Drinfeld Twist and General Relativity with Fuzzy Spaces

TL;DR

This paper introduces a simplified star product formula on fuzzy complex projective spaces, enabling the definition of a Drinfeld twist that leads to a covariant tensor calculus and a twisted diffeomorphism-invariant Einstein-Hilbert action.

Contribution

It provides a new simplified formula for the star product on fuzzy CP^n, constructs a Drinfeld twist, and formulates a twisted diffeomorphism-invariant gravity action on fuzzy spaces.

Findings

The twist of the coproduct is trivial for isometry generators.

A covariant tensor calculus on fuzzy CP^n is developed.

An Einstein-Hilbert action invariant under twisted diffeomorphisms is formulated.

Abstract

We give a simplified formula for the star product on CP^n_L, which enables us to define a twist element suited for discussing a Drinfeld twist like structure on fuzzy complex projective spaces. The existence of such a twist will have several consequences for field theories on fuzzy spaces, some of which we discuss in the present paper. As expected, we find that the twist of the coproduct is trivial for the generators of isometries on CP^n_L. Furthermore, the twist allows us to define a covariant tensor calculus on CP^n_L from the perspective of the standard embedding of CP^n in flat Euclidean space. That is, we find a representation of a truncated subgroup of the diffeomorphisms on CP^n on the algebra of functions on CP^n_L. Using this calculus, we eventually write down an Einstein-Hilbert action on the fuzzy sphere, which is invariant under twisted diffeomorphisms.