N $$ \mathcal{N} $$ =2 gauge theories and quantum phases

The partition function of general N $$ \mathcal{N} $$ = 2 supersymmetric SU(2) Yang-Mills theories on a four-sphere localizes to a matrix integral. We show that in the decompactification limit, and in a certain regime, the integral is dominated by a saddle point. When this takes effect, the free energy is exactly given in terms of the prepotential, F = − R 2 Re(4 πi ℱ), evaluated at the singularity of the Seiberg-Witten curve where the dual magnetic variable a D vanishes. We also show that the superconformal fixed point of massive supersymmetric QCD with gauge group SU(2) is associated with the existence of a quantum phase transition. Finally, we discuss the case of N $$ \mathcal{N} $$ = 2 * SU(2) Yang-Mills theory and show that the theory does not exhibit phase transitions