Discretized Yang–Mills and Born–Infeld Actions on Finite Group Geometries

Abstract

Discretized non-Abelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ *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 four-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 MD×G, and we find the corresponding field theories on MD after Kaluza–Klein reduction on the G discrete internal spaces. We examine in detail the case G=ZN, and discuss the limit N→∞. A self-contained review on the noncommutative differential geometry of finite groups is included.