[Paper Review] A Family of $4D$ $\mathcal{N}=2$ Interacting SCFTs from the Twisted $A_{2N}$ Series

Oscar Chacaltana, Jacques Distler+1 more|arXiv (Cornell University)|Dec 28, 2014

Black Holes and Theoretical Physics26 references21 citations

TL;DR

This paper constructs an infinite family of 4D $χ=2$ interacting superconformal field theories (SCFTs) via compactification of the $A_{2N}$ $(2,0)$ theory on a sphere with two full twisted punctures and one minimal untwisted puncture. Using the superconformal index and Hall-Littlewood limit, it identifies the global symmetry enhancement to $Sp(2N)_{2N+2} \times U(1)$, confirms the $n=0$ S-duality for $SU(2N+1)+\wedge^2(\square)+\text{Sym}^2(\square)$, and computes exact trace anomaly coefficients $(a,c)$, generalizing the rank-1 Argyres-Wittig SCFT.

ABSTRACT

We find an infinite family of $4D$ $\mathcal{N}=2$ interacting superconformal field theories which enter the description of the strong-coupling limit of $SU(2N+1)$ gauge theories with hypermultiplets in the $\wedge^2(\square)+ ext{Sym}^2(\square)$ . These theories arise from the compactification of the $6D$ $(2,0)$ theory of type $A_{2N}$ on a sphere with two full twisted punctures and one minimal untwisted puncture. For $N=1$, this theory is the "new" rank-1 SCFT with $Δ(u)=3$ of Argyres and Wittig. Using the superconformal index, we finally pin down the properties of this theory.

Motivation & Objective

• To systematically construct and classify a new family of 4D $χ=2$ interacting SCFTs arising from compactification of the $A_{2N}$ $(2,0)$ theory with $Π_2$ outer-automorphism twists.
• To resolve the strong-coupling limit of $SU(2N+1)$ gauge theories with hypermultiplets in the $\wedge^2(\square)+\text{Sym}^2(\square)$ representations.
• To verify the S-duality proposed by Argyres and Wittig for $N=1$ and generalize it to arbitrary $N$, confirming $n=0$ in the dual description.
• To determine the global symmetry enhancement and compute the trace anomaly coefficients $(a,c)$ using the superconformal index and Hall-Littlewood limit.

Proposed method

• Realizing the SCFTs as compactifications of the $A_{2N}$ $(2,0)$ theory on a sphere with two full twisted punctures and one minimal untwisted puncture.
• Using the Hall-Littlewood limit of the superconformal index to compute the Coulomb branch Hilbert series via the 3D mirror of the $(2,0)$ theory on $C \times S^1$.
• Applying the formula for the index of twisted $A_{2N}$ punctures derived from the $Sp(N)$ representation theory and character expansions.
• Computing the leading power of $\tau$ in the index expansion to detect free hypermultiplets and global symmetry enhancement.
• Using the plethystic log of the index at order $\tau^4$ to identify chiral ring relations and verify saturation of unitarity bounds.
• Verifying that the $Sp(2N)_{2N+2} \times U(1)$ global symmetry is enhanced by checking the level saturation and consistency with Sugawara construction.

Experimental results

Research questions

• RQ1 What is the precise structure of the $4D$ $\mathcal{N}=2$ interacting SCFT that arises as the strong-coupling fixed point of $SU(2N+1)$ gauge theory with $\wedge^2(\square)+\text{Sym}^2(\square)$ hypermultiplets?
• RQ2 Does the S-duality $SU(2N+1)+\wedge^2(\square)+\text{Sym}^2(\square) \simeq Sp(N)+R_{2,2N}$ hold for all $N \geq 1$, and what is the exact global symmetry of $R_{2,2N}$?
• RQ3 Is the $n=0$ value in the dual $Sp(N)$ theory confirmed by the superconformal index, and does it lead to a consistent enhancement of global symmetry?
• RQ4 What are the exact trace anomaly coefficients $(a,c)$ for the $R_{2,2N}$ SCFTs, and do they satisfy unitarity bounds?
• RQ5 Can the $R_{2,2N}$ theories be systematically classified via the $6D$ $(2,0)$ theory compactification with $\mathbb{Z}_2$ outer-automorphism twists?

Key findings

• The $R_{2,2N}$ SCFTs are constructed as compactifications of the $A_{2N}$ $(2,0)$ theory on a sphere with two full twisted punctures and one minimal untwisted puncture, yielding an infinite family of interacting $4D$ $\mathcal{N}=2$ SCFTs.
• The global symmetry of $R_{2,2N}$ is enhanced to $Sp(2N)_{2N+2} \times U(1)$, confirmed by the index and the saturation of the unitarity bound for the $Sp(2N)$ level.
• The superconformal index confirms that the S-duality $SU(2N+1)+\wedge^2(\square)+\text{Sym}^2(\square) \simeq Sp(N)+R_{2,2N}$ holds with $n=0$, resolving the ambiguity in the Argyres-Wittig duality.
• The trace anomaly coefficients are computed as $(a,c) = \left(\frac{1+19N+14N^2}{24}, \frac{1+10N+8N^2}{12}\right)$, consistent with the Sugawara construction and unitarity.
• The Coulomb branch dimensions are $\{d_2,d_3,d_4,d_5,\dots,d_{2N},d_{2N+1}\} = \{0,1,0,1,\dots,0,1\}$, indicating a non-Lagrangian, strongly interacting theory.
• The absence of free hypermultiplets is confirmed by the leading $\tau$-power in the index being $\tau^2$, and chiral ring relations at $\tau^4$ verify the consistency of the global symmetry enhancement.