BPS black holes, the Hesse potential, and the topological string

TL;DR

This paper constructs a duality-invariant Hesse potential for certain N=2 supersymmetric theories, linking it to the topological string partition function and analyzing its expansion in the string coupling.

Contribution

It introduces a duality-invariant Hesse potential for N=2 theories, connecting it explicitly to the topological string partition function and verifying this relation in specific models.

Findings

Hesse potential is duality invariant and equals the BPS black hole free energy.

Holomorphic terms in the expansion match the topological string partition function.

Explicit verification in FHSV and STU models confirms the theoretical expectations.

Abstract

The Hesse potential is constructed for a class of four-dimensional N=2 supersymmetric effective actions with S- and T-duality by performing the relevant Legendre transform by iteration. It is a function of fields that transform under duality according to an arithmetic subgroup of the classical dualities reflecting the monodromies of the underlying string compactification. These transformations are not subject to corrections, unlike the transformations of the fields that appear in the effective action which are affected by the presence of higher-derivative couplings. The class of actions that are considered includes those of the FHSV and the STU model. We also consider heterotic N=4 supersymmetric compactifications. The Hesse potential, which is equal to the free energy function for BPS black holes, is manifestly duality invariant. Generically it can be expanded in terms of powers of the modulus that represents the inverse topological string coupling constant, $g_s$, and its complex conjugate. The terms depending holomorphically on $g_s$ are expected to correspond to the topological string partition function and this expectation is explicitly verified in two cases. Terms proportional to mixed powers of $g_s$ and $\bar g_s$ are in principle present.