The N=4 Superconformal Bootstrap (1304.1803v2)

Abstract: We implement the conformal bootstrap for N=4 superconformal field theories in four dimensions. Consistency of the four-point function of the stress-energy tensor multiplet imposes significant upper bounds for the scaling dimensions of unprotected local operators as functions of the central charge of the theory. At the threshold of exclusion, a particular operator spectrum appears to be singled out by the bootstrap constraints. For large values of the central charge, this extremal spectrum is compatible with that of supergravity in AdS_5 x S^5. For finite central charge, we conjecture that the extremal spectrum is that of N=4 SYM at an S-duality invariant value of the complexified gauge coupling.

• The paper establishes numerical bounds on the scaling dimensions of unprotected operators as functions of the central charge in 4D N=4 SCFTs.
• It utilizes conformal blocks, sum rules, and linear programming techniques to analyze the four-point function of the stress-energy tensor multiplet.
• The study indicates that the derived bounds align with holographic predictions and saturate at self-dual points of N=4 SYM, enhancing nonperturbative insights.

Analysis of the ${\mathcal N}=4$ Superconformal Bootstrap

The paper articulated in "The ${\mathcal N}=4$ Superconformal Bootstrap" explores the application of conformal bootstrap methods to ${\mathcal N}=4$ superconformal field theories (SCFTs) in a four-dimensional framework. Through rigorous implementation of these methods, the authors, Beem, Rastelli, and van Rees, delineate numerical bounds on the scaling dimensions of unprotected local operators as functions of the central charge within these theories. The research highlights notable outcomes, most prominently the potential identification of an extremal spectrum corresponding to ${\mathcal N}=4$ supersymmetric Yang-Mills (SYM) theory at an S-duality invariant value of the gauge coupling.

Summary of Methods

The conformal bootstrap strategy leverages consistency conditions derived from conformal symmetry, unitarity, and the associativity of the operator product expansion (OPE). The analysis is centered around the four-point function of the stress-energy tensor multiplet, a crucial component uniquely dictated by ${\mathcal N}=4$ superconformal symmetry. By considering the constraints imposed by superconformal invariance and the spectrum of operators emerging in OPE decompositions, the authors provide a path to derive upper bounds for scaling dimensions.

Key aspects of their approach involve the use of conformal blocks and sum rules. The authors specifically examine the unprotected component $(u, v)$, simultaneously engaging in a detailed analysis of SU(4)$_R$ representations and superconformal primary operators. They utilize a discretized methodology to navigate the conformal block space and employ linear programming techniques, ultimately achieving precise exclusion plots that illustrate permissible regions for operator dimensions based on different values of the central charge.

Significant numerical bounds on scaling dimensions are reported with precision, offering benchmark constraints for theories like ${\mathcal N}=4$ SYM and providing substantial evidence about which theories saturate these bounds.

Numerical Outcomes and Conjectures

The numerical investigation yields intriguing findings. For different gauge groups, the paper sets constraints on the leading twist operators associated with varying spins. Notably, as the central charge approaches infinity, the estimated bounds closely align with the dimensions predicted by ${\mathcal N}=4$ SYM in the large $N$ limit. These bounds coincide with expected dimensions derived from both holographic descriptions via Type IIB supergravity and planar Yang-Mills calculations with large 't Hooft coupling.

The authors posit that these bounds may correspond to solutions present at the extremal points of the theory's moduli space, specifically the $S$ and $S \cdot T$ self-dual points of ${\mathcal N}=4$ SYM’s conformal manifold. This supposition is anchored on the observed cubic nature of exclusion plots, which intimates symmetric maximization across anomalous dimensions at specific coupling constants.

Implications and Future Prospects

The findings document a nonperturbative pathway to elucidate the attributes of ${\mathcal N}=4$ SYM at strong coupling, offering bounds that are concordant with existing theoretical predictions for various gauged groups. This work represents a significant stride in applying the bootstrap approach to highly symmetric theories, underscoring its utility in bridging gaps left by perturbative and Planar technique.

Practically, these insights augment our comprehension of SCFT landscapes, providing groundwork for future analyses investigating whole moduli spaces of ${\mathcal N}=4$ SYM. The evidence suggesting the saturation of bounds at self-dual points could pave the way for future work, potentially extending numeric bootstrap to extract physical observables across this conformal manifold.

The authors’ use of technical methods grounded in the principles of the conjectured S-duality presents intriguing potential for discovery, particularly if these methods can be applied to explore unexplored or enigmatic points within conformal theories. In essence, as numerical constraints become finer and more comprehensive data are incorporated, there's a promising avenue to further unlock and analyze $=4$ superconformal dynamics beyond traditional perturbative bounds.