Unified Theories from Fuzzy Extra Dimensions

TL;DR

This paper integrates Coset Space Dimensional Reduction with Non-commutative Geometry to reduce high-dimensional gauge theories on fuzzy spaces, providing a novel approach to model compactification in theoretical physics.

Contribution

It extends CSDR methods to fuzzy coset spaces, enabling dimensional reduction on finite matrix approximations of homogeneous spaces like the fuzzy sphere.

Findings

Demonstrates dimensional reduction on fuzzy spaces.

Shows gauge symmetry reduction to subgroups.

Provides a framework for finite approximations of extra dimensions.

Abstract

We combine and exploit ideas from Coset Space Dimensional Reduction (CSDR) methods and Non-commutative Geometry. We consider the dimensional reduction of gauge theories defined in high dimensions where the compact directions are a fuzzy space (matrix manifold). In the CSDR one assumes that the form of space-time is M^D=M^4 x S/R with S/R a homogeneous space. Then a gauge theory with gauge group G defined on M^D can be dimensionally reduced to M^4 in an elegant way using the symmetries of S/R, in particular the resulting four dimensional gauge is a subgroup of G. In the present work we show that one can apply the CSDR ideas in the case where the compact part of the space-time is a finite approximation of the homogeneous space S/R, i.e. a fuzzy coset. In particular we study the fuzzy sphere case.