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.
Highlights
One case that can be computed exactly is SU(N ) N = 2 supersymmetric gauge theories in the limit of large N
We show that in the decompactification limit, and in a certain regime, the integral is dominated by a saddle point
We 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
Summary
The supersymmetric vacuum in pure N = 2 SU(N ) gauge theories is characterized by the expectation value of the scalar field of the vector multiplet, given by. The low-energy effective action in N = 2 gauge theory is fully determined in terms of the prepotential F(ai). It represents the renormalized coupling in the vacuum (2.1), 8a τ (a) = 2τUV − 2πi ln ΛUV +. The exact expression for the coupling at a given vacuum parametrized by a is obtained from the Seiberg-Witten (SW) solution. For pure SU(2) SYM, the SW curve is given by [1]. The curve (2.7) has singularities at u = ±Λ2. These integrals can be expressed in terms of elliptic functions.
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