Outline Basic facts Modular group actions Finite ¯ Λ orbits Conclusions Algebraic solutions of the sixth Painlev´ e equation Oleg Lisovyy LMPT, Tours, France December 16, 2008 In collaboration with Yu. Tykhyy, arXiv:0809.4873 Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Modular group actions Finite ¯ Λ orbits Conclusions 1 Basic facts 2 Modular group actions Finite ¯ 3 Λ orbits Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI equation : � 1 � � dw � 1 � dw d 2 w � 2 dt 2 = 1 1 1 1 1 w + w − 1 + − t + t − 1 + dt + 2 w − t dt w − t � θ 2 � ( θ ∞ − 1) 2 − θ 2 ( w − 1) 2 + (1 − θ 2 y ( t − 1) + w ( w − 1)( w − t ) x t z ) t ( t − 1) w 2 + 2 t 2 ( t − 1) 2 ( w − t ) 2 Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI equation : � 1 � � dw � 1 � dw d 2 w � 2 dt 2 = 1 1 1 1 1 w + w − 1 + − t + t − 1 + dt + 2 w − t dt w − t � θ 2 � ( θ ∞ − 1) 2 − θ 2 ( w − 1) 2 + (1 − θ 2 y ( t − 1) + w ( w − 1)( w − t ) x t z ) t ( t − 1) w 2 + 2 t 2 ( t − 1) 2 ( w − t ) 2 4 parameters θ x , y , z , ∞ most general equation of type w ′′ = F ( t , w , w ′ ) without movable critical points (Painlev´ e property) w ( t ) is meromorphic on the universal cover of P 1 \{ 0 , 1 , ∞} Okamoto affine F 4 Weyl symmetry group P I – P V are obtained as limiting cases applications in nonlinear physics, classical and quantum integrable systems, random matrix theory, differential geometry... Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Solutions of Painlev´ e VI According to Watanabe (1998): Riccati solutions or solutions of P VI are either ‘new’ transcendental functions or algebraic functions Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Solutions of Painlev´ e VI According to Watanabe (1998): Riccati solutions or � solutions of P VI are either ‘new’ transcendental functions or algebraic functions Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Solutions of Painlev´ e VI According to Watanabe (1998): Riccati solutions or � solutions of P VI are either ‘new’ transcendental functions or algebraic functions ??? Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Solutions of Painlev´ e VI According to Watanabe (1998): Riccati solutions or � solutions of P VI are either ‘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) Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Solutions of Painlev´ e VI According to Watanabe (1998): Riccati solutions or � solutions of P VI are either ‘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) no complete classification as yet Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Schwarz list Question : When does Gauss hypergeometric function 2 F 1 ( a , b , c , λ ) become algebraic? (Schwarz, 1873) d Φ � A x A y � d λ = + Φ , λ − u x λ − u y standard choice u x = 0, u y = 1 Φ ∈ Mat 2 × 2 , A x , y ∈ sl 2 ( C ) monodromy matrices M x , y ∈ SL (2 , C ) algebraic solutions lead to finite monodromy → 15 classes Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Isomonodromy approach Painlev´ e VI describes monodromy preserving deformations of Fuchsian systems d Φ � A x A y A z � d λ = + + Φ , Φ ∈ Mat 2 × 2 . λ − u x λ − u y λ − u z A ν ∈ sl 2 ( C ) are independent of λ , with eigenvalues ± θ ν / 2 4 regular singular points u x , u y , u z , ∞ ∈ P 1 � − θ ∞ / 2 � 0 def A x + A y + A z = − A ∞ = 0 θ ∞ / 2 monodromy matrices M x , M y , M z ∈ SL (2 , C ), defined up to overall conjugation (3 × 3 − 3 = 6 parameters) Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI ↔ linear system dictionary : P VI independent variable t = ( u x − u y ) / ( u x − u z ); w ( t ) is a combination of matrix elements of A x , y , z Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI ↔ linear system dictionary : P VI independent variable t = ( u x − u y ) / ( u x − u z ); w ( t ) is a combination of matrix elements of A x , y , z to each branch of a solution of P VI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of M x , M y , M z , M ∞ = M z M y M x give P VI parameters θ x , y , z , ∞ ; the other two correspond to integration constants Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI ↔ linear system dictionary : P VI independent variable t = ( u x − u y ) / ( u x − u z ); w ( t ) is a combination of matrix elements of A x , y , z to each branch of a solution of P VI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of M x , M y , M z , M ∞ = M z M y M x give P VI parameters θ x , y , z , ∞ ; the other two correspond to integration constants analytic continuation induces an action of the pure braid group P 3 on the space M = G 3 / G , G = SL (2 , C ) of conjugacy classes of G -triples Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI ↔ linear system dictionary : P VI independent variable t = ( u x − u y ) / ( u x − u z ); w ( t ) is a combination of matrix elements of A x , y , z to each branch of a solution of P VI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of M x , M y , M z , M ∞ = M z M y M x give P VI parameters θ x , y , z , ∞ ; the other two correspond to integration constants analytic continuation induces an action of the pure braid group P 3 on the space M = G 3 / G , G = SL (2 , C ) of conjugacy classes of G -triples algebraic PVI solutions → finite orbits Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
Outline Basic facts Solutions of Painlev´ e VI Modular group actions Schwarz list Finite ¯ Λ orbits Isomonodromy approach Conclusions Painlev´ e VI ↔ linear system dictionary : P VI independent variable t = ( u x − u y ) / ( u x − u z ); w ( t ) is a combination of matrix elements of A x , y , z to each branch of a solution of P VI corresponds a (conjugacy class of) triple of monodromy matrices; eigenvalues of M x , M y , M z , M ∞ = M z M y M x give P VI parameters θ x , y , z , ∞ ; the other two correspond to integration constants analytic continuation induces an action of the pure braid group P 3 on the space M = G 3 / G , G = SL (2 , C ) of conjugacy classes of G -triples algebraic PVI solutions → finite orbits Main question : classify these orbits Oleg Lisovyy Algebraic solutions of the sixth Painlev´ e equation
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