What are we quantizing in integrable field theory?

TL;DR

This paper explores how the classical limit of integrable asymptotically free field theories relates to finite-gap solutions, linking particle momenta to Riemann surface moduli and local operators to measures on phase space.

Contribution

It provides a new understanding of the classical limit of local operators through measures induced by embeddings into the space of classical fields.

Findings

Particle momenta correspond to Riemann surface moduli

Isotopic structures relate to period lattices

Classical limits of local operators are measures on phase space

Abstract

We continue study of the connection of classical limit of integrable asymptotically free field theory to the finite-gap solutions of classical integrable equations. In the limit the momenta of particles turn into the moduli of Riemann surfaces while their isotopic structure is related to the period lattices. In this paper we explain that the classical limit of the local operators can be understood as a measure induced on the phase space by embedding into the projective space of "classical fields".