Twist accumulation in conformal field theory. A rigorous approach to the lightcone bootstrap

TL;DR

The paper proves that in any unitary CFT in $d\ge 2$, a twist gap in the $\phi\times\phi$ OPE forces a tower of high-spin quasiprimary operators with twists approaching $2\Delta_{\phi}$, and analogously in 2D Virasoro CFTs with $c>1$ forces a tower with $h\to (c-1)/24$ and $\bar h\to\infty$. It achieves this through a mathematically rigorous lightcone bootstrap, introducing a precise DLC_* lightcone limit to bound the direct-channel and crossed-channel contributions, and then mirrors this analysis in the modular bootstrap to obtain a twist upper bound $\tau_{gap} \le (c-1)/12$ and a parallel accumulation in $h$. A central technical device is an approximate factorization of conformal blocks (via Hogervorst’s dimensional reduction), along with two-sided block bounds and a robust treatment of high- and low-twist sectors, ensuring that only near-threshold blocks contribute in the DLC_* and M_* limits. The work further provides generalized statements to continuous spectra and non-identical/spinning external operators, and it sketches a unified dictionary linking the lightcone and modular bootstrap problems, paving the way toward VMFT universality in 2D and broader spectral constraints in higher dimensions.

Abstract

We prove that in any unitary CFT, a twist gap in the spectrum of operator product expansion (OPE) of identical scalar primary operators (i.e. $φ\times φ$) implies the existence of a family of primary operators $\mathcal{O}_{τ, \ell}$ with spins $\ell \rightarrow \infty$ and twists $τ\rightarrow 2 Δ_φ$ in the same OPE spectrum. A similar twist-accumulation result is proven for any two-dimensional Virasoro-invariant, modular-invariant, unitary CFT with a normalizable vacuum and central charge $c > 1$, where we show that a twist gap in the spectrum of Virasoro primaries implies the existence of a family of Virasoro primaries $\mathcal{O}_{h, \bar{h}}$ with $h \rightarrow \infty$ and $\bar{h} \rightarrow \frac{c - 1}{24}$ (the same is true with $h$ and $\bar{h}$ interchanged). We summarize the similarity of the two problems and propose a general formulation of the lightcone bootstrap.

Figures (1)

• Figure 1: To show that the low twists vanish in the DLC$_{\ast}$ limit (red curve) we show that they are (a) bounded on the blue curve $z = (1 - \bar{z})^{2 \Delta_{\phi} / \tau_{\rm gap}}$ and (b) get suppressed when moving from the blue to the red curve.

Theorems & Definitions (37)

• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Proposition 2.5
• proof
• Remark 2.6
• proof : Proof of Theorem \ref{['theorem:DLCtwist']}(a)