Drinfeld twist and general relativity with fuzzy spaces

We give a simplified formula for the star product on , 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 . Furthermore, the twist allows us to define a covariant tensor calculus on from the perspective of the standard embedding of in flat Euclidean space. That is, we can—in principle—find a representation of a truncated subgroup of the diffeomorphisms on on the algebra of functions on . Using this calculus, we eventually write down an Einstein–Hilbert action on the fuzzy sphere, which is invariant under twisted diffeomorphisms.