Singularity Structure of N=2 Supersymmetric Yang-Mills Theories: A Review

Abstract

In this review, we consider the case where electrons, magnetic monopoles, and dyons become massless. Here we consider the $\mathcal{N}=2$ supersymmetric Yang-Mills (SYM) theories with classical gauge groups with a rank r, SU(r+1), SO(2r), Sp(2r), and SO(2r+1). which are studied by Riemann surfaces called Seiberg-Witten curves. We discuss physical singularity associated with massless particles, which can be studied by geometric singularity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particular attention to the cases where mutually non-local states become massless (Argyres-Douglas theories), which corresponds to Riemann surfaces degenerating into cusps. We discuss non-trivial topology on the moduli space of the theory, which is reflected as monodromy associated to vanishing 1-cycles. We observe how dyon charges get changed as we move around and through singularity in moduli space.

45 pages, 19 figures, a version accepted by International Journal of Modern Physics A (IJMPA)

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