Closure of the Operator Product Expansion in the Non-Unitary Bootstrap

Abstract

We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

arXiv comment11 figures, 24+19 pages

References in corpus (19)

• Bounding scalar operator dimensions in 4D CFT
• Liouville theory revisited
• Liouville Field Theory -- A decade after the revolution
• The Bootstrap Program for Boundary CFT_d
• Entanglement Entropy at Large Central Charge
• OPE Convergence in Conformal Field Theory
• Correlation functions for M^N/S_N orbifolds
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
• Indecomposable Fusion Products
• Dijet Cross Sections and Parton Densities in Diffractive DIS at HERA
• More constraining conformal bootstrap
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• A lecture on the Liouville vertex operators
• Critical exponents of the 3d Ising and related models from Conformal Bootstrap
• Virasoro conformal blocks in closed form
• Truncatable bootstrap equations in algebraic form and critical surface exponents
• Conformal field theory on the plane
• The universal Racah-Wigner symbol for Uq(osp(1|2))
• Some remarks on D-branes and defects in Liouville and Toda field theories

Cited by in corpus (42)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
• Quantum Regge Trajectories and the Virasoro Analytic Bootstrap
• Lightcone Modular Bootstrap and Pure Gravity
• Crossing Symmetry in Alpha Space
• The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap
• Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity
• Free box^k Scalar Conformal Field Theory
• Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block
• Constraints on Flavored 2d CFT Partition Functions
• The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points
• On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles
• Bootstrapping boundary-localized interactions
• Rigorous bounds on irrelevant operators in the 3d Ising model CFT
• OPE statistics from higher-point crossing
• A conformal block Farey tail
• Solving Conformal Field Theories with Artificial Intelligence
• Finite-size and finite bond dimension effects of tensor network renormalization
• Inverse Bootstrapping Conformal Field Theories
• Conformal bootstrap for percolation and polymers
• Universality in asymptotic bounds and its saturation in $2$D CFT
• The unitary subsector of generalized minimal models
• Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits
• New method for the conformal bootstrap with OPE truncations
• The Most Irrational Rational Theories
• Improving the five-point bootstrap
• Lining up a Positive Semi-Definite Six-Point Bootstrap
• Anomalous dimensions of spinning operators from conformal symmetry
• Ising model close to $d=2$
• Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods
• Testing the RG-flow $M(3,10)+{\varphi }_{1,7}\to M(3,8)$ with Hamiltonian Truncation
• Bootstrapping Pure Quantum Gravity in AdS3
• Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic ${\varphi }^{2n}$ theories
• Aspects of the S transformation Bootstrap
• Quantum Groups as Global Symmetries
• Conformal Bootstrap with Reinforcement Learning
• Universal Dynamics in Non-Orientable CFT${}_2$
• Quantum Groups as Global Symmetries II. Coulomb Gas Construction
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Accurate boundary bootstrap for the three-dimensional O($N$) normal universality class
• A Monte Carlo approach to the conformal bootstrap
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations