Matrix Theory in Curved Space

We study curved-space versions of matrix stringtheory taking as a definition of the theory a gaugedmatrix sigma model. By computing the contribution to theone-loop divergent terms in the effective action coming from the diagonal matrix elements weshow that these versions of matrix theory in curvedspace reproduce the string equations of motion and theR⁴ correction to the Hilbert-Einstein action.It is then demonstrated that the divergences due tothe nondiagonal elements induce terms in the effectiveaction that cannot be removed by appropriatecounterterms. This implies that the model can only beconsistent for Ricci flat manifolds with vanishingsix-dimensional Euler density.