The superconformal bootstrap for structure constants

TL;DR

This work extends the conformal bootstrap to ${\cal N}=4$ SYM to derive non-perturbative bounds on structure constants from the four-point function of protected operators. It implements a numerical superconformal bootstrap using a linear-functional approach to obtain exclusion regions for OPE coefficients that hold for any coupling, and it analyzes special cases at the free theory and at duality-invariant points, comparing with S-duality constrained interpolating functions. The results show consistency between the bootstrap bounds and the interpolants, with particularly strong agreement for $SU(2)$ and spin-0 leading-twist operators, supporting the conjecture that bounds saturate at duality-invariant couplings. Overall, the paper demonstrates that combining bootstrap techniques with duality symmetry yields non-perturbative, exact information about a highly nontrivial CFT and points toward a practical route to determine coupling-dependent structure constants in ${\cal N}=4$ SYM.

Abstract

We report on non-perturbative bounds for structure constants on N=4 SYM. Such bounds are obtained by applying the conformal bootstrap recently extended to superconformal theories. We compare our results with interpolating functions suitably restricted by the S-duality of the theory. Within numerical errors, these interpolations support the conjecture that the bounds found in this paper are saturated at duality invariant values of the coupling. This extends recent conjectures for the anomalous dimension of leading twist operators.

Figures (4)

• Figure 1: Exclusion regions for the structure constants (or rather, their square) involving two protected operators and one non-protected operator. We show the results for the leading twist operator of spin 0 (top) and spin 2 (bottom), for gauge groups $SU(2)$ (left) and $SU(5)$ (right)
• Figure 2: Threshold values for structure constants as calculated from the conformal bootstrap (assuming the spectrum) vs the actual value (solid line). Different points for a given central charge $c$ denote different values of $\Lambda$. In all cases, as $\Lambda$ increases the threshold values approach the actual value. The figure shows the result for the leading twist operators (left) and twist 6 operator (right) both with zero spin.
• Figure 3: Interpolating functions vs. exclusion region, for the structure constant of leading twist operator with $\ell=0$ as a function of the anomalous dimension
• Figure 4: Interpolating functions vs. exclusion region, for the structure constant of leading twist operator with $\ell=2$ as a function of the anomalous dimension