Deformations of special geometry: in search of the topological string

TL;DR

This paper identifies a sector within the effective action of superstring theory that behaves like a topological string free energy, using real special geometry and the Hesse potential, and verifies its properties up to genus 3.

Contribution

It introduces the Hesse potential as a duality covariant function linking effective actions to topological string free energies, providing a new conceptual framework.

Findings

Constructed genus g≤3 contributions explicitly.

Demonstrated satisfaction of holomorphic anomaly equations.

Established a theorem on non-holomorphic deformations.

Abstract

The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $g\leq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-K\"ahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.