N=2 gauge theories and quantum phases

TL;DR

This paper analyzes N=2 SU(2) gauge theories on a four-sphere, revealing saddle point dominance in the decompactification limit, connections to quantum phase transitions, and the absence of phase transitions in N=2* theories.

Contribution

It demonstrates the saddle point approximation for the partition function and links the free energy to the Seiberg-Witten prepotential, also exploring phase transition phenomena in these theories.

Findings

Free energy given by the prepotential at the Seiberg-Witten singularity

Quantum phase transition at the superconformal fixed point of massive SQCD

No phase transitions in N=2* SU(2) Yang-Mills theory

Abstract

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. Finally, we discuss the case of N=2* SU(2) Yang-Mills theory and show that the theory does not exhibit phase transitions.