Higgs branch, HyperKahler quotient and duality in SUSY N=2 Yang-Mills theories

Abstract

Low--energy limits of N=2 supersymmetric field theories in the Higgs branch are described in terms of a non--linear 4--dimensional sigma--model on a \hk target space, classically obtained as a \hk quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low--energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg--Witten $SU(2)$ theory with $N_f$ flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on $\RR^4$. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a $U(1)$ N=2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg--Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of $U(N_c)N_f$ flavors and $U(N_f-N_c)N_f$ flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality $N_c\leftrightarrow N_f-N_c$ in N=1 SUSY $SU(N_c)$ gauge theories.

32 pages, latex2e, no figures, uses amsfonts package

References in corpus (5)

• Small Instantons in String Theory
• The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD
• Spontaneous Breaking of N=2 Global Supersymmetry
• Lectures on Special Kahler Geometry and Electric--Magnetic Duality Rotations
• Dual N=2 SUSY Breaking

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