Gluino Condensate and Magnetic Monopoles in Supersymmetric Gluodynamics

TL;DR

This work introduces a semi-classical analysis of ${\cal N}=1$ SU(${\it N}$) gauge theories on ${\mathbb R}^3\times S^1$ to compute the four-dimensional gluino condensate. By identifying $N$ monopole varieties (including a KK-monopole) as the dominant configurations at finite $\beta$, it derives a monopole-induced superpotential that lifts the classical moduli and yields a pair of SUSY vacua; the dual photon becomes massive, signaling confinement. Summing monopole contributions to the gluino condensate and taking the decompactification limit $\beta\to\infty$ reproduces the weak-coupling instanton result $\langle {\rm tr}\lambda^2\over 16\pi^2\rangle = \Lambda^3$ for SU(2) and, more generally, $\Lambda^{3N}$ for SU(${\it N}$), in agreement with the constrained instanton (WCI) approach and resolving historical SCI-WCI discrepancies. The analysis also illustrates a confinement mechanism via a monopole plasma that generates a mass for the dual photon, thereby removing massless modes in the low-energy spectrum. The results support the view that instanton physics at strong coupling can be understood through constituent monopole configurations and holomorphy, providing a concrete cross-check of non-perturbative dynamics in supersymmetric gauge theories.

Abstract

We examine supersymmetric SU(N) gauge theories on R^3*S^1 with a circle of circumference beta. These theories interpolate between four-dimensional N=1 pure gauge theory for beta=infinity and three-dimensional N=2 gauge theory for beta=0. The dominant field configurations of the R^3*S^1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R^3*S^1 theory and hence determine its vacuum structure. We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit beta=infinity. In this way we obtain a value for the gluino condensate in the four-dimensional N=1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known `weak coupling' expression but not with the `strong coupling' expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.