Topological wave functions and heat equations

We give a new, purely holomorphic description of the holomorphic anomaly equations of the topological string, clarifying their relation to the heat equation satisfied by the Jacobi theta series. In cases where the moduli space is a Hermitian symmetric tube domain G/K, we show that the general solution of the anomaly equations is a matrix element 〈Ψ|g|Ω〉 of the Schrödinger-Weil representation of a Heisenberg extension of G, between an arbitrary state 〈Ψ| and a particular vacuum state |Ω〉. We argue that these results suggest the existence of a one-parameter generalization of the usual topological amplitude, transforming in the smallest unitary representation of the symmetry group G that appears upon compactification of N = 2 supergravity from four to three dimensions. We speculate on its relations to corrections to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies. Email: [email protected], [email protected], [email protected]