Theories of Class S and New N=1 SCFTs

TL;DR

This work introduces the infinite two-parameter family of class ${\cal S}$ theories $T_{N,k}$, generalizing Gaiotto’s $T_N$ and connecting to $N^2$ free hypermultiplets at endpoints, and uses the reduced superconformal index to extract operator content. It then constructs new ${\cal N}=1$ SCFTs from these building blocks and studies RG flows between them, employing $a$-maximization, unitarity checks, and Higgsing analyses to assess conformal behavior. The results illuminate the landscape of ${\cal N}=1$ theories arising from class ${\cal S}$, uncover potential SCFTs and their flows, and raise important questions about conformal manifolds and possible AdS duals. Overall, the paper advances understanding of how ${\cal N}=1$ fixed points can be engineered from non-Lagrangian ${\cal S}$-theory ingredients and highlights both promising directions and significant obstacles in establishing robust SCFTs in this framework.

Abstract

We describe an infinite two-parameter subfamily of theories of class S where dialing one of the parameters interpolates between Gaiotto's T_N theory and a theory of N^2 free hypermultiplets. After using the reduced superconformal index to study the operator content, we use these theories to construct new N=1 SCFTs and then examine the flows between them.

Figures (17)

• Figure 1: A Young tableau for a regular puncture; the flavor symmetry associated to the puncture is $S\left(U(3) \times U(2)^2 \times U(1) \right)$.
• Figure 2: The two degeneration limits of a punctured sphere with two maximal punctures and two minimal punctures. The picture on the left corresponds to the degeneration limit corresponding to an $SU(3)$ gauge theory with 6 fundamental hypermultiplets. The picture on the right corresponds to an $SU(2)$ gauge theory with one fundamental hypermultiplet and the $E_6$ SCFT, where an $SU(2)$ subgroup of the $E_6$ flavor symmetry is gauged.
• Figure 3: Two different degeneration limits of a Riemann surface with two maximal punctures and $N-1$ minimal punctures. On the top is an $SU(N)^{N-2}$ gauge theory with bifundamental hypermultiplets. Each fixture by itself corresponds to $N^2$ free hypermultiplets and each cylinder corresponds to an $SU(N)$ gauge group which weakly gauges the flavor symmetries of the hypermultiplets. On the bottom is the $T_N$ coupled to a superconformal tail.
• Figure 4: Duality between $\mathcal{N}=2$ linear quiver gauge theories (left) and $T_{N,k}$ theories coupled to an $\mathcal{N}=2$ superconformal tail (right). Circles represent gauge symmetries, boxes represent flavor symmetries, and lines represent bifundamental hypers. Trivalent vertices represent $T_{N,k}$ theories. $\supset$ represents gauging of a subgroup of a flavor symmetry. In one duality frame, we have an $SU(N)^{k-1}$ gauge theory with bifundamental hypermultiplets. In the other frame, we have a $T_{N,k}$ coupled to a quiver theory with gauge groups of decreasing rank. In the case of $k=1$, we have $N^2$ free hypermultiplets in both duality frames, and for the case of $k=N-1$ we have a $T_N$ coupled to a superconformal tail. For all $k>N-1$ we have a $T_N$ coupled to $SU(N)^{k-N+1} \times SU(N-1) \times SU(N-2) \times \cdots \times SU(2)$ gauge theory.
• Figure 5: Top: the degeneration limit of a surface with two maximal punctures and $N-1$ minimal punctures into thrice-punctured spheres connected by cylinders. The maximal punctures appear on the $k_1$-th sphere from the left and the $k_2$-th sphere from the right. Bottom: the quiver diagram for the corresponding theory, which contains a $T_{N,k_1}$ and a $T_{N,k_2}$.