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

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

sdiff s2 and the orbit method the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

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

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

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 phases reduction and lie poisson structure for the resonant three wave interaction hamiltonian lie poisson structures of the three wave equations associated with the lie algebras su 3 and su 2 1 are derived and shown to be compatible poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated these results can be applied to applications of nonlinear waves in for instance nonlinear optics some of the general structures presented in the latter part of this paper are implicit in the literature our purpose is to put the three wave interaction in the modern setting of geometric mechanics and to explore some new things such as explicit geometric phase formulas as well as some old things such as integrability in this context

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

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

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

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

complete decoupling of linear multivariable systems by means of linear static and differential state feedback the concept of complete decoupling is introduced in this paper we show in particular that for any linear controllable system with non singular transfer matrix there always exists a static and differential state feedback such that the closed loop system is completely decoupled

from the classical to the generalized von karman and marguerre von karman equations in this work we describe and analyze two models that were recently proposed for modeling generalized von karman plates and generalized marguerre von karman shallow shells first we briefly review the classical von karman and marguerre von karman equations their physical meaning and their mathematical justification we then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von karman type while the remaining portion is free using techniques from formal asymptotic analysis we obtain in each case a two dimensional boundary value problem that is analogous to but is more general than the classical equations in particular it is remarkable that the boundary conditions for the airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data following recent joint works we then reduce these more general equations to a single cubic operator equation which generalizes an equation introduced by berger and fife and whose sole unknown is the vertical displacement of the shell we next adapt an elegant compactness method due to lions for establishing the existence of a solution to this operator equation

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

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

eigenvalue distributions of reduced density matrices given a random quantum state of multiple distinguishable or indistinguishable particles we provide an effective method rooted in symplectic geometry to compute the joint probability distribution of the eigenvalues of its one body reduced density matrices as a corollary by taking the distribution s support which is a convex moment polytope we recover a complete solution to the one body quantum marginal problem we obtain the probability distribution by reducing to the corresponding distribution of diagonal entries i e to the quantitative version of a classical marginal problem which is then determined algorithmically this reduction applies more generally to symplectic geometry relating invariant measures for the coadjoint action of a compact lie group to their projections onto a cartan subalgebra and can also be quantized to provide an efficient algorithm for computing bounded height kronecker and plethysm coefficients

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

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

a normal form theorem around symplectic leaves we prove the poisson geometric version of the local reeb stability from foliation theory and of the slice theorem from equivariant geometry which is also a generalization of conn s linearization theorem

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

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

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

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

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

cohomology of symplectomorphism groups and critical values of hamiltonians let p co be a compact simply connected symplectic manifold sym p co its group of automorphisms the period group f p co c r is defined to be the image of co x h2 p z under the integration pairing h 2 p r x h2 p z r since co 4 0 the quotient g p co r f p co is either a circle or the quotient of a circle by a countable dense subgroup the purpose of this paper is to define and study a natural homomorphism a from 7h sym p co to g p co simply stated a measures the average over p with respect to the symplectic measure of the action integral around the trajectories of a loop of symplectomorphisms in terms of hamiltonians we give a formula for a which becomes especially simple when the loop is a 1 parameter subgroup in which case we can make some concrete computations these are related to the work of guillemin sternberg 9 and duistermaat heckmann 7 on momentum mappings we also give an interpretation of a in terms of lifting symplectomorphisms from p to a prequantization i e a g p co bundle over p with a connection whose curvature form is co here we extend the theory from the usual case where g p co is a circle by using the concept of diffeological structure introduced by souriau 16 in a sequel to this paper 19 taking as a starting point the observation that the graphs of symplectomorphisms are lagrangian submanifolds in a product manifold we shall study the possible extension of a to an invariant on loops in the space of lagrangian submanifolds in a symplectic manifold it turns out that this invariant should be defined as the holonomy of a connection whose definition involves choosing a probability measure on each lagrangian submanifold the connection is flat only for certain ways of choosing these measures and the curvature in the general case is closely related to the geometric phase introduced by berry 4 which is a subject of current interest among theoretical and experimental physicists as well as mathematicians 1 6 i0 11 13 173

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

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

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

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

2 matrix versus complex matrix model integrals over the unitary group as triangular integrals we prove that the 2 hermitian matrix model and the complex matrix model obey the same loop equations and as a byproduct we find a formula for itzykzon zuber type integrals over the unitary group integrals over u n are rewritten as gaussian integrals over triangular matrices and then computed explicitly that formula is an efficient alternative to the former shatashvili s formula

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

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

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

the duistermaat heckman formula with application to circle actions and poincare q polynomials in twisted equivariant k theory in this paper we deduce the sketch of proof of the duistermaat heckman formula and investigate how the known duistermaat heckman result could be specialized to the symplectic structure on the orbit space the theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics we present the elliptic genera and the characteristic q series for the circle actions and twisted equivariant k theory with the case of the symmetric group of n symbols separately analyzed we show that the poincare q polynomials admit presentation in terms of the patterson selberg or the ruelle type spectral functions in this paper we deduce the sketch of proof of the duistermaat heckman formula and investigate how the known duistermaat heckman result could be specialized to the symplectic structure on the orbit space the theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics we present the elliptic genera and the characteristic q series for the circle actions and twisted equivariant k theory with the case of the symmetric group of n symbols separately analyzed we show that the poincare q polynomials admit presentation in terms of the patterson selberg or the ruelle type spectral functions

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

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

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

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

proportional and derivative state feedback decoupling of linear systems we consider the row by row decoupling of linear time invariant systems by proportional and derivative state feedback our contribution with respect to previous results is that our procedure is based on simple operator say matrices manipulations without any need to use a canonical form the only assumptions for applying such a decoupling strategy are that the system is right invertible which is a necessary condition to ensure solvability and minimum phase an illustrative example is proposed

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

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

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

a vector partition function for the multiplicities of slkc abstract we use gelfand tsetlin diagrams to write down the weight multiplicity function for the lie algebra sl k c type ak 1 as a single partition function this allows us to apply known results about partition functions to derive interesting properties of the weight diagrams we relate this description to that of the duistermaat heckman measure from symplectic geometry which gives a large scale limit way to look at multiplicity diagrams we also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors using symplectic geometry we prove that the partition of the permutahedron into domains of polynomiality of the duistermaat heckman function is the same as that for the weight multiplicity function and give an elementary proof of this for sl 4 c a3

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

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

brief numerically reliable design for proportional and derivative state feedback decoupling controller in this paper we continue the work in malabre and velasquez decoupling of linear systems by means of proportional and derivative state feedback proceedings of the sci 94 wuhan china 1994 and bonilla estrada and malabre ieee trans automat control 45 2000 730 and study the row by row decoupling problem of linear time invariant systems by proportional and derivative state feedback our contribution with respect to previous results is that we develop a numerical method to compute the desired feedback matrices our numerical method is only based on orthogonal transformations and hence is numerically reliable

a littelmann type formula for duistermaat heckman measures given a compact hamiltonian t manifold m some component of whose moment map is a palais smale morse function we give a formula for the duistermaat heckman measure the fourier transform of the equivariant symplectic volume as a sum of volumes of simplices unlike previous formulae for this measure this sum has only positive terms in the case of m a flag manifold our results are an asymptotic version of littelmann s multiplicity result in representation theory 9

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system