A family quantization formula for symplectic manifolds with boundary
type: Item
instance of: scholarly article
title: A family quantization formula for symplectic manifolds with boundary
zbMATH Open document ID: 1008.53076
DOI: 10.1007/BF02876708
author: Huitao Feng
author: Wenchuan Hu
author: Weiping Zhang
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: Let \(M\) be a compact symplectic manifold and let \(L\to M\) be a Hermitian line bundle with a Hermitian connection whose curvature \(\omega\) is the symplectic form on \(M\). For an almost complex structure on \(M\) compatible with \(\omega\) one can define a \(\text{Spin}^c\)-Dirac operator \(D^L\) on \(\Omega^{0,*}(M,L)\). Then the geometric quantization \(Q(M,L)\) of \((M,\omega)\) is the virtual vector space \( Q(M,L) = \Omega^{0,{\text{even}}}\cap\ker D^L - \Omega^{0,{\text{odd}}}\cap\ker D^L\). Now, suppose that a compact connected Lie group \(G\) with Lie algebra \({\mathfrak g}\) acts symplectically on \(L\to M\) and with moment map \(\mu\colon M\to\mathfrak g^*\). If \(0\in\mathfrak g^*\) is a regular value of \(\mu\) and \(G\) acts freely on \(\mu^{-1}(0)\), then \(M_G=\mu^{-1}(0)/G\) is a symplectic manifold, known as the Marsden-Weinstein or symplectic reduction of \(M\) by \(G\) at 0, and it inherits a prequantum line bundle, \(L_G\to M_G\), and the geometric quantization \(Q(M_G,L_G)\) can be constructed. The famous Guillemin-Sternberg conjecture, which has become known as ``quantization commutes with reduction'', or \([Q,R]=0\) theorem, is that \(\dim Q(M,L)^G = \dim Q(M_G,L_G)\), where \(Q(M,L)^G\) is the \(G\)-fixed part of \(Q(M,L)\) which is now a virtual representation of \(G\). The conjecture has been proved by several different methods under various different hypotheses. \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 515-538 (1982; Zbl 0503.58018)] themselves proved it in a special case when \(M\) is Kähler. \textit{Y. Tian} and \textit{W. Zhang} [Invent. Math. 132, No. 2, 229-259 (1998; Zbl 0944.53047)] gave a direct analytic proof of the conjecture and of some other related results, using a deformation of the \(\text{Spin}^c\)-Dirac operator by using the Clifford action of the Hamilton vector field, of the norm-squared of the moment map, following ideas of Kirwan and Witten. In [Geom. Funct. Anal. 9, No. 3, 596-640 (1999…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 19L47
zbMATH DE Number: 1782113
zbMATH Keywords: symplectic manifold with boundary
zbMATH Keywords: reduction
zbMATH Keywords: family quantization
zbMATH Keywords: Hamiltonian action
zbMATH Keywords: higher spectral flow
zbMATH Keywords: \(\text{Spin}^c\)-Dirac operator
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02876708
OpenAlex ID: W1491774649
description: scientific article; zbMATH DE number 1782113