Quantization of symplectic cobordisms
type: Item
instance of: scholarly article
title: Quantization of symplectic cobordisms
zbMATH Open document ID: 0955.53049
author: Levente Korpás
author: A. Uribe
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: For \(2n\)-dimensional compact symplectic manifolds \((M_1,w_1)\) and \((M_2,w_2)\) an oriented symplectic cobordism is considered, i.e., a \((2n+1)\)-dimensional manifold \(X\) equipped with a two-form \(\sigma\) such that \(\partial X=M_1\bigsqcup M_2^-\), the pull-back of \(\sigma\) on \(M_i\) is equal to \(w_i\) (\(i=1,2\)), and the kernel of \(\sigma\) is a rank one-subbundle \(\nu\) in \(TX\). A symplectic cobordism between cobordant symplectic manifolds can be viewed as a generalization of an extended phase space conception: in the case when \(M_1=M_2\), \(w_1=w_2\), \(X=M\times [0,T]\) and \(\sigma=w+dt\wedge dH\) then the integral curves of the subbundle \(\nu\) can be identified with the integral curves of the Hamiltonian flow of \(H\) on \(M\). The trajectories of \(\nu\) in \(X\) define a characteristic relation \(\Gamma\), which is a generalization of the canonical transformation \(M_1\rightarrow M_2\). In the case when \(\sigma\) has integral cohomology class in \(H^2(X,\mathbb{R})\), the authors consider the quantization of the symplectic cobordism as a unitary operator between Spin\(^c\) quantizations of cobordant symplectic manifolds \(M_1\) and \(M_2\), which in the semiclassical limit corresponds to the characteristic relation \(\Gamma\). In the end of the paper, using these constructions, the authors consider the following two examples: the correspondence between theta functions on a torus and holomorphic differentials on a genus 2 surface, and a quantized ``Baker's transformation''. The technique developed in the paper can be applied for simplifying the quantization of complicated objects using cobordism relations.
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 1489483
zbMATH Keywords: symplectic cobordism
zbMATH Keywords: extended phase space
zbMATH Keywords: Spin\(^c\) quantization
zbMATH Keywords: Dirac operators
zbMATH Keywords: unitary operator
zbMATH Keywords: symbol calculus of Hermite-Fourier distributions
reviewed by: Hovhannes M. Khudaverdian
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n1.a9
OpenAlex ID: W2318370432
DOI: 10.4310/MRL.1999.V6.N1.A9
description: scientific article; zbMATH DE number 1489483