Representation theory of super Yang-Mills algebras

TL;DR

This paper explores the representation theory of super Yang-Mills algebras, revealing that certain Clifford-Weyl super algebras can be realized as quotients, thus providing a new family of representations relevant to supersymmetric gauge theories.

Contribution

It demonstrates that Clifford-Weyl super algebras are quotients of super Yang-Mills algebras, offering new insights into their representations and connections to physics.

Findings

Clifford-Weyl super algebras appear as quotients of super Yang-Mills algebras.

Provides a family of representations for super Yang-Mills algebras.

Connects super algebra representations to supersymmetric gauge theories.

Abstract

We study in this article the representation theory of a family of super algebras, called the \emph{super Yang-Mills algebras}, by exploiting the Kirillov orbit method \textit{\`a la Dixmier} for nilpotent super Lie algebras. These super algebras are a generalization of the so-called \emph{Yang-Mills algebras}, introduced by A. Connes and M. Dubois-Violette in \cite{CD02}, but in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras $\Cliff_{q}(k) \otimes A_{p}(k)$, for $p \geq 3$, or $p = 2$ and $q \geq 2$, appear as a quotient of all super Yang-Mills algebras, for $n \geq 3$ and $s \geq 1$. This provides thus a family of representations of the super Yang-Mills algebras.