Non-Perturbative Dynamics in Supersymmetric Gauge Theories

TL;DR

This work surveys nonperturbative dynamics in four-dimensional supersymmetric gauge theories, emphasizing exact results derived from holomorphy and SUSY tools. It develops a framework around moduli spaces, dualities (electric–magnetic), and conformal windows to map infrared behavior across theories, including s-confinement and oblique confinement phenomena. The approach yields precise structures such as the NSVZ beta function, gluino condensates, and dynamically generated superpotentials, illuminating how strong coupling can be tamed in controlled settings. The insights offer qualitative lessons for non-supersymmetric QCD, suggesting strategies to relate SUSY dynamics to real-world strong interactions through soft SUSY breaking and careful phase analyses, with potential implications for confinement and chiral symmetry breaking in QCD.

Abstract

I give an introductory review of recent, fascinating developments in supersymmetric gauge theories. I explain pedagogically the miraculous properties of supersymmetric gauge dynamics allowing one to obtain exact solutions in many instances. Various dynamical regimes emerging in supersymmetric Quantum Chromodynamics and its generalizations are discussed. I emphasize those features that have a chance of survival in QCD and those which are drastically different in supersymmetric and non-supersymmetric gauge theories. Unlike most of the recent reviews focusing almost entirely on the progress in extended supersymmetries (the Seiberg-Witten solution of N=2 models), these lectures are mainly devoted to N=1 theories. The primary task is extracting lessons for non-supersymmetric theories.

Figures (4)

• Figure 1: The grids of the electric-magnetic charges in the $SU(2)$ gauge theory. The charges $\{q,m\}$ are measured with respect to the $U(1)$ subgroup of $SU(2)$, as in Sect. 1.5. The closed circles indicate the charges in the theory with the adjoint ($SU(2)$ triplet) matter. The crosses indicate the electric charge of the matter field in the fundamental ($SU(2)$ doublet) representation. ($a$) The rectangular grid corresponding to $\vartheta =0$. The point denoted by $C$ is a bound state of the triplet matter quantum with the monopole. Bound states of the doublet matter quanta with monopoles are possible too (but not indicated). ($b$) The oblique grid corresponding to $\vartheta =\pi +\varepsilon$, $0<\varepsilon \ll \pi$. The state denoted by $D$ is a dyon with a small value of the electric charge, which presumably condenses. The point denoted by $F$ is an unconfined bound state of the quark and dyon $B$. Its external quantum numbers are those of the quark.
• Figure 2: Interaction vertices in QED and its supergeneralization, SQED. ($a$) $\bar{e} e\gamma$ vertex; ($b$) selectron coupling to photon; ($c$) electron-selectron-photino vertex. All vertices have the same coupling constant. The quartic self-interaction of selectrons is also present, but not shown.
• Figure 3: One-particle reducible graphs which might lead to $1/\sigma$ terms. This mechanism is not (and must not be) included in the superpotential.
• Figure 4: A typical two-loop supergraph for the vacuum energy.