localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

quality adjusted survival estimation with periodic observations quality adjusted survival is a measure that integrates both longevity and quality of life information the analysis of quality adjusted survival in a clinical study with data collected at periodic intervals encounters difficulties due to incomplete information based on observed time points the time axis is partitioned into a set of disjoint time intervals and under a markovian assumption on patient s health status the expected quality adjusted survival is estimated as the summed product of the quality of life and its mean sojourn time of each health state within partitioned intervals it is shown that the estimator is asymptotically normal with a simple variance calculation a simulation study is conducted to investigate the behavior of the estimator and a stroke study illustrates the use of the estimator

n 2 topological yang mills theory on compact kahler surfaces we study a topological yang mills theory withn 2 fermionic symmetry our formalism is a field theoretical interpretation of the donaldson polynomial invariants on compact kahler surfaces we also study an analogous theory on compact oriented riemann surfaces and briefly discuss a possible application of witten s non abelian localization formula to the problems in the case of compact kahler surfaces

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

efficient estimation of the distribution of quality adjusted survival time quality of life is an important aspect in evaluation of clinical trials of chronic diseases such as cancer and aids quality adjusted survival analysis is a method that combines both the quantity and quality of a patient s life into one single measure in this paper we discuss the efficiency of weighted estimators for the distribution of quality adjusted survival time using the general representation theorem for missing data processes we are able to derive an estimator that is more efficient than the one proposed in zhao and tsiatis 1997 biometrika 84 339 348 simulation experiments are conducted to assess the small sample properties of this estimator and to compare it with the semiparametric efficiency bound the value of this estimator is demonstrated from an application of the method to a data set obtained from a breast cancer clinical trial

yang mills measure and the master field on the sphere we study the yang mills measure on the sphere with unitary structure group in the limit where the structure group has high dimension we show that the traces of loop holonomies converge in probability to a deterministic limit which is known as the master field on the sphere the values of the master field on simple loops are expressed in terms of the solution of a variational problem we show that given its values on simple loops the master field is characterized on all loops of finite length by a system of differential equations known as the makeenko migdal equations we obtain a number of further properties of the master field on specializing to families of simple loops our results identify the high dimensional limit in non commutative distribution of the brownian bridge in the group of unitary matrices starting and ending at the identity

bohr sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula we show how the moduli space of flatsu 2 connections on a two manifold can be quantized in the real polarization of 15 using the methods of 6 the dimension of the quantization given by the number of integral fibres of the polarization matches the verlinde formula which is known to give the dimension of the quantization of this space in a kahler polarization

a quantum group version of quantum gauge theories in two dimensions for the special case of the quantum group slq 2 c q exp pi i r r or 3 the authors present an alternative approach to quantum gauge theories in two dimensions they exhibit the similarities to witten s combinatorial approach which is based on the ideas of migdal 1970 the main ingredient is the turaev viro combinatorial construction of topological invariants of closed compact 3 manifolds and its extension to arbitrary compact 3 manifolds as given by the authors in collaboration with w muller 1992

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

matrix group integrals surfaces and mapping class groups i u n since the 1970 s physicists and mathematicians who study random matrices in the gue or goe models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces we establish a new aspect of this theory for random matrices sampled from the group u n of unitary matrices more concretely we study measures induced by free words on u n let fr be the free group on r generators to sample a random element from u n according to the measure induced by w fr one substitutes the r letters in w by r independent haar random elements from u n the main theme of this paper is that every moment of this measure is determined by families of pairs f where is an orientable surface with boundary and f is a map from to the bouquet ofr circles which sends the boundary components of to powers of w a crucial role is then played by euler characteristics of subgroups of the mapping class group of as corollaries we obtain asymptotic bounds on the moments we show that the measure on u n bears information about the number of solutions to the equation u1 v1 ug vg w in the free group and deduce that one can hear the stable commutator length of a word through its unitary word measures

variance and sample size calculations in quality of life adjusted survival analysis q twist the quality adjusted time without symptoms or toxicity q twist statistic previously introduced by glasziou simes and gelber 1990 statistics in medicine 9 1259 1276 combines toxicity disease free survival and overall survival information in assessing the impact of treatments on the lives of patients this methodology has received positive reviews from clinicians as intuitive and useful but to date the variance of this statistic has remained unspecified we review aspects of the q twist method for analyzing clinical trial data extend the method to accommodate multiple treatment arms and provide closed form asymptotic variance formulas we also provide a framework for designing q twist clinical trials with sample sizes determined using the derived asymptotic variance formulas trials currently collecting quality of life data did not have the benefit of these sample size calculation techniques in designing their studies

gauge theories and macdonald polynomials we study the mathcal n 2 four dimensional superconformal index in various interesting limits such that only states annihilated by more than one supercharge contribute extrapolating from the su 2 generalized quivers which have a lagrangian description we conjecture explicit formulae for all a type quivers of class mathcal s which in general do not have one we test our proposals against several expected dualities the index can always be interpreted as a correlator in a two dimensional topological theory which we identify in each limit as a certain deformation of two dimensional yang mills theory the structure constants of the topological algebra are diagonal in the basis of macdonald polynomials of the holonomies

partition functions of n 2 2 gauge theories on s 2 and vortices we apply localization techniques to compute the partition function of a two dimensional mathcal n 2 2 r symmetric theory of vector and chiral multiplets on s2 the path integral reduces to a sum over topological sectors of a matrix integral over the cartan subalgebra of the gauge group for gauge theories which would be completely higgsed in the presence of a fayet iliopoulos term in flat space the path integral alternatively reduces to the product of a vortex times an antivortex partition functions weighted by semiclassical factors and summed over isolated points on the higgs branch for applications we evaluate the partition function for some u n gauge theories showing equality of the path integrals for theories conjectured to be dual by hori and tong and deriving new expressions for vortex partition functions

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

operator algebraic construction of gauge theories and jones actions of thompson s groups using ideas from jones lattice gauge theory and loop quantum gravity we construct 1 1 dimensional gauge theories on a spacetime cylinder given a separable compact group g we construct localized time zero fields on the spatial torus as a net of c algebras together with an action of the gauge group that is an infinite product of g over the dyadic rationals and using a recent machinery of jones an action of thompson s group t as a replacement of the spatial diffeomorphism group adding a family of probability measures on the unitary dual of g we construct a state and obtain a net of von neumann algebras carrying a state preserving gauge group action for abelian g we provide a very explicit description of our algebras for a single measure on the dual of g we have a state preserving action of thompson s group and semi finite von neumann algebras for g mathbf s the circle group together with a certain family of heat kernel states providing the measures we obtain hyperfinite type iii factors with a normal faithful state providing a nontrivial time evolution via tomita takesaki theory kms condition in the latter case we additionally have a non singular action of the group of rotations with dyadic angles as a subgroup of thompson s group t for geometrically motivated choices of families of heat kernel states

sewing yang mills measures and moduli spaces over compact surfaces consider a closed surface obtained by pasting two surfaces 1and 2along some of their boundary components a formula is derived expressing quantum yang mills functional integrals for in terms of the corresponding integrals for the surfaces i using this a formula is derived which expresses the symplectic volume measure on the moduli space of flatsu 2 connections over a closed riemann surface in terms of the corresponding volume measures for the surfaces 1and 2 when the latter have one boundary component each

similarity classes of integral p adic matrices and representation zeta functions of groups of type a2 we compute explicitly dirichlet generating functions enumerating finite dimensional irreducible complex representations of various p adic analytic and adelic profinite groups of type a2 this has consequences for the representation zeta functions of arithmetic groups h k where k is a number field and h is a k form of sl3 assuming that possesses the strong congruence subgroup property we obtain precise uniform estimates for the representation growth of our results are based on explicit uniform formulae for the representation zeta functions of the p adic analytic groups sl3 o and su3 o where o is a compact discrete valuation ring of characteristic 0 these formulae build on our classification of similarity classes of integral p adic 3 3 matrices in gl3 o and gu3 o where o is a compact discrete valuation ring of arbitrary characteristic organising the similarity classes by invariants which we call their shadows allows us to combine the kirillov orbit method with clifford theory to obtain explicit formulae for representation zeta functions in a different direction we introduce and compute certain similarity class zeta functions our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form sl3 o su3 o gl3 o and gu3 o arising from the respective congruence filtrations these formulae are valid in case that the characteristic of o is either 0 or sufficiently large analysis of some of these formulae leads us to observe p adic analogues of ennola duality

regularization method for the application of k space data extrapolation to near field acoustical holography a regularization method for applying the k space data extrapolation method to near field acoustical holography nah is proposed the influence of the finiteness of measurement aperture is a serious impediment to the nah reconstruction of structural vibration when the size of the measurement aperture is smaller than that of the structure the k space data extrapolation method has been proposed to reduce the influence of this finiteness an appropriate regularization method is necessary to apply the k space data extrapolation method if the measured pressure is contaminated by noise some regularization methods for nah have been proposed however it is difficult to determine an appropriate k space filter if the finiteness of the measurement aperture becomes an issue an estimation method of appropriate regularization filter for applying the k space data extrapolation method is proposed to solve this difficulty the effectiveness of this method is demonstrated by simulations the simulation results confirm

quantum yang mills on the two sphere we obtain the quantum expectations of gauge invariant functions of the connection on a principalg su n bundle overs 2 we show that the spacea g m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over g based loops in the symmetry group the fiber ina g m is an affine linear space quantum expectations are iterated path integrals first over this fiber then over g each with respect to the push forward toa g m of the measure s s a da s a denotes the yang mills action ona there is a global section ofa g m on which the first integral is a gaussian the resulting measure on g is the conditional wiener measure we explicitly compute the expectations of a special class of wilson loops

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

nonparametric estimation of the joint distribution of survival time and mark variables summary in many applications variables of interest are marks of the endpoint which are not observed when the survival time is censored this paper focuses on nonparametric estimation of the joint distribution and summaries of survival time and mark variables we establish a representation of the joint distribution function through the cumulative markspecific hazard function which is analogous to the product integral representation of univariate survival function we identify a basic data structure common to various applications propose nonparametric estimators and show that they maximise the likelihood we formulate the problem in the marked point process framework and study both finite and large sample properties of the estimators we show that the joint distribution function estimator is nearly unbiased uniformly strongly consistent and asymptotically normal we also derive asymptotic variances for the estimators and propose sample based variance estimates numerical studies demonstrate that both the estimators and their variance estimates perform well for practical sample sizes we outline an application strategy

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

two dimensional lattice gauge theory based on a quantum group in this article we analyse a two dimensional lattice gauge theory based on a quantum group the algebra generated by gauge fields is the lattice algebra introduced recently by a yu alekseev h grosse and v schomerus in 1 we define and study wilson loops this theory is quasi topological as in the classical case which allows us to compute the correlation functions of this theory on an arbitrary surface

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

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

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

an explicit description of the symplectic structure of moduli spaces of flat connections the moduli space of flat connections on a principal g bundle over a compact oriented surface of genus g 1 is considered herein using the holonomies around noncontractible loops the moduli space is described as a quotient of a submanifold of g2g an explicit expression is obtained for the symplectic form on the smooth part of moduli space and several properties of this form are established

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

macdonald s identities and the large n limit of ym 2 on the cylinder the purpose of this paper is to determine the large n asymptotics of the free energy fn a u a of ym2 two dimensional yang mills theory with gauge group gn su n on a cylinder where a is a so called principal element of type mathematically where hgn is the central heat kernel of gn we find that where is an explicit quadratic functional in the limit distribution d of eigenvalues of un depending only on the integral geometry of su 2 the factor of n contradicts some predictions in the physics literature on the large n limit of ym2 on the cylinder due to gross matytsin kazakov wynter and others

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

on complex zeros off the critical line for non monomial polynomial of zeta functions in this paper we show that any polynomial of zeta or l functions with some conditions has infinitely many complex zeros off the critical line this general result has abundant applications by using the main result we prove that the zeta functions associated to symmetric matrices treated by ibukiyama and saito certain spectral zeta functions and the euler zagier multiple zeta functions have infinitely many complex zeros off the critical line moreover we show that the lindelof hypothesis for the riemann zeta function is equivalent to the lindelof hypothesis for zeta functions mentioned above despite of the existence of the zeros off the critical line next we prove that the barnes multiple zeta functions associated to rational or transcendental parameters have infinitely many zeros off the critical line by using this fact we show that the shintani multiple zeta functions have infinitely many complex zeros under some conditions as corollaries we show that the mordell multiple zeta functions the euler zagier hurwitz type of multiple zeta functions and the witten multiple zeta functions have infinitely many complex zeros off the critical line

half density volumes of representation spaces of some 3 manifolds and their application in this paper we compute half density volumes of the irreducible su 2 representation spaces for seifert fibred manifolds and graph manifolds these representation spaces are not discrete so that the half density derived from the reidemeister torsion is used as a measure as an application of our result we get the exact value of the jeffrey weitsman witten invariant of a seifert fibred manifold with non discrete irreducible su 2 representation space

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

survival estimation using splines a nonparametric maximum likelihood procedure is given for estimating the survivor function from right censored data it approximates the hazard rate by a simple function such as a spline with different approximations yielding different estimators a special case is that proposed by nelson 1969 journal of quality technology 1 27 52 and altshuler 1970 mathematical biosciences 6 1 11 the estimators are uniformly consistent and have the same asymptotic weak convergence properties as the kaplan meier 1958 journal of the american statistical association 53 457 481 estimator however in small and in heavily censored samples the simplest spline estimators have uniformly smaller mean squared error than do the kaplan meier and nelson altshuler estimators the procedure is extended to estimate the baseline hazard rate and regression coefficients in the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model and is illustrated using experimental carcinogenesis data

commutator maps measure preservation and t systems let g be a finite simple group we show that the commutator map g g g is almost equidistributed as g 1 this somewhat surprising result has many applications it shows that a for a subset x g we have 1 x g 2 x g o 1 namely is almost measure preserving from this we deduce that almost all elements g 2 g can be expressed as commutators g x y where x y generate g this enables us to solve some open problems regarding t systems and the product replacement algorithm pra graph we show that the number of t systems in g with two generators tends to infinity as g 1 this settles a conjecture of guralnick and pak a similar result follows for the number of connected components of the pra graph of g with two generators some of our results apply for more general finite groups and more general word maps our methods are based on representation theory combining classical character theory with recent results on character degrees and values in finite simple groups in particular the so called witten zeta function g s p 2irr g 1 s plays a key role in the proofs

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

topological sectors and measures on moduli space in quantum yang mills on a riemann surface previous path integral treatments of yang mills on a riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations this paper extends the path integral formulation to treat separately each topological sector the formulation is sufficiently explicit to calculate wilson line expectations exactly further it suggests two new measures on the moduli space of flat connections one of which proves to agree with the small volume limit of the yang mills measure

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

group valued equivariant localization we prove a localization formula for group valued equivariant de rham cohomology of a compact g manifold this formula is a non trivial generalization of the localization formula of berline vergne and atiyah bott for the usual equivariant de rham cohomology we derive from this result a duistermaat heckman formula for group valued moment maps as an application we prove part of witten s conjectures about intersection pairings on moduli spaces of flat connections on 2 manifolds

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

the yang mills measure for s2 abstract the yang mills measure for gauge fields over the two sphere is constructed using a conditioned white noise process stochastic differential equations are used to study parallel translation under this measure expectation values of a broad class of wilson loop configurations are computed explicitly and it is shown that these values are invariant under area preserving diffeomorphisms of the sphere