Direct integration for general Omega backgrounds

TL;DR

The paper extends the direct integration technique for holomorphic anomaly equations to general Omega backgrounds, applying it to pure SU(2) N=2 SYM and to topological strings on non-compact Calabi-Yau threefolds. By combining generalized holomorphic anomaly equations with modular constraints and a conifold gap boundary condition, the authors solve refined amplitudes across diverse local geometries, including toric Calabi-Yaus and K3 fibrations, and provide explicit genus-2 and genus-3 results. They interpret refined amplitudes via Schwinger-loop BPS counting, derive universal gap structures at the conifold, and demonstrate consistency with heterotic/type II dualities and with refined Göttsche counts. The work lays a framework for extending holomorphic anomaly methods to β-deformed settings and local Calabi-Yau geometries, with potential links to matrix models and open refinement problems.

Abstract

We extend the direct integration method of the holomorphic anomaly equations to general Omega backgrounds for pure SU(2) N=2 Super-Yang-Mills theory and topological string theory on non-compact Calabi-Yau threefolds. We find that an extension of the holomorphic anomaly equation, modularity and boundary conditions provided by the perturbative terms as well as by the gap condition at the conifold are sufficient to solve the generalized theory in the above cases. In particular we use the method to solve the topological string for the general Omega backgrounds on non-compact toric Calabi-Yau spaces. The conifold boundary condition follows from that the N=2 Schwinger-Loop calculation with BPS states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the decompactification limit of Calabi-Yau spaces with regular K3 fibrations and half K3s embedded in Calabi-Yau backgrounds.

Figures (2)

• Figure 1: Resolved Moduli Space of $\mathds{F}_0$
• Figure 2: Conifold coordinates