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Talalaev D.V.

Quantum metho d of the sp ectral curve

Contents
Intro duction 1 The classic metho d of the sp ectral curve 1.1 Lax representation . . . . . . . . . . . . 1.2 The Hitchin description . . . . . . . . . 1.2.1 Spectral curve . . . . . . . . . . 1.2.2 Line bundle . . . . . . . . . . . 1.3 The Hitchin system on singular curves . 1.3.1 Generalizations . . . . . . . . . . 1.3.2 Scheme points . . . . . . . . . . 1.4 The Gaudin model . . . . . . . . . . . . 1.4.1 The Lax operator . . . . . . . . . 1.4.2 R-matrix bracket . . . . . . . . . 1.4.3 The integrals . . . . . . . . . . . 1.4.4 Algebraic-geometric description . 1.5 Separated variables . . . . . . . . . . . . 1.5.1 sl2 -Gaudin model . . . . . . . . . 2 4 4 5 5 6 6 6 7 9 9 10 10 11 13 13 14 14 14 15 15 15 15 16 16 18 18 19 19 20 21 22 22 22 24 25 25 26

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2 The quantization problem 2.1 The deformation quantization . . . . . . . . . . . 2.1.1 Correspondence . . . . . . . . . . . . . . . 2.1.2 Quantization of an integrable system . . . 2.1.3 The Gaudin model quantization problem 2.2 Quantum spectral curve . . . . . . . . . . . . . . 2.2.1 Noncommutative determinant . . . . . . . 2.2.2 Quantum spectral curve . . . . . . . . . 2.2.3 Yangian . . . . . . . . . . . . . . . . . . . 2.2.4 The Bethe subalgebra . . . . . . . . . . . 2.2.5 The commutativity proof . . . . . . . . 2.3 Traditional solution methods . . . . . . . . . . . 2.3.1 Bethe ansatz . . . . . . . . . . . . . . . . 2.3.2 Quantum separated variables . . . . . . . 2.3.3 The monodromy of Fuchsian systems . . . 2.4 Elliptic case . . . . . . . . . . . . . . . . . . . . . 2.4.1 The notations . . . . . . . . . . . . . . . . 2.4.2 Felder algebra . . . . . . . . . . . . . . . . 2.4.3 Commutative algebra . . . . . . . . . . . 2.4.4 Characteristic polynomial . . . . . . . . . 2.4.5 The limit and the Gaudin model . . . . . 2.4.6 The explicit form of the sl2 elliptic Gaudin

.... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... model

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3 Solution for quantum integrable systems 3.1 Monodromic formulation . . . . . . . . . . . . . . 3.1.1 A scalar and a matrix Fuchsian equations 3.1.2 Dual equation . . . . . . . . . . . . . . . . 3.1.3 Backup . . . . . . . . . . . . . . . . . . . 3.2 Schlesinger transformations . . . . . . . . . . . . 3.2.1 Action on bundles . . . . . . . . . . . . . 3.2.2 The action on connections . . . . . . . . . 3.3 Elliptic case . . . . . . . . . . . . . . . . . . . . . 3.3.1 Separated variables . . . . . . . . . . . . . 3.3.2 Bethe ansatz . . . . . . . . . . . . . . . . 3.3.3 Matrix form of the Bethe equations . . . 3.3.4 Hecke transformations . . . . . . . . . . . 4

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27 27 27 29 29 33 33 34 35 35 36 36 38 38 39 39 41 42 42 44 44 45

Applications 4.1 Geometric Langlands correspondence . . . . . . . . . . 4.1.1 The center of U (gln ) on the critical level . . . . 4.1.2 Explicit description of the center of Ucrit (gln )) 4.1.3 The Beilinson-Drinfeld scheme . . . . . . . . . 4.1.4 Correspondence . . . . . . . . . . . . . . . . . . 4.2 Non-commutative geometry . . . . . . . . . . . . . . . 4.2.1 The Drinfeld-Sokolov form of the quantum Lax 4.2.2 Caley-Hamilton identity . . . . . . . . . . . . .

..... ..... ..... ..... ..... ..... operator .....

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Introduction
The main focus of these lectures is of direct relevance to the two most important directions of development of geometry and topology of the 20th century, the applications of the theory of Integrable systems and applications of the ideas of quantum physics. The most visible result of the first direction is the solution of the Schottky problem [1], based on the hypothesis of S.P. Novikov. The challenge of characterizing Jacobians among other principally-polarized abelian varieties has been resolved in terms of non-linear equations: a functional expression satisfies the KP equation if and only if the corresponding Abelian variaty is the Jacobian of an algebraic curve. Development of this activity was the proof of the Welters conjecture [2] on Jacobian matrices in terms of triple secant of Kummer's manifolds. The second ma jor layer of results associated with the applications of the quantum field theory in a challenge of constructing topological invariants. JonesWitten invariants or more broadly - quantum topological field theory generalizes the traditional invariants: Alexander polynomial and Jones polynomial. The invariants in this case are constructed as correlation functions for some quantum field theory [3]. The theory of the Donaldson invariants [4] and its development by Seiberg and Witten is another important example of applications of the quantum physics in topology. This work is devoted to constructing quantum analogues of algebraic-geometric methods applicable in solving classical integrable systems. These methods are based on the spectral curve concept and the Abel transform. In addition to applications in the topology, explicit description of solutions for quantum integrable systems is directly linked to such problems as calculation the cohomology of the -divisor for Abelian varieties [6], calculation of cohomology and characteristic classes of moduli spaces of stable holomorphic bundles [7] and some generalizations [8]. In these lectures we propose a quantum analog of the spectral curve method for rational and elliptic Gaudin model [9]. These cases correspond to the genus 0 and 1 base curve in the Hitchin classification. The material is related to topological invariants of the quantum field theory type, as well as closely connected with the geometric properties of moduli spaces partially in task of the description of the spectrum for quantum systems. The results concern methodological approach based on the concept of the quantum spectral curve. They crystallize in the explicit discrete group symmetry construction on the corresponding spectrum systems.

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Classic Integrable systems. Interaction between the theory of Integrable systems and algebraic geometry appeared quite early. Pioneer work, linking these areas of mathematics, was due to k. Jacoby [12] where the problem of geodesics on ellipsoid was solved in terms of the Abel transfrom for some algebraic curve. The extent of this observation was conceived in the 1970s by S.P. Novikov school [10, 13]. Later the universal geometric description of the phase space of a wide class of finite-dimensional Integrable systems in terms of the cotangent bundle to some moduli space of holomorphic bundles on an algebraic curve was given in the work of N. Hitchin [14]. The algebraic point of view at Integrable systems evolved parallelly was based on principles of Hamiltonian Dynamics and Poisson geometry. Significant progress in the classical theory of Integrable systems was related with the invention of the inverse scattering problem in 60-ies of the last century [16]. It turned out that the Lax representation is extremely effective description of dynamical systems [17]. This language relates Hamiltonian flows with the related Lie algebra action. This point of view allows to introduce the notion of the spectral curve and use methods of algebraic geometry to construct explicit solutions [18], to solve dynamical systems in algebraic terms by the pro jection method [19] or by a little more general construction of the Sato grassmanian and the corresponding -function [20]. Further, we use the term "`the spectral curve method"' for the method of solving dynamical systems having Lax representation in terms of Abel transform for the curve defined by the characteristic polynomial of the Lax operator. The first part of the work is dedicated to construction of a generalization for the Hitchin type systems in case of the base curves with singularities and fixed points. The main example of the proposed here quantization technic, the Gaudin model, is a particular case of the Hitchin system of the generalized nature. Quantization. Examples of quantum integrable models discussed here have an independent physical meaning as spin chain quantum-mechanical systems describing one-dimensional magnets. However, the main focus of these lectures is to study the structural role of Integrable systems including the quantum level, where their role as Symmetries of more complex ob jects is also evident. In particular, spin chains that describe very one-dimensional physical systems are associated with 2D problems of statistical physics [9]. The main method of quantum systems called quantum inverse scattering method (QISM) was established in the 70s of the 20th century by the school of L. D. Faddeev [21]. In many aspects this method relies on the classical method of the inverse problem, in particular with regard to the Hamiltonian description. Using QISM there were constructed several examples of quantum integrable systems: quantum nonlinear Schroedinger equation, the Heisenberg magnet and the sine-Gordon model (this is equivalent to the massive Tirring model). The asymptotic correlation functions for these model were found in [47]. Many of the results regarding QISM was aware of earlier framework of the Bethe ansatz method discovered in 1931 [22]. QISM was much generalized by the theory of quantum groups imposed by Drinfeld [23]. The language of Hopf algebras is exclusively convenient for working with algebraic structures of the theory of quantum integrable systems specifically for the generalization of the ring of invariant polynomials on the group. One can consider QISM as the quantum analog of the algebraic part of the integrable systems theory. At the same time, the role of spectral curve and methods of algebraic geometry was out of the QISM paradigm. The second part of the lectures concerns the quantum spectral curve method, whose central ob ject is the quantum characteristic polynomial for the quantum Lax operator. We propose a construction for the sln Hitchin-type systems for the base curve of genus 0 and 1 with marked points. The elliptic spin Kalogero-Moser system is a particular case of the considered family. The quantum characteristic polynomial is a generating function for quantum Hamiltonians. The construction is based on the methods of the theory of quantum groups, in particular the theory of Yangians and the Felder dynamic elliptic quantum algebra. As noted above QISM has not provided substantial progress in solving quantum systems on the finite scale level. Despite the fact that separated variables were found for some models the analogue of the Abel transform as the transition from the divisor space to the Jacobian in the quantum case has not been found. In part 3 a family of the geometric symmetries on the set of the quantum system solutions is constructed significantly using the quantum characteristic polynomial of the model. The alternating formulation of the Bethe system is used to construct this family, the formulation in terms of a family of special Fuchsian systems with restricted monodromy representation. In turn, these differential operators are scalar analogue

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of the quantum characteristic polynomial. This permits to realize quantum symmetries in terms of very known Schlesinger transformation in theory of isomonodromic deformations [24] and apply known solutions of differential equations of the Painleve type to describe variations of the spectrum of quantum systems changing the inhomogeneity parameters. In a sense to build a family of symmetries of the spectrum is an analog of the Abel transform. Quantum metho d of the sp ectral curve and other areas of mo dern mathematics. Study of quantum characteristic polynomial for the Gaudin models has systematized and made much more efficient methods of solving quantum integrable systems. The constructed discrete symmetries of the spectrum systems provide a generalized angle operators, meaning that you can build eigenvectors of the model recurrently. The significance of the results in geometry and topology is the possibility to apply this technique to field theoretic models arising in topological quantum field theories and theories of fields used in constructing the Donaldson and Seiberg-Witten invariants. In addition, the results in the problem of solving quantum systems have direct application in the description of cohomologies of moduli spaces of holomorphic bundles, analogues of the Laumon spaces, as well as affine Jacobians. The method have got issue in numerous relations and application in other areas of modern mathematics and mathematical physics. In the representation theory of Lie algebras the results are related to the effectivization of the multiplicity formula. Applications of this type occur thanks to special limits of the Gaudin commutative subalgebras which are interpreted as subalgebras of central elements in U (sln )N [25]. Another result of this technique is an explicit descriptions of the centre of the universal enveloping algebra of the affine algebra on the critical level for sln . It is also worth noting the importance of quantum spectral curve method in geometrical Lenglends program over C [26] in the booming field of Noncommutative Geometry, mathematical physics and the condensed matter. Some of the applications are presented in section 4. Thanks The author is very grateful to the staff of the Chair of higher geometry and topology of Mechanics and Mathematics Faculty of Moscow State University for fruitful atmosphere and valuable observations during the preparation of the lectures. The author is grateful to 170-th and 197-th laboratories of the Institute for theoretical and experimental physics for stimulating conversations. The author expresses special thanks to O. Babelon, V.M. Buhshtaber, A.P. Veselov, A. M. Levin, S.A. Loktev, M.A. Olshanetsky, I.E. Panov, V.N. Rubtsov, A.V. Silantiev, A.V. Chervov, G.I. Sharygin. This work is partially supported by the Foundation "` Dynasty" ', RFBR grant 09-01-00239 and the grant for support of scientific school 5413.2010.1.

1
1.1

The classic metho d of the sp ectral curve
Lax representation

This topic describes the classic method of the spectral curve for finite-dimensional Integrable systems. The explanation begins with Lax representation [17], which have led to the formation of the inverse problem method in the theory of Integrable systems. It turns out that the very wide class of Integrable systems is of the Lax type L(z ) = [M (z ), L(z )] (1.1)

where M (z ), L(z ) are matrix-valued functions of the formal variable z , those matrix elements are in turn the functions on the phase space of a model. In other words, the phase space of a system may be embedded into some space of matrix-valued functions where the dynamics is described by the Lax equation (1.1). Locally, this property is fulfilled for all integrable systems due to the existence of local action-angle variables ([28], 2 4 Example 1). Globally the Lax expression is known for: harmonic oscillator, integrable tops, the Newman model, the problem of geodesics on ellipsoid, the open and periodic Toda chain, the CalogeroMoser system for all types of root systems, the Gaudin model, nonlinear hierarchies: KdV, KP, Toda, as well as their famous matrix generalizations. The Lax representation demonstrates that the Hamiltonian vector field L = {h, L} can be expressed in terms of the lie algebra structure on the space of matrices. This property is at the heart of many of algebraic analytic techniques, in particular of the r-matrix approach and the decomposition problem [29].

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The Lax representation means that the characteristic polynomial of the Lax operator is preserved by the dynamics. The spectral curve is defined by the equation det(L(z ) - ) = 0. (1.2)

It turns out that the solution of equations that allow the Lax representation simplifies using the so-called linear problem L(z )(z ) = (z ). The Lax equation is equivalent to the compatibility condition of following equations: (z ) = L(z )(z ), (z ) = M (z )(z ). If now we interpret the auxiliary linear problem as a way of specifying a line bundle on a spectral curve, the system can be solved by means of linear coordinates on the moduli space of line bundles on the spectral curve identified with an associate Jacobian. Further a Hitchin scheme and some of its generalization sets out pretending to the classification description in the theory of finite-dimensional Integrable systems. In this section we also determine the Gaudin model and give details of the classical method of the spectral curve for the system and separated variables technique. (1.3)

1.2

The Hitchin description

Let 0 be an algebraic curve and M = Mr,d (0 ) be the moduli space of holomorphic stable bundles over 0 of rank r and the determinant bundle d [30]. Let us consider the canonical holomorphic simplectic form on the cotangent bundle to the moduli space T M The deformation theory [31] allows to explicitly describe fibers of the cotangent bundle. A tangent vector to the moduli space at E corresponding to the infinitesimal deformation in terms of the Cech cocycle can be realized by an element of H 1 (End(E )), in turn the cotangent vector at E to the moduli space M through the Serre duality is an element of the cohomology space H 0 (End(E ) K); here K denotes the canonical class of 0 . In this description the following family of functions can be defined on T M hi : T M H 0 (Ki ); The direct sum of the collection of mappings h
i r i=1

hi (E , ) =

1 tri . i

(1.4)

h : T M -

H 0 (Ki )

is called the Hitchin map [14] and defines a Lagrangian fibration of the phase space of the integrable system. 1.2.1 Sp ectral curve

The spectral curve method implies an explicit method of solution in terms of some geometric ob jects on a certain algebraic curve. Consider the (nonlinear) bundle map char() : K K defined by the expression char()(µ) = det( - µ I d) (1.6)
r

,

(1.5)

where µ defines a fiber point of K, and the expression I d - the tautological section of End(E ). The spectral curve is defined as the preimage of the zero section of Kr . The preimage defines an algebraic curve in the pro jectivization of the total space of K.

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1.2.2

Line bundle

Solution for the Hitchin type system can be constructed in terms of the following line bundle. Consider the pro jection map corresponding to the canonical bundles K :K and the inverse images map E - (E K), where µ is the tautological section K. Let us also consider the quotient F , corresponding to the inclusion ~ 0 - E - (E K) - F - 0.
-µI d -µI d ~ 0

(1.7)

The support of F coincides with the spectral curve defined below by the reason that the solution of the eigen problem exists only for eigen values of a linear operator. Let us restrict the exact sequence (1.7) to 0 - L - E | - (E K)| - F | - 0. It turns out that L specifies a line bundle on the spectral curve associated with eigenvectors of the Lax operator. Let us define the Abel transform as follows: let {a1 , . . . , ag , b1 , . . . , bg } be a basis in H1 (0 , Z) with the intersection indexes (ai , bj ) = ij , {i } be the basis of holomorphic differentials in H 0 (K) normalized by the condition ai j = ij , and let Bij = bi j be the matrix of b-periods. Then we define the lattice in Cg generated by the Zg and the lattice generated by the columns of the matrix B . Fixing a point P0 one can define the Abel transform by the formula P P0 1 . . . A : J ac = Cg /; A(P ) = (1.8) . P P0 g This definition does not depend on the integrating path due to the factorization and generalizes to the map from the space of divisor classes to the moduli space of line bundles. Theorem 1.1 ([14]). The linear coordinates on the Jacobian J ac() applied to the image of the Abel transform A(L) are the "`angle"' variables for the Hitchin system.
-µI d

1.3
1.3.1

The Hitchin system on singular curves
Generalizations

The Hitchin construction can be generalized to the case of singular curves and curves with fixed points [32], [33]. This generalization permits to give explicit parametrization to the wide class of integrable systems preserving the geometric analogy with the intrinsic ingredients of the original Hitchin system. · Fixed points: It can be considered the moduli space of holomorphic bundles on an algebraic curve with additional structures, namely with trivializations at fixed points. This moduli space can be obtained as the quotient of the space of gluing functions by the trivialization change group with the condition of preserving trivializations at fixed points. Let us denote this moduli space by Mr,d (z1 , . . . , zk ). The tangent vector to the space Mr,d (z1 , . . . , zk ) at the point E in an element of the space
k

TE Mr,d (z1 , . . . , zk )

H 1 (End(E ) O(-
i=1

zi )).

The cotangent vector can be identified with the following element
k

H 0 (End(E ) K O(
i=1

zi )).

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· Singular points: The moduli space of bundles can be considered on curves with singularities of the types: double point, cusp or the so-called scheme double point. In this situation the consistent Hitchin system formalism can be established. This results in constructing a large class of interesting integrable systems. The description of the dualizing sheaf and the moduli space of bundles in this case turns out to be more explicit then in the case of nonsingular curve of the same algebraic genus. 1.3.2 Scheme p oints

Let us describe in detail the Hitchin formalism on curves with double scheme points. The singularities class Let us consider the curve proj obtaining by gluing 2 subscheme A( ), B ( ) of CP 1 (i.e. a curve obtained by adding a point to the affine curve af f = S pec{f C[z ] : f (A( )) = f (B ( ))}, where N = 0 ). Calculating the algebraic genus (dimH 1 (O)) we obtain: · Nilpotent elements: A( ) = , B ( ) = 0, g = N - 1. · Roots of unity: A( ) = , B ( ) = , where k = 1. g = N - 1 - [(N - 1)/k ]. · Different points: A( ) = a0 + a1 + ... + aN supposing a0 = b0 . g = N . Holomorphic bundles The most convenient way to describes the moduli space of holomorphic bundles for singular curves is an algebraic language, due to the duality between a bundle and the sheaf of its sections, which is a sheaf of locally-free and thus pro jective modules over the structure sheaf of an algebraic curve. The geometrical characterisation of a pro jective module in the affine chart without of the normalized curve is made in terms of the submodule M of rank r in the trivial module of vector-functions s(z ) on C satisfying the condition: s(A( )) = ( )s(B ( )), where ( ) = i=0,...,N is expressed as follows:
-1 -1 N -1

, B ( ) = b0 + b

1

+ ... + b

N -1

N -1

,



i

i

is a matrix-valued polynomial. The pro jectivity condition of this module M



· Nilpotent elements: A( ) = , B ( ) = 0, condition: 0 = I d. · Roots of unity: A( ) = , B ( ) = , where k = 1, condition: ( )( )...( · Different points: A( ) = a0 + a1 + ... + aN condition: 0 is invertible. The open sell of the moduli space of holomorphic bundles for proj is the quotient space of the space of ( ) in general position satisfying the condition above with respect to the adjoint action of GLr . Dualizing sheaf and global section In smooth situation the canonical class K is determined by the line bundle of the highest order forms on complex analytical variety M . To reconstruct this ob ject in the singular case we axiomatize the Serre duality condition H n (F ) â H
m-n -1 N -1

k-1

) = I d.

, B ( ) = b0 + b

1

+ ... + b

N -1

N -1

,

(F K) C

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for a coherent sheaf F . In the present case the dualizing sheaf can be defined by its global sections. The global sections of the dualizing sheaf on proj can be described in terms of meromorphic differentials on C of the form = Res for an element ( ) = progression:
i=0,...,N -1

( )dz ( )dz - z - A( ) z - B ( )

,

(1.9)

i

1
i+1

. In this expression fractions should be understood as geometric 1 (z - a0 )(1 -

1 1 = z - A( ) z - a0 - a1 - a2 =

2

- ...
2

=

1 a1 + a2 + ... a1 + (1 + +( (z - a0 ) z - a0 z - a0

a1 +a2 2 +... z -a0 a2 2 + ... 2

)

) + ...).

The symbol Res means the coefficient at 1 . It turns out that for an arbitrary ( ) the expression above gives a holomorphic differential on the singular curve proj , and in addition any differential is obtained in this way. Let us describe the Serre pairing for the structure sheaf. Let us consider the covering consisting of two opens U0 = af f and U - an open disk centered at . The intersection U0 U can be identified · · with the punctured disk U also centered at . Let s OU - be a representative of H 1 (O). The pairing is determined by the formula: < , s >=
U0 U


s.

(1.10)

It is easy to see that the pairing is correctly defined on classes of cohomology. The endomorphisms of the mo dule M are described by polynomial matrix-valued functions (z ) satisfying the condition (A( )) = ( )(B ( ))( )-1 . The action of (z ) on a section s(z ) is given by the formula : s(z ) (z )s(z ). The space H 1 (End(M )) is described as the quotient of the space of matrix-valued polynomial functions by two subspaces: Endout = {(z ) M atn [z ]|(z ) = const} and Endin = {(z ) M atn [z ]|(A( ))) = ( )(B ( ))( )-1 }. Elements of H 1 (End(M )) are treated as tangential vectors to the moduli space of holomorphic bundles at M . The infinitesimal deformation corresponding to an element (z ) is defined by the formula
(z )

( ) = (A( ))( ) - ( )(B ( )).

(1.11)

Global section H 0 (End(M ) K) are described by the expressions: (z ) = Res where Res (( )( )( )-1 - ( )) = 0 and ( ) = i i i1 is a polynomial matrix-valued function. This expression also implies a decomposition +1 of the denominator in the geometric progression. It turns out that all global section of H 0 (End(M ) K) are of this form. ( )-1 ( )( ) ( ) dz - dz , z - A( ) z - B( ) (1.12)

8


Symplectic form on the cotangent bundle to the moduli space of holomorphic bundles can be described in terms of Hamiltonian reduction with respect to the adjoint action of GLn of the symplectic form on the space of pairs ( ), ( ), given by the expression: Res T rd(( )-1 ( )) d( ). (1.13)

Integrability The Hitchin system on proj in now defined as a system with phase space which is the Hamiltonian quotient of the space of pairs ( ), ( ). The simplectic form is given by the formula 1.13. The reduction is considered with respect to the adjoint action of the group GLn . The Lax operator is defined by the formula 2.42. The Hamiltonians are defined by the coefficients of the function T r((z )k ) sub ject to some basis of holomorphic k -differentials (i.e. sections of H 0 (Kk )). Let us remark that z , w, k , l the following commutativity condition is fulfilled: T r((z )k ) and T r((w)l ) commute on the nonreduced space. The integrability proof realizes the r-matrix technique. Example 1.1. Consider a rational curve with a double point z1 z2 (the ring of rational functions on a curve is a subring of rational functions f on CP 1 satisfying the condition f (z1 ) = f (z2 )) with one marked 1 1 point z3 . The dualizing sheaf has the global section dz ( z-z1 - z-z2 ). Consider the moduli space M of holomorphic bund les E of rank n on node with fixed trivialization at z3 . There is the fol lowing isomorphism of linear spaces TE M = H 1 (End(E ) O(-p)). Let us restrict ourself to the open sel l of the moduli space of equivalence classes of matrices with different eigenvalues. The cotangent space is isomorphic to the space of holomorphic sections of End (E ) K O(p). This space can be realized by the space of rational matrix-valued functions on z of the fol lowing type (z ) = with the fol lowing conditions on residues 1 =
2

1 2 3 - + z - z1 z - z2 z-z

dz ,
3

1 - 2 + 3 = 0 .

The phase space of the system is parameterized by elements of U GL(n) giving a trivialization at z3 , matrix describing the projective module over O(mode ), residues of the Higgs field i . In these coordinates the canonical simplectic form on T M can be expressed as fol lows = T r(d(
-1

1 ) d) + T r(d(U

-1

3 ) dU ).

After Hamiltonian reduction with respect to the group GL(n) action (the right action on U and the adjoint action on i , ) one obtains the space parameterized by the matrix elements (3 )ij = fij , i = j ; eigenvalues e2xi of the matrix and the diagonal elements of the matrix (1 )ii = pi with the fol lowing Poisson structure {xi , pj } = ij , {fij , fkl } = j k fil - il fkj .

The Hamiltonian of the trigonometric spin Calogero-Moser system related with the finite-zone solutions of the matrix generalization for the KP equation [34] can be obtained as the coefficient of T r2 (z ) at 1/(z - z1 )2
n

H = T r 2 = 1
i=1

p2 - 4 i
i=j

fij fj i . sinh2 (xi - xj )

1.4
1.4.1

The Gaudin mo del
The Lax op erator

The Gaudin model was proposed in [9] (section 13.2.2) as a limit of the XXX Heisenberg magnet. It describes a one-dimensional chain of interacting particles with spin. The Gaudin model can be considered

9


as a generalization of the Hitchin system for the rational curve = CP 1 with N marked points z1 , . . . , zN . The Higgs field (the Lax operator) can be represented by a rational section = L(z )dz where L(z ) =
i=1...N

i . z - zi

(1.14)

The residues of the Lax operator i are matrices n â n those matrix elements lie in gln . . . gln . (i )kl coincides with the k l-th generator of the i-th copy of gln . The generators of the Lie algebra are interpreted as functions on the dual space gl . The symmetric algebra S (gln )N C[gl . . . gl ] is equipped with n n n the Poisson structure given by the Kirillov-Kostant bracket: {(i )kl , (j )mn } = ij ( 1.4.2 R-matrix bracket
lm

(i )kn - nk (i )ml ).

R- matrix representations of Poisson structures turned out to be a key element of the theory of quantum groups. In some sense the existence of an R-matrix structure is equivalent to integrability. It should be noted that, in theory of quantum groups [35] the important concept is the so-called quasitriangular or braided bialgebra. Let us introduce the notations: · {ei } - the standard basis in Cn ;
j · {Eij } - the standard basis in End(Cn ), (Eij ek = k ei );

· eij - generators of the s-th copy gln N gln . The Lax operator can be represented as
N

( s)

L(z ) =
ij

Eij
s=1

eij . z - zs

(s)

The Poisson structure can be described in terms of generating functions: {L(z ) L(u)} = [R12 (z - u), L(z ) 1 + 1 L(u)] End(Cn )2 S (gln )N , with the classical Yang R-martix R(z ) = P12 , z P12 v1 v2 = v2 v1 , P
12

=
ij

Eij Ej i .

1.4.3

The integrals

The integrals of motion can be retrieved as the characteristic polynomial coefficients
n

det(L(z ) - ) =
k=0

Ik (z )

n-k

.

(1.15)

It is often used the alternative basis of symmetric functions of eigenvalues of the Lax operators Jk (z ) = T rLk (z ), k = 1, . . . , n.

Traditional quadratic Hamiltonians can be obtained as follows H
2,k

= Resz

=zk

T rL2 (z ) =
j =k

2T rk j =2 (zk - zj )

(k ) (j ) lm elm eml j =k

zk - z

.

j

They describe the magnet model that consists of a set of pair interaction particles. It is known that 10


Prop osition 1.2. The coefficients of the characteristic polynomial of L(z ) commute with respect to the Kiril lov-Kostant bracket {Ik (z ), Im (u)} = 0. Let us present here the baseline of the proof. Pro of Let L1 (z ) = L(z ) 1 and L2 (u) = 1 L(u). {Jk (z ), Jm (u)} = T r12 {Lk (z ) Lm (u)} = Tr = Tr + Tr - Tr - Tr In particular, (1.16) + (1.17) = T r12 [
ij 12 ij 12 ij 12 ij 12 ij 12 ij m Li (z )Lj (u)R12 (z - u)Lk-i (z )L2 -j 1 2 1 -1 m (z )L2 -j -1

Li (z )Lj (u){L(z ) L(u)}Lk 1 2 1

-i-1

(u) (1.16)

(u)

Li (z )Lj (u)R12 (z - u)Lk 1 2 1

-i-1

(z )Lm-j (u) 2 (z )Lm-j 2
-1

Li+1 (z )Lj (u)R12 (z - u)Lk 1 2 1

-i-1

(u) (u).

(1.17)

Li (z )Lj +1 (u)R12 (z - u)Lk 1 2 1

-i-1

(z )Lm-j 2

-1

Li (z )Lj (u)R12 (z - u)Lk 1 2 1

-i-1

m (z )L2 -j

-1

(u), L1 (z )].

The last expression is zero because it is trace of a commutator. 1.4.4 Algebraic-geometric description

This section describes the basic algebraic-geometrical components of the generalized Hitchin system for curves with marked points. Namely it is constructed a pair {, L} - the spectral curve and the line bundle on it, which allows to solve the classical Gaudin model. Sp ectral curve ~ The spectral curve of the Gaudin system is described by the equation det(L(z ) - ) = 0. (1.18)

To build a nonsingular compactification of that curve one should consider the total space of the bundle where the Lax operator takes values (z ) = L(z )dz H 0 (CP 1 , End(On ) ) where = K(k ) = O(k - 2). We define a compactification of by the equation det((z ) - ) = 0. (1.20) (1.19)

This curve is a subvariety of the rational surface Sk-2 , obtained by compactification of the total space of the line bundle O(k - 2) over CP 1 , or as a pro jectivisation P (O(k - 2) O) over the rational curve. This rational surface contains three types of divisors: E - the infinite divisor, C - the fiber of the bundle and E0 - the base curve with the following intersections E0 · E0 = k - 2, E0 · C = 1 , C · C = 0, E · C = 1 . 11


To determine the genus of the curve we use the adjunction formula. First let us calculate the canonical class of Sk-2 . It corresponds to the class of divisors KS
k -2

= -2E0 + (k - 4)C.

Let the class of be equal to [] = n1 E0 + n2 C. is n-folded covering of CP 1 . Hence [] · C = n and n1 = n. To calculate n2 it is sufficient to use the fact that is a spectral curve of a holomorphic section of End(On ) and hence does not intersect E . We obtain [] = nE0 . By the adjunction formula we have 2g - 2 = KS
k -2

· [] + [] · []
0

= (-2E0 + (k - 4)C ) · nE0 + n2 E0 · E = -2(k - 2)n + (k - 4)n + (k - 2)n2 . This allows to calculate the genus of the spectral curve g () = Line bundle (k - 2)n(n - 1) - (n - 1). 2

(1.21)

Let us recall the sequence defining the line bundle on the spectral curve
n n 0 L O OS ((k - 2)C + E )| F 0,

(1.22)

where L and F are line bundles. We also obtain the following
n n (L) = (O ) - (OS ((k - 2)C + E )| ) + (F )

Let us denote the divisor (k - 2)C + E



S as D. Then

n (O ) = n (OS (D)| ) = = (F ) =

n(1 - g ), nD · [] + n(1 - g ) n2 (k - 2) + n(1 - g ), n ( On (k - 2) OS (E )) - (OS ) 1 = n (D · D - D · KS ) = n(k - 1). 2

(1.23)

Hence (L) = -n2 (k - 2) + n(k - 1). Calculating the number of branching points = 2(g + n - 1) = (k - 2)(n2 - n) we obtain Lemma 1.3. deg (L) = g + n - 1 - .

The dimension of the commutative family is given by the equation R(z , ) = 0, where

On the affine chart without {zi } and the spectral curve
n-1

R(z , ) = (-1)n n +
m=0

m Rm (z ),

(1.24)

k n-m

Rm (z ) =
i=1 l=1

Rm,i . (z - zi )l

(l )

12


+1) The number of free coefficients is equal m=0 k (n - m) = k n(n2 . The central functions (the symmetric polynomials of eigenvalues for corresponding orbits) are the highest coefficients of Rm (z ) of the total number k n - 1. The Lax operator has double zero at infinity

n-1

L(z ) =

1 z2

i zi + O (
i

1 ). z3

It follows that Rm (z ) has zero of order 2(n - m) at infinity. This observation, in turn, imposes additional
n-1

(2(n - m) - 1) = n
m=0

2

conditions on the values of Hamiltonians. Thus, the dimension of the commutative family is k n(n - 1) n(n + 1) - k n + 1 - n2 = k - n2 + 1 = g . 2 2

1.5

Separated variables

For the wide class of integrable systems the separated variables are associated with the divisor of the line bundle L on the spectral curve. Namely pairs of coordinates of the divisor points are separated. Typically, the divisor is the divisor for the Baker function. A construction of separated variables for some class of integrable systems is given in [36]. In the case of sl2 -Gaudin model separated variables were known before [37], and can be found even more explicitly. 1.5.1 sl2 -Gaudin mo del

Let us remind that sl2 -Gaudin model is obtained from the gl2 model (1.14) choosing orbits with tr = 0. The Lax operator in this case is: L= A(z ) B (z ) C (z ) -A(z ) .

We will consider the characteristic polynomial as a function of parameters z , and values of the Hamiltonians: det(L(z ) - ) = R(z , , h1 , . . . , hd ). Let us define the variables yj as zeroes of C (z ). For dual variables we take wj = A(yj ). This set of variables defines the Darboux coordinates of the phase space: {yi , wj } = ij . Let us consider the generating function S (I , y ) of the canonical transformation from the variables yj , wj to the "`action-angle"' variables Ij , j wj = yj S, j = Ij S.

The point with coordinates (yj , wj ) is a point of the spectral curve by definition. The fact that "`action"' variables are functions of Hamiltonians allows to separate variables in the problem of finding the canonical transformation S S (I , y1 , . . . , yd ) =
i

s(I , yi ),

where each factor s(I , z ) solves the equation R(z , z s, h1 , . . . , hd ) = 0. 13


2

The quantization problem

The quantization problem has physical motivation, it is related to the quantum paradigm in modern physics. In mathematical context this problem can be formulated by different manners: in [38] it was considered the problem of deformation of an algebra of functions on a simplectic manifold satisfying the so-called "`correspondence principle"'. The particular case of the deformation quantization for the cotangent bundle to a Lie group used in these lectures was considered in [39]. Further the methods of deformation quantization, -product, the Moyal product and the geometric quantization was generalized to wider class of examples. One of the structure results in this field was the formality theorem by M. Kontsevich [40] which demonstrates the existence of the quantization. Another ensemble of important results in this domain are due to Fedosov [41]. In this work it is proposed radically more strong quantization problem, demanding not only the deformation of an algebra of functions but of a pair: Poisson algebra + Poisson commutative subalgebra, representing an integrable system. Let us call this task the algebraic part of integrable system quantization. Moreover it is stated a problem of constructing quantum analogs of the essential geometric ob jects from the point of view of algebraic-geometric methods in integrable systems. In general the problem is to find an associative deformation of a Poisson algebra such that the Poisson-commutative subalgebra remains commutative, and moreover the deformation of the spectral curve provides quantum separated variables. The last part of the quantization problem is called "`algebraic-geometric"' quantization.

2.1
2.1.1

The deformation quantization
Corresp ondence

The traditional scheme of deformation quantization supposes a construction of an associative algebra starting with a Poisson algebra. A Poisson algebra is a commutative algebra Acl with multiplication denoted by ·, furnished by an antisymmetric bilinear operation called the Poisson bracket {, }, such that Acl is a Lie algebra and both structures are compatible by the Leibniz rule: {f , g · h} = {f , g } · h + g · {f , h}. A Poisson algebra is an infinitesimal version of an associative algebra. Due to the so-called Drinfeld construction it is not hard to note that the space Acl []/2 with multiplication f g = f · g + {f , g } is an associative algebra. The quantization of the Poisson algebra Acl with the structure defined by operations (·,{, }) which is called the algebra of classical observables is an associative algebra A with multiplication (), satisfying the following conditions: A Acl [[h]] as linear spaces.

Moreover if identified the algebra of classical observables and the space of constants in A the following structure compatibility is required: a b = a · b + O(h), a b - b a = h{a, b} + O(h2 ). The map lim : A - Acl : is called the classical limit. Example 2.1. Let us consider the Poisson algebra S (gln ), on the space of symmetric algebra of the Lie algebra gln defined by the Kiril lov-Kostant bracket. This has a canonical quantization, realizing the concept of the deformation quantization: let Uh (gln ) be the deformed universal enveloping algebra Uh (gln ) = T (gln )[[h]]/{x y - y x - h[x, y ]}. The classical limit is defined as the limit h 0 which is correctly defined on the family of algebras Uh (gln ). The existence of a limit fol lows from the common Poincare-Birkhof-Witt basis for this family. 14 h0


2.1.2

Quantization of an integrable system

An integrable system is a pair: a Poisson algebra Acl and a Poisson commutative subalgebra Hcl of the dimension dim(S pec(Hcl )) = 1/2dim(S pec(Acl )). An algebraic problem of quantization is the following correspondence Hcl Acl H A satisfying the conditions ·A Acl [[h]] as linear spaces, the map lim : A Acl is called the classical limit;

· H is commutative; · lim : H = H
cl

Remark 2.1. In the case of quantization for the symmetric algebra of the Lie algebra gln the correspondence can be simplified. Let us consider U (gln ), which is a filtered algebra (the filtration is given by degree) {Fi }. The projection map to the associated graded algebra induces a Poisson structure: U (gln ) Gr(U (gln )) = i Fi /Fi
-1

= S (gln ).

(2.1)

We wil l associate this map with the classical limit operation. On generators a Fi and b Fj the induced commutative multiplication and the Poisson bracket are given by the fol lowing expressions: a · b = a b mod Fi
+j -1

,

{a, b} = a b - b a mod Fi

+j -2

.

2.1.3

The Gaudin mo del quantization problem

The classical part is defined by the following ob jects A H
cl cl

= S (gln )N C[gl . . . gl ], n n - the subalgebra generated by the Gaudin Hamiltonians(1.15).

The algebraic part of the quantization problem is reduced to constructing a pair with the quantum observables algebra coinciding with the tensor power of the universal envelopping algebra: A = U (gln )N , such that the commutative subalgebra H is a deformation of the subalgebra generated by the classical Gaudin Hamiltonians.

2.2
2.2.1

Quantum sp ectral curve
Noncommutative determinant

Let us consider a matrix B = ij Eij Bij those elements are elements of some generally speaking not commutative associative algebra Bij A. We will use the following definition for the noncommutative determinant in this case det(B ) = 1 n! (-1) B
,
n

(1), (1)

...B

(n), (n)

.

This definition is the same as the classical one for matrices with commuting elements. There is an equivalent definition. Let us introduce the operator An of the antisymmetrization in (Cn )n An v1 . . . vn = 1 n! (-1) v
S
n

(1)

. . . v

( n)

.

15


The definition above is equivalent to the following det(B ) = T r
1...n

An B1 . . . Bn ,

where Bk denotes an operator in End(Cn )n A given by the inclusion Bk =
ij

1 ... E

ij

. . . 1 Bij ,

k

the trace is taken on End(Cn )n . 2.2.2 Quantum sp ectral curve

Let us call a quantum Lax operator for the Gaudin system the following expression:
N

L(z ) =
ij

Eij
s=1

eij . z - zs

(s)

L(z ) is a rational function on a variable z with values in End(Cn ) U (gln )N . Let us define a quantum characteristic polynomial of the quantum Lax operator by the formula
n

det(L(z ) - z ) =
k=0

n QIk (z )z

-k

.

(2.2)

The following theorem says that this generalization of the classic characteristic polynomial (1.15) allows to construct quantum Hamiltonians. Theorem 2.1 ([42]). The coefficients QIk (z ) commute [QIk (z ), QIm (u)] = 0 and quantize the classical Gaudin Hamiltonians in the fol lowing sense lim(QIk ) = Ik . The proof of this fact uses significant results of the theory of quantum groups such as the construction of the Yangian, the Bethe subalgebras and generally fits into the concept of quantum inverse scattering method. The following sections introduce the necessary definitions and provides an outline of the proof of the theorem of quantization of the Gaudin model. 2.2.3 Yangian

This Hopf algebra was constructed in [23] and plays an important role in the problem of description of rational solutions of the Yang-Baxter equation. Y (gln ) first and foremost is an associative algebra generated (k ) by the elements tij (in this section i = 1, . . . , n; j = 1, . . . , n; k = 1, . . . , ). Let us introduce the generating function T (u, h) Y (gln ) End(Cn )[[u-1 , h]], which takes the form T (u, h) =
i,j

Eij tij (u, h),

tij (u, h) = ij +
k

t

(k) k -k ij h u

,

where Eij are the matrix unities in End(Cn ). The relations can be written with the help of the Yang R-matrix R(u) = 1 - h u Eij E
i,j ji

16


and take the form R(z - u, h)T1 (z , h)T2 (u, h) = T2 (u, h)T1 (z , h)R(z - u, h). Both parts are regarded as elements of End(Cn )2 Y (gln )[[z the rational function
1 z -u -1

(2.3)

, z, u

-1

, u, h]],

in the R-matrix formula has an expansion 1 = z-u


u z

l

l+1

.

l=0

We use the following notations T1 (z , h) =
i,j

Eij 1 tij (z , h),

T2 (u, h) =
i,j

1 Eij tij (u, h).

The Yangian is a Hopf algebra those comultiplication is given in terms of the generating function by the following formula (id )T (z , h) = T 1 (z , h)T 2 (z , h), where we use the notations T 1 (z , h) =
i,j

Eij tij (z , h) 1,

T 2 (z , h) =
i,j

Eij 1 tij (z , h).

The evaluation representation Let us remind the construction of the so-called evaluation homomorphism : Y (gln ) U (gln ). To do this we consider a rational function on u, h with values in End(Cn ) U (gln ) given by the formula h h def Tev (u, h) = 1 + , (2.4) Eij eij = : 1 + u i,j u where eij are the generators of gln . Tev (u, h) satisfy RTT relations (2.3), hence the map {tij eij ; tij 0 with k > 1} determines an algebra homomorphism. Let us consider the tensor product U (gln )N [[h, h-1 ]] and the generating function (2.4) for the evaluation l representation to the l-th component of the product Tev (u - zl , h). It turns out that for an arbitrary set of complex numbers (z1 , . . . , zN ), the expression
1 2 k T (u, h) = Tev (u - z1 , h)Tev (u - z2 , h) . . . Tev (u - zN , h), (1) (k )

(2.5)

which is a rational function on u and h with values in End(Cn ) U (gln )N , determines a homomorphism : Y (gln ) U (gln )N [[h, h-1 ]]. More precisely, the following lemma is true. Lemma 2.2. The map, defined on the Yangian generators tij as the ij -th matrix element of the expansion coefficient of T (u, h)h-k at u-k in u = gives an algebra homomorphism : Y (gln ) U (gln )N [[h, h-1 ]]. This lemma follows from the properties of the comultiplication homomorphism and the evaluation homomorphism.
(k )

17


2.2.4

The Bethe subalgebra

This subalgebra is closely related with Quantum inverse scattering Method (QISM) [21, 44, 45], namely its generators are quantum integrals of the Heisenberg XXX model [44, 43]. Here we use the description from [46] (section 2.14): let us consider an n â n-matrix C and T (u, h) - a generating function for the Yangian n generators Y (gln ). Let us also use the notation An for the antisymmetrization operator in (Cn ) and the n following elements of End(Cn ) Y (gln )[[u, u-1 , h]] Tm (u, h) =
ij

1 . . . 1 Eij 1 . . . 1 tij (u, h).

m

It turns out [46] (section 2.14), that the expressions of the form k (u, h) = T rAn T1 (u, h)T2 (u - h, h) . . . Tk (u - h(k - 1), h)Ck
+1

. . . Cn

(2.6)
-1

for k = 1, . . . n, which are called the Bethe generators, constitute a commutative family in Y (gln )[[u, u in the following sense: [i (u, h), j (v , h)] = 0.

, h]]

In addition, this family is maximal if the matrix C has simple spectrum. The trace in the formula 2.6 is n meant over matrix components End(Cn ) , the series expansion of Tm (u - h(m - 1), h) is realized at u = , for example hm 1 = , u - h m=0 um+1 Next we will consider an identity matrix C and images of the Bethe generators with the evaluation homomorphism. For simplicity, we refer to the same letters
k (u, h) = T rAn T1 (u, h)T2 (u - h, h) . . . Tk (u - h(k - 1), h) k = 1, . . . n.

(2.7)

2.2.5

The commutativity pro of

The presence of the comultiplication structure in the theory of quantum groups allows to use the so-called "`fusion"' method to construct non-trivial integrable systems. Literally, the method is as follows: let us consider the image T (z ) by the evaluation homomorphism in composition with enough comultiplication operations z1 . . . zN N -1
1 T (z ) = Tz1 (z ) . . . T N zN

(z ) End(Cn ) U (gln )N .

The image of the Bethe subalgebra raises to some commutative subalgebra which can be described by the generating function:
Q(z , h) = T rAn (e-hz T1 (z , h) - 1) . . . (e-hz Tn (z , h) - 1) n

=
j =0

j (z - h, h)(-1)n

-j

j Cn e-j hz

= det(e-hz T (z , h) - 1).

(2.8)

The expression (2.8) can be represented as a series of z . From the commutativity of the Bethe generators it follows that the coefficients of this series which are rational functions on u with values in U (gln )N [[h]] also commute at different values of the parameter u. Hence the lowest coefficients on h also commute. These are exactly the coefficients of the characteristic polynomial of the Gaudin model. It turns out that the highest coefficient of the expression (2.8) on h has the form det(e-hz T (z , h) - 1) = hn det(L(z ) - z ) + O(hn in virtue of the expansion: e-hz T (z ) - 1 = h(L(z ) - z ) + O(h2 ). 18
+1

)


Remark 2.2. It should be noted that the independence of the quantum Hamiltonians directly fol lows from the independence of their classic limits, since the algebraic relations on the constructed operators in U (gln )N induces a nontrivial relation on their symbols. The maximality fol lows from the maximality on the classical level.

2.3

Traditional solution metho ds

The traditional methods of solving quantum integrable system on finite scale are reduced to the Bethe ansatz method or the method of separated variables which in turn allow to express the condition on the quantum model spectrum in terms of the solutions of some system of algebraic equations or the monodromy properties of some Fuchsian system. Those methods do not suppose any way of solving the substituting problems. However there is quite rich material in solving quantum integrable systems in various limits. Further we explain two basic methods in the case of the simplest Gaudin model. 2.3.1 Bethe ansatz

Let us consider the quantum sl2 Gaudin model. The Lax operator in this case takes the form L= where i = h i /2 ei fi -hi /2 . A(z ) B (z ) C (z ) -A(z ) =
i

i , z - zi

The quantum characteristic polynomial is a differential operator of the second order with values in the algebra of quantum observables:
2 det(L(z ) - z ) = z -

1 2

i

ci - (z - zi )2

(2)

i

Hi . z - zi

The Gaudin Hamiltonians are the residues Hi =
i=j

hi hj /2 + ei fj + ej fi . zi - zj commutative subalgebra but of trivial model but fits well for a wide class of in [9]. Let us observe the construction. . . . VN where Vi are the finite

The coefficients at the poles of second order are also elements of the nature - they are central in the quantum algebra. The Bethe ansatz method was firstly proposed for the Heisenberg systems. The Bethe ansatz method for the Gaudin model was realized We consider the sl2 Gaudin model in fixed representation V = V1 dimensional irreducible representations of highest weights i . Lemma 2.3. The vector =
j =1 M

C (µj )|v ac >

is the common eigenvector for the ensemble of Gaudin Hamiltonanians if the set of parameters µj (cal led the Bethe roots) satisfies the system of Bethe equations - 1 2 i + µj - zi 1 = 0, µj - µk j = 1, . . . , M . (2.9)

i

k=j

The eigenvalues of Hi on the vector are expressed as follows 1 1 j Hi = -i - . zi - µj 2 zi - zj j
j =i

19


Pro of In this case the quantum characteristic polynomial takes the form:
2 2 det(L(z ) - z ) = z - A2 (z ) - C (z )B (z ) + A (z ) = z - H (z ).

The following commutation relations on the matrix elements of the Lax operator are also true: [A(z ), B (z )] = -B (z ), [A(z ), C (z )] = C (z ), Using this relation and the condition: H (z )|v ac >= we obtain: H (z ) = h0 (z ) + 2
j =1 M M

[A(z ), C (u)] =

1 (C (z ) - C (u)), z-u 2 [B (z ), C (u)] = (A(z ) - A(u)). u-z

1 ( 4

i

i 2 1 )- z - zi 2

i

i (z - zi )2

|v ac >= h0 (z )|v ac >

1 A(z ) + µj - z
l=j

+ 2C (z )
j =1

1 z - µj

1 (µj - z )(µk - z ) j =k 1 C (µl ) + A(µj ) . µk - µj
k=j

Let us remark that the Bethe equations can be rephrased in the form: 1 + A(µj ) = 0. µk - µj

k=j

This proves the lemma. 2.3.2 Quantum separated variables

Let us consider the quantum sl2 Gaudin model as in the previous section. An irreducible representation of this type can be realized as the quotient of the Verma module C[ti ]/ti +1 , such that the generators of sl2 i act as differential operators: h
( s)

= -2t

s

+ s , e( ts

s)

= -t

s

2 + s ,f t2 ts s

( s)

= ts .

Let us explore the problem in the tensor product of the Verma modules which is realized in this case on the space of polynomials on N variables C[t1 , . . . , tN ]. Let us introduce the set of variables yj , defined by the formula: C (z ) = C0
j i

(z - yj ) (z - zi )

.

They are elements of some algebraic extension of the ring C[t1 , . . . , tN ]. Let us denote by the same symbols functions and operators of multiplication by those functions. Let be a common eigenvector for the Gaudin Hamiltonians in C[t1 , . . . , tN ] H (z ) = h(z ). (2.10)

Considering both parts of 2.10 as rational functions on z and substituting z = yj from the left we obtain: H (yj ) = A2 (yj ) - A (yj ) 1 1 1 = hi hk + 4 (yj - zi )(yj - zk ) 2
i,k

k

1 hk . (yj - zk )2

(2.11)

20


Using the definition of the separated variables let us express the partial derivatives: yj =
k

tk t = yj k

k

tk t . y j - zk k

(2.12)

Substituting 2.12 in 2.11 we obtain: -y 1 + 2 k yj - z
2

j

= h(yj ).
k

k

Hence the common eigenfunction for the Gaudin Hamiltonians factorizes, its dependence on yj is separated: =
j

(yj ).

Each of the factors (z ) is related to the solution for the Sturm-Liouville equation
2 (z - h(z )) (z ) = 0

as follows: (z ) =
i

(z - zi )-i /2 (z ).

2.3.3

The mono dromy of Fuchsian systems

The results of traditional separation of variables in quantum integrable systems discussed above demonstrate that the spectrum description is closely related with the families of Fuchsian equations obeying special monodromy properties. These properties are quite natural in the Heisenberg approach explained in [47], and correspond to existence of globally defined wave-functions. In the considered sl2 Gaudin model it was obtained that if is a common Bethe eigenvector with values Hi then the equation 2 - has a solution of the form (z ) =
i

1 4

i

i (i + 2) - (z - zi )2

i

Hi z-z

(z ) = 0
i

(2.13)

(z - zi )-i

/2 j

(z - µj ),

where the set of parameters µj satisfy the system of Bethe equations. This observation was generalized in [48]. Let us consider the quantum characteristic polynomial:
2 det(L(z ) - z ) = z - i

Ci - (z - zi )2

(2)

i

Hi . z - zi

Let H be the algebra generated by the coefficients of the quantum characteristic polynomial. A character (2) 1 of the algebra H is called admissible if it takes values (Ci ) = 4 (i + 2)i on central elements. Theorem 2.4 ([48]). There is a one-to-one correspondence between the set of "`admissible"' characters for those the differential equation (det(L(z ) - z ))(z ) = 0 has monodromy ±1, and the set of common eigenvectors of the Gaudin model in the representation V . In contrast with the traditional Bethe ansatz and separation of variables methods this spectrum characterization can be generalized to the sln case. 21


2.4

Elliptic case

It turns out that the elements of the algebraic-geometric part of the quantization problem can the constructed also in the case of the Elliptic Gaudin model: the quantum spectral curve and the quantum separated variables. Let us remark that the elliptic Gaudin model can be obtained in generalized Hitchin system framework. This corresponds to the moduli space of holomorphic semistable bundles with the trivial determinant bundle over an elliptic curve with a set of marked points. A modified algebraic structure is applicable for this problem, namely the dynamical gln RLL equation corresponding to the "`elliptic quantum group"' E , (gln ), defined in [50]. The commutativity in this case is meant modulo the Cartan subalgebra. To obtain an integrable system one should restrict the constructed family to the zero weight subspace with respect to the diagonal action of the Lie algebra. 2.4.1 The notations

Let us define the so-called odd Riemann -functions on an elliptic curve. Let C, Im > 0 be the parameter of elliptic curve C/, where = Z + Z - is the periods lattice. The odd -function (u) = -(-u) is defined by the relations (u + 1) = -(u), (u + ) = -e-2
iu- i

(u),

(0) = 1.

(2.14)

Let us also introduce some matrix notations. Let T=
j

tj · a1,j . . . aN

,j

be a tensor over an algebra R, where tj R and ai,j are elements of the space End Cn . Then the notation T (k1 ,...,kN ) corresponds to the following element of R (End C n )M for numbers M N : T
(k1 ,...,kN )

=
j

tj · 1 . . . a1,j . . . aN ,j . . . 1. ki -th tensor component, the numbers ki are pairwise different and M. = F (1 , . . . , n ) be a function on n parameters k , taking values in case we define special shifts

Here each element ai,j is placed in the the following condition fulfills 1 ki We need also the notation: let F () an algebra R: i.e. F : Cn R. In this

F ( + P ) = F (1 + P1 , . . . , n + Pn )


=
i1 ,...,in =0

1 i1 ! · · · in !

i1 +...+i

F (1 , . . . , n ) P · · · in n
n

i1 1

i 1

1

i · · · Pnn

(2.15)

for some set P = (P1 , . . . , Pn ), Pk R. We do not discuss here the convergency questions, in our context all such expressions will be well defined. 2.4.2 Felder algebra

Let us introduce the notion of the elliptic L-operator, corresponding to the Felder R-matrix. We use the notations {ei }, {Eij } from the section 1.4.2. Let h be a commutative algebra of dimension n. In [50] it was constructed an element of End Cn End Cn , meromorphly depending on the parameter u and n dynamical parameters 1 , . . . , n : R(u; ) = R(u; 1 , . . . , n ) = (u + ) (u)
n

Eii Eii +
i=1

+
i=j

(ij + ) Eii E (ij )

jj

+

(u - ij )( ) Eij E (u)(-ij )

ji

, (2.16)

22


where ij = i - j . This element is called the dynamical Felder R-matrix. This satisfy the dynamical Yang-Baxter equation R =R
(12) (23)

(u1 - u2 ; )R(13) (u1 - u3 ; + E (2) )R (u2 - u3 ; + E (1) )R(13) (u1 - u3 ; )R

(23) (12)

(u2 - u3 ; ) = (u1 - u2 ; + E

(3)

),

and the additional conditions R
(21)

(-u; )R +E +

(12)

(u; ) =

(u + )(u - ) , (u)2
(1) ii (1) )(D

(1) ii (1) (D

(E

(2) ii (2) D

)R(u; ) = R(u; )(E )R(u; ) = R(u; ( i) , D = k

+E +

(2) ii ), (2) D ),

where D =

n

n

E
k=1

kk

E
k=1

( i) kk

. k

We should mention that the expression in formulas above denotes a vector 1 , . . . , n , and the expression ( s) + E (s) implies a shift of the type 2.15 with the parameters values Pi = Eii . Let R be a C[[ ]]-algebra, L(u; ) an invertible n â n matrix over R depending on the spectral parameter u and n dynamical parameters 1 , . . . , n . Let h1 , . . . , hn be a set of pairwise commuting elements of R. L(u; ) is called an elliptic dynamical L-operator corresponding to the set of Cartan elements hk if L(u; ) satisfies the dynamical RLL relation R(12) (u - v ; )L(1) (u; + E = L(2) (v ; + E (1) )L(1) (u; )R and a condition of the form (Eii + hi )L(u; ) = L(u; )(Eii + hi ). Let us introduce an equivalent but more symmetric form of RLL relations. For an L-operator we define an expression: LD (u) = e- D L(u; ). (2.18) The equation (2.17) can be rewritten in new notations as follows: R
(12) (2)

(12)

)L(2) (v ; ) (u - v ; + h),

(2.17)

(u - v ; )LD (u)LD (v ) = LD (v )LD (u)R

(1)

(2)

(2)

(1)

(12)

(u - v ; + h).

(2.19)

The next lemma plays the role analogous to the fusion method in the rational case, namely this describes a method of elliptic L-operators construction. Lemma 2.5. If L1 (u; ) End(Cn ) R1 and L2 (u; ) End(Cn ) R2 are two el liptic dynamical Loperators with respect to two sets of Cartan elements: h1 = (h1 , . . . , h1 ) and h2 = (h2 , . . . , h2 ), then the n n 1 1 product L2 (u; )L1 (u; + h2 ) End(Cn ) R1 R2 is also an el liptic dynamical L-operator with respect to the set h = h1 + h2 = (h1 + h2 , . . . , h1 + h2 ). Hence, if L1 (u; ), . . . , Lm (u; ) are el liptic dynamical n n 1 1 L-operators with the sets of Cartan elements h1 , . . . , hm , then the matrix - Lj u; +
mj1 l=j +1 m m

h

l

(2.20)

is also an el liptic dynamical L-operator with the fol lowing set of Cartan elements h =
i=1

hi .

Remark 2.3. The arrow in the product notation above denotes the order of multipliers with growing indexes: - for example, the expression 3 i 1 Ai means A3 A2 A1 . 23


The main example of elliptic dynamical L-operator is given by the Felder R-matrix: L(u) = R(u - v ; ). In this case the second space End(Cn ) takes the role of the algebra R. Here v is a complex number and (2) the Cartan elements coincide with the diagonal matrices hk = Ekk . Lemma 2.5 allows to generalize this example: let v1 , . . . , vm be a set of complex numbers, then the matrix R
(0)

- (u; {vj }; ) =
mj1

m

R

(0j )

u - vj ; +
l=j +1 m (l )

E

(l )

(2.21)

is a dynamical elliptic L-operator with the Cartan elements hk =
l=1

Ek k .

A more general class of dynamical elliptic L-operators is related with the so-called smal l el liptic quantum ~ group e , (gln ) constructed in [51]. This represents a C[[ ]]((1 , . . . , n ))-algebra generated by tij and hk with relations ~ ~ tij hk = (hk - ik + j k )tij , tij k - (k - ik )tij = 0, tij tik - tik tij = 0, tik t (
{2} jl + {2 } (j l ) jk

(2.22) i = j, () )

-

( (

{1 } ij {1 } ij

+) -) )

tj k t

ik

= 0, (

)

tij tkl -

(

{1} ik + {1} (ik )

tkl tij -

{1} {2 } ik + j l ) {1} {2} (ik )(j l

til t

kj

= 0,

{1} {2} ~ with i = k , j = l, where tij = ij + tij , ij = i - j , ij = i - j - hi + hj , it is also supposed that h1 , . . . , hk , 1 , . . . , k commute. One constructs a generating function for these generators T (-u)

Tij (-u) = (-u + ij - hi )tj i . Representing this matrix in the form T (-u) = (-u)e-
n k=0

(2.23)

(hk +Ekk )

k

L0 (u; )e

n k=0

hk

k

(2.24)

we obtain a dynamical elliptic L-operator L0 (u; ) for the algebra T = e , (gln )[[ ]] with the Cartan elements h = (h1 , . . . , hn ), where C[[ ]] = C[[1 , . . . , n ]]. The elements k = k commute with hi and ~ do not commute with tij . 2.4.3 Commutative algebra

Let us consider a dynamical elliptic L-operator L(u; ) with a set of Cartan elements hk . This function takes values in the algebra End Cn R. Let us introduce the operators L[
m,N ]

({ui }; ) = e-

D

(m+1)

L(

m+1)

(u

m+1

; ) · · · e-

D

(N )

L(

N)

(uN ; ),

(2.25)

where m < N . Let us consider a particular case with the parameters values ui = u + (i - 1), L[
a,b]

(u; ) = L[

a,b]

({ui = u + (i - a - 1)}; )

for a < b. Let An = C((1 , . . . , n )) be the completed function space. The operators D act on the space An Cn , in turn the operators L[a,b] (u; ) act from An (Cn )(b-a) to the space An (Cn )(b-a) R: fixing u we obtain L[a,b] (u; ) End(Cn )(b-a) An , where An = An [e± ] R. Let us consider a subalgebra h R An generated by the elements hk and its normalizer An : Nn = N
An

(h) = {x An | hx An h}.

(2.26)

Let us remark that An h is a two-sided ideal in Nn . In [49] the following statement is proved 24


Theorem 2.6. Let us define An -valued functions tm (u) = tr A[0
,m]

L[0

,m]

(u; ) ,

(2.27)

where we suppose the trace operation over m spaces Cn . These expressions commute with the Cartan elements hk : hk tm (u) = tm (u)hk . (2.28)

Hence they are elements of the subalgebra Nn . Moreover these generators commute modulo the ideal An h Nn : tm (u)ts (v ) = ts (v )tm (u) 2.4.4 Characteristic p olynomial mod An h. (2.29)

As in the rational case the generators tm (u) can be organized into a generating function called the quantum characteristic polynomial. This generating function is constructed as a "`determinant"' of the corresponding L-operator. Prop osition 2.7. Let us consider the matrix M = e- generate the family tm (u) in the fol lowing sense: P (u, e
u D


L(u; )e
n

u

. Then the determinant of 1 - M

) = det(1 - e

-D



L(u; )e

u

)=
m=0

(-1)m tm (u)em

u

,

(2.30)

where t0 (u) = 1. This property induces the commutativity of the quantum characteristic polynomial with elements hk , and the pairwise commutativity modulo An h of the generating functions: [P (u, e 2.4.5

u

), hk ] = 0,

[P (u, e

u

), P (v , e

v

)] = 0

mod An h.

(2.31)

The limit and the Gaudin mo del

Let us consider degenerated elliptic dynamical RLL relations at 0. This limit describes the elliptic quantum Gaudin model. To do this we use a shift of the L-operator. The limit of the generating function for the generic family gives the generating function for the Hamiltonians of the elliptic Gaudin model. The result obtained generalizes the works [52],[53]. Let L(u; ) be a dynamical elliptic L-operator of the form L(u; ) = 1 + (u; ) + o( ), (2.32)

those matrix elements are elements of the algebra R0 = R/ R. The matrix (u; ) is called a classical dynamical elliptic L-operator. This satisfy the rLL-relations
n

[

(1)

(u; ) - D ,

(1)

(2)

(v ; ) - D ] -
k=1

(2)

h

k

r(u - v ; ) = = [
(1)

(u; ) +

(2)

(v ; ), r(u - v ; )]

(2.33)

with the classical dynamical elliptic r-matrix r(u; ) = +
i=j

(u) (u)

n

Eii E
i=1

ii

(ij ) Eii E (ij )

jj

+

(u - ij ) Eij E (u)(-ij )

ji

.

(2.34)

The matrix (2.34) is related with the Felder R-matrix (2.16) by the formula R(u; ) = 1 + r(u; ) + o( ). 25 (2.35)


Theorem 2.8. Let An = R0 An [ ] Nn = NAn (h) = {x An | hx An h}, where An = C((1 , . . . , n )). Let us define a set of Nn -valued functions sm (u) by the formula - D + (u; ) = Q(u, u ) = det u where s0 (u) = 1. They commute with the Cartan elements hk : hk sm (u) = sm (u)h and moreover pairwise commute modulo An h: sm (u)sl (v ) = sl (v )sm (u) mod An h. (2.38)
k n

sm (u)
m=0

u

n-m

,

(2.36)

(2.37)

The values of the functions s1 (u), s2 (u), . . . , sn (u) generate a commutative subalgebra in Nn on the level hk = 0. This means that the images of these elements with respect to the canonical homomorphism Nn Nn /An h pairwise commute. The quantum elliptic Gaudin model is defined with the help of the Lax operator ij (u; ) = ej i (u; ), those coefficients are expressed by the formulas: eii (u) = eij (u; ) = (u - z ) eii = (u - z ) (-1)m (u) m! (u)
(m)

ii (u; ) = eii (u; ) +
k=i

(ik ) hk , (ik )

(2.39)

eii z m ,
(m)

(2.40) eij z m . (2.41)

m0

(u - z + ij ) eij = (u - z )(ij )

m0

(-1)m (u + ij ) m! (u)(ij )

An analog of the evaluation representation is the homomorphism to the small elliptic quantum group defined by the generating function (2.24). Let us consider an expansion on the parameter of the dynamical Loperator corresponding to the tensor power of the small elliptic group. It turns out that the coefficient at of this expansion coincides with the elliptic Gaudin model L-operator. 2.4.6 The explicit form of the sl2 elliptic Gaudin mo del

The L-operator of the elliptic sl2 Gaudin model considered in [52, 53, 54] has the form (u; ) = where =
12

h(u)/2 f (u) e (u) -h(u)/2

,

(2.42)

= 1 - 2 and the currents are expressed by the formulas
N

h(u) = e11 (u) - e22 (u) =
s=1 N

(u - vs ) (s) ( s) (e - e22 ), (u - vs ) 11

e (u) = e12 (u; ) =
s=1 N

(u - vs + ) (s) e, (u - vs )() 12 (u - vs - ) (s) e. (u - vs )(-) 21

f (u) = e21 (u; ) =
s=1

The fact that the L-operator depends only on the difference = 1 - 2 allows to restrict the generating function of the commutative subalgebra Q(u, u ) to the space A = {a A2 | (1 + 1 )a = 0} A2 coinciding with C((12 )). Let A = R0 A[ ] then the values of sm (u) are elements of N = NA (h) = {x 26


A | hx Ah}. In virtue of the representation : h1 + h2 0 the operator D has the form H , where H = E11 - E22 . Let us find the quantum characteristic polynomial in this case: Q(u, u ) = det = det = u
u

() h = - D + (u; ) - u () 2
() ()

- + h+ (u)/2 - e+ (u)

h/2
u

+ f (u)

+ - h+ (u)/2 -

() ()

h/2 (2.43)

2

-

() h - S (u), () u

where h = h1 - h2 . S (u) is an N -valued function S (u) = - h(u)/2
2

+ u h(u)/2 + e (u)f (u)

mod Ah.

The commutativity condition can be formulated in terms of this generating function as follows: [S (u), S (v )] = 0 mod Ah. Using the commutation relations
+ [e+ (u), f (u)] = -

+ () h (u) + h u ()

one can simplify this generating function: S (u) = - h(u)/2
2

+ e (u)f (u) + f (u)e (u) /2

mod Ah.

(2.44)

3

Solution for quantum integrable systems

As was mentioned above the traditional methods of solving quantum integrable systems on the finite scale in some cases allow to solve the Hamiltonian diagonalization problem in terms of solutions of a system of algebraic equations (the Bethe system). However, the system of equations itself, in cases where it can be deduced, turns out to be quite complicated and hypothetically prevents algebraic solutions. In this section we use an equivalent formulation for quantum eigenproblem in terms of Fuchsian systems with special monodromy representation. In turn the construction of relevant Fuchsian systems uses the quantum characteristic polynomial of a model. This observation also emphasizes the quantum characteristic polynomial among others generating function for the commutative subalgebra.

3.1
3.1.1

Mono dromic formulation
A scalar and a matrix Fuchsian equations

Consider a Fuchsian system defined by a connection in trivial bundle of rank 2 on the disk with punctures: A(z ) = with residues satisfying the conditions: T r(Ai ) = 0; Det(Ai ) = -d2 ; i
i

a11 (z ) a12 (z ) a21 (z ) a22 (z )

k

=
i=1

Ai z-z

(3.1)
i

Ai =

0

0 -

.

(3.2)

The Fuchsian system is written by the equation (z - A(z ))(z ) = 0. 27 (3.3)


The components of this system may be represented as follows
1 2

= a11 1 + a12 2 , = a21 1 + a22 2 .

The first vector component satisfies the second order equation 1 = (a12 /a12 )1 + u1 , where u = a11 + a2 - a11 (a12 /a12 ) + a12 a21 . 11 With the following variable change: = 1 /, where = + U = 0, those potential is defined by the formula U = / - (a12 /a12 ) / - u. Introducing the expression for to U we obtain: U= 1 2 a12 a12 - 1 4 a12 a12
2



a12 , we obtain the equation

(3.4)

+ a11

a12 - a11 - a2 - a12 a21 . 11 a12

(3.5)

Let us suppose that a12 (z ) has no multiple poles a12 (z ) = c
k-2 j =1 (z k i=1 (z

- wj ) - zi )

.

We should note that the number of zeros agrees with the normalization (3.2). The expression for the logarithmic derivative can be simplified: a12 = a12 The potensial U taked the form
k-2 k-2

j =1

1 - z - wj

k

i=1

1 . z - zi

(3.6)

U=
j =1

-3/4 + (z - wj )2

k

i=1

1/4 + detAi + (z - zi )2

k-2

j =1

Hw j + z - wj

k

i=1

Hzi , z - zi

(3.7)

in which H 1 = a11 (wj ) + 2 = 1 + ai 1 2
1 j

w

j

i=j

1 - wj - wi

i

1 ; wj - zi

H

zi

1 - zi - wj

j =i

T r(Ai Aj ) + ai + aj + 1/2 11 11 . zi - zj

Let us remark that the coefficients at (z - zi )-2 take values 1/4 + detAi = (1/2 - di )(1/2 + di ). (3.8)

In what follows we identify these factors with the values of the quadratic Casimir elements of the Lie algebras sl2 in the representations of highest weights i (i = 2di - 1 in our case). 28


3.1.2

Dual equation

As was shown in previous calculations, the matrix form connection leads to the Sturm-Liouville operator with additional poles at points wj . A consideration of the second vector component of a solution of the matrix equation 2 leads to another scalar differential operator with poles at points zi and additional points wj , determined by the formula a21 (z ) = c
k-2 j =1 (z k i=1 (z

- wj ) - zi )

.

Let us call the corresponding Sturm-Liouville operator
2 z - U

the dual sl2 -oper. In this case, the potential is expressed by the formula
k-2

U=
j =1

-3/4 + (z - wj )2

k

i=1

1/4 + detAi + (z - zi )2

k-2

j =1

Hw j + z - wj

k

i=1

Hzi . z - zi

(3.9)

3.1.3

Backup

In this section we construct an inverse map, namely for a Sturm-Liouville operator that has trivial monodromy we construct a rank 2 connection of the form (3.3) with the monodromy representation in the subgroup Z/2Z GL(2) of scalar matrices ±1. Let us consider an ansatz for the solution of the matrix linear equation (3.3) (z - A(z )) = 0 of the type
k

l =
i=1

(z - zi )-si l (z ),

l = 1, 2;

(3.10)

which satisfy
M

1 =
j =1

(z - j ),
M

2 /1 =
j =1

j . z - j

(3.11)

Let us rewrite the system (3.3) taking into account the new parameterization (3.11) z 1 /1 = a11 + a12 2 /1 , (z 1 /1 )(2 /1 ) + z (2 /1 ) = a21 + a22 2 /1 . Let us represent these equations more precisely: -
i

(3.12) (3.13)

si + z - zi

-
i

1 = z - j j si 1 + z - zi z - j j -
i i

i

ai 11 + z - zi j - z - j

i

ai 12 z-z

i

j

j , z - j

(3.14)

j

j

j = (z - j )2 (3.15)

ai 21
i

ai 11 z-z
i j

z-z

j . z - j 29


The comparison of residues of both part of (3.14), (3.15) at points z = zi gives: j , -si = ai + ai 11 12 zi - j j -
j

(3.16) (3.17)

j si zi - j

= ai - ai 21 11
j

j . zi - j

These equations coupled with the condition of zero trace ai + ai = 0 lead to a condition that si must be one 11 22 of the eigenvalues of Ai , in particular, can be adopted as si = di . Let us consider the behavior at the poles z = j . Let us note that the second order poles of the equation (3.15) at these points cancel. Calculating residues of both sides of the equations (3.14) and (3.15) we obtain 1 j -
i

= j
i

ai 12 , j - zi ai 11 . j - zi

(3.18)

si + j - zi
i=j

1 + j - i

i=j

i j - i

= -j
i

(3.19)

Let us recall that one of the normalization condition controls the diagonal form of the residue at
k

ai 12
i=1 k

=

0,

(3.20)

ai 21
i=1

=

0.

(3.21)

We also note that the choice of Sturm-Liouville operator poles captures zeros of the rational function a12 (z ), which is determined up to a constant: a12 (z ) = c
k-2 j =1 (z k i=1 (z

- wj ) - zi )

.

Then the condition (3.20) will be satisfied automatically. The coefficients ai are expressed by the formula 12 ai = c 12
j

(zi - wj ) (zi - zj )

.

(3.22)

j =i

The coefficients ai are expressed by the following formula in virtue of (3.16) 11 ai = -si - c 11
j =1

(zi - wj ) (zi - zj )
l

j =i

l . zi - l

(3.23)

Let us substitute the expressions for ai and ai to the equations (3.18), (3.19). Then expressing j from 12 11 the first and substituting to the second we obtain: -
k

2sk + j - zk
i

k=j

1 + j - k
i

l k ,m l=k

(zk - wl )
s

s=k

(m - zs )

(zk - zl )

(m - ws )(j - zk )

+

(j - wi ) i (j - zi )

k=j

i

(k - zi ) = 0. (k - wi )(j - k )
i i

An equivalent form can be obtained if one divides both sides by -
k

(j -zi ) (j -wi ) s=k

2sk + j - zk

k=j

1 + j - k
m

l k ,m l=k

(zk - wl )
s

(m - zs )

(zk - zl )

(m - ws )(j - zk )

+
k=j i

i (k - zi ) (k - wi )(j - k )

(j - wm ) = 0. m (j - zm ) 30


Let us consider the left-hand side of equality as a rational function F (j ) and calculate its primitive fractions decomposition at poles zk , wk , k and . It turns out that this decomposition will look like: F (j ) = -
k

2sk - 1 - j - zk

k

1 +2 j - wk

i=j

1 . j - i

(3.24)

Thus, the equality is equivalent to an equation of the Bethe system. Let us demonstrate, for example, the residue calculating at the point j = wi Res
=w

j

i

F (j ) =
k l=k s

l

(zk - wl )
s=i

s=k

(wi - zs )

(zk - zl )

(wi - ws )(wi - zk ) (zk - wl ) . (zk - zl )(wi - zk )2
l

=
s

(wi - zs ) =i (wi - ws )

(3.25)

k

l=k

Let us write down the expesion on the right side of the equality (Resz where (z ) =
l l (z - wl ) , (z - zl )(z - wi )2 =w
i

(z ))-1
k

Resz

=zk

(z ),

and therefore is -1. The sufficiency condition was proved in [55]. Theorem 3.1. If the set of numbers i where i = 1, . . . , M satisfies the system of Bethe equations (2.9) with the parameters: the set of poles z1 , . . . , zk and w1 , . . . , wk-2 and the set of highest weights 2s1 - 1, . . . , 2sk - 1 and 1, . . . , 1 correspondingly, then the vector
k

=
i=1

(z - zi )-s

i

1 (z ) 2 (z )

,

(3.26)

where
M

1 =
j =1

(z - j ),
M

2 /1 =
j =1

j , z - j

(3.27)

and the coefficients j are given by the expressions j =
i i

(j - zi ) , (j - wi )

(3.28)

solves the matrix linear problem (3.3), where the connection coefficients are given by ai 12 =
j

(zi - wj ) (zi - zj )

,

(3.29)

j =i

and the coefficients ai and ai are determined form (3.16), (3.17). The conditions of normalization (3.2) 21 11 are fulfil led.

31


Pro of. Actually, we should prove just that normalization condition (3.21) does not depend on the choice of the parameter c, in particular, it may be taken 1. Indeed, on the basis of (3.16), (3.17) we obtain: 2 (zi - wj ) j j j (3.30) ai = - 2si + . 21 (zi - zj ) zi - j zi - j j =i j j We need to prove that j + zi - j
j i

2si
i j

(zi - wj ) (zi - zj )


j

2 j = 0. zi - j (3.31)

j =i

The first summand of (3.31) then using the Bethe equations can be converted to the following: 2si
i j

j = zi - j

j
j i

2si zi - j 1 . j - i (3.32)

=
j

j -
i

1 + j - zi

i

1 -2 j - wi

i=j

Now we will simplify the second summand (3.31) changing the order m l
m=l i j =i j

(zi - wj )

(zi - zj )(zi - m )(zi - l )
j

+
m

(m )2
i j =i

(zi - wj )

(zi - zj )(zi - m )2

.

(3.33)

Considering the second summand (3.33) let us note that
j i j =i

(zi - wj )

(zi - zj )(zi - m )2

= -m 1 (m ),

(3.34)

where 1 (m ) =
i j =i j

(zi - wj )

(zi - zj )(zi - m )

=

j

(m - wj ) (m - zj )

.

(3.35)

j

Therefore, the expression (3.34) becomes: -
j

(m - wj ) (m - zj )
s

j

1 - m - ws

s

1 m - z

,
s

(3.36)

which is reduced with the relevant part of (3.32). Let us consider the first summand (3.33), this also can be simplified:
j i j =i j j

(zi - wj ) (m - wj )

(zi - zj )(zi - m )(zi - l ) -
j j

=

(l - wj )

(l - zj )(m - l )

(m - zj )(m - l )

.

(3.37)

Substituting the expression in (3.33) we finish the proof 32


3.2

Schlesinger transformations

There is a discrete group of transformations that preserve the connection form (3.3) and, moreover, do not change the class of monodromy representation. However, these changes shift characteristic exponents at fixed points by half-integer values. Such transformations are called Schlesinger, Hecke or Backlund transformations depending on the context. They have simple geometric interpretation explained in the beginning of this section. 3.2.1 Action on bundles

Let us consider a curve C, a holomorphic bundle F on it, the corresponding sheaf of sections F , the additional set of data x C and a point of the dual space to the fiber l Fx . Then the lower Hecke transform T(x,l) E is defined by the subsheaf F = {s F : (s(x), l) = 0}, which in turn corresponds to a certain holomorphic bundle on the curve C. The equivalent definition can be defined in terms of gluing functions. Let us consider the action on holomorphic bundles on CP 1 . In virtue of the Birkhoff-Grothendieck theorem [56] any holomorphic bundle on CP 1 of rank n is isomorphic to the sum of line bundles O(k1 ) . . . O(kn ) for a specific set of integers (k1 , . . . , kn ) called the type of a bundle and determined up to the symmetric group action. Let us consider the open covering of CP 1 consisting in: U - a disk around which does not contain z = zi , i = 1, . . . , N and the domain U0 = CP 1 \{}. We consider holomorphic rank 2 bundles and parameterize them by gluing function G(z ) which is a holomorphically invertible function on U0 U with values in GL(2). Let us say that a pair S (z ) O(2) (U ) and S0 (z ) O(2) (U0 ) defines a global section if S0 (z ) = G(z )S (z ). We describe the transformation on bundles in terms of actions on corresponding gluing functions defined as a multiplication on the left by an element Gs (z ) = G
s

z-z 0

s

0 1

G

-1 s

(3.38)

for some constant matrix Gs and some point zs U0 Remark 3.1. The action classes of holomorphic bun change also the matrix Gs We will investigate the on the space of gluing functions can be reduced to the action on the isomorphism d les if one chooses Gs appropriately, if changing a trivialization in U0 by T (z ) we as fol lows T (zs )Gs . This is obviously referring to the invariant definition above. composition of these changes applied at two points.
-1 j

Lemma 3.2. A composition of two transformations specified by an expression Gi (z )G choice of matrices Gi , Gj preserves the trivial bund le.

(z ), for a generic

Pro of It is sufficient to find a decomposition for G(z ) = Gi (z )G-1 (z ) with G(z ) = Gij (z )G (z ), where j Gij (z ), G (z ) are invertible respectively at U0 , U . The thought-consideration for this evidence is the cohomological dimension count in families at a generic point. Indeed, for a particular choice G-1 Gj = 1 we i get a trivial bundle which is semistable and hence minimizes the dimension of H 0 (End(V )) for V of degree 0. In this context, the trivial bundle is generic in the family of bundles for different G. Despite the general argument here we propose a proof in spirit of the decomposition lemma in [56]. Let us introduce the notations Gi = We can decompose the product G(z ) = G into the alternative product G(z ) = Gij (z )G (z ),
i

1 yi

xi 1

.

(3.39)

z 0

0 1

G

-1 i

G

j

(z - 1)-1 0

0 1

G

-1 j

(3.40)

33


where Gij (z ), G (z ) are holomorphically invertible functions on U0 , U respectively. The traditional calculations allows to find a decomposition in the form G (z ) =
z (1-xj yi )(1-xj yj )-xj (yi -2yj -xj yi yj ) (1-2xj yi +xj yj )(1-xj yi )(1-xj yj )(z -1) yi -2yj +xj yi yj (1-xj yi )(1-2xj yi +xj yj )

-

xj (1-xj yi )(1-xj yj )(z -1) 1 1-xj yi

.

3.2.2

The action on connections

The action of Hecke transformations on classes of holomorphic bundles can be extended to the space of pairs: (bundle, connection) when certain conditions are satisfied. Let us describe in detail the induced action. A connection is a sheave map satisfying Leibniz rule with respect to the action of the structure sheaf: : F F 1 . Hecke transformations can be defined on the space of connections preserving the space Annl = {v Fx :< l, v >= 0} x : Annl Annl 1 . x In our case we consider the composition of pairs of Hecke transformations localized at zi , zj , preserving the trivial rank 2 bundle. As is mentioned above, the action can be defined by using the gluing functions language. Let us consider the trivial bundle specified by the gluing function 1. Hecke transformation change the bundle structure, the global section is defined by the pair S0 , S , such that S0 = GS , where G = Gij G . One can define the action on connections as follows: let z - A is a connection in the trivial bundle, determined by this expression on both opens, the transformed ob ject is the pair of connection forms: z - A G(z - A)G-1 After the basis change in U of the type S


over over


U , U0 .

= G S

we obtain the connection of the form
-1

z - A G (z - A)G The trivialization change in U0 of the kind S0 = G G(z - A)G
-1 -1 ij S0

.

(3.41)

gives the following
-1

G

-1 ij

G(z - A)G

Gij = G (z - A)G

-1

.

(3.42)

Therefore, the transformed connection is of the same type as the initial one. The analytic properties at are preserved in virtue of the fact that G is holomorphically invertible in U . Using the results of the previous sections we calculate explicitly the Hecke action. To preserve the normalization condition A(z ) at it is necessary to consider transformations of the kind G(z ) = G =
-1

()G (z ) z-
x1 (y0 -2y1 +x1 y0 y1 ) (1-x1 y0 )(1-x1 y1 ) y0 -2y1 +x1 y0 y1 (1-x1 y0 )(1-x1 y1 ) x1 (1-2x1 y0 +x1 y1 ) (1-x1 y0 )(1-x1 y1 ) 1-2x1 y0 +x1 y1 (1-x1 y0 )(1-x1 y1 )

1 z-1

.

(3.43)

~ Then one just needs to apply the gauge transformation G(z ) to the connection ~ ~ A G(z )AG
-1

~ ~ (z ) + z G(z )G

-1

(z ).

The complete family of Hecke transformations in the case of 3 points associated with the analysis of the Painleve VI equation was described in [57].

34


Remark 3.2. The choice of the highest weights 1 in moving poles wi is not obligatory, but in some ways, the most general. One can consider a potential of the form
m

U=
j =1

-1/4(j + 2)j + (z - wj )2

k

i=1

1/4 + detAi + (z - zi )2

k-2

j =1

Hw j + z - wj

k

i=1

Hzi z-z

(3.44)
i

with the higher values of weights. It can be implemented if one requires that a12 (z ) have zeroes wj with m multiplicities j satisfying the condition j =1 j = k - 2. The local analysis at poles shows that the eigenvalues of residues Ai transform due to the 4 following rules depending on the choice of the low and upper Hecke transformations subspaces: (. (. (. (. . . . . . . . . , , , ,
i i i i

, , , ,

. . . .

. . . .

. . . .

, , , ,



j j j j

, , , ,

. . . .

. . . .

. . . .

) ) ) )

- (. . . , - (. . . , - (. . . , - (. . . ,



i i i i

+ + - -

1, 1, 1, 1,

. . . .

. . . .

. . . .

, , , ,



j j j j

- + - +

1, 1, 1, 1,

. . . .

. . . .

.), .), .), .).

The result obtained allows to treat recurrent relations on the space of solutions for the Bethe equation system. The most interesting in the program of explicit solving of quantum systems is the set of transformations lowering the highest weight values in both points. The consecutive application of these transformations could reduce the highest weight to zero, which corresponds to the trivial representation of the quantum algebra and hence the trivial quantum problem.

3.3

Elliptic case

The elliptic sl2 Gaudin model is provided by a similar technique of quantum model solution including the quantum spectral curve, quantum separated variables and Hecke symmetries on the spectrum. 3.3.1 Separated variables

Let us recall the traditional method of separation of variables for this system [53], [59]. As in the rational case we consider the sl2 Gaudin model with fixed representation V = V1 . . . Vk of the quantum algebra U (sl2 )k , where Vi is the finite dimensional irreducible representation of the highest weight i . Vi can be realized as the quotient of the Verma module C[ti ]/ti +1 , such that the generators of sl2 act by differential i operators: h
( s)

= -2t

s

+ s , ts

e(

s)

= -t

s

2 + s , 2 ts ts

f

( s)

= ts .

Let us start with the study of quantum problem on the tensor product of Verma modules W = C[t1 , . . . , tk ]. We introduce the variables C, {yj } defined by:
k

s=1

(u - us - ) ts = C (u - us )(-)

k

s=1

(u - ys ) . (u - us )

Let us now represent the elliptic Gaudin model eigenvector as a function of introduced variables: S (u)(C, y1 , . . . , yk ) = s (u)(C, y1 , . . . , yk ). In this formula s (u) is a scalar functions on u of the form s (u) = ci (u - ui ) + d
i

(3.45)

(u - ui ) ; (u - ui )

ci = 2 /4 + i /2. i

(3.46)

Introducing further u = yj by the left in formula (3.45) we obtain: 1 - yj 2
k

s=1

(yj - us ) (yj - us )

2 s

(C, y1 , . . . , yk ) = s (yj )(C, y1 , . . . , yk ).

35


This equation induces a factorization of an eigenvector: (C, y1 , . . . , yk ) = C moreover it may be argued that the expression w(u) = ponent of the eigenvector satisfies the equation:
a j k s=1

(yj ), (u - us )-s /2 (u) associated with the com(3.47)

2 u - s (u) w(u) = 0

Therefore, each equation 3.47 of the form 3.46, having solution s (u) with half-integer exponents at {u1 , . . . , uk } and meromorphic outsides these points corresponds to an eigenvector for the elliptic Gaudin Hamiltonians in representation V , obtained pro jecting the vector . Hyp othesis 3.3. There is a one-to-one mapping between this kind of differential operators and the eigenvectors of the model in the representation V . Through the following sections we will consider only such eigenvectors for the Gaudin model that correspond to elliptic Sturm-Liouville operators with the described analytic properties. 3.3.2 Bethe ansatz

The traditional Bethe ansatz method in the elliptic case [59] can be obtained considering the following particular solution with simple zeroes (u) =
i



-i /2

(u - ui )
j

(u - j )

(3.48)

for the elliptic Sturm-Liouville equation
2 u - i

ci (u - ui ) -
i

d

i

(u - ui ) (u - ui )

(u) = 0.

(3.49)

This condition is equivalent to the following system of equations: ci d
i

= 2 /4 + i /2, i (ui - j ) = i - (ui - j ) j i /2
i

j =i

j (ui - uj ) , 2(ui - uj ) (j - i ) , (j - i ) (3.50)



0=

(j - ui ) - (j - ui )

i=j

the latter is called the elliptic Bethe system. 3.3.3 Matrix form of the Bethe equations

In this section we find a matrix Fuchsian system equivalent to the elliptic Sturm-Liouville equation (3.49), (u - A(u))(u) = 0, where (u) = 1 (u) , 2 (u)
(u-zi ) (u-zi ) i (u-zi +) a21 (u-zi )()

(3.51)

A(u) =

a11 (u) a12 (u) a21 (u) a22 (u)

=

ai 11

ai 12

(u-zi -) (u-zi ) (-) u-z ai ((u-zii)) . 11

36


The equivalence relation of the matrix and scalar systems is the following: the function w = 1 / a12 solves the equation w - U w = 0, of the same form as 3.47, with the potential whose highest term is given by the formula: U (u) = - (1/4 + det(Ai ))(u - zi ) + 3/4(u - wi ) + ...

Here the points wj are defined by the condition a12 (u) = c In turn Ai are defined as residues of A(u) at zi . Remark 3.3. Note that the sets of poles of the Sturm-Liouvil le operator and of the matrix problem do not match, the first one is compiled from two subsets {u1 , . . . , u2l } = {z1 , . . . , zl , w1 , . . . , wl }. It turns out that method of solution construction for the matrix problem from a solution for the SturmLiouville equation is also explicit. Let us consider a scalar problem that corresponds to the set of marked points {z1 , . . . , zl , w1 , . . . , wl }, the set of highest weights 2s1 - 1, . . . , 2sk - 1, 1, . . . , 1} and the set of Bethe roots {1 , . . . , }. Then the 2-vector function with components:
k

(u - wi ) . (u - zi )



1

=
i=1

(u - ui )s

i

(u - j )
j =1



2

=
j =1

j (u - j + ) 1 , (u - j )

(3.52)

those coefficients j are given by the formula j =
i i

(j - wi ) , (j - ui )

satisfy the matrix equation 3.51. An explicit calculation shows that the equation (3.51) for given by the expression (3.52) is equivalent to the following system of equations: det -
k

ai - si 11 ai 21

ai 22

ai 12 - si

= =

0, 0,

(2sk - 1)

(j - uk ) - (j - uk )

k

(j - wk ) +2 (j - wk )

i=j i i

(j - i ) (j - i ) (j - wi ) (j - ui )

= j .

The system of equations means that exponents are eigenvalues of the residues of the connection and the set of j satisfy the elliptic Bethe system (3.50) corresponding to the set of marked points {u1 , . . . , uk , w1 , . . . , wk } and the set of highest weights {2s1 - 1, . . . , 2sk - 1, 1, . . . , 1}.

37


3.3.4

Hecke transformations

Let us describe in more details how the Hecke transformations are calculated over an elliptic curve. The most suitable way of parameterization of holomorphic bundles for an elliptic curve implies the lift of a bundle to the universal covering C ([58] (2,6)). The monodromy group Z2 acts by homomorphisms on the sheaf of sections E corresponding to the bundle E . In case of the degree 0 line bundle the only way to define the multiplier set up to equivalence is the set of quasiperiodic factors of the expression: f (z ) = (z - ) (z )

for . Let us denote the corresponding line bundle by O . The Hecke transform at a point w supposes considering the subsheaf of O taking values 0 at w. This sheaf is isomorphic to the sheaf of sections of some line bundle of degree 1 s(z ) s(z ) . (z - w)

This map is an isomorphism due to the property that (z ) has a unique zero at z = 0. The Hecke transformations on connections on the rank 2 bundle O/2 O-/2 construct connections on a bundle Oµ/2 O-µ/2 as follows. Let the residues of the connection have the decomposition: Ai = a11 ( i) a21
( i)

a12 ( i) -a11

( i)

=G

i

di 0

0 -d

G
i

-1 i

where Gi are constant matrices. Then the connection is transformed by the gauge transformation with the group element ~ Gij (z ) = G where ~ Gi = G
j -1 i

1 0 0 (z - zi )

~ G

-1 i

G


j

-1

(z - zj ) 0 0 1

G

-1 j

,

(zi - zj ) 0 0 1

G

-1 j

Gi .

As well as in the rational case we consider a pair of Hecke transformations at various points ui , uj with different signs Tij = T(-1,li ) T(uj ,lj ) acting on rank 2 bundles with trivial determinant. Depending on the ui choice of subspaces of upper and lower transformations we get the following action of Tij on a variety of highest weights of the Gaudin model (. (. (. (. . . . . . . . . , , , ,
i i i i

, , , ,

. . . .

. . . .

. . . .

, , , ,



j j j j

, , , ,

. . . .

. . . .

. . . .

) ) ) )

- - - -

(. (. (. (.

. . . .

. . . .

, , , ,



i i i i

+ + - -

1, 1, 1, 1,

. . . .

. . . .

. . . .

, , , ,



j j j j

- + - +

1, 1, 1, 1,

. . . .

. . . .

. . . .

), ), ), ).

As in the rational case it is of particular interest the family of transformations that lower the weights of all representations hence simplifying the diagonalization problem.

4

Applications

This section is devoted to two main applications of the quantum spectral curve method. The first application is related with the geometric Langlands correspondence and mainly consists in an effective description of the center Ucrit (gln ) which in turn plays a key role in the Beilinson-Drinfeld quantization of the Hitchin system. Let us note that this problem is closely related to the representation theory of affine Lie algebras.

38


4.1
4.1.1

Geometric Langlands corresp ondence
The center of U (gln ) on the critical level

We introduce the following notation Ucrit (gln ) for the local completion U (gln )/{C -crit}, where C is a central element and crit = -h = -n is the critical level inverse to the dual Coxeter number of the Lie algebra sln . It was proved in [61] that Ucrit (gln ) has a center isomorphic to the polynomial ring of the Cartan algebra as a linear space. Despite the geometric description of the center there was absent an explicit construction for the generators of this commutative algebra. For this purpose we use the Adler-Costant-Symes scheme [60]. This approach has an important place in the theory of integrable systems: can be exploited to construct a wide family of commutative algebras, allows to make relation of integrable systems with decomposition problems and provide an algebraic interpretation for the Lax representation, r-matrix structures. The AKS scheme can be generalized to the quantum level and takes important role in description, solution and classification of quantum integrable models. The most simple case is that of finite dimensional Lie algebra allowing a decomposition g = g+ g- into the sum of two Lie subalgebras. To each choice of normal ordering one can attach an isomorphism of linear spaces : U (g) U (g+ ) U (g- ). Let us introduce a notation gop for the inverse Lie algebra structure to the space g- defining by the formula - -{, }. Let us denote the Lie algebra g+ gop by the symbol gr . The corresponding enveloping algebras - can be identified as linear spaces with the help of the Poincare-Birkhoff-Witt basis: U (gop ) - U (g- ).

Lemma 4.1. The center of z(U (g)) is mapped to a commutative subalgebra in U (g+ ) U (gop ) by . - Pro of Let is denote the commutator in U (g+ ) U (gop ) as follows [, ]R . Let c1 , c2 be two central elements - in U (g) represented as follows ci =
j

xj y

( i ) ( i) j

xj U (g+ ), yj U (g- ).

( i)

( i)

The result of calculating the modified commutator is as follows [(c1 ), (c2 )]R =[
j

xj y
(1) [xj

(1) (1) j

,
k

xk y

(2) (2) k ]R

=
j,k

,

(2) (1) (2) xk ]R yj yk

+ xj xk [y

(1) (2)

(1) j

,y

(2) k ]R

.

In virtue of the definition above we have [xj , xk ]R = [xj , xk ] [(c1 ), (c2 )]R =
k (1) (2) (1) (2)

[y
(2)

(1) j

,y

(2) k ]R

= -[y
(1)

(1) j

,y

(2) k

]

[c1 , xk ]y

(2) k

-
j

xj [y
2

(1) j

, c2 ]

The last expression is zero due to the centrality of elements c1 , c

Remark 4.1. In what fol lows we wil l be interested in applying this scheme for Ucrit (gln ). To use the result of the AKS lemma in the infinite dimensional case one should choose an appropriate completion of an algebra. In our case we use the completion corresponding to the bigrading deg (g tk ) = (k , 0), deg (g t-k ) = (0, k ) for · k 0. One needs to prove that the considering central elements belong to this completion Ucrit (gln ). This is a matter of fact due to the classical limit argument. In what fol lows we omit the completion in notation Ucrit (gln ), U (gr ) and the tensor products for the sake of the simplicity. 39


One considers also the linear space map : U (g) U (g+ ) U (g)g- defining by the direct sum decomposition for the Lie algebra. Let us denote by the pro jector to the first subspace U (g+ ). Lemma 4.2. The image of z(U (g)) with respect to is a commutative subalgebra of U (g+ ). Pro of Let c1 , c2 Z. [c1 - (c1 ), c2 - (c2 )] = [(c1 ), (c2 )] The r.h.s. belongs to U (g+ ); the l.h.s in an element of U (g)g- ; this takes issue in vanishing of both sides We will identify Ucrit (gln ) with the loop algebra as linear spaces. Let us list several important facts about the loop algebra. Prop osition 4.3. Let us consider g = gln [t, t represented by the generating series Lf where s =
ij ul l -1

] = gln [t

-1

] tgln [t] those generators eij = eij tk can be
-s-1

(k )

(z ) =
s=-,

s z

(4.1)

Eij eij .

( s)

Here as above eij are generators of the Lie algebra gln , and Eij are matrix unities in M atn . The Lie algebra structure on gr can be described as the fol lowing commutation relations {Lf
ul l

(z ) Lf

ul l

(u)} = [

P , Lf z-u

ul l

(z ) 1 + 1 Lf

ul l

(u)].

(4.2)

Let us remark that these relations are the same as for the Gaudin Lax operator (2.42). The center of (Ucrit (gln )) and a commutative subalgebra in U (tgln [t]) "`positive"' Lax operator: L(z ) which satisfies the following R-matrix relations: {L(z ) L(u)} = [ P , L(z ) 1 + 1 L(u)]. z-u (4.3) =
k >0

Let us also introduce the

k z

-k-1

,

Theorem 4.4. The commutative subalgebra tum characteristic polynomial det(L(z ) - z ) : Ucrit (gln ) U (tgln [t]). Pro of The proof is based on the results of i quadratic Gaudin Hamiltonians H2 in U (tgln [t

in U (tgln [t]) defined by the set of coefficients of the quancoincide with the image of z(Ucrit (gln )) by the projection [62] where it was proved that the centralizer of the set of ]) coincide with the pro jection of U (gln ) on the critical level.

Remark 4.2. This particular property, namely the fact that the quadratic generators determine the complete commutative subalgebra is known also in the theory of Fomenko-Mishenko subalgebras [63] and in the theory of the Calogero-Moser system [64]. Following the proposed logic and using the fact that the subalgebra defined by the coefficients of det(L(z )- i z ) commute with H2 , one can show that this subalgebra is a subalgebra of the algebra obtained from the center. In order to prove their coincidence it is sufficient ot consider the classical limit Remark 4.3. The analogous strategy is applicable in the case of projection to U (gln [t]). One needs to take into account that both algebras are invariant with respect with the GL(n) action. 40


4.1.2

Explicit description of the center of Ucrit (gln ))

Theorem 4.5. The center of Ucrit (gln ) is isomorphic to a subalgebra in U (gln [t-1 ] tglop [t]) defined by the n coefficients of th quantum characteristic polynomial det(Lf ull (z ) - z ). The isomorphism is induced by the mapping I : U (gln [t
-1

]) U (tglop [t]) Ucrit (gln ), n

I : h1 h2 h1 h

2

(4.4)

Pro of follows the same lines as those of [62]. Let us firstly show that the algebra generated by the coefficients of the characteristic polynomial of the Lax operator Lf ull (z ) coincide with the centralizer of its quadratic elements. Further using the Sugawara formula for the quadratic center generators we prove that their image in U (gln [t-1 ]) U (tgln [t]) coincide with the quadratic elements of the quantum characteristic polynomial. For proving the first statement we consider a special limit of the commutative family. Using the commutation relations 4.2, 4.3 and the traditional r-matrix calculations we show that T rLm ll (z ) are central in the symmetric algebra S (gln [t, t-1 ]) and moreover T rLm ll (z ) generate the comfu fu mutative Poisson subalgebra in S (gln [t-1 ] tglop [t]). n Let us consider the family of automorphisms of the algebra U (gln [t-1 ]) U (tglop [t]) defined in terms of n the Lax operator as follows: let K is a generic diagonal n â n matrix. The Lax operator Lf
ul l

(z ) = Lf

ul l

(z ) + K

also satisfies the r-matrix reletions (4.2). This automorphism family is parameterized by the parameter . Let us consider the family of commutative subalgebras M U (gln [t
-1

]) U (tglop [t]) n

defined by the generating function det(Lf ull (z ) - z ). M centralizes the set of quadratic generators QI2 (Lf ull (z )). QIk (z , ) has the following leading term in expansion on QIk (z , ) = Changing the basis QIk (z , ) QIk (z , ) = (QIk (z , ) - and considering the limit QIk (z , ) T r(Lf
ul l k k

T r An K 1 K 2 . . . K k + O (

k-1

).

T r An K 1 K 2 . . . K k )

-k+1

(z )K

k-1

)

we obtain that these expressions generate the Cartan subalgebra H = H- H+ = U (h[t
-1

]) U (th[t]).

Let us demonstrate that this subalgebra coincide with the centralizer of its quadratic generators
H2 (z ) = lim

QI 2 (z , ) =
i=-,

T r(i K )z

-i-1

.

Obviously H Z (H2 (z )). Let us introduce the notations (k1 , . . . , kn ) for the diagonal elements of K. Let us also denote by hi H the sum of the form n

hi =
s=1

(i )ss ks ,

-i-1 then H2 (z ) = . The centralizer elements should commute with h1 and h-1 . Let i=-, hi z -1 ]), yi U (tg[t]). We also suppose that this sei=- xi yi b e the infinite series such that xi U (g[t ries is an element of the considered completion, i.e. such that it contains only finite number of elements of

41


each bigrading. The operators [h1 , ] and [h-1 , ] are homogeneous of bigrading (0, 1) and (1, 0). Hence the centralizer description question is reduced to the analogous question in the polynomial algebra. The answer is given by the formulas Z (h1 ) = U (gln [t
-1

]) H+ ,

Z (h

-1

) = H- U (tglop [t]). n

An intersection of these subspaces in a completed sense coincides with the Cartan subalgebra H. Summarizing we obtain that in the generic point of the family the commutative subalgebra M belongs to the centralizer of the set QI2 and in the limit generates the centralizer. From the arguments analogous to those of [62] at the generic point M should coincide with the centralizer of the quadratic generators. To finish the proof let us remind the Sugawara formula Ucrit (gln ) c2 (z ) =: T r(L2 f
ull

(z )) :

This uses the normal ordering symbol : : for currents in sl2 . These elements pro ject to QI2 (z ) up to a central elements in U (gln [t-1 ]) U (tgln [t]) 4.1.3 The Beilinson-Drinfeld scheme

In [65] it was proposed a universal construction for the Hitchin system quantization. Let be the connected smooth pro jective curve over C of genus g > 1, G - a semisimple Lie group, g - the corresponding Lie algebra, B unG - the moduli stack of principal G-bundles on . Let us also define the Langlands dual group L G as a group determined by the dual root data, namely such that its root lattice coincides with the dual lattice for G. The main result of [65] can be reduced to the following: · There exists a commutative ring of differential operators on z(, G), acting on sections of the canonical bundle KB unG such that the symbol map produces the commutative subalgebra of classical Hitchin Hamiltonians on T B unG . · The spectrum of the ring z(, G) is canonically isomorphic to the moduli space of L g-opers (for the G = S L2 case an L g-oper is just the Sturm-Liouville operator on S ; in general case this is a flat connection in a principal L G bundle with a parabolic structure). · To each L g-per one can correspond a D-module on B unG by fixing eigenvalues of the Hitchin Hamiltonians. This D-module is an eigensheaf for the Hecke action defined naturally on the moduli stack of bundles. Moreover the eigenvalue in this case coincide with the corresponding L g-oper. The basement of this construction is the natural action of the center of Ucrit (g) on the loop group of the corresponding Lie group. This action can induce an action by differential operators on B unG () in virtue of one of the realizations of the moduli stack of principal bundles B unG () G(F )\G(AF )/G(OF ) Gin \G[[z , z
-1

]/G

out

here Gin and Gout denote the subgroups of function converging in Uin and Uout , where Uin and Uout determine a covering of of the type: Uin is an open disk centered in P with the local parameter z , Uout = \P. The middle part of the equality represents the so-called adelic realization of the moduli stack of principal G-bundles for an algebraic group G. The construction uses the adel group G(AF ) for the field F of rational functions on , the group of entire adels G(OF ) and the group of principal adels G(F ). This realization is convenient for describing the geometric complex analogy of the arithmetic Langlands correspondence and the quantum Gaudin model. 4.1.4 Corresp ondence

Historically the Langlands hypothesis generalize the field-class theory [66, 67], one of those principal results is the following statement in the case of a number field. Namely let F be a number field (this means a finite

42


¯ extension of Q), F - its maximal algebraic extension, F of an extension F F is

ab

- its maximal abelian extension. The Galois group

Gal(F , F ) = { Aut(F ) : (x) = x x F }. The ab elian recipro city law There exists a group isomorphism Gal(F ab , F ) The group of connectivity components of F â \A
â F

where Aâ is the idel group of the ring F, F â is the group of invertible elements of F. The topology of of the F completion product is considered. The Langlands hypothesis is formulated as an n-dimensional (non-commutative) generalization of the abelian reciprocity law. Namely it is assumed the isomorphism between the cathegory of the Galois group representations of the maximal algebraic extension of a ring and the category of automorphic representations for the corresponding idel group. By an automorphic representation we mean a GLn (AF ) - representation realized on the space of functions on GLn (F )\GLn (AF ), meet some additional conditions [68, 69]. The right part is traditionally called automorphic for the following reason. For n = 2 these representations are related with the theory of modular functions. It should be reminded that modular functions are functions on the upper-half Siegel plain matching the condition f ((az + b)/(cz + d)) = (a)(cz + d)k f (z ) ab cd S L2 (Z).

In particular, the modular functions can be represented as functions on the following quotient space S L2 (R)/S L2 (Z) K \GL2 (AQ )/GL2 (Q)

The Langlands program covers the following types of fields F : · A number field. · Field of functions on an algebraic curve over the finite field Fq (In this case, the hypothesis was proven in [70]). · Field of functions on an algebraic curve over C. This is called the geometrical case over C. The following papers are on the sub ject [71]. The corresp ondence over C: In this case on the Galois side one considers classed of representations of the fundamental group or classes of flat connections in a holomorphic bundle of rank n. The automorphic side deals with the Hitchin D-module on GL(F )\GL(AF )/GL(OF ) B unn (). The results of [65] and [61] ensures the correspondence between Hitchin D-modules and flat connections related to L g-opers. Due to the construction of the quantum characteristic polynomial for the loop algebra, as well as an explicit construction for the center of Ucrit (gln ) in theorem 4.5 the correspondence for the Lie algebra gln can be realized in a more effective way. The scheme demonstrate the correspondence Hitchin D-module
F F, B D



Character on z(Ucrit (gln )) det(Lf

CT

ul l

- z ).

Remark 4.4. The construction of a character on z(Ucrit (gln )) by a Hitchin D-module is a corol lary of the Feigin and Frenkel theorem on existence of the center and the Beilinson and Drinfeld quantization. To obtain the explicit description for the corresponding flat connection [26] one should exploit the identification of commutative algebras: the commutative subalgebra in U (gln [t-1 ]) U (tgln [t]) defined by the coefficients of the quantum characteristic polynomial on one side and the image of the center of z(Ucrit (g)) by the AKS map on another side. 43


4.2

Non-commutative geometry

The main plot of these lectures is relevant to the emerging field of Noncommutative Geometry, substantive issues of which consist in geometric interpretation of algebraic structures in which the commutativity property is weakened. In this context the quantum characteristic polynomial is a natural generalization of classical one. Some properties of this ob ject, in particular the role played by quantum characteristic polynomial in the program of effective solution of the quantum integrable models, suggest it to be a natural noncommutative generalization of an algebraic curve the spectral curve of an integrable system. This section describes some linear algebraic properties of the quantum characteristic polynomial obtained in [27]. 4.2.1 The Drinfeld-Sokolov form of the quantum Lax op erator

Let L(z ) M atn U (gln )N F un(z ) be the quantum Lax operator for the Gaudin model (1.14), here and further F un(z ) means the space of rational functions on a parameter z . Let us denote by L[i] (z ) quantum powers of the Lax operator defined by the formula: L[0 L[i
] ]

= I d, = L[i-1] L + z L[

i-1]

.

Theorem 4.6. The expression C (z ) M atn U (gln )N v vL C (z ) = ... v L[n-1]

F un(z ) defined by the formula , (4.5)

where v Cn is a generic vector defines a gauge transformation 0 1 0 ... 0 0 1 ... ... ... ... C (z )(L(z ) - z ) = ... 0 0 ... 0 QHn QHn-1 ... QH2

0 0 ... 1 QH1



(4.6)

- z C (z ),

where the r.h.s. lower line coefficients are determined by the coefficients of the quantum characteristic polynomial det(L(z ) - z ) = T rAn (L1 (z ) - z ) . . . (Ln (z ) - z ) =
n (-1)n (z - i i QHn-i z ).

(4.7)

Knizhnik-Zamolo dchikov equation Here and further we denote by V a finite-dimensional representation of U (gln )N . It was shown in [72] that there exists a relation between solutions of the KnizhnikZamolodchikov (KZ) equation [73] (L(z ) - z )S (z ) = 0, where S (z ) is a function with values in Cn V , solutions of the Baxter equation det(L(z ) - z )(z ) = 0 (4.8)

where (z ) is a function with values in V . To make this relation clear it is sufficient to take the antisymmetric pro jection of U (z ) = v1 . . . vn-1 S (z ) where vi are some vectors in Cn . For special choice of such vectors one obtains that vector components of S (z ) solves the equation (4.8).

44


Pro of of theorem 4.6 Let us consider both sides of (4.6)applied to a function S (z ) Cn V F un(z ), < v , LS - z S > < v , L[1] (LS - z S > , L.H.S = C (L - z )S = (4.9) ... [n-1] < v, L (LS - z S ) > < v , LS - z S > ... . - z )C S = < v , L[n-1] S - z (L[n-2] S ) > n-1 < v , i=0 QHn-i L[i] S - z (L[n-1] S ) > L[k] S - z (L[ The difference (4.10) - (4.9) takes the form < v,
k-1]

R.H.S = (LD

S

(4.10)

Using the definition for quantum powers we obtain S ) = L[
k-1]

(LS - z S ).

n-1 i=0

0 ... . 0 [ i] [ n] QHn-i L S - L S >

(4.11)

Let us now consider this expression if S (z ) is a solution for the KZ equation L(z )S (z ) = z S (z ). Let (z ) = C (z )S (z ), where C (z ) is given by the formula (4.5). Then 1 (z ) = < v 2 (z ) = < v ... k (z ) = < v = L(z ), S (z ) >=< v , z S (z ) > (L[k-1] L(z ) + z L[k-1] ), S (z ) > L[k-1] , z S (z ) > + < v z L[k-1] , S (z ) >= z

k-1

(z )

One of the consequences of [72] is that 1 (z ) =< v , S (z ) > solves the Baxter equation
n-1 i n QHn-i z 1 (z ) - z 1 (z ) = 0 i=0

(4.12)

for each solution S (z ) of the KZ equation and each vector v Cn . The general position argument allows to claim that the n-th element of (4.11) vanishes identically on S (z ) Cn V F un(z ). Theorem 2.5.7 [74] induces the equality of universal differential operators with values in the quantum algebra. 4.2.2 Caley-Hamilton identity

Corollary 4.6.1. The quantum powers of the Lax operator satisfy the quantum version of the Caley-Hamilton identity
n

L (z ) =
i=1

[ n]

QHi (z )L[

n-i]

(z ).

(4.13)

Pro of Let us consider the last line of the equation (4.6)
n

v L[

n-1]

(z )(L(z ) - z ) =
i=1

v QHi (z )L[

n-i]

(z ) - z v L[

n-1]

(z ).

The result follows from the general choice of the vector v Cn . 45


References
[1] Shiota T., Characterization of Jacobian varieties in terms of soliton equations, Invent. Math., 83(2):333382, 1986. [2] Krichever I., Integrable linear equations and the Riemann-Schottky problem, in: Algebraic Geometry and Number Theory, Birkh¨ r, Boston, 2006. ause Krichever I., Characterizing Jacobians via trisecants of the Kummer Variety, arXiv:math/0605625. [3] Atiyah M., The geometry and physics of knots. CUP, 1990 [4] Donaldson S., An application of gauge theory to four- dimensional topology, J. Differential Geometry 18 A983, 279-315. [5] Kronheimer P.B., Mrowka T.S., The genus of embedded surfaces in the protective plane, Math. Research Letters 1 A994, 797-808. [6] Nakayashiki, A., Structure of Baker-Akhiezer modules of principal ly polarized abelian varieties, commuting partial differential operators and associated integrable systems, Duke Math. J.62-2 (1991) 315 358, Nakayashiki, A. and Smirnov, F., Cohomologies of affine hyperel liptic Jacobi varieties and integrable systems, Comm. Math. Phys. 217 (2001), 623-652. [7] Thaddeus M. Conformal field theory and the cohomology of the moduli space of stable bund les, J. Differential Geom. 1992. V. 35, 1. P. 131-149. [8] Feigin B., Finkelberg M., Negut A., Rybnikov L., Yangians and cohomology rings of Laumon spaces, arXiv:0812.4656. Feigin B., Finkelberg M., Frenkel I., Rybnikov L., Gelfand-Tsetlin algebras and cohomology rings of Laumon spaces, arXiv:0806.0072. [9] Gaudin M., La Fonction d' Onde de Bethe, Masson, Paris (1983). [10] Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of the KdV type, finite zone linear operators and Abelian varieties. Uspekhi Mat. Nauk, V. 31, 1976, P. 55-136. English transl. in Russian Math.Surveys V. 31, 1976. Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys, 32 6 (1977), 185213 [11] Kowalevski S., Sur le probl` e de la rotation d'un corps solide autour d'un point fixe. Acta Mathematica em 12 (1889),177-23. [12] Jacobi C.G.J., Vorlesungen uber Dynamik. Georg Reimer, Berlin, 1866. [13] Novikov S.P., Veselov A.P., On Poisson brackets compatible with algebraic geometry and Korteweg-de Vries dynamics on the set of finite-zone potentials, Sov. Math., Dokl. 26, 357-362 (1982) [14] Hitchin N., Stable bund les and integrable systems. Duke Math. Journal 1987 V 54 N1 91-114. [15] Krichever I. M., Rational solutions of the KadomtsevPetviashvili equation and integrable systems of N particles on a line, Funkts. Anal. Prilozh., 12, 1 (1978), 7678 [16] Gardner C.S., Green J.M., Kruskal M.D., Miura R.M., Method for solving the Korteweg-de Vries equation, Phys.Rev.Lett. 1967. V. 19. P. 1095-1097. [17] Lax P., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 1968. V. 21. 5. P. 467-490. [18] Dubrovin B.A., Krichever I.M., Novikov S.P., Integrable systems. I, Dynamical systems 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, VINITI, Moscow, 1985, 179277 46


[19] Perelomov A. M., Integrable Systems of Classical Mechanics and Lie Algebras, Birkhauser-Verlag, 1990. [20] Demidov E.E., The KadomtsevPetviashvili hierarchy and the Schottky problem, Fundam. Prikl. Mat., 4, 1 (1998), 367460. [21] Faddeev L.D., Sklyanin E.K., Quantum mechanical approach to completely integrable field theory models, Soviet Phys. Dokl., 23 (1978), 902904. Faddeev L.D., Sklyanin E.K., Takhtadzhyan L.A., Quantum inverse problem method. I Theoretical and Mathematical Physics, 1979, 40 2, 688706. ulish P.P., Sklanin E.K. Quantum inverse scattering method and the Heisenberg ferromagnet. Phys. Lett. 1979. V. 70 A, 59.P. 461463. [22] Bethe H., Zur Theorie der Metal le. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain), Zeitschrift fur Physik A, Vol. 71, pp. 205-226 (1931). [23] Drinfeld V.G., Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl., 1985, V.32, [24] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painleve, A modern theory of special functions, 1991. [25] Chervov A., Falqui G., Rybnikov L., Limits of Gaudin Systems: Classical and Quantum Cases, arXiv:0903.1604v1 [math.QA]. [26] Chervov A., Talalaev D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence. hep-th/0604128. [27] Chervov A.V., Talalaev D.V., KZ equation, G-opers, quantum DrinfeldSokolov reduction, and quantum CayleyHamilton identity, Zap. Nauchn. Sem. POMI, 360 (2008), 246259. [28] Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge University Press 2003. [29] Reyman A.G., Semenov Tian-Shanski M.A., Integrable Systems (group- theoretical approach), Sovremennaya matematika, Izhevsk, 2003. [30] Ramanan S., The moduli space of vector bund les over an algebraic curve, Math. Ann. 200 (1973) 69-84. [31] Kodaira K., Complex Manifolds and Deformation of Complex Structures 1986 by Springer-Verlag. [32] Nekrasov N., Holomorphic bund les and many-body systems. PUPT-1534, Comm. Math. Phys.,180 (1996) 587-604; hep-th/9503157. [33] Chervov A.V., Talalaev D.V., Hitchin System on Singular Curves, Theoretical and Mathematical Physics, 2004, 140, 2, 10431072. Chervov A., Talalaev D., Hitchin systems Int.J.Geom.Meth.Mod.Phys.4:751-787,2007. on singular curves II. Gluing subschemes,

[34] Krichever I., Babelon O., Billey E., Talon M., Spin generalization of the Calogero-Moser system and the Matrix KP equation, arXiv:hep-th/9411160. [35] Kassel K., Quantum groups, Springer, 1995, Graduate Texts in Mathematics, 155, Berlin, New York. [36] Krichever I., Phong D., Symplectic forms in the theory of solitons, hep-th/9708170. D'Hoker E., Krichever I., Phong D., Seiberg-Witten Theory, Symplectic Forms, and Hamiltonian Theory of Solitons, hep-th/0212313. [37] Sklyanin E., Separation of variables in the Gaudin model. J.Sov.Math. 47: Zap.Nauchn.Semin. 164: 151-169, 1987. 2473-2488, 1989,

47


[38] Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61110. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111151. [39] Gutt S., An explicit -product on the cotangent bund le of a Lie group, Lett. Math. Phys. 7 (1983), 249258. [40] Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157216. [41] Fedosov B.V., A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213238. Fedosov B.V., Deformation quantization and index theory, Mathematical Topics, Vol. 9, Akademie Verlag, Berlin, 1996. [42] Talalaev D.V., The Quantum Gaudin System, Functional Analysis and Its Applications, 2006, 40, 1, 7377. [43] Sklyanin E., Separations of variables: new trends. Progr. Theor. Phys. Suppl. 118 (1995), 35­60. solvint/9504001. [44] Kulish P., Sklyanin E., Quantum spectral transform method. Recent developments, Lect. Notes in Phys. 151 (1982), pp. 61-119. [45] Kirillov A., Reshetikhin N., The Yangians, Bethe Ansatz and combinatorics, Lett. Math. Phys. 12, 3 (1986), pp. 199-208. [46] Molev A.I., Yangians and their applications. math.QA/0211288. [47] Bogolubov N.M., Izergin A.G., Korepin V.E., Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1993). [48] Mukhin E., Tarasov V., Varchenko A., Schubert calculus and representations of general linear group arXiv:0711.4072. [49] Rubtsov V., Silantiev A., Talalaev D., Manin Matrices, Quantum El liptic Commutative Families and Char-acteristic Polynomial of El liptic Gaudin model, SIGMA 5 (2009), 110-131. [50] Felder G., Conformal field theory and integrable systems associated to el liptic curves. Proc. ICM Zurich ¨ 1994, 1247-55, Birkh¨ auser (1994); Elliptic quantum groups, Proc. ICMP Paris 1994, 211-8, International Press (1995). [51] Tarasov V., Varchenko A., Smal l El liptic Quantum Group e , (slN ), Mosc. Math. J., 1, no. 2, (2001), 243­286, 303­304. [52] Enriquez B., Feigin B., Rubtsov , Separation of variables for Gaudin-Calogero systems, Compositio Math. 110, no. 1, (1998), 1­16. [53] Felder G., Schorr A., Separation of variables for quantum integrable systems on el liptic curves, math.QA/9905072, J. Phys., A 32, (1999), no. 46, 8001­8022. [54] Gould M. D., Zhang Y.-Z., Zhao S.-Y., El liptic Gaudin models and el liptic KZ Equations nlin/0110038. [55] Talalaev D.V., Bethe ansatz and isomondromy deformations, Theoretical and Mathematical Physics, 159, 2, 627-639. [56] Bolibrukh A.A., Holomorphic vector bund les on the Riemann sphere and the 21 st Hilbert problem, Journal of Mathematical Sciences (New York), 82 6 (1996), 3759 [57] Mu U., Sakka A., Schlesinger transformations for the Painlev´ VI equation, J. Math. Phys. 36 (3) gan e 1995. 48


[58] Griffiths Ph., Harris J., Principles of Algebraic Geometry, Wiley-Interscience, 1994, ISBN 0-471-05059-8. [59] Schorr A., Separation of variables for the 8-vertex SOS model with antiperiodic boundary conditions, PhD Thesis, ETH, Zurich, 2000. [60] Adler M., On a trace functional for formal pseudodifferential operators and the symplectic structure for the KdV type equations, Invent. Math. (1979) 50, 219-248. Kostant B., Quantization and Representation Theory, in: Representation Theory of Lie Groups, Proc. SRC/LMS Res. Symp., Oxford 1977. London Math. Soc. Lecture Notes Series, 34, 287-316, 1979. Symes W.W., Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 59, 13-51 (1980). [61] B. Feigin, E. Frenkel, Int. J. Mod. Phys. A7, Suppl. 1A 1992, 197-215. [62] Rybnikov L., Uniqueness of higher Gaudin hamiltonians, archiv:math/0608588. [63] Rybnikov L., Centralizers of some quadratic elements in Poisson-Lie algebras and a method for the translation of invariants. Russ. Math. Surv. 60, No.2, 367-369 (2005); translation from Usp. Mat. Nauk 60, No.2, 173-174 (2005) math.QA/0608586. [64] Ochiai, H., Oshima, T., and Sekiguchi, H. Commuting families of symmetric differential operators , Proc. of the Japan Acad. 70, Ser. A 2 1994 62­68. [65] Beilinson A., Drinfeld V., Quantization of Hitchin's integrable system and Heche eigensheaves. Preprint 1991. [66] Serre J.-P., Groupes algebriques et corps de classes. Paris, Hermann, 1959. [67] Iwasawa K., Local Class Field Theory, Oxford Univ. Press, 1986. [68] Frenkel E., Lectures on the Langlands Program and Conformal Field Theory, hep-th/0512172. [69] Vergne M., Al l what I wanted to know about the Langlands program and was afraid to ask, math.GR/0607479. [70] Lafforgue L., Chtoucas de Drinfeld et correspondance de Langlands, Inv. Math. 147 1-241 (2002). [71] Hausel T., Thaddeus , Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003) 197-229. Frenkel E., Gaitsgory D., Vilonen K., On the geometric Langlands conjecture, Journal of AMS 15 (2001) 367­417. Frenkel E., Gaitsgory D., Local geometric Langlands correspondence and affine Kac-Moody algebras, math.RT/0508382 Bezrukavnikov R., Braverman A., Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case, math.AG/0602255. [72] Talalaev D., Chervov A., Universal G-oper and Gaudin eigenproblem hep-th/0409007. [73] Knizhnik V.G., Zamolodchikov A.B., Nucl.Phys.B 247, (1984) 83-103. [74] Dixmier J., Enveloping algebras. Providence: American Mathematical Soc., 1991. [75] Frenkel E., Affine Algebras, Langlands Duality and Bethe Ansatz, q-alg/9506003.

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