More ${\mathcal N}=4$ superconformal bootstrap

TL;DR

This paper advances the conformal bootstrap program for ${\mathcal N}=4$ superconformal field theories, providing analytic and numerical bounds on operator dimensions and OPE coefficients, and exploring the structure of the conformal manifold and S-duality.

Contribution

It develops new analytic and numerical bootstrap techniques for ${\mathcal N}=4$ SCFTs, including bounds on operators and insights into the conformal manifold and S-duality.

Findings

Central charge $c \geq 3/4$ for interacting theories.

Numerical bounds on scaling dimensions and OPE coefficients.

Conjectures on the embedding of the conformal manifold and self-dual points.

Abstract

In this long overdue second installment, we continue to develop the conformal bootstrap program for ${\mathcal N}=4$ superconformal field theories in four dimensions via an analysis of the correlation function of four stress-tensor supermultiplets. We review analytic results for this correlator and make contact with the SCFT/chiral algebra correspondence of arXiv:1312.5344. We demonstrate that the constraints of unitarity and crossing symmetry require the central charge $c$ to be greater than or equal to $3/4$ in any interacting ${\mathcal N}=4$ SCFT. We apply numerical bootstrap methods to derive upper bounds on scaling dimensions and OPE coefficients for several low-lying, unprotected operators as a function of the central charge. We interpret our bounds in the context of ${\mathcal N}=4$ super Yang-Mills (SYM) theories, formulating a series of conjectures regarding the embedding of the conformal manifold --- parametrized by the complexified gauge coupling --- into the space of scaling dimensions and OPE coefficients. Our conjectures assign a distinguished role to points on the conformal manifold that are self-dual under a subgroup of the S-duality group. This paper contains a more detailed exposition of a number of results previously reported in arXiv:1304.1803 in addition to new results.