Integral Transforms for Finite Gauge Theory
Abstract
This paper shows that quantization of -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 -finite quantization procedure considered herein and the geometric quantization of a symplectic vector space.
Cite
@article{arxiv.2312.00117,
title = {Integral Transforms for Finite Gauge Theory},
author = {Jackson Van Dyke},
journal= {arXiv preprint arXiv:2312.00117},
year = {2026}
}
Comments
27 pages. some text overlap with arXiv:2311.01637 In sec. 2.3.1 Hyp. Q was refined and exm. 2.13 was expanded. In sec. 5, the projective action of the orthogonal group was corrected, more emphasis on the Fourier-type transform. Fixed typos in sec. 3.2, 4.1, and app. B. Removed unneeded background information (sec. 2 and app. B). This is available in arXiv:2311.01637