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.
Highlights
We examine the partition function of N = 2∗ supersymmetric SU(N ) Yang-Mills theory on the four-sphere in the large radius limit
We have argued that the partition function of the N = 2∗ theory with SU(2) gauge group, on a large four-sphere, is computed by a single saddle point and the system cannot exhibit any non-analyticities as a function of the gauge coupling
We have shown that one solution to the large volume saddle-point equations following from Nekrasov’s partition function is a point of maximal degeneration on the Coulomb branch satisfying Re(aD j) = Re(aj) = 0
Summary
We have argued that the partition function of the N = 2∗ theory with SU(2) gauge group, on a large four-sphere, is computed by a single saddle point (the dyon singularity C ) and the system cannot exhibit any non-analyticities as a function of the gauge coupling This was the expectation in [18]. A subset of the dual periods are degenerate and lead to massless dyons, but aD 12 and aD N,N−1 are required to be (non-integer) linear combinations of cycles with non-zero intersection This picks out a particular point on the wall/surface of marginal stability in the N = 2∗ Coulomb branch.
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