  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.

High Energy Physics - Theory