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1. Introduction GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN January Introduction Gaudins mo del describ es a completely integrable quantum spin chain Originally it was formulated as a spin mo del related to the Lie algebra sl Later it was realized cf Sect and that one can asso ciate such a mo del to any semisimple complex Lie algebra g and a solution of the corresp onding classical Yang Baxter equation In this work we will fo cus on the mo dels corresp onding to the rational solutions Denote by V the nitedimensional irreducible representation of g of dominant integral highest weight Let b e a set of dominant integral N V and asso ciate with weights of g Consider the tensor pro duct V V 1 N of this tensor pro duct a complex numb er z The hamiltonians of each factor V i i Gaudins mo del are mutually commuting op erators z z i N i i N acting on the space V Denote by h i the invariant scalar pro duct on g normalized as in Let fI g a a a d dim g b e a basis of g and fI g b e the dual basis For any element A g i denote by A the op erator A which acts as A on the ith factor i of V and as the identity on all other factors Then d i aj X X I I a i z z i j a j i One of the main problems in Gaudins mo del is to nd the eigenvectors and the eigenvalues of these op erators Bethe ansatz metho d is p erhaps the most eective metho d for solving this problem for g sl As general references on Bethe ansatz cf Applied to Gaudins mo del this metho d consists of the following There is an obvious eigenvector s One constructs in V the tensor pro duct of the highest weight vectors of the V i other eigenvectors by acting on this vector by certain elementary op erators dep ending on auxiliary parameters w w The vectors obtained this way are called Bethe m vectors Such a vector is an eigenvector of the Gaudin hamiltonians if a certain BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN system of algebraic equations involving the parameters z s and w s and the highest i j weights s is satised These equations are called Bethe ansatz equations i One usually constructs eigenvectors in statistical mo dels asso ciated to a general simple Lie algebra g by cho osing a sequence of emb eddings of Lie algebras of lower rank into g and inductively solving diagonalization problems for these subalgebras This leads to a rather complicated combinatorial algorithm which very much dep ends on the structure of g Another metho d was recently prop osed by Babujian and Flume They con sider as analogues of Bethe vectors the rational V valued functions which enter SchechtmanVarchenko solutions of the KnizhnikZamolo dchikov KZ equa tion asso ciated to g cf also Varchenko and one of us show elsewhere that such a vector can b e obtained as a quasiclassical asymptotics of a solution of the KZ equation This establishes a remarkable connection b etween a mo del of statistical mechanics and correlation functions of a conformal eld theory In this work we prop ose a new metho d of diagonalization of Gaudins hamiltonians b which is based on Wakimoto mo dules over the ane algebra g at the critical level and the concept of invariant functionals correlation functions on tensor pro ducts b of gmo dules This will allow us to treat the diagonalization problem and the KZ equation on equal fo oting and thus explain the connection b etween them The critical level is the level k h where h is the dual Coxeter numb er of g The p eculiarity of this value of level is that it is only for k h that the lo cal b b completion U g of the universal enveloping algebra of g contains central elements k lo c b _ g was describ ed in cf also Elements of The center of U lo c h b the center act on the space of functionals on tensor pro ducts of representations of g at the critical level which are invariant with resp ect to a certain Lie algebra One can interpret the hamiltonian as the action of a quadratic central element from i b _ g on the ith factor of invariant functional Other central elements give rise U lo c h to other op erators which commute with each other and with Gaudins hamiltonians We call them higher Gaudins hamiltonians Wakimoto mo dules at the critical level are sp ecial b osonic representations b of g which are essentially parametrized by functions on the circle with values in the dual space of the Cartan subalgebra h of g We will construct eigenvectors of Gaudins hamiltonians by restricting certain invariant functionals on tensor pro ducts of Wakimoto mo dules In conformal eld theory language those are certain b osonic correlation functions Analogues of Bethe ansatz equations will naturally app ear as KacKazhdan typ e equations on the existence of certain singular vectors in Wakimoto mo dules The existence of these singular vectors will ensure that the vector constructed this way is an eigenvector of the Gaudin hamiltonians and their generalizations We will express the eigenvalues of these hamiltonians in terms of the Miura trans formation and show that Bethe ansatz equations app ear as certain analytic conditions GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL on the eigenvalues The formulas we obtain for the eigenvalues generalize Sklyanins formula for g sl The app earance of the Miura transformation in this con b _ g is isomorphic to the classical text is not surprising b ecause the center of U lo c h W algebra asso ciated to g the Langlands dual Lie algebra to g The space of invariant functionals makes sense for an arbitrary level k For non critical values of k it coincides with the space of genus correlation functions or con formal blo cks of the corresp onding WessZuminoNovikovWitten WZNW mo del This denition is equivalent to the more common denition of correlation functions as matrix elements of certain vertex op erators acting b etween b representations of g It is known that for noncritical k these functions satisfy the KZ equation One can obtain solutions of the KZ equation by restricting certain invariant func tionals on tensor pro ducts of Wakimoto mo dules but those of noncritical level The rational function which enters such a solution cf also then app ears as a b osonic correlation function cf which coincides with the formula for an eigevector obtained from Wakimoto mo dules at the critical level This gives an explanation of the connection b etween the eigenvectors of the Gaudin mo del and solutions of the KZ equation asso ciated to g which was observed in Gaudins hamiltonians can b e obtained from the transfermatrix of a quantum b spin chain related to ane quantum groups U g in a certain limit when q q b A generalization of the results presented here to the case of U g will b e published q separately We also want to remark that our construction of eigenvectors naturally ts into a program of geometric Langlands corresp ondence prop osed by Drinfeld This corre sp ondence relates equivalence classes of G bundles over a complex algebraic curve E with at connections and D mo dules on the mo duli space of Gbundles over E The pap er is organized as follows In Sect we recall the denition of Gaudins mo del and Bethe ansatz pro cedure in the case g sl Sect contains an inter pretation of the mo del in terms of invariant functionals at the critical level We show how singular vectors of imaginary weight give rise to Gaudins hamiltonians and their generalizations In Sect we recall the denition of Wakimoto mo dules b _ g We use these results in Sect and discuss the structure of the center of U lo c h to construct eigenvectors of Gaudins hamiltonians and to compute their eigenval ues In Sect we derive SchechtmanVarchenko solutions of the KZ equation using Wakimoto mo dules We show that the rational function entering a solution coincides with the formula for an eigenvector In the App endix we prove some technical results concerning Wakimoto mo dules which are used in the main text Gaudins model Let g b e a simple Lie algebra over C of rank l and dimension d and let U g b e its universal enveloping algebra For a dominant integral weight denote by V BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN the irreducible representation of g of highest weight Denote by h i the invariant scalar pro duct on g normalized as in Let I a d b e a ro ot basis of a a g and I b e the dual basis Denote by the quadratic Casimir op erator from the center of U g d X a I I a a The op erator acts on V by multiplication by a numb er which we will denote by Now x a p ositive integer N and a set of dominant highest N weights Denote by V the tensor pro duct V V 1 N Let u z z b e a set of distinct complex numb ers Intro duce linear op erator N S u which we will call the Gaudin hamiltonian on V by the formula N N X X i i S u u z u z i i i i where is given by formula i One can check directly that the op erators commute with each N other Therefore the op erators S u commute with each other for dierent values of u In the next section we will give a new pro of of this fact One of the main problems in Gaudins mo del is to diagonalize op erators S u This is equivalent to simultaneous diagonalization of the op erators N For g sl this problem can b e solved by algebraic Bethe ansatz Denote by v the highest weight vector of the mo dule V The tensor pro duct of the highest weight vectors ji v v V 1 N a is
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