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Conformal field theory, solitons, and elliptic Calogero--Sutherland models

Bjorn K. Berntson, Edwin Langmann, Jonatan Lenells

TL;DR

This work constructs a non-chiral CFT on the torus whose central operator ${ moldsymbol{H}}_{3, u}$ provides a unified second-quantization framework for the elliptic Calogero–Sutherland system and its two-component ncILW soliton equation. By introducing a two-copy Fock space with anyons and a Bogoliubov-transformed basis, the authors derive a quantum ncILW equation and show that a generalized eCS Hamiltonian with four particle types emerges from the same operator, with rational coupling $g$ ensuring a four-fold quantization. The results connect CFT, vertex-operator techniques, and quantum integrable many-body systems, offering a pathway to eigenfunctions and integrability of the generalized models. They also illuminate non-relativistic limits and potential connections to sine-Gordon/ Coleman-type dualities, suggesting broader physical relevance in condensed matter and particle physics contexts.

Abstract

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

Conformal field theory, solitons, and elliptic Calogero--Sutherland models

TL;DR

This work constructs a non-chiral CFT on the torus whose central operator provides a unified second-quantization framework for the elliptic Calogero–Sutherland system and its two-component ncILW soliton equation. By introducing a two-copy Fock space with anyons and a Bogoliubov-transformed basis, the authors derive a quantum ncILW equation and show that a generalized eCS Hamiltonian with four particle types emerges from the same operator, with rational coupling ensuring a four-fold quantization. The results connect CFT, vertex-operator techniques, and quantum integrable many-body systems, offering a pathway to eigenfunctions and integrability of the generalized models. They also illuminate non-relativistic limits and potential connections to sine-Gordon/ Coleman-type dualities, suggesting broader physical relevance in condensed matter and particle physics contexts.

Abstract

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.
Paper Structure (28 sections, 14 theorems, 336 equations)

This paper contains 28 sections, 14 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Construction of anyons
  4. Quantum ncILW equation
  5. Second quantization of a generalized eCS model
  6. Prerequisites
  7. Fock space, Klein factors and, Heisenberg algebras
  8. Normal ordering
  9. Bogoliubov transformation and vertex operators
  10. Anyons and the eCS model
  11. Anyons
  12. Second quantizations of eCS model
  13. Derivation of the ncILW equation
  14. Second quantization of a generalized eCS model
  15. Proof of Theorem \ref{['thm:eCSgen']}
  16. ...and 13 more sections

Key Result

Lemma 3.3

For arbitrary $\mu_\pm\in {\mathbb Z}$, $\alpha_\pm \in{\mathbb C}$, and $S=\; \stackrel { \times}{ \times} \! S\! \stackrel { \times}{ \times}$ as in Sdef, defines a sesquilinear form on $\mathcal{D}$.

Theorems & Definitions (44)

  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6: Vertex operators
  • Lemma 3.7
  • proof : Proof of Lemma \ref{['lem:Phi']}
  • Definition 4.1: Anyons
  • ...and 34 more