A Rigorous Functional-Integral Construction of Toral Chern-Simons Theory

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb{T}=\mathfrak{t}\mathrm{/}\mathrm{\Lambda }\cong U(1{)}^n$ at level $K$, where $K:\mathrm{\Lambda }\times \mathrm{\Lambda }\to \mathbb{Z}$ is an even, integral, nondegenerate symmetric bilinear form, by exact zeta-regularized Gaussian evaluation of the formal quotient integral over connections modulo gauge. For closed $3$-manifolds, this yields a topological invariant; for manifolds with boundary, the relative functional integral produces the canonical boundary state. The resulting theory satisfies the required axioms of a $(2+1)$-dimensional TQFT.