  We analyze large gauge invariance in combined nonabelian and thermal QFT and
their physical consequences for D=3 effective actions. After briefly reviewing
the structure of bundles and large gauge transformations that arise in
non-simply connected 3-manifolds and gauge groups, we discuss their connections
to Chern-Simons terms and Wilson-Polyakov loops. We then provide an invariant
characterization of the ``abelian'' fluxes encountered in explicit computations
of finite temperature effective actions. In particular we relate, and provide
explicit realizations of, these fluxes to a topological index that measures the
obstruction to global diagonalization of the loops around compactified time. We
also explore the fate of, and exhibit some everywhere smooth, large
transformations for non-vanishing index in various topologies.
