Signs, growth and admissibility of quasi-characters and the holomorphic modular bootstrap for RCFT

TL;DR

This paper analyzes the signs and growth of quasi-characters in 2D rational conformal field theories, providing methods to classify admissible characters and construct RCFT partition functions.

Contribution

It introduces a rigorous approach using Frobenius recursion relations to estimate quasi-character coefficients' growth and signs, aiding RCFT classification.

Findings

Quasi-character coefficients have alternating signs that stabilize at high levels.

Growth estimates are obtained in the region inaccessible to Cardy asymptotics.

Provides a practical method to generate RCFT partition functions at arbitrary Wronskian index.

Abstract

Rational conformal field theories in 2d have partition functions built from holomorphic characters, whose classification can be addressed via the holomorphic modular bootstrap. This is facilitated by a special basis of ``quasi-characters'' that has been completely classified for rank-2. Suitably combining these to form admissible characters with non-negative integral coefficients $a_n$ depends crucially on the signs and growth of the quasi-character coefficients. We use Frobenius recursion relations for Modular Linear Differential Equations to estimate the growth with $c$ of these coefficients in the region $n\sim\frac{c}{12}$ that is inaccessible to Cardy asymptotics, and to prove rigorously that they have alternating signs that stabilise to a fixed sign at this order. This provides a practical path to obtain candidate RCFT partition functions at arbitrary Wronskian index.