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Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories

Daniel Galviz

Abstract

We prove a natural isomorphism between toral Chern-Simons theory with gauge group $\mathbb T=\mathcal t/Λ\cong U(1)^n$ and the Reshetikhin-Turaev theory associated with the finite quadratic module determined by an even, integral, nondegenerate symmetric bilinear form $K:Λ\timesΛ\to\mathbb Z.$ More precisely, let $G_K=Λ^*/KΛ$ be the discriminant group of $K$, equipped with its induced quadratic form $q_K$, and let $C(G_K,q_K)$ be the corresponding pointed modular category. Using the geometric quantization formulation of toral Chern-Simons theory, we show that the resulting toral TQFT is naturally isomorphic to the Reshetikhin-Turaev TQFT determined by $C(G_K,q_K)$. The comparison is established both for closed $3$-manifold invariants and for bordisms with boundary, yielding an isomorphism of extended $(2+1)$-dimensional TQFTs.

Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories

Abstract

We prove a natural isomorphism between toral Chern-Simons theory with gauge group and the Reshetikhin-Turaev theory associated with the finite quadratic module determined by an even, integral, nondegenerate symmetric bilinear form More precisely, let be the discriminant group of , equipped with its induced quadratic form , and let be the corresponding pointed modular category. Using the geometric quantization formulation of toral Chern-Simons theory, we show that the resulting toral TQFT is naturally isomorphic to the Reshetikhin-Turaev TQFT determined by . The comparison is established both for closed -manifold invariants and for bordisms with boundary, yielding an isomorphism of extended -dimensional TQFTs.

Paper Structure

This paper contains 12 sections, 45 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Toral U(1)n Chern--Simons Extended TQFT
  3. Boundary phase space and real-polarized quantization
  4. Torsion sectors, boundary states, and the extended theorem
  5. General Abelian Reshetikhin–Turaev Theory
  6. Reshetikhin–Turaev invariants for general Abelian groups
  7. Surgery formulas and finite quadratic modules
  8. Reshetikhin–Turaev boundary theory
  9. Equivalence of Toral Chern--Simons and Reshetikhin–Turaev Theories
  10. Closed comparison for T isomorphic to U(1)n
  11. Boundary equivalence for U(1)n
  12. Extended equivalence Theorem

Key Result

Proposition 2.1

Galviz2 For every closed oriented surface $\Sigma$, the pair $(\mathcal{M}_\Sigma(\mathbb T),\omega_{\Sigma,K})$ is a compact symplectic torus. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (84)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 74 more