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

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. $\varphi \times \varphi$) implies the existence of a family of primary operators ${\mathcal{O}}_{\tau ,\ell }$ with spins $\ell \to \infty$ and twists $\tau \to 2{\Delta }_{\varphi }$ 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 $h\to \infty$ and $\bar{h}\to \frac{c-1}{24}$ (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.

arXiv comment35+10 pages, 1 figure. v2: minor changes, typos corrected. v3: version accepted for publication by Comm.Math.Phys., with minor changes, typos corrected and references added

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