Question

Why is the semiclassical approximation of the abelian Chern-Simons theory exact?

Answer 1

I was told that in abelian Chern-Simons theory (say, with a general level matrix K), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the perturbative series factorially divergent. Why should the converging perturbative series in this case give us an exact result? For instnace, there might be some terms which vanish to all order of perturbation, which is actually the case for non-abelian complex Chern-Simons theory where we consider trans-series, taking into account all the instanton effects, and then do the resurgent analysis. Why is this not the case for abelian Chern-Simons theory?

I am also curious if the exactness of semiclassical approximation in the abelian Chern-Simons theory is somehow related to that of supersymmetric field theories where the effect of bosons and fermions cancel each other out, making the situation localized near the vacua.