The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

TL;DR

This work advances the analytic understanding of the 3d Ising CFT by marrying numerical bootstrap with a robust lightcone bootstrap framework. It introduces new techniques for summing infinite SL(2,R) block families and for diagonalizing mixing among double-twist towers via a twist Hamiltonian, achieving all-orders large-spin results and precise comparisons to numerics for the Ising data. The approach yields analytic approximations for dimensions and OPEs of several infinite families in terms of a small set of initial data, and demonstrates how biregular constraints from low-twist operators can constrain key parameters like $\Delta_\sigma$, $\Delta_\epsilon$, $f_{\sigma \\sigma \\epsilon}$, $f_{\\epsilon \\epsilon \\epsilon}$, and $c_T$. The work lays a concrete path toward analytics in the conformal bootstrap, offering a hybrid paradigm that combines lightcone analytic control with targeted numerics to push beyond the large-spin regime and to illuminate the structure of crossing in strongly coupled CFTs.

Abstract

We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2,R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data $\{Δ_σ,Δ_ε,f_{σσε},f_{εεε},c_T\}$. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.

Figures (21)

• Figure 1: Estimates of $\mathbb{Z}_2$-even operators in the 3d Ising model. Larger circles represent "stable" operators whose dimensions and OPE coefficients have small errors in our computation. We plot the twist $\Delta-\ell$ versus spin $\ell$. The grey dashed lines are $\tau=2\Delta_\sigma+2n$ and $\tau=2\Delta_\epsilon+2n$ for nonnegative integer $n$.
• Figure 2: Estimates of $\mathbb{Z}_2$-odd operators in the 3d Ising model. Larger circles represent "stable" operators. We plot the twist $\Delta-\ell$ versus spin $\ell$. The grey dashed lines are $\tau=\Delta_\sigma+\Delta_\epsilon+2n$ for nonnegative integer $n$.
• Figure 3: A diagram representing the contribution of the exchange of $\mathcal{O}_0$ in one channel (left to right) to anomalous dimensions and OPE coefficients of double-twist operators $[\phi\phi]_n$ in the other channel (bottom to top). In the physical picture of Alday:2007mf, this diagram shows the exchange of virtual $\mathcal{O}_0$-particles between $\phi$-particles separated by a distance $\log \ell$ over time $\log \overline{z}$.
• Figure 4: Exponentiation of the contribution of $\mathcal{O}_0$ in the bottom-to-top channel becomes an exchange of multi-twist operators $[\mathcal{O}_0\dots\mathcal{O}_0]$ in the left-to-right channel.
• Figure 5: Large-spin diagrams for the contribution of $\mathcal{O}$ to $\lambda_{12[12]_n}\lambda_{43[12]_n}$ in (\ref{['eq:productsoflargespinope']}).

Theorems & Definitions (4)

• Theorem 1.1: Existence of double-twist operators Fitzpatrick:2012yxKomargodski:2012ek
• Lemma 1
• proof
• Claim