The Omega deformed B-model for rigid N=2 theories

TL;DR

This work provides a comprehensive framework to compute refined topological amplitudes in Omega-deformed B-models for rigid $N=2$ theories by extending BCOV holomorphic anomaly equations to a generalized setting. By interpreting the Omega deformation as a worldsheet insertion $\phi$ and employing a direct integration method, the authors derive modular, almost-holomorphic expressions for $F^{(n,g)}$ across conformal and asymptotically free Seiberg–Witten theories, including non-Lagrangian cases via local CY geometries. The approach yields explicit results for SU(2) theories with $N_f=0$–$3$ and the conformal $N_f=4$ and $N=4$ cases, with precise Nekrasov–Shatashvili limits and comparisons to Nekrasov’s instanton sums, and extends to refined motivic Donaldson–Thomas invariants for local ${\cal O}(-3)\to\mathbb{P}^2$ and to orbifold/conifold regimes. The findings highlight the role of modularity, gap boundary conditions, and UV/IR coupling ambiguities in organizing refined BPS data, and point to deep connections with 2d theories via the 4d/2d correspondence and geometric engineering. The framework thus unifies a broad class of rigid $N=2$ theories under a common holomorphic anomaly structure, providing computational tools and structural insights with potential extensions to E-type SCFTs and related geometries.

Abstract

We give an interpretation of the Omega deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four dimensional rigid N=2 theories explicitly in general Omega-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N=2 supersymmetric theories. The rigid N=2 field theories we focus on are the conformal rank one N=2 Seiberg-Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N=2 theories arising from compactifications on local Calabi-Yau manifolds, we consider the theory of local P2. We calculate motivic Donaldson-Thomas invariants for this geometry and make predictions for generalized Gromov-Witten invariants at the orbifold point.