Towards Gauge theory for a class of commutative and non-associative fuzzy spaces

TL;DR

This paper develops gauge theories for a class of commutative, non-associative fuzzy spaces, revealing how non-associativity necessitates generalized gauge fields and suggesting string-inspired spacetime deformations that preserve Lorentz invariance.

Contribution

It introduces a gauge theory framework for commutative, non-associative fuzzy spaces related to SO(2k+1) algebras, highlighting the role of non-associativity in gauge field generalization.

Findings

Gauge fields depend on coordinates and derivatives due to non-associativity.

Partial gauge fixing recovers usual gauge fields.

Deformation parameter linked to rotation group suggests Lorentz-invariant spacetime deformations.

Abstract

We discuss gauge theories for commutative but non-associative algebras related to the $ SO(2k+1)$ covariant finite dimensional fuzzy $2k$-sphere algebras. A consequence of non-associativity is that gauge fields and gauge parameters have to be generalized to be functions of coordinates as well as derivatives. The usual gauge fields depending on coordinates only are recovered after a partial gauge fixing.The deformation parameter for these commutative but non-associative algebras is a scalar of the rotation group. This suggests interesting string-inspired algebraic deformations of spacetime which preserve Lorentz-invariance.