Drinfeld twist and general relativity with fuzzy spaces

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