Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions)

TL;DR

This work applies numerical conformal bootstrap to six-dimensional ${\cal N}=(1,0)$ SCFTs with $E_8$ flavor symmetry by analyzing the four-point function of flavor current multiplets. Using superconformal blocks for ${\cal D}[2]\times{\cal D}[2]$ and a semidefinite program, the authors derive universal bounds on the central charges $C_T$ and $C_J$, and they conjecture that the rank-one E-string theory saturates the lower bound on $C_J$, enabling a partial bootstrap solution of its flavor current OPE data. The results also reveal a spectrum of long multiplets and suggest a path to solving higher-rank E-string theories, potentially linking bootstrap data to M-theory on $AdS_7\times S^4/\mathbb{Z}_2$. Overall, the paper demonstrates that the conformal bootstrap can constrain and, in favorable cases, nearly solve a rich class of 6d ${\cal N}=(1,0)$ SCFTs with nontrivial flavor symmetry. The approach provides concrete, testable predictions for spectra and central charges that can inform both field theory and holographic duals.

Abstract

We bootstrap ${\cal N}=(1,0)$ superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming $E_8$ flavor group, we present universal bounds on the central charge $C_T$ and the flavor central charge $C_J$. Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on $C_J$, and numerically determine the spectrum of long multiplets in the rank-one E-string theory. We comment on the possibility of solving the higher-rank E-string theories by bootstrap and thereby probing M-theory on AdS${}_7\times{\rm S}^4$/$\mathbb{Z}_2$.

Figures (10)

• Figure 1: The lower bounds on $C_T$ and $C_J$ at different derivative orders $\Lambda$, assuming SU(2) flavor group and allowing higher spin conserved currents in the trivial or adjoint representation. Also shown are the values for a free hypermultiplet, $C_T = {84 \over 5}$ and $C_J = {5\over 2}$. Also shown are the extrapolations to $\Lambda \to \infty$ using the ansatz \ref{['GapAnsatz']}, for $\Lambda \in 4\mathbb{Z}$ and $\Lambda \in 4\mathbb{Z}+2$, separately.
• Figure 2: Left: The extremal functional optimizing the lower bound on $C_J$ acted on the contribution of the spin-zero long multiplet to the crossing equation, ${\alpha}_E[{\cal K}^{{\cal L}[0]_{\Delta,0}}]$, in the 1 and 5 channels of the SU(2) flavor, plotted in logarithmic scale. Increasing derivative orders $\Lambda = 24, 26, \dotsc, 48$ are shown from green to red. Right: The gap (lowest scaling dimension) in the spectrum of long multiplets in each channel at different $\Lambda$. Also shown are the extrapolations to $\Lambda \to \infty$ using the ansatz \ref{['ExtrapAnsatz']}, for $\Lambda \in 4\mathbb{Z}$ and $\Lambda \in 4\mathbb{Z}+2$, separately.
• Figure 3: The lower bounds on $C_T$ and $C_J$ at different derivative orders $\Lambda$ for interacting theories with $E_8$ flavor group. Also shown are the extrapolations to infinite derivative order using the quadratic ansatz \ref{['ExtrapAnsatz']}, as well as the values in the rank-one E-string theory.
• Figure 4: The allowed region in the $C_T^{-1}-C_J^{-1}$ plane for interacting theories with $E_8$ flavor group, at derivative orders $\Lambda = 24, 28, \dotsc, 40$, shown from green to red. Also plotted are the points corresponding to the E-string theories.
• Figure 5: Left: The upper and lowers bounds on the inverse of the central charge $C_T^{-1}$ when the value of the flavor central charge $C_J$ is set close to saturating the lower bound, $C_J = (1+10^{-4}) \, {\rm min}\,C_J$, at different derivative orders $\Lambda$, for interacting theories with $E_8$ flavor group. Also shown is the value for the rank-one E-string theory. Right: The same plot zoomed in on high derivative orders, showing a trend that the value of $C_T$ at min $C_J$ potential approaches the rank-one E-string value as $\Lambda \to \infty$.

Theorems & Definitions (3)

• Conjecture 1
• Conjecture 2
• Conjecture 3