Exact conserved quantities on the cylinder I: conformal case

TL;DR

This paper derives and analyzes nonlinear integral equations for the spectra of quantum KdV equations on a cylinder, connecting lattice models, Bethe Ansatz, and conformal field theory, with explicit calculations at the free fermion point.

Contribution

It introduces a new mapping between integral equations of twisted spin chains and derives explicit eigenvalues for local integrals of motion in conformal models.

Findings

Derived nonlinear integral equations from lattice field theories.

Mapped equations to more common spin chain integral equations.

Explicit eigenvalues calculated at the free fermion point.

Abstract

The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin $+1/2$ chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers \cite{BLZ, BLZ2} by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point.