Quantization of polysymplectic manifolds (Q2331501)

This paper studies geometric quantizations of polysymplectic manifolds. By definition, a polysymplectic manifold is a smooth manifold equipped with a closed and nondegenerate 2-form with values in a given vector space. Polysymplectic manifolds bear many properties similar to those of symplectic manifolds; for example, the author introduces the notion of Hamiltonian vector fields, Hamiltonian functions and symplectic reductions for them. Suppose \(M\) is a polysymplectic manifold, then one of the key concepts in this paper is the prequantization of \(M\), which consists of a Hermitian vector bundle \(E\to M\) and a Lie algebra representation of the observables on the space of sections of \(E\) which preserves the inner product and the Hamiltonian vector fields. A polysymplectic symplectic manifold is said to be quantizable if it admits a prequantization. In the main body of the paper, the author studies various geometric structures relating to prequantizations of a polysymplectic manifold, and in particular, he shows that the polysymplectic Guillemin-Sternberg conjecture, i.e., the Kähler reduction commuting with the quantization, does not hold in general. The author concludes his paper with a discussion of potential interactions with Chern-Simons theory, multisymplectic quantization, and quantum field theory.