Exact results for perturbative Chern–Simons theory with complex gauge group

Abstract

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G C , sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G C invariants turn out to be interesting topological invariants, which are very different from the finite-type (Vassiliev) invariants usually studied in a theory with compact gauge group G and a trivial flat connection. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic topological quantum field theory" and conjecture (with supporting numerical evidence) that SL(2, C) Chern-Simons theory is an example of such a theory.