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 $SL(2,\mathbb{R})$ conformal block plays a critical role. We prove the $SL(2,\mathbb{R})$ conformal block is totally positive for large scaling dimension $\Delta$ and show that the total positivity is violated below a critical value ${\Delta }_{\text{TP}}^{\ast }\approx 0.32315626$. 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 $\Delta =0.1627$! We construct a series of analytical functionals $\{{\alpha }_M\}$ which satisfy the bootstrap positive conditions up to a range $\Delta ⩽{\Lambda }_M$. The functionals $\{{\alpha }_M\}$ 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 ${\Lambda }_M$, 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.

arXiv commentv1: 48 pages, 9 figures; v2: refs added, typos corrected, added an estimation of the critical value for total positivity

References in corpus (46)

• Gauge Theory Correlators from Non-Critical String Theory
• Bounding scalar operator dimensions in 4D CFT
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Holography from Conformal Field Theory
• The Amplituhedron
• Conformal Partial Waves and the Operator Product Expansion
• Writing CFT correlation functions as AdS scattering amplitudes
• The Analytic Bootstrap and AdS Superhorizon Locality
• Convexity and Liberation at Large Spin
• Bootstrapping the O(N) Vector Models
• A Natural Language for AdS/CFT Correlators
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
• The EFT-Hedron
• Radial Coordinates for Conformal Blocks
• Unitarity and the Holographic S-Matrix
• The S-matrix Bootstrap I: QFT in AdS
• Bootstrapping the 3d Ising twist defect
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• Conformal Bootstrap in Mellin Space
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• Bootstrapping the half-BPS line defect
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• Dispersive CFT Sum Rules
• Harmonic Analysis and Mean Field Theory
• AdS Bulk Locality from Sharp CFT Bounds
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• Integrability and Conformal Bootstrap: One Dimensional Defect CFT
• Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ SYM at strong coupling
• Analytic Functional Bootstrap for CFTs in $d>1$
• Bootstrapping line defects with $O(2)$ global symmetry
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• On the Positive Geometry of Conformal Field Theory
• Searching for gauge theories with the conformal bootstrap
• Cut-touching linear functionals in the conformal bootstrap
• Dispersion Relation for CFT Four-Point Functions
• Towards Bootstrapping RG flows: Sine-Gordon in AdS
• Twist accumulation in conformal field theory. A rigorous approach to the lightcone bootstrap
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Notes on Resonances and Unitarity from Celestial Amplitudes
• Positive geometry in the diagonal limit of the conformal bootstrap
• Dispersive Sum Rules in AdS${}_2$
• Celestial Mellin Amplitude
• Charging Up the Functional Bootstrap

Cited by in corpus (1)

• Bootstrap Method in Theoretical Physics