Geometrical effective action and Wilsonian flows

TL;DR

The paper develops a gauge-invariant Wilsonian flow equation for the geometrical Vilkovisky-DeWitt effective action by employing geodesic normal fields and horizontal projections in configuration space. The resulting flow is gauge invariant and regulator-flexible, but the base-point dependence introduces modified Nielsen identities that constrain truncations and relate to modified Slavnov-Taylor identities. A key challenge is ensuring infrared regularity of truncations, which the authors address with strategies such as choosing the base point to stabilize the flow (e.g., $\varphi_* = \bar{\varphi}$) and by leveraging heat-kernel techniques. The framework is positioned as a practical, self-contained approach with potential extensions to quantum gravity and 2PI formalisms, offering an alternative to traditional gauge-fixed formulations while preserving gauge symmetry at the level of observables.

Abstract

A gauge invariant flow equation is derived by applying a Wilsonian momentum cut-off to gauge invariant field variables. The construction makes use of the geometrical effective action for gauge theories in the Vilkovisky-DeWitt framework. The approach leads to modified Nielsen identities that pose non-trivial constraints on consistent truncations. We also evaluate the relation of the present approach to gauge fixed formulations as well as discussing possible applications.