FieldsX -- An extension package for the xAct tensor computer algebra suite to include fermions, gauge fields and BRST cohomology

TL;DR

FieldsX extends the xAct tensor algebra suite to facilitate complex computations in field theory involving fermions, gauge fields, and BRST cohomology, aiding researchers in analyzing supersymmetric and gauge theories.

Contribution

The paper introduces FieldsX, a new package that adds tools for fermions, gauge fields, and cohomology calculations to the xAct tensor algebra suite, enabling advanced field theory computations.

Findings

Supports curved-space gamma matrices and Fierz identities.

Includes functions for nilpotent differentials and cohomology computations.

Demonstrates application with $ abla=1$ Super-Yang--Mills theory.

Abstract

I present the tensor computer algebra package FieldsX, which extends the xAct suite of tensor algebra packages to perform computations in field theory with fermions and gauge fields. This includes the standard tools of curved-space $\gamma$ matrices, Fierz identities, invariant tensors on Lie algebras, arbitrary gradings and left and right variational derivatives, as well as the decomposition of spinor products into irreducible components following the approach of d'Auria, Fr\'e, Maina and Regge [Annals Phys. 139 (1982) 93]. Lastly, it also includes functions to work with nilpotent differentials such as the BV-BRST differential for (supersymmetric) gauge theories and to compute their (relative) cohomologies, from which anomalies and gauge-invariant operators can be determined. I illustrate the use of the package with the example of $\mathcal{N}=1$ Super-Yang--Mills theory.