Toral Chern-Simons TQFT via Geometric Quantization in Real Polarization

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