Novikov-Veselov Symmetries of the Two-Dimensional $O(N)$ Sigma Model

TL;DR

This paper demonstrates that the Novikov-Veselov hierarchy acts as a complete set of symmetries for the 2D $O(N)$ sigma model, leading to new insights into algebraic spectral curves and harmonic maps.

Contribution

It establishes the role of Novikov-Veselov symmetries in the sigma model and extends the construction to reducible Fermi curves, connecting to elliptic Calogero-Moser systems.

Findings

Fermi spectral curve for double-periodic sigma model is algebraic.

Construction of complexified harmonic maps is complete for irreducible Fermi curves.

Solutions are parameterized by spectral curves of elliptic Calogero-Moser system.

Abstract

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional $O(N)$ sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system.