Gauge Invariant Perturbation Theory via Homotopy Transfer

Abstract

We show that the perturbative expansion of general gauge theories can be expressed in terms of gauge invariant variables to all orders in perturbations. In this we generalize techniques developed in gauge invariant cosmological perturbation theory, using Bardeen variables, by interpreting the passing over to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie algebras encoding the gauge theory. This is illustrated for Yang-Mills theory, gravity on flat and cosmological backgrounds and for the massless sector of closed string theory. The perturbation lemma yields an algorithmic procedure to determine the higher corrections of the gauge invariant variables and the action in terms of these.

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

“…Similarly, one may subject any double field theory to a time/space split (see, e.g., sec. 3 of [16]).…”

“…• They can be pure gauge modes. Those can be dealt with by the usual field-theoretical gauge-fixing procedure or by homotopy transfer to gauge invariant variables [16]. In our context, the Batalin-Vilkovisky field-antifield formalism [17,18] is particularly well suited for gauge-fixing, given its natural link with L ∞ -algebras; see e.g.…”

“…Since above we have seen that, as was to be expected, the L ∞ formulation of integrating out fields works well in the presence of gauge symmetries we can now simplify by eliminating the gauge redundancy, say by fixing a gauge or by passing over to gauge invariant variables. In fact, the passing over to gauge invariant variables can also be interpreted as homotopy transfer [16]. In the present case this homotopy transfer is defined by the projection and inclusion maps…”