N=2 gauge theories and quantum phases

Understand

The partition function of general 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\pi i {\cal 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.