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We develop several methods that allow us to compute all-loop partitionfunctions in perturbative Chern-Simons theory with complex gauge group G_C,sometimes in multiple ways. In the background of a non-abelian irreducible flatconnection, perturbative G_C invariants turn out to be interesting topologicalinvariants, which are very different from finite type (Vassiliev) invariantsobtained in a theory with compact gauge group G. We explore various aspects ofthese invariants and present an example where we compute them explicitly tohigh loop order. We also introduce a notion of "arithmetic TQFT" and conjecture(with supporting numerical evidence) that SL(2,C) Chern-Simons theory is anexample of such a theory.

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group