  We analyze and give explicit representations for the effective abelian vector
gauge field actions generated by charged fermions with particular attention to
the thermal regime in odd dimensions, where spectral asymmetry can be present.
We show, through $\zeta-$function regularization, that both small and large
gauge invariances are preserved at any temperature and for any number of
fermions at the usual price of anomalies: helicity/parity invariance will be
lost in even/odd dimensions, and in the latter even at zero mass. Gauge
invariance dictates a very general ``Fourier'' representation of the action in
terms of the holonomies that carry the novel, large gauge invariant,
information. We show that large (unlike small) transformations and hence their
Ward identities, are not perturbative order-preserving, and clarify the role of
(properly redefined) Chern-Simons terms in this context. From a powerful
representation of the action in terms of massless heat kernels, we are able to
obtain rigorous gauge invariant expansions, for both small and large fermion
masses, of its separate parity even and odd parts in arbitrary dimension. The
representation also displays both the nonperturbative origin of a finite
renormalization ambiguity, and its physical resolution by requiring decoupling
at infinite mass. Finally, we illustrate these general results by explicit
computation of the effective action for some physical examples of field
configurations in the three dimensional case, where our conclusions on finite
temperature effects may have physical relevance. Nonabelian results will be
presented separately.
