1. Large N analytical functional bootstrap. Part I. 1D CFTs and total positivity.
- Author
-
Li, Zhijin
- Subjects
CONFORMAL field theory ,OPTIMISM - 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 numerical conformal bootstrap bound is saturated by the generalized free field theories, while its extremal functional actions do not converge to any non-vanishing limit. 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 in the limits with large ∆ or small 1 − z and show that the total positivity is violated below a critical value ∆ TP ∗ ≈ 0.32315626. The SL(2, ℝ) 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 {αM } which satisfy the bootstrap positive conditions up to a range ∆ ⩽ ΛM . The functionals {αM } have a trivial large M limit. However, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high ΛM . Moreover, in the region ∆ ⩽ ΛM , the analytical functional actions are consistent with the numerical bootstrap results, therefore it clarifies the positive structure in the crossing equation 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. [ABSTRACT FROM AUTHOR]- Published
- 2023
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