Toral Chern-Simons TQFT via Geometric Quantization in Real Polarization

TL;DR

This paper constructs a (2+1)-dimensional toral Chern-Simons topological quantum field theory using geometric quantization with real polarization, revealing connections to Abelian topological order.

Contribution

It introduces a novel geometric quantization approach to toral Chern-Simons theory, explicitly constructing the TQFT and analyzing its boundary states and algebraic structures.

Findings

The theory recovers finite quadratic data for bosonic Abelian topological order at genus one.

The boundary state spaces and operators satisfy the cylinder and gluing axioms.

The finite discriminant group G_K controls the genus-g state spaces.

Abstract

We construct toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/\Lambda\cong U(1)^n$ from an even, integral, nondegenerate symmetric bilinear form $K:\Lambda\times\Lambda\to\mathbb Z$ 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 $G_K=\Lambda^*/K\Lambda$ 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.