AP08_2.vp 1 Introduction Some interrelations between classical integrable systems and field theories in dimensions 3 and 4 were proposed by N. Hitchin twenty years ago [1, 2]. This approach to inte- grable systems has some advantages. It immediately leads to the Lax representation with a spectral parameter, to prove in some cases the algebraic integrability and to find separated variables [3, 4]. It was found later that some well-known integrable systems can be derived in this way [5, 6, 7, 8, 9, 10, 11, 12]. It was demonstrated in [13] that there exists an integrable regime in � � 2 supersymmetric Yang-Mills theory in four di- mension, which is described by Sieberg and Witten [14]. A general picture of the interrelations between integrable mod- els and gauge theories in dimensions 4, 5 and 6 was presented in [15]. Some new aspects of the interrelations between integrable systems and gauge theories have recently recently been found in the framework of four-dimensional reformulation of the geometric Langlands program [16, 17, 18]. These lectures take into account this approach, but they are also based on [1, 2, 11, 12, 19, 20, 21, 22]. The derivation of integrable systems from field theories is based on symplectic or Poisson reduction. This construction is familiar in gauge field theories. The physical degrees of free- dom in gauge theories are defined upon imposing first and second the class constraints. First class constraints are analogs of the Gauss law generating gauge transformations. A combi- nation of the Gauss law and constraints coming from the gauge fixing yields second class constraints. We start with gauge theories that have some important properties. First, they have at least a finite number of inde- pendent conserved quantities. After reduction they will play the role of integrals of motion. Next, we assume that after gauge fixing and solving the constraints, the reduced phase space becomes a finite-dimensional manifold and its dimen- sion is twice the number of integrals. This property provides complete integrability. It is, for example, the theory of Higgs bundles describing Hitchin integrable systems [1]. This the- ory corresponds to a gauge theory in three dimension. On the other hand, a similar type of constraints arises in reducing the self-duality equations in four-dimensional Yang-Mills theory [1], and in four-dimensional � � 4 supersymmetric Yang-Mills theory [16] after reducing them to a space of di- mension two. We also analyze the problem of classifying integrable sys- tems. Roughly speaking two integrable systems are called equivalent, if the original field theories are gauged equiva- lent. We extend gauge transformations by allowing singular gauge transformations of a special kind. On the field the- ory side these transformations correspond to monopole configurations, or, equivalently, the inclusion of the t’Hooft operators [23, 24]. For some particular examples we establish in this way an equivalence of integrable systems of particles (Calogero-Moser systems) and integrable Euler-Arnold tops. It turns out that this equivalence is the same as equivalence of two types of R-matrices of dynamical and vertex type [25, 26]. Before considering concrete cases we point out the main definitions of completely integrable systems [20, 21, 27]. 2 Classical integrable systems Consider a smooth symplectic manifold � of dim( )� � 2l. It means that there exists a closed non-degenerate two-form w, and the inverse bivector � � � �( ),a b bc a c� , such that the space C�( )� becomes a Lie algebra (Poisson algebra) with re- spect to the Poisson brackets � �F G dF dG F Ga ab b, � �� � � � . Any H C� �( )� defines a Hamiltonian vector field on � � �H dH H Ha ab b� � �� � � � , . A Hamiltonian system is a triple ( , , )� � H with the Hamil- tonian flow � � � � � �t a a b bax H x H� �, A Hamiltonian system is called completely integrable, if it satisfies the following conditions � there exist l Poisson commuting Hamiltonians on � (inte- grals of motion) I1, …, Il. � Since the integrals commute the set �T I cc j j� �{ } is in- variant with respect to the Hamiltonian flows� �I j , . Then being restricted on Tc, Ij(x) are functionally independent for almost all x Tc� , i.e. det( )( )�a bI x 0. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 5 Acta Polytechnica Vol. 48 No. 2/2008 Lectures on Classical Integrable Systems and Gauge Field Theories M. Olshanetsky In these lectures we consider Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimension. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N, �) and explain the Symplectic Hecke correspondence for these systems. Keywords: Integrable systems, gauge theories, monopoles. In this way we come to the hierarchy of commuting flows on � � ��t jj Ix x x� ( ), . (2.1) Tc � is a submanifold Tc � �. It is is a Lagrangian sub- manifold, i.e. � vanishes on Tc. If Tc is compact and con- nected, then it is diffeomorphic to an l-dimensional torus. To- rus Tc is called the Liouville torus. In a neighborhood of Tc there is a projection p B: � � , (2.2) where the Liouville tori are generic fibers and the base of fibration B is parameterized by the values of the integrals. The coordinates on a Liouville torus (“the angle” variables) along with dual variables on B (“the action” variables) de- scribe a linearized motion on the torus. Globally, the picture can be more complicated. For some values of cj Tc ceases to be a submanifold. In this way the action-angle variables are local. Here we consider a complex analog of this picture. We assume that � is a complex algebraic manifold and the symplectic form � is a (2,0) form, i.e. locally in the coordinates ( , , , , )z z z zl l1 1 � the form is represented as � �� �a b a bdz dz, . General fibers of (2.2) are abelian subvarieties of �, i.e. they are complex tori �l �, where the lattice � satisfies the Riemann conditions. Integrable systems in this situation are called alge- braically integrable systems. Let two integrable systems be described by two isomorphic sets of the action-angle variables. In this case the integrable systems can be considered as equivalent. Establishing equiva- lence in terms of angle-action variables is troublesome. There is a more direct way based on Lax representation. Lax represen- tation is one of the commonly accepted methods for con- structing and investigating integrable systems. Let L x z( , ), M x z M x zl1( , ), , ( , )� be a set of l 1 matrices depending on x � � with a meromorphic dependence on the spectral param- eter z � �, where � is a Riemann surface. (It will be explained below that L and M are sections of some vector bundles over �.) This is called a basic spectral curve. Assume that the com- muting flows (2.1) can be rewritten in the matrix form � ��t jj L z L z M z( , ) ( , ), ( , )x x x� . (2.3) Let f be a non-degenerate matrix of the same order as L and M. The transformation � � �L f L f1 , � � � �M f f f M fj t jj 1 1� , (2.4) is called the gauge transformation because it preserves the Lax form (2.3). The flows (2.3) can be considered as special gauge transformations L t t f t t L f t tl l l( , , ) ( , , ) ( , , ),1 1 1 0 1� � �� � where L0 is independent on times and defines an initial data, and M f fj t j� �1� . Moreover, it follows from this representa- tion that the quantities tr L z j( ( , ))x are preserved by the flows and thereby can produce, in principle, the integrals of mo- tion. As we mentioned above, it is reasonable to consider two integrable systems to be equivalent if their Lax matrices are related by a non-degenerate gauge transformation. We relax the definition of the gauge transformations and assume that can have poles and zeroes on the basic spectral curve � with some additional restrictions on f. This equiva- lence is called the Symplectic Hecke Correspondence. This exten- sion of equivalence will be considered in details in these lec- tures. The following systems are equivalent in this sense: EXAMPLES � 1. Elliptic Calogero-Moser system � Elliptic GL( , )N � Top, [11]; � 2. Calogero-Moser field theory � Landau-Lifshitz equa- tion, [11, 10]; � 3. Painlevé VI � Zhukovsky-Volterra gyrostat, [12]. The first example will be considered in Section 4. The gauge invariance of the Lax matrices allows one to define the spectral curve � �� � � � � �( , ) det( ( , ))� �� z L x z� 0 (2.5) The Jacobian of � is an abelian variety of dimension g, where g is the genus of �. If g l� � 1 2 dim � then � plays the role of the Liouville torus and the system is algebraical- ly integrable. In generic cases g > l and to prove algebraic integrability one should find additional reductions of the Jacobians, leading to abelian spaces of dimension l. Finally we formulate two goals of these lectures � to derive the Lax equation and the Lax matrices from a gauge theory; � to explain the equivalence between integrable models by inserting t’Hooft operators in gauge theory. 3 1d field theory The simplest integrable models, such as the rational Calogero-Moser system, the Sutherland model, and the open Toda model can be derived from matrix models of a finite order. Here we consider a particular case – the rational Calogero-Moser system (RCMS) [32, 33]. 3.1 Rational Calogero-Moser System (RCMS) The phase space of the RCMS is � �� RCM NC� �2 ( , )v u , v � ( , , )v vN1 � , u � ( , , )u uN1 � with the canonical symplectic form � RCM j j j N d v d u� � � � 1 , � �v uj k jk, � � . (3.1) The Hamiltonian describes interacting particles with complex coordinates u � ( , , )u uN1 � and complex momenta v � ( , , )v vN1 � H v u u RCM j j N j kj k � � � � � �12 12 1 2 2 ( ) . 6 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 The Hamiltonian leads to the equations of motion �t j ju v� , (3.2) � t j j kj k v u u � � � � �2 3 1 ( ) . (3.3) The equations of motion can be put in the Lax form �t L L M( , ) [ ( , ), ( , )]v u v u v u� . (3.4) Here L, M are the N×N matrices of the form L P X� , M D Y� , P diag v vN� ( , , )1 � , X u ujk j k� � � ( ) 1, (3.5) Y u ujk j k� � � � ( ) 2, D d dN� diag( , , )1 � , (3.6) d u uj j k k j � � � � ( ) 2. The diagonal part of the Lax equation (3.4) implies �t diagP X Z� [ , ] . It coincides with (3.3). The non-diagonal part has the form � �t nondiagX P Y X Y X D� �[ , ] [ , ] [ , ] . It can be found that [ , ] [ , ]X Y X Dnondiag � and the equa- tion �t X P Y� [ , ] coincides with (3.2). The Lax equation produces the integrals of motion Im � 1 m Lmtr( ), �t mLtr( ) � 0, m N�1 2, , ,� . (3.7) It will be proved later that they are in involution � �I Im n, � 0. In particular, I H RCM2 � . Eventually, we come to the RCMS hierarchy � ��j jf I f( , ) , ( , )v u v u� . (3.8) 3.2 Matrix mechanics and RCMS This construction was proposed in Ref. [35, 36]. Consider a matrix model with the phase space � � �gl gl( , ) ( , )N N� � � � ( , ) A , , ( , )A N�gl � (These notations will be justified in next section) dim � � 2 2N . The symplectic form on � is � � � � ��tr( ) , d dA d dAj,k j,k j k . (3.9) The corresponding Poisson brackets have the form � � j,k j,k ki jlA, � � � . Choose N commuting integrals Im � 1 m mtr( ) , � �I Im n, � 0, m N�1 2, , ,� . Take as a Hamiltonian H I� 2. Then we come to the free mo- tion on � � ��t H � �, 0 , (3.10) � ��t A H A� �, . (3.11) Generally, we have a free matrix hierarchy � j � 0, � j jA � ��, (�j jI� { , }). 3.2.1 Hamiltonian reduction for RCMS The form � and the integrals Im are invariant with resect to the action of the gauge group � � GL( , )N � , � �f f1 , A f A f� �1 , f N�GL( , )� . The action of gauge Lie algebra Lie N( ) ( , )� � gl � is repre- sented by the vector fields V� � [ , �], V� A A� [ , �]. (3.13) Let � be the contraction operator with respect to the vector field V� ( � �� (, Vj k �)jk �� jk ) and �� = d �+ �d is the corresponding Lie derivative. The invariance of the sym- plectic form and the integrals means that �� � � 0 , �� Im � 0 . Since the symplectic form is closed d� � 0, we have d �� � 0. Then on the affine space � the one-form �� is exact ��� dF A( , , �). (3.14) The function F A( , , �) is called the momentum Hamiltonian. The Poisson brackets with the momentum Hamiltonian gen- erate the gauge transformations: � �F f A, ( , ) � �� f A( , ) . The explicit form of the momentum Hamiltonian is F A( , , �)� tr( �[ , A]). Define the moment map � : *( ) ~ ( , )� � Lie gauge group Ngl � � ��( , ) , A A� , � �( , ) , A A� . (3.15) Let us fix its value as � �� , A J� , (3.16) J � � � � � � � � � � � � � 0 1 1 1 0 1 1 1 1 0 � � � � � � � � � � . (3.17) It follows from the definition of the moment map that (3.16) is the first class constraints. In particular, �F A( , , �), F A( , , �' �) ( , ,� F A [�, �']). Note, that matrix J is degenerate and is conjugated to the diagonal matrix diag( , , )N � � �1 1 1� . Let �0 be a subgroup of the gauge group preserving the moment value � �� �0 1� � ��f f Jf J , (dim( ) ( ) )�0 21 1� � N . In other words �0 preserves the surface in � � �F J A J� � �1( ) [ , ] . (3.18) Let us fix a gauge on this surface with respect to the �0 action. It can be proved that generic matrices A can be diago- nalized by �0 f Af u un � � �1 1u diag( , , )� , f ��0. (3.19) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 7 Acta Polytechnica Vol. 48 No. 2/2008 In other words, we have two conditions – first class con- straints (3.16) and gauge fixing (3.19). The reduced phase space �red is the result of putting both types of constraints � � � � red F J� � �/ / ( ) /1 0 . It has dimension dim( ) dim( ) dim( ) dim( ) ( ) � � � � red N N N N � � � � � � � � 0 2 2 2 12 2 2 1 � � � Let us prove that � �red RCM� and that the hierarchy (3.12) being restricted to �RCM coincides with the RCMS hi- erarchy (3.8). Let f ��0 diagonalize A in (3.19). Define L f f� �1 (3.20) Then it follows from (3.10) that L satisfies the Lax equation � �t t L L M � � �0 [ , ], (M f ft� � �1� ). The moment constraints (3.18) allow one to find the off-diagonal part of L. Evidently, it coincides with X (3.5). The diagonal elements of L are free parameters. In a similar way, the off-diagonal part Y (3.6) of M can be derived from the equation of motion for A (3.11). Thereby, we come to the Lax form of the equations of motion for RCMS. Since � L and A � u, the symplectic form � (3.9) coincides on �RCM with �RCM (3.1). It follows from (3.20) that the integrals (3.7) Pois- son commute. Therefore, we obtain RCMS hierarchy. The same system can be derived starting with matrix mechanics based on SL( , )N � . In this case I1 0� �tr and thereby in the reduced system v j� � 0. 3.2.2 Hamiltonian reduction for 4d Yang-Mills theory In this subsection we take a step aside to illustrate the Hamiltonian reduction in terms of the familiar phase space of the Yang-Mills theory. For this purpose consider 4d YM the- ory with a group G in the Hamiltonian formalism [37, 38]. The phase space is generated by the space components on � 3 1 2 3� ( , , )x x x of the vector potential and the electric field � �� � � �A E( , , ), ( , , )A A A E E E1 2 3 1 2 3 . where E F A A A Aj j j j j� � � 0 0 0 0� � [ , ]. Here we have suppressed the Lie algebra indices. It is a symplectic space with the canonical form � � � � �� �� � d d d E d Aj j j E A � � 3 3 1 3 tr( ) . The Hamiltonian is quadratic in fields and has the form H � � 1 2 2 2 3 E B � , where B � ( , , )B B B1 2 3 is the magnetic field B j � �jklFkl ��jkl( [ , ])� �k j j k k jA A A A� . We assume that the fields are smooth and vanish at infinity such that the Hamiltonian and the symplectic form are well defined. The Hamiltonian defines the classical equations of motion �t j jA E� , � �t j k k kj k E A F� �[ , ]. The Hamiltonian and the form are invariant with respect to the gauge transformations A A A� � � �f f df f f1 1 , e E E� � �f f f1 . where d � ( , , )� � �1 2 3 . We assume that f is a smooth map f C G� � ��� ( )�3 , vanishing at infinity and at some marked points f a( )y � 0, y a a a ay y y� ( , , )1 2 3 , ( , , )a n�1 � . Infinitesimal gauge transformations defines the vector field on the phase space V� E E� [ , �], V� A � d� [ ,A �], � � Lie( )� . The corresponding momentum Hamiltonian is F( , ,E A �) � � � 3 �� ( [ , ])�j j j j j E A E � � 1 3 , (compare with (3.14)). Therefore the moment takes the form � �( , [ , ]E A) = j j j j j E A E � � 1 3 . This is an element of the gauge co-algebra Lie * ( )� . In other words the moment belongs to the map of the phase space � to the distributions on �3 with values in Lie G* ( ) . Let us fix it as � � �j j j j j a a a E A E � � � � �[ , ] ( ) 1 3 x y , (3.21) where �a Lie� * ( )� . The moment constraints (3.21) are nothing else than the Gauss law, and �a are electric charges. To come to the reduced phase space � � �red � / / we should add a gauge fixing condition to the Gauss law. Note that the gauge transformations vanish at the points ya, and in this way they preserves the right hand side of (3.21). Starting with six fields ( , )A Ej j defining �, we put two types of con- straints – the Gauss law and gauge fixing. Roughly speaking, they kill two fields and the reduced phase space describes the “transversal degrees of freedom”. 4 3d field theory 4.1 Hitchin systems 4.1.1 Fields Define a field theory on (2 1) dimensional space-time of the form � � g n, , where is a Riemann surface of genus g with a divisor D x xn� ( , , )1 � of n marked points. The phase space of the theory is defined by the following field content: 1) Consider a vector bundle E of rank N over �g,n equipped with the connection �� �! "d dzz . It acts on the sections 8 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 s s sT N� ( , , )1 � of E as �� � d s s As� . The vector fields A z z( , ) are C� maps � g n N, ( , )� gl � . 2) The scalar fields (the Higgs fields) ( , )z z dz" , �: ( , ),g n N� gl � . The Higgs field is a section of the bundle � �( , ) ,( , ) 1 0 g n EEnd . This means that acts on the sections s s dzj kj j� " . We assume that has holo- morphic poles at the marked points ~ a az x� � Let ( , , , , , )� � � �1 1� �g g be a set of fundamental cycles of � g n, , ( � � � �j j j jj � �# �1 1 1). The bundle E is defined by the monodromy matrices ( , )Q j j� � j js Q s: � �1 , � j j s:� �1 Similarly, for A and we have � �j j j j jA Q Q Q AQ: � � �1 1, � �j j j j jA A: � � �� � � �1 1 �j j jQ Q: � �1, bj j j: � �� �1. (4.1) 3) The spin variables are attributed to the marked points S Na � gl( , )� , a n�1, ,� , S g S ga a� �1 0( ) , where S a ( )0 is a fixed element of gl( , )N � . In other words, S a belong to coadjoint orbits a of gl( , )N � . They play the role of non-abelian charges located at the marked points. Let � �T� , ( , , )� �1 2 � N be a basis in the Lie algebra gl( , )N � , [ , ] ,T T C T� � � � � �� . Define the Poisson structure on the space of fields: 1) Darboux brackets of the fields ( , )A : A z z A z z T( , ) ( , )�� � � � , ( , ) ( , )w w w w T�� � � � . � � � � � � �( , ), ( , ) ( , )w w A z z T T z w z w� � � , (��= trace in ad) 2) Linear Lie brackets for the spin variables: S a aS T�� �� � � �S S C Sa b a b a� � ���� ��, ,� . In this way we have defined the phase space � � ( , , )A a S . (4.2) The Poisson brackets are non-degenerate and the space � is symplectic with the form � � � �� � � ��� � 0 1 a a a a n z x z x g n ( , ) ,� , (4.3) �0 � �� D DA g n � , , (4.4) �a aD S g Dg� ��( )1 . (4.5) The last form is the Kirillov-Kostant form on the coadjoint orbits. The fields ( , ) A are holomorphic coordinates on � and the form �0 is the (2,0)-form in this complex structure on �. Similarly, ( , )S g ga �1 are holomorphic coordinates on the orbit a, and �a is also (2,0) form. 4.1.2 Hamiltonians The traces j ( j N�1, ,� ) of the Higgs field are peri- odic (j, 0)-forms� �( , ) ,( ) j g n 0 with holomorphic poles of order j at the marked points. To construct integrals from j one should integrate them over � g n, and to this end prepare (1, 1)-forms from the (j, 0)-forms. For this purpose consider the space of smooth (1�j, 1)-differentials � �( , ) ,( \ ) 1 1� j g n D vanishing at the marked points. Locally, they are represented as � � � �j j z jz z dz� "�( , )( ) 1 . In other words �j are (0,1)-forms taking values in degrees of vector fields on � g n D, \ . For example, �2 is the Beltrami differential. The product j j� can be integrated over the surface. We explain below that �j can be chosen as elements of the ba- sis in the cohomology space H Dg n j1 1( \ , ),� " � . This space has dimension n H D j g jn j g jj g n j� � � � � � � � � " �dim ( \ , ) ( )( ) , 1 1 2 1 1 1 1 � (4.6) Let �j,k be a basis in H Dg n j1 1( \ , ),� " � , (k n j�1, ,� ). The product � j k j , can be integrated to define the Hamiltonians I jj k j k j g n , , , � � 1 � � , J N�1, ,� . (4.7) It follows from (4.6) that the number of independent integrals n j� for GL( , )N � is d n g N n N N N g n j j N , , ( ) ( ) � � � � � � 1 1 12 2 1 . (4.8) Since � 0 for SL( , )N � the number of independent integrals is d n g N n N N N g n j j N , , ( )( ) ( ) � � � � � � � 1 1 12 2 2 . (4.9) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 9 Acta Polytechnica Vol. 48 No. 2/2008 Fig. 1: ( , , )x x1 4� –marked points The integrals I(j, k) are independent and Poisson commute � �I Ij k j k( , ) ( , ),1 1 2 2 0� . (4.10) Thus we come to dN g n, , commuting flows on the phase space �( , , )A a S � �� �t I j,k j,k � ! �, 0, (4.11) � � � t A j,k j,k j� � 1, (4.12) � �tj,k aS � 0.Res z x a a S� � (4.13) 4.1.3 Action and gauge symmetries The same theory can be described by the action � � � � � �� � � ����� �� � � � j k k n j N a a a A z x z x g nj k j , ,, ( , ) � 12 S g g I dta j k a j k a n j k � � � � � � �� 1 1 � , , , where the time-like Wilson lines at the marked points are in- cluded. The action is gauge invariant with respect to the gauge group � ��� �� �smooth maps : GL� g n N, ( , ) The elements f ��� are smooth and have the same monodromies as the Higgs field (4.1). The action is invariant with respect to the gauge transformations A f f f Af� � �1 1� , � �f f1 , g g fa a a� , S Sa a a af f� �( ) 1 , f f z za z xa � �( , ) . Consider the infinitesimal gauge transformations V A A� �� �� [ , ], V� � � [ , ], V g g xa a a� �� ( ) , V S S x a a a� �� [ , ( ) ], � � Lie( )�� . The Hamiltonian F generating the gauge vector fields �� � DF has the form F A z x z xa a a a n g n � � � � � �� � � �( [ , ] ( , )) , � S 1 . The moment map � : ( , , ) * ( )� �A Liea S � � , � � �� � � � � � [ , ] ( , )A z x z xa a a a n S 1 . The Gauss law (the moment constraints) takes the form � � � � � � �[ , ] ( , )A z x z xa a a a n S 1 . (4.14) Upon imposing these constraints the residues of the Higgs fields become equal to the spin variables by analogy Res z x a a S� � with the Yang-Mills theory, where the Higgs field corresponds to the electric field and Sa are the analog of the electric charges. The reduced phase space � � red aA +� ( , , ) / ( S Gauss law) (gauge fixing) defines the physical degrees of freedom, and the reduced phase space is the symplectic quotient � � � red aA� ( , , ) / / S � (4.15) 4.1.4 Algebra-geometric approach The operator ��d acting on sections defines a holomorphic structure on the bundle E. A section s is holomorphic if ( )� �A s 0 The moment constraint (4.14) means that the space of sections of the Higgs field over � g D\ is holomorphic. Consider the set of holomorphic structures � �� � dA on E. Two holomorphic structure are called equivalent if the cor- responding connections are gauge equivalent. The moduli space of holomorphic structures is the quotient � ��. Generi- cally the quotient has very singular structure. To have a rea- sonable topology one should consider the so-called stable bundles. The stable bundles are generic and we consider the space of connection � stable corresponding to the stable bun- dles. The quotient is called the moduli space of stable holomorphic bundles � � �( , , )N g n stable� . It is a finite-dimensional manifold. The tangent space to �( , , )N g n is isomorphic to H Eg n 1( , ),� End . Its dimension can be extracted from the Riemann-Roch theorem and for curves without marked points ( )n � 0 dim ( , ) dim ( , ) ( ) dimH E H E g G0 1 1� �End End� � � . For stable bundles and g �1 and dim ( , )H E0 1� End � and dim ( , , ) ( )� N g g N0 1 12� � for GL( , )N � , and dim ( , , ) ( )( )� N g g N0 1 12� � � for SL( , )N � . Thus, in the absence of the marked points we should con- sider bundles over curves of genus g $ 2. But the curves of ge- nus g � 0 and 1 are important for applications to integrable systems. Including the marked points improves the situation. We extend the moduli space by providing an additional data at the marked points. Consider an N-dimensional vector space V and choose a flag Fl V V V VN� � � �( )1 2 � , where Vj is a subspace in Vj 1. Note that a flag is a point in a homoge- neous space called the flag variety Fl N B�GL( , )� , where B is a Borel subgroup. If ( , , )e eN1 � is a basis in V and Fl is a flag � �Fl V a e V a e a e V VN� � � �1 11 1 2 21 1 22 2{ }, { },� , 10 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 then B is the subgroup of lower triangular matrices. The flag variety has the dimension 1 2 1N N( )� . The moduli space �( , , )N g n is the moduli space �( , , )N g 0 equipped with maps g Na �GL( , )� of V to the fibers over the marked points V E xa � , preserving Fl in V. In other words ga are defined up to the right multiplication of B and therefore we supply the moduli space �( , , )N g 0 with the structure of the flag variety GL( , )N B� at the marked points. We have a natural “forget- ting” projection � : ( , , ) ( , , )� �N g n N g� 0 . The fiber of this projection is the product of copies of the flag varieties. Bundles with this structure are called the quasi-parabolic bun- dles. The dimension of the moduli space of quasi-parabolic holomorphic bundles is dim ( , , ) ( , , ) ( )� �N g n N g nN N� �0 1 2 1 . For curves of genus g �1, dim ( , , )� N g n is independent on the degree of the bundles d E c E� �deg( ) (det )1 . In fact, we have a disjoint union of components labeled by the corre- sponding degrees of the bundles � �� ( )d� . For elliptic curves ( )g �1 one has dim ( , ) dim ( , )H E H E1 0� �End End� , and dim ( , )H E0 � End does depend on deg(E). Namely, dim( ( , , , )) ( , )� N d N d1 0 � g.c.d. . (4.16) In this case the structure of the moduli space for trivial bundles (i.e. with deg( )E � 0) and, for example, for bundles with deg( )E �1 is different. Now consider the Higgs field . As we already mentioned, defines an endomorphism of the bundle E � � � �: ( , ) ( , )( ) , ( , ) , 0 1 0 g n g nE E� , s s dz� " . Similarly, they can be described as sections of � � C g n D E K� " ( ) ,( , ) 0 End . Here KD is the canonical class on � \ D that locally apart from D is represented as dz. Re- memeber that has poles at D. On the other hand, as it fol- lows from the definition of the symplectic structure (4.4) on the set of pairs ( , ) A , that the Higgs field plays the role of a “covector” with respect to vector A. In this way the Higgs field is a section of the cotangent bundle T stable* � . The pair of the holomorphic vector bundle and the Higgs field (E, ) is called the Higgs bundle. The reduced phase space (4.15) is the moduli space of the quasi-parabolic Higgs bundles. It is the cotangent bundle � � red T N g n d� * ( , , , ) . (4.17) Due to the Gauss law (4.14), the Higgs fields are holo- morphic on � \ D. Then on the reduced space �red �� "H E Kg n D 0( , ), End* . (4.18) A part of T N g n d* ( , , , )� comes from the cotangent bun- dle to the flag varieties T G B a*( ) located at the marked points. Without the null section T G B a*( ) is isomorphic to a unipotent coadjoint orbit, while the null section is the trivial orbit. Generic coadjoint orbits passing through a semi-simple element of gl( , )N � are affine spaces over T G B a*( ) . In this way we come to the moduli space of the quasi-parabolic Higgs bundles [29]. It has dimension dim ( ) ( )� red N g N N n� � �2 1 2 12 (4.19) This formula is universal and valid also for g � 0 1, and does not depend on deg(E). At first glance, for g �1 this formula appears to contradict to (4.16). In fact, we have a residual gauge symmetry generated by a subgroup of the Cartan group of GL( , )N � . The symplectic reduction with re- spect to this symmetry kill these degrees of freedom and we come to dim ( )� red N N n� �2 1 (see (4.6). We explain this mechanism on a particular example in Section 4.2.2. Formula (4.6) suggests that the phase spaces corresponding to bundles of different degrees may be symplectomorphic. We will see soon this is the case. It follows from (4.18) that �j g n D jH K� 0( , ), . In other words j are meromorphic forms on the curve with poles of the order j at the divisor D. Let �jk be a basis of H Kg n D j0( , ),� . Then 1 1 j Ij jk jk k n j � � � � . (4.20) The basis �jk in H Dg n j1 1( \ , ),� " � introduced above is dual to the basis �jk � � � �jk lm j l k m g n� , � � . Then the coefficients of the expansion (4.20) coincide with the integrals (4.7). The dimensions nj (4.6) can be calcu- lated as dim ( , ),H Kg n D j0 � . Symplectic reduction preserves the involutivity (4.10) of the integrals (4.7). Since (see(4.8), (4.9)) we come to integrable systems on the moduli space of the quasi-parabolic Higgs bundles �red. For GL( , )N � the Liouville torus is the Jacobian of the spectral curve � (2.5). Consider bundles with the structure group replaced by a reductive group G. The algebraic inte- grability for g �1and G is a classical simple group was proved in [1]. The case of exceptional groups was considered in [30, 31]. 4.1.5 Equations of motion on the reduced phase space Let us fix a gauge A A� 0. For an arbitrary connection A define a gauge transform © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 11 Acta Polytechnica Vol. 48 No. 2/2008 1 2 dim ( , , )* � N g n � number of integrals f A A A[ ] : � 0, A f f A f A Af A0 1 1� � �( )[ ] [ ] [ ]� Then f A[ ] is an element of the coset space � �� 0, where the subgroup �0 preserves the gauge fixing � ��0 0 0� �f f A f� [ , ] . The same gauge transformation brings the Higgs field to the form L f A f A� �1[ ] [ ] . The equations of motion for (4.11) in terms of L take the form of the Lax equation (4.21) where M f A f Aj k j k, ,[ ] [ ]� �1 � . Therefore, after reduction the Higgs field becomes the Lax matrix. Equations (4.21) de- scribes the Hitchin integrable hierarchy. The matrix M j k, can be extracted from the second equa- tion (4.12) � � �M M A A Lj k j k j k j j k, , , ,[ , ]� � � � 0 0 1 . (4.22) The Gauss law restricted to �red takes the form � �L A L x xa a a a n � � �[ , ] ( , )0 1 S (4.23) Thus, the Lax matrix is the matrix Green function of the op- erator � A0 on � g n, acting in the space� � ( , ) ,( , ) 1 0 g n EEnd . The linear system corresponding to the integrable hierar- chy takes the following form. Consider a section � of the vec- tor bundle E. The section is called the Baiker-Akhiezer func- tion if it is a solution of the linear system for 1 0 2 0 3 0 0. ( ) , . ( ) , . ( ) ., , � � � � � � � � � � � � % � % A L Mj k j k The first equation means that � is a holomorphic section. Compatibility of the first equation and the second equation is the Gauss law (4.23) and the first equation and the last equa- tion is the Lax equations (4.21). In terms of the Lax matrix the integrals of motion Ij,k are expressed by the integrals (4.7) I j L x zjk jk j g n � � 1 � tr( ( , )) ,� (4.24) or by the expansion (4.20) 1 1 j L Ij jk jk k n j � � � � . (4.25) The moduli space of the Higgs bundles (4.17) is para- meterized by the pairs ( , )A 0 L . The projection (2.2) T N g n B H D Kg n D j j N * ,( , , ) ( \ , )� � � � � 0 1 � is called the Hitchin fibration. An illustrative example of the Hitchin construction is the Higgs bundles over elliptic curves. These cases will be de- scribed explicitly in following subsections. 4.2 N-body Elliptic Calogero-Moser System (ECMS) 4.2.1 Description of the system Let C� be an elliptic curve � � �( ) � , Im� � 0. The phase space �ECM of ECMS is described by N complex coordinates and their momenta u v � � � � ( , , ) ( ) ( u u u C v N j1 � � coordinates of particles, 1, , ) ( )� v uN j � � � � � � momentum vector with the Poisson brackets { , }v uj k jk� � . The Hamiltonian takes the form H u uCM j k j k � & � ' �12 2 2v ( ). (4.26) Here 2 is a coupling constant and&( )z – is the Weiersch- trass function. It is a double periodic meromorphic function & �& &( ) ( ) ( )z z z1 � with a second order pole & �( )~z z 2, z � 0. The system has the Lax representation [39] with the Lax matrix L iV XCM � , V v vN� diag( , , )1 � , (4.27) X z z u u u u z x ix jk j k j k� � � �� � � � � � � � � � � � e e ( ) ( , ), ( ) exp 2 (4.28) where � � � � � ( , ) ( ) ( ) ( ) ( ) u z u z u z � � 0 , (4.29) and � � � � ( ) ( ) exp ( ) , exp z q n n nz q i n n � � � � � � � � � � � 1 8 1 2 1 2 1 2 Z � (4.30) is the standard theta-function with a simple zero at z � 0 and the monodromies � �( ) ( )z z � �1 , � � ��( ) ( )z q e ziz � � � � 1 2 2 . (4.31) Then from (4.31) that � �( , ) ( , )u z u z �1 , � � �( , ) ( ) ( , )u z u u z � �e , (4.32) and �( , )u z has a simple pole at z � 0 Res u z z�( , ) � �0 1 (4.33) 12 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 �j k j kL L M, ,[ , ]� Fig. 2 4.2.2 ECMS and Higgs bundles [6, 7] To describe ECMS as a Hitchin system consider a vector bundle E of rank N and degree 0 over an elliptic curve �1,1 with one marked point. We assume that the curve is isomor- phic to C� �� � � �( ). The quasi-parabolic Higgs bundle T E* has coordinates � �� 0 � ( , ), ( , ),z z A z z S , , ( , )A N�gl � , S � , where is a degenerate orbit at the marked point z � 0 � � � � � ��S g S g g N S J1 0 0GL( , ),� , and J is the matrix (3.17). The orbit has dimension dim( ) � �2 2N . For degree zero bundles the monodromies around the two fundamental cycles can be chosen as Q Id1 � and �1 � e u( ), where e u( ) ((exp , , exp )� diag iu iuN2 21� �� . A section with these monodromies is s s sT N� ( , , )1 � , s u zj j� �( , ) . (4.34) where �( , )u zj is (4.29). It follows from (4.32) that the section has the prescribed monodromies. For the fields and the gauge group we have the same monodromies A z A z( ) ( ) �1 , ( ) ( )z z �1 , A z A z( ) ( ) ( ) ( ) � �� e u e u , ( ) ( ) ( ) ( )z z � �� e u e u , f z z f z z( , ) ( , ), �1 1 f z z f z z( , ) ( ) ( , ) ( ) � �� � e u e u . It can be proved that for bundles of degree zero generic connections A f f� � �� 1 is trivial and therefore A A� �0 0. (4.35) This means that stable bundles E of rank N are decom- posed into the direct sum of line bundles E j N j� � �1� , with the sections (4.34). The elements uj are the points of the Jacobian Jac( )�� . They play the role of coordinates, and thereby, C Jac� �~ ( )� . This gauge fixing is invariant with respect to the constant diagonal subgroup D0. It acts on the spin variables S � . This action is Hamiltonian. The moment equation of this action is diag( ) � 0. This condition dictates the form of S J0 � . Gauge fixing allows one to kill the degrees of freedom related to the spin variables, because dim( ) ( ) � �2 1N and dim( )D N0 1� � . Thus, the symplectic quotient is a point (dim( / / ) ) D0 0� . Remark 4.1 One can choose an arbitrary orbit �. In this case we come to the symplectic quotient �//D0. It has di- mension dim( ) ( ) � �2 1N . Now consider solutions the moment equation (4.23) with the prescribed monodromies and prove that becomes the Lax matrix � � L V XCM (4.27). Since A0 0� , V does not contribute in (4.23) and its elements are free parame- ters. We identify them with momenta of the particles V v vN� diag( , , )1 � . Due to the term with the delta-function in (4.23) the off-diagonal part should have a simple pole with the residue J and the prescribed monodromies. It follows from (4.32) and (4.33) that Xjk satisfies these conditions. They uniquely fix its matrix elements. The reduced space is described by the variables v and u. The symplectic form on the reduced space L A D v D uCM j j, , 0 1 1� � �� � leads to the brackets { , }v uj k jk� � . From the general construction the integrals of motion come from the expansion of tr( ) ( , , )L zCM j v u . They are dou- ble periodic meromorphic functions with poles at z � 0. This is a finite-dimensional space generated by a basis of derivative of the Weierschtrass functions. They are elements of the basis � jk in (4.25). 1 0 2j L z I I z I zCM j j CM j CM jj CM jtr( ) ( , , ) ( ) ( )( )v u � & &� . (4.36) There are N N( ) �1 2 1 integrals. Due to a special choice of the orbit only N �1 integrals are independent. In particular, 1 2 tr( ) ( , , ) ( )L z H zCM CM2 2v u � � & . For generic orbits (see Remark 4.1) the Hamiltonian take the form H S S u uCM jk kj j k j k � & � ' �12 2v ( ) . This is ECMS with spin [34]. Note, that Ij, j are the Casimir functions defining a generic orbit . Therefore we have N N N NN( ) ( )( ) �� � � �1 2 1 2 1 1 commuting integrals of motion. The number of independent commuting integrals is always equal to 1 2 dim( ) . 4.3 Elliptic Top (ET) on GL( , )N � 4.3.1 Description of the system The elliptic top is an example of Euler-Arnold top related to the group GL( , )N � . Its phase space is a coadjoint orbit of GL( , )N � . The Hamiltonian is a quadratic form on the co- algebra g* *� gl( , )N � . The ET is an integrable Euler-Arnold top. Before defining the Hamiltonian introduce a special ba- sis in the Lie algebra gl( , )N � . Define the finite set � � � � �N N N ( ) ( )2 � � , ~ ( ) \ ( , )( )� � � � �N N N 2 0 0� � and let eN i N x x( ) exp� 2� . Then a basis is generated by N2 1� matrices © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 13 Acta Polytechnica Vol. 48 No. 2/2008 T N i QN� � � � � � � � � � � � � 2 2 1 2 1 2e � , � � �� �( , ) ~( ) 1 2 2 �N , where Q NN N� �diag( , ( ), , ( ))1 1 1e e� , (4.37) � � � �Ej j j N N , , ,(mod ) 1 1 (4.38) The commutation relations in this basis have a simple form [ , ] sin ( )T T N N T� � � � � � � �� . Let S � � �� S TN �� ��( ) \( , ) * 2 0 0 g . The Poisson brackets for the linear functions S� come from the Lie brackets � �S S N N S� � � � � � � �, sin ( )� . The phase space �ET of the ET is a coadjoint orbit � �� ET g g g N~ , ( , )*� � � ��S S Sg 0 1 GL � . A particular orbit passes through S 0 � J, as for the spinless ECMS. The Euler-Arnold Hamiltonian is defined by the qua- dratic form H ET � � ( 1 2 tr( ( ))S J S , where J is diagonal in the basis T� J S( ) : ,S S� � ��& & �& � � � � � �� � � � �1 2 N , � � �N ( )2 . The equations of motion corresponding to this Hamil- tonian take the form �t ETHS S J S S� �{ , } [ ( ), ], � � � � �� � � � � � tS N S S N N � & � � � sin ( ) ~( ) � 2 . 4.3.2 Field theory and Higgs bundles The curve �1,1 is the same as for the Calogero-Moser sys- tem. Consider a vector bundle E of a rank N and degree one over �1,1. It is described by its sections s s z z s z zN� ( ( , ), , ( , ))1 � with monodromies s z z Q s z zT T( , ) ( , ) � �1 1 1 , s z z s z zT T( , ) ~ ( , ) � �� � � 1 , (4.39) where Q is (4.37), ~ ( ) � �� � eN z � 2 , and � is (4.38). Since det Q � )1 and det ~ ( ) � � ) � e1 2 z � the determinants of the transition matrices have the same quasi-periods as the Jacobi theta-functions. The theta-functions have a simple pole on �1,1. Thereby, the vector bundle EN has degree one. The Higgs bundle has the same field content as ECMS � �� � A, , S , A N, ( , ) �gl � , S � . The orbit � � ��{ , ( , )}S Sg g g N1 0 GL � is located at the marked point z � 0. It follows from (4.39) that the fields , A have the monodromies A z Q A z Q( ) ( ) � �1 1, ( ) ( )z Q z Q � �1 1, A z A z( ) ( ) � �� � � 1, � �( ) ( )z z � �� 1. The group of the automorphisms �� � { }f of E should have the same monodromies f z Q f z Q( ) ( ) � �1 1, f z f z( ) ( ) � �� � � 1. Due to the monodromy conditions the generic field A is gauge equivalent to the trivial f Af f f� � �1 1 0� . Therefore A f A f A� � �� [ ] [ ]1 . (4.40) This allows us to choose A � 0 as an appropriate gauge. It means that there are no moduli of holomorphic vector bundles. More precisely, the holomorphic moduli are related only to the quasi-parabolic structure of E related to the spin variables S. The monodromies of the gauge matrices prevent the existence of nontrivial residual gauge symmetries. Let f A z z[ ]( , ) be a solution of (4.40). Consider the transformation of by solutions of (4.40) L A g z z f A z z f A z zET [ , ]( , ) [ ]( , ) [ ]( , )� � 1 (4.41) The moment constraints (4.14) take the form � �L z zET � ( , )S The solution takes the form where � � �� � � ( ) ( ) ( , )z z zN t N � e 2 1 2 . The Lax matrix was found in [40] using another approach. It is the Lax matrix of the vertex spinchain. The Lax matrix is meromorphic on �1,1 with a simple pole with ResLET z� �0 S. The monodromies of ��(z) are read off from (4.32) � � � �� �( ) ( ) )z zN �1 2e , � � � �� �( ) ( ) ( )z zN � �e 1 Then LET have the prescribed monodromies. The reduced phase space �ET is the coadjoint orbit: � �� ET g g� � � �S S 0 1 , S � � �� S TN � �� g * \( , )( )� 2 0 0 . The symplectic form on �ET is the Kirillov-Kostant form (4.5). For a particular choice of the orbit passing through J (ref J) its dimension coincide with the dimension of the phase of the spinless ECMS dim dim� �ET CMS N� � �2 2. This is not occasional and we prove below that �CM is sym- plectomorphic to �ET. Since the traces tr( )LET j are double periodic and have poles at z � 0 the integrals of motion come from the expan- sion (see (4.36)) 14 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 L S z TET N � �� � � �� � ( )( ) \( , )� 2 0 0 , tr( ( )) ( ) ( ), , , ( )L z I I z I zET k k k k k k� & & �0 2 2 � In particular, tr( ) ( )L H C zET ET2 2� & . The coefficients Is,k are in involution � �I s k j, ,, Im � 0. In particular, all functions Is, k Poisson commute with the Hamiltonian HET. Therefore, they play the role of conserva- tion laws of elliptic rotator hierarchy on GL(N, �). We have a tower of N N( ) 1 2 independent integrals of motion I I I I I I I In N N N 0 2 2 2 0 3 2 3 3 3 0 2 , , , , , , , , � � � � � � There are no integrals I1,k because there are no double periodic meromorphic functions with one simple pole. The integrals I k Nk k, , , , , ,� 0 2 3 � are the Casimir functions cor- responding to the coadjoint orbit � �� ET S N g g� � � �gl( , ), ( )� S S1 0 . The conservation laws Is,k generate commuting flows on � rot � �� s k s kI, , ,S S� 1, ( : ), ,� �s k t s k� . 4.4 Symplectic Hecke correspondence Let E and ~ E be two bundles over � of the same rank. As- sume that there is a map � �: ~ E E (more precisely a map of the space of sections � �( ) ( ~ )E E� ) such that it is an iso- morphism on the complement to z0 and it has one-dimen- sional cokernel at x � �: 0 0 0 � * �** � � E E C z � ~ . The map �+ is called upper modification of the bundle E at the point z0. Let w z z� � 0 be a local coordinate in a neighbor- hood of z0. We represent locally E as a sum of line bundles E j N j� � �1� with holomorphic sections s s s s N� ( , , , )1 2 � . (4.42) After modification we come to the bundle ~ ( )E zj N j� � "�1 0� . The sections of ~ E are represented locally as ~ ( ( ) , , ( ) )s g w s w g w sN N� � 1 1 1 � , where g j ( )0 0 . In this basis the upper modification at the point z0 is represented by the matrix � � � � � � � � � Id 0 0 N w 1 . This is a modification of order 1, since it increase the degree of E deg( ~ ) deg( ) deg( ( )) deg( )E E z E� � 0 1 (4.43) On the complement to the point z0 consider the map E E� � + *** ~ such that � �� � Id. It defines the lower modification at the point z0. The upper modification � is represented by the vector (0,…, 1) and �� by (0,…, �1). For Higgs bundles the modification acts as ( ) ( ~~ )E E �* �* � �� ~ , � � � ~ A A� � . (4.44) The Higgs fields and ~ should be holomorphic with prescribed simple poles at the marked points. The holo- morphity of the Higgs field put restrictions on its form. Consider the upper modification � ~ ( , , )0 1� and assume that in the above-defined basis takes the form � � � � � � � a b c d , where a is a matrix of order N �1. Then � � a b c d a bw cw d � � � � � � � � � � � � � � � �1 . We see that a generic Higgs field acquires a first order pole af- ter the modification. To escape this, we assume that there exists an eigen-vector � � such that it belongs to the K er . Let � ( , , , )0 0 1� and � � � � � � � a c d 0 . Then the Higgs field ~ does not have a pole ~ � � � � � � � a cw d 0 . In other words the matrix elements ( ) jN should have first order null. In this way the upper modification is lifted from E to the Higgs bundle. After the reduction we come to the map (see (4.17)) T N g n d T N g n d* *( , , , ) ( , , , )� �� 1 . We call this the upper Symplectic Hecke Correspondence (SHC). Generically the modified bundle ~ E is represented locally as a sum of line bundles ~ ( ( ) ) ( )E z mj N j j m j� � " ��1 0� � with holomorphic sections ~ (~, ,~ ) ( , , , )s s s w g s w g s w g sN m m m N N N� � � � �1 1 1 2 21 2� � . (4.45) It has the degree deg( ~ ) deg( )E E m j j N � � � 1 . This modification is represented by the vector ( , , )m mn1 � . Rememeber that the Higgs field is an endomorphism of E s s� and near z0 it acts as ( �s sj j k k( ) . © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 15 Acta Polytechnica Vol. 48 No. 2/2008 Similarly the modified Higgs field acts on sections of the modified bundle ~ E ~ ~~s s� . Then it follows from (4.45) that ~ ~ ~ ~ ( �s sj j k k , ~ ( ) ( ) j k m m k j j kw g w g wk j� � �1 . Since ~ is holomorphic and g j ( )0 0 , j k m mz z k j( )� � 0 must be regular at z z� 0. If we order m m mN1 2$ $ $� then the number of parameters of the endomorphisms is ( )m mj kj k � '� . In general case T N g n d T N g n d m j j N * *( , , , ) ( , , , )� �� � � 1 . If m jj N �� �1 0 the SHC does not change the topological type of the bundle. Therefore, such SHC defines a Bcklund transformation of integrable hierarchy. 4.5 Symplectic Hecke correspondence � � CM ET� [11] We work directly with the Lax matrices L LET CM � � � The modification matrix should intertwine the multipliers corresponding to the fundamental cycles � �( , ) ( , )z Q z � 1 � � , (4.46) � � �( , ) ~ ( , ) ( , ) ( ( ))z z z u j � � � � � diag e . (4.47) Consider the modification at z � 0. The Lax matrix of the CMS has the first order pole L z JCM ~ 1 . Its residue has an eigen-vector t � ( , , )1 1� with the eigen- -value N �1. The matrix � satisfying (4.46) and (4.47) that an- nihilates the vector has the form � �( ) ~ ( ) ( ) ( , ) ; , z z u ul k j j k j k l � � � � � � �� � � � ��' #diag 1 � � ~ ( , , , , ) ( , )�ij N jz u u i N N z Nu N1 1 2 2 � � � �� �, - . . . / 0 1 1 1 � . Here � � i N N jz Nu N �, - . / 0 1 � 1 2 2 ( , ) is the theta-function with a char- acteristic. The determinant of � can be calculated explicitly det ~ ( , , , , ) ( ) ( ) ( ) (�ij N l kz u u i z i u u1 � � ! � � ! � �, - . . / 0 1 1 � � ) ( )i k l N ! � 12 ' 2 # , where ! �( ) ( )� � �#q q n n 1 24 1 0 is the Dedekind function. It has a simple pole at z � 0 and therefore � is degenerate. We use the modification to write down the interrelations between the coordinates and momenta of the Calogero- -Moser particles and the orbit variables of the Elliptic Top in the SL( , )2 � case S v u u1 10 10 10 2 01 0 0 2 2 0 0 0 � � � � � � � � � � � � "" ( ) ( ) ( ) ( ) ( ) ( ) ( ) "" "" "" � � � � � � ( ) ( ) ( ) , ( ) ( ) ( ) ( ) 2 2 2 0 0 2 2 01 2 2 u u u S v u u � � � � � � � � � � � "" 2 10 01 10 01 2 3 01 0 0 0 2 2 2 0 ( ) ( ) ( ) ( ) ( ) ( ) , ( u u u S v� � ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( � � � � � � � � � � "" "" 0 2 2 0 0 0 201 01 2 10 10u u u 2 22 u u ) ( ) . � (4.48 ) Here � � � � 1 0 1 2 0 0 1 2 1 2 2 2 2 1 2 , , exp ( ) , exp � � � � � � � � � � � � q n z q n n n � nz q nz n n n n � � � � � � � � � � �� � � � , ( ) exp .,� �0 1 1 2 1 21 2 2 These relations describe the Darboux coordinates ( , )v u � �2 the coadjoint SL( , )2 � -orbit S� 2 2� � It turns out that this modification is equivalent to the twist of R-matrices. Namely, it describes the passage from the dy- namical R matrix of the IRF models to the vertex R-matrix [25, 26]. We do not discuss this aspect of SHC here. 5 4d theories 5.1 Self-dual YM equations and Hitchin equations 5.1.1 2-d self-dual equations Consider a rank N complex vector bundle E over �4 with coordinates x � ( , , , )x x x x1 2 3 4 . Assume that the space of sections is equipped with a nondegenerate Hermitian metric h, ( )h h � . It satisfies the following condition dh x y h x y h x y( , ) ( , ) ( , )� ! ! , where ! is a connection on E. If dh x y( , ) � 0 for vectors in fibers y V� , x V t� ~ , then there exist connections ! � j x jj A� such that A A � � �� �h dh h h1 1 , ( )A � � � A dxj j j 0 3 . In this situation the transition functions are reduced to the unitary group SU GL( ) ( , )N N� � . Let F su N( ) ( , ( ))( )A �� 2 4� be the curvature Fij i j� ! ![ , ] or F d( )A A A� 2. Here su( ) { }N x x h xh� � � �1 . The self-duality equation F F�3 , where 3 is the Hodge operator in �4 takes the form F F F F F F 01 23 02 31 03 12 � � � � � % � % (5.1) 16 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 Assume that Aj depend only on (x1, x2). This means that the fields are invariant under the shifts in directions x0, x3. Then (A0, A3) become adjoint-valued scalar fields which we denote as ( , )� �1 2 . They are called the Higgs fields. In fact, they will be associated below with the Higgs field . In this way we come to the self-dual equations on the plane � 2 1 2� ( , )x x � �F12 1 2� � �, , (5.2) � � � �! � !1 1 2 2, ,� � , (5.3) � � � �! � !1 2 2 1, ,� � . (5.4) Introduce complex coordinates z x ix� 1 2, z x ix� �1 2 and let � �!d z , �� �!d z . Consider the fields, taking values in the Lie algebra sl( , )N � � � z z i dz i dz � � � � � 1 2 1 2 1 0 2 1 2 1 2 0 ( ) ( , ), ( ) ( , ) ( � � � � � ad E , )( , ).1 2� ad E � � % �% They are not independent since the Hermitian conjugation acts as z zh h �� � 1 . (5.5) Similarly, A A iA A A iA z z � � � � � % � % 1 2 1 2 1 2 1 2 ( ) ( ) A h dh h A hz z � �� � �1 1 . (5.6) In terms of fields � ( , , , )A A F Fz z z (5.7) (5.2) – (5.4) can be rewritten in the coordinate invariant way: 1 0 2 0 3 0 . [ , ] , . , . , F d d z z z z � �� � � � � � % � % (5.8) where [ , ] z z z z z z� . Due to (5.5) and (5.6) the third equation is not independent. Thus, we have two equa- tions with the left side of type (1, 1) for two complex valued fields ( ) z zA and the hermitian matrix h. Equations (5.8) are conformal invariant and thereby can be defined on a complex curve �g. In this case � �z g N� ( , )( , ( )),1 0 su , � �z g N� ( , )( , ( )),0 1 su �� � d N Nj k g j k g: ( , ( )) ( , ( )). ( , ) ( , )� � � �su su1 The self-duality equations (5.9) on �g are called the Hitchin equations. In fact, instead of (5.8) we will consider further a modified system 1 0 2 0 3 0 . [ , ] , . , . . F d d z z z z � � �� � � � � � % � % (5.9) This comes from the self-duality on �4 with a metric of signa- ture (2, 2). Consider the gauge group action on solutions of (5.9) � �� � �f Ng� �0( , ( ))SU , (5.10) z z z zf f f f� � � �1 1, , (5.11) �� � ���d f d f1 . (5.12) If ( , , , )A Az z z are solutions of (5.9), then the transformed fields are also solutions. If f takes values in GL( , )N � then it again transforms solutions to solutions. As above we denote this gauge group as ��. Define the moduli space of solutions of (5.9) as a quotient under the gauge group action � �H gS( ) � solutions of (5.9) / . Now look at the second equation in (5.9). It is the moment constraint equation for the Higgs bundles in the absence of marked points (4.14). The gauge group �� transforms solu- tions of (5.9) to solutions but breaks (5.5), (5.6). Now we will restrict ourself with the second equation in (5.9). Dividing the space of its solution on the gauge group �� we come to the moduli space of the Higgs bundles T N g d* ( , , , )� 0 (4.17). There exists a dense subset of moduli space of stable Higgs bundles T N g d T N g dstable* *( , , , ) ( , , , )� �0 0� . The mod- uli space of stable Higgs bundles parameterize the smooth part of �H g( )� (5.13) [2]. Consider a Higgs bundle with data ( , ) A satisfying eq. 2 in (5.9) and reconstruct from it solutions ( , , , )A Az z z z of (5.9). Define them as z z h h� � � � , 1 , A A A h h h A hz z� � � � � � , 1 1� . Then ( , ) z zA satisfy eq. 3. (5.9). Equation 1. (5.9) takes the form � � � �( ) [ ,( )] [ , ] h h h A h A A h h h A h h h � � � � � � � � 1 1 1 1 1 0 . For almost all ( , ) A there exists a solution h of this equation (see appendix of Donaldson in [2]). In this way we pass from the holomorphic data to solutions of the system (5.9). Summarizing, to define �H g( )� one can acts in two ways: 1. Divide the space of solutions of (5.9) on the SU(N)-valued gauge group �. 2. Consider the moduli space of stable Higgs bundles. 5.1.2 Hyper-Kahler reduction In this section we explain how to derive the moduli space �H g( )� (5.13) via an analog of the symplectic reduction. This is the so-called Hyper-Kahler reduction [41]. We prove that infinite-dimensional space (5.7) is a Hyper-Kahler © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 17 Acta Polytechnica Vol. 48 No. 2/2008 manifold, and �H is its Hyper-Kahler quotient, where (5.9) play the role of the moment equations. To define a Hyper-Kahler manifold we need three com- plex structures and a metric satisfying certain axioms. Define a flat metric on depending on the complex structure on � ds A A A Az z z z z z z z 2 1 4 � � " " " " �� � � � � � � � � Tr( ). � (5.14) Introduce three complex structures I, J, K on . The corresponding operators act on the tangent bundle T , such that they obey the imaginary quaternion relations I J K Id2 2 2� � � � , IJ K� ,�. The complex structures are integrable because is flat. Introduce a basis of one-forms in T * V A Az z z z� ( , , , )� � � � Then the action of the conjugated operators on T * in this basis takes the form I i i i i T � � � � � � � � � � � � � � � 0 0 0 0 0 0 0 0 0 0 0 0 , JT � � � � � � � � � � � � � � � 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 , K i i i i T � � � � � � � � � � � � � � � 0 0 0 0 0 0 0 0 0 0 0 0 . Linear functions on are holomorphic with respect to a complex structure, if they are transformed under the action of the corresponding operator with eigen-value i. Thus Az, z are holomorphic in the complex structure I, A iz z , Az z are holomorphic in the complex structure J, and Az z� , Az z are holomorphic in the complex structure K. To be hyper-Kahler on the metric ds2 should be of type (1,1) in each complex structure. This means that ds I I ds J J ds K K dsT T T T T T2 2 2 2~ ( ) ( ) ( )" � " � " . In this way we have described a flat hyper-Kahler metric on . A linear combination of the complex structures produces a family of complex structures, parameterized by ��1. We define three symplectic structures associated with the complex structures on as �I TI Id ds� "( ) 2, � J TJ Id ds� "( ) 2, �K TK Id ds� "( ) 2. � � I z z z z i DA DA D D g � � � � ��2 tr( ) � , � � �J z z z zD DA D A g � � �� 1 2 tr( ) � , (5.15) � � K z z z z i D DA D DA g � � � ��2 tr( ) � . These forms are closed and of type (1,1) with respect to the corresponding complex structures. Now consider the gauge transformations (5.10) of the fields (5.11), (5.12). Since the gauge transform takes values in SU( )N , the forms (5.15) are gauge invariant. Therefore we can proceed as in the case of standard symplectic reduc- tion (3.14). But now we obtain three generating momentum Hamiltonians with respect to the three symplectic forms F i trI g � �2� (� �( [ , ])),Fz z z z� , (�� Lie( )� ), F trJ g � � � 1 2� ( � �( ))� ��d dz z F i trK g � � �2� (� �( ))� � ��d dz z . and the three moment maps �� Lie *( ) �I z ziF i� � [ , ] , � J z zd d� � �� �K z zi d d� � � ��( ) . The zero-valued moments coincide with the Hitchin systems. The hyper-Kahler quotient �/ / / is defined as � �/ / / ( ) ( ) ( ) /� 4 4� � �� � �I J K 1 1 10 0 0 To come to system (5.9) consider the linear combination � �I J K zi d� � �� . (5.16) This moment map is derived from the symplectic form � � I J K z zi D DA� � ��� � � 1 tr( ) . This is a (2,0)-form in the complex structure I. Thus we have the holomorphic moment map I in the complex structure I. Vanishing of the holomorphic moment map I and the real moment map �I is equivalent to the Hitchin equations. Divid- ing their solutions on the gauge group � we come to the moduli space �H(�g) (5.13). Now consider an analog of (5.16) corresponding to the complex structure J � �J K I z zi� � � , , � � � �z z z z, ( , )� , �z z zA i� , �z z zA i� . This moment map comes from the symplectic form � � J D D g � �� 1 2� tr( )� � . It is (2,0) form in the complex structure J. Putting J � 0 we come to the flatness condition of the bundle E. Dividing the set of solutions �z z, � 0 on the GL( , )N � valued gauge transformations �� we come to the space 18 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 � � �� �( ),z z 0 � (5.17) of homomorphisms �1( ) ( , )� g N� GL � defined up to conju- gations. In accordance with [42] and Donaldson (the appen- dix in Ref.[2]) generic flat bundles parameterize �H(�g) (5.13) in the complex structure J. This space is a phase space of non-autonomous Hamiltonian systems leading to monodromy preserving equations (see Section 6.3). Thus, the space �H(�g) describes phase spaces of integrable systems �red (4.17) in the complex structure I and phase spaces of monodromy preserving equations � (5.17) in the complex structure J. 5.2 � � 4 SUSY Yang-Mills in four dimension and Hitchin equations Here we consider a twisted version of � � 4 super Yang- -Mills theory in four dimension. This theory was analyzed in detail in [16, 17, 18] to develop a field-theoretical approach to the Geometric Langlands Program. Quantum Hitchin sys- tems are one side of this construction and here we use only a minor part of [16]. The twisted theory is a topological theory that contains a generalization of the Hitchin equations (5.9) as a condition of BRST invariance. Our goal is to describe the Hecke transformations in terms of the theory. In section 4 we have defined the Hecke transformations as an instant sin- gular gauge transformation. The four-dimensional theory allows to consider gauge transformations varying along a space coordinate x3. They become singular at some point, say x3 0� , where a singular t’Hooft operator is located. This gives a natural description of the symplectic Hecke correspondence in terms of a monopole configuration in the twisted theory. 5.2.1 Twisting of SUSY Yang-Mills theory � � 4 SUSY SU( )N Yang-Mills action in four dimension can be derived from the � � 4 SUSY SU( )N Yang-Mills ac- tion in ten dimensions by dimensional reduction. We need only the bosonic part of the reduced theory. The bosonic fields of the 4d Yang-Mills theory are the four-dimensional gauge potential A A A A A� ( , , , )0 1 2 3 , and six scalar fields coming from six extra dimensions � � � � � � �� ( , , , , , )0 1 2 3 4 5 . The bosonic part of the action has the form I e d x F F D Di i� � � � � � �� � ��� 1 2 4 0 3 1 6 0 3 tr 1 2 � � � � � � � � , 1 2 [ ] . , � �i j i j 2 0 6 � � � � � �� The symmetry of the action is Spin(4) × Spin(6) (or Spin(1,3) × Spin(6) in the Lorentz signature). The sixteen generators of the 4d supersymmetry are transformed under Spin(1,3) × Spin(6)~SL(2)×SL(2)×Spin(6) as ( , , ) ( , , )2 1 4 1 2 4� : { } { }Q QAX A Y� , ( , ; , , )A X� �1 2 1 4� , ( � , ; , , )A Y� �1 2 1 4� . They satisfy the super-symmetry algebra { , } �Q Q PAX A Y X Y AA � � �� � � � � 0 3 , { , }Q Q � 0, { , }Q Q � 0. (5.18) The action of Q on a field X takes the form �X Q X� { , }. Let # be a map Spin(4) � Spin(6) and set Spin Id Spin� � ( ) ( ) ( )4 4# . Define # in such a way that the action of Spin�( )4 on the chiral spinor � has an invariant vector. Let Q be the corresponding supersymmetry. It follows from (5.18) that it obeys Q 2 0� . The twisted theory is defined by the physical observables from the cohomology groups H Q5( ). The twisted four scalar fields � � �� ( , , )0 3� are reinterpreted as adjoint-valued one-forms on �4, while untwisted $ $ � �, � )4 5i remain adjoint-valued scalars. In fact there is a family of topological theories para- meterized by t � ��1. To be invariant under Q the bosonic fields should satisfy the equations 1 0 2 0 3 0 1 ) ( ) , ) ( ) , ) , F tD F t D D � � � � � � � 3 3 � � � � � � � � � � (5.19) where ) denote the self-dual and the anti-self-dual parts for four-dimensional two-forms, D d� [ , ]A and3 is the Hodge operator in four dimension. We are interested in solutions of this system up to gauge transformations. This theory defined on flat �4 can be extended to any four-manifold M in such a way that it preserves the Q-symme- try and the contributions of the metric come only from Q-ex- act terms. The bosonic part of the theory is described by connections A � ( , , , )A A A A0 1 2 3 in a bundle E over M in the presence of the adjoint-valued one-forms � � � � �� ( , , , )0 1 2 3 satisfying (5.19). The important thing for the case of integrable systems is M g� � 2 � , where �2 0 3� � �( ) ( )time x x y and �g will play the role of the basic spectral curve. �2 is not involved in the twisting and the fields ( , )� �0 3 remain scalars, while � �1 2, become one-forms on �g. It turns out that after reduction the system (5.19) becomes equivalent to the Hitchin equations (5.9). 5.2.2 Hecke correspondence and monopoles The system (5.19) for t �1 can be replaced by F D� 3 �� � � 0 (5.20) * *D � � 0 (5.21) Assume that the fields are time independent and consider the system on the three-dimensional manifold W I x g� ( )3 � , where �� 2 2 �x3 . In terms of the tree-dimensional fields © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 19 Acta Polytechnica Vol. 48 No. 2/2008 and ~ ( ( , ~ ) A � A A0 , � � ( , ~ ) 0 0dx , the equations take the form [16] ~ ~ ~ ( [ , ~ ]), ~ [ , ~ ] , ~ [ F D A D DA D A � � �3 � 3 � 3 3 0 0 0 0 0, ] . 0 0� (5.22) Here the Hodge operator 3 is taken in the three-dimensional sense. Replace the coordinates x x x x� ( , , )1 2 3 on x y3 � and ( , ) ( , )x x z z2 3 � , where ( , )z z are local coordinates on �g. Let g z z dz( , ) 2 be a metric on �g. The metric on W is ds g dz dy2 2 2� . Then the Hodge operator takes the form 3 � � 3 � � � 3 � �dy ig dz dz dz idz dy dz idz dy 1 2 , , . It is argued in [16] that y � 0 and A0 0� are solutions of the system. Then we come to the equations (5.23) where as before z zh h �� � 1 . The system is simplified in the gauge Ay � 0. In particular, for 0 0� the system (5.23) becomes essentially two-dimensional and coincides with the Hitchin equations (5.9). Let �g be an elliptic curve ( )g �1 . This case is important for application to integrable systems. The nonlinear system (5.23) can be rewritten as a compatibility condition for the lin- ear system depending on the spectral parameter � � � ( ) , ( � � � � � � � � � � z y z z z y z z a A i i a a A i � � � � 1 2 1 0 2 0 � � � � % �% i a� � 0 0) Here a2 1� �� � . This linear system allows to apply the methods of the Inverse Scattering Problem or the Whitham approxi- mation to find solutions of (5.23). Define the complex connection �z z zA i� . In terms of ( , , )� �z z yA the systems (5.23) assumes the form of the Bogo- molny equation F D�3 0 . (5.24) Consider a monopole solution of this equation. Let ~ ( \ ( , ))W W x y z z� � � � 0 00 . The Bianchi identity DF � 0 in the space ~ W implies that 0 is the Green function for the operator 3 3D 3 3 � �D D M x x 0 0�( ) , where M N�gl( , )� . Consider first the abelian case G � U( )1 . Then F A Az z( , ) is a curvature of a line bundle �. Locally near x y z z z z0 0 00� � � �( , , ) 0 has a singularity 0 02 ~ im x x � , (5.25) where m is a magnetic charge. Due to 1. in (5.23), F takes the form F A A gz z y( , ) � 1 2 0 � , F A A mg z z y x x z z( , ) ~ ( , ) 1 2 0 3 2 � . Consider a small sphere S2 enclosing x0. Due to (5.24) and (5.25) F m S 2 � � . This solution describes the Dirac monopole of charge m cor- responding to a line bundle over S2 of degree m. Let � �g g ) � )�( ) and �) be the line bundles over � g ) . The two-dimensional cycle C describing the boundary C W I xg� � �(( ) \ )� 0 is � �g g S �� � 2. Taking the integral over C we find that F C � � 0. In other words, for the Chern classes of the bundles c( ) deg( )� �� we have deg( ) deg( )� � �� m , or � � "� ( )z m0 . Here ( )z m 0 is a line bundle whose holo- morphic sections are holomorphic functions away from z0 with a possible single pole of degree m at z0. The line bundles over �g are topologically equivalent for y ' 0 or y � 0. The gauge transformation 0 is smooth away from x0. The singu- larity change the degree of the bundle. 20 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 1. F A A igDz z z z Ay( , ) [ , ]� � 1 2 0 , 2. DAz z � 0 , 3. F A A iDy z Az( , ) ,� 0 4. D iA z zy � � [ , ].0 Fig. 3: Monopole, located at y0 changes the Chern class The monopole increase the Chern class: c c m1 1� . In four-dimensional abelian theory we have the Dirac monopole singular along the time-like line L x x� ( , )0 0 . This corresponds to including the t’Hooft operator in the theory say- ing that the connections have monopole singularity along the line L. A generic vector bundles E near x0 splits E y N~ � � �1 2� � �� . Consider the gauge transformation 0 0 12 ~ ( , , ) i x x m mN � � diag . (5.26) This causes the transformation � � j j m z j� " ( )0 . The de- gree of the bundles E changes after crossing y � 0 by m j� , as it was described for bundles over � in Section 4.4. To be more precise we specify the boundary conditions of solutions on the ends y � �� and y � �. Since 0 0� for y � )� system (5.23) coincides with the Hitchin system (5.9). If �H N g n m( , , , ) ) is the moduli space of solutions on the boundaries y � )� the gauge transformation with the monopole singularity stands that m m m j �� � . It de- fines the SHC between two integrable systems related to �H N g n m( , , , ) ) . In particular, we have described it at the point y � 0 for �H N n( , , , )1 0 and �H N n( , , , )1 1 . 6 Conclusion Here we will briefly discus some related issues have not in- cluded in the lectures. 6.1 Solutions of the Hitchin equations (5.9) corresponding to quasi-parabolic Higgs bundles were analyzed in Ref. [17]. In the three-dimensional gauge theory considered in Section 4.3 we have the Wilson lines located at the marked points. In the four-dimensional Yang-Mills theory they corresponds to sin- gular operators along two-dimensional surfaces. Locally on a punctured disc around a marked point the Hitchin system (5.9) assumes the form of the Nahm equations [43]. It was proved in Ref. [46] that the space of its solutions after dividing on a special gauge group is symlectomorphic to a coadjoint orbit of SL( , )N � . A hyper-Kahler structure on the space of solutions induces a hyper-Kahler structure on the orbits. It es- tablishes the interrelations between the Hitchin equations and the Higgs bundles with the marked points (the quasi-par- abolic Higgs bundles). 6.2 There exists a generalization of this approach to Higgs bundles of infinite rank. In other words, the structure group G N� GL( , )� or SL( , )N � of the bundles is replaced by an in- finite-rank group. One way is to consider the central ex- tended loop group S G1 � . Then the Higgs field depends on additional variable x S� 1 and instead of the Lax equation we come to the Zakharov-Shabat equation � �j x j jL M M L� �[ , ] 0. This equation describes an infinite-dimensional integrable hierarchy like the KdV hierarchy. The two-dimensional ver- sion of ECMS was constructed in [10,11]. In particular, SHC establishes an equivalence of the two-particles (N � 2) elliptic Calogero-Moser field theory with the Landau-Lifshitz equa- tion [44, 45]. The latter system is the two-dimensional version of the SL( , )2 � elliptic top. Relations (4.48) work in the two-di- mensional case. Another way is to consider GL( )� bundles. In Ref. [48] ECMS for infinite number of particles N � � was analyzed. The elliptic top on the group of the non-commutative torus was considered in Ref. [47]. It is a subgroup of GL( )� . This construction describes an integrable modification of the hy- drodynamics of the ideal fluid on a non-commutative two-di- mensional torus. 6.3 Consider dynamical systems, where the role of times is played by parameters of complex structures of curves � g n, . In this case we come to monodromy preserving equations, like the Schlesinger system or the Painlevé equations. They can be constructed in the similar fashion as the integrable Hitchin systems [8]. To this purpose the one should replace the Higgs bundles by the flat bundles and afterwards use the same symplectic reduction (see (5.17)). In this situation the Lax equations takes the form � �j z j jL M M L� �[ , ] 0. An analysis of this system is more complicated than the stan- dard Lax equations due to the presence of derivative with re- spect to the spectral parameter. Note that Mj corresponds only to quadratic Hamiltonians, since they responsible for the deformations of complex structures. Concrete examples of this construction was given in [8, 52, 53]. Interrelations with Higgs bundles were analyzed in [8, 49]. It is remarkable that the Symplectic Hecke correspondence works in this case. It establishes an equivalence of the Painlevé VI equation and a non-autonomous Zhukovski-Volterra gyrostat [12]. 6.4 A modification of the Higgs bundles allows one to con- struct relativistic integrable systems [50]. The role of the Higgs field is played by a group element g cK� �exp( )1 where K is a canonical class on � and c is the relativistic pa- rameter. This construction works only for curves of genus © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 21 Acta Polytechnica Vol. 48 No. 2/2008 g 21. This approach was realized in Ref. [28] to derive the elliptic Rujesenaars system and in Ref. [51, 53] to derive the elliptic classical r-matrix of Belavin-Drinfeld [54] and a qua- dratic Poisson algebra of the Sklyanin-Feigin-Odesski type [55, 56]. The inclusion of the relativistic systems allows to define a du- ality in integrable systems [57, 58] (see [59] for recent devel- opments). This type of dualities has a natural description for the corresponding quantum integrable systems in terms of Hecke algebras [60]. There, it is called the Fourier transform and takes the form of S-duality. Another form of duality in the classical Hitchin system considered in [61, 62, 63]. It is related to Langlands duality and similar to T-duality of fibers in the Hitchin fibration. 6.5 There is an useful description of the moduli space of holomorphic vector bundles closely related to the modifica- tion described in Section 4.4. It is so-called Tyurin parametri- zation [64]. This construction was applied to describe Higgs bundles and integrable systems related to a curve of arbitrary genus in Ref. [10, 65, 66]. Using this approach classical r-ma- trices with a spectral parameter living on curves of arbitrary genus were constructed in Ref. [67]. References [1] Hitchin, N.: Stable Bundles and Integrable Systems. Duke Math. Jour. Vol. 54 (1987), p. 91–114. [2] Hitchin, N.: The Self-Duality Equations on a Riemann surface. Proc. London Math. Soc,. Vol. 55 (1987), 59–126. [3] Hurtubise, J.: Integrable Systems and Algebraic Sur- faces. Duke Math. Journ., Vol. 83 (1996), 19–50. [4] Gorsky, A., Nekrasov, N., Rubtsov, V.: Hilbert Schemes, Separated Variables, and D-Branes. 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Mikhail Olshanetsky e-mail: olshanet@itep.ru Chern Institute of Mathematics Nankai University Tianjin, China 24 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends false /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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