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B The Background Effective Action

The effective action summarizes the content of a field theory in a way which is technically convenient and physically instructive. It is familiar from perturbation theory as the generating functional of the “1-particle irreducible” (1-PI) Green’s functions. It can however be given a meaning independent of perturbation theory and its functional derivatives can be used to reconstruct all the correlation functions of the field theory under consideration. Since the latter can be viewed as the moments of the functional measure, this replaces the “cone of measures” (or the space of “Wilsonian effective actions Sl”) in the Wilsonian setting by the “space of effective action functionals” as the arena on which the renormalization group acts.

The standard effective action admits two fruitful generalizations, discussed in Appendices B and C, respectively. The first one is adapted to theories with symmetries, which can be field reparameterization symmetries, gauge symmetries, or both. These effective actions are known as “background effective actions”. In Appendix B.2 we provide a concise summary of these constructions. The background reparameterization and/or gauge invariance also provides an, often crucial, simplification of the renormalization. In Appendix B.3 we describe this for the case of field reparameterization symmetries in Riemannian sigma-models, which also provides some of the renormalization prerequisites for Section 3. In a gravitational context the background field formalism also provides a setting which, despite the name, can reasonably be regarded as “background independent”.

The second generalization of the standard effective action is one where, roughly, the bare action SΛ is replaced by SΛ + C Λ,k, where CΛ,k effectively suppresses field modes with momenta less than k. This leads to the effective average action reviewed in Appendix C.

For definiteness we assume a Euclidean setting throughout these Appendices.

 B.1 Standard effective action and its perturbative construction
 B.2 Survey of background field formalisms
  B.2.1 Linear split, non-gauge theories
  B.2.2 Geodesic split, non-gauge theories
  B.2.3 Linear split, gauge theories
  B.2.4 Geodesic split, gauge theories
 B.3 Renormalization of Riemannian sigma-models

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