Localization and the quantization conjecture
type: Item
instance of: scholarly article
title: Localization and the quantization conjecture
zbMATH Open document ID: 0876.55007
DOI: 10.1016/S0040-9383(96)00015-8
author: Lisa C. Jeffrey
author: Frances Kirwan
published in: Topology
publication date: 1997-10-09T00:00:00Z
review text: Let \(M\) be a compact quantizable symplectic manifold acted on in a Hamiltonian fashion by a compact connected Lie group \(K\) preserving the structure and suppose that \(0\) is a regular value of the moment map so that the reduced space \(M_{\text{red}}\) acquires a structure of a symplectic orbifold. Pick a compatible almost complex structure on \(M\) --- this is always possible; it descends to a compatible almost complex structure on \(M_{\text{red}}\). Let \(\mathcal L\) be a \(K\)-equivariant prequantum bundle for \(M\), the \(K\)-action being compatible with the moment map, and denote by \(\mathcal L_{\text{red}}\) the corresponding orbifold line bundle on the reduced space. When the almost complex structure is a genuine complex structure and \(\mathcal L\) a holomorphic line bundle, geometric quantization endows the cohomology groups \(\text H^j(M,\mathcal L)\) with structures of representation spaces for \(K\), and the Riemann-Roch numbers \(RR^K(\mathcal L)\) and \(RR(\mathcal L_{\text{red}})\) are defined as usual by \(RR^K(\mathcal L) = \sum (-1)^j \dim \text H^j(M,\mathcal L)^K\) and \(RR(\mathcal L_{\text{red}}) =\sum (-1)^j \dim \text H^j(M_{\text{red}},\mathcal L_{\text{red}})\). When the structure is only almost complex, Riemann-Roch numbers \(RR^K(\mathcal L)\) and \(RR(\mathcal L_{\text{red}})\) can still be defined by means of a direct image construction in \(K\)-theory involving a Spin\(^c\) Dirac operator, cf. [\textit{J. J. Duistermaat}, The heat kernel Lefschetz fixed point formula for the Spin\(^c\) Dirac operator (1996; Zbl 0858.58045); \textit{V. Guillemin}, Progr. Math. 123, 305-334 (1994; Zbl 0869.58017); \textit{M. Vergne}, Duke Math. J. 82, No. 1, 143-179 (1996; Zbl 0855.58033); 181-194 (1996; Zbl 0855.58034)]. This direct image construction has been used already by \textit{M. F. Atiyah} and \textit{F. Hirzebruch} in the proof of the differentiable Riemann-Roch theorem [Bull. Am. Math. Soc. 65, 276-281 (1959; Zbl 0142.40901)]; it invol…
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 966984
zbMATH Keywords: geometric quantization
zbMATH Keywords: classical systems with symmetry
zbMATH Keywords: moment map
zbMATH Keywords: differentiable Riemann-Roch theorem
zbMATH Keywords: Spin\(^ c\) Dirac operator
zbMATH Keywords: localization
Wikidata QID: Q123274572
reviewed by: J. Huebschmann
MaRDI profile type: Publication
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description: scientific article