the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

intertwining operators into cohomology representations for semisimple lie groups abstract letgbe a semisimple lie group with finite center andlthe centralizer of a torus ing an explicit intertwining operator is constructed from a nonunitary principal series forginto dolbeault cohomology ong l the image lies in the space of strongly harmonic forms this paper generalizes results in l barchini a w knapp and r zierau j funct anal 107 1992 302 341 by replacing the one dimensional fiber overg lwith certain langlands quotients

adapted complex structures and the geodesic flow in this paper we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact real analytic riemannian manifold motivated by the complexifier approach of t thiemann as well as certain formulas of v guillemin and m stenzel we obtain the polarization associated to the adapted complex structure by applying the imaginary time geodesic flow to the vertical polarization meanwhile at the level of functions we show that every holomorphic function is obtained from a function that is constant along the fibers by composition with the imaginary time geodesic flow we give several equivalent interpretations of this composition including a convergent power series in the vector field generating the geodesic flow

direct images fields of hilbert spaces and geometric quantization geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces and the question arises if the spaces h s are canonically isomorphic axelrod et al j diff geo 33 787 902 1991 and hitchin commun math phys 131 347 380 1990 suggest viewing h s as fibers of a hilbert bundle h introduce a connection on h and use parallel transport to identify different fibers here we explore to what extent this can be done first we introduce the notion of smooth and analytic fields of hilbert spaces and prove that if an analytic field over a simply connected base is flat then it corresponds to a hermitian hilbert bundle with a flat connection and path independent parallel transport second we address a general direct image problem in complex geometry pushing forward a hermitian holomorphic vector bundle e to y along a non proper map y to s we give criteria for the direct image to be a smooth field of hilbert spaces third we consider quantizing an analytic riemannian manifold m by endowing tm with the family of adapted kahler structures from lempert and sz ke bull lond math soc 44 367 374 2012 this leads to a direct image problem when m is homogeneous we prove the direct image is an analytic field of hilbert spaces for certain such m but not all the direct image is even flat which means that in those cases quantization is unique

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

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

quantization of abelian varieties distributional sections and the transition from kahler to real polarizations abstract we study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization real and mixed polarizations are interpreted as degenerate complex structures using a weak version of the equations of covariant constancy and the weil brezin expansion to describe distributional sections we give a unified analytical description of the quantization spaces for all non negative polarizations the blattner kostant sternberg bks pairing maps between half form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of sp 2 g r moreover these maps are shown to be unitary

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

n symplectic algebra of observables in covariant lagrangian field theory n symplectic geometry on the adapted frame bundle l e e of an n m k dimensional fiber bundle e m is used to set up an algebra of observables for covariant lagrangian field theories using the principle bundle l e j1 we lift a lagrangian l j1 r to a lagrangian l l l e r and then use l to define a modified n symplectic potential l on l e the cartan hamilton poincare chp rn valued 1 form if the lifted lagrangian is nonzero then l e d l is an n symplectic manifold to characterize the observables we define a lifted legendre transformation l from l e into le the image ql l l e is a submanifold of le and ql d ql is shown to be an n symplectic manifold we prove the theorem that l l ql and pull back the reduced canonical n symplectic geometry on ql to l e to define the algebras of observables on the n symplectic manifold l e d l to find the reduced n symplectic algebra on ql we set up the equations of n symplectic reduction and apply the general theory to the mo

boundary value maps szego maps and intertwining operators we consider one series of unitarizable representations the cohomological induced modules aq a with dominant regular infinitesimal character the minimal k type ir v of aq a determines a homogeneous vector bundle v g k the derived functor modules can be realized on the solution space of a first order differential operator da on v barchini knapp and zierau gave an explicit integral map s from the derived functor module realized in the langlands classification into the space of smooth sections of the vector bundle v g k in this paper we study the asymptotic behavior of elements in the image of s we obtain a factorization of the standard intertwining opeartors into the composition of the szego map s and a passage to boundary values 0 introduction let g be a linear connected semisimple lie group and let k be a maximal compact subgroup of g we assume that rank g rank k in this situation the cohomologically induced representation aq a with dominant regular infinitesimal character can be realized on solution spaces of invariant differential equations in g k see wo b 2 the minimal k type r v of aq a determines an homogeneous vector bundle vtr g k corresponding to certain choices of positive roots there is defined a certain homogeneous first order differential operator d on vtu in bkz and b 1 the authors gave an explicit nonzero integral map s from the derived functor module realized in the langlands classification to the space of smooth sections of the vector bundle vt g k the image of s is annihilated by the differential operator d when the representation under consideration is in the discrete series of g the differential operator vd coincides with schmid s differential operator s in this case the s transform was introduced by knapp and wallach in kw the realization of irreducible unitary representations in the g k picture generalizes bargmann s formulation of holomorphic discrete series of su 1 1 bar in terms of semisimple lie groups functions on the circle suggest non unitary principal series and holomorphic functions on the disk suggest discrete series representations the s transform in kw reduces in limiting cases limits of discrete series for su 1 1 to the szeg6 kernel for the disk see k 1 thus we call s a szeg6 map received by the editors february 17 1995 and in revised form october 9 1995 1991 mathematics subject classification primary 22c05 22e45 22e46 1997 american mathematical society 1877 this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 42 59 utc all use subject to http about jstor org terms

virasoro model space the representations of a compact lie groupg can be studied via the construction of an associated model space this space has the property that when geometrically quantized its hilbert space contains every irreducible representation ofg just once we construct an analogous space for the group diffs1 it is naturally a complex manifold with a holomorphic free action of diffs1 preserving a family of pseudo kahler structures all of the good coadjoint orbits are obtained from our space by hamiltonian constraint reduction we briefly discuss the connection to the work of alekseev and shatashvili

harmonic spinors on semisimple symmetric spaces abstract let g h be a semisimple symmetric space we consider a dirac operator d on g h twisted by a finite dimensional h representation we give an explicit integral formula for certain solutions of the equation d 0 in particular some quotients of standard principal series representations are seen to occur in the kernel of d

geometric quantization parallel transport and the fourier transform in quantum mechanics the momentum space and position space wave functions are related by the fourier transform we investigate how the fourier transform arises in the context of geometric quantization we consider a hilbert space bundle mathcal h over the space mathcal j of compatible complex structures on a symplectic vector space this bundle is equipped with a projectively flat connection we show that parallel transport along a geodesic in the bundle mathcal h to mathcal j is a rescaled orthogonal projection or bogoliubov transformation we then construct the kernel for the integral parallel transport operator finally by extending geodesics to the boundary for which the metaplectic correction is essential we obtain the segal bargmann and fourier transforms as parallel transport in suitable limits

covariant field theory on frame bundles of fibered manifolds we show that covariant field theory for sections of e m lifts in a natural way to the bundle of vertically adapted linear frames l e our analysis is based on the fact that l e is a principal fiber bundle over the bundle of 1 jets j1 on l e the canonical soldering 1 forms play the role of the contact structure of j1 a lifted lagrangian l l e r is used to construct modified soldering 1 forms which we refer to as the cartan hamilton poincare 1 forms these 1 forms on l e pass to the quotient to define the standard cartan hamilton poincare m form on j1 we derive generalized hamilton jacobi and hamilton equations on l e and show that the hamilton jacobi and canonical equations of caratheodory rund and de donder weyl are obtained as special cases

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

on the n symplectic structure of faithful irreducible representations each faithful irreducible representation of an n dimensional vector space v1 on an n dimensional vector space v2 is shown to define a unique irreducible n symplectic structure on the product manifold v1 v2 the basic details of the associated poisson algebra are developed for the special case n n2 and 2n dimensional symplectic submanifolds are shown to exist

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

partial coherent state transforms g t invariant kahler structures and geometric quantization of cotangent bundles of compact lie groups abstract in this paper we study the analytic continuation to complex time of the hamiltonian flow of certain g t invariant functions on the cotangent bundle of a compact connected lie group g with maximal torus t namely we will take the hamiltonian flows of one g g invariant function h and one g t invariant function f acting with these complex time hamiltonian flows on g g invariant kahler structures gives new g t invariant but not g g invariant kahler structures on t g we study the hilbert spaces h corresponding to the quantization of t g with respect to these non invariant kahler structures on the other hand by taking the vertical schrodinger polarization as a starting point the above g t invariant hamiltonian flows also generate families of mixed polarizations p 0 c im 0 each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a kahler structure on the leaves of a foliation of t g the geometric quantization of t g with respect to these mixed polarizations gives rise to unitary partial coherent state transforms corresponding to ksh maps as defined in 11 12

coherent state transforms and the mackey stone von neumann theorem mackey showed that for a compact lie group k the pair k c 0 k has a unique non trivial irreducible covariant pair of representations we study the relevance of this result to the unitary equivalence of quantizations for an infinite dimensional family of k k invariant polarizations on t k the kahler polarizations in the family are generated by complex time hamiltonian flows applied to the schrodinger vertical real polarization the unitary equivalence of the corresponding quantizations of t k is then studied by considering covariant pairs of representations of k defined by geometric prequantization and of representations of c 0 k defined via heisenberg time evolution followed by time geometric quantization induced evolution we show that in the semiclassical and large imaginary time limits the unitary transform whose existence is guaranteed by mackey s theorem can be approximated by composition of the time geometric quantization induced evolution with the time evolution

principal series representations and harmonic spinors abstract let g be a real reductive lie group and g h a reductive homogeneous space we consider kostant s cubic dirac operator d on g h twisted with a finite dimensional representation of h under the assumption that g and h have the same complex rank we construct a nonzero intertwining operator from principal series representations of g into the kernel of d the langlands parameters of these principal series are described explicitly in particular we obtain an explicit integral formula for certain solutions of the cubic dirac equation d 0 on g h

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

complex time evolution in geometric quantization and generalized coherent state transforms abstract for the cotangent bundle t k of a compact lie group k we study the complex time evolution of the vertical tangent bundle and the associated geometric quantization hilbert space l 2 k under an infinite dimensional family of hamiltonian flows for each such flow we construct a generalized coherent state transform cst which is a unitary isomorphism between l 2 k and a certain weighted l 2 space of holomorphic functions for a particular set of choices we show that this isomorphism is naturally decomposed as a product of a heisenberg type evolution for complex time within l 2 k followed by a polarization changing geometric quantization evolution for complex time in this case our construction yields the usual generalized segal bargmann transform of hall we show that the infinite dimensional family of hamiltonian flows can also be understood in terms of thiemann s complexifier method which generalizes the construction of adapted complex structures

on a kind of noether symmetries and conservation laws in k cosymplectic field theory this paper is devoted to studying symmetries of certain kinds of k cosymplectic hamiltonian systems in first order classical field theories thus we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of noether s theorem

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

szego mappings harmonic forms and dolbeault cohomology abstract let g l be the quotient of a semisimple lie group g by the centralizer l of a torus the author constructs an explicit intertwinig operator from derived functor modules realized in the langlands classification into dolbeault cohomology on g l this operator produces strongly harmonic forms this paper generalizes the results in l barchini a w knapp and r zierau j funct anal 107 1992 302 341 by dropping the condition real rank l real rand g

classical mechanics and geometric quantization on an infinite dimensional disc and grassmannian we discuss the classical mechanics on the grassmannian and the disc modeled on the ideal l 2 we apply methods of geometric quantization to these systems their relation to a flat symplectic space is also discussed

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

schouten nijenhuis brackets the poisson and graded poisson schouten nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields respectively on an n dimensional manifold m are shown to be n symplectic this is accomplished by showing that both brackets may be defined in a unified way using the n symplectic structure on the bundle of linear frames lm of m new results in n symplectic geometry are presented and then used to give globally defined representations of the hamiltonian operators defined by the schouten nijenhuis brackets

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

strongly harmonic forms for representations in the discrete series abstract letgbe a semisimple connected lie group and letkbe a maximal compact subgroup assume that rankg rank k and lett kbe a cartan subgroup ofg the quotientg tcarries an indefiniteg invariant hermitian form the standard dolbeault operator has a formal adjoint differential operator invwith respect to the invariant hermitian form letsdenote the complex dimension ofk t we form the indefinite harmonic space h s g t l 2 0 s l 2 valued forms in ker ker inv in this paper we show that under some positivity conditions on the cohomology space h s g t l contains a copy of the representation in the discrete series ofgwith parameter