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Chapter 6 Lax integrable systems and conformal field theory This chapter brings together the material from all the previous chapters. We address here the following problem: given a Lax integrable system of the type discussed above, construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields. For the Lax equations in question, we pro- pose a way to represent Hamiltonian vector fields by covariant derivatives with re- spect to the Knizhnik–Zamolodchikov connection. This is a Dirac-type prequantiza- tion from the

New Results and Applications

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 Hamiltonian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.7 Examples: Calogero–Moser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6 Lax integrable systems and conformal field theory 129 6.1 Conformal field theory related to a Lax integrable system . . . . . . . . . . 129 6.2 From Lax operator algebra to commutative Krichever–Novikov algebra

Index A Admissible module, 20 Affine connection, 12 Almost graded – algebra, 7, 91 – central extension, 11 – module, 7 C Canonically associated commutative algebra, 132 Casimir element, 35 Central extension, 9 – equivalence, 10 Coboundary, 10 Cocycle, 10 – V -invariant, 11, 14 – due to Virasoro–Gelfand–Fuchs, 11 – geometric, 12, 100 – local, 10 – mixing, 12 – standard, 12 Cohomological cocycles, 10 Cohomology class – V -invariant, 100 – local, 100 Coinvariants, 63 Conformal blocks, 63 Conformal Field Theory (CFT), 55, 130 – related to a Lax integrable system, 130

Preface The present book is an introduction into the new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory and contemporary mathematical physics. It presents the theory of Krichever–Novikov algebras, Lax operator algebras, their interaction, elements of their representation theory, relations to moduli spaces of Riemann surfaces and holomorphic vector bundles on them, to Lax integrable systems and Conformal Field Theory. The previous book ([68], in Russian) by the author of this volume is focused on Krichever–Novikov algebras

transformations for nonlinear evolution equations in (2+1)-dimensions”, Phys. Lett A., Vol. 81(8), (1981), pp. 419–423. http://dx.doi.org/10.1016/0375-9601(81)90401-1 [45] Liu Wen: Darboux transformations for a Lax integrable systems in 2n-dimensions, arXive:solve-int/9605002 v1 15 May 1996. [46] C.H. Gu: Generalized self-dual Yang-Mills flows, explicit solutions and reductions. Acta Applicandae Mathem., Vol. 39, (1995), pp. 349–360. http://dx.doi.org/10.1007/BF00994642

1991 [24] W. Oevel, Symplectic Runge-Kutta schemes. Symmetries and integrability of difference equations. 299–310, London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge, (1999). [24] Oevel W. Symplectic Runge-Kutta schemes. Symmetries and integrability of difference equations 299–310, London Math. Soc. Lecture Note Ser. 255, Cambridge Univ. Press, Cambridge, (1999) [25] W. X. Ma, Darboux transformations for a Lax integrable system in 2n dimensions. Lett. Math. Phys. 39 (1997), 33. 10.1007/s11005-997-3049-3 [25] Ma W. X. Darboux transformations

]. Most recently, Lou and Ma [ 8 ] applied the direct method of symmetry transformation group for Lax integrable system in place of the traditional method of solving symmetry transformation group. As we all know, the nonlocal symmetry has closed relation with the integrable model and is beneficial to the enlarge the class of the symmetry that provides the chance of obtaining the exact solution. However, the nonlocal symmetry canot be used to construct solution directly. In other words, only nonlocal symmetry is not enough unless we localised nonlocal symmetry into the

( r 1 2 − r 2 2 ) 9 ≠ 0. $$\det(F_{1})=4096(\alpha^{2}\eta_{1}-\alpha\eta_{2}+2\eta_{3})^{6}\eta_{3}^{12}(r_{1}^{2}-r_{2}^{2})^{9}\neq 0.$$ In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form { a, b } on R 18 as follows: (18) { a , b } = a T