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

TL;DR

This work advances the nonperturbative study of four-dimensional ${ m N}=4$ SCFTs via the conformal bootstrap by analyzing the four-point function of stress-tensor multiplets, exploiting the protected chiral algebra to constrain meromorphic data and drastically reduce the space of effective observables. The authors derive a universal unitarity bound $c\ge\tfrac{3}{4}$ for interacting theories, and then deploy linear programming and semidefinite programming to bound leading-twist operator dimensions and OPE coefficients as functions of $c$, including the mean-field limit at $c\to\infty$ and the free limit at $c=1/4$. They introduce a three-dimensional cuboid exclusion structure for the first unprotected spins (0,2,4) on the conformal manifold, argue for an extremal spectrum saturated by a single solution at fixed $c$, and conjecture that for simply-laced gauge groups the extremal solution corresponds to a self-dual point on the conformal manifold, with large-$c$ behavior matching AdS$_5$ supergravity corrections. The results illuminate how conformal data mosaic together the conformal manifold, S-duality, and holographic expectations, offering a nonperturbative window into the strong coupling regime of ${ m N}=4$ SYM through bootstrap constraints.

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.

Figures (13)

• Figure 1: Upper bounds for the dimension of the LTUSO of spin $\ell=0$, $2$, and $4$, respectively, as a function of the (square root of the) central charge.
• Figure 2: Upper bounds for the dimensions of the LTUSOs of spins $\ell=0,2,4$ for $c = 3/4$ as a function of $1/\Lambda$ for $\Lambda=14,15,\ldots,38$. These bounds were derived using the semi-definite programming method with sdpb Simmons-Duffin:2015qma. The red line denotes the "corner estimate" of the extremal value for these operator dimensions from Beem:2013qxaBeem:2013hha -- see also Section \ref{['subsec:combining_bounds']} below.
• Figure 3: Numerical upper bounds for spin-zero and spin-two LTUSOs at infinite central charge. The curves in the two cases are a two-parameter linear fit and a three-parameter exponential fit. The mean field theory values of $\Delta_0=4$ and $\Delta_2=6$ are shown in red.
• Figure 4: Combined-channel exclusion plots for the LTUSOs of spin $\ell=0,2,4$. The plots are for central charges $c=\frac{3}{4}$, $c=\frac{15}{4}$, and $c=\infty$, as labelled.
• Figure 5: Detailed view of the corner of the cube corresponding to $c = \infty$. The superimposed axes intersect at the point $(\hat{\Delta}_0,\hat{\Delta}_2,\hat{\Delta}_4)=(4,6,8)$, which corresponds to the mean-field solution.

Theorems & Definitions (4)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Conjecture 4