This paper considers the enhanced symplectic "category" for purposes of quantizing quasi-Hamiltonian $G$-spaces, where $G$ is a compact simple Lie group. Our starting point is the well-acknowledged analogy between the cotangent bundle $T^*G$ in Hamiltonian geometry and the internally fused double $D(G)=G\times G$ in quasi-Hamiltonian geometry. Guillemin and Sternberg consider the former, studing half-densities and phase functions on its so-called character Lagrangians $Λ_{\mathcal{O}}\subseteq T^*G$. Our quasi-Hamiltonian counterpart replaces these character Lagrangians with the universal centralizers $Λ_{\mathcal{C}}\longrightarrow\mathcal{C}$ of regular, $\frac{1}{k}$-integral conjugacy classes $\mathcal{C}\subseteq G$. We show each universal centralizer to be a "quasi-Hamiltonian Lagrangian" in $D(G)$, and to come equipped with a half-density and phase function.
At the same time, we consider a Dehn twist-induced automorphism $R:D(G)\longrightarrow D(G)$ that lacks a natural Hamiltonian analogue. Each quasi-Hamiltonian Lagrangian $R(Λ_{\mathcal{C}})$ is shown to have a clean intersection with every $Λ_{\mathcal{C}'}$, and to come equipped with a half-density and phase function of its own. This leads us to consider the possibility of a well-behaved, quasi-Hamiltonian notion of the BKS pairing between $R(Λ_{\mathcal{C}})$ and $Λ_{\mathcal{C}'}$. We construct such a pairing and study its properties. This is facilitated by the nice geometric fearures of $R(Λ_{\mathcal{C}})\capΛ_{\mathcal{C}'}$ and a reformulation of the classical BKS pairing. Our work is perhaps the first step towards a level-$k$ quantization of $D(G)$ via the enhanced symplectic "category".
In the context of the two dimensional sigma model, we show that classical field theory naturally defines a functor from Segal's category of Riemann surfaces to the Guillemin-Sternberg/Weinstein category of canonical relations in symplectic geometry, following ideas of Cattaneo, Mnev, and Reshetikhin. This is an expository article, based on a research tutorial.
We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quant…
We study the formal geometric quantization of $b^m$-symplectic manifolds equipped with Hamiltonian actions of a torus $T$ with nonzero leading modular weight. The resulting virtual $T$-modules are finite dimensional when $m$ is odd, as in [GMW2]; when $m$ is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight.
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schrödinger equation. We also comment on applications to…
A symplectic manifold gives rise to a triangulated A-infinity category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from and what symplectic topologists want to do with it.
We expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poisson manifolds with non regular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.
Given a compact symplectic manifold whose symplectic form has integer periods, one associates to it a dg category based on the microlocal analysis according to Kashiwara-Schapira
K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for two results of D. McDuff about the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating set of …
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
On any symplectic manifold of dimension greater than 2, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to 1, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast with the dimension 2 case, where by a partial case of a result of Zapolsky, an opposite statement holds.
In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold.
We study a stabilization of the symplectic category introduced by A. Weinstein as a domain for the geometric quantization functor. The symplectic category is a topological category with objects given by symplectic manifolds, and morphisms being suitable lagrangian correspondences. The main drawback of Weinstein's symplectic category is that composition of morphisms cannot always be defined. Our stabilization procedure rectifies this problem while remaining faithful to the original notion of composition. The stable symplectic category is enriched over the category of spectra (in particular, its…
Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim-Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second au…
We study a notion of pre-quantization for $b$-symplectic manifolds. We use it to construct a formal geometric quantization of $b$-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional $T$-modules.
Deformation quantization algebroids over a complex symplectic manifold X are locally given by rings of WKB operators, that is, microdifferential operators with an extra central parameter τ. In this paper, we will show that such algebroids are classified by H^2(X;k^*), where k^* is a subgroup of the group of invertible formal Laurent series in τ^-1.
On a complex symplectic manifold, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter, a substitute to the lack of homogeneity. We also quantize involutive submanifolds of contact manifolds.
Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered…
We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kaehler case we show that the linear representation of a pair of compact Lie groups on the geometric quantization of the manifold is then equipped…
We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on the fact that one can realize the core of a symplectic double …
Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a symplectic hopfoid, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that symplectic hopfoids provide a characterization of symplectic double groupoids in these terms. Then, we show how such structures may be used to produce examples of symplectic orbifolds, and conjecture that all symplectic orbifolds arise via a similar construction. The symplectic st…
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relation) in terms of Z_2-graded chain complexes. It is a straightforward consequence of the theory of BV…
A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.
In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This extends versions of a statement well-known in the algebraic category to the symplectic category, where it appears to be less familiar. The latter's inherent flexibility then leads to a confirmation of Pandharipande's Gopakumar-Vafa prediction for GW-invariants of Fano classe…
The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.
This is a lecture note from a seminar course given at Tel Aviv University in Spring 2018. Part of the preliminary section is built from Kazhdan's seminar organized in the Hebrew University of Jerusalem in Fall 2017. The main topic of this note is a detailed introduction of Tamarkin category theory from a quantitative perspective, followed by a demonstration of various applications in symplectic topology. Many examples are provided in order to obtain certain geometric intuitions of those abstract algebraic constructions in Tamarkin category. In this note, we try to explain how standard symplect…
We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.
We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Di…
In this paper, we show the spectral convergence result of $\overline{\partial}$-Laplacians when $(X,ω)$ is a compact toric symplectic manifold equipped with the natural prequantum line bundle $L$. We consider a family $\{ J_s\}_s$ of $ω$-compatible complex structures tending to the large complex structure limit, and obtain the spectral convergence of $\overline{\partial}$-Laplacians acting on $L^k$.
We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.
We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDuff (a six dimensional symplectic non-Hamiltonian circle action with fixed tori) and its products with copies of a two dimensional sphere with the usual rotation, the condition on the dimension of the acting torus is optimal to obtain the result.
We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a s…
We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino and allows us to obtain the most general convexity statement available in the literature for momentum maps associated to a symplectic Lie group action.
We introduce a persistence-like pseudo-distance on Tamarkin's category and prove that the distance between an object and its Hamiltonian deformation is at most the Hofer norm of the Hamiltonian function. Using the distance, we show a quantitative version of Tamarkin's non-displaceability theorem, which gives a lower bound of the displacement energy of compact subsets of cotangent bundles.
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.
Given a closed symplectic manifold (M,ω) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric, we look at the first eigenvalue λ_1 of the Laplacian associated with it. We show that λ_1 can be made arbitrarily large, when we vary the metric. This generalizes previous results of Polterovich, and of Mangoubi.
In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kahler and non-Kahler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the co…
In this paper we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian resolutions 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 …
We construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization. Our proof uses the formalism of Gelfand-Kazhdan descent, whose foundations we establish in the super-symplectic setting.
Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (…
In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which takes into account the Atiyah-Singer indices of all possible line bundles. We apply this method to generalized flag varieties. In order to find the zeros of the corresponding generalized Hilbert pol…
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of t…
A. Weinstein has conjectured a nice looking formula for a deformed product of functions on a hermitian symmetric space of non-compact type. We derive such a formula for symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional symplectic manifolds: the euclidean plane, the sphere and the hyperbolic plane. For the euclidean plane we obtain the well known Moyal-Weyl product. The other cases show that Weinstein's original idea should be interpreted with care. We conclude with comm…
We present a formal, algebraic treatment of Fedosov's argument that the coordinate algebra of a symplectic manifold has a deformation quantization. His remarkable formulas are established in the context of affine symplectic algebras.
In this article, we construct a $2$-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a $2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a $2$-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of $0$-shifted symplectic derived stacks where the objects and morphisms in $Symp^0$ are enhanced with orientation data. Using this, we define a parti…
We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the "category" of symplectic manifolds and canonical relations obtained by localizing them around lagrangian submanifolds in the spirit of Milnor's microbundles.
In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class $[α]$. The same condition was considered before by Crainic and Zhu but in a different context. In the case when $[α]$ is in the image of the sharp map we reproduce the Vaisman's condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show…
This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…
We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.
