The perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the minimal models $M(p,p')$ and the trinomial analogue of Bailey's lemma

TL;DR

This paper derives fermionic polynomial character formulas for specific perturbations of minimal conformal field theories using a trinomial Bailey's lemma, providing new insights into their structure and associated TBA equations.

Contribution

It introduces fermionic polynomial character formulas for $ ext{phi}_{2,1}$ and $ ext{phi}_{1,5}$ perturbations using a trinomial Bailey's lemma, and conjectures TBA equations for certain models.

Findings

Fermionic character formulas for $ ext{phi}_{2,1}$ and $ ext{phi}_{1,5}$ perturbations derived.

Multiple fermionic polynomials can correspond to the same bosonic polynomial, indicating ambiguity in perturbation specification.

Conjectured TBA equations for the unitary case $M(p,p+1)$ based on incidence matrices.

Abstract

We derive the fermionic polynomial generalizations of the characters of the integrable perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the general minimal $M(p,p')$ conformal field theory by use of the recently discovered trinomial analogue of Bailey's lemma. For $\phi_{2,1}$ perturbations results are given for all models with $2p>p'$ and for $\phi_{1,5}$ perturbations results for all models with ${p'\over 3}<p< {p'\over 2}$ are obtained. For the $\phi_{2,1}$ perturbation of the unitary case $M(p,p+1)$ we use the incidence matrix obtained from these character polynomials to conjecture a set of TBA equations. We also find that for $\phi_{1,5}$ with $2<p'/p < 5/2$ and for $\phi_{2,1}$ satisfying $3p<2p'$ there are usually several different fermionic polynomials which lead to the identical bosonic polynomial. We interpret this to mean that in these cases the specification of the perturbing field is not sufficient to define the theory and that an independent statement of the choice of the proper vacuum must be made.