On Matrix Painleve Systems

1. On Matrix Painleve Systems Yoshihiro Murata Nagasaki University 20 September 2006 Isaac Newton Institute

2. Contents 1. Introduction 2. Dimensional Reductions of ASDYM eqns and Matrix Painleve Systems (Reconstruction of the result of Mason & Woodhouse) 3. Degenerations of Painleve Equations and Classification of Painleve Equations 4. Generalized Confluent Hypergeometric Systems included in MPS (joint work with Woodhouse)

3. 1. Introduction Basic Motivation Can we have new good expressions of Painleve eqns to achieve further developments? (Background) We often use various expressions to find features of Painleve: e.g. Painleve systems (Hamiltonian systems) 3-systems of order 1 (Noumi-Yamada system) Answer: New possively good expressions exist.

4. Grassmann Var concepts Painleve group Jordan group New reduction relationship Detailed investigation common framework Overview１ 1993,1996 1980’～1990’ Mason&Woodhouse Gelfand et al, H.Kimura et al ASDYM eqs Theory of GCHS (Generalized Confluent Symmetric Hypergeometric Systems) reduction reduced eqs (Matrix ODEs) Painleve eqns = Matrix Painleve Systems

5. Overview2 (Common framework) GCHS ASDYM MPS

6. Matrix Painleve Systems Overview3 Young diagram ASDYang-Mills eq + Constraints = symmetry & region Painleve eqns degenerated eqns × 3 type constant matrix ⇒15 type MPS

7. Dimmensional Reductions of ASDYM eqns and Matrix Painleve Systems 2.1 Preliminaries Painleve III’ Third Painleve eqn has two expressions: These are transformed by PIII’ is often better than PIII

8. Young diagrams and Jordan groups (Basic concepts in the theory of GCHS) We express a Young diagram by the symbolλ If λ consists l rows and boxes, we write as e.g. For , we define Jordan group Hλas follows:

9. e.g. If , Jordan group H(2,1,1)is a group of all matrices of the form:

10. So, on the case , we have Jordan groups Hλ as :

11. Subdiagrams and Generic stratum of M(r,n) (Basic concepts in the theory of GCHS) e.g. If , then subdiagrams μof weight 2 are (2,0,0), (1,1,0), (1,0,1), (0,1,1)

12. e.g. On the case M(2,4), λ = (2,1,1), μ= (1,0,1) e.g.

13. 2.2 New Reduction Process We consider the ASDYM eq defined on a Grassmann variety. Reduction Process (1) Take where Let then (2) Take a metric on Let sl(2,C) gauge potential satisfies ASD condition

14. ASDYM eqn (3) We consider projective Jordan group whereλare Young diagrams of weight 4. PHλacts (4) Restrict ASDYM onto PHλinvarinatregions

15. Let . Then gives 3-dimensional fibration

16. Change of variables. There exists a mapping St : orbit of PHλ t

17. (7) PHλ–invariant ASDYM eqn Lλ (8) Calculations of three first integrals

18. 2.3 Matrix Painleve Systems We call the combined system (Lλ＋1st Int) Matrix Painleve Sytem Mλ

19. We obtained 5 types of Matrix Painleve Systems: By gage transformations, constant matrix P is classified into 3 cases: Thoerem1By the reduction process (1),…,(8), we can obtain Matrix Painleve Systems MλfromASDYM eqns. Mλare classified into 15cases by Young diagram λand constant matrix P. A B C

20. Degenerations of Painleve Equations • and Classification of Painleve Equations 3.1 Degenerations and Classification of Painleve eqns e.g.PVI PV PIVPIII’ PII PI

21. PV is divided into two different classes: PV(δ≠0) can be transformed into Hamiltonian system SV PV(δ＝0) can’t use SV； and equivalent to PIII’(γδ≠0) PIII’ isdivided into four different classes: (by Ohyama-Kawamuko-Sakai-Okamoto) PIII’(γδ≠0) type D6 generic case of PIII’ can be transformed into Hamiltonian sytem SIII’ PIII’(γ＝0,αδ≠0) type D7 PIII’(γ＝0,δ=0, αβ≠0) type D8 PIII’(β＝δ＝0) type Q solvable by quadrature PI :If we change eq to , it is solvable by Weierstrass

22. For special values of parameters, PJ (J=II…VI) have classical solutions. Equations which have classical solcan be regardedas degenerated ones. This type degeneration is related to transformation groups of solutions. Question 1: Can we systematically explain these degenerations? Question 2: Can we classify Painleve eqns by intrinsic reason?

23. 3.2 Correspondences between Matrix Painleve Systems and Painleve Equations Theorem2 λ P Correspondences

24. ì ü L 0 í ý L î þ - I Riccati linear

25. PVI(D4) (α≠1/2) PVI(D4) (α＝1/2) PVI(D4) (α＝1/2) PV(D5) PIII’(D6) ? PIV(E6) PII(E7) Riccati PIII’(D6) PIII’(D7) PIII’(Q) PII(E7) PI (E8) Linear

26. (1) Nondeg cases of MPS correspond to PII,…,PVI. (2) All cases of Painleve eqns are written by Hamiltonian systems. (3) Degenerations of Painleve eqns are characterized by Young diagramλand constant matrix P. Degenerations of Painleve eqn are classified into 3 levels: 1st level: depend on λonly 2nd level: depend on λand P 3rd level: dependon transformation group of sols (4) Parameters of Painleve Systems are rational functions of parameters (k,)l,m,n. (5) On , numbers of parameters are decreased at the steps of canonical transformations between NJ and SJ.

27. 4. Generalized Confluent Hypergeometric Systems included in MPS(joint work with Woodhouse) Summary1 (general case) Theory of GCHS is a general theory to extend classical hypergeometric and confluent hypergeometric systems to any dimension paying attention to symmetry of variables and algebraic structure. Original GCHS is defined on the space . Factoring out the effect of the group , , we obtain GCHS on and GCHS on Concrete formula of is obtained (with Woodhouse)

28. Summary 2 (On the case of MPS) Painleve System SJ (J=II～VI) contains Riccati eqn RJ. RJ is transformed to linear 2-system LSλ contained in Mλ(k,l,m,n). LSλhas 3-parameters. Let denote lifted up systems of onto , then we have following diagram.

29. From these, Matrix Painleve Systems may be good expressions of Painleve eqns.

30. References: H.Kimura, Y.Haraoka and K.Takano, The Generalized Confluent Hypergeometric Functions, Proc. Japan Acad., 69, Ser.A (1992) 290-295. Mason and Woodhouse, Integrability Self-Duality, and Twistor Theory, London Mathematical Society Monographs New Series 15, Oxford University Press, Oxford (1996). Y.Murata, Painleve systems reduced from Anti-Self-Dual Yang-Mills equation, DISCUSSION PAPER SERIES No.2002-05, Faculty of Economics, Nagasaki University. Y.Murata, Matrix Painleve Systems and Degenerations of Painleve Equations, in preparation. Y.Murata and N.M.J.Woodhouse, Generalized Confluent Hypergeometric Systems on Grassmann Variety, DISCUSSION PAPER SERIES No.2005-10, Faculty of Economics, Nagasaki University. Y.Murata and N.M.J.Woodhouse, Generalized Confluent Hypergeometric Systems included in Matrix Painleve Systems, in preparation.