Quantization of Polysymplectic Manifolds

Abstract: We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in an essential way the action of the space of coefficients. We define quantization with respect to a polarization and to a spin$^\text{c}$ structure. In the presence of a complex polarization, it is shown that the polysymplectic Guillemin-Sternberg conjecture is false. We conclude with potential extensions and applications.