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Toral Chern-Simons TQFT via Geometric Quantization in Real Polarization

Daniel Galviz

Abstract

We construct toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ from an even, integral, nondegenerate symmetric bilinear form $K:Λ\timesΛ\to\mathbb Z$ by geometric quantization via real polarization. We obtain a unitary extended $(2+1)$-dimensional TQFT by constructing the boundary state spaces and canonical operators and proving that they satisfy the cylinder and gluing axioms. The finite discriminant group $G_K=Λ^*/KΛ$ arises naturally in the theory and controls the genus-$g$ state spaces. At genus one, the theory recovers the finite quadratic data underlying bosonic Abelian topological order.

Toral Chern-Simons TQFT via Geometric Quantization in Real Polarization

Abstract

We construct toral Chern-Simons theory with gauge group from an even, integral, nondegenerate symmetric bilinear form by geometric quantization via real polarization. We obtain a unitary extended -dimensional TQFT by constructing the boundary state spaces and canonical operators and proving that they satisfy the cylinder and gluing axioms. The finite discriminant group arises naturally in the theory and controls the genus- state spaces. At genus one, the theory recovers the finite quadratic data underlying bosonic Abelian topological order.

Paper Structure

This paper contains 22 sections, 40 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Boundary Phase Space and Geometric Quantization
  3. Classical Boundary Phase Space
  4. Boundary Polarizations and Lagrangians
  5. Canonical Toral Prequantum Line Bundle
  6. Real Polarizations and Bohr--Sommerfeld Leaves
  7. The Toral Quantum Hilbert Space
  8. BKS Operators for Toral Real Polarizations
  9. Torsion Sectors and Canonical Boundary States
  10. Moduli of Flat Toral Bundles
  11. Boundary leaves in each torsion component
  12. Classical Toral Chern--Simons Section
  13. Torsion half-densities on torsion components
  14. Canonical Toral Boundary Vector
  15. Extended Toral Chern–Simons TQFT
  16. ...and 7 more sections

Key Result

Proposition 2.1

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

Theorems & Definitions (99)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 89 more