Quantum A-infinity Structures for Open-Closed Topological Strings

We study factorizations of topological string amplitudes on higher genus Riemann surfaces with multiple boundary components and find quantum A-infinity relations, which are the higher genus analog of the (classical) A-infinity relations on the disk. For topological strings with $\hat c=3$ the quantum A-infinity relations are trivially satisfied on a single D-brane, whereas in a multiple D-brane configuration they may be used to compute open higher genus amplitudes recursively from disk amplitudes. This can be helpful in open Gromov--Witten theory in order to determine open string higher genus instanton corrections. Finally, we find that the quantum A-infinity structure cannot quite be recast into a quantum master equation on the open string moduli space.