Towards 3D CFT Cartography with the Stress Tensor Bootstrap

Abstract

We present new numerical results on the space of local, unitary, parity-preserving conformal field theories (CFTs) in three dimensions from the stress tensor bootstrap. In bounds maximizing certain OPE coefficients, we find a plethora of sharp features, such as kinks and ridges, as a function of scaling dimensions. We show that some of these features correspond to known theories, but there are many others that are equally strong but do not match known CFTs. We argue that these features are robust to raising numerical order and could then correspond to numerous as yet unknown CFTs. We conclude in proposing a program of "CFT cartography": the systematic exploration of the landscape of CFTs without individual theory targets in mind.

Figures (10)

• Figure 1: Allowed values of the parity-even and parity-odd singlet scalar gaps for all 3d CFTs, computed by bootstrapping on the $\langle TTTT \rangle$ correlator. See main text for the labeling of points, and table \ref{['tab:params1']} for our solver parameters.
• Figure 2: Results from maximizing the OPE coefficient $\lambda_{TT+}$, which corresponds to the lightest parity even scalar, reproduced from Chang:2024whx. In the left-hand plot, we vary the gap $\Delta'_+$ to the next parity even scalar, while in the right-hand plot, we show a slice for $\Delta'_+ \ge 3$ to more clearly illustrate the kinks---one corresponding the free scalar, another corresponding to the Ising model, and a third, which remains unidentified.
• Figure 3: Contour plot, maximizing the coefficient $\lambda_{TT+}$ across the entire allowed region. Several kinks are visible, running between identified theories (see figure \ref{['fig:max-lttp-dl']}).
• Figure 4: Discrete Laplacian of the maximal $\lambda_{TT+}$, corresponding to the curvature of the surface in figure \ref{['fig:max-lttp-contour']}. We see a kink running from the free Majorana fermion, to the free Dirac fermion/QED$_3$, to the chiral Ising model---we call this the "Majorana ridge." Another, which we call the "Dirac ridge," runs from the free Dirac fermion to a point at $(4,5)$, while a third, which we call the "MFT ridge," runs through this point, up to the tensor mean field theory at $(6,7)$. (While it doesn't quite pass through this point, this is due to the slow convergence of the $\Delta_-$ upper bound with numerical order.)
• Figure 5: Higher-resolution Laplacian plot, zoomed in on the Majorana, Dirac, and MFT ridges.