Generalization of Okamotos equation to arbitrary $2times 2$ Schlesinger systems

Generalizati o n of Ok amoto’s equation to arbitrary 2 × 2 Sc hlesing er systems D. Korotkin 1 and H. Sam tleb en 2 1 Departmen t of Ma thematics and Statistic s, Concordia Univ ersit y , 7141 Sherbro oke W est, Mon treal H4B 1R6, Q u eb ec, Canada 2 Univ ersit ´ e de L y on, Lab oratoire de Physique, Ecole Normale Sup´ erieure de Lyo n, 46, all ´ ee d’Italie, F-6936 4 Lyon CEDEX 07, F r ance korotkin @mathstat .concordia.ca , henning. samtleben @ens-lyon.fr Abstract The 2 × 2 Schlesinge r system for the case of four regular s ingularities is equiv- alen t to the P ainlev ´ e VI equatio n. The P ainlev ´ e VI equation can in turn b e rewritten in the symm etric form of Ok amoto’s equation; the dep enden t v ariable in Ok amoto’s form of th e PVI equ ation is the (sligh tly transformed ) logarithmic deriv ativ e of the Jim b o-Miw a tau-fu nction of the Schlesinger system. The goal of th is note is t w ofold. First, we find a symmetric uniform form u lation of an arbitrary Schlesinge r system w ith regular singularities in terms of app ropriately defined Virasoro generators. Second, we fin d analogues of O k amoto’s equation for the case of the 2 × 2 Sc hlesinger system with an arbitrary num b er of p oles. A new set of scalar equations for the logarithmic deriv ativ es of the Jimb o-Miw a tau-function is derived in terms of generators of the Virasoro algebra; these gen- erators are expressed in terms of deriv ativ es w ith resp ect to singularities of th e Sc hlesinger sy s tem. 1 In tro duction The Schlesinger system is the f ollo wing non-autonomous system of differential equations for N un k n o wn matrices A j ∈ sl ( M ) dep ending on N v ariables { λ j } : ∂ A j ∂ λ i = [ A j , A i ] λ j − λ i , i 6 = j , ∂ A j ∂ λ j = − X i 6 = j [ A j , A i ] λ j − λ i . (1.1) 1 The system (1.1) determines isomono dr omic deformations of a solution of matrix ODE with meromorphic co efficien ts ∂ Ψ ∂ λ = A ( λ ) Ψ ≡ N X j =1 A j λ − λ j Ψ . (1.2) The solution of this system normalized at a fix ed p oin t λ 0 b y Ψ( λ 0 ) = I s olv es a matrix Riemann-Hilb ert pr oblem with some mono dr om y matrices aroun d the sin gularities λ j . The Sc hlesinger equations were disco vered almost 100 years ago [1]; ho wev er, they contin ue to pla y a k ey role in many areas of mathematical physics: th e theory of random matrices, in - tegrable systems, theory of F rob en iu s manifolds, etc. . Th e system (1.1) is a non -autonomous hamiltonian system with resp ect to the P oisson b rac k et { A a j , A b k } = δ j k f ab c A c j , (1.3) where f ab c are stru cture constants of sl ( M ); δ j k is the Kr onec k er sy mb ol. Ob vious ly , th e traces tr A n j are integ rals of the Schlesinger sys tem for any v alue of n . The commuting Hamiltonians defin ing ev olution with resp ect to the times λ j are giv en by H j = 1 4 π i I λ i tr A 2 ( λ ) dλ ≡ 1 2 X k 6 = j tr A j A k λ k − λ j . (1.4) The generating f unction τ JM ( { λ j } ) of the h amiltonians H j , defined by ∂ ∂ λ j log τ JM = H j , (1.5) w as introd uced by Jimb o, Miwa and th eir co-authors [2, 3]; it is called the τ -fu nction of the Sc hlesinger system. The τ -function pla ys a k ey role in th e theory of the Sc h lesinger equatio ns; in particular, the divisor of zeros of th e τ -function coincides with th e divisor of singularities of the sol ution of the Sc hlesinger system; on the same divisor th e underlying Riemann-Hilbert problem lo oses its solv ability . In the simplest non -trivial case when the matrix d imension equals M = 2 and the num b er of singularities equals N = 4, the Sc hlesinger system can equiv alen tly b e rewritten as a s ingle scalar differentia l equation of order tw o — the Painlev ´ e VI equation d 2 y dt 2 = 1 2  1 y + 1 y − 1 + 1 y − t   dy dt  2 −  1 t + 1 t − 1 + 1 y − t  dy dt + y ( y − 1)( y − t ) t 2 ( t − 1) 2  α + β t y 2 + γ t − 1 ( y − 1) 2 + δ t ( t − 1) ( y − t ) 2  , (1.6) where t is th e cross-ratio of the four singularities λ 1 , . . . , λ 4 , and y is the p osition of a zero of the upp er righ t corner elemen t of the matrix P 4 k =1 A k λ − λ k . Let us d enote the eigenv alues of the matrices A j b y α j / 2 and − α j / 2. T hen the constan ts α, β , γ and δ from the Painlev ´ e VI equation (1.6) are related to the constan ts α j as follo ws: α = ( α 1 − 1) 2 2 , β = − α 2 2 2 , γ = α 2 3 2 , δ = 1 2 − α 2 4 2 . (1.7) 2 It w as furth er observed by Ok amoto [4, 5], th at the Pa inlev ´ e VI equ ation (and, therefore, the original 2 × 2 Sc hlesinger system with four singularities) can b e r ewritten alternativ ely in a simple form in terms of the so-called au x iliary hamiltonian fu nction h ( t ). T o define this function w e need to introduce fir st four constan ts b j , which are expressed in terms of the eigen v alues of the matrices A j as follo ws: b 1 = 1 2 ( α 2 + α 3 ) , b 2 = 1 2 ( α 2 − α 3 ) , b 3 = 1 2 ( α 4 + α 1 ) , b 4 = 1 2 ( α 4 − α 1 ) . (1.8) The auxiliary h amiltonian function h ( t ) is d efi ned in terms of solution y of equation (1.6 ) and the constants b j as follo ws: h = y ( y − 1)( y − t )  dy dt  2 − { ( b 1 + b 2 )( y − 1)( y − t ) + ( b 1 − b 2 ) y ( y − t ) + ( b 3 + b 4 ) y ( y − 1) } dy dt +  1 4 (2 b 1 + b 3 + b 4 ) 2 − 1 4 ( b 3 − b 4 ) 2  ( y − t ) + σ ′ 2 [ b ] t − 1 2 σ 2 [ b ] , (1.9) where σ ′ 2 [ b ] := b 1 b 3 + b 1 b 4 + b 3 b 4 , σ 2 [ b ] := 4 X j,k =1 j 2, or the Schlesinge r system f or orthogonal, symplectic, and exceptional group s ) w here the num b er of indep en d en t tensors ma y b e larger. It w ould b e highly interesting to u nderstand if equations analogo us to (3.1), (3.2) can b e deriv ed from suc h higher rank algebraic identitie s. As those ident ities will b e built from a larger num b er of in v ariant tensors (structur e constants, etc.), the corresp onding differen tial equ ations would b e of h igher order in deriv ativ es. • Is it p ossible to com bine ou r p resen t constru ction app licable to Schlesinger s ystems with simple p oles only with construction of [8 ] whic h requires the presence of higher order p oles? What would b e the full set of equations for th e tau-function with r esp ect to th e full set of deformation parameters in presence of higher order p oles? • The Sc hlesinger system (1.1) has also b een constructed for v arious higher genus Rie- mann su rfaces [10, 11, 12, 13]. It would b e interesting to fi rst of all fi nd the pr op er generalizat ion of the symmetric form (2.11), (2.12) of the Schlesinge r sys tem to higher gen us surfaces wh ic h in tu r n should allo w to deriv e by an analogous construction the non-trivial differentia l equations satisfied by the asso ciated τ -fun ction. W e conjecture that in some sense the form (2.11) s hould b e universal: it sh ould remain the same, al- though the defin ition of the Virasoro generators L m and the v ariables B m ma y c hange. 12 • As we hav e ment ioned ab o v e, the extra term n B m + n in the Hamiltonian dynamics of the sym metrised S c hlesinger system (2.17 ) can b e absorb ed into the sy m plectic action up on replacing the standard affine Lie-P oisson b rac k et (2.16) by its cen trally extended v ersion. Ho we v er, this cen tral extension is not seen in any of th e fi nite- N Sc hlesinger systems. This s eems to su ggest th at the s y s tem (2.11) shou ld b e considered not just as a symm etric form of the u s ual Schlesinge r system with fi n ite num b er of p oles, bu t as a “univ ersal” S c hlesinger s ystem which inv olves an infinite set of indep end en t v ariables B n . Pr esumably , this full system inv olv es the generators L n and coefficients B n not only for p ositiv e, b ut also f or negativ e n . In this setting, the cen trally extended ve rsion of the brac k et (2.16) should app ear nat- urally . The m ost interesting problem would b e to find the geometric origin of such a generalized system; a p ossible candidate could b e the isomono dr omic deformations on higher genus cur v es. Ac kno w ledgemen ts: The w ork of H.S. is su pp orted in part by the Agence Nationale de la Rec herc he (ANR). The work of D.K. w as su pp orted by NSER C, NA TEQ and Concordia Univ ersit y Researc h Ch air grant. References [1] L. Schlesinger, ¨ Ub er e ine Klasse von D i ffer entialsystemen b eliebiger O r dnung mit festen kritischen Punkten. J. Reine u. An gew. Math. 141, 96 (1912). [2] M. Jim b o, T. Miwa, Y. Mˆ o ri, and M. S ato, Density matrix of an imp enetr able Bose gas and the fifth Painlev´ e tr ansc endent. P hysica 1D (1980) 80. [3] M. Jimb o, T. Miwa, and K. Ueno, M ono dr omy pr eserving deforma tion of line ar or dinary differ ential e quations with r ationa l c o efficients. Physica 2D (1981) 306. [4] K. O k amoto, Isomono dr omic deformatio n and the Painlev´ e e qu ations and the Garnier system. J. F ac. Sci. Univ. T oky o IA 33 (1986) 575. [5] K. O k amoto, Studies on the Painlev´ e e qu ations I, si xth Painlev´ e e quation P VI . Ann. Mat. Pu ra Appl. 146 (1987) 337. [6] B. Dubr o vin and M. Mazzocco, Canonic al structur e and symmetries of the Schlesinger e quations. Comm. Math. Phys. 271 (2007) 289. [7] M. Mazzo cco, Irr e gular isomono dr omic deformations for Garnier systems and Okamoto’s c anonic al tr ansformations. J . London Math. So c. 70 (2004) 405. [8] J. Harn ad , Vir asor o gener ators and biline ar e quations for isomono dr omic tau functions. In : Isomono d r omic deformations and applications in physics, CRM Pro c. Lecture Notes, 31, Amer. Math. S o c., Providence, RI, 2002. [9] M.L. Kontsevic h, Vi r asor o algebr a and T eichmul ler sp ac es , F unctional Analysis and its applications, 21 N.2 (1987) 13 [10] D. Korotkin and H. Samtle b en, On the quantization of isomono dr omic deformat ions on the torus, Int. J. Mo d. P hys. A 12 (1997) 2013. [arXiv:hep-th/95110 87 ] [11] S . Kaw ai, Isomono dr omic deformation of F uchsian-typ e pr oje ctive c onne ctions on el liptic curve s. RIMS 1022 (1997) 53. [12] A.M. Levin and M.A. Olshanetsky , Hier ar chies of isomono dr omic deformat ions and Hitchin systems. In: Mo sco w Seminar in Mathematical P h ysics, Amer. Math. So c., Pro vidence, RI (1999) p. 223. [arXiv:hep-th/97092 07 ] [13] K. T ak asaki, Gaudin mo del, KZB e quation, and isomono dr omic pr oblem on torus. Lett. Math. Ph ys. 44 (1998) 143. [arXiv:hep-th/97110 58 ] 14