Testing Seiberg-Witten Solution

TL;DR

This work develops a nonperturbative program to test Seiberg-Witten solutions of ${\cal N}=2$ SYM via instanton calculus on twisted theories, focusing on the Coulomb-branch effective action and its modular structure. It constructs the low-energy abelian framework, analyzes the UV/IR map of observables through $Q$-cohomology, and derives universal contact terms that encode short-distance physics, using blow-up arguments and theta-function anomalies to fix pair and higher-order deformations of the action. A key result is the explicit expression for pair contact terms ${\cal C}({\cal P}_{1},{\cal P}_{2}) = \frac{\partial {\cal P}_{1}}{\partial a^{i}} \frac{\partial {\cal P}_{2}}{\partial a^{j}} \frac{\partial}{\partial \tau_{ij}} \log \Theta(\tau)$ (for suitable theories), with complementary derivations from 0- and 4-observables that yield ${\cal C}(u_{1},u_{k})$ in terms of the beta-function coefficient $\beta_{1}$. The paper also outlines ADHM localization on ${\cal M}_{k,N}$, providing an explicit contour integral for equivariant integrals that connects instanton counting to the Seiberg-Witten data. Overall, the results give concrete, modularly constrained checks of SW duality and illuminate how UV deformations reflect in the low-energy geometry of the Coulomb branch.

Abstract

We propose a few tests of Seiberg-Witten solutions of $\mathcal{N}=2$ supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We re-examine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems.