Dirac Matrices for Chern-Simons Gravity

TL;DR

This paper derives a general formula and efficient algorithms for computing traces of symmetrized products of Dirac Gamma matrices, crucial for constructing Chern-Simons gravity theories in higher dimensions.

Contribution

It introduces a new explicit formula and two algorithms, including a minimal recurrence-based method, to efficiently compute Gamma matrix traces in high-dimensional spacetimes.

Findings

The trace formula is expressed as a sum over integer partitions with specific coefficients.

A recurrence relation between coefficients simplifies calculations.

The minimal algorithm computes coefficients in under a minute for n=25.

Abstract

A genuine gauge theory for the Poincar\'e, de Sitter or anti-de Sitter algebras can be constructed in (2n-1)-dimensional spacetime by means of the Chern-Simons form, yielding a gravitational theory that differs from General Relativity but shares many of its properties, such as second order field equations for the metric. The particular form of the Lagrangian is determined by a rank n, symmetric tensor invariant under the relevant algebra. In practice, the calculation of this invariant tensor can be reduced to the computation of the trace of the symmetrized product of n Dirac Gamma matrices \Gamma_{ab} in 2n-dimensional spacetime. While straightforward in principle, this calculation can become extremely cumbersome in practice. For large enough n, existing computer algebra packages take an inordinate long time to produce the answer or plainly fail having used up all available memory. In this talk we show that the general formula for the trace of the symmetrized product of 2n Gamma matrices \Gamma_{ab} can be written as a certain sum over the integer partitions s of n, with every term being multiplied by a numerical coefficient \alpha_{s}. We then give a general algorithm that computes the \alpha-coefficients as the solution of a linear system of equations generated by evaluating the general formula for different sets of tensors B^{ab} with random numerical entries. A recurrence relation between different coefficients is shown to hold and is used in a second, "minimal" algorithm to greatly speed up the computations. Runtime of the minimal algorithm stays below 1 min on a typical desktop computer for up to n=25, which easily covers all foreseeable applications of the trace formula.