### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 006, 37 pages      arXiv:2106.14201      https://doi.org/10.3842/SIGMA.2022.006
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

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

Igor Krichever acd and Nikita Nekrasov bce
a) Department of Mathematics, Columbia University, New York, USA
b) Simons Center for Geometry and Physics, Stony Brook University, Stony Brook NY, USA
c) Center for Advanced Studies, Skoltech, Russia
d) Higher School of Economics, Moscow, Russia
e) Kharkevich Institute for Information Transmission Problems, Moscow, Russia

Received October 19, 2021; Published online January 24, 2022

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.

Key words: Novikov-Veselov hierarchy; sigma model; Fermi spectral curve.

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