Worldsheet Realization of the Refined Topological String

TL;DR

This work provides a concrete worldsheet realization of the refined topological string by constructing one-loop amplitudes in Heterotic and Type I string theory on $K3\times T^2$ that compute generalized $\mathcal{F}_{g,n}$ couplings of the form $F_{g,n} W^{2g} \Upsilon^{2n}$. By identifying $\epsilon_-$ with the anti-self-dual graviphoton background and $\epsilon_+$ with the self-dual gauge field of the $\bar{T}$-modulus vector multiplet, the authors show that, near an $SU(2)$ enhancement point, these amplitudes reproduce the perturbative Nekrasov partition function and its radius deformations (Nekrasov-Okounkov) within a perturbative string framework. The Type I realization corroborates the universality of the result across perturbative corners of the moduli space. The work also discusses holomorphic anomaly considerations and outlines non-perturbative extensions via dualities and connections to broader Omega-background formalisms, highlighting open questions about vector multiplet selection and integrable-system implications.

Abstract

A worldsheet realization of the refined topological string is proposed in terms of physical string amplitudes that compute generalized N=2 F-terms of the form F_{g,n} W^{2g}Y^{2n} in the effective supergravity action. These terms involve the chiral Weyl superfield W and a superfield Y defined as an N=2 chiral projection of a particular anti-chiral T-bar vector multiplet. In Heterotic and Type I theories, obtained upon compactification on the six-dimensional manifold K3xT2, T is the usual Kähler modulus of the T2 torus. These amplitudes are computed exactly at the one-loop level in string theory. They are shown to reproduce the correct perturbative part of the Nekrasov partition function in the field theory limit when expanded around an SU(2) enhancement point of the string moduli space. The two deformation parameters epsilon_- and epsilon_+ of the Omega-supergravity background are then identified with the constant field-strength backgrounds for the anti-self-dual graviphoton and self-dual gauge field of the T-bar vector multiplet, respectively.