Exploiting a notion of Kaehler structure on a stratified space introduced elsewhere we show that, in the Kaehler case, reduction after quantization coincides with quantization after reduction: Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kaehler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kaehler spaces arise from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces, and from representations of compact Lie groups. For illustration, we carry out Kaehler quantization on various spaces of that kind including singular Fock spaces.
We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kaehler manifold which respects the almost-Kaehler structure to construct a Fedosov-type deformation quantization on this manifold.
Simple classical mechanical systems and solution spaces of classical field theories involve singularities. In certain situations these singularities can be understood in terms of stratified Kaehler spaces. We give an overview of a research program whose aim is to develop a holomorphic quantization procedure on stratified Kaehler spaces.
For phase-space manifolds which are compact Kaehler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are studied. First the results on the Berezin-Toeplitz quantization of arbitrary compact Kaehler manifolds due to Bordemann, Meinrenken and Schlichenmaier are recalled. It is shown that the covariant symbol map is adjoint to the Toeplitz map. The Berezin transform for compact Kaehler manifolds is discussed. (Talk presented at the XVII workshop on geometric methods in physics, Bialowieza, Poland, July 3 --Ju…
For compact quantizable Kähler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the Berezin-Toeplitz, Berezin, and star product of geometric quantization are treated in detail. It is shown that all three are equivalent. A prominent role is played by the Berezin transform and its asymptotic expansion. A few ideas on two general constructions of star products of separation of variables type by Karabegov and by Bordemann--Waldmann respectively are gi…
We study Hamiltonian actions of compact Lie groups K on Kaehler manifolds which extend to a holomorphic action of the complexified group K^C. For a closed normal subgroup L of K we show that the Kaehlerian reduction with respect to L is a stratified Hamiltonian Kaehler K^C/L^C-space whose Kaehlerian reduction with respect to K/L is naturally isomorphic to the Kaehlerian reduction of the original manifold with respect to K.
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces; in particular, in the world of singular Poisson geometry, the closures of principal holom…
We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamishima and the second author, in the Vaisman case (that is, when a metric in the conformal class has parallel Lee form) if the manifold is compact its universal cover comes equipped with the structure of Kaehler cone over a Sasaki compact manifold. We show the compatibility between…
The first obstacle in building a Geometric Quantization theory for nilpotent orbits of a real semisimple Lie group has been the lack of an invariant polarization. In order to generalize the Fock space construction of the quantum mechanical oscillator, a polarization of the symplectic orbit invariant under the maximal compact subgroup is required. In this paper, we explain how such a polarization on the orbit arises naturally from the work of Kronheimer and Vergne. This occurs in the context of hyperkaehler geometry. The polarization is complex and in fact makes the orbit into a (positive) Kaeh…
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and…
Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive improvements of several classical results known for more general Hamiltonian reductions. Among these are a forms-level proof of the Duistermaat-Heckman theorem; an elementary proof of Atiyah's proof of the convexity of the moment image of a complexified T-orbit; another formula due to Biquard-Gauduchon for the Kaehler potential; and a formula in terms of mom…
We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle of this complex structure is holomorphically trivial by explicitly constructing a nowhere vanishing holomorphic global section. Then we construct a Hilbert space consisting of a certain class of holomorphic functions on the punctured cotangent bundle by the method of pairing polar…
By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantiz…
Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kaehler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kaehler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular S…
Let M be a compact Kaehler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no.~3, 515--538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M//G. This map, though, is not in general unitary, even to leading order in h-bar. In [Comm. Math. Phys. 275 (2007), no.~2, 401--422], Hall and the author showed that when the metaplectic correction is included, one does obtain a map which, while not in general u…
We propose that geometric quantization of symplectic manifolds is the arrow part of a functor, whose object part is deformation quantization of Poisson manifolds. The `quantization commutes with reduction' conjecture of Guillemin and Sternberg then becomes a special case of the functoriality of quantization. In fact, our formulation yields almost unlimited generalizations of the Guillemin--Sternberg conjecture, extending it, for example, to arbitrary Lie groups or even Lie groupoids. Technically, this involves symplectic reduction and Weinstein's dual pairs on the classical side, and Kasparov'…
In this paper, we begin a quantization program for nilpotent orbits of a real semisimple Lie group. These orbits and their covers generalize the symplectic vector space. A complex structure polarizing the orbit and invariant under a maximal compact subgroup is provided by the Kronheimer-Vergne Kaehler structure. We outline a geometric program for quantizing the orbit with respect to this polarization. We work out this program in detail for minimal nilpotent orbits in the non-Hermitian case. The Hilbert space of quantization consists of holomorphic half-forms on the orbit. We construct the repr…
This note describes the functional-integral quantization of two-dimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the Batalin-Vilkovisky formalism is included.
We discuss the Quantum-Koszul method for constructing star products on reduced phase spaces in the symplectic, regular case. It is shown that the reduction method proposed by Kowalzig, Neumaier and Pflaum for cotangent bundles is a special case of the Quantum-Koszul method. We give sufficient conditions that reduction by stages is possible in the Quantum-Koszul framework and show that the star product obtained by two steps is identical to that obtained by one step. In order to do so, we prove an equivariant version of the compatible tubular neighborhood theorem.
In the presence of classical phase space singularities the standard methods are insufficient to attack the problem of quantization.In certain situations the difficulties can be overcome by means of Kähler quantization on stratified Kähler spaces. Such a space is a stratified symplectic space together with a complex analytic structure which is compatible with the stratified symplectic structure; in particular each stratum is a Kähler manifold in an obvious fashion. Holomorphic quantization on a stratified Kähler space then yields a costratified Hilbert space, a quantum object having the classic…
In this paper, we prove that the "quantization commutes with reduction" phenomenon of Guillemin-Sternberg applies in the context of the metaplectic correction.
In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo "Quantization commutes with reduction".
For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on $G$. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $G = \operatorname{SU} (2)$ the main hypotheses …
Let M be a compact Kahler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M//G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515--538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M//G and the G-invariant subspace of the quantum Hilbert space over M. We prove that in general the natural map of Guillemin and Sternberg is not unitary, \textit{even to leading order in Planck's constant}. We then modify the quan…
Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-homological approach to the quantization commutes with reduction theorem.
We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples including symplectic groupoids, Hamiltonian spaces and moduli spaces of flat connections. In the case of Hamiltonian spaces we prove a derived analog of the "quantization commutes with reduction" principle.
This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L-infinity and A-infinity algebras, and bimodule structures are recalled. As an application, an "almost" functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson-Lie groups.
Quantization commutes with reduction for coisotropic A-branes
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学宇宙物质起源研究机构（WPI） Kavli宇宙物理数学研究所）
A new look at unitarity in quantization commutes with reduction for toric manifolds
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；Instituto Superior de Economia e Gestão, Universidade de Lisboa（里斯本大学经济与管理高等学院）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）
For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization to the case where the underlying set is a subset of a not necessarily regular Poisson manifold which can be written as the quotient of a regular Poisson manifold on which a compact Lie group acts freely by Poisson maps. Finally, if the quotient Poisson manifold is regular as well, we show a "quantization commutes with reduction" type result. For the proofs, we use methods stemming from both singul…
We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2].
We compute explicitly the equivariant Hirzebruch $χ_y$-characteristic of an equivariant complex line bundle over a toric manifold and state a weighted version of the quantization commutes with reduction principle in symplectic geometry. Then, we give a weighted decomposition formula for any simple polytope in $\R^n$. This formula generalizes a polytope decomposition due to Lawrence [10] and Varchenko [14] and extends a previous weighted version obtained by Karshon, Sternberg and Weitsman [9].
We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra $\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on $\mathfrak{g}$ instead …
We show that the results of the paper Symplectic Reduction and Riemann-Roch for Circle Actions of Duistermaat, Guillemin, Meinrenken and Wu can be expressed entirely in K-theory. We show that their quantization is simply a pushforward in K-theory, and use Lerman's symplectic cutting and the localization theorem in equivariant K-theory to prove that quantization commutes with reduction. Only the case where the action is free on the zero level set of the moment map is addressed.
Let $M$ be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=ϕ^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $ϕ$. The space $M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over $M_0$ and the $G$-invariant subspace of the quantum Hilbert space over $M$. In this paper, without any regularity assumption on the quotient $M_0$, we discuss the relation between the inner products of these two quantum Hilbert …
In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.
