On the partition sum of the NS five-brane

TL;DR

This paper calculates the partition function of the NS five-brane on a Calabi-Yau, linking it to topological string amplitudes and holomorphic anomaly equations, revealing a deep connection between little strings and topological strings.

Contribution

It provides a novel computation of the five-brane partition function incorporating flux sums and quantum corrections via topological string theory, establishing a new link between these areas.

Findings

Partition function expressed as a theta-function over fluxes.

Quantum corrections given by B-model topological string amplitudes.

Partition function satisfies holomorphic anomaly equations.

Abstract

We study the Type IIA NS five-brane wrapped on a Calabi-Yau manifold X in a double-scaled decoupling limit. We calculate the euclidean partition function in the presence of a flat RR 3-form field. The classical contribution is given by a sum over fluxes of the self-dual tensor field which reduces to a theta-function. The quantum contributions are computed using a T-dual IIB background where the five-branes are replaced by an ALE singularity. Using the supergravity effective action we find that the loop corrections to the free energy are given by B-model topological string amplitudes. This seems to provide a direct link between the double-scaled little strings on the five-brane worldvolume and topological strings. Both the classical and quantum contributions to the partition function satisfy (conjugate) holomorphic anomaly equations, which explains an observation of Witten relating topological string theory to the quantization of three-form fields.