Bootstrapping Conformal Field Theories with the Extremal Functional Method

TL;DR

This paper introduces the Extremal Functional Method (EFM) to bootstrap conformal field theories by analyzing extremal linear functionals at the boundary of the allowed crossing-symmetric region. It demonstrates that the extremal functional encodes detailed spectrum information, enabling the extraction of operator dimensions and OPE coefficients for many primaries from a single known scalar dimension, without appealing to Virasoro symmetry in 2d. Applying EFM to the 2d Ising model, the authors recover the low-lying spectrum and OPEs with high accuracy (Δ_ε ≈ 1 and central charge c ≈ 0.5 to six-digit precision), validating the method as a practical benchmark for more complex CFTs in higher dimensions. The work highlights both practical and theoretical limits of boundary-based bootstrap and outlines promising directions for extending EFM to additional correlators, higher dimensions, and potential connections to semidefinite programming approaches.

Abstract

The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and OPE coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the Extremal Functional Method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the 2d Ising model based solely on the dimension of a single scalar quasi-primary -- no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.

Figures (9)

• Figure 1: The plot above depicts a crossing symmetry bounds plot. The shaded blue region corresponds to values of ($\Delta_\sigma, \Delta_\epsilon$) consistent with crossing symmetry. Note here $\Delta_\epsilon$ is defined as the first scalar appearing in the $\sigma \, \sigma$ OPE. Note the kink at the value $\Delta_\epsilon \approx 1.000003$ corresponding to the two-dimensional Ising model. Some other minimal models are marked with crosses (in red).
• Figure 2: Convex hull formed by the $F^{(\sigma)}_{\Delta,L}$ vectors. In blue and purple the spin-0 and spin-2 vectors, respectively. The lines start at the unitarity bounds ($\Delta=0,2$ respectively on the left and $\Delta$ increases to the right). Higher spin curves lie outside the plot. The red curve shows part of the boundary of the polytope, the other being the spin-0 line in blue. A solution to crossing symmetry exists whenever the origin (in blue) is contained inside the polytope. In the critical case (b) a solution must involve the vectors marked with black dots. The dashed line is the linear functional separating the origin from the polytope, and it overlaps with a polytope face in the critical case.
• Figure 3: The extremal functional $\phi$ acting on the spin-0 (a), and spin-2 (b) vectors. Notice the logarithmic scale. There are two clear zeroes, corresponding to operators $(1.03,0)$ and $(2,2)$. The functional is positive when acting on all other vectors.
• Figure 4: Evolution of the critical dimension $\Delta_{\epsilon}$ as the number of derivatives is increased.
• Figure 5: The extremal functional acting on the spin-2 vectors.