Closure of the operator product expansion in the non-unitary bootstrap

• Published in: The journal of high energy physics Vol. 2016; no. 11; pp. 1 - 34
• Main Authors: Esterlis, Ilya, Fitzpatrick, A. Liam, Ramirez, David M.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2016; Springer Nature B.V; Springer Berlin
• Subjects: Classical and Quantum Gravitation; Conformal Field Theory; Coulomb friction; Elementary Particles; Field Theories in Lower Dimensions; Formalism; High energy physics; Mathematical analysis; Mathematical models; Matrices (mathematics); Matrix methods; Operators; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Field Theories in Lower Dimensions; Conformal Field Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2016)030

A bstract 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.