The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices

Abstract

We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large $AdS$ radius limit of gapped theories in $Ad{S}_2$. In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large $Ad{S}_2$. In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.

39 pages + appendices, version 2: references added

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• Beurling-Selberg Extremization and Modular Bootstrap at High Energies
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• Dispersion relations and exact bounds on CFT correlators
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Excited States of One-Dimensional Defect CFTs from the Quantum Spectral Curve
• On the Positive Geometry of Conformal Field Theory
• Bounds on the density of states and the spectral gap in CFT${}_2$
• Dispersion Relation for CFT Four-Point Functions
• Bootstrapping line defects in $\mathcal{N}=2$ theories
• Distributions in CFT I. Cross-Ratio Space
• Towards Bootstrapping RG flows: Sine-Gordon in AdS
• Analytic Bootstrap in 2D Boundary Conformal Field Theory: Towards Braneworld Holography
• Universality in asymptotic bounds and its saturation in $2$D CFT
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• Loops in AdS: From the Spectral Representation to Position Space II
• EFT Asymptotics: the Growth of Operator Degeneracy
• An invitation to the principal series
• Crossing antisymmetric Polyakov blocks + Dispersion relation
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• Mellin amplitudes for 1$d$ CFT
• Dispersive Sum Rules in AdS${}_2$
• Positive geometry in the diagonal limit of the conformal bootstrap
• Conformal partial waves in momentum space
• Lightcone expansions of conformal blocks in closed form
• Bounding 3d CFT correlators
• Renormalization group flows in AdS and the bootstrap program
• Yangian Symmetry in Holographic Correlators
• Exploring Confinement in Anti-de Sitter Space
• Mixed Correlator Dispersive CFT Sum Rules
• More Analytic Bootstrap: Nonperturbative Effects and Fermions
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• Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
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• The Parisi-Sourlas Uplift and Infinitely Many Solvable 4d CFTs
• 1d Conformal Field Theory and Dispersion Relations
• Bound on the central charge of CFTs in large dimension
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$
• Charging Up the Functional Bootstrap
• Sparsity in the numerical six-point bootstrap
• A Large-$N$ Expansion for Minimum Bias