The N=4 Superconformal Bootstrap

TL;DR

The conformal bootstrap for N=4 superconformal field theories in four dimensions is implemented and it is conjecture that this extremal spectrum is that of N= 4 supersymmetric Yang-Mills theory at an S-duality invariant value of the complexified gauge coupling.

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.

Figures (4)

• Figure 1: Exclusion plots in the space of leading twist gaps $\Delta_0$, $\Delta_2$, and $\Delta_4$. Central charges $a=3/4$, $a=15/4$, and $a=\infty$ are shown, corresponding to ${\cal N}=4$ SYM with gauge group $SU(2)$, $SU(4)$, and $SU(\infty)$, respectively. The area outside of a cube-shaped region is excluded.
• Figure 2: Bounds for the scaling dimension of the leading twist unprotected operator of spin $\ell=0, 2, 4$. The bounds are displayed as a function of the (square root of the) central charge $a$. The best bound is shown in blue, corresponding to $\Lambda=17$, while the lighter lines represent bounds for lower values of $\Lambda$.
• Figure 3: Estimates for twist gap $\Delta_\ell$ for $\ell=0, 2, 4$ that characterize the corners of the exclusion cubes at large central charge. Uncertainty is due to the smoothing of the cube. The improvement relative to the single spin bounds of Fig. \ref{['fig:plots']} is apparent. Superimposed in red are the results for planar ${\cal N}=4$ SYM in the limit of infinite 't Hooft coupling; $\Delta_0\approx4-\frac{4}{a}$, $\Delta_2\approx6-\frac{1}{a}$, and $\Delta_4\approx8-\frac{12}{25a}$.
• Figure 4: A neighborhood of the corner of the exclusion boundary for infinite central charge. The origin of the superimposed axes sits at $\Delta_\ell = \ell+4$.