Probing the Moduli Dependence of Refined Topological Amplitudes

TL;DR

This paper explores the moduli dependence of refined topological string amplitudes in type II string theory, deriving differential equations and testing their consistency with dual heterotic calculations to better understand their worldsheet description.

Contribution

It formulates conditions under which the differential equations for refined amplitudes simplify into generalized holomorphic anomaly equations, advancing the understanding of their moduli dependence.

Findings

Derived differential equations relating $F_{g,n}$ to other couplings.

Identified conditions for simplifying these equations into holomorphic anomaly forms.

Validated the approach through explicit heterotic string theory calculations.

Abstract

With the aim of providing a worldsheet description of the refined topological string, we continue the study of a particular class of higher derivative couplings $F_{g,n}$ in the type II string effective action compactified on a Calabi-Yau threefold. We analyse first order differential equations in the anti-holomorphic moduli of the theory, which relate the $F_{g,n}$ to other component couplings. From the point of view of the topological theory, these equations describe the contribution of non-physical states to twisted correlation functions and encode an obstruction for interpreting the $F_{g,n}$ as the free energy of the refined topological string theory. We investigate possibilities of lifting this obstruction by formulating conditions on the moduli dependence under which the differential equations simplify and take the form of generalised holomorphic anomaly equations. We further test this approach against explicit calculations in the dual heterotic theory.