Topological wave functions and heat equations

It is generally known that the holomorphic anomaly equations in topologicalstring theory reflect the quantum mechanical nature of the topological stringpartition function. We present two new results which make this assertion moreprecise: (i) we give a new, purely holomorphic version of the holomorphicanomaly equations, clarifying their relation to the heat equation satisfied bythe Jacobi theta series; (ii) in cases where the moduli space is a Hermitiansymmetric tube domain $G/K$, we show that the general solution of the anomalyequations is a matrix element $\IP{\Psi | g | \Omega}$ of theSchr\"odinger-Weil representation of a Heisenberg extension of $G$, between anarbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$.Based on these results, we speculate on the existence of a one-parametergeneralization of the usual topological amplitude, which in symmetric casestransforms in the smallest unitary representation of the duality group $G'$ inthree dimensions, and on its relations to hypermultiplet couplings, nonabelianDonaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification adde