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

TL;DR

This paper interprets the Omega deformed B-model to derive generalized holomorphic anomaly equations, enabling explicit calculation of topological amplitudes and refined BPS spectra in four-dimensional rigid N=2 theories using modular forms.

Contribution

It introduces a formalism that computes topological amplitudes in N=2 theories via anomaly equations, applicable regardless of Lagrangian existence, and extends to local Calabi-Yau compactifications.

Findings

Explicit formulas for topological amplitudes in Omega-backgrounds

Calculation of motivic Donaldson-Thomas invariants for local P2

Predictions for generalized Gromov-Witten invariants at orbifold points

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.