Noncommutative gravity: fuzzy sphere and others

TL;DR

This paper develops finite-dimensional matrix models for gravity on noncommutative spaces like the fuzzy sphere and ${f CP}^2$, exploring their actions and large N limits to connect with classical gravity.

Contribution

It introduces explicit matrix actions for gravity on noncommutative ${f S}^2$ and ${f CP}^2$, providing a finite mode truncation approach.

Findings

Derived matrix actions for noncommutative gravity on ${f S}^2$ and ${f CP}^2$

Discussed the large N limit connecting to classical gravity

Showed finite mode truncation yields a finite-dimensional model

Abstract

Gravity on noncommutative analogues of compact spaces can give a finite mode truncation of ordinary commutative gravity. We obtain the actions for gravity on the noncommutative two-sphere and on the noncommutative ${\bf CP}^2$ in terms of finite dimensional $(N\times N)$-matrices. The commutative large $N$ limit is also discussed.