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Extended Equivalence of $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs

Daniel Galviz

Abstract

We establish the equivalence between $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs associated with finite quadratic modules. For gauge group $U(1)$ and even level $k$, we prove that the corresponding Chern--Simons TQFT is naturally isomorphic to the Reshetikhin-Turaev TQFT determined by the pointed modular category $\mathrm{C}(\mathbb{Z}_k,q_k)$. The equivalence holds both for closed $3$-manifolds and for bordisms with boundary, so that the two constructions define naturally isomorphic extended $(2+1)$-dimensional TQFTs. In particular, the finite quadratic module $(\mathbb Z_k,q_k)$ completely determines the even-level $U(1)$ Chern--Simons theory.

Extended Equivalence of $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs

Abstract

We establish the equivalence between Chern-Simons and Reshetikhin-Turaev TQFTs associated with finite quadratic modules. For gauge group and even level , we prove that the corresponding Chern--Simons TQFT is naturally isomorphic to the Reshetikhin-Turaev TQFT determined by the pointed modular category . The equivalence holds both for closed -manifolds and for bordisms with boundary, so that the two constructions define naturally isomorphic extended -dimensional TQFTs. In particular, the finite quadratic module completely determines the even-level Chern--Simons theory.

Paper Structure

This paper contains 18 sections, 21 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Geometric Quantization of U(1) Chern--Simons Theory
  3. The Abelian Chern--Simons Functional
  4. Geometric Quantization of the Boundary Phase Space
  5. Abelian Chern--Simons TQFT
  6. Surgery Invariants from Abelian Gauge Groups
  7. The Abelian RT invariant for G=Zk
  8. The case G=R.
  9. Reciprocity formulas and the relation with U(1) Chern--Simons theory
  10. The obstruction for G=U(1)
  11. Abelian Reshetikhin–Turaev Theory with Boundary
  12. Turaev's Extended RT Formalism
  13. Abelian RT Boundary State Spaces
  14. Extended Equivalence of U(1) Chern--Simons and RT Theories)
  15. Closed Comparison of Abelian RT and U(1) Chern-Simons Theory
  16. ...and 3 more sections

Key Result

Theorem 2.1

The assignments define a unitary extended $(2+1)$-dimensional TQFT. More precisely:

Theorems & Definitions (42)

  • Theorem 2.1: Manoliu2, Theorem VI.11
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1: Turaev1994, Theorem 9.2.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • ...and 32 more