Curved $A_{\infty}$-algebras and gauge theory

TL;DR

This paper develops a general algebraic framework for gauge theories using curved $A_{ abla}$-algebras, enabling the transfer of gauge structures to discrete settings like simplicial cochains.

Contribution

It introduces a novel algebraic approach to gauge theories based on curved $A_{ abla}$-algebras and demonstrates transfer techniques to discrete models.

Findings

Established a transfer method for algebraic gauge theories along chain contractions.

Derived a discrete simplicial gauge theory with notions of connection, curvature, and gauge invariance.

Provided a framework for discretizing gauge theories on triangulated manifolds.

Abstract

We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved $A_{\infty}$-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Specializing to the case of the contraction from differential forms to cochains, we obtain a simplicial gauge theory on the matrix-valued simplicial cochains of a triangulated manifold. In particular, one obtains discrete notions of connection, curvature, gauge transformation and gauge invariant action.