Quantum gauge symmetries in Noncommutative Geometry

TL;DR

This paper extends gauge symmetry concepts to noncommutative geometry using quantum groups, providing universal constructions and examples relevant to particle physics models like the Standard Model.

Contribution

It introduces universal quantum group objects for gauge symmetries in noncommutative geometry and applies them to physically relevant algebras, advancing the mathematical framework for particle physics.

Findings

Existence of universal quantum gauge groups established.

Examples include quantum symmetries for matrix algebras relevant to physics.

Identification of a free version of the symplectic group Sp(n).

Abstract

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M_n(R), M_n(C) and M_n(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A_F=C+H+M_3(C) and A^{ev}=H+H+M_4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).