Quantization of symplectic cobordisms (Q1574694)

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.