Bethe ansatz and Isomonodromic deformations

ITEP-TH-03/08 Bethe ansatz and Isomono dromic deformations D. T alalae v 1 Institute fo r Theo re tical and Exp erimental Physics 2 Abstract W e study symmetries of the Bethe equations for the Gaudin mo del a ppear ed naturally in the framework of the geometr ic Langla nds corresp ondence under the name of Heck e o pera tors and under the name o f Schlesinger transforma tions in the theor y of isomo nodr omic deformations, and particularly in the theor y of Painlev´ e transcendents. 1 E-mail: talalaev@itep.ru 2 ITEP , 25 B. Cheremushkinsk a ya, Mosco w, 117259, Russia. In tro duc tion The Gaudin mo del is one of th e first n on trivial examples of quant um integ r able sys tems wh ic h shows deep connection of prob lems of mathematical physics and s u c h mo dern f undamen tal mathematical questions as the Langlands corresp ondence. F or C P 1 with mark ed p oin ts the geometric Lan glands corresp ondence is an interpretati on of the separation of v ariables in qu an tum Gaudin mo del. The basic p urp ose of th is w ork is to sp ecify the Langlands corresp ondence concerning the p roblem of solution of a quantum m od el. W e use the form of the Bethe equ atio ns whic h expresses a condition for the sp ectrum of the Gaudin mo del in terms of the mono dromy pr op erties of certain differen tial op erator with regular sin gularities (called the G -op er). F urther w e constru ct the d iscrete mono drom y preserving transformations known as Sc hlesinger or Hec ke transf ormations. It app ears, that they determin e also an action on rational solutions of the G-op er, and therefore on eigen v ectors of the Gaudin mo del. It is necessary to men tion that suc h transformations act by c h anging the highest w eigh ts in repr esen tations and also the inhomogeneit y parameters of the mo del. The exp osition is organized as follo ws: in section 1 the Gaud in mo del is defined, the traditional v ersion of th e Bethe ansatz and its formulat ion in terms of mono drom y prop er ties of sp ecific differen tial equations is describ ed. In section 2 the ”matrix” form of the mono dromy condition equiv alen t to the Bethe equ atio ns is constructed. In section 3 the Schlesinge r tr an s formations are in vestig ated. The last section is devot ed to some relations of this su b j ect with other prob lems. Ac kno wledgmen ts. The p art of this w ork h as b een done d uring the sta y of th e author at LPTHE, the author is grateful for th e s timulating atmosphere. The pro of of the theorem 1 w as obtained during the sta y of the auther at LAREMA. T h is w ork was p artially supp orted b y th e F ederal Nuclear Energy Agency of Russia, the RFBR gran t 07-02 -00645, the grant of Supp ort for the S cien tific Schools 303 5.200 8.2, and the fu nd ”Dynast y”. Auth or w ould like to thank O. Bab elon, B. F eigin, A. Gorod en tsev, I. K r ic hev er and V. R u btso v for us eful remarks. 1 Bethe ansatz for the Gaud in mo del 1.1 Gaudin mo del The Lax op erator of this mo d el is a rational fun ction with simple p oles: L ( z ) = N X i =1 Φ i z − z i , with the residues Φ i ∈ M at n ⊗ L gl n ⊂ M at n ⊗ U ( gl n ) ⊗ N defined b y the f orm ula: Φ i = X k l E k l ⊗ e ( i ) k l (1) where E k l form the standard basis in M at n and e ( i ) k l form a b asis in the i -th cop y of the Lie algebra gl n . It is the same as to sa y th at the quantum algebra is A = U ( gl n ) ⊗ N and the commutatio n r elations for th e Lax op erator are [ L ( z ) ⊗ 1 , 1 ⊗ L ( u )] = [ R 12 ( z − u ) , L ( z ) ⊗ 1 + 1 ⊗ L ( u )] ∈ M at ⊗ 2 n ⊗ A 1 Let us no w define the “quant um” determinan t of a matrix w ith non commutativ e en tries as the completely symmetrized determinant ” det ”( B ) = 1 n ! X τ ,σ ∈ Σ n ( − 1) τ σ B τ (1 ) ,σ (1) . . . B τ ( n ) ,σ ( n ) Then th e qu an tum sp ectral curv e is defined as follo ws ” det ”( L ( z ) − ∂ z ) = n X k =0 QI k ( z ) ∂ n − k z (2) Theorem [1] The c o efficients QI k ( z ) c ommute [ QI k ( z ) , QI m ( u )] = 0 and quantize the classic al Gaudin hamiltonians. Remark 1 By the same formula in [3] it was c onstructe d a c ommutative sub algebr a in U ( gl n )[ t ] /t N , U ( gl n )[ t ] and the c enter of the universal enveloping affine algebr a on the critic al level . 1.2 Bethe ansatz 1.2.1 sl 2 case Let u s consider the Gaudin mod el for the sl 2 case. Th e Lax operator can b e expr essed by the form u la L =  A ( z ) B ( z ) C ( z ) D ( z )  = N X i =1 Φ i z − z i where Φ i =  h i / 2 e i f i − h i / 2  Let u s also fi x a fi nite dimens ional represen tation V λ = ⊗ V λ i whic h is the tensor pro du ct of highest weig h t r epresen tations for eac h factor U ( sl 2 ) of the quantum algebra. The v acuum v ector is | v ac > = ⊗ i | 0 > i where the vec tors | 0 > i are defined by e i | 0 > i = 0 and h i | 0 > = λ i | 0 > . Th e quan tum c haracteristic p olynomial in this case is ” det ”( L ( z ) − ∂ z ) = − 1 2 T r L 2 ( z ) + ∂ 2 z T r L 2 ( z ) = X i T r Φ 2 i ( z − z i ) 2 + 2 X i for some v alues of p arameters µ j . T h is condition implies the follo win g sy s tem of algebraic equation on p arameters µ j (the Bethe system) − 1 2 X i λ i µ j − z i + X k 6 = j 1 µ j − µ k = 0 j = 1 , . . . , M (4) The eigen v alues tak e the form χ Ω ( H i ) = − λ i   X j 1 z i − µ j − 1 2 X j 6 = i λ j z i − z j   1.3 “Null-mono dromic” form Let Ψ( z ) = Y ( z − z i ) − λ i / 2 Y ( z − µ j ) (5) b e a solution of the equation ” det ”( L ( z ) − ∂ z )Ψ( z ) = ∂ 2 z − 1 2 X i c (2) i ( z − z i ) 2 − X i H i z − z i ! Ψ( z ) = 0 (6) supp osing that H i are scalars. This is equiv alen t to the follo w ing s y s tem of equations 1 z − z i : − λ i   X j 6 = i − λ i / 2 z i − z j + X j 1 z i − µ j   = H i 1 z − µ j : − 1 2 X i λ i µ j − z i + X k 6 = j 1 µ j − µ k = 0 whic h is exactly the sy s tem of Bethe equations and the conditions on eigen v alues. Remark ably the second solution of the equation 6 has the same b eha v ior at z = z i , th at is at most half inte ger exp onen ts. This imp ly that the mono dr om y of the fund amen tal solution lies in Z / 2 Z . T here is also an inv erse statemen t Theorem [4] The e quation ∂ 2 − 1 4 X i λ i ( λ i + 2) ( z − z i ) 2 − X i ν i z − z i ! Ψ( z ) = 0 (7) with ν i ∈ C has mono dr omy ± 1 iff ν i ar e e qual to χ Ω ( H i ) for some c ommon eigenve ctor Ω of the Gaudin hamiltonians in the r epr esentation V λ . 3 2 Matrix form 2.1 Scalar v er sus matrix G -op ers Let us no w consid er a connection in the trivial rank 2 b u ndle on the pu n ctured disc of the form: A ( z ) =  a 11 ( z ) a 12 ( z ) a 21 ( z ) a 22 ( z )  = k X i =1 A i z − z i (8) where the r esidues satisfy the follo wing conditions T r ( A i ) = 0; D et ( A i ) = − d 2 i ; X i A i =  κ 0 0 − κ  . (9) Let us no w consid er the F uc hsian sy s tem ( ∂ z − A ( z ))Ψ( z ) = 0 (10) whic h is equiv alen t to the follo win g ψ ′ 1 = a 11 ψ 1 + a 12 ψ 2 ψ ′ 2 = a 21 ψ 1 + a 22 ψ 2 This system implies ψ ′′ 1 = a ′ 12 /a 12 ψ ′ 1 + uψ 1 where u = a ′ 11 + a 2 11 − a 11 a ′ 12 /a 12 + a 12 a 21 Then if one p asses to Φ = ψ 1 /χ where χ = √ a 12 one obtains th e equation Φ ′′ + U Φ = 0 where U = χ ′′ /χ − a ′ 12 /a 12 χ ′ /χ − u (11) More precisely this implies U = 1 2  a ′ 12 a 12  ′ − 1 4  a ′ 12 a 12  2 + a 11 a ′ 12 a 12 − a ′ 11 − a 2 11 − a 12 a 21 (12) Supp ose th at a 12 ( z ) h as distinct zeros a 12 ( z ) = c Q k − 2 j =1 ( z − w j ) Q k i =1 ( z − z i ) Let us note that the n u m b er of zeros is in agreemen t with th e n ormalizat ion condition 9. Hence a ′ 12 a 12 = k − 2 X j =1 1 z − w j − k X i =1 1 z − z i (13) 4 The expression for the p oten tial tak es the follo wing form U = k − 2 X j =1 − 3 / 4 ( z − w j ) 2 + k X i =1 1 / 4 + detA i ( z − z i ) 2 + k − 2 X j =1 H w j z − w j + k X i =1 H z i z − z i (14) where H w j = a 11 ( w j ) + 1 2   X i 6 = j 1 w j − w i − X i 1 w j − z i   H z i =  1 2 + a i 11  X j 1 z i − w j − X j 6 = i T r ( A i A j ) + a i 11 + a j 11 + 1 / 2 z i − z j Let us note that the co efficients at ( z − z i ) − 2 can b e rewritten in the form 1 / 4 + detA i = (1 / 2 − d i )(1 / 2 + d i ) (15) F urther w e will iden tify this with the v alues of C asimir elemen ts. In this wa y λ i = 2 d i − 1 . 2.2 Dual G -op er As it w as sh o wn in pr evious calculat ion th e m atrix conn ectio n is related to the Stur m-Liouville op erator with additional regular sin gularities at p oin ts w j . The similar consideration for the seco nd comp onen t Ψ 2 can p ro duce another scalar differenti al op erator of the s econd order with p oles at z i and additional p oin ts e w j defined b y the f orm ula a 21 ( z ) = e c Q k − 2 j =1 ( z − e w j ) Q k i =1 ( z − z i ) W e will call the corresp ondin g Sturm-Liouville op erator ∂ 2 z − e U the dual G -op er w h ere e U = k − 2 X j =1 − 3 / 4 ( z − e w j ) 2 + k X i =1 1 / 4 + detA i ( z − z i ) 2 + k − 2 X j =1 H e w j z − e w j + k X i =1 e H z i z − z i (16) 2.3 Pull bac k In th is section w e construct the inv erse map, namely from the scalar G -oper, which has trivial mono drom y in the sense of the Bethe equations, we constru ct a r an k 2 connection of the form 10 , suc h th at its h orisontal sections h a v e mono drom y in Z / 2 Z . Let us consider the solution for the matrix linear equation 10 ( ∂ z − A ( z ))Ψ = 0 in the follo wing ansatz ψ l = k Y i =1 ( z − z i ) − s i φ l ( z ) (17) 5 where φ 1 = M Y j =1 ( z − γ j ) φ 2 /φ 1 = M X j =1 α j z − γ j (18) Let us rewrite the system 10 w ith resp ect to the n ew parameterization 18 ∂ z ψ 1 /ψ 1 = a 11 + a 12 φ 2 /φ 1 (19) ∂ z ψ 1 /ψ 1 ( φ 2 /φ 1 ) + ∂ z ( φ 2 /φ 1 ) = a 21 + a 22 φ 2 /φ 1 (20) Let us represent these equations in a more explicit form: − X i s i z − z i + X j 1 z − γ j = X i a i 11 z − z i + X i a i 12 z − z i X j α j z − γ j (21)   − X i s i z − z i + X j 1 z − γ j   X j α j z − γ j − X j α j ( z − γ j ) 2 = X i a i 21 z − z i − X i a i 11 z − z i X j α j z − γ j (22) F urther w e find the cond itions, th at the r esidues of b oth sides of 19,20 at z = z i coincide: − s i = a i 11 + a i 12 X j α j z i − γ j (23) − X j α j z i − γ j s i = a i 21 − a i 11 X j α j z i − γ j (24) These equations together w ith the trace condition a i 11 + a i 22 = 0 imply that s i are the eigen v alues of A i , in p articular the c h oice s i = d i is app ropriate. Let us consider the b eha vior at p oles z = γ j . Remark ably the p oles of th e second ord er at these p oin ts in equation 24 redu ce. Let us calculate teh residues of b oth sid es of the equ ations 23 and 24 1 = α j X i a i 12 γ j − z i (25) α j   − X i s i γ j − z i + X i 6 = j 1 γ j − γ i   + X i 6 = j α i γ j − γ i = − α j X i a i 11 γ j − z i (26) Let us recall the normalizing condition on the residue at ∞ k X i =1 a i 12 = 0 (27) k X i =1 a i 21 = 0 (28) Let us note th at the choice of p oles for the Stur m-Liouville op erator fixes the zeros of th e rational function a 12 ( z ) , whic h in turn is defined up to a constan t: a 12 ( z ) = c Q k − 2 j =1 ( z − w j ) Q k i =1 ( z − z i ) 6 Then th e condition 27 fu lfills automatically . Th e co efficien ts a i 12 can b e expressed by the formula a i 12 = c Q k − 2 j =1 ( z i − w j ) Q k j 6 = i ( z i − z j ) (29) The co efficien ts a i 11 can also b e exp ressed as follo ws in virtue of 23 a i 11 = − s i − c Q k − 2 j =1 ( z i − w j ) Q k j 6 = i ( z i − z j ) X l α l z i − γ l (30) Let us sub stitute the expressions f or a i 12 and a i 11 in equations 25, 26, and express then α j from the first one an d substitute to th e other: Q i ( γ j − z i ) Q i ( γ j − w i )   − X k 2 s k γ j − z k + X k 6 = j 1 γ j − γ k + X k ,m Q l ( z k − w l ) Q s 6 = k ( γ m − z s ) Q l 6 = k ( z k − z l ) Q s ( γ m − w s )( γ j − z k )   + X k 6 = j Q i ( γ k − z i ) Q i ( γ k − w i )( γ j − γ k ) = 0 Let us pass to the equ iv alent f orm d ividing by Q i ( γ j − z i ) Q i ( γ j − w i ) − X k 2 s k γ j − z k + X k 6 = j 1 γ j − γ k + X k ,m Q l ( z k − w l ) Q s 6 = k ( γ m − z s ) Q l 6 = k ( z k − z l ) Q s ( γ m − w s )( γ j − z k ) + X k 6 = j Q i ( γ k − z i ) Q i ( γ k − w i )( γ j − γ k ) Q m ( γ j − w m ) Q m ( γ j − z m ) = 0 Consider the left hand side of th e equ alit y as a rational function F ( γ j ) and calculate its simple fraction decomp osition at p oles z k , w k , γ k and ∞ . It turns out th at this decomp osition take s the form: F ( γ j ) = − X k 2 s k − 1 γ j − z k − X k 1 γ j − w k + 2 X i 6 = j 1 γ j − γ i (31) Hence, the exp lored equalit y is equiv alen t to one of the Bethe equation. Let us demonstr ate, for example, the calculation of the residu e at γ j = w i Res γ j = w i F ( γ j ) = X k Q l ( z k − w l ) Q s 6 = k ( w i − z s ) Q l 6 = k ( z k − z l ) Q s 6 = i ( w i − w s )( w i − z k ) = Q s ( w i − z s ) Q s 6 = i ( w i − w s ) X k Q l ( z k − w l ) Q l 6 = k ( z k − z l )( w i − z k ) 2 (32) Let us note that the expression on th e r igh t is of the form ( Res z = w i Φ( z )) − 1 X k Res z = z k Φ( z ) where Φ( z ) = Q l ( z − w l ) Q l ( z − z l )( z − w i ) 2 and hence is equal to − 1 . Then the sufficien t condition expresses in the follo wing 7 Theorem 1 If the set of nu mb ers γ i wher e i = 1 , . . . , M satisfy the system of Bethe e quations with p ar ameters: the set of p oles is z 1 , . . . , z k and w 1 , . . . , w k − 2 with the highest we ghts 2 s 1 − 1 , . . . , 2 s k − 1 and 1 , . . . , 1 c orr esp ondingly, then the ve ctor Ψ = k Y i =1 ( z − z i ) − s i  φ 1 ( z ) φ 2 ( z )  (33) wher e φ 1 = M Y j =1 ( z − γ j ) φ 2 /φ 1 = M X j =1 α j z − γ j (34) and the c o efficie nts α j given by the e xpr essions α j = Q i ( γ j − z i ) Q i ( γ j − w i ) (35) solves the matrix line ar pr oblem 10 wher e the c onne ction c o efficients ar e define d by a i 12 = Q j ( z i − w j ) Q j 6 = i ( z i − z j ) (36) the c o efficie nts a i 11 and a i 21 ar e define d fr om 23 , 24. The normalizing c onditions 9 hold. Pro of. Prop erly , w e need to that the normalization condition 28 do es not d ep end on the c h oice of the parameter c , in particular can b e taken to b e equal 1. Indeed, in virtue of 23, 24 we obtain: a i 21 = −   2 s i X j α j z i − γ j + Q j ( z i − w j ) Q j 6 = i ( z i − z j )   X j α j z i − γ j   2   (37) W e need to pro v e that X i 2 s i X j α j z i − γ j + X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )   X j α j z i − γ j   2 = 0 (38) The first summand of 38 can b e transformed to the follo wing exp ression using the Bethe equations: X i 2 s i X j α j z i − γ j = X j α j X i 2 s i z i − γ j = X j α j   − X i 1 γ j − z i + X i 1 γ j − w i − 2 X i 6 = j 1 γ j − γ i   (39) Let us simplify th e second su m mand of 38 c hanging the summ ation order X m 6 = l α m α l X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )( z i − γ m )( z i − γ l ) + X m ( α m ) 2 X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )( z i − γ m ) 2 (40) 8 Considering the second summand of 40 let us emphasize that X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )( z i − γ m ) 2 = − ∂ γ m Φ 1 ( γ m ) (41) where Φ 1 ( γ m ) = X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )( z i − γ m ) = Q j ( γ m − w j ) Q j ( γ m − z j ) (42) Hence the expression 41 tak es the form: − Q j ( γ m − w j ) Q j ( γ m − z j ) X s 1 γ m − w s − X s 1 γ m − z s ! (43) whic h reduces th e corresp ond ing p art of 39. L et us consider the first su mmand of 40, it can also b e s implified: X i Q j ( z i − w j ) Q j 6 = i ( z i − z j )( z i − γ m )( z i − γ l ) = Q j ( γ l − w j ) Q j ( γ l − z j )( γ m − γ l ) − Q j ( γ m − w j ) Q j ( γ m − z j )( γ m − γ l ) (44) Inserting this expression in 40 we finish the pro of  3 Sc hlesinger transformations There is a discrete group of transform atio ns whic h p r eserv e th e form of the connection 10 and more- o ve r do not c h ange the class of m on o d r om y representati on. How ev er they c hange the charac teristic exp onen ts at sin gu lar p oin ts by half-in tegers in a sp ecific m an n er. Such tr ansformations are called Sc h lesinger, Hec ke or B¨ aklund transformations in d ifferen t context s. This t yp e of transf orm atio n has simple geometric in terpretation. 3.1 Action on bundles Let us first of all consider holomorphic bund les on C P 1 . Due to the Birkh of-Grotendiec k theorem an y h olomorphic bu ndle on C P 1 of r ank n is isomorphic to O ( k 1 ) ⊕ . . . ⊕ O ( k n ) for some tuple ( k 1 , . . . , k n ) which is called the type of the b undle and is defin ed up to the symmetric group action. Let us n o w consider a co v ering of C P 1 whic h consist of U ∞ - a d isc around ∞ not con taining z = z i , i = 1 , . . . , N and U 0 = C P 1 \{∞} . W e will work with r k = 2 holomorphic bund les and parameterize them by the gluing fu nction G ( z ) - holomorphic in vertible function in U 0 ∩ U ∞ with v alues in GL (2) . W e say that the pair S ∞ ( z ) ∈ O (2) ( U ∞ ) and S 0 ( z ) ∈ O (2) ( U 0 ) d efines a global section if S 0 ( z ) = G ( z ) S ∞ ( z ) . All transformations will b e d escrib ed by some c hange of th e gluing fu nction. Let us consider suc h a tr ansformation w hic h multiplies the gluing fun ction on the left by G s ( z ) = G s  z − z s 0 0 1  G − 1 s (45) for some constan t matrix G s and some p oin t z s ∈ U 0 . 9 Remark 2 The action on g luing functions c an b e br ought to the action on isomorphism classes of bund les making the choic e of c onstant matrix G s dep e ndent on the trivialization, mor e pr e cisely if one change the trivialization in U 0 by T ( z ) one should change the matrix G s by T ( z s ) G s . This is cle ar fr om the r emark 3 on the invariant definition. The case of our in terest will b e the comp osition of tw o such transformations at differen t p oint s. Lemma 1 The c omp osition of two tr ansformations given by G i ( z ) G − 1 j ( z ) applie d to the trivial bund le give also the trivial one for generic choic e of c onstant matric es G i , G j . Pro of Our goal is to p erform the d ecomposition G ( z ) = G i ( z ) G − 1 j ( z ) = G ij ( z ) G ∞ ( z ) where G ij ( z ) , G ∞ ( z ) are inv ertible resp ectiv ely in U 0 , U ∞ . T h e general argument here is the observ ation ab out the dimension of cohomology spaces in families at generic p oin t. Indeed, for particular c hoice of G − 1 i G j = 1 one obtains the trivial bu ndle wh ic h is semistable an d hence m inimizes the d imension of H 0 ( E n d ( V )) for V of d egree 0 which is the case. Due to this the trivial b undle is generic in the family of b undles give n by different G . Ho we v er we provide here an exp licit d emonstration in spirit of the decomp osition lemma of [5]. Let us decomp ose the pro duct G ( z ) = G i  z 0 0 1  G − 1 i G j  ( z − 1) − 1 0 0 1  G − 1 j (46) where w e use th e follo wing notations G i =  1 x i y i 1  (47) in to the pro duct G ( z ) = G ij ( z ) G ∞ ( z ) where G ij ( z ) , G ∞ ( z ) are h olomorphic in vertible functions in U 0 , U ∞ resp ectiv ely . T raditional calculatio ns p ro vide a d ecomposition of the form G ∞ ( z ) = z (1 − x j y i )(1 − x j y j ) − x j ( y i − 2 y j − x j y i y j ) (1 − 2 x j y i + x j y j )(1 − x j y i )(1 − x j y j )( z − 1) − x j (1 − x j y i )(1 − x j y j )( z − 1) y i − 2 y j + x j y i y j (1 − x j y i )(1 − 2 x j y i + x j y j ) 1 1 − x j y i ! 3.2 T ransforma t ions on connections Let us d efine an action of the group of Sc h lesinger transformations T ij on the sp ace of connections with singularities at z = z i . As men tioned the action on bund les c hanges the gluing function. Let us start with the trivial bun dle giv en by the gluing fu nction 1 . The S c hlesinger transformation giv es another bund le structure, a global section is d efined by a pair S 0 , S ∞ suc h that S 0 = GS ∞ with G = G ij G ∞ . O n e can d efine an action on the space of connections as follo w s : let ∂ z − A b e an initial conn ectio n in th e trivial bundle defin ed by the same exp ression o ver tw o considered op en sets, the transformed connection is then defined as follo ws: ∂ z − A o ve r U ∞ G ( ∂ z − A ) G − 1 o ve r U 0 After c h an ging b asis in U ∞ of the form f S ∞ = G ∞ S ∞ w e ob tain the expr ession for the connection ∂ z − A → G ∞ ( ∂ z − A ) G − 1 ∞ (48) 10 If one c hanges th e trivialization in U 0 as follo w s f S 0 = G − 1 ij S 0 one obtains G ( ∂ z − A ) G − 1 → G − 1 ij G ( ∂ z − A ) G − 1 G ij = G ∞ ( ∂ z − A ) G − 1 ∞ (49) Hence th e trans formed connection can b e repr esen ted as a global r ational connection of the same t yp e as th e initial one, the same b eha vior at ∞ is guarantie d by the fact that G ∞ is holomorphic in vertible at U ∞ . Remark 3 This definition is c oher ent with the invariant c onstruction, the He cke tr ansforma tion on the she af language is the op er ation of taking a su b she af, define d by a line ar c ondition on values of se ctions at fixe d p oint. T o define an action on the sp ac e of singular c onne ctions at a singular p oint one has to cho ose as a c ondition the dual c ove ctor to one of the eigen dir e ction for the r esidue of the c onne ction. Using the results of the pr evious section let u s calculate the actio n of th e S c hlesinger tr ans- formation on the connection. T o obser ve the normalizat ion condition of A ( z ) at ∞ one should consider the transformation of the form e G ( z ) = G − 1 ∞ ( ∞ ) G ∞ ( z ) = 1 z − 1 z − x 1 ( y 0 − 2 y 1 + x 1 y 0 y 1 ) (1 − x 1 y 0 )(1 − x 1 y 1 ) x 1 (1 − 2 x 1 y 0 + x 1 y 1 ) (1 − x 1 y 0 )(1 − x 1 y 1 ) y 0 − 2 y 1 + x 1 y 0 y 1 (1 − x 1 y 0 )(1 − x 1 y 1 ) 1 − 2 x 1 y 0 + x 1 y 1 (1 − x 1 y 0 )(1 − x 1 y 1 ) !! (50) The wh ole set of S chlesinger transformations for the 3-p oint case relev an t to the analysis of the P ainlev´ e VI equ atio n was calculated in [6] 3.3 Action on Bethe v ectors Let us introd uce th e notation Z = { z i } , W = { w j } , Λ = { λ i } . Let us also in tr o d u ce sym b ols for the follo wing ob jects: • O S ( Z, W, Λ) - a solution for the system of Bethe equations, that is the set µ i whic h solv es the system 4 for the fix ed p oin ts (the p oles of the Lax op erator) z 1 , . . . , z k and w 1 , . . . , w k − 2 and the corresp ondin g highest we igh ts λ 1 , . . . , λ k and 1 , . . . , 1; • O L ( Z , W , Λ) - a solution of the linear pr oblem 6 of the form 5 with the given set of p arameters; • M L ( Z, W , Λ) - a s olution of the matrix linear p roblem 10 of the form 17 w ith th e giv en set of parameters, it has t wo comp onen ts M L ( Z, W , Λ) 1 , 2 ; Remark 4 The choic e of the highest weights 1 at “moving p oles” w i is not ne c essary but in a sense generic. One c an take the p otential U = m X j =1 − 1 / 4( η j + 2) η j ( z − w j ) 2 + k X i =1 1 / 4 + detA i ( z − z i ) 2 + k − 2 X j =1 H w j z − w j + k X i =1 H z i z − z i (51) with generic values of the highest weights . This r e alizes if one demand a 12 ( z ) to have zer os w j with multiplicities η j subje ct to the c ondition P m j =1 η j = k − 2 . The resu lts of our pr evious considerations can b e summ arized in the follo wing: for a solution of the Bethe system O S ( Z, W, Λ) one can consider a solution M L ( Z, W, Λ) in virtu e of the theorem 1. Then one can d efine a family of transform ations: 11 • Dual solution S ym b olically this transformation expresses vie th e diagram O S ( Z , W , Λ) O S ( Z , f W , Λ) ↓ ↑ O L ( Z, W , Λ) → M L 1 ( Z, W , Λ) → M L 2 ( Z, W , Λ) → O L ( Z, f W , Λ) • Schlesing er transformation Using the n otation b elo w one has O S ( Z , W , Λ) O S ( Z , f W , e Λ) ↓ ↑ O L ( Z, W , Λ) → M L ( Z, W, Λ) → M L ( Z, f W , e Λ) → O L ( Z , f W , e Λ) Dep ending on the c hoice of the sub s pace of th e upp er an d lo wer Sc hlesinger transform atio ns one obtains the follo wing action of T ij on the set of highest w eights ( . . . , λ i , . . . , λ j , . . . ) 7− → ( . . . , λ i + 1 , . . . , λ j − 1 , . . . ) ( . . . , λ i , . . . , λ j , . . . ) 7− → ( . . . , λ i + 1 , . . . , λ j + 1 , . . . ) ( . . . , λ i , . . . , λ j , . . . ) 7− → ( . . . , λ i − 1 , . . . , λ j − 1 , . . . ) ( . . . , λ i , . . . , λ j , . . . ) 7− → ( . . . , λ i − 1 , . . . , λ j + 1 , . . . ) Discussion P ainlev´ e equations The idea of d iscreet transformations preserving the condition of “minimalit y” of the mono dr omy exploited h ere lies in the middle of the theory of isomono dromic transformations. The same group of transformations act on th e sp ace of solutions of the Pa in lev ´ e VI equation realized as the mono dromy preserving condition for the F uc h sian system (the references can b e f ound for example in [7]): ( ∂ z − A ( z ))Ψ( z ) = 0 where A ( z ) =  a 11 ( z ) a 12 ( z ) a 21 ( z ) a 22 ( z )  = A 0 z + A 1 z − 1 + A t z − t One can exp ect that the p roblem of con tin uous deformation of the sp ectrum for the quant um Gaudin mo del w ith r esp ect to the parameters of the mod el - the marked p oints, is equiv alen t to the Sc h lesinger system of equations, hence for some particular case of the mo del is just the P ainlev´ e VI equation. This is not a priori evident b ecause of the essenti ally r esonan t case of the connection app earing in th e isomono dr omic formulati on of the Bethe equation for the Gaudin mo del. Langlands corresp ondence The qu an tum Gaudin mo del pla ys an imp ortan t role in the geometric Langlands corresp onden ce [8, 3 ], n amely the corresp ond ence is pro vided b y the r elatio n : “Gaudin-Hitc hin” D -mo dule ⇔ flat connection ( G -op er) In th e arithmetic case moreo ve r one has an imp ortant prop er ty of coi ncidence of the corre- sp onding L -functions. In the geometric case o ver C there is no n atural n otion of the F rob enius automorphism and hence of the Galois-side L -function. P r esumably , the symmetry elaborated here is r elate d n amely to the Galois side of the corresp ondence. Recen tly in [9] it was observed another app earence of the arithmetic L -fun ctions in quantum in tegrable systems, namely in the theory of quan tum affine T o da c hain. 12 The Bibliograph y [1] D. T alalaev Q u antizatio n of the Gaudin system , hep -th/0404153 F unctional Analysis and Its application V ol. 40 No. 1 p p.86-91 (2006) [2] A.Cherv ov, D. T alalaev U ni v ersal G-op er and Gaudin eigenpr oblem , hep-th/040900 7 [3] A.Cherv ov, D. T alalaev Qu antum sp e ctr al curves, qu antum inte gr able systems and the ge omet- ric L anglands c orr esp ondenc e , hep-th/06041 28 [4] E. Mukhin , V. T arasov, A. V arc henk o, The B. and M. Shapir o c onje ctur e in r e al algebr aic ge ometry and the Bethe ansatz , math.A G/051 2299 [5] A. A. Bolibrukh , F uchsian differ ential e quations and holomor phic bund les , Mosco w. C en ter of Con tinuous Mathematical Ed ucation, 2000, R u ssian [6] U. Mu˘ gan, A. Sakk a, Sc hlesinger tr ansformations for the Painlev´ e VI e quation , J. Math. Ph ys. 36 (3) 1995 [7] K. Iwa saki, H. K im ura, S. Sh imom ura, M. Y oshida, F r om Gauss to Painleve A mo dern theory of sp ecial fun ctions, 1991 [8] E. F renkel, R e c ent A dvanc es in the L angland s P r o gr am , math/0303074 [9] A. Gerasimo v, D. Lebedev, S. Oblezin, Baxter op er ator a nd Ar chime de an He cke algebr a arXiv:0706 .3476 A. Gerasimo v, D. L eb edev, S. Oblezin, On Baxter Q-op er ators And Their Arithmetic Impli- c ations arXiv:0711 .2812 13