Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD

Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD

17 Aug 1994 | N. Seiberg, E. Witten
The paper by Seiberg and Witten explores four-dimensional $N = 2$ supersymmetric gauge theories with matter multiplets, focusing on the $SU(2)$ gauge group. They derive the exact metric on the moduli space of quantum vacua and the spectrum of stable massive states for these models. New physical phenomena are observed, such as chiral symmetry breaking driven by the condensation of magnetic monopoles carrying global quantum numbers. For theories where conformal invariance is broken only by mass terms, the formalism automatically yields results invariant under electric-magnetic duality. In some cases, this duality is mixed with $SO(8)$ triality. The authors start by discussing an abelian theory with $N = 2$ supersymmetry as a warm-up example, followed by a detailed analysis of the classical and quantum moduli spaces of the $SU(2)$ gauge theory with matter. They explore the behavior of BPS-saturated states and the breaking of $N = 2$ to $N = 1$ supersymmetry. The paper then delves into the quantum theory, examining symmetries, the quantum moduli space, and the spectrum of stable particles. They also analyze duality transformations and the structure of the moduli space for theories with different numbers of flavors. A key finding is that in theories with $N_f = 2, 3, 4$, chiral symmetry breaking can occur through the condensation of magnetic monopoles, which can be continuously transformed into elementary quanta. This allows for a continuous interpolation between the confining phase and the Higgs phase. The authors also discuss the implications of electric-magnetic duality in strongly interacting gauge theories, showing that theories with mass terms breaking conformal invariance exhibit $SL(2,\mathbf{Z})$ duality. They provide examples of this duality in $N = 4$ super Yang-Mills theory and a theory with $N_f = 4$.The paper by Seiberg and Witten explores four-dimensional $N = 2$ supersymmetric gauge theories with matter multiplets, focusing on the $SU(2)$ gauge group. They derive the exact metric on the moduli space of quantum vacua and the spectrum of stable massive states for these models. New physical phenomena are observed, such as chiral symmetry breaking driven by the condensation of magnetic monopoles carrying global quantum numbers. For theories where conformal invariance is broken only by mass terms, the formalism automatically yields results invariant under electric-magnetic duality. In some cases, this duality is mixed with $SO(8)$ triality. The authors start by discussing an abelian theory with $N = 2$ supersymmetry as a warm-up example, followed by a detailed analysis of the classical and quantum moduli spaces of the $SU(2)$ gauge theory with matter. They explore the behavior of BPS-saturated states and the breaking of $N = 2$ to $N = 1$ supersymmetry. The paper then delves into the quantum theory, examining symmetries, the quantum moduli space, and the spectrum of stable particles. They also analyze duality transformations and the structure of the moduli space for theories with different numbers of flavors. A key finding is that in theories with $N_f = 2, 3, 4$, chiral symmetry breaking can occur through the condensation of magnetic monopoles, which can be continuously transformed into elementary quanta. This allows for a continuous interpolation between the confining phase and the Higgs phase. The authors also discuss the implications of electric-magnetic duality in strongly interacting gauge theories, showing that theories with mass terms breaking conformal invariance exhibit $SL(2,\mathbf{Z})$ duality. They provide examples of this duality in $N = 4$ super Yang-Mills theory and a theory with $N_f = 4$.
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