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

We analyse the phase structure of an $\mathcal{N}=2$ massive deformation of $\mathcal{N}=4$ SYM theory on an four-dimensional ellipsoid using recent results on supersymmetric localisation. Besides the 't~Hooft coupling $\lambda$, 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 $Q\sim 2M$ 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 $\lambda=\infty$. Quite interestingly, the threshold points at which transitions occur can be pushed towards the weak coupling region by letting $Q$ approach $2M$.