Gauge Invariant Perturbation Theory via Homotopy Transfer

TL;DR

This paper develops a homotopy transfer approach to express gauge theory perturbations in gauge-invariant variables at all orders, generalizing cosmological techniques to broader theories including gravity and string theory.

Contribution

It introduces a novel homotopy transfer method to systematically obtain gauge-invariant variables in perturbative gauge theories, extending previous cosmological methods.

Findings

Provides an algorithmic procedure for higher-order corrections.

Demonstrates the approach on Yang-Mills, gravity, and string theory.

Establishes a unifying framework for gauge invariance in perturbation theory.

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.