equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

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

one dimensional electromagnetic inverse reflection problem formulation as a riemann hilbert problem and imaging of discontinuities this article deals with the one dimensional electromagnetic inverse problem of geophysics the problem is shown to have the structure of a riemann hilbert problem analogously to the quantum mechanical inverse scattering problem a trace type formula for the electric conductivity is derived further it is shown that the born approximation of this formula contains all the information of the possible discontinuities of the conductivity profile the proof is based on the idea of singularity expansions of pseudodifierential operators

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

isotropic angle domain elastic reverse time migration multicomponent data usually are not processed with specifically designed procedures but with procedures analogous to those used for single component data in isotropic media the vertical and horizontal components of the data commonly are taken as proxies for the p and s wave modes which are imaged independently with the acoustic wave equations this procedure works only if the vertical and horizontal components accurately represent p and s wave modes which generally is not true therefore multicomponent images constructed with this procedure exhibit artifacts caused by incorrect wave mode separation at the surface an alternative procedure for elastic imaging uses the full vector fields for wavefield reconstruction and imaging thewavefieldsarereconstructedusingthemulticomponentdata as a boundary condition for a numerical solution to the elastic wave equation the key component for wavefield migration is theimagingcondition whichevaluatesthematchbetweenwavefields reconstructed from sources and receivers for vector wave fields a simple component by component crosscorrelation between two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes we can separate elastic wavefields after reconstruction in the subsurface and implement theimagingconditionascrosscorrelationofpurewavemodesinstead of the cartesian components of the displacement wavefield thisapproachleadstoimagesthatareeasiertointerpretbecause they describe reflectivity of specified wave modes at interfaces of physical properties as for imaging with acoustic wavefields the elastic imaging condition can be formulated conventionally crosscorrelation with zero lag in space and time and extendedtononzerospaceandtimelags theelasticimagesproduced by an extended imaging condition can be used for angle decomposition of primary pp or ss and converted ps or sp reflectivity angle gathers constructed with this procedure have applicationsformigrationvelocityanalysisandamplitude variation with angleanalysis

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

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

on localization and regularization different regularizations are studied in localization of path integrals we discuss the effect of the choice of regularization by evaluating the partition functions for the harmonic oscillator and the weyl character for su 2 in particular we solve the weyl shift problem that arises in path integral evaluation of the weyl character by using the atiyah patodi singer invariant and the borel weil theory

robust source independent elastic full waveform inversion in the time domain abstractaccurate estimation of source wavelet is crucial in a successful full waveform inversion fwi however it cannot be guaranteed in the case of real seismic data we have developed time domain source independent elastic fwi using the convolution based objective function that was originally developed for acoustic fwi we have applied a new time window on the reference traces in the objective function to suppress the noises induced by the convolution and crosscorrelation operations also we have adopted l1 huber and hybrid norm objective functions to improve the antinoise ability of our algorithm we implemented a multiscale inversion strategy to conduct the tests with a quasi newton limited memory broyden fletcher goldfarb shanno l bfgs method to reduce the sensitivity to initial models and to improve the quality of inversion results synthetic tests verified that the new added time window can not only improve the inversion results but also accelerate the convergence rate our method can be

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

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

recovery of jumps and singularities of an unknown potential from limited data in dimension 1 we prove that in dimension 1 the jumps and singularities of an unknown potential appearing in the schrodinger equation can be recovered using the born approximation the result is based on an accurate determination of the first nonlinear term of the born series we shall also work in wider function space than the previous publications on this topic numerical examples illustrate the feasibility of this technique

gauged vortices in a background we discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism at critical self coupling and for low temperatures it has been argued that the configuration space for vortex dynamics in each topological class of the abelian higgs model approximately truncates to a finite dimensional moduli space with a kahler structure for the case where the vortices live on a 2 sphere we explain how localization formulae on the moduli spaces can be used to compute exactly the partition function of the vortex gas interacting with a background potential the coefficients of this analytic function provide geometrical data about the kahler structures the simplest of which being their symplectic volume computed previously by manton using an alternative argument we use the partition function to deduce simple results on the thermodynamics of the vortex system in particular the average height on the sphere is computed and provides an interesting effective picture of the ground state

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

probability distributions and coherent states of and algebras a new approach to probability theory based on quantum mechanical and lie algebraic ideas is proposed and developed the underlying fact is the observation that the coherent states of the heisenberg weyl su 2 and algebras in certain symmetric bosonic representations give the probability amplitudes or the square roots of the well known poisson binomial multinomial negative binomial and negative multinomial distributions in probability theory new probability distributions are derived based on coherent states of the classical algebras and in symmetric representations these new probability distributions are simple generalization of the multinomial distributions with some added new features reflecting the quantum and lie algebraic construction as byproducts simple proofs and interpretation of addition theorems of hermite polynomials are obtained from the coordinate representation of the negative multinomial states in other words these addition theorems are higher rank counterparts of the well known generating function of hermite polynomials which is essentially the coordinate representation of the ordinary heisenberg weyl coherent state

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

exactness in the wentzel kramers brillouin approximation for some homogeneous spaces analysis of the wentzel kramers brillouin wkb exactness in some homogeneous spaces is attempted cpn as well as its noncompact counterpart dn 1 is studied u n 1 or u n 1 based on the schwinger bosons leads us to cpn or dn 1 path integral expression for the quantity tr e iht with the aid of coherent states the wkb approximation terminates in the leading order and yields the exact result provided that the hamiltonian is given by a bilinear form of the creation and the annihilation operators an argument on the wkb exactness to more general cases is also made

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

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

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

coherent states path integral and semiclassical approximation using the generalized coherent states it is shown that the path integral formulas for su 2 and su 1 1 in the discrete series are wkb exact if it is started from the trace of e it where h is given by a linear combination of generators in this case the wkb approximation is achieved by taking a large spin limit j k under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result it is further pointed out that the discretized form of path integral is indispensable in other words the continuum path integral expression sometimes leads to a wrong result therefore great care must be taken when some geometrical action would be adopted even if it is so beautiful as the starting ingredient of path integral discussions on generalized coherent states are also presented both from geometrical and simple oscillator schwinger boson points of view

quotient spaces and critical points of invariant functions for c actions consider a linear action of the group c on x c n 1 we study the fundamental algebraic properties of the sheaves of invariant and basic differential forms for such an action and use these to define an algebraic notion of multiplicity for critical points of functions which are invariant under the c action we also prove a theorem relating the cohomology of the milnor fibre of the critical point on the quotient space with this algebraic multiplicity we also include an appendix showing how to use cech complexes to compute local cohomology

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

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

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

non hamiltonian actions with isolated fixed points we construct a non hamiltonian symplectic circle action on a closed connected six dimensional symplectic manifold with exactly 32 fixed points

an inverse born approximation for the general nonlinear schrodinger operator on the line this work deals with the inverse scattering problems for the one dimensional schrodinger equation of the most general form with general nonlinearity f and with a real parameter k under some assumptions on f we prove that all singularities and jumps of f x u0 where u0 x k eikx can be recovered by the reflection coefficient with arbitrary large k

multicomponent joint migration velocity analysis in the angle domain for pp waves and ps waves employing the vector processing of multicomponent seismic data elastic kirchhoff migration is used to conduct a multicomponent joint migration velocity analysis mva of pp and ps waves in the angle domain in vector wavefield imaging the elastic imaging condition has been extended to nonzero time and space shifts we apply the extended imaging condition to elastic kirchhoff migration to extract angle domain common image gathers adcigs of pp and ps waves this method which is derived from 3d wave propagation theory directly operates on the vector wavefields and automatically resolves the problem of ps wave polarity reversal in the migration sections based on the kinematic characteristics of pp and ps waves in the incident angle domain the velocity updating functions for p and s waves are derived by combining the adcigs flatness criterion and the pp and ps wave image depth consistency principle the accuracy of p and s wave velocities can be assessed the method was tested on a 2d thin inter

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 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

nonlinear microstrain theories a hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes rotation and straining of an underlying microstructure an increase in the number of degrees of freedom represents a refinement of the material description in addition to available nonlinear cosserat and micromorphic theories general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua a microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element in each case balance equations and boundary conditions are derived decompositions of the finite strain measures into elastic and plastic parts are provided the formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials thus complying with requirements of continuum thermodynamics some guidelines for the selection of a suitable higher order model for a given material close the discussion

estimation principal components and hamiltonian systems in this paper we show how symplectic geometry can be used to analyse a fundamental problem in estimation theory the fitting of lines and planes to a data set in n dimensions symplectic methods yield information not only on the critical point structure of the maximum likelihood estimation mle function see byrnes and willems 7 but also on the asymmetries in the distribution of the data points in n space

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

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

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

numerical simulation of ductile fracture with the rousselier constitutive law this contribution aims at developing robust and reliable tools dedicated to predictive numerical simulation of ductile fracture description of damage mechanisms relies here on the rousselier constitutive law to achieve robustness and objectivity several points need then to be addressed reduction of the set of internal variables so as to recast the model into the framework of generalised standard materials proper expression of the normal plastic flow rule at the vertex of the yield surface finite strain formulation based on a multiplicative split of the deformation gradient fully implicit integration of the constitutive equations leading to a single scalar equation that admits a unique root use of mixed finite elements to avoid volumetric locking strain gradient based nonlocal model to control strain localisation and remedy spurious mesh dependency the capabilities of the resulting numerical formulation are demonstrated through simulating the cup cone fracture of a notched tensile specimen

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

on path integral localization and the laplacian we introduce a new localization principle which is a generalized canonical transformation it unifies brst localization the non abelian localization principle and a special case of the conformal duistermaat heckman integration formula of paniak semenoff and szabo the heat kernel on compact lie groups is localized in two ways first using a non abelian generalization of the derivative expansion localization of palo and niemi and second using the brst localization principle and a configuration space path integral in addition we present some new formulas on homogeneous spaces which might be useful in a possible localization of selberg s trace formula on locally homogeneous spaces

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

witten s formulas for intersection pairings on moduli spaces of flat g bundles abstract in a 1992 paper j geom phys 9 1992 303 witten gave a formula for the intersection pairings of the moduli space of flat g bundles over an oriented surface possibly with markings in this paper we give a general proof of witten s formula for arbitrary compact simple groups and any markings for which the moduli space has at most orbifold singularities

extension of the barut girardello coherent state and path integral we extend the barut girardello coherent state for the representation of su 1 1 to the coherent state for a representation of u n 1 and construct the measure we also construct a path integral formula for some hamiltonian

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

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

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

bordism of semi free s1 actions we calculate geometric and homotopical bordism rings associated to semi free s1 actions on complex manifolds giving explicit generators for the geometric theory the classification of semi free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry

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

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