Gravity and generalised geometry from a Lie 2-algebroid perspective

Abstract

We revisit the problem of defining natural torsion and curvature tensors for generalised connections and of using them to reconstruct the effective action for the massless modes of the closed string from generalised geometry. The starting point is the description of a Courant algebroid as a differential graded (dg) symplectic manifold. A systematic procedure for split dg manifolds singles out a generalised Lie bracket as part of a larger Lie 2-algebroid structure, with the right properties to unambiguously define torsion and Riemann curvature tensors together with their corresponding Bianchi identities. This framework eliminates the structural circularity and ambiguity problems of previous approaches in the literature. We show that a unique generalised metric compatible generalised connection with fixed (con)torsion exists for any choice of ordinary connection that splits the dg manifold, while at the same time all Lie 2-algebroids obtained in this way are equivalent up to $L_{\infty}$ quasi-isomorphisms. For a given Lie 2-algebroid we demonstrate that the unique compatible generalised connection reproduces the effective closed string action from a geometric action based on the new generalised Ricci tensor upon making a canonical choice of torsion.