Four dimensional superconformal theories from M5 branes

TL;DR

The paper develops a framework to extract scaling dimensions of chiral operators in four-dimensional N=1 SCFTs arising from M5 branes on a Riemann surface, using a spectral curve and a holomorphic three-form. It generalizes ideas from class S, showing how to determine trial R-charges via a-maximization and then obtain exact dimensions after maximizing the trial a central charge, including handling decoupling of free fields. The approach recovers known N=2 (class S) results in appropriate limits and extends to novel N=1 theories, such as deformations of Argyres-Douglas models, providing consistency checks across Lagrangian and non-Lagrangian cases. This method offers a geometric, six-dimensional origin for N=1 SCFT data and enables systematic exploration of non-Lagrangian IR fixed points and their operator spectra with potential broad applicability to other puncture types and ADE setups.

Abstract

We study N=1 superconformal theories in four dimensions obtained wrapping M5 branes on a Riemann surface. We propose a method to determine from the spectral curve the scaling dimension of chiral operators in the SCFT. Whenever the R-symmetry has to be determined via a-maximization, our procedure allows us to determine the charge of chiral operators under the "trial" R-symmetry. Our proposal reduces to the correct prescription in the special case of N=2 theories of class S. We perform several consistency checks and apply our method to study some new SCFT's such as N=1 deformations of Argyres-Douglas theories.

Figures (6)

• Figure 1: We have a linear quiver with two $SU(N)$ gauge groups (in this case $N=5$). One vectormultiplet is ${\mathcal{N}}=1$, since we connect spheres of different kind, whereas the other is ${\mathcal{N}}=2$. We indicated two full punctures with the corresponding Young diagram. The cross denotes a simple puncture and the dashed lines denote the tubes connecting the various three-punctured spheres. The dots indicate full punctures whose $SU(N)$ flavor symmetry is gauged. All the three-punctured spheres in the figure describe bifundamentals of $SU(N)\times SU(N)$.
• Figure 2: On the left we have the three-punctured sphere representing $T_N$ theory. On the right we turned one full puncture of black type into one of red type (we call this process "rotation"). The resulting sphere describes $T_N$ coupled to a chiral multiplet $M$ in the adjoint of $SU(N)$. There is also the superpotential term ${\rm Tr}\mu M$.
• Figure 3: The three-punctured sphere describing the theory we obtain starting from $T_N$ plus a chiral multiplet $M$ and giving to $M$ a nilpotent vev with a Jordan block of dimension $N-1$. The cross again denotes a simple puncture.
• Figure 4: Sphere with a full and two simple punctures. Starting from a bifundamental of $SU(N)$ plus a chiral multiplet $M$ in the adjoint of one $SU(N)$ group (again with superpotential ${\rm Tr}\mu M$), we can obtain this theory giving a nilpotent vev to $M.$
• Figure 5: On the left we have the linear quiver of $SU(2N)$ gauge groups (in this case $N=3$). One gauge group is ${\mathcal{N}}=1$ and the other ${\mathcal{N}}=2$. The boxes denote the chiral multiplets in the fundamental and the line between gauge groups the bifundamental multiplets. We denote the fundamentals as $Q$ and $P$ and the bifundamental as $q$ (indeed we also have $\widetilde{Q}$, $\widetilde{P}$ and $\widetilde{q}$ multiplets). Giving the nilpotent vev we get the theory on the right. Each matter field is denoted with the same letter as the parent matter field on the left. We denote as $M$ the chiral multiplet in the adjoint of $SU(N)$.