Towards a loop representation of connection theories defined over a super Lie algebra

TL;DR

This paper reviews loop space formulations of gauge theories over Lie algebras and introduces initial results for super Lie algebra connections, aiming to extend Mandelstam identities to supersymmetric contexts.

Contribution

It presents the first steps towards formulating Mandelstam identities for super Lie algebra connections, relevant for supergravity and super Chern-Simons theories.

Findings

Extended Mandelstam identities to super Lie algebra connections

Initial results for supersymmetric gauge theories

Potential applications in supergravity and super Chern-Simons theories

Abstract

The purpose of this contribution is to review some aspects of the loop space formulation of pure gauge theories having the connection defined over a Lie algebra. The emphasis is focused on the discussion of the Mandelstam identities, which provide the basic constraints upon both the classical and the quantum degrees of freedom of the theory. In the case where the connection is extended to be valued on a super Lie algebra, some new results are presented which can be considered as first steps towards the construction of the Mandelstam identities in this situation, which encompasses such interesting cases as supergravity in $3+1$ dimensions together with $2+1$ super Chern-Simons theories, for example. Also, these ideas could be useful in the loop space formulation of fully supersymmetric theories.