Topological wave functions and heat equations

TL;DR

This paper refines the understanding of holomorphic anomaly equations in topological string theory by linking them to heat equations and Schrödinger-Weil representations, revealing new mathematical structures and potential physical implications.

Contribution

It introduces a holomorphic form of the anomaly equations and connects solutions to matrix elements of the Schrödinger-Weil representation in symmetric moduli spaces.

Findings

Holomorphic anomaly equations relate to heat equations of Jacobi theta series.

Solutions are expressed as matrix elements of Schrödinger-Weil representations.

Speculation on a new topological amplitude linked to duality groups and black hole physics.

Abstract

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) 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 $\IP{\Psi | g | \Omega}$ of the Schr\"odinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.