Sicilian gauge theories and N=1 dualities

TL;DR

The paper develops a general framework to study 4d SCFTs built from non-Lagrangian sectors by extending the NSVZ beta-function and by systematically counting exactly marginal deformations. It identifies Sicilian gauge theories as N=1 mass-deformed T_N blocks arising from 6d (2,0) A_N compactifications with punctures and SU(2) Wilson lines, and verifies the holographic duals via Maldacena–Nuñez solutions. Through detailed case studies, it computes central charges and conformal-manifold dimensions, demonstrates NSVZ beta-function vanishing after deformations, and shows a universal M5-brane origin for the conformal manifold across maximal and simple puncture configurations. The results provide a powerful, geometry-inspired approach to constructing and testing new non-Lagrangian SCFTs and their holographic descriptions.

Abstract

In theories without known Lagrangian descriptions, knowledge of the global symmetries is often one of the few pieces of information we have at our disposal. Gauging (part of) such global symmetries can then lead to interesting new theories, which are usually still quite mysterious. In this work, we describe a set of tools that can be used to explore the superconformal phases of these theories. In particular, we describe the contribution of such non-Lagrangian sectors to the NSVZ beta-function, and elucidate the counting of marginal deformations. We apply our techniques to N=1 theories obtained by mass deformations of the N=2 conformal theories recently found by Gaiotto. Because the basic building block of these theories is a triskelion, or trivalent vertex, we dub them "Sicilian gauge theories." We identify these N=1 theories as compactifications of the six-dimensional A_N (2,0) theory on Riemann surfaces with punctures and SU(2) Wilson lines. These theories include the holographic duals of the N=1 supergravity solutions found by Maldacena and Nunez.

Figures (6)

• Figure 1: a) The $T_N$ theory is shown as a triskelion on the left. The box with $N$ inside stands for an $\mathrm{SU}(N)$ flavor symmetry. It is obtained by wrapping $N$ M5-branes on a sphere with three maximal punctures, as shown on the right. b) A hypermultiplet in the bifundamental of $\mathrm{SU}(N)\times \mathrm{SU}(N)$ is shown as an edge with two boxes attached. It is obtained by wrapping $N$ M5-branes on a sphere with two maximal punctures and one simple puncture, denoted by $\bullet$.
• Figure 2: a) Two copies of the $T_N$ theory. b) Two copies of $T_N$ with one $\mathrm{SU}(N)$ gauge group, which couples to one from three $\mathrm{SU}(N)$ flavor symmetries for each of the $T_N$ theories. It arises from a sphere with four maximal punctures. c) One $T_N$ theory and a bifundamental multiplet. It arises from a sphere with three maximal punctures and one simple puncture.
• Figure 3: The Sicilian flag. The central symbol is a triskelion. Image taken from Wikipedia.
• Figure 4: The Sicilian gauge theory corresponding to $N$ M5-branes wrapped on a genus-$g$ Riemann surface $\Sigma_g$ without any punctures, for $g=2$. The left column shows the Sicilian diagrams, and the right column shows the corresponding degenerations of the Riemann surfaces.
• Figure 5: The Sicilian $\mathcal{T}_g$ theory in a particular S-dual frame. In this particular picture, $T_N$ blocks are emphasized as triangles, and numbered. The combination $Y = J^{(1)} - J^{(2)} + J^{(3)} - J^{(4)} + \dots$, where each $J^{(a)}$ term is the $\mathrm{U}(1)$ flavor symmetry of a $T_N$ theory, is the anomaly free $\mathrm{U}(1)_F$.