Part 1: Algebraic solutions of Painlev£© VI Modular group actions Finite orbits Part 1:...

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Transcript of Part 1: Algebraic solutions of Painlev£© VI Modular group actions Finite orbits Part 1:...

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Part 1: Algebraic solutions of Painlevé VI

    Oleg Lisovyy

    LMPT, Tours, France

    October 26th, 2012

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Painlevé VI equation:

    d2w

    dt2 =

    1

    2

    ( 1

    w +

    1

    w − 1 +

    1

    w − t

    )( dw

    dt

    )2 − (

    1

    t +

    1

    t − 1 +

    1

    w − t

    ) dw

    dt +

    + w(w − 1)(w − t)

    2t2(t − 1)2

    ( (θ∞ − 1)2 −

    θ2x t

    w2 + θ2y (t − 1) (w − 1)2

    + (1− θ2z )t(t − 1)

    (w − t)2

    )

    4 parameters θx,y,z,∞

    most general equation of type w ′′ = F (t,w ,w ′) without movable critical points (Painlevé property)

    w(t) is meromorphic on the universal cover of P1\{0, 1,∞} Okamoto affine F4 Weyl symmetry group

    PI –PV are obtained as limiting cases

    applications in nonlinear physics, classical and quantum integrable systems, random matrix theory, differential geometry...

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Painlevé VI equation:

    d2w

    dt2 =

    1

    2

    ( 1

    w +

    1

    w − 1 +

    1

    w − t

    )( dw

    dt

    )2 − (

    1

    t +

    1

    t − 1 +

    1

    w − t

    ) dw

    dt +

    + w(w − 1)(w − t)

    2t2(t − 1)2

    ( (θ∞ − 1)2 −

    θ2x t

    w2 + θ2y (t − 1) (w − 1)2

    + (1− θ2z )t(t − 1)

    (w − t)2

    )

    4 parameters θx,y,z,∞

    most general equation of type w ′′ = F (t,w ,w ′) without movable critical points (Painlevé property)

    w(t) is meromorphic on the universal cover of P1\{0, 1,∞} Okamoto affine F4 Weyl symmetry group

    PI –PV are obtained as limiting cases

    applications in nonlinear physics, classical and quantum integrable systems, random matrix theory, differential geometry...

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Example: 2D Ising model

    (Jimbo, Miwa ’81) Diagonal two-point correlation functions are Painlevé VI τ -functions:

    τ(t) = (1− t)− N2

    2 〈σ(0, 0)σ(N,N)〉T

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Solutions of Painlevé VI

    According to Watanabe (1998):

    solutions of PVI are either

     Riccati solutions or

    X

    ‘new’ transcendental functions or

    algebraic functions

    ???

    Lot of examples of algebraic solutions: Hitchin (1995); Dubrovin (1995); Dubrovin & Mazzocco (1998); Andreev & Kitaev (2001); Kitaev (2003–2005); Boalch (2003–2007)

    is complete classification possible?

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Solutions of Painlevé VI

    According to Watanabe (1998):

    solutions of PVI are either

     Riccati solutions or X

    ‘new’ transcendental functions or

    algebraic functions

    ???

    Lot of examples of algebraic solutions: Hitchin (1995); Dubrovin (1995); Dubrovin & Mazzocco (1998); Andreev & Kitaev (2001); Kitaev (2003–2005); Boalch (2003–2007)

    is complete classification possible?

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Solutions of Painlevé VI

    According to Watanabe (1998):

    solutions of PVI are either

     Riccati solutions or X

    ‘new’ transcendental functions or

    algebraic functions ???

    Lot of examples of algebraic solutions: Hitchin (1995); Dubrovin (1995); Dubrovin & Mazzocco (1998); Andreev & Kitaev (2001); Kitaev (2003–2005); Boalch (2003–2007)

    is complete classification possible?

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Solutions of Painlevé VI

    According to Watanabe (1998):

    solutions of PVI are either

     Riccati solutions or X

    ‘new’ transcendental functions or

    algebraic functions ???

    Lot of examples of algebraic solutions: Hitchin (1995); Dubrovin (1995); Dubrovin & Mazzocco (1998); Andreev & Kitaev (2001); Kitaev (2003–2005); Boalch (2003–2007)

    is complete classification possible?

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Schwarz list

    Question: When does Gauss hypergeometric function 2F 1(a, b, c, λ) become algebraic? (Schwarz, 1873)

    dλ =

    ( Ax

    λ− ux +

    Ay

    λ− uy

    ) Φ,

    standard choice ux = 0, uy = 1

    Φ ∈ Mat2×2, Ax,y ∈ sl2(C) monodromy matrices Mx,y ∈ SL(2,C) algebraic solutions lead to finite monodromy → 15 classes

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Isomonodromy approach

    Painlevé VI describes monodromy preserving deformations of Fuchsian systems

    dλ =

    ( Ax

    λ− ux +

    Ay

    λ− uy +

    Az

    λ− uz

    ) Φ, Φ ∈ Mat2×2.

    Aν ∈ sl2(C) are independent of λ, with eigenvalues ±θν/2 4 regular singular points ux , uy , uz ,∞ ∈ P1

    Ax + Ay + Az def = − A∞ =

    ( −θ∞/2 0

    0 θ∞/2

    ) monodromy matrices Mx ,My ,Mz ∈ SL(2,C), defined up to overall conjugation (3× 3− 3 = 6 parameters)

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Painlevé VI ↔ linear system dictionary: PVI independent variable t = (ux − uy )/(ux − uz ); w(t) is a combination of matrix elements of Ax,y,z

    to each branch of a solution of PVI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of Mx , My , Mz , M∞ = MzMyMx give PVI parameters θx,y,z,∞; the other two correspond to integration constants

    analytic continuation induces an action of the pure braid group P3 on the space M = G3/G , G = SL(2,C) of conjugacy classes of G -triples

    u x

    u y

    u z

    u x uzuy

    ux

    Mx M My x

    uzux uy uz

    -1

    My

    uy

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    Painlevé VI ↔ linear system dictionary: PVI independent variable t = (ux − uy )/(ux − uz ); w(t) is a combination of matrix elements of Ax,y,z

    to each branch of a solution of PVI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of Mx , My , Mz , M∞ = MzMyMx give PVI parameters θx,y,z,∞; the other two correspond to integration constants

    analytic continuation induces an action of the pure braid group P3 on the space M = G3/G , G = SL(2,C) of conjugacy classes of G -triples

    u x

    u y

    u z

    u x uzuy

    ux

    Mx M My x

    uzux uy uz

    -1

    My

    uy

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    algebraic PVI solutions → finite P3 orbits

    Main question: classify these orbits

    Geometric viewpoint:

    nonlinear action of OutG on Hom(G ,H)/H

    here G = π1(P1\4 points), Out G ∼= MCG∗(P1\4 points), H = SL(2,C)

    Oleg Lisovyy Part 1: Algebraic solutions of Painlevé VI

  • Basic facts Modular group actions

    Finite Λ̄ orbits

    Solutions of Painlevé VI Schwarz list Isomonodromy approach Reconstruction

    algebraic PVI solutions → finite P3 orbits

    Main question: classify these orbits

    Geometric viewpoint:

    nonlinear action of OutG on Hom(G ,H)/H

    here G = π1(P1\4 points), Out G ∼= MCG∗(P1\4 points), H =