the propagator for quantum mechanics on a group manifold from an infinite dimensional analogue of the duistermaat heckman integration formula an exact expression for the quantum mechanical propagator for a particle moving on a group manifold is shown to arise from the application of an infinite dimensional version of the duistermaat heckman integration formula to a suitable path integral over the based loop group of the group in an appendix the equivalence of this expression to a spectral representation of the propagator is demonstrated by means of a poisson resummation

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

the duistermaat heckman integration formula on flag manifolds an exposition is given of various geometrical properties of flag manifolds and of the duistermaat heckman integration formula as applied to flag manifolds

the existence of current vortex sheets in ideal compressible magnetohydrodynamics we prove the local in time existence of solutions with a surface of current vortex sheet tangential discontinuity of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity this paper is a natural completion of our previous analysis trakhinin in arch ration mech anal 177 331 366 2005 where a sufficient condition for the weak stability of planar current vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities the original nonlinear problem is a free boundary hyperbolic problem since the free boundary is characteristic the functional setting is provided by the anisotropic weighted sobolev spaces h m the fact that the kreiss lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates therefore we prove our existence theorem by a suitable nash moser type iteration scheme

a priori estimates for 3d incompressible current vortex sheets we consider the free boundary problem for current vortex sheets in ideal incompressible magneto hydrodynamics it is known that current vortex sheets may be at most weakly neutrally stable due to the existence of surface waves solutions to the linearized equations the existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms however under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front we prove an a priori estimate in sobolev spaces for smooth solutions with no loss of derivatives the result of this paper gives some hope for proving the local existence of smooth current vortex sheets without resorting to a nash moser iteration such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instabilities which is well known in astrophysics

the hypoelliptic laplacian on a compact lie group abstract let g be a compact lie group and let g be its lie algebra in this paper we produce a hypoelliptic laplacian on g g which interpolates between the classical laplacian of g and the geodesic flow this deformation is obtained by producing a suitable deformation of the dirac operator of kostant we show that various poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory the paper was motivated by papers by atiyah and frenkel in connection with localization formulas in equivariant cohomology and with kac s character formulas for affine lie algebras in a companion paper we will use similar methods in the context of selberg s trace formula

wall crossing for symplectic vortices and quantum cohomology we derive a wall crossing formula for the symplectic vortex invariants of toric manifolds as an application we give a proof of batyrev s formula for the quantum cohomology of a monotone toric manifold with minimal chern number at least two

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

existence of compressible current vortex sheets variable coefficients linear analysis we study the initial boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the rankine hugoniot relations about an unsteady piecewise smooth solution this solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity current vortex sheet under some assumptions on the unperturbed flow we prove an energy a priori estimate for the linearized problem since the tangential discontinuity is characteristic the functional setting is provided by the anisotropic weighted sobolev space w21 despite the fact that the constant coefficients linearized problem does not meet the uniform kreiss lopatinskii condition the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current vortex sheets the result of this paper is a necessary step in proving the local in time existence of current vortex sheet solutions of the nonlinear equations of magnetohydrodynamics

the kirwan map equivariant kirwan maps and their kernels consider a hamiltonian action of a compact lie group k on a compact symplectic manifold we find descriptions of the kernel of the kirwan map corresponding to a regular value of the moment map k we start with the case when k is a torus t we determine the kernel of the equivariant kirwan map defined by goldin in go corresponding to a generic circle s t and show how to recover from this the kernel of t as described by tolman and weitsman in to we in the situation when the fixed point set of the torus action is finite similar results have been obtained in our previous papers je je ma for a compact nonabelian lie group k we will use the non abelian localization formula of je ki1 and je ki2 to establish relationships some of them obtained by tolman and weitsman in to we between ker k and ker t where t k is a maximal torus in the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of t

complex equivariant intersection excess normal bundles and bott chern currents the purpose of this paper is to establish an intersection formula in equivariant complex geometry in the presence of an excess normal bundle the contribution of the excess normal bundle to the formula appears through an additive genus k r in a forthcoming paper an infinite dimensional analogue of this formula will be shown to be the result of bismut lebeau on the behaviour of quillen metrics under complex immersions

index theorem and equivariant cohomology on the loop space in this paper extending ideas of witten and atiyah we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the dirac operator on a twisted spin complex in particular the natural extension of the chern character to the whole loop space is described also it is shown that the non zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem

equivariant de rham torsions the purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de rham theory this difference is the sum of the integral of a chern simons current and of a new invariant the v invariant of an odd dimensional manifold equipped with an action of a compact lie group the v invariant localizes on the critical manifolds of invariant morse bott functions the results in this paper are shown to be compatible with results of bunke and also our with previous results on analytic torsion forms

stability of incompressible current vortex sheets abstract we revisit the study in y trakhinin on the existence of incompressible current vortex sheets study of a linearized free boundary value problem math methods appl sci 28 2005 917 945 where an energy a priori estimate for the linearized free boundary value problem for planar current vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in s i syrovatskij the stability of tangential discontinuities in a magnetohydrodynamic medium zh eksper teor fiz 24 1953 622 629 in russian w i axford note on a problem of magnetohydrodynamic stability canad j phys 40 1962 654 655 in this paper we derive an a priori estimate in the whole stability domain the crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure this symmetrizer is an analogue of kreiss type symmetrizers as in hyperbolic theory the failure of the uniform lopatinski condition i e the fact that current vortex sheets are only weakly neutrally stable yields loss of derivatives in the energy estimate the result of this paper is a necessary step to prove the local in time existence of stable nonplanar incompressible current vortex sheets by a suitable nash moser type iteration scheme

thom polynomials of morin singularities we prove a formula for thom polynomials of a d singularities in any codimension we use a combination of the test curve model of porteous and the localization methods in equivariant cohomology our formulas are independent of the codimension and are computationally effective up to d 6

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

nonlinear stability of the current vortex sheet to the incompressible mhd equations in this paper we solve a long standing open problem nonlinear stability of the current vortex sheet in the ideal incompressible magnetohydrodynamics under the syrovatskij stability condition this result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instability 2017 wiley periodicals inc

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

bott chern currents and complex immersions in the whole paper we will say that i m m is an immersion of smooth manifolds if m is a submanifold of m and if i is the corresponding injection map in particular the topology of m is the topology induced by the topology of m in differential geometry such maps i m m are also called embeddings

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

on certain infinite dimensional aspects of arakelov intersection theory letk y x be an embedding of compact complex manifolds bismut and lebeau have calculated the quillen norm of the canonical isomorphism identifying the determinant of the cohomology of a holomorphic vector bundle overy and the determinant of the cohomology of a resolution by a complex of holomorhic vector bundles overx the purpose of this paper is to show that the formula of bismut lebeau can be viewed as an equivariant intersection formula over the loop space of the considered manifolds in the presence of an infinite dimensional excess normal bundle this excess normal bundle is responsible for the appearance of the additive genusr of gillet and soule in the formula of bismut and lebeau

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

quotient of a loop group and witten genus we define a dirac ramond operator over the quotient lg h of a loop group by a subgroup h of the compact lie group g we state the conjecture that its equivariant index with respect of the natural circle action over the quotient of the loop group is equal to the witten genus of the homogeneous manifold g h we motivate the conjecture by a short time argument which allows to come back to a dirac ramond operator at the manner of taubes on a limit model where all the computations can be done by hand

intersection pairings in the n fold reduced product of adjoint orbits in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object

well posedness of the free boundary problem in incompressible elastodynamics abstract in this paper we prove the local well posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition which ensures that the evolution equation describing the free boundary is strictly hyperbolic our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the rayleigh taylor instability

twisted n 2 supersymmetry with central charge and equivariant cohomology we present an equivariant extension of the thom form with respect to a vector field action in the framework of the mathai quillen formalism the associated topological quantum field theories correspond to twisted n 2 supersymmetric theories with a central charge we analyze in detail two different cases topological sigma models and non abelian monopoles on four manifolds

path integrals for spinning particles stationary phase and the duistermaat heckmann theorem we examine the problem of the evaluation of both the propagator and of the partition function of a spinning particle in an external field at the classical as well as the quantum level in connection with the asserted exactness of the stationary phase approximation at the classical level we argue that exactness of this approximation stems from the fact that the dynamics on the two sphere s2 of a spinning particle in a magnetic field is the reduction from r4 to s2 of a linear dynamical system on r4 at the quantum level however and within the path integral approach the restriction inherent to the use of the stationary phase approximation to regular paths clashes with the fact that no regulators are present in the action that enters the path integral this is shown to lead to a prefactor for the path integral that is strictly divergent except in the classical limit a critical comparison is made with the various approaches that have been presented in the literature the validity of a formula given i

well posedness of the linearized problem for mhd contact discontinuities abstract we study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics mhd they are characteristic discontinuities with no flow across the discontinuity for which the pressure the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump under the rayleigh taylor sign condition p n 0 on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity we prove the well posedness in sobolev spaces of the linearized problem for 2d planar mhd flows

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

canonical partition functions of hamiltonian systems and the stationary phase formula we show that there is a symplectic action of a torus associated to harmonic flows on the cotangent bundle of a semi simple lie algebra this allows to obtain a completely classical proof of the gallavotti marchioro formula by the method of the stationary phase

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

douglas sebestyen s lemmas a tool for solving an operator equation problem abstract the existence of a positive solution of the equation a x b c was considered in different settings but only under additional conditions including that of regularity as well as under certain range conditions such as r b r a in this paper we consider this open question and present some equivalents of the existence of a positive solution of the operator equation a x b c without any additional range assumptions using two well known results of douglas and sebestyen also we consider a general form of a positive solution