  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 finite type (Vassiliev) invariants
obtained in a theory with compact gauge group G. 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 TQFT" and conjecture
(with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an
example of such a theory.
