem 3.2.2 of @xcite shows that there is a unique maximal local solution @xmath25 on @xmath278 , where the stopping time @xmath279 . by it s formula we obtain that for @xmath280 @xmath281 where the condition ( [ mc ] ) was used . thus @xmath282 now , raising both sides of ( [ ito2 ] ) to the power @xmath283 and using hlder s inequality yield @xmath284 by the burkholder - davis - gundy inequality @xcite , one computes that , with @xmath285 , @xmath286 & & \leq c_1\left\{1+\mathbb{e}\|\psi\|^p + \int_0^t \mathbb{e}\sup_{0 \leq r \leq s}|x(r\wedge\rho_r)|^p\mbox{d}s\right . \nonumber \\ & & \left . + \mathbb{e}\left[\int_0^{t\wedge\rho_r }    \label{ito5}\end{aligned}\ ] ] next , by an elementary inequality , @xmath287 + \frac{c_1}{2}t^{p/2 - 1}\mathbb{e}\int_0^{t\wedge\rho_r}|g(x(s),x(s-\tau(s)))|^p \mbox{d}s \nonumber \\ & \leq & \frac{1}{2c_1}\mathbb{e}\left[\sup\limits_{0 \leq s \leq t}|x(s\wedge\rho_r)|^{p}\right]+\frac{c_1}{2}(3t)^{p/2 - 1}k^{p/2}\int_0^t ( 1 + \mathbb{e}\sup_{0 \leq r \leq s}|x(r\wedge\rho_r)|^p+\mathbb{e}\|\psi\|^p ) \mbox{d}s . \nonumber\end{aligned}\ ] ] inserting it into ( [ ito5 ] ) , for proper constants @xmath288 we have that @xmath289 the gronwall inequality gives @xmath290 this implies @xmath291 letting @xmath292 leads to @xmath293 since @xmath294 is arbitrary , we must have @xmath295 a.s . and hence @xmath296 a.s . the existence and uniqueness of the global solution is justified . finally , the desired moment bound follows from ( [ ito6 ] ) by letting @xmath292 and setting @xmath297 . baker , e. buckwar , exponential stability in pth mean of solutions , and of convergent euler - type solutions , to stochastic delay differential equations , j. comput . , 184 ( 2 ) ( 2005 ) , pp.404 - 427 . k.burrage , p.m.burrage , t.tian , numerical methods for strong solutions of stochastic differential equations : an overview , proceedings : mathematical , physical and engineering , royal society of london 460(2004 ) , pp.373 - 402 . y.hu , semi - implicit euler - maruyama scheme for stiff stochastic equations , in stochastic analysis and related topics v : the silvri workshop , progr . 38 , h. koerezlioglu , ed . , birkhauser , boston , 1996 , pp.183 - 202 . a.jentzen , p.e.kloeden , a.neuenkirch , pathwise approximation of stochastic differential equations on domains : higher order convergence rates without global lipschitz coefficients , numer . 112 , 1 ( 2009 ) , 41 - 64 ., transport of molecules across cell membranes is an essential mechanism for life processes . these molecules are often long and flexible , and the pores in the membranes are too narrow to allow them to pass through as a single unit . in such circumstances , the passage of a molecule through the pore  i.e.  its translocation  proceeds through a random process in which polymer segments sequentially move through the pore . dna , rna and proteins are naturally occurring long molecules @xcite subject to translocation in a variety of biological processes . translocation is used in gene therapy @xcite , in delivery of drug molecules to their activation sites @xcite , and as an efficient means of single molecule sequencing of dna and rna @xcite . understandably , the process of translocation has been an active topic of current research : both because it is an essential ingredient in many biological processes and for its relevance in practical applications . translocation is a complicated process in living organisms  its dynamics may be strongly influenced by various factors , such as the presence of chaperon molecules , ph value , chemical potential gradients , and assisting molecular motors . it has been studied empirically in great variety in the biological literature @xcite . studies of translocation as a _ biophysical _ process are more recent . in these , the polymer is simplified to a sequentially connected string of @xmath0 monomers . quantities of interest are the typical time scale for the polymer to leave a confining cell or vesicle , the `` escape time '' @xcite , and the typical time scale the polymer spends in the pore or `` dwell time '' , @xcite as a function of chain length @xmath0 and other parameters like membrane thickness , membrane adsorption , electrochemical potential gradient , etc . @xcite . ( large compared to the polymer s typical size ) . the polymer repeatedly moves back and forth from one cell to the other through the pore . at any time , exactly one monomer can be within the pore . the kuhn length of the polymer and the lattice spacing are also set to unity . polymers can be in three different states ( i ) state a : all monomers are in cell a ; ( ii ) state t ( threaded ) : some monomers are in cell a and the rest in cell b ; ( iii ) state b : all monomers are in cell b. the dwell time @xmath7 is defined as the pore - blockade time in experiments , i.e. , as how long the polymer spends in state t during a translocation event.[fig1 ] ]    these have been measured directly in numerous experiments @xcite . experimentally , the most studied quantity is the dwell time @xmath5 , i.e. , the pore blockade time for a translocation event . for theoretical descriptions of @xmath5 , during the last decade a number of mean - field type theories @xcite have been proposed , in which translocation is described by a fokker - planck equation for first - passage over an entropic barrier in terms of a single `` reaction coordinate '' @xmath8 . here @xmath8 is the number of the monomer threaded at the pore ( @xmath9 ) . these theories apply under the assumption that translocation is slower than the equilibration time - scale of the entire polymer , which is likely for high pore friction . in ref . @xcite , this assumption was questioned , and the authors found that for a self - avoiding polymer performing rouse dynamics , @xmath10 , the rouse time . using simulation data in 2d , they suggested that the inequality may actually be an equality , i.e. , @xmath11 . this suggestion was numerically confirmed in 2d in ref . however , in a publication due to two of us , @xmath5 in 3d was numerically found to scale as @xmath12 @xcite . additionally , in a recent publication @xcite @xmath5 was numerically found to scale as @xmath13 in three dimensions [ a discussion on the theory of ref . @xcite appears at the end of sec . [ sec3 ] ] .    amid all the above results on @xmath5 mutually differing by @xmath14 , the only consensus that survives is that @xmath10 @xcite . simulation results alone can not determine the scaling of @xmath5 : different groups use different polymer models with widely different criteria for convergence for scaling results , and as a consequence , settling differences of @xmath14 in @xmath15 , is extremely delicate . an alternative approach that can potentially settle the issue of @xmath5 scaling with @xmath0 is to analyze the dynamics of translocation at a microscopic level . indeed , the lower limit @xmath16 for @xmath5 implies that the dynamics of translocation is anomalous @xcite . we know of only two published studies on the anomalous dynamics of translocation , both using a fractional fokker - planck equation ( ffpe ) @xcite . however , whether the assumptions underlying a ffpe apply for polymer translocation are not clear . additionally , none of the studies used ffpe for the purpose of determining the scaling of @xmath5 . in view of the above , such a potential clearly has not been thoroughly exploited . the purpose of this paper is to report the characteristics of the anomalous dynamics of translocation , _ derived from the microscopic dynamics of the polymer _ , and the scaling of @xmath5 obtained therefrom . translocation proceeds via the exchange of monomers through the pore : imagine a situation when a monomer from the left of the membrane translocates to the right . this process increases the monomer density in the right neighbourhood of the pore , and simultaneously reduces the monomer density in the left neighbourhood of the pore . the local enhancement in the monomer density on the right of the pore _ takes a finite time to dissipate away from the membrane along the backbone of the polymer _ ( similarly for replenishing monomer density on the left neighbourhood of the pore ) . the imbalance in the monomer densities between the two local neighbourhoods of the pore during this time implies that there is an enhanced chance of the translocated monomer to return to the left of the membrane , thereby giving rise to _ memory effects _ , and consequently , rendering the translocation dynamics subdiffusive . more quantitatively , the excess monomer density ( or the lack of it ) in the vicinity of the pore manifests itself in reduced ( or increased ) chain tension around the pore , creating an imbalance of chain tension across the pore ( we note here that the chain tension at the pore acts as monomeric chemical potential , and from now on we use both terms interchangeably ) . we use well - known laws of polymer physics to show that in time the imbalance in the chain tension across the pore relaxes as @xmath4 @xcite . this results in translocation dynamics being subdiffusive for @xmath17 , with the mean - square displacement @xmath18 of the reaction co - ordinate @xmath19 increasing as @xmath2 ; and diffusive for @xmath20 . with @xmath21 , this leads to @xmath22 . this paper is divided in four sections . in sec . [ sec1 ] we detail the dynamics of our polymer model , and outline its implications on physical processes including equilibration of phantom and self - avoiding polymers . in sec . [ sec2 ] we elaborate on a novel way of measuring the dwell time that allows us to obtain better statistics for large values of @xmath0 . in sec . [ sec3 ] we describe and characterize the anomalous dynamics of translocation , obtain the scaling of @xmath7 with @xmath0 and compare our results with that of ref . @xcite . in sec . [ sec4 ] we end this paper with a discussion .    ] over the last years , we have developed a highly efficient simulation approach of polymer dynamics . this approach is made possible via a lattice polymer model , based on rubinstein s repton model @xcite for a single reptating polymer , with the addition of sideways moves ( rouse dynamics ) and entanglement . a detailed description of this lattice polymer model , its computationally efficient implementation and a study of some of its properties and applications can be found in refs . @xcite .    in this model , polymers consist of a sequential chain of monomers , living on a fcc lattice . monomers adjacent in the string are located either in the same , or in neighboring lattice sites . the polymers are self - avoiding : multiple occupation of lattice sites is not allowed , except for a set of adjacent monomers . the polymers move through a sequence of random single - monomer hops to neighboring lattice sites . these hops can be along the contour of the polymer , thus explicitly providing reptation dynamics . they can also change the contour `` sideways '' , providing rouse dynamics . time in this polymer model is measured in terms of the number of attempted reptation moves . a two - dimensional version of our three - dimensional model is illustrated in fig . [ fig2 ] .      from our experience with the model we already know that the dynamical properties are rather insensitive to the ratio of the rates for rouse vs. reptation moves ( i.e. , moves that alter the polymer contour vs. moves that only redistribute the stored length along the backbone ) . since the computational efficiency of the latter kind of moves is at least an order of magnitude higher , we exploit this relative insensitivity by attempting reptation moves @xmath23 times more often than rouse moves ; typical values are @xmath24 , @xmath25 or 10 , which correspond to a comparable amount of computational effort invested in both kinds of moves . certainly , the interplay between the two kinds of moves is rather intricate @xcite . recent work by drzewinski and van leeuwen on a related lattice polymer model @xcite provides evidence that the dynamics is governed by @xmath26 , supporting our experience that , provided the polymers are sizable , one can boost one mechanism over the other quite a bit ( even up to @xmath27 ) before the polymer dynamics changes qualitatively .        in order to further check the trustworthiness of this model , we use it to study the equilibration properties of polymers with one end tethered to a fixed infinite wall ( this problem relates rather directly to that of a translocating polymer : for a given monomer threaded into the pore , the two segments of the polymer on two sides of the membrane behave simply as two independent polymer chains ; see fig . [ fig1 ] ) . this particular problem , wherein polymer chains ( of length @xmath0 ) undergo pure rouse dynamics ( i.e. , no additional reptation moves ) is a well - studied one : the equilibration time is known to scale as @xmath28 for self - avoiding polymers and as @xmath29 for phantom polymers . to reproduce these results with our model we denote the vector distance of the free end of the polymer w.r.t . the tethered end at time @xmath30 by @xmath31 , and define the correlation coefficient for the end - to - end vector as @xmath32 the angular brackets in eq . ( [ eq1 ] ) denote averaging in equilibrium . the @xmath33 quantities appearing in fig . [ fig3 ] have been obtained by the following procedure : we first obtain the time correlation coefficients @xmath34 for @xmath35 independent polymers , and @xmath33 is a further arithmatic mean of the corresponding @xmath35 different time correlation coefficients . for different values of @xmath0 we measure @xmath33 for both self - avoiding and phantom polymers . when we scale the units of time by factors of @xmath36 and @xmath37 respectively for phantom and self - avoiding polymers , the @xmath33 vs. @xmath30 curves collapse on top of each other . this is shown in fig . note here that @xmath38 , which is sufficiently close to @xmath39 , and in simulations of self - avoiding polymers [ fig . [ fig2](b ) ] we can not differentiate between @xmath40 and @xmath39 . our translocation simulations are carried out only for self - avoiding polymers . for long polymers , full translocation simulations , i.e. , having started in one of the cells ( fig . [ fig1 ] ) , finding the pore and distorting their shapes to enter the pore to finally translocate to the other side of the membrane are usually very slow . an uncoiled state , which the polymer has to get into in order to pass through the pore is entropically unfavourable , and therefore , obtaining good translocation statistics for translocation events is a time - consuming process . to overcome this difficulty , in ref . @xcite , we introduced three different time scales associated with translocation events : translocation time , dwell time and unthreading time . for the rest of this section , we refer the reader to fig . [ fig1 ] .      in states a and b ( fig . [ fig1 ] ) , the entire polymer is located in cell a , resp . b. states m and @xmath41 are defined as the states in which the middle monomer is located exactly halfway between both cells . finally , states t and @xmath42 are the complementary to the previous states : the polymer is threaded , but the middle monomer is not in the middle of the pore . the finer distinction between states m and t , resp . @xmath41 and @xmath42 is that in the first case , the polymer is on its way from state a to b or vice versa , while in the second case it originates in state a or b and returns to the same state . the translocation process in our simulations can then be characterized by the sequence of these states in time ( fig . [ fig4 ] ) . in this formulation , the dwell time @xmath7 is the time that the polymer spends in states t , while the translocation time @xmath43 is the time starting at the first instant the polymer reaches state a after leaving state b , until it reaches state b again . as found in ref . @xcite , @xmath7 and @xmath43 are related to each other by the relation @xmath44 where @xmath45 , @xmath46 and @xmath47 is the volume of cell a or b ( see fig . [ fig1 ] ) . the unthreading time @xmath48 is the average time that either state a or b is reached from state m ( not excluding possible recurrences of state m ) . notice that in the absence of a driving field , on average , the times to unthread to state a equals the the times to unthread to state b , due to symmetry . the advantage of introducing unthreading time is that when one measures the unthreading times , the polymer is at the state of lowest entropy at the start of the simulation and therefore simulations are fast and one can obtain good statistics on unthreading times fairly quickly . additionally , the dwell and unthreading times are related to each other , as outlined below , and using this relation one is able to reach large values of @xmath0 for obtaining the scaling of the dwell time .    the main point to note that the dwell time can be decomposed into three parts as @xmath49 whereas mean unthreading time can be decomposed into two parts as @xmath50 here @xmath51 , @xmath52 and @xmath53 respectively are the mean first passage time to reach state m from state a , mean time between the first occurance at state m and the last occurance of state m with possible reoccurances of state m , and the mean first passage time to reach state b from state m without any reoccurance of state m. since on average @xmath54 due to symmetry , and all quantities on the r.h.s . of eqs . ( [ eq2 ] ) and ( [ eq3 ] ) are strictly positive , we arrive at the inequality @xmath55 since eq . ( [ eq4 ] ) is independent of polymer lengths , on average , the dwell time scales with @xmath0 in the same way as the unthreading time @xcite . the reaction coordinate [ the monomer number @xmath56 , which is occupying the pore at time @xmath57 a convenient choice for the description of the microscopic movements of the translocating polymer , since the important time - scales for translocation can be obtained from the time evolution of the reaction coordinate @xmath56 as shown below . to delve deeper into its temporal behavior , we determine @xmath58 , the probability distribution that at time @xmath30 a polymer of length @xmath0 is in a configuration for which monomer @xmath59 is threaded into the pore , and the polymer evolves at time @xmath60 into a configuration in which monomer @xmath61 is threaded into the pore . the subscript @xmath62 denotes our parametrization for the polymer movement , as it determines the frequency of attempted reptation moves and the sideways ( or rouse ) moves of the polymer . [ sec1.1 ] for details .    to maintain consistency , all simulation results reported here in this section are for @xmath63 , so we drop @xmath23 from all notations from here on . this value for @xmath23 is used in view of our experience with the polymer code : @xmath63 yields the fastest runtime of our code ; this was the same value used in our earlier work @xcite . as discussed in sec . [ sec1.1 ] , this value of @xmath23 does not change any physics . first , we investigate the shape of these probability distributions for various values of @xmath59 , @xmath61 and @xmath30 , for different sets of @xmath0 and @xmath62 values . we find that as long as the @xmath30 values are such that neither @xmath59 nor @xmath61 are too close to the end of the polymer , @xmath64 depends only on @xmath65 , but the centre of the distribution is slightly shifted w.r.t . the starting position @xmath59 by some distance that depends on @xmath59 , @xmath0 and @xmath23 . this is illustrated in fig . [ fig5 ] , where ( on top of each other ) we plot @xmath64 for @xmath66 and @xmath67 at @xmath68 time units , for @xmath69 , as well as the gaussian distribution . notice in fig . [ fig5 ] that the distribution @xmath64 differs slightly from gaussian ( the parameter for the gaussian is calculated by least - square optimization ) . we find that this difference decreases with increasing values of @xmath30 ( not shown in this paper ) . we now define the mean @xmath70\rangle(s_1,t)$ ] and the variance @xmath71 of the distribution @xmath64 as @xmath72\rangle(s_1,t)\,=\,\int ds_2\,p_{n}(s_1,s_2,t)\,(s_2-s_1 ) ; \nonumber \\ & & \hspace{-7.85cm}\langle\delta s^2(s_1,t)\rangle\,=\,\int ds_2\,p_{n}(s_1,s_2,t)\times\nonumber\\ & & \hspace{-5.8cm}\{(s_2-s_1)^2-\,[\langle s_2-s_1\rangle(s_1,t)]^2\}\ , , \label{eq5}\end{aligned}\ ] ] where the quantities within parenthesis on the l.h.s . ( [ eq5 ] ) denote the functional dependencies of the mean and the standard deviation of @xmath65 . we also note here that we have checked the skewness of @xmath64 , which we found to be zero within our numerical abilities , indicating that @xmath64 is symmetric in @xmath65 .     for @xmath69 , at @xmath68 . note the data collapse when plotted as a function of @xmath65 . the distribution differs slightly from gaussian . [ fig5 ] ]    in principle , both the mean and the standard deviation of @xmath65 can be used to obtain the scaling of @xmath7 with @xmath0 , but the advantage of using @xmath71 for this purpose is that , as shown in fig . [ fig5 ] , it is independent of @xmath59 , so from now on , we drop @xmath59 from its argument . since unthreading process starts at @xmath73 , the scaling of @xmath7 with @xmath0 is easily obtained by using the relation @xmath74 in combination with the fact that the scalings of @xmath7 and @xmath48 with @xmath0 behave in the same way [ inequality ( [ eq4 ] ) ] . note here that eq . ( [ eq6 ] ) uses the fact that for an unthreading process the polymer only has to travel a length @xmath75 along its contour in order to leave the pore .      the key step in quantitatively formulating the anomalous dynamics of translocation is the following observation : a translocating polymer comprises of _ two polymer segments tethered at opposite ends of the pore _ that are able to exchange monomers between them through the pore ; so _ each acts as a reservoir of monomers for the other . _ the velocity of translocation @xmath76 , representing monomer current , responds to @xmath77 , the imbalance in the monomeric chemical potential across the pore acting as `` voltage '' . simultaneously , @xmath77 also adjusts in response to @xmath78 . in the presence of memory effects , they are related to each other by @xmath79 via the memory kernel @xmath80 , which can be thought of as the ( time - dependent ) ` impedance ' of the system . supposing a zero - current equilibrium condition at time @xmath81 , this relation can be inverted to obtain @xmath82 , where @xmath83 can be thought of as the ` admittance ' . in the laplace transform language , @xmath84 , where @xmath85 is the laplace variable representing inverse time . via the fluctuation - dissipation theorem , they are related to the respective autocorrelation functions as @xmath86 and @xmath87 .     following monomer injection at the pore corresponding to @xmath88 , with @xmath89 . see text for details . red circles : @xmath90 , blue circles : @xmath91 , green circles : @xmath92 , solid black line : @xmath93 with @xmath94 for self - avoiding polymers . the steeper drop at large times correspond to the exponential decay @xmath95 ( we have checked this , but have not shown in this letter ) . [ fig6 ] ]    the behaviour of @xmath80 may be obtained by considering the polymer segment on one side of the membrane only , say the right , with a sudden introduction of @xmath96 extra monomers at the pore , corresponding to impulse current @xmath88 . we then ask for the time - evolution of the mean response @xmath97 , where @xmath98 is the shift in chemical potential for the right segment of the polymer at the pore . this means that for the translocation problem ( with both right and left segments ) , we would have @xmath99 , where @xmath100 is the shift in chemical potential for the left segment at the pore due to an opposite input current to it . we now argue that this mean response , and hence @xmath80 , takes the form @xmath101 . the terminal exponential decay @xmath95 is expected from the relaxation dynamics of the entire right segment of the polymer with one end tethered at the pore [ see fig . [ fig3](b ) ] . to understand the physics behind the exponent @xmath102 , we use the well - established result for the relaxation time @xmath103 for @xmath104 self - avoiding rouse monomers scaling as @xmath105 . based on the expression of @xmath103 , we anticipate that by time @xmath30 the extra monomers will be well equilibrated across the inner part of the chain up to @xmath106 monomers from the pore , but not significantly further . this internally equilibrated section of @xmath107 monomers extends only @xmath108 , less than its equilibrated value @xmath109 , because the larger scale conformation has yet to adjust : the corresponding compressive force from these @xmath107 monomers is expected by standard polymer scaling @xcite to follow @xmath110\sim t^{-(1+\nu)/(1 + 2\nu)}$ ] . this force @xmath111 must be transmitted to the membrane , through a combination of decreased tension at the pore and increased incidence of other membrane contacts . the fraction borne by reducing chain tension at the pore leads us to the inequality @xmath112 , which is significantly different from ( but compatible with ) the value @xmath113 required to obtain @xmath114 . it seems unlikely that the adjustment at the membrane should be disproportionately distributed between the chain tension at the pore and other membrane contacts , leading to the expectation that the inequality above is actually an equality . we have confirmed this picture by measuring the impedance response through simulations . in ref . @xcite , two of us have shown that the centre - of - mass of the first few monomers is an excellent proxy for chain tension at the pore and we assume here that this further serves as a proxy for @xmath115 . based on this idea , we track @xmath97 by measuring the distance of the average centre - of - mass of the first @xmath25 monomers from the membrane , @xmath116 , in response to the injection of extra monomers near the pore at time @xmath81 . specifically we consider the equilibrated right segment of the polymer , of length @xmath117 ( with one end tethered at the pore ) , adding @xmath89 extra monomers at the tethered end of the right segment at time @xmath81 , corresponding to @xmath89 , bringing its length up to @xmath75 . using the proxy @xmath116 we then track @xmath97 . the clear agreement between the exponent obtained from the simulation results with the theoretical prediction of @xmath118 can be seen in fig . we have checked that the sharp deviation of the data from the power law @xmath93 at long times is due to the asymptotic exponential decay as @xmath95 , although this is not shown in the figure . having thus shown that @xmath119 , we can expect that the translocation dynamics is anomalous for @xmath17 , in the sense that the mean - square displacement of the monomers through the pore , @xmath120 for some @xmath121 and time @xmath17 , whilst beyond the rouse time it becomes simply diffusive . the value @xmath122 follows trivially by expressing @xmath18 in terms of ( translocative ) velocity correlations @xmath123 , which ( by the fluctuation dissipation theorem ) are given in terms of the time dependent admittance @xmath124 . and hence inversely in terms of the corresponding impedance .     as a function of time @xmath30 , for @xmath125 ( orange ) , 200 ( red ) and 500 ( blue ) . the thick black line indicates the theoretically expected slope corresponding to @xmath126 . the dashed black line corresponds to @xmath127 , which would have been the slope of the @xmath128 vs. @xmath30 curve in a double - logarithmic plot , if @xmath7 were to scale as @xmath129 . [ fig7 ] ]    indeed , as shown in fig . [ fig7 ] , a double - logarithmic plot of @xmath18 vs. @xmath30 is consistent with @xmath130 . the behaviour of @xmath18 at short times is an artifact of our model : at short times reptation moves dominate , leading to a transport mechanism for `` stored lengths '' @xcite along the polymer s contour in which individual units of stored length can not pass each other . as a result , the dynamics of @xmath19 , governed by the movement of stored length units across the pore , is equivalent to a process known as `` single - file diffusion '' on a line , characterized by the scaling @xmath131 ( not shown here ) . at long times the polymer tails will relax , leading to @xmath132 for @xmath20 . the presence of two crossovers , the first one from @xmath133 to @xmath130 and the second one from @xmath126 to @xmath134 at @xmath135 , complicates the precise numerical verification of the exponent @xmath136 . however , as shown in fig . [ fig7 ] , there is an extended regime in time at which the quantity @xmath137 is nearly constant . the subdiffusive behaviour @xmath138 for @xmath17 , combined with the diffusive behaviour for @xmath139 leads to the dwell time scaling as @xmath22 , based on the criterion that @xmath140 . the dwell time exponent @xmath141 is in acceptable agreement with the two numerical results on @xmath5 in 3d as mentioned in the introduction of this letter , and in table i below we present new high - precision simulation data in support of @xmath22 , in terms of the median unthreading time . the unthreading time @xmath48 is defined as the time for the polymer to leave the pore with @xmath142 and the two polymer segments equilibrated at @xmath143 . both @xmath48 and @xmath7 scale the same way , since @xmath144 [ see eq . ( [ eq4 ] ) ] .    [ cols="<,<,<",options="header " , ]     table i : median unthreading time over 1,024 runs for each @xmath0 . our results have two main significant implications :    * even in the limit of small ( or negligible ) pore friction , equilibration time scale of a polymer is smaller than its dwell time scale . yet , quasi - equilibrium condition can not be assumed as the starting point to analyze the dynamics of unbiased translocation , as has been done in the mean - field theories . * since @xmath145 , the diffusion in reaction - coordinate space is anomalous . this means that the dynamics of the translocating polymer in terms of its reaction co - ordinate _ can not _ be captured by a fokker - planck equation in the limit of small ( or negligible ) pore friction . + in view of our results , a fokker - planck type equation [ such as a fractional fokker - planck equation ( ffpe ) ] to describe the anomalous dynamics of a translocating polymer would definitely need input from the physics of polymer translocation . it therefore remains to be checked that the assumptions underlying a ffpe does not violate the basic physics of a translocating polymer . we now reflect on the theory presented in ref . @xcite . we have defined @xmath5 as the pore - blockade time in experiments ; i.e. , if we define a state of the polymer with @xmath146 as ` 0 ' ( polymer just detached from the pore on one side ) , and with @xmath147 as ` n ' , then @xmath5 is the first passage time required to travel from state 0 to state n _ without _ possible reoccurances of state 0 . in ref . @xcite , the authors attach a bead at the @xmath148 end of the polymer , preventing it from leaving the pore ; and their translocation time ( @xmath149 hereafter ) is defined as the first passage time required to travel from state 0 to state n _ with _ reoccurances of state 0 . this leads them to express @xmath149 in terms of the free energy barrier that the polymer encounters on its way from state 0 to @xmath150 , where the polymer s configurational entropy is the lowest . below we settle the differences between @xmath149 of ref . @xcite and our @xmath5 . consider the case where we attach a bead at @xmath148 and another at @xmath151 , preventing it from leaving the pore . its dynamics is then given by the sequence of states , e.g. , @xmath152 where the corresponding times taken ( @xmath149 and @xmath5 ) are indicated . at state x and x@xmath153 the polymer can have all values of @xmath8 except @xmath154 and @xmath0 ; and at states m and m@xmath153 , @xmath150 . the notational distinction between primed and unprimed states is that a primed state can occur only between two consecutive states 0 , or between two consecutive states n , while an unprimed state occurs only between state 0 and state n. a probability argument then leads us to @xmath155 where @xmath156 , @xmath157 and @xmath158 are the probabilities of the corresponding states , @xmath159 and @xmath160 . since the partition sum of a polymer of length @xmath104 with one end tethered on a membrane is given by @xmath161 with @xmath162 a non - universal constant and @xmath163 @xcite , we have @xmath164\sim1/n$ ] . similarly , @xmath165 @xcite . finally , @xmath166 @xcite yields @xmath167 . we have thus shown that the free energy barrier does not play a role for @xmath149 , implying that the theoretical expression for @xmath149 in ref . @xcite can not be correct . the numerical result @xmath168 in ref . @xcite , however , confirms our theoretical expression @xmath22 . to conclude , we have shown that for the swollen rouse chain , translocation is sub - diffusive up the configurational relaxation time ( rouse time ) of the molecule , after which it has a further fickian regime before the longer dwell time is exhausted : the mean square displacement along the chain is described by @xmath169 up to @xmath170 , after which @xmath171 . consequently , the mean dwell time scales as @xmath172 .    in future work , we will study the role of hydrodynamics . rouse friction may be an appropriate model for the dynamics of long biopolymers in the environment within living cells , if it is sufficiently gel - like to support screened hydrodynamics on the timescale of their configurational relaxation . however , we should also ask what is expected in the other extreme of unscreened ( zimm ) hydrodynamics . for our theoretical discussion the key difference is that , instead of the rouse time @xmath16 , in the zimm case the configurational relaxation times scale with @xmath0 according to @xmath173 in 3d , which upon substitution into our earlier argument would gives the lower bound value @xmath174 for the time exponent of the impedance , leading to @xmath175 ( whose resemblance to the rouse time is a coincidence  note that with hydrodynamics rouse time loses all relevance ) . these results , however , do need to be verified by simulations incorporating hydrodynamics . j. kasianowicz _ et al . _ , pnas usa * 93 * , 13770 ( 1996 ) ; e. henrickson _ et al . _ , lett . * 85 * , 3057 ( 2000 ) ; a meller _ et al . lett . * 86 * , 3435 ( 2001 ) ; m. akeson _ et al . _ , j. * 77 * , 3227 ( 1999 ) ; a. meller _ et al . _ , pnas usa * 97 * , 1079 ( 2000 ) ; a. j. storm _ et al . _ , nanoletters * 5 * , 1193 ( 2005 ) . strictly speaking , @xmath16 in this expression should be replaced by the characteristic equilibration time of a tethered polymer with length of @xmath177 ; since both scale as @xmath28 , we use @xmath16 here , favouring notational simplicity .            in ref . @xcite we overlooked the @xmath178 term of eq . ( [ eq2 ] ) to ( mistakenly ) deduce @xmath179 . nevertheless , due to the independence of eq . ( [ eq4 ] ) on @xmath0 our results for the scaling of @xmath7 in ref . @xcite remain unaffected . @xmath180 is a number slightly smaller than @xmath181 : if the polymer reaches @xmath150 from state 0 , due to the memory effects , it will have slightly higher chance to go back to state 0 rather than to proceed to state n @xcite ., source_sentence: <S> we follow the approach of induced  matter theory for a five  dimensional ( @xmath0 ) vacuum brans  dicke theory and introduce induced  </S> <S> matter and induced potential in four dimensional ( @xmath1 ) hypersurfaces , and then employ a generalized frw type solution . we confine ourselves to the scalar field and scale factors be functions of the cosmic time . </S> <S> this makes the induced potential , by its definition , vanishes , but the model is capable to expose variety of states for the universe . in general situations , in which the scale factor of the fifth dimension and scalar field are  not constants , </S> <S> the @xmath0 equations , for any kind of geometry , admit a power  </S> <S> law relation between the scalar field and scale factor of the fifth dimension . hence </S> <S> , the procedure exhibits that @xmath0 vacuum frw  like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new scalar field and a new coupling constant , @xmath2 . </S> <S> we show that the @xmath0 vacuum frw  like equations , or its equivalent @xmath1 vacuum ones , admit accelerated solutions . for a constant scalar field , </S> <S> the equations reduce to the usual frw equations with a typical radiation dominated universe . </S> <S> for this situation , we obtain dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . for non  constant scalar fields and spatially flat geometries , solutions are found to be in the form of power  law and exponential ones . </S> <S> we also employ the weak energy condition for the induced  matter , that gives two constraints with negative or positive pressures . </S> <S> all types of solutions fulfill the weak energy condition in different ranges . </S> <S> the power  </S> <S> law solutions with either negative or positive pressures admit both decelerating and accelerating ones . </S> <S> some solutions accept a shrinking extra dimension . by considering non  ghost scalar fields and appealing the recent observational measurements , </S> <S> the solutions are more restricted . </S> <S> we illustrate that the accelerating power  law solutions , which satisfy the weak energy condition and have non  ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor . </S> <S> these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum brans  dicke equations . </S> <S> the results are presented in a few tables and figures .    </S> <S> -2.7 cm    keywords : brans  dicke theory ; induced  matter theory ; frw cosmology . </S>
sentences: the lhc run 1 results , and especially the measured higgs boson properties @xcite , reveal a sm - like picture of the particle physics at the electroweak scale . nevertheless this situation is compatible with a potentially very rich scalar sector , hidden from direct inspection due to some decoupling or alignment arguments . assuming that this is the situation at the tev scale , one can ask what are the best experimental shortcuts to this hidden sector , and how one should probe it at colliders . of course , the answer strongly depends on the higgs sector chosen , and hundreds of papers have investigated it in various specific circumstances . also , the full parameter space is usually huge which renders it impossible to systematically explore the model in its full complexity , neither with algebra nor with numerical methods . in these cases , one usually resorts to specific models in particular corners of the parameter space .    without any direct clue from experiment , one usually tries to uncover all physically interesting situations within a given model and to check which of them offer the most attractive description of the data . this undertaking requires a systematic exploration of the entire parameter space of a model , which for most cases is beyond the reach of traditional methods . one of the main reasons is that , with multiple higgs fields , the scalar potential , even the renormalizable one , becomes very complicated . even an efficient description of the full parameter space is a challenge , let alone its investigation . the two - higgs - doublet model ( 2hdm ) @xcite is a hallmark example of the bsm higgs sector , whose phenomenology exhibits a variety of bsm effects and which , at the same time , is still amenable to systematic treatment in the entire parameter space , although with somewhat non - standard mathematical methods . for more involved higgs sectors , the complexity of the analysis skyrockets , making the systematic study impossible . as a result , extensions of the higgs sector beyond 2hdm , such as @xmath0-higgs - doublet models ( nhdm ) , are nowhere near in the detail of their investigation , despite hundreds of publications ( for a few early publications , see @xcite ) . it all makes systematic investigation of nhdm a task which is very challenging but worth pursuing . how should one attack this problem ? experience in conservative bsm model - building shows that models , which are phenomenolgically viable and theoretical attractive , often arise from additional symmetries , either exact or approximate . nhdms can have much richer ( discrete ) symmetries than 2hdm but usually these exact symmetries lead to either unrealistic or very sm - like phenomenology . it is therefore natural to systematically explore nhdms _ in the vicinity of large discrete symmetry group_.    this task splits into two parts , each of them being challenging on its own . first , one needs to know which highly symmetric nhdms one can construct for a given @xmath0 and what are their phenomenological consequences . if we assume that higgs doublets transform under irredicible representation of a large symmetry group @xmath1 , then the renormalizable higgs potential takes the form @xmath2 with the quadratic part being symmetric under all linear unitary transformations of @xmath0 doublets and the quartic part @xmath3 encoding the desired symmetry . second , one should explicitly break this symmetry group in a way that does not lead to dramatic consequences in order not to violate existing data . a natural procedure is to introduce soft breaking terms in the quadratic potential , @xmath4 , keeping the quartic part unchanged . in this way , the symmetric model serves as a starting point , and one needs to systematically trace the evolution of phenomenology as one shifts away from the symmetry .    the model still remains highly multidimensional , and the direct scans of the entire parameter space is of little use . i propose instead to search for analytically calculable _ robust quantities _ which would be applicable to a wide class of models and not be sensitive to specific numerical values of free parameters . it would be especially encouraging if these quantities are basis - invariant . below i will outline some directions in 3hdm , along which this task can be pursued . the first step is to understand the symmetric situations possible within a given scalar sector . here , we use for illustration the three - higgs - doublet model ( 3hdm ) , whose scalar symmetries have been recently investigated in much detail in @xcite . the effect of these symmetries and their breaking on the fermionic sector were investigated in @xcite for specific groups and , in general terms , in @xcite which completes the old analysis @xcite . the mere fact that we have at our disposal only three doublets , which interact via a renormalizable potential , restricts the list of symmetry groups @xmath1 which can be implemented in such scalar sectors . limiting ourselves only to discrete groups , one obtains the following list @xcite : @xmath5 imposing any other discrete symmetry group on the 3hdm scalar sector will unavoidably lead to an accidental continuous symmetry . some of these groups , namely , @xmath6 , @xmath7 , @xmath8 , @xmath9 , and @xmath10 , automatically lead to explicit @xmath11-conservation in the scalar sector ; the others are compatible with explicit @xmath11-violation . .the amount of residual symmetry possible after ewsb for each discrete symmetry group of the 3hdm scalar potential ( see text for details ) . @xmath12 signals the presence of a ( generalized ) @xmath11 symmetry in the model . [ cols=">,^,^,^,^",options="header " , ]     all possible symmetry breaking patterns for each of these groups were listed in @xcite ; see also @xcite for results in specific groups . these findings are summarized in table  [ table ] . the strongest symmetry breaking of a given group @xmath1 , in which we also include ( generalized ) @xmath11-symmetries when they are present , corresponds to the smallest residual symmetry group @xmath13 , and its order is denoted by @xmath14 . the weakest breaking corresponds to the largest residual symmetry group , with order @xmath15 . the groups in the upper block allow for all types of symmetry breaking : complete , partial , or no breaking at all . the groups in the middle block can remain intact at the global minimum , but if they are broken , their breaking is only partial . the last block contains groups which can neither remain unbroken nor break completely . they are always broken to a proper subgroup . the fact the large discrete symmetry groups can not be broken completely with tree - level potentials has several phenomenological consequences . in the last column of table  [ table ] we indicate whether the @xmath11-symmetry present in the scalar sector can spontaneously break . one sees that highly symmetric models prevent not only explicit but also spontaneous @xmath11-violation in the scalar sector . it is curious to note that these two types of @xmath11-violations always come in pairs , at least in the 3hdm . whether this is just a coincidence or reveals a generic fact in nhdm is not yet known . incomplete symmetry breaking has also consequences for the fermion sector . working within pure nhdm ( no bsm fields beyond several higgs doublets ) and extending the symmetry group @xmath1 to the full theory , one can prove that the lack of complete breaking leads to unphysical quark sector . the presence of a residual symmetry among higgses coupled to quarks leads either to massless quarks , or to block - diagonal form of the ckm matrix , or to the absence of @xmath11-violation . this detrimental role of residual family symmetries were noted long ago @xcite but the accurate statement and its proof were presented only recently @xcite .    nhdm with large symmetry groups contain many fields but very few free parameters , which leads to degeneracies among some of the physical higgses . examples of this situation appeared in many papers , and , for the charged higgs bosons , could be traced from general nhdm analysis @xcite . residual symmetries can also stabilize some of the higgs bosons making them dark matter candidates . this is a well - known feature of multi - scalar models , the inert doublet model @xcite and the simple extensions with ew singlets @xcite being the most studied examples . here we want to stress that , for large discrete groups , such a symmetry - protected stabilization can be automatic and does not rely on one s choice of free parameters .    in short , nhdms equipped with large discrete symmetry groups have predictive phenomenological consequences , which are not sensitive to the exact numerical values of the free parameters . these consequences reflect robust structural properties of these models and especially of the scalar potential . it is true that some of nhdms with large discrete symmetry group are already incompatible with experiment , especially in the fermon sector . however if one wants to softly break these symmetries later , it is not an obstacle but rather a good starting point in understanding phenomenology . the second step is to softly break the large symmetry group and to track down how the phenomenological picture changes . certainly , for concrete models , one can perform this analysis numerically . however understanding in this way the entire spectrum of physical possibilities is a hopeless task due to the huge number of free parameters in multi - higgs sectors . if we want to gain some insight into these models in the entire parameter space , we need to devise a method beyond simple numerical scans or case - by - case investigation . a promising direction is to find robust and calculable quantities , which would not be too sensitive to the exact numerical values of the free parameters but which would reflect the structural features of the whole class of models . here i describe one example of such quantities which i call `` critical exponents '' . consider a model with large multi - dimensional parameter space @xmath16 , see fig . [ fig - space ] . models possessing ( large ) symmetry groups can be contructed by imposing relations among these free parameters ; thus , they occupy certain low - dimensional manifolds in this space . their phenomenology in the scalar and fermionic sector often features quantities which are zero . depending on the specific construction , they can reflect degenerate extra higgs bosons , their stability due to residual symmetry after ewsb , massless quarks , absence of the @xmath11-violation in the quark mixing matrix , etc . symmetry - based nhdm examples with these properties can be easily constructed . we call these symmetry - protected quantities `` order parameters '' and denote them generically as @xmath17 .     in the multi - dimensional parameter space , symmetric models occupy lower - dimensional manifolds . these manifolds are embedded into one another dependeing on their symmetrry groups @xmath1 . the phenomenology of a non - symmetric model in the vicinity of a symmetric situation can bear resemblance to the symmetric case , up to corrections which are powerlike in the distance @xmath18 . ]    now , a generic point @xmath19 in the parameter space does not correspond to any exact symmetry , hence it leads to non - zero order parameters @xmath20 . however this point can lie close to a symmetric manifold , within small distance @xmath18 , where `` small '' means comparable or smaller than typical sizes of the symmetric manifold structures . one can then expect the order parameters to exhibit a powerlike behaviour : @xmath21 . the index @xmath22 is called the critical exponent for the quantity @xmath17 . a set of critical exponents describes how the phenomenology evolves in the vicinity of a symmetric situation . there are two important aspects of these critical exponents . first , they are robust upon variation of free parameters and are , therefore , calculable . they do not depend on the exact position of the model in the parameter space , but only reflect its degrees of proximity to a certain symmetric situation . they can depend , though , on the symmetry group and on the geometric shape of its manifold in the parameter space . in short , critical exponents reveal certain structural properties of the whole family of models , and should be calculable analytically . second , having these critical exponents at hand will provide qualitative understanding of multi - parametric models in the vicinity of symmetries , and in particular of nhdm with large softly broken non - abelian groups . indeed , knowing whether certain critical exponent is @xmath23 and another is @xmath24 would immediately provide insight on which effects are phenomenologically important and how they can be related . in short , when working in hugely multidimensional parameter spaces , critical exponents can guide the search for models with desired phenomenology .    as an illustration of critical exponents displaying non - trivial behaviour , consider the scalar sector in the general 2hdm in the vicinity of critical points in the parameter space at which the lightest higgs mass @xmath25 . the parameter space can be fully reconstructed @xcite , and locus of critical points corresponds to a certain ellipse in it . in its vicinity , the mass is non - zero and the critical exponent @xmath22 was found to be either @xmath26 or @xmath27 , depending on how one approaches the ellipse @xcite . these two cases differ by a residual discrete symmetry of the model . this example , not pretending to be phenomenologically relevant , provides a taste of how easily the things can become non - trivial even in rather simple situations . systematric derivation of critical exponents for all order parameters in nhdm will already provide a new insight into its phenomenology . 99 g.  aad _ et al . _ [ atlas collaboration ] , science * 338 * , 1576 ( 2012 ) ; 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( 2 ) these sources were not detected by _ iras _ ( _ infrared astronomical satellite _ ) , indicating a lack of circumstellar material such as extended disks and/or envelopes . ( 3 ) the jets show [ ] /h@xmath5 line ratio decreasing from the base outward , indicating that the dominant excitation mechanism changes from shocks at the base to photoionization at the end of the jet . ( 4 ) the jet systems all have a highly asymmetric or even unipolar morphology , indicating perhaps different jet forming conditions in the launch and collimation regions . the rosette hh jets show subtle differences from other externally photoionized hh jets because of different degrees of hardness in the uv radiation field or strength of fast stellar winds . both the rosette hh1 and hh2 jets show high excitation @xcite , as the rosette nebula contains an o4 star and an o5 star @xcite . in the orion nebula , hh jets with [ ] emission are found only within 30 , or @xmath20.06 pc , from @xmath9oric , an o4 - 6 star , the earliest o star in the orion nebula @xcite . the high excitation of these hh jets results from both the harsh uv radiation and strong fast stellar wind of @xmath9oric @xcite . @xcite propose that the rosette jets provide evidence for efficient dissipation of circumstellar disks and envelopes in the close vicinity of massive ob stars . this uv dissipation of pre - existing protostellar systems may lead to the formation of isolated brown dwarfs ( bds ) and free - floating giant planets . such a formation mechanism for single sub - stellar objects has indeed been shown to be effective by theoretical studies @xcite . it is therefore important to explore the nature of jet formation and disk dissipation of low - mass ysos in close vicinity of massive ionizing ob stars , as the occurrence of such ob clusters and associations is common in the galaxy , and the solar system may have been formed in such environments @xcite . furthermore , there has been an on - going debate whether weak - lined ttauri stars ( wttss ) evolve from classical ttauri stars ( cttss ) through gradual dissipation of circumstellar material , or wttss are formed through rapid disk dissipation due to external forces after the formation of the protostar . a detailed study of the rosette jet systems may provide insight on the rapid evolution of cttss to wttss due to external photoionization of their protostellar disks in massive star forming regions . wttss formed in this way have indistinguishable evolutionary ages from those of ctts that originated from the same episode of star formation . @xcite presented a kinematical study of the rosette hh1 jet and confirmed the jet nature of the system . here we investigate in detail the physical nature of the jet system using high - resolution imaging and echelle spectroscopy , as well as data from a simultaneous photometric and spectroscopic monitoring of the jet - driving source . narrow - band h@xmath5 images of the rosette nebula were obtained with the 8k@xmath108k mosaic ccd camera on the mayall 4  m telescope at the kitt peak national observatory on 2001 october 13 . a set of five 600 s exposures was taken , with each image slightly offset to fill in physical gaps between the mosaic ccds . the pixel scale is 0258 pixel@xmath0 , resulting in roughly a 36@xmath11 field of view . we obtained high - dispersion spectroscopic observations of rosette hh1 with the echelle spectrograph on the blanco 4  m telescope at the cerro tololo inter - american observatory on 2004 january 9 and 12 . in each observation a 79 line mm@xmath0 echelle grating was used . the observations on 2004 january 9 were made in a multi - order mode , using a 226 line mm@xmath0 cross - disperser and a broad - band blocking filter ( gg385 ) . the spectral coverage is roughly 40007000  , so that nebular lines of a range of excitation can be examined . in the case of the [ ] @xmath126717 , 6731 doublet , the line ratio has been used to estimate the electron densities within the jet . the observations on 2004 january 12 were made in a single - order mode , using a flat mirror and a broad h@xmath5 filter ( central wavelength 6563   with 75   fwhm ) to isolate the order containing the h@xmath5 and [ ] @xmath126548 , 6583 lines . the exposure time used for both instrumental setups was 1,200 s.    for each observation the long - focus red camera was used to obtain a reciprocal dispersion of 3.5   mm@xmath0 at h@xmath5 . the spectra were imaged using the site2k # 6 ccd detector . the 24 @xmath13 m pixel size corresponds to 026 pixel@xmath0 along the slit and @xmath20.08 pixel@xmath0 along the dispersion axis . both observations used a 16 slit oriented roughly along the jet direction , at position angles of 312@xmath14 ( multi - order ) and 318@xmath14 ( single - order ) . the resultant instrumental resolution , as measured by the fwhm of the unresolved telluric emission lines , was 0.29   or 13 km  s@xmath0 at h@xmath5 . the observations were reduced following standard procedures in the iraf ( ver . 2.12 ) software package . this included bias correction , flat - fielding and gain - jump removal between the chips . cosmic - ray hits were manually rejected from the 2d spectrograms . wavelength calibration of the data was carried out based on th - ar lamp exposures and further improved by comparison with night sky emission lines , which resulted in an accuracy of @xmath21 km  s@xmath0 before converting to the heliocentric frame . we have carried out a simultaneous photometric and spectroscopic monitoring campaign of the rosette hh1 source between 2004 december 31 and 2005 january 7 . the time - series photometric observations , unaccompanied by spectroscopy , were further extended from january 8 to january 13 . the photometric observations were made in @xmath15 and @xmath16 filters with the 0.8  m telescope of the hsing - hua university , located at the xing - long station of the national astronomical observatory of the chinese academy of sciences ( naoc ) . differential photometry of the jet - driving source was obtained through comparisons with two slightly brighter stars in the same field at @xmath5(j2000 ) = 06@xmath17 , @xmath18(j2000 ) = @xmath19 and @xmath5(j2000 ) = @xmath20 , @xmath18(j2000 ) = @xmath21 . the @xmath16 band photometry of the reference stars is found to be constant within 0.04 mag throughout the monitoring campaign . many of the @xmath15 band exposures were affected by charge bleeding from the saturated o9.5 star hd46241 . these unreliable data are not presented here . low - resolution spectroscopy of the jet source was obtained with the 2.16  m telescope of naoc during this monitoring campaign . two different spectrographs were used . from 2004 december 31 to 2005 january 4 , the beijing faint object spectrograph and camera ( bfosc ) , a copy of efosc in service at the european southern observatory , and a thinned back - illuminated orbit 2k @xmath10 2k ccd were used . the g4 grating was employed , which gave a two - pixel resolution of 8.3   . on 2005 january 57 , an omr ( optomechanics research inc . ) spectrograph and a tecktronix 1024 @xmath10 1024 ccd were used . these spectroscopic data have a higher resolution , with a 100    mm@xmath0 reciprocal dispersion and a two - pixel resolution of 4.8  . both sets of observations used a 2@xmath22 slit . the spectroscopic data were reduced using standard procedures and packages in iraf . the ccd reductions included bias and flat - field correction , nebular background subtraction , and cosmic rays removal . wavelength calibration was performed using he - ar lamp exposures at both the beginning and the end of the observations every night . flux calibration of each spectrum was based on observations of at least 2 of the kpno spectral standards @xcite per night . figure 1 presents our new h@xmath5 image of the rosette hh1 jet . this high - quality image reveals that the jet does not trace back through the exact center of the jet - driving source . the morphology of the jet appears to indicate episodic or nonsteady mass ejection . a close inspection shows a split at the end of the collimated jet , with one branch remaining straight while the other bending north possibly as a result of an interaction with the stellar wind of the o4 star , hd46223 .    in figure 2 , we present the echelle spectrograms of the rosette hh1 jet . the continuum emission at the origin of each spectrogram is from the jet - driving source . the single - order observations cover only the h@xmath5 and [ ] lines ( the two panels to the left in fig .  2 ) . the multi - order observations detected the h@xmath5 , h@xmath23 , h@xmath24 , @xmath255876 , [ ] @xmath126548 , 6583 , [ ] @xmath124959 , 5007 , and [ ] @xmath126716 , 6731 lines . the three panels to the right in figure 2 show the [ ] @xmath256583 , [ ] @xmath256731 , and [ ] @xmath255007 lines ( the brighter component of each doublet ) . these lines have different thermal widths and require different excitation energies , and thus appear different and can be intercompared to gain physical insight . the [ ] lines have smaller thermal widths than the h@xmath5 line and thus resolve the velocity structures of the hh jet and the superposed nebula more clearly . the [ ] lines detect a prominent irregular component at the location of the jet . this jet component is blue - shifted with respect to two nebular components that have nearly uniform velocity and surface brightness throughout the slit . these two nebular components , at heliocentric velocities ( @xmath26 ) of @xmath213 and 40 km  s@xmath0 , arise from the approaching and receding sides of the rosette nebula s expanding shell . these velocities imply a systemic velocity of @xmath27 27 km  s@xmath0 and an expansion velocity of @xmath214 km  s@xmath0 . the extreme velocity of the jet reaches @xmath26 = @xmath130 km  s@xmath0 , which is blue - shifted from the rosette s sytemic velocity by 57 km  s@xmath0 . these results are consistent with those reported by @xcite .    in the single - order observation along pa = 318@xmath14 , the [ ] emission of the jet shows two bright knots and two faint knots , with the outermost knot being the faintest . no emission from a counterjet is detected . the multi - order observation has a shorter slit along a slightly different position angle , pa = 312@xmath14 , and thus shows a slighly different velocity structure in the [ ] line . the [ ] emission shows velocity structure and surface brightness similar to those of the [ ] emission . the [ ] emission , on the other hand , shows a smooth surface brightness distribution , in contrast to the knots seen in the other lines . the observed velocity fwhm of the [ ] line ranges from @xmath216 to @xmath223 km  s@xmath0 , with the fainter knots showing broader velocity widths . these widths are not much larger than the observed fwhm of 18 - 20 km  s@xmath0 in the rosette expanding shell components . for an instrumental fwhm of 12 km  s@xmath0 and a thermal width of 5.7 km  s@xmath0 for [ ] at 10@xmath7 k , the observed fwhm of the jet implies an intrinsic turbulent fwhm of 9 to 19 km  s@xmath0 . one interesting spectral feature suggested by @xcite for the jet - driving source is an inverse p cygni profile at the h@xmath5 line based on a low - dispersion spectrum . an inverse p cygni profile , if confirmed , indicates that material is being accreted onto the star . our new high - dispersion echelle observations clearly resolve both spatially and spectrally the nebular and stellar components of the h@xmath5 line profile , and thus allow a critical assessment of this suggested inverse p cygni profile . as seen in figure 2 and shown below , the bright nebular emission from the rosette nebula makes it difficult to accurately extract a clean stellar spectrum . the nebular spectrum varies along the slit . to assess the nebular contribution , we have extracted seven h@xmath5 line profiles using 1@xmath28-wide windows and 05 intervals stepping across the stellar spectrum . these h@xmath5 profiles are shown in figure 3 . the spectrum j is extracted from the jet side of the star , and the blue - shifted jet component is clearly seen . seeing spreads the jet emission into the stellar spectra s1s4 . the contribution from the rosette nebula is better represented by the spectra n1 and n2 , extracted outside the star on the side opposite to the jet . the average of these nebular spectra is subtracted from the four stellar spectra s1s4 . the nebula - subtracted stellar spectra s1@xmath29s4@xmath29 , displayed in figure 4 , show a narrow , blue - shifted emission component and a broad , red - shifted absorption component superposed on a continuum . to determine the origin of the blue - shifted emission component , we use the [ ] @xmath256583 forbidden line that is expected only from low - density gas , such as the rosette nebula and the hh1 jet . the seven [ ] line profiles extracted in a similar manner are displayed in the right panel of figure 3 , and they indeed show the two components from the rosette nebula throughout the slit . the [ ] profiles in the four nebula - subtracted stellar spectra , displayed in the right panel of figure 4 , show that the nebula - subtraction satisfactorily removes the emission from the rosette nebula and that the remaining [ ] emission is from the hh1 jet . the comparison between the nebula - subtracted h@xmath5 and [ ] profiles suggests that the blue - shifted h@xmath5 emission in the stellar spectra predominantly arises from the jet .    to obtain a clean stellar spectrum , we scale and subtract the nebula - subtracted jet spectrum ( j@xmath29 ) from the nebula - subtracted stellar spectra ( s1@xmath29s4@xmath29 ) by trial and error until the [ ] emission is minimized . to better show the stellar continuum , the spectra from each step of this procedure are shown in figure 5 over a larger wavelength range . the final clean stellar spectra s2@xmath28 and s3@xmath28 show h@xmath5 absorption with little or no blue - shifted emission . the lack of strong stellar h@xmath5 emission implies the absence of a significant disk , making it difficult to be associated with a jet . this will be discussed further in section 7 . our final clean stellar h@xmath5 line profile is quite different from the previously reported inverse p - cygni profile @xcite . as we have illustrated above , the h@xmath5 emission is dominated by contributions from the rosette nebula and the hh1 jet . these emission components can be resolved and subtracted accurately only if high - dispersion spectra are used . the apparent difference between the final clean stellar h@xmath5 profiles s2@xmath28 and s3@xmath28 probably results from imperfect subtraction of the jet component , as the [ ] /h@xmath5 ratio may vary along the jet . the previously reported inverse p - cygni profile is most likely an artifact caused by difficulties in subtracting the nebular background and jet contribution using low - dispersion spectra . the uv radiation in the rosette nebula is predominantly provided by the massive stars hd 46223 and hd 46150 . hd 46223 , of spectral type o4v(f ) , is the hottest star in ngc 2244 , and produces lyman photons at a rate of 10@xmath30 s@xmath0 @xcite . it is located at 277@xmath22 , or 2.0 pc for a distance of 1.5 kpc @xcite , from the hh1 jet source . hd 46150 is an o5v star projected at 433@xmath22 , or 3.1 pc , from the jet source . it produces ionizing photons at a rate of 10@xmath31 s@xmath0 @xcite . the combined lyman continuum emission from these two exciting stars renders 1 - 2 orders of magnitudes higher impact on the rosette hh1 jet than that on similar jets discovered in the vicinity of @xmath8 orionis @xcite and the trapezium stars @xcite . the rosette nebula is therefore among the most extreme environments in which photoionized jets are found . although immersed in a photoionized medium , the presence of highly collimated jets strongly suggests the existence of at least a relic disk as a sustained feed to the surviving jet . in the case of the rosette hh1 jet , we expect a photoevaporating disk with a configuration resembling that of hh527 in the orion nebula , as resolved by the _ hubble space telescope _ @xcite . this could serve as a schematic impression of the appearance of the disk - jet system . their configuration of the disk subject to photoevaporation induced dissipation is believed to be similar , although the jet associated with hh527 may be oriented at a different direction with respect to the incident uv radiation and has a low excitation , being located in the outskirts of the orion nebula . the electron density of the hh1 jet was derived from the [ ] doublet ratios of @xmath256716/@xmath256731 measured with our multi - order echelle observation along the jet . the @xmath256716/@xmath256731 ratios are 0.85@xmath320.1 in the jet and 1.3@xmath320.1 in the rosette nebula . the corresponding electron densities are @xmath21000 @xmath3 in the jet and @xmath4100 @xmath3 in the backgound region . if the hh1 jet is indeed within the cavity of the rosette nebula , the medium between the ionizing stars and the hh1 jet is hot and ionized with a density of @xmath20.1 h - atom @xmath3 @xcite . the stellar ionizing flux at the hh1 jet would be nearly unattenuated , at a level of @xmath33 photons @xmath34 s@xmath0 . for a medium of 1000 @xmath3 density , this flux can ionize gas to a thickness of 0.37 pc . the width of the hh1 jet ( measured from fig . 1 ) is @xmath351.5@xmath28 , or @xmath35 0.01 pc ; thus the hh1 jet can be fully photoionized by the radiation from hd 46223 and hd 46150 .    using the h@xmath5 surface brightness of the hh1 jet , @xmath36 ergs s@xmath0 @xmath34 sr@xmath0 , and a density of 1000 @xmath3 , we find that the width ( or the depth for a cylindrical geometry ) of the jet is @xmath20.8@xmath28 , or 0.0056 pc . if we assume a flow velocity of 200 km s@xmath0 , as did @xcite , the mass loss rate would be @xmath37 yr@xmath0 . it ought to be noted that when the disk - jet systems are exposed to photoionizing environments , the jet production is probably no longer a dominant role of mass loss from the circumstellar disk . photoionization and dissipation of the disk then takes place , or at least consumes the circumstellar materials at a comparable rate as the mass ejection in the form of a jet . if we assume that evaporated flows associated with the dissipating disk of the jet source govern a comparably effective mass erosion as those of the proplyds in the orion nebula @xcite , then for a mean mass loss rate of 4.1 x 10@xmath38 @xmath39 yr@xmath0 @xcite and a disk mass of 0.006 @xmath39 associated with the rosette hh1 source @xcite , the estimated photodissipation timescale of the relic disk is @xmath2 10@xmath40 yrs . given the more extreme environment the rosette hh1 source faces , a mass loss rate an order of magnitude higher may be more likely and the disk dissipation time would be reduced to @xmath41 yrs . photometric results of the jet driving source are presented in figure 6 , which shows irregular variations around the mean with an amplitude as large as @xmath20.2 mag in the r band . this amplitude of variation is about one magnitude lower than that detected for the energy source of the rosette hh2 jet , which amounts to as large as 1.4 mag in r @xcite . variations in the r band are primarily attributed to erratic fluctuations of the h@xmath5 emission of the source , which relies on a time - variable mass accretion rate , disk inhomogeneity , or otherwise chromospheric activity of the central yso . the rosette hh1 source s low amplitude of variation is believed to be due to a lack of circumstellar material and subsequently a subtle mass accretion rate , as shown by the nearly absent h@xmath5 emission from the jet source , although the irregular variation itself is consistent with a young status of evolution of the central source . we thus suggest that the rosette hh1 source may well represent a transient phase of ysos evolving rapidly from a ctts to a wtts by fast photodissipation of their circumstellar disks . based on the time series photometric data achieved , we find no evidence of a binary origin of the jet source , which otherwise could imply a different mechanism of jet production . as noted in section 2.3 , the b band observations do not give very good results , although reminiscent irregular variations with a comparable magnitude of up to @xmath20.25 mag are indeed indicated . the spectral monitoring observations are not very useful for the h@xmath5 line profile because of the difficulty in background nebular subtraction , as discussed in section 4 . nevertheless , the overall spectral characteristics of the star can be determined . we find that the spectral type appears to vary between f8v and f9v during the period of observations . this spectral change may be related to photo - erosion of the rotational disk with an inhomogeneous configuration . being immersed in the fierce uv radiation field of the rosette , the optical jets associated with ysos indicate either a jet production timescale of as long as 1 - 2 myr , comparable to the evolutionary age of the main cluster ngc 2244 , or that the ysos have a much younger age and the cocoons associated with their protostars had , in some way , been successfully shielded from the strong ionization fields . @xcite investigated the ysos with near infrared excesses , an indicator of the existence of circumstellar disks , of the young open cluster ngc 2244 based on the 2mass database . the jet - driving sources in the rosette , however , show infrared colors commensurate with those of wttss , which have spectral energy distributions indistinguishable from main - sequence dwarfs . see the color - color and color - magnitude diagrams in figures 4 and 7 of @xcite . this , along with the fact that none of the rosette jet sources and their only rivals found near @xmath8 orionis were detected by iras , suggests a lack of circumstellar material as compared to conventional ysos driving outflows . this is in agreement with the estimated mass of 0.006 @xmath39 for the relic disk associated with the rosette hh1 source @xcite , far below the typical value of @xmath20.1 @xmath39 around ctts . given the emerging nature of the young open cluster with a turnoff age of 1.9 myr @xcite , fast disk dissipation is suggested . @xcite suggest that this provides indirect observational evidence for the formation of isolated bds and free - floating giant planets , as discovered in orion by @xcite , by uv dissipation of unshielded protostellar systems . this can be very important to our understanding of the formation of such sub - stellar and planetary mass objects , particularly in regions of massive star formation . such uv dissipation could , on the other hand , impose strong effects on the formation of and hence the search for extra - solar planets around low - mass stars , the circumstellar disks of which could otherwise be potential sites of terrestrial planet formation . this alternatively introduces a viable solution to the long puzzle of how wtts were formed as a consequence of fast ctts evolution and the rapid dissipation of circumstellar disks under particular forming conditions near massive ob stars or in cluster environments . the spatial distribution of the extreme jets with respect to the dozens of exciting ob stars of the spectacular region is presented in figure 7 , superimposed on which is the relic shell structure as delineated by the apparent congregation of excessive emission sources in the near infrared @xcite . this suggests the existence of a former working interface layer of the region with its ambient molecular clouds . the projected location of the energy source of the rosette hh1 jet near the relic arc provides evidence of a triggered origin of its formation in or near the swept - up layer . the rosette hh2 source has a similar radial distance from the statistical center of ngc 2244 @xcite and introduces a similar origin . in this scenario , the jet sources should have a much younger age than the main cluster ngc 2244 . molecular gas and dust in the shell could have played an important role in shielding new generation protostellar objects from the harsh uv evaporation and ionization from the massive ob stars . it is therefore reasonable to infer that the jet sources have been directly exposed to the harsh photoionization fields recently . @xcite first noted the existence of ionized knots and filaments in the southeastern quadrant of the rosette s central cavity . the rosette hh jets are also located in this region . all are prominent features in narrowband [ ] images , indicating a high excitation . among these , knot c shows high - velocity components and is in association with a high excitation bow - shock at its tip @xcite . this feature is believed to be a hh flow , though no apparent energy source has yet been identified . however , a giant shock - like structure to the west of the rosette hh1 jet can be easily identified ( figure 8) . at a distance of @xmath21.5 kpc , its large - scale appearance and lack of a potential exciting source seem to exclude the possibility of a herbig - haro origin . the preferential distribution of these high - excitation structures to the southeast edge of the region suggests a possible association with the large monoceros loop supernova remnant ( snr ) projected to the northeast of the rosette nebula . while there is no morphological evidence for dynamical interactions between these two objects , it is possible that some of the high - excitation structures in the rosette nebula are caused by the monoceros loop snr s ballistic ejecta that proceeds ahead of the snr shock front . to test this scenario , proper motion or abundance measurements of the high - excitation structures in the rosette nebula are needed .    alternatively , we propose that these structures are globules or former dust pillars , similar to those around the working surface of the region , that have been overrun by the ionization front and are now in the process of photodissipation , as is the fate of the hh jets in this region . at least one high - excitation structure is likely associated with a neutral cometary knot ( see figure 8) , the tip of which is highly ionized and has an appearance resembling those in the orion nebula @xcite , but with a physical size roughly 4 times larger . we present follow - up high - qaulity imaging and echelle spectroscopic observations of the rosette hh1 jet . the high angular and spectral resolution allow us to determine accurately the contributions from the region , jet , and star . the expansion of the region and the kinematics of the jet are consistent with the previous measurements by @xcite . using the [ ] doublet ratios , we further determined the electron density of the jet , @xmath21000 @xmath3 .    with a careful subtraction of the nebular and jet components , we find the stellar h@xmath5 line is dominated by a broad absorption profile with little or no emission component , indicating a lack of substantial circumstellar material . the circumstellar material has most likely been photo - evaporated by the strong uv radiation field in the rosette nebula . the evaporation time scale is 10@xmath6  10@xmath7 yr . the rosette hh1 jet source provides evidence for an accelerated evolution from a ctts to a wtts due to the strong uv radiation field ; therefore , both cttss and wttss can be spatially mixed in regions with massive star formation . finally , we suggest that the giant high - excitation structures residing at the center of the rosette nebula may be globules or former dust pillars in the midst of uv dissipation . further observations of the nebular kinematics are needed to determine whether these are dissipating interstellar structures or related to the supernova ejecta associated with the monoceros loop snr . we greatly appreciate the helpful comments and suggestions from the referee of the paper , john meaburn . thanks to the team working with the hsing - hua 80 cm telescope for their help on coordinating the photometric observations . this project is supported by the 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2001 , , 372 , 516 whitworth , a.  p. , & zinnecker , h.  2004 , , 427 , 299 zapatero osorio , m.  r. , bejar , v.  j.  s. , martin , e.  l. , rebolo , r. , barrado y navascus , d. , bailer - jones , c.  a.  l. , & munct , r. 2000 , science , 290 , 103, attempts to geometrical unification of gravity with other interactions , using higher dimensions other than our conventional @xmath1 space  time , began shortly after invention of the special relativity ( * sr * ) . nordstrm was the first who built a unified theory on the base of extra dimensions  @xcite . tight connection between sr and electrodynamics , namely the lorentz transformation , led kaluza  @xcite and klein  @xcite to establish @xmath0 versions of general relativity ( * gr * ) in which electrodynamics rises from the extra fifth dimension . since then , considerable amount of works have been focused on this idea either using different mechanism for compactification of extra dimension or generalizing it to non  compact scenarios ( see e.g. ref . @xcite ) such as brane  world theories  @xcite , space  time  matter or induced  matter ( * i m * ) theories  @xcite and references therein . the latter theories are based on the campbell  magaard theorem which asserts that any analytical @xmath6dimensional riemannian manifold can locally be embedded in an @xmath7dimensional ricci  flat riemannian manifold  @xcite . this theorem is of great importance for establishing @xmath1 field equations with matter sources locally to be embedded in @xmath0 field equations without _ priori _ introducing matter sources . indeed , the matter sources of @xmath1 space  times can be viewed as a manifestation of extra dimensions . this is actually the core of i m theory which employs gr as the underlying theory . on the other hand , jordan  @xcite attempted to embed a curved @xmath1 space  time in a flat @xmath0 space  time and introduced a new kind of gravitational theory , known as the scalar  tensor theory . following his idea , brans and dicke  @xcite invented an attractive version of the scalar  tensor theory , an alternative to gr , in which the weak equivalence principle is saved and a non  minimally scalar field couples to curvature . the advantage of this theory is that it is more machian than gr , though mismatching with the solar system observations is claimed as its weakness  @xcite . however , the solar system constraint is a generic difficulty in the context of the scalar  tensor theories  @xcite , and it does  not necessarily denote that the evolution of the universe , at all scales , should be close to gr , in which there are some debates on its tests on cosmic scales  @xcite .    although it is sometimes desirable to have a higher dimensional energy  momentum tensor or a scalar field , for example in compactification of extra curved dimensions  @xcite , but the most preference of higher dimensional theories is to obtain macroscopic @xmath1 matter from pure geometry . in this approach , some features of a @xmath0 vacuum brans  dicke ( * bd * ) theory based on the idea of i m theory have recently been demonstrated  @xcite , in where the role of gr as fundamental underlying theory has been replaced by the bd theory of gravitation . actually , it has been shown that @xmath0 vacuum bd equations , when reduced to four dimensions , lead to a modified version of the @xmath1 brans  dicke theory which includes an induced potential . whereas in the literature , in order to obtain accelerating universes , inclusion of such potentials has been considered in _ priori _ by hand . a few applications and a @xmath8dimensional version of this approach have been performed  @xcite . though , in refs . @xcite , it has also been claimed that their procedure provides explicit definitions for the effective matter and induced potential . besides , some misleading statements and equations have been asserted in ref . @xcite , and hence we have re  derived the procedure in section @xmath9 . actually , the reduction procedure of a @xmath0 analogue of the bd theory , with matter content , on every hypersurface orthogonal to an extra cyclic dimension ( recovering a modified bd theory described by a 4metric coupled to two scalar fields ) has previously been performed in the literature  @xcite . however , the key point of i m theories are based on not introducing matter sources in @xmath0 space  times . in addition , recent measurements of anisotropies in the microwave background suggest that our ordinary @xmath1 universe should be spatially flat  @xcite , and the observations of type ia  supernovas indicate that the universe is in an accelerating expansion phase  @xcite . hence , the universe should mainly be filled with a dark energy or a quintessence which makes it to expand with acceleration  @xcite . then after an intensive amount of work has been performed in the literature to explain the acceleration of the universe .    in this work , we explore the friedmann  robertson  walker ( * frw * ) type cosmology of a @xmath0 vacuum bd theory and obtain solutions and related conditions . this model has extra terms , such as a scalar field and scale factor of fifth dimension , which make it capable to present accelerated universes beside decelerated ones . in the next section , we give a brief review of the induced modified bd theory from a @xmath0 vacuum space  time to rederive the induced energy  momentum tensor , as has been introduced in ref . @xcite , for our purpose to employ the energy density and pressure . in section  @xmath10 , we consider a generalized frw metric in the @xmath0 space  time and specify frw cosmological equations and employ the weak energy condition ( * wec * ) to obtain the energy density and pressure conditions . then , we probe two special cases of a constant scale factor of the fifth dimension and a constant scalar field . in section @xmath11 , we proceed to exhibit that @xmath0 vacuum bd equations , employing the generalized frw metric , are equivalent , in general , to the corresponding vacuum @xmath1 ones . this equivalency can be viewed as the main point within this work which distinguishes it from refs . @xcite . in section @xmath12 , we find exact solutions for flat geometries and proceed to get solutions fulfilling the wec while being compatible with the recent observational measurements . we also provide a few tables and figures for a better view of acceptable range of parameters . finally , conclusions are presented in the last section . following the idea of i m theories  @xcite , one can replace gr by the bd theory of gravitation as the underlying theory  @xcite . for this purpose , the action of @xmath0 brans  dicke theory can analogously be written in the jordan frame as @xmath13=\int\sqrt{|{}^{_{(5)}}g| } \left ( \phi \ ^{^{(5)}}\!r-\frac{\omega}{\phi}g^{_{ab}}\phi_{,_{a}}\phi_{,_{b}}+ 16\pi l_{m } \right ) d^{5}x\ , , \ ] ] where @xmath14 , the capital latin indices run from zero to four , @xmath15 is a positive scalar field that describes gravitational coupling in five dimensions , @xmath16 is @xmath0 ricci scalar , @xmath17 is the determinant of @xmath0 metric @xmath18 , @xmath19 represents the matter lagrangian and @xmath20 is a dimensionless coupling constant . the field equations obtained from action ( 1 ) are @xmath21 and @xmath22 where @xmath23 , @xmath24 is @xmath0 einstein tensor , @xmath25 is @xmath0 energy  momentum tensor , @xmath26 . also , in order to have a non  ghost scalar field in the conformally related einstein frame , i.e. a field with a positive kinetic energy term in that frame , the bd coupling constant must be @xmath27  @xcite .    as explained in the introduction , we propose to consider a @xmath0 vacuum state , i.e. @xmath28 , where equations ( 2 ) and ( 3 ) read @xmath29 and ) for later on convenient . ] @xmath30 for cosmological purposes one usually restricts attention to @xmath0 metrics of the form , in local coordinates @xmath31 , @xmath32 where @xmath33 represents the fifth coordinate , the greek indices run from zero to three and @xmath34 . it should be noted that this ansatz is restrictive , but one limits oneself to it for reasons of simplicity . assuming the @xmath0 space  time is foliated by a family of hypersurfaces , @xmath35 , defined by fixed values of the fifth coordinate , then the metric intrinsic to every generic hypersurface , e.g. @xmath36 , can be obtained when restricting the line element ( [ 6 ] ) to displacements confined to it . thus , the induced metric on the hypersurface @xmath37 can have the form @xmath38 in such a way that the usual @xmath1 space  time metric , @xmath39 , can be recovered . hence , equation ( [ 4 ] ) on the hypersurface @xmath37 can be written as @xmath40,\ ] ] where @xmath41 is an induced energy  momentum tensor of the effective @xmath1 modified bd theory , which is defined as @xmath42 with @xmath43 \bigg \}\end{aligned}\ ] ] and @xmath44-\frac{1}{2}g'_{\alpha\beta}\phi ' \bigg \}.\ ] ] also , the induced potential has been defined in the formal identification as  @xcite @xmath45\equiv -\epsilon \frac{\omega}{b^{2}}\frac{\phi^{'2}}{\phi } \big |_{_{\sigma_{0 } } } , \ ] ] where the prime denotes derivative with respect to the fifth coordinate . such an identification has been claimed  @xcite to be valid depending on metric background and considering separable scalar fields . however , this definition is different from what has been used in ref . @xcite . reduction of equation ( [ 5 ] ) on the hypersurface @xmath37 gives @xmath46-\frac{b_{,\mu}}{b}\phi^{,\mu}\ , , \ ] ] which after manipulation resembles the other field equation of a modified bd theory in four dimensions with induced potential . the definition @xmath41 and equation ( [ freduction ] ) are all we need for our purpose in this work and an interested reader can consult refs . @xcite for further details .    in the next section we assume a generalized frw metric in a vacuum @xmath0 universe to find its cosmological implications . for a @xmath0 universe with an extra space  like dimension in addition to the three usual spatially homogenous and isotropic ones , metric ( [ 6 ] ) can be written as @xmath47+b^2(t , y)dy^2\,,\ ] ] that can be considered as a generalized frw solution . the scalar field @xmath15 and the scale factors @xmath48 and @xmath49 , in general , are functions of @xmath50 and @xmath33 . however , for simplicity and physical plausibility , we assume the extra dimension is cyclic , i.e. the hypersurface  orthogonal space  like is a killing vector field in the underlying @xmath0 space  time  @xcite . hence , all fields are functions of the cosmic time only , and definition ( [ 6.3 ] ) makes the induced potential vanishes . in this case , we will show that such a universe can have accelerating and decelerating solutions . note that , the functionality of the scale factor @xmath49 on @xmath33 , either can be eliminated by transforming to a new extra coordinate if @xmath49 is a separable function , and or makes no changes in the following equations if @xmath49 is the only field that depends on @xmath33 . besides , in the compactified extra dimension scenarios , all fields are fourier  expanded around @xmath51 , and henceforth one can have terms independent of @xmath33 to be observable , i.e. physics would thus be effectively independent of compactified fifth dimension  @xcite . considering metric ( [ 7 ] ) , equations ( [ 4 ] ) and ( [ 5 ] ) result in cosmological equations @xmath52 @xmath53 @xmath54 and @xmath55 which are  not independent equations and where @xmath56 , @xmath57 and @xmath58 . by employing relation ( [ 6.4 ] ) , one can interpret the right hand side of equations ( [ 8 ] ) and ( [ 9 ] ) as energy density and pressure of the induced effective perfect fluid , i.e. @xmath59 and @xmath60 where @xmath61 or @xmath9 or @xmath10 without summation on it . the latter equality in ( [ 11.2 ] ) comes from equation ( [ 22 ] ) which will be derived in the next section . therefor , the equation of state is @xmath62    the usual matter in our universe has a positive energy density , this basically has been demanded by the wec , in which time  like observers must obtain positive energy densities . actually , the complete wec is  @xcite @xmath63 now , let us consider that the scale factor of the fifth dimension and the scalar field are  not constant values , i.e. @xmath64 and @xmath65 . then , by applying conditions ( [ 11.3 ] ) into relations ( [ 11.1 ] ) and ( [ 11.2 ] ) , one gets @xmath66 or @xmath67 where we also have assumed expanding universes , i.e. @xmath68 . using conditions ( [ 11.4 ] ) and ( [ 11.5 ] ) in relation ( [ 11.6 ] ) gives @xmath69 or @xmath70 in where the effective dust matter can be achieved when @xmath71 goes to negative or positive infinity , respectively .    in section  @xmath12 , we explore characteristic of the corresponding universes for the above results . meanwhile , in the following , we consider two special cases of a constant scale factor of the fifth dimension and a constant scalar field . +   + * constant scale factor of fifth dimension * + when @xmath49 is a constant , equations ( [ 8])([11 ] ) reduce to @xmath72 these are exactly the ordinary vacuum bd equations in @xmath1 space  time , with @xmath73 , as expected . +   + * constant scalar field * + when @xmath15 is a constant , action ( [ 1 ] ) reduces to a @xmath0 einstein gravitational theory that has been considered in ref . @xcite in general situation ( i.e. the extra dimension is  not cyclic ) . in this case , equations ( [ 8])([11 ] ) become @xmath74 and , the usual frw equations are equipped with @xmath75 , which refers to a radiation  like dominated universe for any kind of geometry without a _ priori _ assumption that the scale factor of the fifth dimension is proportional to the inverse of the usual scale factor , i.e. @xmath76 . actually , the radiation  like result is expected . for where there is no dependency on the extra dimension , the usual four dimensional part of metric ( [ 7 ] ) and the third equation ( [ 15 ] ) give a wave equation for the scale factor of fifth dimension . hence , definitions ( [ 6.5 ] ) and ( [ 6.6 ] ) yield a traceless induced energy  momentum tensor , as mentioned in ref.@xcite .    exact solution of the second equation of ( [ 15 ] ) is @xmath77 substituting solution ( [ 19 ] ) into the first or third equation of ( [ 15 ] ) gives @xmath78 where @xmath79 and @xmath80 are constants of integration , and we have assumed that @xmath1 space  time has originated from a big bang . for a closed geometry , solution ( [ 19 ] ) admits @xmath81 and predicts a big crunch at @xmath82 for the usual spatial coordinates while the fifth dimension tends to infinite size and is always real , for the maximum value of the usual scale factor is @xmath83 . but , a flat geometry expands for ever and accepts @xmath81 . an open geometry also expands for ever and admits @xmath84 . in this case , @xmath85 results in @xmath86 and @xmath87 . time evolution of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 with constant values of @xmath89 and @xmath90 as an example .    in the next two sections , we again consider a more general situation in which the scale factor of the fifth dimension and the scalar field are  not constants .    [ cols="^,^,^ " , ] analogous to the approach of i m theories , one can consider the bd gravity as the underlying theory . hence , extra geometrical terms , coming from the fifth dimension , are regarded as an induced  matter and induced potential . we have followed , with some corrections , the procedure of ref . @xcite for introducing the induced potential and have employed a generalized frw type solution for a @xmath0 vacuum bd theory . hence , the scalar field and scale factors of the @xmath0 metric can , in general , be functions of the cosmic time and the extra dimension . however , for simplicity , we have assumed the scalar field and scale factors to be only functions of the cosmic time , where this makes the induced potential , by its definition , vanishes . we then have revealed that in general situations , in which the scale factor of the fifth dimension and scalar field are  not constants , the @xmath0 equations , for any kind of geometry , admit a power  law relation between the scalar field and scale factor of the fifth dimension . hence , the procedure exhibits that @xmath0 vacuum frw  like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new ( or modified ) scalar field and a new coupling constant . this equivalency can be viewed as the distinguished point of this work from refs . indeed , through investigating the @xmath0 vacuum frw  like equations , we have shown that its equivalent @xmath1 vacuum equations admit accelerated scale factors , contrary to what one may have expected from a vacuum space  time . conclusions of the complete investigation of the induced @xmath1 equations are as follows .    following our investigations for cosmological implications , we have shown that for the special case of a constant scale factor of the fifth dimension , the @xmath0 vacuum frw  like equations reduce to the corresponding equations of the usual @xmath1 vacuum bd theory , as expected . in the special case of a constant scalar field , the action reduces to a @xmath0 einstein gravitational theory and the equations reduce to the usual frw equations with a typical radiation dominated universe . for this situation , we also have obtained dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . solutions predict a limited life time for closed geometries and unlimited one for flat and open geometries . a typical time evolutions of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 . then , we have focused on spatially flat geometries and have obtained exact solutions of scale factors and scalar field . solutions are found to be in the form of power  law and exponential ones in the cosmic time . we also have employed the wec for the induced  matter of the @xmath1 modified bd gravity , that gives two conditions ( [ 49.1 ] ) and ( [ 49.2 ] ) . we then have pursued properties of these solutions and have indicated mathematically and physically acceptable ranges of them , and the results have been presented in a few tables and figures . all types of solutions fulfill the wecs in different ranges , where the exponential solutions are more restricted . the solutions fulfilling the wec ( [ 49.1 ] ) have negative pressures , but the figures illustrate that for the power  law results there are decelerating solutions beside accelerating ones . for this condition , both @xmath91 and @xmath92 decrease with the cosmic time , but the extra dimension grows . on the other hand , the solutions satisfying the wec ( [ 49.2 ] ) have positive pressures , where the power  law results accept accelerating solutions in addition to decelerating ones . for this condition , again decreasing energy density and pressure with the time can occur for some solutions , however all with shrinking extra dimension . the homogeneity between the extra dimension and the usual spatial dimensions , i.e. @xmath93 , can take place in the solutions , but for the power  law ones the wecs exclude it .    by considering non  ghost scalar fields and appealing the recent observational measurements , the solutions have been more restricted . actually , we have illustrated that the accelerating power  law solutions , which satisfy the wec and have non  ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the bd coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor . these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum bd equations . incidentally , this range is more restricted than the one obtained in ref . @xcite , i.e. @xmath94 , where the difference may have been caused by the distinct definition of the induced potential in two approaches of ref . @xcite and ref . however , we should remind 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