Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics

Abstract

1. Krichever–Novikov algebras and their place in the theory of Lie algebras, geometry and topology of moduli spaces, the theory of integrable systems, and conformal quantum field theory. In the 1960s–1980s, the theory of infinite-dimensional Lie algebras experienced a period of fierce growth related to the appearance of affine Kac–Moody algebras and Virasoro algebras, which combine direct inheritance from the classical theory of semisimple Lie algebras with successful applications. In this period, two important disciplines of mathematical physics were formed, the theory of integrable systems and conformal field theory, characterized by the deep penetration of methods of the theory of Riemann surfaces into theoretical physics. The discovery of the role of Kac–Moody and Virasoro algebras as the algebras of formally gauge and conformal infinitesimal symmetries of these theories and an important tool for integrating nonlinear equations has united both objects. The passage from formal types of symmetries to more geometric ones required considering current algebras and vector fields on Riemann surfaces. In the late 1980s, several works on this topic appeared, which were mainly related to elliptic curves, but not only to them. In 1987, in relation to the study of solitons and conformal field theory, I. M. Krichever and S. P. Novikov introduced the central extensions of current algebras and algebras of vector fields on Riemann surfaces (with complex structure and marked points), which later acquired the name Krichever–Novikov algebras [1–3]. These algebras naturally generalize affine (untwisted) Kac– Moody algebras and Virasoro algebras, respectively. Among other algebras of currents and vector fields on Riemann surfaces the Krichever–Novikov algebras are distinguished by the important property of carrying an almost graded structure, which is weaker than grading but stronger than filtration and leads to many important results; in particular, this makes it possible to consider analogues of highest weight representations. As applied to central extensions, being almost graded is equivalent to the locality property of the corresponding cocycles; moreover, the cocycles are uniquely determined by this property in themost important cases. Affine (untwisted) Kac–Moody algebras andVirasoro algebras are special cases of Krichever–Novikov algebras; from which point of view, they correspond to the Riemann sphere with two marked points. Krichever–Novikov algebras are related in many ways to fundamental problems of geometry, analysis, and mathematical physics. Being introduced as algebras of Fourier–Laurent series on a Riemann surface, they are a part of harmonic analysis. The representation theory of Krichever–Novikov algebras is closely related to the theory of holomorphic bundles on Riemann surfaces. In particular, holomorphic bundles play the key role in parameterizing fermion representations. The this class of representations is most important for the theory among those known at present; it was introduced by the author in [4]. The construction of fermion representations also demonstrates another important relationship, namely, to integrable systems: this construction uses essentially objects of the theory of commutative rings of difference operators constructed in [5–7].