on kawai theorem for orbifold riemann surfaces we prove a generalization of kawai theorem for the case of orbifold riemann surface the computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the teichmuller space to the mathrm psl 2 mathbb c character variety which allows to evaluate explicitly the pullback of goldman symplectic form in the spirit of riemann bilinear relations as a corollary we obtain a generalization of goldman s theorem that the pullback of goldman symplectic form on the mathrm psl 2 mathbb r character variety is a symplectic form of the weil petersson metric on the teichmuller space

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

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

analysis on generalized superspaces the analysis over commutative algebras generalized supercommutative algebras that is differentiation and integration for functions defined on superspace over a commutative algebra is studied

the kauffman skein algebra of a surface at sqrt 1 we study the structure of the kauffman algebra of a surface with parameter equal to sqrt 1 we obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat su 2 connections over the surface we analyse the asymptotics of traces of curve operators in tqft in non standard regimes where the root of unity parametrizing the tqft accumulates to a root of unity we interpret the case of sqrt 1 in terms of parallel transport operators

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

group cohomology and the symplectic structure on the moduli space of representations the space of equivalence classes of irreducibleb representations of the fundamental group of a compact oriented surface of genus at least 2 in a lie group has a natural symplectic form in lab atiyah and bott described this symplectic structure using methods from gauge theory goldman g constructed the skew symmetric pairing algebraically using methods from group cohomology using poincar6 duality goldman showed that the pairing is nondegenerate and identified it with the symplectic structure given by gauge theory

an extreme value theorem on closed convex subsets of lp spaces and two applications abstract in this paper we prove that each r valued bounded below lower semicontinuous function defined on e reaches its infimum where e is a closed convex subset of l p x a and each element of e is dominated by some h l p x a p 1 as a consequence of this extreme value theorem a version of the minimax theorem is proved in the context of dominated p integrable functions as above one more consequence is that certain euler lagrange equation has a weak solution in a dominated sobolev subspace of w 0 1 1

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

superfiber bundle structure of gauge theories with faddeev popov fields in this work the mathematical structure of principal superfiber bundle ps is used to give a geometrical description of gauge theories the base space of ps is an s4 2 supermanifold xs four commuting and two anticommuting variables and the structure group an sn 0 supergroup gs where n is the dimension of the gauge group for the classical theory the body of ps is the usual principal fiber bundle p of gauge theories gauge and faddeev popov fields arise as superfields components of the connections in ps in a local coordinate system brs becchi rouet and stora and anti brs transformations are gauge transformations in ps of parameters the ghost and antighost superfields respectively in the case of soul flat connections which are connections in ps coming from connections in p the brs and anti brs transformations are finite translations along the anticommuting directions of xs and generate an s0 2 supergroup

n 2 topological gauge theory the euler characteristic of moduli spaces and the casson invariant we discuss gauge theory with a topologicaln 2 symmetry this theory captures the de rham complex and riemannian geometry of some underlying moduli space and the partition function equals the euler number of we explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions to motivate our constructions we explain the relation between the mathai quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the gauss codazzi equations we interpret the gauge theory actions from the atiyah jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections as a consequence of these considerations we propose the euler number of the moduli space of flat connections as a generalization to arbitrary three manifolds of the casson invariant we also comment on the possibility of constructing a topological version of the penner matrix model

some minimax theorems of set valued maps and their applications abstract in the paper we consider two types of scalarization functions of sets and investigate their properties moreover based on two set relations we propose two kinds of minimax and maximin values of set valued maps respectively and show some minimax theorems of set valued maps with respect to those minimax and maximin values by using several properties of the above two functions as an application of these results we give common saddle point theorems of vector valued functions

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

a knot invariant via representation spaces a beautiful construction of a casson on the representation spaces corresponding to a heegard splitting of an oriented homology 3 sphere m gives rise to an integer invariant m of m this invariant generalizes the rohlin invariant and gives stricking corollaries in low dimension al topology defined as an intersection number of appropriate subspaces this invariant m can be roughly thought of as the number of conjugacy classes of irreducible representations of n m into su2 counted with signs a detailed discussion of this invariant can be found in an expose by s akbulut and j mccarthy 1 further works on casson s invariant include the generalizations by k walker as well as s boyer and a nicas to rational homology 3 spheres 14 3 and by s cappell r lee and e miller to representatio ns into sun 4 the works of c taubes 13 and a flore 6 interpret casson s invariant as the euler number of the instanton homology or flore homology of m in this paper analogous to casson s original construction we will define an intersection number of the representation spaces corresponding to a braid representative of a knot k in s this intersection number turns out to be an integer knot invariant see theorem 1 8 the representations of the knot group s k used in our construction seem to be mysterious they are representations of s3 k into su2 such that all meridians of k are represented by trace zero matrices call such a representation of the knot group a trace free representation then roughly speaking our knot invariant h k is the number of conjugacy classes of irreducible trace free representations of s3 k counted with signs our knot invariant h k can be computed via knot diagrams it turns out that our algorithm of computing h k by using the skein model is the same as the algorithm of computing sgn k given by j h conway 5

characteristic classes and representations of discrete subgroups of lie groups a volume invariant is used to characterize those representations of a countable group into a connected semisimple lie group g which are injective and whose image is a discrete cocompact subgroup of g let it be a discrete cocompact subgroup of g and consider the analytic variety hom 7r g consisting of homomorphisms 0 tt g denote by k a maximal compact subgroup of g and x k g the associated symmetric space let m be the orbit space x tt for convenience we shall henceforth assume that n is torsionfree by selberg s lemma 12 this may be accomplished by replacing tt by a subgroup of finite index this insures that m is a compact smooth manifold having tt as its fundamental group the case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here to every representation 0 g hom 7r g we associate a foliated bundle e over m with fibre x and structure group g see e g 6 if co is a closed ginvariant differential fc form on x then we may spread co over the fibres of e copies of x to obtain a closed fc form co on e we define co 0 fj f where is any section of e moreover co 0 is independent of the choice of section for example taking co to be the g invariant volume form on x we obtain a real number u 0 which depends on 0 when x is even dimensional the chern gauss bonnet theorem implies that u 0 may be described as an euler number i e the self intersection number of any section which is a topological invariant of e when x is odd dimensional this volume invariant is related to a more recent kind of topological invariant based on bounded cohomology and due to gromov 3 and is constant on the connected components of hom 7r g the volume invariant satisfies an inequality

character varieties of abelian groups we prove that for every reductive group g with a maximal torus mathbb t and the weyl group w mathbb t n w is the normalization of the irreducible component x g 0 mathbb z n of the g character variety x g mathbb z n of mathbb z n containing the trivial representation we also prove that x g 0 mathbb z n mathbb t n w for all classical groups additionally we prove that even though there are no irreducible representations in x g 0 mathbb z n for non abelian g the tangent spaces to x g 0 mathbb z n coincide with h 1 mathbb z n ad rho consequently x g 0 mathbb z 2 has the goldman symplectic form for which the combinatorial formulas for goldman bracket hold

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

integration on supermanifolds and a generalized cartan calculus a suggestion by berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection the relevant objects for integration are identified as suitable horizontal vertical projections of hyperforms the latter are generalizations of differential forms having both covariant and contravariant indices the exterior calculus of these projected hyperforms is developed analogously to the cartan calculus by introducing appropriate derivations and determining their commutators respectively anticommutators

weil petersson volumes and intersection theory on the moduli space of curves in this paper we establish a relationship between the weil petersson volume vgin b of the moduli space mg n b of hyperbolic riemann surfaces with geodesic boundary components of lengths b bn and the intersection numbers of tauto logical classes on the moduli space mg n of stable curves as a result by using the recursive formula for vg n b obtained in 22 we derive a new proof of the virasoro constraints for a point this result is equivalent to the witten kontsevich formula 14

the volume of the moduli space of flat connections on a nonorientable 2 manifold we compute the riemannian volume of the moduli space of flat connections on a nonorientable 2 manifold for a natural class of metrics we also show that witten s volume formula for these moduli spaces may be derived using haar measure and we give a new proof of witten s volume formula for the moduli space of flat connections on a 2 manifold using haar measure

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

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

how different are the supermanifolds of rogers and dewitt a dewitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold whereas a rogers supermanifold is not so restricted corresponding to the vector space fibers of the dewitt supermanifold a rogers supermanifold has a foliation by submanifolds or leaves parametrized by soul coordinates only we show that the universal covering space of any leaf always admits a flat metric if the covering space is complete in this metric it must in fact be a vector space we combine this result with known theorems about foliations to give conditions under which a compact rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace we also show that a supermanifold defined by a polynomial equation in flat superspace is always of the dewitt type finally we exhibit new supermanifold structures forr2 and the 2 torus which show that the foliation of a rogers supermanifold can be quite exotic

left and right centers in quasi poisson geometry of moduli spaces abstract we introduce left central and right central functions and left and right leaves in quasi poisson geometry generalizing central or casimir functions and symplectic leaves from poisson geometry they lead to a new type of quasi poisson reduction which is both simpler and more general than known quasi hamiltonian reductions we study these notions in detail for moduli spaces of flat connections on surfaces where the quasi poisson structure is given by an intersection pairing on homology

on the body of supermanifolds the problem of constructing the body of a g manifold is considered it is shown that any such manifold is foliated and the body is defined to be the space of the leaves of this foliation under certain regularity conditions on the foliation the body is a smooth finite dimensional real manifold

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

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

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

berezin integration on general fermionic supermanifolds recent results on the global structure of supermanifolds are used to define a notion of berezin integration on any purely fermionic rogers supermanifold this leads to an integration theory on a large class of supermanifolds having both bosonic and fermionic coordinates the existence of global functions and forms on such supermanifolds is discussed as is some elementary cohomology of supermanifolds

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

supersymmetric field theoretic models on a supermanifold we propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general space time manifold so as to include superfield models on a supermanifold we only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure our considerations are based on the catenacci reina teofillatto bryant approach to supermanifolds in particular we show that the class of supermanifolds constructed by bonora pasti tonin satisfies the criteria which guarantee that a supermanifold admits a hausdorff body manifold this construction is the closest to the physicist s intuitive view of superspace as a manifold with some anticommuting coordinates where the odd sector is topologically trivial the paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects moreover a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions which is equivalent to the requirement that all of the component fields satisfy on the body manifold a microlocal spectral condition proposed by brunetti fredenhagen kohler

generalized supermanifolds iii ρ supermanifolds supermanifolds are characterized as supermanifolds modeled in a superspace s having a well behaved body tangent cotangent and tensor bundles to supermanifolds are discussed the elements of the theory of lie supergroups and principal superfiber bundles as supermanifolds are given

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities