Conformal Bootstrap in Three Dimensions?

TL;DR

This work advocates a nonperturbative conformal bootstrap program to determine 3D Ising and O(N) critical exponents directly from CFT data. By leveraging crossing symmetry and conformal blocks, it outlines how tight bounds on operator dimensions and OPE coefficients can be obtained, discusses results in 2D and 4D, and proposes extending the approach to 3D using general D-dimensional blocks and targeted constraints (e.g., on Δε′) to locate the Ising fixed point. The paper emphasizes that such a bootstrap-based method avoids resummation ambiguities of the ε-expansion and, with sufficient computational effort, can yield high-precision, model-independent information about the 3D critical theory and its operator content. It also highlights the technical challenges and proposes a program to compute or continue blocks in D, enabling a pathway to the 3D Ising and O(N) universality classes from first principles.

Abstract

We discuss an idea of how 3D critical exponents can be determined by Conformal Field Theory techniques.

Figures (5)

• Figure 1: The accepted phase diagram of the $O(N)$ fixed points for $3\le D \le 4$.
• Figure 2: A bound on $\Delta_\varepsilon$ as a function of $\Delta_\sigma$, shown here schematically.
• Figure 3: Bound analogous to Fig. \ref{['fig:d4']}, but for $D=2$.
• Figure 4: Lower curve: maximal possible value of $\Delta_\varepsilon$ as a function of $\Delta_\sigma$, computed from the crossing symmetry constraint by using the algorithm of VR.${}^{\ref{['note:1']}}$Upper curve: maximal possible value of $\Delta_{\varepsilon'}$ as a function of $\Delta_\sigma$ and $\Delta_\varepsilon$ (the latter fixed to the maximal value allowed by the first bound). The dots are computed; the dashed lines are interpolated.
• Figure 5: Shaded: the region of the $(\Delta_{\sigma},\Delta_{\varepsilon})$ plane consistent with the assumed constraint $\Delta_{\varepsilon'}\ge 3$ and the crossing symmetry in 2D CFT. Computed via the algorithm of usVR with derivatives up to order 10. The tip of the allowed region is at the point $\Delta_\sigma\approx0.124, \Delta_{\varepsilon}\approx0.996$.