On spectral theory of quantum vertex operators

TL;DR

This paper proves a conjecture about the asymptotic behavior of quantum vertex operators in quantum affine algebra, by analyzing eigenvalues and eigenvectors of their compositions.

Contribution

It establishes the asymptotic properties of quantum vertex operators and computes key eigenvalues and eigenvectors, confirming a conjecture in the field.

Findings

Proved the Davies-Foda-Jimbo-Miwa-Nakayashiki conjecture.

Computed leading eigenvalues and eigenvectors of quantum vertex operator products.

Analyzed asymptotics of quantum vertex operator compositions.

Abstract

In this note we prove the Davies-Foda-Jimbo-Miwa-Nakayashiki conjecture on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra U_q(\hat sl_2), as n goes to infinity. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo. The results of his computation are given in Section 4.