Massive-Conformal Dictionary

TL;DR

This paper explores the relationship between the finite-volume spectrum of perturbed conformal field theories and their conformal limits, providing conjectures for UV conformal dimensions based on combinatorial identities.

Contribution

It introduces explicit conjectures linking IR Bethe Ansatz quantum numbers to UV conformal dimensions in specific minimal models, supported by combinatorial and numerical evidence.

Findings

Conjectured UV conformal dimensions match numerical results.

Explicit formulas are provided for $M(2,5)$ and $M(3,5)$ models.

Combinatorial interpretation based on Rogers-Ramanujan-Schur identities.

Abstract

The finite-volume spectrum of an integrable massive perturbation of a rational conformal field theory interpolates between massive multi-particle states in infinite volume (IR limit) and conformal states, which are approached at zero volume (UV limit). Each state is labeled in the IR by a set of `Bethe Ansatz quantum numbers', while in the UV limit it is characterized primarily by the conformal dimensions of the conformal field creating it. We present explicit conjectures for the UV conformal dimensions corresponding to any IR state in the $\phi_{1,3}$-perturbed minimal models $M(2,5)$ and $M(3,5)$. The conjectures, which are based on a combinatorial interpretation of the Rogers-Ramanujan-Schur identities, are consistent with numerical results obtained previously for low-lying energy levels.