Holomorphicity and Modularity in Seiberg-Witten Theories with Matter

TL;DR

The paper extends the modular-holomorphic anomaly gap-program of Seiberg-Witten theory to SU(2) gauge theories with fundamental matter (N_f = 1, 2, 3). It shows that gravitational corrections F^{(g)} are completely fixed across the moduli space by modularity, gap conditions at dyon/Conifold points, and dual expansions, for both massless and massive hypermultiplets. The authors derive explicit modular expressions, perform dual expansions, and verify agreement with Nekrasov's instanton results, while also connecting to non-compact Calabi–Yau limits and the Eynard–Orantin matrix-model formalism for open and closed amplitudes. The work demonstrates integrability of the topological sector and lays groundwork for extensions to higher rank, conformal points, and potential dual string descriptions, with open amplitudes offering additional structural insight.

Abstract

We calculate the gravitational corrections to the effective action of N=2 SU(2) Seiberg-Witten theory with matter using modularity, the holomorphic anomaly equation and expected behavior at the boundaries of the moduli space. As in pure gauge theory we show that the gap condition at the dyon singularities completely fixes the gravitational corrections. We discuss the behavior of the gravitational corrections at the conformal points. We compare our results with the recursive solution of the loop equation in the matrix model approach, which provides in addition open amplitudes.