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Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics. (English. Russian original) Zbl 1330.17031
Proc. Steklov Inst. Math. 274, Suppl. 1, S85-S161 (2011); translation from Sovrem. Probl. Mat. 10, 5-140 (2007).
From the text: “In this paper, we present a result obtained by the author jointly with M. Schlichenmaier, which is a generalization of the Sugawara construction to currents on a Riemann surface with any number of marked points taking values in any finite-dimensional reductive Lie algebra. The case of noncommutative currents on Riemann surfaces with two marked points was first considered in the physical literature [L. Bonora, M. Rinaldi, J. Russo, and K. Wu, “The Sugawara construction on genus $$g$$ Riemann surfaces,” Phys. Lett. B 208, 440–446 (1988), see also Zbl 0697.30041]. That paper, which essentially solves the problem, contains serious mathematical gaps (they are analyzed in Section 4.2). The joint paper by the author and M. Schlichenmaier [J. Math. Sci., New York 92, 3807–3834 (1998; Zbl 1006.17503)], which considers the case of currents taking values in a semisimple Lie algebra, was written independently but later. This paper uses different methods of proof, which make it possible to fill the gaps of [Bonora et al., loc. cit.], and considers the case of Riemann surfaces with many marked points. An easy generalization of the construction to the reductive case was described by the author in [Geometry, topology, and mathematical physics. Selected papers from S. P. Novikov’s seminar held in Moscow, Russia, 2002–2003. Transl. Ser. 2, Am. Math. Soc. 212, Adv. Math. Sci. 55, 297–316 (2004; Zbl 1081.17014)]. On the whole, our proofs model the proof for Kac-Moody algebras given in [V. G. Kac and A. K. Raina [Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Singapore: World Scientific (1987; Zbl 0668.17012)], but they are much more complicated technically. Because of the importance of the Sugawara construction, it is desirable to have a simpler and more transparent proof.”
“This work consists of an introduction and six sections.
In Section 1, we introduce basic definitions related to Krichever-Novikov algebras and review their main properties. We also introduce Krichever-Novikov algebras of currents, vector fields, and differential operators and Krichever-Novikov tensor spaces on Riemann surfaces. Then, we define the Krichever-Novikov duality between tensors of complementary weights (valencies). After that, we introduce Krichever-Novikov bases and the corresponding almost graded structure, which play the key role in the further exposition. We also describe central extensions and the 2-cohomology of the introduced algebras and define objects being the subject-matter of this paper, namely, affine Krichever-Novikov algebras and Virasoro-type algebras.
Section 2 contains a description of the coadjoint orbit space of an affine Krichever-Novikov algebra.
Section 3 is mainly devoted to the construction of fermion representations of affine Krichever-Novikov algebras. First, following [I. M. Krichever and S. P. Novikov, Russ. Math. Surv. 55, No. 3, 586–588 (2000); translation from Usp. Mat. Nauk 55, No. 3, 181–182 (2000; Zbl 0978.35066), Russ. Math. Surv. 58, No. 3, 473–510 (2003); translation from Usp. Mat. Nauk 58, No. 3, 51–88 (2003; Zbl 1060.37068), Funct. Anal. Appl. 12, 276–286 (1979); translation from Funkts. Anal. Prilozh. 12, No. 4, 41–52 (1978; Zbl 0393.35061), Russ. Math. Surv. 35, No. 6, 53–79 (1980); translation from Usp. Mat. Nauk 35, No. 6(216), 47–68 (1980; Zbl 0501.35071)], we describe holomorphic bundles on Riemann surfaces in terms of the Tyurin parameters and introduce Krichever-Novikov bases in sections of holomorphic bundles with poles at two marked points. Then, we give a similar description of bases for the case of many marked points. The central result of this section is the construction of a class of representations of affine Krichever-Novikov algebras, which we call fermion representations. We also describe equivalence classes of these representations. In conclusion, we give a more traditional (but less general for the class of Lie algebras under consideration) construction of Verma modules in the case of many points. The main results of this section were published by the author in [Zbl 1013.17020; Zbl 1081.17014; Zbl 1106.17307; Zbl 1123.17300].
Section 4 is devoted to representations of Virasoro-type algebras. Developing results obtained in the preceding section, we consider fermion representations of these algebras. Then, we pass to the Sugawara construction, which, given any admissible representation of an affine algebra, yields a representation of a Virasoro-type algebra in the same space. A significant part of this section is occupied by detailed proofs of theorems concerning the Sugawara construction, which are very lengthy and can be skipped in the first reading. The main results of this section were published in [M. Schlichenmaier and O. K. Sheinman, Zbl 1066.17014; Zbl 0943.17019; Zbl 1006.17503 and O. K. Sheinman, Zbl 1081.17014; Zbl 1123.17300].
Section 5 considers applications of Krichever-Novikov algebras to the geometry of moduli spaces and Knizhnik-Zamolodchikov equations. We describe the Kuranishi tangent space to the moduli space of Riemann surfaces with marked points and fixed, up to a certain order, jets of local coordinates at these points in terms of Virasoro-type algebras and Krichever-Novikov bases in them. Using this description, we find formulas for the deformation of Krichever-Novikov functions and vector fields under a deformation of moduli. Then, we define conformal blocks as coinvariants of regular subalgebras of affine Krichever-Novikov algebras, introduce a sheaf of conformal blocks and a generalized Knizhnik-Zamolodchikov connection on it, and prove that this connection is projectively flat (by using already obtained results on deformations). We define Knizhnik-Zamolodchikov equations on Riemann surfaces of positive genus with marked points as equations of the horizontal sections of this connection and explicitly write these equations in terms of Krichever-Novikov bases. Finally, we show that, for genus 0, our approach gives the usual Knizhnik-Zamolodchikov equations and obtain an explicit form of these equations for genus 1. The main results of this section were published in [M. Schlichenmaier and O. K. Sheinman, Zbl 1066.17014; Zbl 0943.17019].
In Section 6, we introduce and describe second-order Casimir operators of affine Krichever-Novikov algebras. We also introduce more general operators, which we call semi-Casimirs, and establish their relation to the tangent spaces to the moduli spaces of Riemann surfaces considered above; namely, we find maps from the tangent spaces of moduli spaces to the space of operators induced by semi-Casimirs on conformal blocks and determine conditions for these maps to be well defined. The main results of this section were published in [M. Schlichenmaier and O. K. Sheinman, Zbl 1006.17503 and O. K. Sheinman, Zbl 1081.17014; Zbl 1123.17300; Zbl 1123.17301].”
This paper is based on the materials of the author’s doctoral dissertation with the same title.
##### MSC:
 17B68 Virasoro and related algebras 17B81 Applications of Lie (super)algebras to physics, etc. 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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