M5 brane and four dimensional N=1 theories I

TL;DR

The work identifies a generalized Hitchin system with two Higgs fields to engineer four-dimensional $\mathcal{N}=1$ theories from M5-brane compactifications on punctured Riemann surfaces, with regular punctures labeled by commuting nilpotent pairs and a two-bundle structure $L_1\oplus L_2$ satisfying $L_1\otimes L_2=K$. It shows that rotating $\mathcal{N}=2$ punctures yields a rich class of $\mathcal{N}=1$ matter content, and that Seiberg-like dualities arise as different degeneration limits of the same punctured surface, with exactly marginal quartic couplings identified with complex-structure moduli. The construction extends to partially rotated punctures and $D_N$ theories, revealing a unified M5-brane perspective on a wide family of $\mathcal{N}=1$ dualities and their gluing rules via three-punctured spheres. This framework provides a concrete geometric handle on IR fixed points, dual frames, and marginal couplings, suggesting broad applicability to a range of $\mathcal{N}=1$ gauge theories and guiding future exploration of irregular punctures and more intricate defect data.

Abstract

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 the Seiberg duality of SU(N) SQCD with Nf=2N and certain superpotential term is realized as different degeneration limits of the same punctured Riemann surface, and also find four more dual theories.

Figures (20)

• Figure 1: The $M5$ brane compactification data for $SU(N)$ SQCD with $N_f=2N$. Black Young Tableaux means that $\Phi_1$ is singular at the puncture while $\Phi_2$ is zero, and the similar interpretation applies to the red Young Tableaux.
• Figure 2: Different duality frames of $SU(N)$ SQCD with $N_f=2N$, and the quartic superpotential couplings are the same for the duality frames in the same column.
• Figure 3: The puncture is represented by half-BPS boundary condition of $\mathcal{N}=4$ SYM theory: left figure represents full puncture and right represents simple puncture.
• Figure 4: Local Higgs and Coulomb branch of four dimensional $\mathcal{N}=2$ compactification can be identified with the Higgs branch of 3d quivers.
• Figure 5: NS and $\text{NS}^{'}$ type three sphere are represented by three junctions with specific boundary condition on three legs, and both lead to same $\mathcal{N}=2$ theory.