Toral Chern-Simons TQFT via Geometric Quantization in Real Polarization

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.