Abstract
We implement the conformal bootstrap for $\mathcal{N}=4$ superconformal field theories in four dimensions. The 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. We conjecture that this extremal spectrum is that of $\mathcal{N}=4$ supersymmetric Yang-Mills theory at an $S$-duality invariant value of the complexified gauge coupling.
Highlights
In this work we initiate the conformal bootstrap program for four-dimensional conformal field theories with N = 2 supersymmetry
The moment map four-point function is related to the flavor symmetry of the theory, and we focus on the cases of su(2) and e6
In appendix B we expanded the superconformal block into a sum of conventional conformal blocks, and with the given coefficient we find the correct contribution of the flavor current conformal block for a four-point function of adjoint fields, see, e.g., [35, 39]
Summary
In this work we initiate the conformal bootstrap program for four-dimensional conformal field theories with N = 2 supersymmetry These theories are extraordinarily rich, both physically and mathematically, and have been studied intensively from many viewpoints. The class S construction of [10, 11] gives rise to an enormous landscape of theories, most of which resist description by conventional Lagrangian field theoretic techniques Despite this abundance, the current catalog seems fairly structured, and one may reasonably suspect that a complete classification of N = 2 superconformal field theories (SCFTs) will be possible. In many examples we know, e.g., the central charges (including flavor central charges), the spectrum of protected operators, and some OPE coefficients associated with protected operators This partial knowledge can be used as input for a numerical bootstrap analysis.
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