Closure of the operator product expansion in the non-unitary bootstrap

Abstract

We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method.In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

Unitary subsector of generalized minimal models

“…This gives us a more straightforward opportunity to use the techniques in [21,22]. The explicit solution for hσσσσi along this line was found in [23], which focused on its special role in the nonunitary (severe truncation) bootstrap of [24][25][26]. 5 This solution exhibits Virasoro symmetry with a central charge given by…”

“…In this case, it is given by which was obtained in [43] via the Coulomb gas formalism. Later reviews are [23,44]. In the following, we will use G 1 , G 2 , and G 3 to denote G ϵϵϵϵ ð1;1Þ , G ϵϵϵϵ ð1;3Þ , and G ϵϵϵϵ ð1;5Þ , respectively.…”

On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles

“…However, obtaining a sharp prediction for the prefactor of the subleading saddles is more challenging. The prefactor is in principle known quasi-analytically for blocks with degenerate external operators from the crossing matrices [43,44,72], but the existence of the non-decaying saddles which do not appear in degenerate conformal blocks suggests that perhaps there are limitations to analytic continuation from degenerate states. Alternatively, it may be possible to extract the prefactor from the results of [73,74] for the braiding matrices of Virasoro conformal blocks.…”

Light-cone modular bootstrap and pure gravity

“…4 Via the holographic correspondence [9], our result has interesting implications on pure Einstein gravity as a quantum gravity theory in AdS 3 . In the strictest sense, pure AdS 3 gravity is dual to a 2d large c, unitary CFT where all non-vacuum Virasoro primary operators have h,h ≥ c−1 24 and are interpreted as Bañados-Teitelboim-Zanelli (BTZ) black holes. The new twist gap bound c−1 16 suggested by our argument would imply that pure AdS 3 gravity does not exist.…”