6D SCFTs, 4D SCFTs, Conformal Matter, and Spin Chains

TL;DR

This work unveils an integrable spin-chain structure governing large-$J$ sectors of 6D SCFTs realized as generalized quivers with conformal matter, with operator scaling dimensions taking the form $\Delta=\Delta_0+\alpha/J^2+\mathcal{O}(J^{-3})$ and mixing captured by a 1D lattice Hamiltonian. Regularization via a 5D KK theory enables controlled perturbative calculations of anomalous dimensions, yielding an open $XXX_{s=1/2}$ spin chain for A-type (6D/4D) cases and a generalized $XXX_s$ chain for D-/E-type conformal matter; Bethe Ansatz provides the spectrum of excitations. The construction extends to 4D $\mathcal{N}=2$ theories obtained on a $T^2$ reduction, and to little string theories via periodic spin chains, linking holographic and integrable structures across dimensions. Altogether, the paper demonstrates that protected operator subsectors in 6D, 4D, and LSTs are governed by explicit integrable spin chains with concrete Hamiltonians and Bethe equations, offering a concrete bridge between higher-dimensional SCFTs and well-developed integrable systems.

Abstract

Recent work has established a uniform characterization of most 6D SCFTs in terms of generalized quivers with conformal matter. Compactification of the partial tensor branch deformation of these theories on a $T^2$ leads to 4D $\mathcal{N} = 2$ SCFTs which are also generalized quivers. Taking products of bifundamental conformal matter operators, we present evidence that there are large R-charge sectors of the theory in which operator mixing is captured by a 1D spin chain Hamiltonian with operator scaling dimensions controlled by a perturbation series in inverse powers of the R-charge. We regulate the inherent divergences present in the 6D computations with the associated 5D Kaluza--Klein theory. In the case of 6D SCFTs obtained from M5-branes probing a $\mathbb{C}^{2}/\mathbb{Z}_{K}$ singularity, we show that there is a class of operators where the leading order mixing effects are captured by the integrable Heisenberg $XXX_{s=1/2}$ spin chain with open boundary conditions, and similar considerations hold for its $T^2$ reduction to a 4D $\mathcal{N}=2$ SCFT. In the case of M5-branes probing more general D- and E-type singularities where generalized quivers have conformal matter, we argue that similar mixing effects are captured by an integrable $XXX_{s}$ spin chain with $s>1/2$. We also briefly discuss some generalizations to other operator sectors as well as little string theories.

Figures (5)

• Figure 1: Depiction of the partial tensor branch of a generic 6D SCFT. These theories resemble generalized quiver gauge theories in which the links consist of conformal matter connecting gauge groups, as denoted by circles. Further decorations at the ends are possible.
• Figure 2: Depiction of the proposed correspondence between spin chain states and 6D operators. Here we consider the special case of a 6D SCFT which has an A-type quiver gauge theory on its tensor branch, in which case the spin excitations are all spin $s = 1/2$ representations of the $SU(2)_{\mathcal{R}}$ R-symmetry group. Here, $X \oplus Y^{\dag}$ denotes the degrees of freedom of a bifundamental hypermultiplet in which $X$ denotes the spin up state and $Y^{\dag}$ denotes the spin down state. These operators are constructed on the partial tensor branch of the 6D SCFT. Actual operators of the 6D SCFT are obtained by imposing a further decoupling constraint which amounts to requiring zero total momentum for quasi-particle excitations.
• Figure 3: Depiction of the partial tensor branch of M5-branes (red vertical lines) filling $\mathbb{R}^{5,1}$ and probing the transverse geometry $\mathbb{R}_{\bot} \times \mathbb{C}^{2} / \Gamma_{ADE}$. Separating the M5-branes along the $\mathbb{R}_{\bot}$ direction generates a 1D lattice. The conformal fixed point corresponds to the limit where all M5-branes coincide. Localized fluctuations on the 6D domain wall are those which are annihilated by the translation operator $P_{\bot}$ and its discretized analog on the partial tensor branch.
• Figure 4: Diagram contributing at leading order to the correlation function of $\mathcal{O}^\dagger_i$ (top) and $\mathcal{O}_{i-1}$ (bottom) via an F-term interaction.
• Figure 5: Orbiting M2-brane wrapped on an $S^2\subset (S^4/\mathbb{Z}_K)$ stretched between constant latitudes $i$ and $i+L$. The associated states give rise to the operators $\mathcal{C}_{i,i+L}$ in the CFT.