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

Daniel Galviz

Abstract

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ at level $K$, where $K:Λ\timesΛ\to\mathbb Z$ 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.

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

Abstract

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group at level , where 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 -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 -dimensional TQFT.

Paper Structure

This paper contains 11 sections, 24 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Functional-Integral Construction of Toral Chern--Simons Theory
  3. Classical setup
  4. Quadratic reduction
  5. Exact Gaussian evaluation
  6. Closed manifolds
  7. Boundary Theory
  8. Boundary phase space and functional integrals
  9. Boundary half-densities and analytic boundary states
  10. The Topological Quantum Field Theory
  11. Functoriality

Key Result

Proposition 2.1

The pair $(\mathcal{M}_\Sigma(\mathbb T),\omega_{\Sigma,K})$ is a compact integral symplectic torus. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (60)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 50 more