M5 brane and four dimensional N=1 theories I

Four dimensional N = 1 theories are engineered by compactifying six dimensional (2, 0) theory on a Riemann surface with regular punctures. A generalized Hitchin's equation involving two Higgs fields is proposed as the BPS equation for N = 1 compactification. The puncture is interpreted as the singular boundary condition of this equation, and regular puncture is shown to be labeled by a nilpotent commuting pair. In this paper, we focus on a subset of regular puncture which is described by rotating branes representing N = 2 puncture. As an application, we show that Seiberg duality of S U (N) SQCD with N-f = 2N and certain superpotential term is realized as different degeneration limits of the same punctured Riemann surface, and find four more dual theories.