Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras

Abstract

Abstract. Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus g are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with N marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector field algebra given by the data of the moduli point. This subspace is complementary to the direct sum of the two subspaces containing the vector fields which vanish at the marked points, respectively which are regular at a fixed reference point. For each representation of the affine algebra 3g − 3 + N equations (∂ k + T [e k ])Φ = 0 are given, where the elements {e k } are a basis of the subspace, and T is the Sugawara representation of the centrally extended vector field algebra. For genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The coefficients of the equations for genus one are found in terms of Weierstraß-σ function.

Global Deformations of the Witt Algebra of Krichever–novikov Type

“…See [28] for a global operator version, and [29] for a sheaf version. In passing to the boundary of the moduli space one obtains the limit objects which are defined over the normalization of curves of lower genus.…”

Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras

“…We obtain families of algebras over the moduli space M g,K+1 of curves of genus g with K + 1 marked points, and we are exactly in the middle of the main subject of this article. In [36] and [37] it is shown that there exists a global operator description of WZWN models with the help of the Krichever Novikov objects at least over a dense open subset of the moduli space. The following is just a very rough outline.…”

“…Tsuchiya, Ueno, and Yamada [38] gave a sheaf version of WZWN models over the moduli space. In [36], [37] Schlichenmaier and Sheinman developed a global operator version. In this context of particular interest is the situation I = {P 1 , .…”

DJKM algebras and non-classical orthogonal polynomials

“…A fair amount of interesting and fundamental work has be done by Krichever, Novikov, Schlichenmaier, and Sheinman on the representation theory of these algebras. In particular Wess-ZuminoWitten-Novikov theory and analogues of the Knizhnik-Zamolodchikov equations are developed for these algebras (see the survey article [She05], and for example [SS99], [SS99], [She03], [Sch03a], [Sch03b], and [SS98]). …”