Solving M-theory with the Conformal Bootstrap

TL;DR

This work combines the conformal bootstrap with exact localization data to study 3d ${ m obreak N}=8$ SCFTs describing M2-brane IR physics, including BLG, ABJM, and ABJ theories. By relating integrated correlators in a 1d topological sector to derivatives of mass-deformed ${S^3}$ partition functions, the authors compute half- and quarter-BPS OPE coefficients for these theories and compare them with numerical bootstrap bounds, uncovering an OPE coefficient minimization pattern in holographic cases. They exploit the extremal functional method to reconstruct low-lying spectra and OPE data for the stress-tensor sector across all $N$, finding that holographic theories saturate the lower bounds at large $c_T$ and that BLG does not generally obey the minimization principle. The results provide a non-perturbative, cross-checked map of operator dimensions and OPE data in M-theory duals, offering a concrete numerical handle on AdS$_4$/CFT$_3$ physics and guiding future refinements including non-perturbative $1/N$ corrections and mixed-correlator analyses.

Abstract

We use the conformal bootstrap to perform a precision study of 3d maximally supersymmetric ($\mathcal{N}=8$) SCFTs that describe the IR physics on $N$ coincident M2-branes placed either in flat space or at a $\C^4/\Z_2$ singularity. First, using the explicit Lagrangians of ABJ(M) \cite{Aharony:2008ug,Aharony:2008gk} and recent supersymmetric localization results, we calculate certain half and quarter-BPS OPE coefficients, both exactly at small $N$, and approximately in a large $N$ expansion that we perform to all orders in $1/N$. Comparing these values with the numerical bootstrap bounds leads us to conjecture that some of these theories obey an OPE coefficient minimization principle. We then use this conjecture as well as the extremal functional method to reconstruct the first few low-lying scaling dimensions and OPE coefficients for both protected and unprotected multiplets that appear in the OPE of two stress tensor multiplets for all values of $N$. We also calculate the half and quarter-BPS operator OPE coefficients in the $SU(2)_k \times SU(2)_{-k}$ BLG theory for all values of the Chern-Simons coupling $k$, and show that generically they do not obey the same OPE coefficient minimization principle.

Figures (5)

• Figure 1: Upper and lower bounds on the $\lambda_{(B,2)}^2$ OPE coefficient in terms of the stress-tensor coefficient $c_T$, where the orange shaded region are allowed, and the plot ranges from the generalized free field theory limit $c_T\to\infty$ to the free theory $c_T=16$. The blue dots denote the exact values in Table \ref{['valList']} in BLG$_k$ for $k\geq1$. The magenta dot denotes the free ABJM$_{1,1}$ theory, the gray and green dots denote the exact values in Table \ref{['valList']} for ABJM$_{N,2}$ and ABJ$_{N}$, respectively, for $N=1,2,\infty$, and the red dots denote ABJM$_{N,1}^\text{int}$ for $N=2,3, \infty$. The red, gray, and green dotted lines show the large $N$ formulae \ref{['cTABJM']} and \ref{['B2ABJM']} for these theories for all $N\geq2$. The black dotted line denotes the numerical point $\frac{16}{c_T}\approx.71$ above which $\lambda_{(B,2)}^2=0$. The solid lines were computed with $\Lambda=43$. To show the level of convergence, the dashed lines are upper and lower bounds that were computed with $\Lambda=19$.
• Figure 2: Bounds on $\lambda_{(B,2)}^2$ in terms of the $\lambda_{(A,+)_0}^2$ OPE coefficients at the ABJM$_{3,1}$ point with $\frac{16}{c_T}\approx0.340$. The orange shaded region is the allowed island, while the red dotted line shows the exactly known value given in Table \ref{['valList']} for $\lambda_{(B,2)}^2$ in this theory. These bounds were computed with $\Lambda=43$.
• Figure 3: The scaling dimensions $\Delta_{(A,0)_{j,n}}$ for the two lowest $n=0,1$ long operators with spins $j=0,2,4$ in terms of the stress-tensor coefficient $c_T$, where the plot ranges from the generalized free field theory limit $c_T\to\infty$ to the numerical point $\frac{16}{c_T}\approx.71$ where $\lambda_{(B,2)}^2=0$. The red dots denote the known values $\Delta_j^{(n),\text{GFFT}}=j+2+2n$ for the generalized free field theory, while the red dotted lines show the linear fit for large $c_T$ given in \ref{['scalLarge']}. These bounds were computed with $\Lambda=43$.
• Figure 4: The $\lambda_{(A,2)_j}^2$ and $\lambda_{(A,+)_j}^2$ OPE coefficients with spins $j=1,3,5$ and $j=0,2,4$, respectively, in terms of the stress-tensor coefficient $c_T$, where the plot ranges from the generalized free field theory limit $c_T\to\infty$ to the numerical point $\frac{16}{c_T}\approx.71$ where $\lambda_{(B,2)}^2=0$. The red dots denotes denote the known values at the generalized free field theory points given in Table \ref{['Avalues']}, while the red dotted lines show the linear fit for large $c_T$ given in \ref{['semiLarge']}. These bounds were computed with $\Lambda=43$.
• Figure 5: The $\lambda_{(A,0)}^2$ OPE coefficients for the three lowest spins in terms of the stress-tensor coefficient $c_T$, where the plot ranges from the generalized free field theory limit $c_T\to\infty$ to the numerical point $\frac{16}{c_T}\approx.71$ where $\lambda_{(B,2)}^2=0$. The red dots denotes denote the known values at the generalized free field theory points given in Table \ref{['Avalues']}, while the red dotted lines show the linear fit for large $c_T$ given in \ref{['longLarge']}. These bounds were computed with $\Lambda=43$.