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.

• Received 24 April 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.071601