flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

higgs bundles for real groups and the hitchin kostant rallis section we consider the moduli space of polystable l twisted g higgs bundles over a compact riemann surface x where g is a real reductive lie group and l is a holomorphic line bundle over x evaluating the higgs field at a basis of the ring of polynomial invariants of the isotropy representation one defines the hitchin map this is a map to an affine space whose dimension is determined by l and the degrees of the polynomials in the basis building up on the work of kostant rallis and hitchin in this paper as a first step in the study of the hitchin map we construct a section of this map this generalizes the section constructed by hitchin when l is the canonical line bundle of x and g is complex in this case the image of the section is related to the hitchin teichmuller components of the moduli space of representations of the fundamental group of x in gsplit a split real form of g in fact our construction is very natural in that we can start with the moduli space for gsplit instead of g and construct the section for the hitchin map for gsplit directly the construction involves the notion of maximal split subgroup of a real reductive lie group

mathrm so p q so p q higgs bundles and higher teichmuller components some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics others are more alluring and unusual either because they are not detected by primary invariants or because they have special geometric significance or both in this paper we describe new examples of such exotic components in moduli spaces of mathrm so p q higgs bundles on closed riemann surfaces or equivalently moduli spaces of surface group representations into the lie group mathrm so p q furthermore we discuss how these exotic components are related to the notion of positive anosov representations recently developed by guichard and wienhard we also provide a complete count of the connected components of these moduli spaces except for mathrm so 2 q with q geqslant 4

on the uniqueness of optimal solutions in continuous location theory abstract this paper deals with the uniqueness of an optimal solution in general continuous single facility minisum and minimax location problems we define the concept of an s norm and obtain general conditions which guarantee the existence of a unique optimal location some consequences for the uniqueness of optimal locations in multi facility location problems are discussed

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

a new assignment rule to improve seed points algorithms for the continuous k center problem abstract due to its complexity the k center problem cannot be solved exactly when the number of points is high enough so heuristic algorithms must be used to find good solutions for this location allocation problem the allocation part of the heuristics given in the literature consists of assigning each given point to its nearest center in this paper we propose a new assignment rule nar from which a new class of seed points algorithms is obtained for the k center problem in r n the new class differs from the usual class of seed points algorithms in the way the points are assigned to each seed point different methods to generate seed points are shown from which different algorithms are given by using the two assignment rules the classical and the new most of these algorithms have linear complexity in the number of points and generate solutions with objective values within two times the optimal value and all of them run very quickly and can be used for any metric it is shown by computational experience that the running times and the quality of the solutions generated are notably improved by using the nar

the moduli of flat pu p p structures with large toledo invariants for a compact riemann surface x of genus g 1 rm hom pi 1 x rm pu p q rm pu p q is the moduli space of flat rm pu p q connections on x there are two invariants the chern class c and the toledo invariant tau associated with each element in the moduli the toledo invariant is bounded in the range 2min p q g 1 le tau le 2min p q g 1 this paper shows that the component associated with a fixed tau 2 max p q 1 g 1 resp tau 2 max p q 1 g 1 and a fixed chern class c is connected the restriction on tau implies p q

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

deformations of twisted harmonic maps and variation of the energy we study the deformations of twisted harmonic maps f with respect to the representation rho after constructing a continuous universal twisted harmonic map we give a construction of every first order deformation of f in terms of hodge theory we apply this result to the moduli space of reductive representations of a kahler group to show that the critical points of the energy functional e coincide with the monodromy representations of polarized complex variations of hodge structure we then proceed to second order deformations where obstructions arise we investigate the existence of such deformations and give a method for constructing them as well applying this to the energy functional as above we prove for every finitely presented group that the energy functional is a potential for the kahler form of the betti moduli space assuming furthermore that the group is kahler we study the eigenvalues of the hessian of e at critical points

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

local rigidity of discrete groups acting on complex hyperbolic space the superrigidity theorem of margulis see zimmer 17 classifies finite dimensional representations of lattices in semi simple lie groups of real rank strictly larger than 1 it is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank i semi simple lie groups that this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or in other words the existence of non trivial deformations in johnson millson 6 and kourouniotis 8 such deformations were constructed for certain representations of lattices in so n 1 based on a construction of thurston called bending the deformation space of the representation of a lattice f so n 1 obtained by restricting an inclusion so n 1 so n l 1 to f is of particular interest if n 2 the space of infinitesimal deformations is hi f r 1 where f acts on r 1 by the restriction of the standard action of so n 1 the space of infinitesimal deformations is non zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable in 6 it is also shown that some are not in this paper we study the complex analogue of the above example we let f be a cocompact torsion free lattice in su n 1 and consider the deformation space of the representation of f obtained by restricting an inclusion su n 1 su n i 1 if n l the space of infinitesimal deformations is h f r h f c 1 in the first summand f acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming f in u n 1 by a curve of homomorphisms into the center of u n 1 observe that the above inclusion factors as su n 1 u n 1 su n i 1 in the second summand f acts by the restriction of the standard action of su n 1 this summand is non zero for the standard arithmetic examples

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

on cyclic higgs bundles in this paper we derive a maximum principle for a type of elliptic systems and apply it to analyze the hitchin equation for cyclic higgs bundles we show several domination results on the pullback metric of the possibly branched minimal immersion f associated to cyclic higgs bundles also we obtain a lower and upper bound of the extrinsic curvature of the image of f as an application we give a complete picture for maximal sp 4 mathbb r representations in the 2g 3 gothen components and the hitchin components

maximal representations of surface groups in bounded symmetric domains let f be the fundamental group of a hyperbolic surface of genus g for 1 2 leave invariant a totally geodesic subspace of dp q isometric to dp p

open sets of maximal dimension in complex hyperbolic quasi fuchsian space let pi 1 be the fundamental group of a closed surface sigma of genus g 1 one of the fundamental problems in complex hyperbolic geometry is to find all discrete faithful geometrically finite and purely loxodromic representations of pi 1 into su 2 1 the triple cover of the group of holomorphic isometries of h c 2 in particular given a discrete faithful geometrically finite and purely loxodromic representation rho 0 of pi 1 can we find an open neighbourhood of rho 0 comprising representations with these properties we show that this is indeed the case when rho 0 preserves a totally real lagrangian plane

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

surface group representations with maximal toledo invariant we develop the theory of maximal representations of the fundamental group 1 of a compact connected oriented surface possibly with boundary into lie groups g of hermitian type for any homomorphism p 1 g we define the toledo invariant t p a numerical invariant which has both topological and analytical interpretations we establish important properties of t among which continuity uniform boundedness on the representation variety additivity under connected sum of surfaces and congruence relations mod we thus obtain information about the representation variety as well as striking geometric properties of maximal representations that is representations whose toledo invariant achieves the maximum value moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle we define a rotation number function for general locally compact groups and study it in detail for groups of hermitian type properties of the rotation number together with the existence of boundary maps lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic

mirror symmetry for moduli spaces of higgs bundles via p adic integration we prove the topological mirror symmetry conjecture by hausel thaddeus for smooth moduli spaces of higgs bundles of type sl n and pgl n more precisely we establish an equality of stringy hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties our proof utilises p adic integration relative to the fibres and interprets canonical gerbes present on these moduli spaces as characters on the hitchin fibres using tate duality furthermore we prove for d prime to n that the number of rank n higgs bundles of degree d over a fixed curve defined over a finite field is independent of d this proves a conjecture by mozgovoy schiffmann in the coprime case

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

asymptotics of higgs bundles in the hitchin component in this paper we pursue a more geometric approach to compactification of the hitchin component our main motivation is wolf s harmonic map interpretation of thurston s compactification of teichmuller space with measured foliations using hitchin s parameterization of the hitchin component by holomorphic differentials we study asymptotics of certain rays of representations more precisely along these rays we solve the hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection the asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space sl n r so n r proves a conjecture of katzarkov noll pandit and simpson 17 on the hitchin wkb problem in our setting

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

unified algebraic method to non hermitian systems with lie algebraic linear structure we suggest a generic algebraic method to solve non hermitian hamiltonian systems with lie algebraic linear structure such method can not only unify the non hermitian hamiltonian and the hermitian hamiltonian with the same structure but also keep self consistent between similarity transformation and unitary transformation to clearly reveal the correctness and physical meaning of such algebraic method we illustrate our method with two different types of non hermitian hamiltonians the non hermitian hamiltonian with heisenberg algebraic linear structure and the non hermitian hamiltonian with su 1 1 algebraic linear structure we obtain energy eigenvalues and the corresponding eigenstates of non hermitian forced harmonic oscillator with heisenberg algebra structure via a proper similarity transformation we also calculate the eigen problems of general non hermitian hamiltonian with su 1 1 structure in terms of the similarity transformation as an application we focus on studying the non hermitian sing

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

remarks on the pseudo norm in symmetric quantum mechanics this paper presents an underlying analytical relationship between the pseudo norm associated with the symmetric hamiltonian h p2 v q and the stokes multiplier of the differential equation corresponding to this hamiltonian we show that the sign alternation of the pseudo norm which has been observed as a generic feature of the inner product is essentially controlled by the derivative of a stokes multiplier with respect to the eigenparameter

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

a note on the m center problem with rectilinear distances abstract given n demand points on a plane the problem we consider is to locate a given number m of facilities on the plane so that the maximum of the set of rectilinear distances of each demand point to its nearest facility is minimized this problem is known as the m center problem on the plane a related problem seeks to determine for a given r the minimum number of facilities and their locations so as to ensure that every point is within r units of rectilinear distance from its nearest facility we formulate the latter problem as a problem of covering nodes by cliques of an intersection graph certain bounds are established on the size of the problem an efficient algorithm is provided to generate this set covering problem computational results with this approach are summarized

ideal triangle groups dented tori and numerical analysis we prove the goldman parker conjecture a complex hyperbolic ideal triangle group is discretely embedded in pu 2 1 if and only if the product of its three standard generators is not elliptic we also prove that such a group is indiscrete if the product of its three standard generators is elliptic a novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

position dependent effective mass dirac equations with pt symmetric and non pt symmetric potentials we present a new procedure to construct the one dimensional non hermitian imaginary potential with a real energy spectrum in the context of the position dependent effective mass dirac equation with the vector coupling scheme in 1 1 dimensions in the first example we consider a case for which the mass distribution combines linear and inversely linear forms the dirac problem with a pt symmetric potential is mapped into the exactly solvable schrodinger like equation problem with the isotonic oscillator by using the local scaling of the wavefunction in the second example we take a mass distribution with smooth step shape the dirac problem with a non pt symmetric imaginary potential is mapped into the exactly solvable schrodinger like equation problem with the rosen morse potential the real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method

maximal representations of uniform complex hyperbolic lattices let rho be a maximal representation of a uniform lattice gamma subset rm su n 1 n geq 2 in a classical lie group of hermitian type h we prove that necessarily h rm su p q with p geq qn and there exists a holomorphic or antiholomorphic rho equivariant map from complex hyperbolic space to the symmetric space associated to rm su p q this map is moreover a totally geodesic homothetic embedding in particular up to a representation in a compact subgroup of rm su p q the representation rho extends to a representation of rm su n 1 in rm su p q

a squeeze like operator approach to position dependent mass in quantum mechanics we provide a squeeze like transformation that allows one to remove a position dependent mass from the hamiltonian methods to solve the schrodinger equation may then be applied to find the respective eigenvalues and eigenfunctions as an example we consider a position dependent mass that leads to the integrable morse potential and therefore to well known solutions

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

toledo invariant of lattices in su 2 1 via symmetric square in this paper we address the issue of the quaternionic toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into su n 2 n 4 we construct four distinct representations to prove that the character variety contains at least seven distinct components we also show the existence of holomorphic horizontal lift to various period domains of su n 2

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

density of zariski density for surface groups we show that a surface group contained in a reductive real algebraic group can be deformed to become zariski dense unless its zariski closure acts transitively on a hermitian symmetric space of tube type this is a kind of converse to a rigidity result of burger iozzi and wienhard

exactly solvable non hermitian jaynes cummings type hamiltonian admitting entirely real spectra from supersymmetry it is shown that for a given hermitian hamiltonian possessing supersymmetry there is always a non hermitian jaynes cummings type hamiltonian jcth admitting entirely real spectra the parent supersymmetric hamiltonian and the corresponding non hermitian jcth are simultaneously diagonalizable the exact eigenstates of these non hermitian hamiltonians are constructed algebraically for certain shape invariant potentials including a non hermitian version of the standard jaynes cummings jc model for which the parent supersymmetric hamiltonian is the superoscillator it is also shown that a non hermitian version of the several physically motivated generalizations of the jc model admits entirely real spectra the positive definite metric operator in the hilbert space is constructed explicitly along with the introduction of a new inner product structure so that the eigenstates form a complete set of orthonormal vectors and the time evolution is unitary

bending fuchsian representations of fundamental groups of cusped surfaces in pu 2 1 we describe a family of representations of 1 in pu 2 1 where is a hyperbolic riemann surface with at least one deleted point which is obtained by a bending process associated to an ideal triangulation of we give an explicit description of this family by describing a coordinate system on it which is in the spirit of fock and goncharov coordi

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

cyclic surgery degrees of maps of character curves and volume rigidity for hyperbolic manifolds this paper proves a theorem about dehn surgery using a new theorem about psl2c character varieties confirming a conjecture of boyer and zhang this p aper shows that a small hyperbolic knot in a homotopy sphere having a non trivial cyclic slope r has an incompressible surface with non integer boundary slope strictly between r 1 and r 1 a corollary is that any small knot which has only integer boundary slopes has property p the proof uses connections between the topology of the complement of the knot m and the psl2c character variety of m that were discovered by culler and shalen the key lemma which should be of independent interest is that for certain components of the character variety of m the map on character varieties induced by m m is a birational isomorphism onto its image this in turn depends on a fancy version of mostow rigidity due to gromov thurston and goldman the proof of the theorem about dehn surgery also gives information about the diameter of the set of boundary slopes of m let me start with the theorem about dehn surgery let k be a knot in a compact closed 3manifold that is a tame embedding s 1 the complement of k m is minus an open regular neighborhood of k so m is a compact 3 manifold whose boundary is a torus suppose is a simple closed curve in m we can create a closed manifold m from m by taking a solid torus and gluing its boundary to m in such a way that bounds a disc in the solid torus m depends only on the isotopy class of the new manifold m is called a dehn filling of m or a dehn surgeryon k recently many people have studied what kinds of manifolds arise when you

moduli spaces of holomorphic triples over compact riemann surfaces a holomorphic triple over a compact riemann surface consists of two holomorphic vector bundles and a holomorphic map between them after fixing the topological types of the bundles and a real parameter there exist moduli spaces of stable holomorphic triples in this paper we study non emptiness irreducibility smoothness and birational descriptions of these moduli spaces for a certain range of the parameter our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary lie groups of indefinite signature 5 7 another application that we study in this paper is to the existence of stable bundles on the product of the surface by the complex projective line

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

the gromov norm of the kaehler class of symmetric domains let d be bounded symmetric domain of rank p and let o9 be the krhler form of its bergmann metric where the metric is normalized so that the minimum holomorphic sectional curvature is 1 i fx is a compact manifold whose universal cover is d to defines a cohomology class to e h 2 x r the purpose of this paper is to compute its sup norm in the sense of gromov 5 we show ii to ii co pn strictly speaking we only prove this for three of the four classes of classical domains cf sect 2 this theorem has the following topological corollary let s be a riemann surface of genus 0 1 and f s x a continuous map then f t o 4 p 1 n

involutions and higher order automorphisms of higgs bundle moduli spaces this research was partially supported by the spanish mineco under the icmat severo ochoa grant no sev 2011 0087 and grant no mtm2013 43963 p and by the european commission marie curie irses moduli programme pirses ga 2013 612534

bihermitian surfaces with odd first betti number compact bihermitian surfaces are considered that is compact oriented conformal four manifolds admitting two distinct compatible complex structures it is shown that if the first betti number is odd then with respect to either complex structure such a manifold belongs to class vii of the enriques kodaira classification moreover it must be either a special hopf or an inoue surface in the strongly bihermitian case or obtained by blowing up a minimal class vii surface with curves in the non strongly bihermitian case

anosov representations domains of discontinuity and applications the notion of anosov representations has been introduced by labourie in his study of the hitchin component for sl n r subsequently anosov representations have been studied mainly for surface groups in particular in the context of higher teichmuller spaces and for lattices in so 1 n in this article we extend the notion of anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties in particular given an anosov representation g we explicitly construct open subsets of compact g spaces on which acts properly discontinuously and with compact quotient as a consequence we show that higher teichmuller spaces parametrize locally homogeneous geometric structures on compact manifolds we also obtain applications regarding non standard compact clifford klein forms and compactifications of locally symmetric spaces of infinite volume

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

pseudo hermiticity and some consequences of a generalized quantum condition we exploit the hidden symmetry structure of a recently proposed non hermitian hamiltonian and of its hermitian equivalent one this sheds new light on the pseudo hermitian character of the former and allows access to a generalized quantum condition special cases lead to hyperbolic and morse like potentials in the framework of a coordinate dependent mass model

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

energy of sections of the deligne hitchin twistor space we study a natural functional on the space of holomorphic sections of the deligne hitchin moduli space of a compact riemann surface generalizing the energy of equivariant harmonic maps corresponding to twistor lines we show that the energy is the residue of the pull back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by hitchin as a byproduct we show the existence of a hyper kahler potentials for new components of real holomorphic sections of twistor spaces of hyper kahler manifolds with rotating s 1 action additionally we prove that for a certain class of real holomorphic sections of the deligne hitchin moduli space the energy functional is basically given by the willmore energy of corresponding equivariant conformal map to the 3 sphere as an application we use the functional to distinguish new components of real holomorphic sections of the deligne hitchin moduli space from the space of twistor lines

transition elements for a non hermitian quadratic hamiltonian the non hermitian quadratic hamiltonian h a a a2 a 2 is analyzed where a and a are harmonic oscillator creation and annihilation operators and and are real constants for the case that 2 4 0 it is shown using operator techniques that the hamiltonian possesses real and positive eigenvalues a generalized bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator the eigenstates are shown to possess an imaginary norm for a large range of the parameter space finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of bender et al for non hermitian hamiltonians transition probabilities governed by h are shown to be manifestly unitary when the complexification procedure is followed a specific transition element between harmonic oscillator states is evaluated for both the hermitian and non hermitian cases to identify the differences in time evolution

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

the quantum effective mass hamilton jacobi problem in this paper the quantum hamilton jacobi theory based on the position dependent mass model is studied two effective mass functions having different singularity structures are used to examine the morse and poschl teller potentials the residue method is used to obtain the solutions of the quantum effective mass hamilton jacobi equation further it is shown that the eigenstates of the generalized non hermitian swanson hamiltonian for morse and poschl teller potentials can be obtained by using the riccati equation without solving a differential equation

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c