# Homological perspective on edge modes in linear Yang-Mills and Chern-Simons theory

## TL;DR

This paper develops a homological framework to construct extended phase spaces for linear Yang-Mills and Chern-Simons theories, clarifying the mathematical structure of edge modes and unifying previous ad hoc approaches.

## Contribution

It provides a rigorous homological construction of extended phase spaces for these gauge theories, formalizing and generalizing earlier heuristic methods.

## Key findings

- Unified homological construction for edge modes in Yang-Mills and Chern-Simons theories.
- Clarified the mathematical structure underlying edge modes.
- Extended the phase space formalism to include boundary effects.

## Abstract

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold $M$ with a time-like boundary $\partial M$ that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern-Simons theory, in which case we obtain the extended phase space introduced by Geiller [Nucl. Phys. B 924, 312 (2017)].