Large N analytical functional bootstrap. Part I. 1D CFTs and total positivity

A bstract We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on 1D O ( ) vector bootstrap. obtain a remarkably simple equation from crossing equations in limit. The numerical bound is saturated by generalized free theories, while its extremal actions do not converge to any non-vanishing functionals equation, for which total positivity SL(2, ℝ) block plays critical role. prove totally positive limits ∆ or small 1 − z and show that violated below value $$ {\Delta }_{\textrm{TP}}^{\ast } ∆ TP ∗ ≈ 0 . 32315626. forms surprisingly sophisticated mathematical structure, instance can violate at order 10 5654 normal = 1627! construct series { α M satisfy conditions up range ⩽ Λ have trivial However, due positivity, they approach limit way consistent arbitrarily high Moreover, region , are results, therefore it clarifies structure analytically. Our result provides concrete example illustrate how properties lead nontrivial bounds. expect work paves higher dimensions.