Phase structure of N $$ \mathcal{N} $$ = 2* SYM on ellipsoids

Abstract

We analyse the phase structure of an $$ \mathcal{N} $$ = 2 massive deformation of $$ \mathcal{N} $$ = 4 SYM theory on a 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. Geometric deformation manifests itself as an effective mass term, thus braking the conformal invariance of the theory with massless hypermultiplets. The structure of perturbative corrections around the spherical geometry is analysed in the details and a systematic computational procedure is given, together with the first few corrections. 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 $$ \mathcal{N} $$ = 2 SU (N) 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 drifting Q to the value 2M.