Matone's Relation in the Presence of Gravitational Couplings

TL;DR

This paper demonstrates that Matone's relation in N=2 SUSY Yang-Mills theories remains valid with gravitational corrections and matter fields, supported by explicit multi-instanton calculations and consistency checks with chiral ring relations.

Contribution

It extends the validity of Matone's relation to include gravitational corrections and matter fields, verified through multi-instanton computations.

Findings

Matone's relation holds at arbitrary instanton winding numbers.

Gravitational corrections can be incorporated with modified correlators.

Results are consistent with chiral ring relations and include first gravitational correction.

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 $u=< \tr \phi^2>$ 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.