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

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. $\phi \times \phi$) implies the existence of a family of primary operators ${\mathcal{O}}_{\tau ,\mathrm{\ell }}$ with spins $\mathrm{\ell }\to \mathrm{\infty }$ and twists $\tau \to 2{\mathrm{\Delta }}_{\phi }$ 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 $h\to \mathrm{\infty }$ 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.