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We analyze and give explicit representations for the effective abelian vectorgauge field actions generated by charged fermions with particular attention tothe thermal regime in odd dimensions, where spectral asymmetry can be present.We show, through $\zeta-$function regularization, that both small and largegauge invariances are preserved at any temperature and for any number offermions at the usual price of anomalies: helicity/parity invariance will belost in even/odd dimensions, and in the latter even at zero mass. Gaugeinvariance dictates a very general ``Fourier'' representation of the action interms of the holonomies that carry the novel, large gauge invariant,information. We show that large (unlike small) transformations and hence theirWard identities, are not perturbative order-preserving, and clarify the role of(properly redefined) Chern-Simons terms in this context. From a powerfulrepresentation of the action in terms of massless heat kernels, we are able toobtain rigorous gauge invariant expansions, for both small and large fermionmasses, of its separate parity even and odd parts in arbitrary dimension. Therepresentation also displays both the nonperturbative origin of a finiterenormalization ambiguity, and its physical resolution by requiring decouplingat infinite mass. Finally, we illustrate these general results by explicitcomputation of the effective action for some physical examples of fieldconfigurations in the three dimensional case, where our conclusions on finitetemperature effects may have physical relevance. Nonabelian results will bepresented separately.Comment: 36 pages, RevTeX, no figure

Effective QED actions: Representations, gauge invariance, anomalies, and mass expansions