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

TL;DR

This paper explores the closure properties of the operator product expansion in two-dimensional conformal field theories using numerical bootstrap methods, revealing connections to minimal models and Coulomb gas formalism.

Contribution

It demonstrates the existence of finite, closed OPE sub-algebras without assuming unitarity, and links solutions to degenerate operators with analytical crossing matrix expressions.

Findings

Identified minimal models as special solutions.

Discovered new solution lines related to Coulomb gas formalism.

Provided a Mathematica tool for calculating crossing matrices and OPE coefficients.

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.