[Paper Review] $\mathcal{N}=2$ SCFT with minimal flavor central charge

Dan Xie|arXiv (Cornell University)|Dec 8, 2017

Organic and Molecular Conductors Research19 references3 citations

TL;DR

This paper identifies 4d $χ=2$ superconformal field theories (SCFTs) with minimal flavor central charge using the 6d $(2,0)$ theory construction. It finds that for $ADE$ and $C_N$ flavor groups, the minimal theories saturate the bootstrap bounds on $k_G$, while $B_N$, $G_2$, and $F_4$ theories exceed the bounds. Notably, it discovers new rank-one SCFTs with $B_3$, $G_2$, $F_4$, $C_4\times U(1)$, and $C_1\times U(1)$ flavor symmetries, many with non-trivial Higgs branches and marginal deformations.

ABSTRACT

We list 4d interacting $\mathcal{N}=2$ SCFT with minimal flavor central charge from the theory space constructed using 6d $(2,0)$ theory. For $ADE$ and $C_N$ flavor groups, our theory saturates the bound found using bootstrap method, but other cases have higher values. We find interesting rank one SCFTs with $B_3, G_2, F_4, C_4 imes U(1), C_1 imes U(1)$ flavor symmetry. Many physical properties of these theories are also studied.

Motivation & Objective

• To identify $χ=2$ SCFTs with minimal flavor central charge $k_G$ within the class of theories constructed from 6d $(2,0)$ theory.
• To determine whether such minimal theories saturate the conformal bootstrap bounds on $k_G$ for various Lie groups $G$.
• To characterize the physical properties—central charges $(a,c)$, Coulomb branch spectra, Higgs branch orbits, and extra $U(1)$ flavor symmetries—of these minimal theories.
• To explore the existence of new rank-one SCFTs with exceptional or non-simply-laced flavor symmetries, including $B_3$, $G_2$, $F_4$, and $C_1\times U(1)$.

Proposed method

• Systematic scanning of the theory space constructed from 6d $(2,0)$ theory using the class $S$ construction.
• Computation of central charges $(a,c)$ and flavor central charge $k_G$ via Lie algebra data: Coxeter number $h$, dual Coxeter number $h^\vee$, and lacity $n$.
• Use of the formula $k_G = h^\vee - \frac{1}{n} \frac{h}{r}$ for $r > 1$, and $k_G \geq k_{\text{min}}$ for $r=1$, to derive bounds.
• Matching the central charge data to Higgs branch dimensions via $(a - c) = -\frac{\dim(\text{Higgs})}{24}$ to infer nilpotent orbit structure.
• Identification of associated vertex operator algebras and their levels $k_{2d} = -k_G$, with admissibility checks for $A_{N-1}$ with odd $N$.
• Use of Coulomb branch operator common denominator $r = k + b$ to classify and constrain possible $k_G$ values.

Experimental results

Research questions

• RQ1 Do $χ=2$ SCFTs constructed from 6d $(2,0)$ theory with minimal flavor central charge $k_G$ saturate the conformal bootstrap bounds on $k_G$ for all Lie groups $G$?
• RQ2 What are the physical properties—central charges, Coulomb branch spectra, Higgs branch orbits, and extra $U(1)$ symmetries—of these minimal $k_G$ theories?
• RQ3 Are there new rank-one SCFTs with non-simply-laced or exceptional flavor symmetries such as $B_3$, $G_2$, $F_4$, $C_4\times U(1)$, and $C_1\times U(1)$?
• RQ4 Why do some minimal $k_G$ theories for $B_N$, $G_2$, and $F_4$ exceed the bootstrap bounds, and what explains the structure of their flavor central charges?
• RQ5 Can the observed pattern of $k_G = h^\vee - \frac{1}{n} \frac{h}{r}$ be generalized to a universal bound for theories with common denominator $r$ on the Coulomb branch?

Key findings

• For $ADE$ and $C_N$ flavor groups, the minimal $k_G$ theories saturate the conformal bootstrap bounds: $k_G = N/2$ for $A_{N-1}$, $k_G = N-2$ for $D_N$, $k_G = 3,4,6$ for $E_6,E_7,E_8$, $k_G = N/2 + 1$ for $C_N$, and $k_G = N - 3/2$ for $B_N$.
• The $B_3$, $G_2$, $F_4$, $C_4\times U(1)$, and $C_1\times U(1)$ theories are new rank-one SCFTs with $k_G = 2$, $k_G = 2$, $k_G = 3$, $k_G = 3$, and $k_G = 1.5$ respectively, and they have the same $(a,c)$ values as known theories but different Coulomb branch spectra.
• The $B_3$ and $G_2$ theories have a dimension-two operator, implying an exact marginal deformation, despite lacking a weakly-coupled gauge theory description.
• The Higgs branch of the $B_3$ theory is the nilpotent orbit $[3,2,2,1,1,1,1]$, and for $G_2$ it is $G_2(a_1)$, both distinct from the minimal orbit.
• The $C_1\times U(1)$ and $B_4\times U(1)$ theories have the same $(a,c)$ as $A_2$ and $E_6$ respectively, but with different Coulomb branch spectra.
• The paper conjectures a general bound $k_G \geq h^\vee - \frac{1}{n} \frac{h}{r}$ for theories with common denominator $r$ on the Coulomb branch, with equality achieved in the minimal cases.