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

2 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) Contents

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 common framework Overview １ 1993, ’ ～ 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 Detailed investigation New reduction relationship concepts Jordan groupPainleve group = Grassmann Var

5 Overview2 (Common framework) MPS GCHS ASDYM

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

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

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

11 Subdiagrams and Generic stratum of M(r,n) (Basic concepts in the theory of GCHS) e.g. If, then subdiagrams μ of 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.

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 where λ are Young diagrams of weight 4. PH λ acts (4) Restrict ASDYM onto PH λ invarinat regions

15 (5)Let. Then gives 3-dimensional fibration

16 (6)Change of variables. There exists a mapping S t : orbit of PH λ t

17 (7) PH λ –invariant ASDYM eqn L λ (8) Calculations of three first integrals

Matrix Painleve Systems We call the combined system (L λ ＋ 1 st Int ) Matrix Painleve Sytem M λ

19 We obtained 5 types of Matrix Painleve Systems: By gage transformations, constant matrix P is classified into 3 cases: Thoerem1 By the reduction process (1),…,(8), we can obtain Matrix Painleve Systems M λ from ASDYM eqns. M λ are classified into 15cases by Young diagram λ and constant matrix P. A B C

Degenerations and Classification of Painleve eqns e.g. P VI P V P IV P III’ P II P I 3.Degenerations of Painleve Equations and Classification of Painleve Equations

21 P V is divided into two different classes: P V ( δ ≠0) can be transformed into Hamiltonian system S V P V ( δ ＝ 0) can’t use S V ； and equivalent to P III’ ( γδ ≠0) P III’ is divided into four different classes: (by Ohyama-Kawamuko-Sakai-Okamoto) P III’ ( γδ ≠0) type D 6 generic case of P III’ can be transformed into Hamiltonian sytem S III’ P III’ ( γ ＝ 0, αδ ≠0) type D 7 P III’ ( γ ＝ 0,δ=0, αβ ≠0) type D 8 P III’ ( β ＝ δ ＝ 0) type Q solvable by quadrature P I : If we change eq to, it is solvable by Weierstrass

22 For special values of parameters, P J (J=II…VI) have classical solutions. Equations which have classical sol can be regarded as 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?

Correspondences between Matrix Painleve Systems and Painleve Equations Theorem2 λ P Correspondences (1,1,1,1) A B C (2,1,1) A B C under calculation

24 (3,1) A B C (2,2) A B C (4) A B C Riccat i linear       I 0 L L

25 P VI (D4) (α≠1/2) P VI (D4) (α ＝ 1/2) P V (D5) P III’ (D6) ? P IV (E6) P II (E7) Riccati P II (E7) P I (E8) Linear P III’ (D6) P III’ (D7) P III’ (Q)

26 (1) Nondeg cases of MPS correspond to P II,…, P VI. (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: 1 st level: depend on λonly 2 nd level: depend on λand P 3 rd level: depend on 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 N J and S J.

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 S J (J=II ～ VI) contains Riccati eqn R J. R J is transformed to linear 2-system LS λ contained in M λ (k,l,m,n). LS λ has 3-parameters. Let denote lifted up systems of onto, then we have following diagram. (1,1,1,1)Gauss Hypergeometric (2,1,1)Kummer (3,1)Hermite (2,2)Bessel (4)Airy

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) 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 , 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 , Faculty of Economics, Nagasaki University. Y.Murata and N.M.J.Woodhouse, Generalized Confluent Hypergeometric Systems included in Matrix Painleve Systems, in preparation.