standard vision of an anomalous quantum field theory in general; here we use this description down to points. The invariant of a 4-manifold in the theory ${\mathscr{A}}$ involves its signature and Euler characteristic. It was first discovered in a combinatorial description [@CY] and Walker [@W] also uses ${\mathscr{A}}$ in his description of Chern-Simons (for a more general class of gauge groups).

Since a torus is an abelian group, the classical Chern-Simons action is quadratic in the connection and so the theory is in some sense “free”. Indeed, one expects that the semi-classical approximation is exact. This is the point of view taken by Manoliu [@Ma], who constructs Chern-Simons for circle groups as a 2-3 theory. The invariant this theory assigns to a closed oriented 3-manifold is made from classical invariants of 3-manifold topology: is the integral over the space of flat connections of the square root of the Reidemeister torsion times a spectral flow phase. (There is an overall “volume” factor as well.) The extension to a 1-2-3 theory is determined by a *modular tensor category* by a theorem of Reshetikhin and Turaev [@T]; it is assigned to the circle by the theory. For toral Chern-Simons this is a well-known 