Algebraic structures in $\kappa$-Poincar\'e invariant gauge theories

TL;DR

This paper investigates $ abla$-Poincaré invariant gauge theories on $ abla$-Minkowski space-time, revealing they only exist in five dimensions and exploring how twisting affects algebraic and gauge structures, with links to twisted spectral triples.

Contribution

It demonstrates the dimensional restriction of $ abla$-Poincaré invariant gauge theories and analyzes the impact of twisting on algebraic and gauge properties, connecting to twisted spectral triples.

Findings

Gauge theories only exist in five dimensions.

Twisting affects hermiticity conditions but not the gauge group.

Connections to algebraic features of twisted spectral triples.

Abstract

$\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $\kappa$-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited.