The Chern-Simons One-form and Gravity on a Fuzzy Space

TL;DR

This paper explores how gravity on fuzzy spaces can be modeled using a matrix Chern-Simons action, connecting finite matrix models to higher-dimensional Chern-Simons theories in the continuum limit.

Contribution

It demonstrates that the matrix Chern-Simons action at finite size encodes gravitational dynamics on fuzzy spaces, bridging discrete matrix models and continuous geometric descriptions.

Findings

Finite matrix Chern-Simons action describes gravity on fuzzy spaces.

Large ${ m N}$ limit yields higher-dimensional Chern-Simons action.

Gravity can be modeled by gauge fields related to space isometries.

Abstract

The one-dimensional ${\cal N}\times {\cal N}$-matrix Chern-Simons action is given, for large ${\cal N}$ and for slowly varying fields, by the $(2k+1)$-dimensional Chern-Simons action $S_{CS}$, where the gauge fields in $S_{CS}$ parametrize the different ways in which the large ${\cal N}$ limit can be taken. Since some of these gauge fields correspond to the isometries of the space, we argue that gravity on fuzzy spaces can be described by the one-dimensional matrix Chern-Simons action at finite ${\cal N}$ and by the higher dimensional Chern-Simons action when the fuzzy space is approximated by a continuous manifold.