witten s invariants of rational homology spheres at prime values of k and trivial connection contribution we establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to witten s invariant of rational homology spheres and the invariants that t ohtsuki extracted from witten s invariant at prime values ofk we also rederive the properties of primek invariants discovered by h murakami and t ohtsuki we do this by using the bounds on taylor series expansion of the jones polynomial of algebraically split links studied in our previous paper these bounds are enough to prove that ohtsuki s invariants are of finite type the relation between ohtsuki s invariants and trivial connection contribution is verified explicitly for lens spaces and seifert manifolds

operads and knot spaces let em denote the space of embeddings of the interval i 1 1 in the cube im with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube equipped with a homotopy through immersions to the unknot see definition 5 1 by proposition 5 17 em is homotopy equivalent to emb i im x i lmm l im in 28 mcclure and smith define a cosimplicial object o associated to an operad with multiplication 0 whose homotopy invariant totalization we denote tot see definition 2 17 and definition 2 5 below let cm denote the rath kontsevich operad introduced in 22 whose entries are compactified configuration spaces and which is weakly equivalent to the little ra disks operad 37 see definition 4 1 and theorem 4 5

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

batalin vilkovisky quantization and the algebraic index abstract into a geometric setting we import the physical interpretation of index theorems via semi classical analysis in topological quantum field theory we develop a direct relationship between fedosov s deformation quantization of a symplectic manifold x and the batalin vilkovisky bv quantization of a one dimensional sigma model with target x this model is a quantum field theory of aksz type and is quantized rigorously using costello s homotopic theory of effective renormalization we show that fedosov s abelian connections on the weyl bundle produce solutions to the effective quantum master equation moreover bv integration produces a natural trace map on the deformation quantized algebra this formulation allows us to exploit a rigorous localization argument in quantum field theory to deduce the algebraic index theorem via semi classical analysis i e one loop feynman diagram computations

globalization for perturbative quantization of nonlinear split aksz sigma models on manifolds with boundary we describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split aksz sigma models on manifolds with and without boundary and show that it captures the change of the quantum state as one changes the constant map around which one perturbs this is done by using concepts of formal geometry moreover we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero this condition is a generalization of the modified quantum master equation as in the bv bfv formalism which we call the modified differential quantum master equation

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

minimal positive solutions to some singular second order differential equations abstract the existence of a minimal c1 0 1 positive solution is established for some second order singular boundary value and initial value problems by new schemes which are related to x our nonlinearity may be singular at t 0 1 x 0 or x 0

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

analysis of the loop length distribution for the negative weight percolation problem in dimensions d 2 through d 6 we consider the negative weight percolation nwp problem on hypercubic lattice graphs with fully periodic boundary conditions in all relevant dimensions from d 2 to the upper critical dimension d 6 the problem exhibits edge weights drawn from disorder distributions that allow for weights of either sign we are interested in the full ensemble of loops with negative weight i e nontrivial system spanning loops as well as topologically trivial small loops the nwp phenomenon refers to the disorder driven proliferation of system spanning loops of total negative weight while previous studies where focused on the latter loops we here put under scrutiny the ensemble of small loops our aim is to characterize using this extensive and exhaustive numerical study the loop length distribution of the small loops right at and below the critical point of the hypercubic setups by means of two independent critical exponents these can further be related to the results of previous finite size scaling analyses carried out for the system spanning loops for the numerical simulations we employed a mapping of the nwp model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms this allowed us to study here numerically exact very large systems with high statistics

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

the trivial connection contribution to witten s invariant and finite type invariants of rational homology spheres we derive an analog of the melvin morton bound on the power series expansion of the colored jones polynomial of algebraically split links and boundary links this allows us to produce a simple formula for the trivial connection contribution to witten s invariant of rational homology spheres we show that then th term in the 1 k expansion of the logarithm of this contribution is a finite type invariant of ohtsuki order 3n and of at most garoufalidis ordern

higher dimensional bf theories in the batalin vilkovisky formalism the bv action and generalized wilson loops this paper analyzes in detail the batalin vilkovisky quantization procedure for bf theories on an n dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables in particular generalized wilson loops for bf theories with additional polynomial b interactions are discussed in any dimensions the paper also contains the explicit proofs to the theorems stated in 16

algebraic characterization of vector supersymmetry in topological field theories a cohomological characterization of a class of linearly broken ward identities is provided the examples of the topological vector supersymmetry and of the landau ghost equation are discussed the existence of these linearly broken ward identities lies in the brst exactness of suitable antifield dependent cocycles with negative ghost number this algebraic set up turns out to be related to the cohomological reformulation of the noether theorem given by g barnich f brandt and m henneaux commun math phys 174 57 93 1995 m henneaux j pure appl algebra 100 3 1995

graph invariants of vassiliev type and application to 4d quantum gravity we consider graph invariants of vassiliev type extended by the quantum group link invariants when they are expanded byx whereq e x the expansion coefficients are known as the vassiliev invariants of finite type in the present paper we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3 space to a category of representations of the quasi triangular ribbon hopf algebra extended byu q sl 2 c which are subject to a quantum group analog of the spinor identity in terms of them we obtain the graph invariants of vassiliev type expressed to be identified with chern simons vacuum expectation values of wilson loops including intersection points we also consider the 4d canonical quantum gravity of ashtekar it is verified that the graph invariants of vassiliev type satisfy constraints of the quantum gravity in the loop space representation of rovelli and smolin

mean field behavior of the negative weight percolation model on random regular graphs we investigate both analytically and numerically the ensemble of minimum weight loops in the negative weight percolation model on random graphs with fixed connectivity and bimodal weight distribution this allows us to study the mean field behavior of this model the analytical study is based on a conjectured equivalence with the problem of self avoiding walks in a random medium the numerical study is based on a mapping to a standard minimum weight matching problem for which fast algorithms exist both approaches yield results that are in agreement on the location of the phase transition on the value of critical exponents and on the absence of any sizable indications of a glass phase by these results the previously conjectured upper critical dimension of d u 6 is confirmed

quantization of the nonlinear sigma model revisited we revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous mathematical point of view our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization the symmetries we consider are twofold i diffeomorphism covariance for a general target manifold ii a transitive group of isometries when the target manifold is a homogeneous space we show that there are no anomalies in case i and that ii is also anomaly free under additional assumptions on the target homogeneous space in agreement with the work of friedan we carry out some explicit computations for the o n model finally we show how a suitable notion of the renormalization group establishes the ricci flow as the one loop renormalization group flow of the nonlinear sigma model

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

singular mixed boundary value problem abstract we study singular boundary value problems with mixed boundary conditions of the form u f t u u 0 u 0 0 u t 0 where 0 t r d 0 0 f is a nonnegative function and satisfies the caratheodory conditions on 0 t d here f can have a time singularity at t 0 and or t t and a space singularity at x 0 and or y 0 we present conditions for the existence of solutions positive on 0 t and having continuous first derivatives on 0 t

feynman integrals and motives of configuration spaces we formulate the problem of renormalization of feynman integrals and its relation to periods of motives in configuration space instead of momentum space the algebro geometric setting is provided by the wonderful compactifications overline conf gamma x of arrangements of subvarieties associated to the subgraphs of a feynman graph with x a quasi projective variety the motive and the class in the grothendieck ring are computed explicitly for these wonderful compactifications in terms of the motive of x and the combinatorics of the feynman graph using recent results of li li the pullback to the wonderful compactification of the form defined by the unrenormalized feynman amplitude has singularities along a hypersurface whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification a regularization of the feynman integrals can be obtained by modifying the cycle of integration by replacing the divergent locus with a leray coboundary the ambiguities are then defined by poincare residues while these residues give periods associated to the cohomology of the exceptional divisors and their intersections the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification

associahedra cyclohedra and a topological solution to the a deligne conjecture abstract we give a topological solution to the a deligne conjecture using associahedra and cyclohedra to this end we construct three cw complexes whose cells are indexed by products of polytopes giving new explicit realizations of the polytopes in terms of different types of trees we are able to show that the cw complexes are cell models for the little discs the cellular chains of one complex in particular which is built out of associahedra and cyclohedra naturally act on the hochschild cochains of an a algebra yielding an explicit topological and minimal solution to the a deligne conjecture along the way we obtain new results about the cyclohedra such as new decompositions into products of cubes and simplices which can be used to realize them via a new iterated blow up construction

on universal vassiliev invariants using properties of ordered exponentials and the definition of the drinfeld associator as a monodromy operator for the knizhnik zamolodchikov equations we prove that the analytic and the combinatorial definitions of the universal vassiliev invariants of links are equivalent

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

positive solutions for singular systems of three point boundary value problems this paper investigates the following singular systems of nonlinear second order three point boundary value problems u f t v email protected 0 1 v g t u email protected 0 1 u 0 v 0 0 u 1 email protected h v 1 email protected h where email protected 0 1 0 email protected 1 f and g may be singular at t 0 and or t 1 under some weaker conditions the existence of positive solutions is obtained by applying the fixed point theorem of cone expansion and compression two examples are then presented to demonstrate the application of our main results

combinatorial expression for universal vassiliev link invariant a general model similar to r matrix type models for link invariants is constructed it contains all r matrix invariants and is a generating function for universal vassiliev link invariants this expression is simpler than kontsevich s expression for the same quantity because it is defined combinatorially and does not contain any integrals except for an expression for the universal drinfeld s associator

existence of positive solutions to some singular and nonsingular second order boundary value problems abstract the topological transversality theorem is employed to study existence of solutions to a variety of singular and nonsingular second order boundary value problems of the form y f t y y 0 0 t t 0 t 1 y 0 and or y 0 the principal tools are the arzela ascoli theorem essential maps and a priori bounds on solutions

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

odd chern simons theory lie algebra cohomology and characteristic classes we investigate the generic 3d topological field theory within the aksz bv framework we use the batalin vilkovisky bv formalism to construct explicitly cocycles of the lie algebra of formal hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes we investigate a toy model which is an odd analogue of chern simons theory and we give some explicit computation of two point functions and show that its perturbation theory is identical to the chern simons theory we give a concrete example of the homomorphism taking lie algebra cocycles to q characteristic classes and we reinterpret the rozansky witten model in this light

positive solutions of singular boundary value problems for nonlinear differential systems in this paper by constructing a special cone and applying the fixed index theory in the cone the existence of positive solutions for nonlinear ordinary differential systems with singularities is investigated and an example is developed to demonstrate our main results

classical bv theories on manifolds with boundary in this paper we extend the classical bv framework to gauge theories on spacetime manifolds with boundary in particular we connect the bv construction in the bulk with the bfv construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners we present several examples including electrodynamics yang mills theory and topological field theories coming from the aksz construction in particular the chern simons theory the bf theory and the poisson sigma model this paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing

on the globalization of the poisson sigma model in the bv bfv formalism we construct a formal global quantization of the poisson sigma model in the bv bfv formalism using the perturbative quantization of aksz theories on manifolds with boundary and analyze the properties of the boundary bfv operator moreover we consider mixed boundary conditions and show that they lead to quantum anomalies i e to a failure of the modified differential quantum master equation we show that it can be restored by adding boundary terms to the action at the price of introducing corner terms in the boundary operator we also show that the quantum grothendieck bfv operator on the total space of states is a differential i e squares to zero which is necessary for a well defined bv cohomology

directed negative weight percolation we consider the negative weight percolation model on directed graphs in particular we study the model on a two dimensional weighted periodic centered square lattice bond weights are taken from a distribution that allows for both positive and negative values for a given realization of the disorder a minimally weighted directed loop and path configuration is determined by performing a nontrivial transformation of the original lattice into a minimum weight perfect matching problem for this problem fast polynomial time algorithms are available thus we could study large systems with high accuracy depending on the fraction of negatively and positively weighted bonds on the lattice a continuous phase transition is identified and its characterizing critical exponents are estimated by means of a finite size scaling analysis we observe a strong change of the universality class with respect to standard directed percolation as well as with respect to undirected negative weight percolation furthermore the relation to directed polymers in random media is illustrated

on operad structures of moduli spaces and string theory we construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy lie algebras gravity algebras and batalin vilkovisky algebras these algebras appeared recently in the context of string theory and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

asymptotic expansion of the one loop approximation of the chern simons integral in an abstract wiener space setting abstract in an abstract wiener space setting we construct a rigorous mathematical model of the one loop approximation of the perturbative chern simons integral and derive its explicit asymptotic expansion for stochastic wilson lines

integral invariants of 3 manifolds ii this note is a sequel to our earlier paper of the same title 4 and describes invariants of rational homology 3 spheres associated to acyclic orthogonal local systems our work is in the spirit of the axelrod singer papers 1 generalizes some of their results and furnishes a new setting for the purely topological implications of their work

two dimensional yang mills theory on surfaces with corners in batalin vilkovisky formalism in this paper we recover the non perturbative partition function of 2d yang mills theory from the perturbative path integral to achieve this goal we study the perturbative path integral quantization for 2d yang mills theory on surfaces with boundaries and corners in the batalin vilkovisky formalism or more precisely in its adaptation to the setting with boundaries compatible with gluing and cutting the bv bfv formalism we prove that cutting a surface e g a closed one into simple enough pieces building blocks and choosing a convenient gauge fixing on the pieces and assembling back the partition function on the surface one recovers the known non perturbative answers for 2d yang mills theory

new perspectives on self linking abstract we initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot which is the induced map on the compactified n point configuration space sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots we compute the homotopy types of the first three mapping space models showing that the third model gives rise to an integer valued invariant we realize this invariant in two ways in terms of collinearities of three or four points on the knot and give some explicit computations we show this invariant coincides with the second coefficient of the conway polynomial thus giving a new geometric definition of the simplest finite type invariant finally using this geometric definition we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot

poisson sigma model with branes and hyperelliptic riemann surfaces we derive the explicit form of the superpropagators in the presence of general boundary conditions coisotropic branes for the poisson sigma model this generalizes the results presented by cattaneo and felder a path integral approach to the kontsevich quantization formula commun math phys 212 591 2000 and cattaneo and felder coisotropic submanifolds in poisson geometry and branes in the poisson sigma model lett math phys 69 157 2004 for kontsevich s angle function kontsevich m deformation quantization of poisson manifolds i e print arxiv hep th 0101170 used in the deformation quantization program of poisson manifolds the relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons pn with particular alternating boundary conditions in the presence of more than three branes we use first order riemann theta functions with odd singular characteristics on the jacobian variety of a hyperelliptic ri

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

reshetikhin s formula for the jones polynomial of a link feynman diagrams and milnor s linking numbers feynman diagrams are used to prove a formula for the jones polynomial of a link derived recently by reshetikhin this formula presents the colored jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components the large k limit of the integral can be calculated with the help of the stationary phase approximation the feynman rules allow one to express the phase in terms of integrals over the manifold and the link components its stationary points correspond to flat connections in the link complement a relation between the dominant part of the phase and milnor s linking numbers is conjectured it is checked explicitly for the triple and quartic numbers by comparing their expression through the massey product with feynman diagram integrals

unbounded solutions for singular boundary value problems on the semi infinite interval upper and lower solutions and multiplicity this paper presents a lower and upper solution technique for singular second order boundary value problems on the half line and discusses sufficient and necessary conditions for the existence of positive solutions also multiplicity is discussed using the fixed point index

positive solutions of some second order nonlinear singular differential equations abstract with new schemes the existence of the positive solutions is proved to some second order nonlinear singular boundary value problems and initial value problems which have higher order singularities

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

2 d physics and 3 d topology invariants of three dimensional manifolds and of framed oriented labeled links in them are rigorously defined using any solution to the moore seiberg axioms for a rational conformal field theory these invariants are generalizations of witten s chern simons path integrals connections are explored with supersymmetry four dimensional manifolds and quantum gravity

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

weyl n algebras we introduce weyl n algebras and show how their factorization complex may be used to define invariants of manifolds in the appendix we heuristically explain why these invariants must be perturbative chern simons invariants

upper critical dimension of the negative weight percolation problem by means of numerical simulations we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d 2 through 7 where edge weights are drawn from a distribution that allows for positive and negative weights we are interested in the appearance of system spanning loops of total negative weight the resulting negative weight percolation nwp problem is fundamentally different from conventional percolation as we have seen in previous studies of this model for the two dimensional case here we characterize the transition for hypercubic systems where the aim of the present study is to get a grip on the upper critical dimension d u of the nwp problem for the numerical simulations we employ a mapping of the nwp model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms we characterize the loops via observables similar to those in percolation theory and perform finite size scaling analyses e g three dimensional hypercubic systems with side length up to l 56 sites in order to estimate the critical properties of the nwp phenomenon we find our numerical results consistent with an upper critical dimension d u 6 for the nwp problem

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

semidensities on odd symplectic supermanifolds we consider semidensities on a supermanifold e with an odd symplectic structure we define a new operator action on semidensities as the proper framework for the batalin vilkovisky bv formalism we establish relations between semidensities on e and differential forms on lagrangian surfaces we apply these results to batalin vilkovisky geometry another application is to 1 1 codimensional surfaces in e we construct a kind of pull back of semidensities to such surfaces this operation and the operator are used for obtaining integral invariants for 1 1 codimensional surfaces

the cyclic bar construction on a h spaces abstract we set up a general framework for enriching a category a over a symmetric monoidal category c using a non operad p in c to do this we require a to come with a functor u a to the category of noncommutative sets by viewing the simplicial indexing category as a category over in two different ways we obtain two generalizations of simplicial objects for the operad given by the stasheff associahedra we obtain a model for the 2 sided bar construction in the first case and the cyclic bar and cobar construction in the second case using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects

perturbative expansion of chern simons theory with non compact gauge group naive imitation of the usual formulas for compact gauge group in quantizing three dimensional chern simons gauge theory with non compact gauge group leads to formulas that are wrong or unilluminating in this paper an appropriate modification is described which puts the perturbative expansion in a standard manifestly unitary format the one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed and their topological invariance is verified

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

link invariants and combinatorial quantization of hamiltonian chern simons theory we define and study the properties of observables associated to any link in r where is a compact surface using the combinatorial quantization of hamiltonian chern simons theory these observables are traces of holonomies in a non commutative yang mills theory where the gauge symmetry is ensured by a quantum group we show that these observables are link invariants taking values in a non commutative algebra the so called moduli algebra when s2 these link invariants are pure numbers and are equal to reshetikhin turaev link invariants

phase transitions in diluted negative weight percolation models we investigate the geometric properties of loops on two dimensional lattice graphs where edge weights are drawn from a distribution that allows for positive and negative weights we are interested in the appearance of spanning loops of total negative weight the resulting percolation problem is fundamentally different from conventional percolation as we have seen in a previous study of this model for the undiluted case here we investigate how the percolation transition is affected by additional dilution we consider two types of dilution either a certain fraction of edges exhibits zero weight or a fraction of edges is even absent we study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms we perform a finite size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary we find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change in the universality class

chern simons theory on r3 in axial gauge a rigorous approach we study pure chern simons models on m r3 using a functional integral quantization approach which is based on axial gauge fixing it is well known see e g comm math phys 126 1989 167 comm math phys 186 1997 563 that in axial gauge the chern simons action function is quadratic and that the faddeev popov determinant of this gauge fixing procedure is a constant function this means that the wilson loop observables wlos of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of gaussian type we demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from white noise analysis with certain regularization techniques like loop smearing and framing for the special case g u 1 g being the group of the model we carry out the details of this approach which is also applicable to the case of non abelian g and find well known linking number expressions for the wlos

quantum groups at odd roots of unity and topological invariants of 3 manifolds topological invariants of three manifolds are constructed using quantum groups associated with thea b c andd series of lie algebras at odd roots of unity these invariants are also explicitly computed for the lens spaces

heat kernels and thom forms abstract in this paper we give an alternate derivation of the gaussian shaped thom form of v mathai and d g quillen superconnections thom classes and equivariant differential forms topology 25 1986 85 110 for the case of the tangent bundle of an oriented compact manifold as a natural scaled limit of the index heat kernel of the laplacian acting on differential forms the precise statement of the result we obtain can be found in theorem 2 3 in this paper the technique we use is simply a variation of key ideas developed by getzler in e getzler a short proof of the local atiyah singer index theorem topology 25 1986 111 117 which he used to give a short and elegant proof of the local index theorem