A Rigorous Functional-Integral Construction of Toral Chern-Simons Theory

TL;DR

This paper rigorously constructs the Abelian Chern-Simons functional integral on toral gauge groups using zeta-regularization, establishing it as a well-defined topological quantum field theory for 3-manifolds.

Contribution

It provides a mathematically rigorous construction of Abelian Chern-Simons theory with toral gauge group via exact Gaussian evaluation, fulfilling TQFT axioms.

Findings

Defines a topological invariant for closed 3-manifolds.

Produces canonical boundary states for manifolds with boundary.

Satisfies the axioms of a (2+1)-dimensional TQFT.

Abstract

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb T=\mathfrak t/\Lambda \cong U(1)^n$ at level $K$, where $K:\Lambda\times\Lambda\to\mathbb Z$ is an even, integral, nondegenerate symmetric bilinear form, by exact zeta-regularized Gaussian evaluation of the formal quotient integral over connections modulo gauge. For closed $3$-manifolds, this yields a topological invariant; for manifolds with boundary, the relative functional integral produces the canonical boundary state. The resulting theory satisfies the required axioms of a $(2+1)$-dimensional TQFT.