Galoisian approach to Monodromy Evolving Deformations Claude - PowerPoint PPT Presentation

Galoisian approach to Monodromy Evolving Deformations Claude Mitschi Institut de Recherche Mathématique Avancée CNRS - Université de Strasbourg Strasbourg, France mitschi@math.unistra.fr Fourth International Workshop on Differential Algebra and Related Topics (DART IV) October 27-30, 2010, Beijing, China

This is joint work with Michael F. Singer North Carolina State University singer@math.ncsu.edu Ref. ArXiv:1002.2005v18

Classical Picard-Vessiot theory and Monodromy ◮ Consider an ordinary differential system dY dx = AY , A ∈ gl ( n , C ( x )) ( S ) ◮ Σ = { x 1 , . . . , x p } ⊂ P 1 ( C ) singular points of ( S ) ◮ Y 0 fundamental at x 0 ∈ C \ Σ , ◮ C ( x )( Y 0 ) = Picard-Vessiot extension of C ( x ) ◮ representation of the differential Galois group ( PV group ) over C ( x ) = group of differential C ( x ) -automorphism of C ( x )( Y 0 ) Gal C ( x ) ( S ) ⊂ GL ( n , C ) as a linear algebraic group.

PV theory and Monodromy (2) � ◮ Analytic continuation of Y 0 along � P 1 ( C ) \ Σ γ (= lifting on of a loop γ from x 0 in P 1 ( C ) ) gives rise to monodromy. ◮ Monodromy representation ◮ ρ π 1 ( P 1 ( C ) \ Σ; x 0 ) → GL ( n , C ) − [ γ ] �− → M γ where analytic continuation of Y 0 along ˜ γ yields Y 0 M γ . Im ρ ⊂ Gal C ( x ) ( S ) ◮ Theorem (Schlesinger) : If all singularities are regular, then the monodromy matrices M γ generate a Zariski-dense subgroup of the PV-group Gal C ( x ) ( S ) .

PV theory and monodromy (3) ◮ Example : Scalar equation ∂ x y = t ( ∂ x = d t ∈ C ( E ) x y , dx ) ◮ Singularities : 0 , ∞ , Fuchsian ( ⇒ regular singular) ◮ Fundamental solution: x t (for fixed t ∈ C ∗ ) ◮ PV-extension: K = C ( x , x t ) ◮ Monodromy: m 0 = e 2 π it = 1 / m ∞ ∈ C ∗ ◮ PV-group: � C ∗ ∈ Q t / if Gal C ( x ) ( E ) = t ∈ Q finite cyclic group if

PV theory and monodromy (4) ◮ Remark : This example with indeterminate parameter t shows there is no Schlesinger-type theorem over the ∂ x -field k = C ( t )( x ) since Gal k ( E ) ⊂ GL ( 1 , C ( t )) m 0 = e 2 π it �∈ C ( t ) ◮ Parametrized approach (tentative): work with differential ∆ -fields, ∆ = { ∂ x , ∂ t } , ◮ Base-field k = C ( t )( x ) ∆ -extension K = k ( x t , log x ) (by sol. x t and its derivatives w.r.t. to both x and t ) ∗ not appropriate as a Galois group: ◮ G = Aut ∆ k ( K ) ⊂ C ( t ) ◮ m 0 �∈ C ( t ) ◮ no Galois correspondence : K G = k ( log x ) � = k ◮ Defect: C ( t ) is algebraically, not differentially closed.

Parametrized Picard-Vessiot Theory References • P . Cassidy, M. F . Singer , Galois theory of parameterized differential equations and linear differential algebraic groups , IRMA Lectures in Mathematics and Theoretical Physics 9 (2006), 113–157. ( Special volume in memory of A. A. Bolibrukh ) • E. R. Kolchin , Differential algebraic groups Academic Press, New York, 1985. • P . Landesman , Generalized differential Galois theory , Trans. Amer. Math. Soc. 360, 8 (2008), 4441–4495.

PPV theory (2) ◮ ∆ = { ∂ 0 , ∂ 1 , . . . , ∂ r } commuting derivations on a field L , ◮ L { y 1 , . . . , y p } ∆ the L -algebra of ∆ -differential polynomials in the indeterminates y 1 , . . . , y p ◮ Definition L is ∆ -closed if for any p ∈ N ∗ and differential polynomials P 1 , . . . , P s , Q ∈ L { y 1 , . . . , y p } ∆ , the system � P 1 = . . . = P s = 0 Q � = 0 has a solution in L whenever it has a solution in some ∆ -extension of L. ( cf. Robinson, Blum, Kolchin...) ◮ Any ∆ -field has a differential closure.

PPV theory (3) ◮ Definition : Over a given ∆ -field L, a linear differential algebraic group G ⊂ GL ( n , L ) is a Kolchin-closed subgroup of GL ( n , L ) . Kolchin-closed = defined by differential polynomial equations f 1 = . . . = f l = 0 , f i ∈ L { y 1 , . . . , y n 2 } ∆ . ◮ Notation C ∆ ′ for the field of ∆ ′ -constants of a ∆ -field L , for L any subset ∆ ′ ⊂ ∆ of derivations.

PPV theory (4) ◮ Consider a parametrized system ( S ) ∂ 0 Y = AY , A ∈ gl ( n , k ) over some ∆ -field k , ∆ = { ∂ 0 , . . . , ∂ r } , with field of ∂ 0 -constants k 0 = C ∂ 0 k . ◮ P . Cassidy and M. F . Singer established an appropriate parametrized Picard-Vessiot theory (PPV) : ◮ PPV extensions, PPV groups, Galois correspondence... ◮ In analogy with classical PV-theory, the condition here is that k 0 be ∆ -differentially closed.

PPV theory (4) ◮ Theorem (Cassidy-Singer): Assume k 0 is ∆ -differentially closed. Then ◮ there is a unique PPV-extension K of k (parametrized Picard-Vessiot extension) = differential ∆ -extension of k such that ◮ K = k ( Z ) ∆ (extension by entries of matrix Z and all their ∆ -derivatives), ◮ Z is a fundamental solution of (S) Z ∈ GL ( n , K ) , ∂ 0 Z = AZ ◮ C ∂ 0 K = C ∂ 0 k = k 0 (no new ∂ 0 -constants)

◮ The parametrized Picard-Vessiot group (PPV-group) Gal ∆ ( S ) = Aut diff k ( K ) is a linear differential algebraic group over k 0 Gal ∆ ( S ) ⊂ GL ( n , k 0 ) ◮ Galois correspondence holds between {intermediate ∆ -fields k ⊂ L ⊂ K} and {Kolchin-closed subgroups of Gal ∆ ( S ) }.

PPV-groups versus PV-groups ◮ k 0 differentially closed ⇒ algebraically closed ◮ Relation betwen PV and PPV extensions: K PV ⊂ K PPV where K PV = k ( Z ) and K PPV = k ( Z ) ∆ . ◮ K PV = k ( Z ) is stable by the PPV-group ◮ Restriction of Gal ∆ ( S ) to K PV is injective Gal ∆ ( S ) ֒ → Gal ∂ 0 ( S ) Gal ∂ 0 ( S ) = Gal ∆ ( S ) (Zariski closure in GL ( n , k 0 ) ) ◮ Example (E) : Take k 0 = ∂ t -closure of C ( t ) . Then t a − ( ∂ t a ) 2 = 0 } Gal ∆ ( E ) = { a ∈ k 0 , ∂ 2 differential subgroup of k ∗ 0 . ◮ Now m 0 , m ∞ ∈ Gal ∆ ( E ) and Galois correspondence holds!

Analytic families of L.O.D.E. Consider analytic parametrized systems of order n ( S ) ∂ x Y = A ( x , t ) Y where A ( x , t ) is analytic in Ω × D , with ◮ Ω ⊂ C open connected such that, for fixed x 0 ∈ Ω , π 1 (Ω; x 0 ) is generated by loops [ γ 1 ] , . . . , [ γ m ] ◮ D ⊂ C r a polydisc in the parameter space ◮ ∂ x = d dx , ∂ t i = d dt i , with t = ( t 1 , . . . , t r ) multiparameter ◮ ∆ = { ∂ x , ∂ t 1 , . . . , ∂ t r }

Isomonodromy ◮ Definition 1 Equation ( S ) is isomonodromic if there are constant matrices G 1 , . . . , G m ∈ GL ( n , C ) such that for each fixed t ∈ D , some fundamental solution Y t ( x ) of ( S ) (at x 0 ) realizes the G i as monodromy matrices along γ i , for all i. ◮ Classically , only Fuchsian systems were considered, with t the moving position of the poles: m m � � B i ( a ) ( F ) ∂ x Y = , B i ( a ) = 0 x − a i i = 1 i = 1 with a = ( a 1 , . . . , a m ) ∈ D ( a 0 ) , neighbourhood of the initial position a 0 .

Schlesinger deformations (Fuchsian case) ◮ Schlesinger (1905) defined isomonodromy by asking that the monodromy representation ρ a π 1 ( P 1 ( C ) \ { a 1 , . . . , a m } ; x 0 ) → GL ( n , C ) − be independant of a for the particular solution ˜ Y a with initial condition ˜ Y a ( x 0 ) = I . ◮ Such families are now called isomonodromic deformations of the Schlesinger type, characterized by the Pfaffian system of Schlesinger equations ( i = 1 , . . . , m ) m � [ B i ( a ) , B j ( a )] d B i ( a ) = − d ( a i − a j ) a i − a j j = 1 , j � = i = compatibility condition of the systems ∂ a i Y = − B i ( a ) Y . x − a i

Fuchsian isomonodromy ◮ Bolibrukh (1995) extended Schlesinger’s definition as follows: ◮ Equation ( F ) is isomonodromic if there is a fundamental solution Y a of ( F ) with initial value Y a ( x 0 ) = C ( a ) analytic in a, such that ρ a is independent of a. ◮ Bolibrukh proved (1997) that for Fuchsian equations this is equivalent to Definition 1 above, and gave examples of non-Schlesinger isomonodromic deformations.

General isomonodromy ◮ Generalization of Schlesinger’s integrability condition : Consider an analytic family as before ( S ) ∂ x Y = A ( x , t ) Y ◮ Theorem (Sibuya) ( S ) is isomonodromic if and only if ( S ) belongs to an integrable system � ∂ x Y = A ( x , t ) Y ∂ t i Y = B i ( x , t ) Y , i = 1 , . . . , r with all B i ( x , t ) analytic in Ω × D . ◮ If moreover ( S ) has regular singularities only (in the parametrized sense) then if A is rational in x, so are the B i . ◮ Example (E) : ∂ x y = ( t / x ) y is indeed non isomonodromic : it can be completed into an integrable system with with ∂ t y = log ( x ) y , which is not rational!

Parametrized regular singularities Consider ◮ U = open connected neighbourhood of 0 in the parameter space C r (parameter t ) ◮ O U = ring of analytic functions of t on U ◮ α ∈ O U with α ( 0 ) = 0 ( → “moving singularity") ◮ O U (( x − α ( t ))) = ring of formal Laurent series in ( x − α ( t )) � a i ( t )( x − α ( t )) i f ( x , t ) = i ≥ m with m independent of t . ◮ O U ( { x − α ( t ) } ) = {series ∈ O U (( x − α ( t ))) that for fixed t ∈ U have convergence radius R t > 0} ◮ Remark For f ∈ O U ( { x − α ( t ) } ) there is, locally in t , a uniform convergence radius R (not depending on the parameter)