Algebraic structure of Yang-Mills theory

TL;DR

This paper explores the algebraic structures underlying Yang-Mills theory, providing rigorous proofs and analyzing two algebraic frameworks, thereby deepening the mathematical understanding of the theory.

Contribution

It offers a detailed comparison of A-infinity algebras and Lie algebra representations in Yang-Mills theory, with new calculations of Hochschild (co)homology.

Findings

Established algebraic interpretations of Yang-Mills theory.

Provided rigorous proofs for non-supersymmetric cases.

Calculated Hochschild (co)homology of relevant algebras.

Abstract

In the present paper we analyze algebraic structures arising in Yang-Mills theory. The paper should be considered as a part of a project started with a paper "On maximally supersymmetric Yang-Mills theories" devoted to maximally supersymmetric Yang-Mills theories. In this paper we collected those of our results which are correct without assumption of supersymmetry and used them to give rigorous proofs of some results of the cited paper. We consider two different algebraic interpretations of Yang-Mills theory - in terms of A_{\infty}-algebras and in terms of representations of Lie algebras (or associative algebras). We analyze the relations between these two approaches and calculate some Hochschild (co)homology of algebras in question.