Phase transitions in 5D super Yang-Mills theory

In this paper we study a phase structure of $5D$ ${\cal N}=1$ super Yang-Mills theory with massive matter multiplets and $SU(N)$ gauge group. In particular, we are interested in two cases: theory with $N_f$ massive hypermultiplets in the fundamental representation and theory with one adjoint massive hypermultiplet. If these theories are considered on $S^5$ their partition functions can be localized to matrix integrals, which can be approximated by their values at saddle points in the large-$N$ limit. We solve saddle point equations corresponding to the decompactification limit of both theories. We find that in the case of the fundamental hypermultiplets theory experiences third-order phase transition when coupling is varied. We also show that in the case of one adjoint hypermultiplet theory experiences infinite chain of third-order phase transitions, while interpolating between weak and strong coupling regimes.