Discretized Yang-Mills and Born-Infeld actions on finite group geometries

TL;DR

This paper develops discretized versions of Yang-Mills and Born-Infeld actions on finite group geometries, exploring their properties, duality invariance, and reductions, with detailed analysis on cyclic groups and their limits.

Contribution

It introduces a geometric discretization framework for nonabelian gauge theories on finite groups, extending to Born-Infeld actions and analyzing duality invariance and Kaluza-Klein reductions.

Findings

Discretized electromagnetism admits duality rotations.

Yang-Mills and Born-Infeld theories are derived on product spaces with finite groups.

Analysis of the limit as the group order N approaches infinity.

Abstract

Discretized nonabelian gauge theories living on finite group spaces G are defined by means of a geometric action \int Tr F \wedge *F. This technique is extended to obtain discrete versions of the Born-Infeld action. The discretizations are in 1-1 correspondence with differential calculi on finite groups. A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a 4-dimensional abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations. Yang-Mills and Born-Infeld theories are also considered on product spaces M^D x G, and we find the corresponding field theories on M^D after Kaluza-Klein reduction on the G discrete internal spaces. We examine in some detail the case G=Z_N, and discuss the limit N -> \infty. A self-contained review on the noncommutative differential geometry of finite groups is included.