[Paper Review] Deconfining $\mathcal{N}=2$ SCFTs, or the Art of Brane Bending

Iñaki García‐Etxebarria, Ben Heidenreich+2 more|arXiv (Cornell University)|Nov 15, 2021

Black Holes and Theoretical Physics4 citations

TL;DR

This paper presents a novel brane construction method to derive $σ=1$ Lagrangian descriptions for a class of interacting $σ=2$ superconformal field theories (SCFTs), specifically the $R_{2,k}$ theories, by exploiting $σ=1$ dualities in orientifolded $Ç^2/\mathbb{Z}_2$ singularities. The approach yields manifestly realized flavor symmetries—full for even $k$, maximal rank for odd $k$—enabling exact computation of the superconformal index and systematic identification of Higgsing flows via puncture reduction.

ABSTRACT

We introduce a systematic approach to constructing $\mathcal{N}=1$ Lagrangians for a class of interacting $\mathcal{N}=2$ SCFTs. We analyse in detail the simplest case of the construction, arising from placing branes at an orientifolded $\mathbb{C}^2/\mathbb{Z}_2$ singularity. In this way we obtain Lagrangian descriptions for all the $R_{2,k}$ theories. The rank one theories in this class are the $E_6$ Minahan-Nemeschansky theory and the $C_2 imes U(1)$ Argyres-Wittig theory. The Lagrangians that arise from our brane construction manifestly exhibit either the entire expected flavour symmetry group of the SCFT (for even $k$) or a full-rank subgroup thereof (for odd $k$), so we can compute the full superconformal index of the $\mathcal{N}=2$ SCFTs, and also systematically identify the Higgsings associated to partial closing of punctures.

Motivation & Objective

• To develop a systematic method for constructing $σ=1$ Lagrangian descriptions of strongly coupled $σ=2$ SCFTs, particularly those arising from compactification of the 6D $(2,0)$ theory.
• To address the longstanding challenge of obtaining Lagrangian descriptions for $σ=2$ SCFTs that are intrinsically strongly coupled and lack known Lagrangian formulations.
• To leverage $σ=1$ duality frameworks—specifically brane bending and deconfinement in orientifolded toric singularities—to access the conformal manifold and dual frames of $σ=2$ SCFTs.
• To compute the superconformal index and identify Higgsing flows (puncture reductions) by realizing the full or maximal-rank flavor symmetry in the $σ=1$ Lagrangian.

Proposed method

• The method constructs $σ=1$ Lagrangians by analyzing D3-branes probing an orientifolded $Ç^2/\mathbb{Z}_2$ singularity, which yields a class of $σ=1$ SCFTs with a conformal manifold parameterized by the string coupling.
• It applies $σ=1$ duality techniques—particularly brane bending and deconfinement—from Garcia-Etxebarria et al. (2015, 2016) to map the $σ=1$ theory to a weakly gauged quiver of $TO_k$ SCFTs.
• The construction realizes the full flavor symmetry group for even $k$ and a maximal-rank subgroup for odd $k$ in the $σ=1$ Lagrangian, ensuring the full $σ=2$ SCFT symmetry is captured.
• The superconformal index is computed directly from the $σ=1$ Lagrangian matter content and R-charge assignments, using fugacities $t$, $J_1$, $J_2$, and $X_{r,s}$ for flavor and R-symmetry.
• The method identifies Higgsing flows by analyzing the closure of punctures in the $σ=2$ class-$\mathcal{S}$ construction through the reduction of gauge groups and matter content in the $σ=1$ Lagrangian.
• The approach is extended to higher $k$ via recursive analysis of the $R_{2,k}$ theories, with explicit index computations provided for $k=2,4,6$.

Experimental results

Research questions

• RQ1 Can $σ=1$ duality frameworks be systematically used to construct Lagrangian descriptions for $σ=2$ SCFTs that are otherwise strongly coupled and non-Lagrangian?
• RQ2 How can the full flavor symmetry of an $σ=2$ SCFT be manifestly realized in a $σ=1$ Lagrangian, particularly when the $σ=2$ theory has enhanced symmetry?
• RQ3 What is the precise relationship between puncture Higgsing in $σ=2$ class-$\mathcal{S}$ theories and the reduction of gauge groups and matter content in the derived $σ=1$ Lagrangians?
• RQ4 Can the superconformal index of $σ=2$ SCFTs be computed directly from $σ=1$ Lagrangians, and does this computation match known results?
• RQ5 Is there a systematic way to generalize this brane bending approach to other $σ=2$ SCFTs beyond the $R_{2,k}$ class, such as those from $Y^{2n,0}$ singularities?

Key findings

• The method successfully constructs $σ=1$ Lagrangians for all $R_{2,k}$ theories, with the rank-one cases including the $E_6$ Minahan-Nemeschansky and $C_2 \times U(1)$ Argyres-Wittig theories.
• For even $k$, the $σ=1$ Lagrangian manifestly realizes the full $SU(2k+1) \times U(1)^{2k}$ flavor symmetry of the $R_{2,k}$ SCFT.
• For odd $k$, the Lagrangian realizes a maximal-rank subgroup of the full flavor symmetry, consistent with the expected enhancement in the $σ=2$ theory.
• The superconformal index for $R_{2,2}$, $R_{2,4}$, and $R_{2,6}$ is computed explicitly from the $σ=1$ Lagrangian, with terms up to $t^3$ and higher-order contributions provided.
• The Higgsing flows corresponding to partial closing of punctures are systematically identified through the reduction of gauge groups and matter content in the $σ=1$ Lagrangian.
• The approach reveals a surprising duality between $σ=1$ and $σ=2$ theories, where $σ=1$ dualities are used to derive $σ=2$ duality frames, challenging the expectation that higher supersymmetry simplifies duality analysis.