Super-Spin Chains for 6D SCFTs

TL;DR

This work shows that a broad class of 6D SCFTs built from M5-branes probing ADE singularities exhibits a protected large-R-charge sector whose operator mixing is captured by an integrable $\mathfrak{osp}(6,2|1)$ super-spin chain. Starting from the rank-$L$ generalized quiver with conformal-matter links, the authors derive open-boundary Bethe Ansatz equations that describe excitations above a ground state $X^{L}$ and prove integrability by relating it to the $\mathfrak{su}(2)_R$ sector and its non-compact $\mathfrak{sl}(2)$ lift, with extensions to D-/E-type matter. The framework yields explicit formulas for anomalous dimensions in closed subsectors (fermionic $\mathfrak{su}(1|1)$, $\mathfrak{sl}(2)$, and $\mathfrak{su}(2)_R$) and generalizes to 6D LSTs and 4D $\mathcal{N}=2$ SCFTs through the same generalized-quiver structure, linking holographic AdS$_7$ geometries to spin-chain dynamics. Overall, the study provides a tractable, integrable approach to computing operator spectra in strongly coupled, higher-dimensional SCFTs and suggests avenues for exploring higher-loop effects and dualities via the Bethe-ansatz data.

Abstract

Nearly all 6D superconformal field theories (SCFTs) have a partial tensor branch description in terms of a generalized quiver gauge theory consisting of a long one-dimensional spine of quiver nodes with links given by conformal matter; a strongly coupled generalization of a bifundamental hypermultiplet. For theories obtained from M5-branes probing an ADE singularity, this was recently leveraged to extract a protected large R-charge subsector of operators, with operator mixing controlled at leading order in an inverse large R-charge expansion by an integrable spin $s$ Heisenberg spin chain, where $s$ is determined by the $\mathfrak{su}(2)_{R}$ R-symmetry representation of the conformal matter operator. In this work, we show that this same structure extends to the full superconformal algebra $\mathfrak{osp}(6,2|1)$. In particular, we determine the corresponding Bethe ansatz equations which govern this super-spin chain, as well as distinguished subsectors which close under operator mixing. Similar considerations extend to 6D little string theories (LSTs) and 4D $\mathcal{N} = 2$ SCFTs with the same generalized quiver structures.

Figures (7)

• Figure 1: Depiction of a Wilson line of 7D super Yang--Mills theory ending on a stack of M5-branes. The endpoint of the Wilson line is given by a conformal matter operator $X^L$ of the corresponding 6D SCFT. Other choices of $\mathfrak{osp}(6,2|1)$ spin chain excitations specify non-supersymmetric boundary conditions for 7D Wilson lines ending on the defect.
• Figure 2: The "Beauty" Dynkin diagram of the 6D superconformal algebra $\mathfrak{osp}(6,2|1)$. Each node is labeled by the corresponding generator, $E^+_I$ satisfying the Serre--Chevalley relations of line \ref{['Serre--Chevalley']}.
• Figure 3: Weight diagram of the free hypermultiplet with respect to the subalgebra of $\mathfrak{osp}(6,2|1)$ generated by $E_I^+=\mathcal{L}^1_2,S_{1-}, J^+$. Each level is reached by applying the corresponding lowering operator, $E^-_I$. The other Lorentz generators can be added straightforwardly. Note that we ignore all overall prefactors for brevity.
• Figure 4: Weight diagram of irreducible Lorentz scalar representations of $\mathfrak{g}_\text{sub}=\left<\mathcal{L}^1_2,S_{1-},J^+\right>$ obtained by taking the tensor product of two hypermultiplets. Each state is reached by applying the lowering operators, $E_I^-$, and the derivatives $D\leftrightarrow P^{12}$ are taken along the $[0,1,0]$$\mathfrak{su}(4)$ direction.
• Figure 5: The "Beauty" Dynkin diagram of the 4D $\mathcal{N}=2$ superconformal algebra, $\mathfrak{su}(2,2|2)$.