Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity

Abstract

We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on the 1D O(N ) vector bootstrap. We obtain a remarkably simple bootstrap equation from the O(N ) vector crossing equations in the large N limit. The bootstrap bound is saturated by the generalized free field theory.We study the analytical extremal functionals of this crossing equation, for which the total positivity of the SL(2, R) conformal block plays a critical role. We prove the SL(2, R) conformal block is totally positive for large scaling dimension ∆ and show that the total positivity is violated below a critical value ∆ * TP ≈ 0.32315626. The conformal block forms a surprisingly sophisticated mathematical structure, which for instance can violate total positivity at the order 10 −5654 for a normal value ∆ = 0.1627! We construct a series of analytical functionals {α M } which satisfy the bootstrap positive conditions up to a range ∆ Λ M . The functionals {α M } have a trivial large M limit. Surprisingly, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high Λ M , therefore proving the bootstrap bound analytically. Our result provides a concrete example to illustrate how the analytical properties of the conformal block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N analytical functional bootstrap in higher dimensions.