Higher Algebraic Structures and Quantization

TL;DR

This paper derives quantum groups from classical actions in 2+1D topological field theories, specifically Chern-Simons theory with finite gauge groups, and explores extending these ideas to lower dimensions and more general theories.

Contribution

It introduces a method to derive (quasi-)quantum groups directly from classical actions and path integrals in topological field theories, with detailed computations for finite gauge groups.

Findings

Path integral over circle yields category of representations of a quasi-quantum group

Extension of classical action to lower-dimensional manifolds is feasible

Finite theories allow reduction of path integral to finite sums

Abstract

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d+1 dimensional topological theory to manifolds of dimension less than d+1. We then ``construct'' a generalized path integral which in d+1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.