The $(2,0)$ superconformal bootstrap

TL;DR

The paper uses the conformal bootstrap to study six-dimensional (2,0) superconformal field theories, focusing on the universal four-point function of stress tensor multiplets. By combining superconformal block decomposition with numerical crossing constraints, it derives rigorous bounds on central charges, OPE coefficients, and operator dimensions, and analyzes their behavior across the full range of central charges. The authors provide strong evidence that the A1 theory realizes the minimal central charge c_min = 25 and that the corresponding four-point function is uniquely fixed by crossing at this value, with dimensions and OPE data aligning with large-c holographic expectations. These results demonstrate the power of bootstrap methods in high-dimensional, highly symmetric theories and offer quantitative spectral data for the (2,0) theories, including precise estimates for the light unprotected operators in the A1 theory. The work leverages a chiral algebra correspondence to fix protected contributions, constructs a detailed superconformal block framework for the D[2,0] multiplet, and employs state-of-the-art numerical optimization to bound the CFT data, revealing smooth c-dependence and holographic saturation at large c. It also outlines a program to bootstraps higher half-BPS correlators and connect to the dual chiral algebras, opening avenues to fully constrain the (2,0) theories through a finite set of crossing equations.

Abstract

We develop the conformal bootstrap program for six-dimensional conformal field theories with $(2,0)$ supersymmetry, focusing on the universal four-point function of stress tensor multiplets. We review the solution of the superconformal Ward identities and describe the superconformal block decomposition of this correlator. We apply numerical bootstrap techniques to derive bounds on OPE coefficients and scaling dimensions from the constraints of crossing symmetry and unitarity. We also derive analytic results for the large spin spectrum using the lightcone expansion of the crossing equation. Our principal result is strong evidence that the $A_1$ theory realizes the minimal allowed central charge $(c=25)$ for any interacting $(2,0)$ theory. This implies that the full stress tensor four-point function of the $A_1$ theory is the unique unitary solution to the crossing symmetry equation at $c=25$. For this theory, we estimate the scaling dimensions of the lightest unprotected operators appearing in the stress tensor operator product expansion. We also find rigorous upper bounds for dimensions and OPE coefficients for a general interacting $(2,0)$ theory of central charge $c$. For large $c$, our bounds appear to be saturated by the holographic predictions obtained from eleven-dimensional supergravity.

Figures (15)

• Figure 1: Bound on the central charge $c$ as a function of $1/\Lambda$, which is a good proxy for the numerical cost of the result. Central charges below the data points are excluded. The dotted line shows a linear extrapolation, which indicates that with infinite numerical power the lower bound converges to $c\simeq25$. This is precisely the value for the $A_1$ theory as indicated by the horizontal line.
• Figure 2: Upper bound on the OPE coefficient squared of the $\mathcal{D}[0,4]$ multiplet as a function of the inverse central charge $c$ for $\Lambda=18,\ldots,22$, with the strongest bound shown in black. The shaded region is excluded by the numerics and unitarity ($\lambda_{\mathcal{D}[0,4]}^2 \geqslant 0$). The red vertical line corresponds to $c=25$, the central charge of the $A_1$ theory. The vertical dashed lines denote the minimum allowed central charge $c_{\rm min}(\Lambda)$ from Fig. \ref{['Fig:cbound']} for the same values of $\Lambda$. The right plot is a magnification of the large central charge region. The dashed green line is the prediction from supergravity including the first $1/c$ correction \ref{['eq:OPE_large_k_small_l']}.
• Figure 3: Upper bounds on the squared OPE coefficients of $\mathcal{B}[0,2]_{\ell}$ multiplets with $\ell=1$ (top) and $\ell = 3$ (bottom) as a function of the inverse central charge $c$. The different curves correspond to different values of $\Lambda=18,\ldots,22$, with the black curve representing the strongest bound. The red vertical line represents the $A_1$ theory. The right plots are magnified at very large central charge. The dashed green line corresponds to the supergravity answer quoted in \ref{['eq:OPE_large_k_small_l']}.
• Figure 4: Upper bound for the dimension of the first long scalar multiplet. The different curves correspond to $\Lambda=18,\ldots,22$, with the black curve representing the strongest bound. The shaded region is excluded by the numerics. The vertical red line is located at the central charge of the $A_1$ theory. On the right we display the bound for very large $c$, with the green dashed line corresponding to the supergravity result \ref{['DTsugra']}.
• Figure 5: Bound on the dimension of the first spin zero long multiplet as a function of the inverse central charge $c$, with the $\mathcal{D}[0,4]$ short multiplet excluded from the spectrum. The different bounds correspond to $\Lambda=14,15,\ldots22$. The vertical red lines marks the central charges of the $A_1$ and $A_2$ theories.

Theorems & Definitions (3)

• Conjecture
• Corollary
• Conjecture