GAMMA: A Mathematica package for performing gamma-matrix algebra and Fierz transformations in arbitrary dimensions

TL;DR

The paper presents GAMMA, a rule-based Mathematica package for performing $\Gamma$-matrix algebra in arbitrary integer dimensions, addressing the intensive computations required in theories like eleven-dimensional supergravity. It introduces a comprehensive set of functions for gamma-algebra (including $\Gamma$-products, expansions, contractions, traces, and canonical ordering) and dedicated tools for Fierz transformations, accommodating both tensor-spinors and hook-tensors. Key contributions include support for dimension and spinor-space specification, efficient contraction routines, and specialized hooks such as $MakeHook$ and $FierzTerm$ to handle complex fierzing scenarios. This package enables reproducible, automated high-dimensional gamma-matrix calculations, significantly aiding research in supergravity and related high-energy theory domains.

Abstract

We have developed a Mathematica package capable of performing gamma-matrix algebra in arbitrary (integer) dimensions. As an application we can compute Fierz transformations.