Unitarity in "quantization commutes with reduction"

对环面流形中‘量子化与约化交换’的幺正性的新审视

A new look at unitarity in quantization commutes with reduction for toric manifolds

针对辛环面流形，研究借助混合极化下的半形式量子化，通过广义相干态变换关联不同极化的希尔伯特空间，新视角分析量子化与约化交换的幺正性，指出初始Kähler极化下该幺正性等价于两种极化量子化的幺正等价且一般不成立。

量子化与约简对共切线A-膜的交换

Quantization commutes with reduction for coisotropic A-branes

本文研究了共切线A-膜在约简下的量子化与约简的交换性，通过Marsden-Weinstein-Meyer型构造，展示了共切线A-膜在约简后的结构，并证明了A-膜的交集与约简的交换性。

量子化与约化再次交换：量子GIT猜想 I

Quantization commutes with reduction again: the quantum GIT conjecture I

本文证明了紧单调辛流形在哈密顿群作用下，其规范化的2维A模型与辛约化空间的A模型等价，验证了量子GIT猜想，平行于量子力学中“量子化与约化交换”定理。

Shifted geometric quantization

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.

Geometric quantization and families of inner products

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 …

Functorial quantization and the Guillemin-Sternberg conjecture

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'…

Spin_c quantization and the K-multiplicities of the discrete series

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.

A K-Theoretic Note on Geometric Quantization

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.

Quantization of Whitney functions and reduction

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…

Geometric quantization of b-symplectic manifolds

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…

Proper maps, bordism, and geometric quantization

Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ with equivariant vector bundle $E$. Besides, suppose $ϕ$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We can define some equivalence relation on the triples $(M, E, ϕ)$ such that the set of equivalence classes form an abelian group. In this paper, we will show that this group is isomorphic to a completion of character ring $R(G)$. In this framework, we provide a geometric proof to the "Quantization Commutes with Reduction" conjecture in the non-compact setting.

关于李群余切丛的Kähler量子化的BKS配对

On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group

本文研究了紧致李群余切丛的Kähler量化，证明其BKS配对映射是酉的且等同于量子联络的平行移动，并揭示了其酉性源于Kähler极化退化时的Hermitian结构连续性。

无挠 CR 流形的量子化与约化

Quantization and reduction for torsion free CR manifolds

本文研究紧致无挠 CR 流形上的量子化与约化，证明在自然假设下群不变 Fourier-Szegő 投影算子有完全渐近展开，并当线丛张量幂足够大时量子化与约化可交换。