Partition function of N=2* SYM on a large four-sphere

TL;DR

This paper analyzes the partition function of N=2* supersymmetric SU(N) Yang-Mills theory on a large four-sphere, revealing saddle points on walls of marginal stability and providing large-N exact results that connect to integrable models.

Contribution

It identifies the saddle points of the partition function on the Coulomb branch and derives explicit large-N formulas for the periods at these points, linking to integrable systems.

Findings

Maximally degenerate saddle point matches elliptic Calogero-Moser model predictions.

Partition function saddle points lie on walls of marginal stability.

The singular saddle point disappears above a critical coupling value.

Abstract

We examine the partition function of N=2* supersymmetric SU(N) Yang-Mills theory on the four-sphere in the large radius limit. We point out that the large radius partition function, at fixed N, is computed by saddle points lying on particular walls of marginal stability on the Coulomb branch of the theory on R^4. For N an even (odd) integer and \theta_YM=0, (\pi), these include a point of maximal degeneration of the Donagi-Witten curve to a torus where BPS dyons with electric charge [N/2] become massless. We argue that the dyon singularity is the lone saddle point in the SU(2) theory, while for SU(N) with N>2, we characterize potentially competing saddle points by obtaining the relations between the Seiberg-Witten periods at such points. Using Nekrasov's instanton partition function, we solve for the maximally degenerate saddle point and obtain its free energy as a function of g_YM and N, and show that the results are "large-N exact". In the large-N theory our results provide analytical expressions for the periods/eigenvalues at the maximally degenerate saddle point, precisely matching previously known formulae following from the correspondence between N=2* theory and the elliptic Calogero-Moser integrable model. The maximally singular point ceases to be a saddle point of the partition function above a critical value of the coupling, in agreement with the recent findings of Russo and Zarembo.