String field theory and brane superpotentials

I discuss tree-level amplitudes in cubic topological string field theory, showing that a certain family of gauge conditions leads to an A∞ algebra of tree-level string products which define a potential describing the dynamics of physical states. Upon using results of modern deformation theory, I show that the string moduli space admits two equivalent descriptions, one given in standard Maurer-Cartan fashion and another given in terms of a ‘homotopy Maurer-Cartan problem’, which describes the critical set of the potential. By applying this construction to the topological A and B models, I obtain an intrinsic formulation of ‘D-brane superpotentials’ in terms of string field theory data. This gives a prescription for computing such quantities to all orders, and proves the equivalence of this formulation with the fundamental description in terms of string field moduli. In particular, it clarifies the relation between the Chern-Simons/holomorphic Chern-Simons actions and the superpotential for A/B-type branes.