Igor Krichever, Nikita Nekrasov, “Novikov–Veselov Symmetries of the Two-Dimensional <nobr>$O(N)$</nobr> Sigma Model”, SIGMA, 2022, Volume 18,006, 37 pp.

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

Igor Krichever^abc, Nikita Nekrasov^cde

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

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.

Keywords: Novikov–Veselov hierarchy, sigma model, Fermi spectral curve.

MSC: 14H70, 17B80, 35J10, 37K10, 37K20, 37K30, 81R12

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

Language: English

DOI: 10.3842/SIGMA.2022.006