Gauge theories on κ-Minkowski spaces: twist and modular operators

Abstract

Abstract We discuss the construction of κ-Poincaré invariant actions for gauge theories on κ-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the κ-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the κ-Poincaré invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the κ-Poincaré invariance still holds.

Keywords

• Non-Commutative Geometry
• Gauge Symmetry
• Models of Quantum Gravity