The ${\cal N} = 8$ Superconformal Bootstrap in Three Dimensions

TL;DR

The authors develop a three-dimensional ${ m N}=8$ superconformal bootstrap for the stress-tensor multiplet, deriving exact ${ m N}=8$ superconformal blocks from Ward identities and identifying differential relations that connect six R-symmetry channels. They show that crossing constraints are largely redundant and implement the independent constraints numerically to bound OPE data and operator dimensions as a function of the central charge ${c_T}$, anchoring the results with exact ${c_T}$ values obtained from localization for ABJM/ABJ(M)/BLG theories. The numerical bounds reveal near-saturation by large-$N$ ABJM at large ${c_T}$ and by the free ABJM theory at ${c_T}=16$, with a notable kink around ${c_T} oughly 22.8$ and informative bounds on protected OPE coefficients. These results provide nonperturbative, model-independent insights into the spectrum of ${ m N}=8$ SCFTs and connect bootstrap data to M-theory via the holographic central charge, while also suggesting parity constraints in the ${f 35}_c imes{f 35}_c$ OPE.

Abstract

We analyze the constraints imposed by unitarity and crossing symmetry on the four-point function of the stress-tensor multiplet of ${\cal N}=8$ superconformal field theories in three dimensions. We first derive the superconformal blocks by analyzing the superconformal Ward identity. Our results imply that the OPE of the primary operator of the stress-tensor multiplet with itself must have parity symmetry. We then analyze the relations between the crossing equations, and we find that these equations are mostly redundant. We implement the independent crossing constraints numerically and find bounds on OPE coefficients and operator dimensions as a function of the stress-tensor central charge. To make contact with known ${\cal N}=8$ superconformal field theories, we compute this central charge in a few particular cases using supersymmetric localization. For limiting values of the central charge, our numerical bounds are nearly saturated by the large $N$ limit of ABJM theory and also by the free $U(1)\times U(1)$ ABJM theory.

Figures (3)

• Figure 5: Upper bounds on $\lambda_{(B, +)}^2$ using only the unitarity assumption (in blue) or a more restrictive assumption on scaling dimensions of long multiplets of spin-$0$ in orange. (See main text.) These bounds are computed with $j_\text{max} = 20$ and $\Lambda = 19$. For the more restrictive bounds, we also show the corresponding values computed with $\Lambda= 17$ (in black) and $\Lambda = 15$ (in light brown). The plot on the right is a zoomed-in version of the plot on the left. The dashed vertical lines correspond to the values of $c_T$ in Table \ref{['cTValues']}.
• Figure 6: Upper bounds on $\lambda_{(B, 2)}^2$ using only the unitarity assumption (in blue) or a more restrictive assumption on scaling dimensions of long multiplets of spin-$0$ in orange. (See main text.) These bounds are computed with $j_\text{max} = 20$ and $\Lambda = 19$. For the more restrictive bounds, we also show the corresponding values computed with $\Lambda= 17$ (in black) and $\Lambda = 15$ (in light brown). The plot on the right is a zoomed-in version of the plot on the left. The dashed vertical lines correspond to the values of $c_T$ in Table \ref{['cTValues']}.
• Figure 7: Upper bounds on $\lambda_{(B, +)}^2$ and $\lambda_{(B, 2)}^2$ computed either using derivatives w.r.t. $x$ and $\bar{x}$ of $d_2$ (see \ref{['dbasis']}) and holomorphic derivatives of $d_1$ (in orange), or using only derivatives of $d_2$ (in green). These bounds are obtained with $j_\text{max} = 20$, $\Lambda = 19$, and a more restrictive set of assumptions on $\Delta_0^*$ only.