Exact Correlators of BPS Operators from the 3d Superconformal Bootstrap

TL;DR

This work shows that a carefully chosen cohomological truncation of the 3d ${\cal N}\ge4$ superconformal algebra yields a 1d topological sector whose crossing constraints produce exact, finite relations among OPE coefficients of 3d ${\cal N}=8$ SCFTs. Applying this to the stress-tensor multiplet, the authors derive an explicit exact relation ${4 \lambda_{\mathrm{Stress}}^2 - 5\lambda_{(B,+)}^2 + \lambda_{(B,2)}^2 + 16 = 0}$, enabling exact determinations of several OPE coefficients in the interacting ${\rm U}(2)_2\times{\rm U}(1)_{-2}$ ABJ theory and constraining others in ABJM-like theories. The analysis predicts large classes of absent multiplets in ${\cal N}=8$ SCFTs with a unique stress tensor, and numerical bootstrap bounds corroborate these constraints, including a kink around ${c_T}\approx22.8$ tied to the potential disappearance of the $(B,2)$ multiplet. The paper also demonstrates that product SCFTs reproduce the observed bound regions, offering a coherent picture that connects exact bootstrap relations, localization results, and numerical bounds. Together, these results provide a powerful, complementary angle on nonperturbative data in highly supersymmetric 3d CFTs and sharpen the landscape of ${\cal N}=8$ theories.

Abstract

We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ${\cal N} \geq 4$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite number of OPE coefficients in the interacting SCFT ($U(2)_2 \times U(1)_{-2}$ ABJ theory) that constitutes the IR limit of $O(3)$ ${\cal N} = 8$ super-Yang-Mills theory. More generally, in ${\cal N} = 8$ SCFTs with a unique stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known ${\cal N} = 8$ SCFTs using the superconformal index. For generic ${\cal N} = 8$ SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coefficients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the $U(2)_2 \times U(1)_{-2}$ ABJ theory, which can be argued to not possess this multiplet.

Figures (3)

• Figure 1: Upper and lower bounds on $\lambda_{(B, +)}^2$ and $\lambda_{(B, 2)}^2$ OPE coefficients, where the orange shaded regions are allowed. These bounds are computed with $j_\text{max} = 20$ and $\Lambda = 19$. The red solid line denotes the exact lower-bound \ref{['microBound']} obtained from the exact relation \ref{['4pntExact']}. The black dotted vertical lines correspond to the kink at $\lambda_\text{stress}^2/16\approx 0.701$ ($c_T\approx22.8$). The brown dashed vertical lines correspond to the $U(2)_2 \times U(1)_{-2}$ ABJ theory at $\lambda_\text{stress}^2/16=.75$ ($c_T=21.333$). The orange horizontal lines correspond to known free (dotted) and mean-field (dashed) theory values listed in Table \ref{['freeMFT']}. The $\lambda_{(B, +)}^2$ bounds can be mapped into the $\lambda_{(B, 2)}^2$ bounds using \ref{['4pntExact']}.
• Figure 2: Upper and lower bounds on $(A,+)$ and $(A,2)$ OPE coefficients for the three lowest spins, where the orange shaded regions are allowed. These bounds are computed with $j_\text{max} = 20$ and $\Lambda = 19$. The red dotted vertical lines correspond to the kink observed at $\lambda_\text{stress}^2/16\approx0.727$ ($c_T\approx22.0$) for bounds on OPE coefficients for the $(A,+)$ and $(A,2)$ multiplets. The black dotted vertical lines that correspond to the kink observed at $\lambda_\text{stress}^2/16\approx0.701$ ($c_T\approx22.8$) for the $(B,+)$ and $(B,2)$ multiplet OPE coefficient bounds and the long multiplet scaling dimension bounds. The brown dashed vertical lines correspond to the $U(2)_2 \times U(1)_{-2}$ ABJ theory at $\lambda_\text{stress}^2/16=0.75$ ($c_T=21.333$). The orange horizontal lines correspond to known free (dotted) and mean-field (dashed) theory values listed in Table \ref{['Avalues']}.
• Figure 3: The region in the $\frac{\lambda_{\text{Stress}}^2}{16}$-$\lambda_{(B, 2)}^2$ plane that corresponds to arbitrary linear combinations of ${\bf 35}_c$ operators ${\cal O}_i$ in (A) mean field theory, (B) $U(2)_2 \times U(1)_{-2}$ ABJ theory, and (C) $U(1)_k \times U(1)_{-k}$ ABJM theory. $\lambda_{(B, 2)}^2$ is the sum of the squared OPE coefficients of all $(B, 2)_{[0200]}$ multiplets that appear in the ${\cal O} \times {\cal O}$ OPE, while $\lambda_{\text{Stress}}^2$ is the sum of the squared OPE coefficients of all stress-tensor multiplets that appear in the ${\cal O} \times {\cal O}$ OPE.