Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves

TL;DR

The paper develops a framework to study ${\cal N}=2$ ${\rm SU}(N)$ gauge theories with fundamental or adjoint matter in a nontrivial ${\boldsymbol \Omega}$-background, by evaluating the instanton partition function through localization and a saddle-point analysis in the limit ${\epsilon_1\to0}$ with ${\epsilon_2=\epsilon}$ finite. Central to the approach is encoding the saddle-point data in a holomorphic function $w(x)$ (and its companion ${\cal Y}(x)$), which leads to an ${\epsilon}$-deformed Seiberg–Witten problem: a non-commutative curve and a deformed SW differential $\lambda=-x\, d\ln w(x)$ that reproduce the prepotential ${\cal F}$ and chiral correlators via contour integrals and generalized Matone relations. For fundamental matter the deformed curve is captured by a finite-difference equation related to a non-commutative SW curve, while for adjoint matter the deformation is expressed through an integral equation for $w(x)$ and a more intricate, modular SW structure; in both cases the non-commutative framework connects the gauge dynamics in the $\Omega$-background to quantum integrable models (e.g., Toda/Calogero-Moser) and to Baxter-type relations. The results provide a concrete method to compute the deformed prepotential and chiral correlators and offer a unified view of ${\epsilon}$-dependent corrections to SW theory across matter contents. The work thereby strengthens the link between localization in deformed backgrounds and integrable systems, offering tools for exploring the quantum geometry of Seiberg–Witten curves."

Abstract

We study the dynamics of a N=2 supersymmetric SU(N) gauge theory with fundamental or adjoint matter in presence of a non trivial Omega-background along a two dimensional plane. The prepotential and chiral correlators of the gauge theory can be obtained, via a saddle point analysis, from an equation which can be viewed as a non commutative version of the "standard" Seiberg and Witten curve.

Figures (2)

• Figure 1: Instanton distribution associated to a Young tableau centered at $a_u$. We display in green the intervals of non trivial instanton density, i.e. $\rho(x)=1$. By blu and red bullets we indicate plus and minus charges in the auxiliary electrostatic problem where crosses stand for points of constant potential.
• Figure 2: Electrostatic problem associated to the $\epsilon_1,\epsilon_2 \to 0$ limit of $SU(2)$ gauge theory with $N_f=3$ fundamental matter (a) or one Adjoint hypermultiplet (b).