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The N = 2 ∗ theory (mass deformation of N = 4 Super-Yang-Mills) un-dergoes an infinite number of quantum phase transitions in the large-N limit. The phase structure and critical behavior can be analyzed with the help of supersymmetric localization, which reduces the problem to an effective matrix model. We study its strong-coupling phase. To Andrei Alexeevich Slavnov on occasion of his 75th birthday 1

Abstract: We analyse the phase structure of an N “ 2 massive deformation of N “ 4 SYM theory on an four-dimensional ellipsoid using recent results on supersymmetric localisation. Besides the ’t Hooft coupling λ, the relevant parameters appearing in the theory and discriminating between the different phases are the hypermultiplet mass M and the deformation (or squashing) parameter Q. The master field approximation of the matrix model associated to the analytically continued theory in the regime Q „ 2M and on the compact space, is exactly solvable and does not display any phase transition, similarly to N “ 2 SUpNq SYM with 2N massive hypermultiplets. In the strong coupling limit, equivalent in our settings to the decompactification of the four-dimensional ellipsoid, we find evidence that the theory undergoes an infinite number of phase transitions starting at finite coupling and accumulating at λ “ 8. Quite interestingly, the threshold points at which transitions occur can be pushed towards the weak coupling region by letting Q approach 2M.

The planar N = 2 ∗ Super-Yang-Mills (SYM) theory is solved at large ’t Hooft coupling using localization on S4. The solution permits detailed investigation of the resonance phenomena responsible for quantum phase transitions in infinite volume, and leads to quantitative predictions for the semiclassical string dual of the N = 2 ∗ theory.