I know two essential points where General Relativity plays a central role in String Theory: i) definition of the theory using a target spacetime with some Riemannian background metric in the Polyakov action and ii) recovery of the Einstein field equations as conditions regarding conformal invariance on the world-sheet, i.e. vanishing beta functions. My question related to (i) is if anybody has ever thought about a construction which is not based on using a Riemann but a Riemann-Cartan manifold? The question related to (ii) would be if that may result in different consistency conditions, i.e. beta functions, effective / low-energy equations, anomaly cancellation, dimensions and geometry of the background spacetime etc.?