$\mathcal N=3$ four dimensional field theories

TL;DR

The work addresses the existence and characterization of four-dimensional $\mathcal{N}=3$ SCFTs, which evade a conventional Lagrangian description and are intrinsically strongly coupled. It constructs them by a discrete quotient $\mathbb{Z}^S_k\cdot\mathbb{Z}^R_k$ of $\mathcal{N}=4$ SYM with gauge group $\mathrm{U}(N)$, yielding genuine $\mathcal{N}=3$ fixed points for $k=3,4,6$ with $a=c$ and no continuous global symmetries. String-theoretic realizations as D3-branes probing S-folds and as F-theory limits of M2-branes on $(\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ are provided, along with holographic dual descriptions via quotients of $AdS_5\times S^5$ with the string coupling frozen at order one. The paper also analyzes the holographic index, ABJM reductions on a circle, and the rank-one moduli space $\mathbb{C}^3/\mathbb{Z}_k$, contributing to the classification and understanding of non-Lagrangian $\mathcal{N}=3$ theories and their S-fold variants.

Abstract

We briefly review a class of four dimensional $\mathcal N=3$ field theories constructed by taking a quotient of $\mathcal N=4$ SYM with gauge group $U(N)$. The quotient involves a discrete symmetry that only exists for specific, order one, values of the coupling constant, so the resulting theories are intrinsically strongly coupled. These theories admit a simple realization in string theory as the worldvolume theory of a stack of D3 branes probing a generalized orientifold plane, or S-fold. Their holographic dual is given by a non-trivial F-theory fibration over $AdS_5 \times S^5/\mathbb Z_k$ which is weakly curved but with the string coupling frozen at an order one value.