the fourier transform of semi simple coadjoint orbits letmbe a closed coadjoint orbit of a real connected semi simple lie groupg and letfm c g gbe its fourier transform in this paper we compute the restriction offmto the lie algebra k of a maximal compact subgroupkofg using our technique of localization in equivariant cohomology developed recently we extend previous results by m duflo g heckman m vergne and i sengupta

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

tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

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

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

coherent states over grassmann manifolds and the wkb exactness in path integral u n coherent states over grassmann manifold gn n u n u n u n n are formulated to be able to argue the wkb exactness in the path integral representation of a character formula the phenomena is the so called localization of duistermaat heckman the exponent in the path integral formula is proportional to an integer k labeling the u n representation thus when k a usual semiclassical approximation by regarding k 1 can be performed to yield a desired conclusion the mechanism of the localization is uncovered by the help of the generalized schwinger boson technique the discussion on the feynman kernel is also presented

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

symplectic manifolds coherent states and semiclassical approximation the symplectic structure and hamiltonian dynamics for a class of grassmannian manifolds are described here using the two dimensional sphere s2 and disc d2 as illustrative cases we write their path integral representations using coherent state techniques these path integrals can be evaluated exactly by semiclassical methods thus providing examples of the localization formula along the way we also give a local coordinate description for a class of grassmannians

a wall crossing formula for the signature of symplectic quotients we use symplectic cobordisrn and the localization result of ginzburg guillemin and karshon to find a wall crossing formula for the signature of regular symplectic quotients of hamiltonian torus actions the formula is recursive depending ultimately on fixed point data in the case of a circle action we obtain a formula for the signature of singular quotients as well we also show how formulas for the poincar6 polynomial and the euler characteristic equivalernt to those of kirwan can be expressed in the same recursive manner

symbolic computing of nonlinear observable and observer forms observer design for highly nonlinear dynamics is an important issue particularly when the locally observable dynamics are not linearly observable in such circumstances the ability to reduce the system to observable or observer form is a useful first step to observer design we describe and illustrate symbolic computing tools to do that

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

nekrasov functions from exact bohr sommerfeld periods the case of su n we suggested in 2009 that the nekrasov function with one non vanishing deformation parameter is obtained by the standard seiberg witten sw contour integral construction the only difference is that the sw differential pdx is substituted by its quantized version for the corresponding integrable system and contour integrals become exact monodromies of the wavefunction this provides an explicit formulation of the earlier guess by nekrasov and shatashvili in 2009 in this paper we successfully check this suggestion in the first order in 2 and the first order in instanton expansion for the su n model where the consistency of the so deformed sw equations is already non trivial

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

centered complexity one hamiltonian torus actions we consider symplectic manifolds with hamiltonian torus actions which are almost but not quite completely integrable the dimension of the torus is one less than half the dimension of the manifold we provide a complete set of invariants for such spaces when they are centered and the moment map is proper in particular this classifies the preimages under the moment map of all sufficiently small open sets which is an important step towards global classification as an application we construct a full packing of each of the grassmannians gr 2 r5 and gr 2 r6 by two equal symplectic balls

reduction in stages and complete quantization of the mic kepler problem the one parameter deformation family of the standard kepler problem known as the mic kepler problem is completely quantized using the explicit momentum mapping of the torus actions on some toric manifolds and some equivariant cohomology theory these manifolds appear as symplectic faces of the system during reduction process

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

weighted pseudometric discriminatory power improvement using a bayesian logistic regression model based on a variational method in this paper we investigate the effectiveness of a bayesian logistic regression model to compute the weights of a pseudometric in order to improve its discriminatory capacity and thereby increase image retrieval accuracy in the proposed bayesian model the prior knowledge of the observations is incorporated and the posterior distribution is approximated by a tractable gaussian form using variational transformation and jensen s inequality which allow a fast and straightforward computation of the weights the pseudometric makes use of the compressed and quantized versions of wavelet decomposed feature vectors and in our previous work the weights were adjusted by the classical logistic regression model a comparative evaluation of the bayesian and classical logistic regression models is performed for content based image retrieval as well as for other classification tasks in a decontextualized evaluation framework in this same framework we compare the bayesian logistic regression model to some relevant state of the art classification algorithms experimental results show that the bayesian logistic regression model outperforms these linear classification algorithms and is a significantly better tool than the classical logistic regression model to compute the pseudometric weights and improve retrieval and classification performance finally we perform a comparison with results obtained by other retrieval methods

integrability of zero dimensional replica field theories at β 1 building on insights from the theory of integrable lattices the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices specifically the fermionic and the bosonic replica partition functions are proven to form a single supersymmetric pfaff kp hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density density eigenvalue correlation function in the infinite dimensional gaussian orthogonal ensemble implications of the formalism outlined are briefly discussed

on the construction of certain 6 dimensional symplectic manifolds with hamiltonian circle actions let m w be a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian s 1 action such that the fixed point set consists of isolated points or surfaces assume dim h 2 m 3 in an earlier paper we defined a certain invariant of such spaces which consists of fixed point data and twist type and we divided the possible values of these invariants into six types in this paper we construct such manifolds with these types as a consequence we have a precise list of the values of these invariants

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

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

correlation functions of harish chandra integrals over the orthogonal and the symplectic groups the harish chandra correlation functions i e integrals over compact groups of invariant monomials prod operatorname tr x p 1 varomega y q 1 varomega dagger x p 2 cdots with the weight exp tr x y are computed for the orthogonal and symplectic groups we proceed in two steps first the integral over the compact group is recast into a gaussian integral over strictly upper triangular complex matrices with some additional symmetries supplemented by a summation over the weyl group this result follows from the study of loop equations in an associated two matrix integral and may be viewed as the adequate version of duistermaat heckman s theorem for our correlation function integrals secondly the gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions

multi periodic coherent states and the wkb exactness we construct the path integral formula in terms of a multi periodic coherent state as an extension of the nielsen rohrlich formula for spin we make an exact calculation of the formula and show that when a parameter corresponding to the magnitude of spin becomes large the leading order term of the expansion coincides with the exact result we also give an explicit correspondence between the trace formula in the multi periodic coherent state and the one in the generalized coherent state

replica field theories painleve transcendents and exact correlation functions exact solvability is claimed for nonlinear replica ensuremath sigma models derived in the context of random matrix theories contrary to other approaches reported in the literature the framework outlined does not rely on traditional replica symmetry breaking but rests on a previously unnoticed exact relation between replica partition functions and painlev e transcendents while expected to be applicable to matrix models of arbitrary symmetries the method is used to treat fermionic replicas for the gaussian unitary ensemble gue chiral gue symmetry classes a and aiii in cartan classification and ginibre s ensemble of complex non hermitian random matrices further applications are briefly discussed