The $(2,0)$ superconformal bootstrap (1507.05637v1)

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.

• The paper develops a numerical bootstrap approach to derive rigorous constraints on the operator spectrum and OPE coefficients of six-dimensional (2,0) SCFTs.
• It establishes a lower bound on the central charge (c > 21.45) that aligns with known values, suggesting the minimal interacting A₁ theory is consistent with these constraints.
• The analysis shows that for large c, the bootstrap predictions match eleven-dimensional supergravity, reinforcing the method's potential to fully characterize these supersymmetric theories.

Overview of the $(2,0)$ Superconformal Bootstrap

The paper develops a numerical conformal bootstrap approach for six-dimensional $(2,0)$ superconformal field theories (SCFTs), which are highly symmetric and theoretically rich CFTs that are not easily accessible through conventional quantum field theory techniques. This class of theories, especially the interacting ones, such as those labeled by the $ADE$ classification, including the mysterious $A_1$ theory, emerges fundamentally from string theory and M-theory constructions. The goal of this research is to derive constraints on the operator spectrum and OPE coefficients of these theories using the bootstrap method, focusing in particular on the four-point function of stress tensor multiplets.

Theoretical Context

Due to the maximal supersymmetry and unique features of six-dimensional CFTs, the $(2,0)$ SCFTs hold a prominent place in theoretical physics. These theories are connected to string theory and M-theory, suggesting they potentially encode insights into quantum gravity. Despite being formulated over two decades ago, incremental advances have been made in the understanding of their conformal data. The paper endeavors to precisely determine these data utilizing the bootstrap approach which imposes constraints based on crossing symmetry and unitarity. This is of interest both for the structure it reveals at strong coupling and as a check against existing holographic dualities at large $N$.

Numerical Bootstrap Analysis

The bootstrap approach focuses on the universal four-point function of stress tensor multiplets. The analysis utilizes a decomposition into superconformal blocks constrained by superconformal symmetry. This four-point function encompasses a single-variable function $h(z)$ determined by the two-dimensional chiral algebra, and a two-variable function $a(z, \bar{z})$, described by unfixed contributions that are tested against numerical crossing symmetry. Extensive numerical techniques, informed by seminal bootstrap strategies, are employed to derive rigorous lower bounds on central charge $c$ and dimensional thresholds on operators.

Key Results and Implications

• Central Charge Bounds: The paper yields a rigorous lower bound for the central charge $c > 21.45$, which, upon extrapolation, aligns with the known central charge $c = 25$ of the $A_1$ theory. This suggests that the $A_1$ theory might be the smallest interacting $(2,0)$ SCFT consistent with these constraints.
• Supergravity Matching: For large $c$, corresponding to the holographic, large $N$ limit, the bounds suggest consistency with eleven-dimensional supergravity results. This highlights the applicability of the bootstrap method in capturing the expected physics dictated by holography.
• Spectrum and OPE Coefficient Bounds: Numerical bounds on the dimensions and OPE coefficients, across various spins, indicate that these parameters are in agreement with known supergravity predictions. Additionally, specific OPE coefficients show non-trivial spectral features, constrained by the presence or absence of certain operators such as the $D[0,4]$ multiplet in the $A_1$ theory.

This research demonstrates that the $A_1$ theory and other $(2,0)$ SCFTs may be robustly cornered through the current bootstrap methodology, suggesting that with further development, complete determination of $(2,0)$ theories through bootstrap equations is plausible. Future work could enhance these frameworks by implementing multi-correlator analysis, thus refining insights into the hidden dynamics of these rich theories.

Conclusion

The paper offers a substantial stride in applying numerical bootstrap techniques to $(2,0)$ SCFTs, providing valuable constraints backed by rigorous numerical computation. This analysis not only supports the conjecture that unique solutions exist at minimal central charge but also paves the way towards a deeper understanding of these enigmatic theories that sit at the crossroads of quantum field theory and string theory-inspired models. The promising results uncovered here foreshadow further refinements that might eventually yield complete nonperturbative solutions for these supersymmetric theories.