Matone's Relation in the Presence of Gravitational Couplings

TL;DR

This work verifies Matone's relation in $N=2$ supersymmetric Yang–Mills theories even in the presence of gravitational couplings, by performing explicit multi-instanton calculations within Nekrasov's localization framework. The authors introduce a deformed scalar field $\tilde{\varphi}$ that remains suitable for localization and show that $\langle \mathrm{Tr}\,\tilde{\varphi}^2\rangle$ encodes the prepotential corrections through $q\partial_q\ln{\cal Z}$, establishing $\mathcal{G}_k=k\mathcal{F}_k$ for all instanton numbers and to all orders in the deformation parameters $\epsilon_1,\epsilon_2$. They extend the analysis to theories with fundamental or adjoint matter and compute the first gravitational corrections to the chiral ring, demonstrating that the ring relations persist in the deformed setting. Overall, the work provides a nontrivial cross-check of Matone's relation under gravitational couplings and clarifies how chiral observables deform with $\epsilon$-parameters, aligning field-theoretic results with localization-based instanton calculus.

Abstract

The prepotential in N=2 SUSY Yang-Mills theories enjoys remarkable properties. One of the most interesting is its relation to the coordinate on the quantum moduli space $u=< \tr φ^2>$ that results into recursion equations for the coefficients of the prepotential due to instantons. In this work we show, with an explicit multi-instanton computation, that this relation holds true at arbitrary winding numbers. Even more interestingly we show that its validity extends to the case in which gravitational corrections are taken into account if the correlators are suitably modified. These results apply also to the cases in which matter in the fundamental and in the adjoint is included. We also check that the expressions we find satisfy the chiral ring relations for the gauge case and compute the first gravitational correction.

Figures (2)

• Figure 1: Two generic Young diagrams denoted by $Y_\alpha$ (dotted line) and $Y_\beta$ (solid line) in the main text.
• Figure 2: Notation for a generic Young diagram.