Integral Transforms for Finite Gauge Theory

Abstract

This paper shows that quantization of $\pi$-finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity. We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the $\pi$-finite quantization procedure considered herein and the geometric quantization of a symplectic vector space.