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

• n

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

(S) dY dx = AY, A ∈ gl(n, C(x))

◮ Σ = {x1, . . . , xp} ⊂ P1(C) singular points of (S) ◮ Y0

fundamental at x0 ∈ C \ Σ,

◮ C(x)(Y0) = Picard-Vessiot extension of C(x) ◮ representation of the differential Galois group (PV group)

• ver C(x)

= group of differential C(x)-automorphism of C(x)(Y0) GalC(x)(S) ⊂ GL(n, C) as a linear algebraic group.

PV theory and Monodromy (2)

◮ Analytic continuation of Y0 along

γ (= lifting on

• P1(C) \ Σ
• f a loop γ from x0 in P1(C)) gives rise to monodromy.

◮ ◮ Monodromy representation

π1(P1(C) \ Σ; x0)

ρ

− → GL(n, C) [γ] − → Mγ where analytic continuation of Y0 along ˜ γ yields Y0Mγ. Im ρ ⊂ GalC(x)(S)

◮ Theorem (Schlesinger): If all singularities are regular,

then the monodromy matrices Mγ generate a Zariski-dense subgroup of the PV-group GalC(x)(S).

PV theory and monodromy (3)

◮ Example: Scalar equation

(E) ∂xy = t x y, t ∈ C (∂x = d dx )

◮ Singularities : 0, ∞, Fuchsian ( ⇒ regular singular) ◮ Fundamental solution: xt (for fixed t ∈ C∗) ◮ PV-extension: K = C(x, xt) ◮ Monodromy: m0 = e2πit = 1/m∞ ∈ C∗ ◮ PV-group:

GalC(x)(E) =

• C∗

if t / ∈ Q finite cyclic group if t ∈ Q

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 Galk(E) ⊂ GL(1, C(t)) m0 = e2πit ∈ C(t)

◮ Parametrized approach (tentative):

work with differential ∆-fields, ∆ = {∂x, ∂t},

◮ Base-field k = C(t)(x)

∆-extension K = k(xt, log x) (by sol. xt and its derivatives w.r.t. to both x and t)

◮ G = Aut∆ k (K) ⊂ C(t) ∗ not appropriate as a Galois group: ◮ m0 ∈ 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{y1, . . . , yp}∆ the L-algebra of ∆-differential polynomials

in the indeterminates y1, . . . , yp

◮ Definition L is ∆-closed if for any p ∈ N∗ and differential

polynomials P1, . . . , Ps, Q ∈ L{y1, . . . , yp}∆, the system P1 = . . . = Ps = Q = 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 f1 = . . . = fl = 0, fi ∈ L{y1, . . . , yn2}∆.

◮ Notation C∆′ L

for the field of ∆′-constants of a ∆-field L, for any subset ∆′ ⊂ ∆ of derivations.

PPV theory (4)

◮ Consider a parametrized system

(S) ∂0Y = AY, A ∈ gl(n, k)

• ver some ∆-field k, ∆ = {∂0, . . . , ∂r},

with field of ∂0-constants k0 = 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 k0 be ∆-differentially closed.

PPV theory (4)

◮ Theorem (Cassidy-Singer):

Assume k0 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), ∂0Z = AZ

◮ C∂0 K = C∂0 k = k0 (no new ∂0-constants)

◮ The parametrized Picard-Vessiot group (PPV-group)

Gal∆(S) = Autdiff

k (K)

is a linear differential algebraic group over k0 Gal∆(S) ⊂ GL(n, k0)

◮ Galois correspondence holds between

{intermediate ∆-fields k ⊂ L ⊂ K} and {Kolchin-closed subgroups of Gal∆(S)}.

PPV-groups versus PV-groups

◮ k0 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, k0))

◮ Example (E): Take k0 = ∂t-closure of C(t). Then

Gal∆(E) = {a ∈ k0, ∂2

t a − (∂ta)2 = 0}

differential subgroup of k∗

0. ◮ Now m0, m∞ ∈ Gal∆(E) and Galois correspondence holds!

Analytic families of L.O.D.E.

Consider analytic parametrized systems of order n

(S) ∂xY = A(x, t)Y

where A(x, t) is analytic in Ω × D, with

◮ Ω ⊂ C open connected such that, for fixed x0 ∈ Ω,

π1(Ω; x0) is generated by loops [γ1], . . . , [γm]

◮ D ⊂ Cr a polydisc in the parameter space ◮ ∂x = d dx , ∂ti = d dti , with t = (t1, . . . , tr) multiparameter ◮ ∆ = {∂x, ∂t1, . . . , ∂tr }

Isomonodromy

◮ Definition 1 Equation (S) is isomonodromic if there are

constant matrices G1, . . . , Gm ∈ GL(n, C) such that for each fixed t ∈ D, some fundamental solution Yt(x) of (S) (at x0) realizes the Gi as monodromy matrices along γi, for all i.

◮ Classically, only Fuchsian systems were considered,

with t the moving position of the poles: (F) ∂xY =

m

• i=1

Bi(a) x − ai ,

m

• i=1

Bi(a) = 0 with a = (a1, . . . , am) ∈ D(a0), neighbourhood of the initial position a0.

Schlesinger deformations (Fuchsian case)

◮ Schlesinger (1905) defined isomonodromy by asking that

the monodromy representation π1(P1(C) \ {a1, . . . , am}; x0)

ρa

− → GL(n, C) be independant of a for the particular solution ˜ Ya with initial condition ˜ Ya(x0) = I.

◮ Such families are now called isomonodromic deformations

• f the Schlesinger type, characterized by the Pfaffian

system of Schlesinger equations (i = 1, . . . , m) dBi(a) = −

m

• j=1,j=i

[Bi(a), Bj(a)] ai − aj d(ai − aj) = compatibility condition of the systems ∂aiY = − Bi(a) x − ai Y.

Fuchsian isomonodromy

◮ Bolibrukh (1995) extended Schlesinger’s definition as

follows:

◮ Equation (F) is isomonodromic if there is a fundamental

solution Ya of (F) with initial value Ya(x0) = C(a) analytic in a, such that ρa 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) ∂xY = A(x, t)Y

◮ Theorem (Sibuya) (S) is isomonodromic if and only if (S)

belongs to an integrable system ∂xY = A(x, t)Y ∂tiY = Bi(x, t)Y, i = 1, . . . , r with all Bi(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 Bi.

◮ Example (E):

∂xy = (t/x)y is indeed non isomonodromic : it can be completed into an integrable system with with ∂ty = log(x)y, which is not rational!

Parametrized regular singularities

Consider

◮ U = open connected neighbourhood of 0 in the parameter

space Cr (parameter t)

◮ OU = ring of analytic functions of t on U ◮ α ∈ OU with α(0) = 0 ( → “moving singularity") ◮ OU((x − α(t))) = ring of formal Laurent series in (x − α(t))

f(x, t) =

• i≥m

ai(t)(x − α(t))i with m independent of t.

◮ OU({x − α(t)}) = {series ∈ OU((x − α(t))) that for fixed

t ∈ U have convergence radius Rt > 0}

◮ Remark For f ∈ OU({x − α(t)}) there is, locally in t, a

uniform convergence radius R (not depending on the parameter)

◮ Consider, for t close to 0, parametrized systems

(S) dY dx = A(x, t)Y with moving singularity α(t) ∈ OU and A ∈ gln(OU({x − α(t)})).

◮ Equivalence of systems is defined by a change of

independent variables Y(x, t) = P(x, t)Z(x, t), P ∈ GLn(OU({x − α(t)})).

◮ Definition: Equation (S) has

• simple singular points near 0 if α(t) is a simple pole of

A(x, t) for all t, locally near t = 0

• regular singular points near 0 if (S) is equivalent to an

equation with simple singularities near 0.

Parametrized regular singularities (2)

Solutions in the neighbourhood of a parametrized regular singularity have moderate growth: Proposition (M. - Singer) If (S) has regular singular points near 0 then ∃ neighbourhood U′ ⊂ U of 0 in the t-space such that

◮ (S) has a fundamental solution of the form

Y(x, t) =  

i≥i0

(x − α(t))iQi(t)   (x − α(t))

• A(t)

with A(t), Qi(t) ∈ gln(OU′)

◮ for any r-tuple (m1, . . . , mr) there is an integer N such that

for all t ∈ U′ and sectors St from α(t) in C lim

x→α(t) x∈St

(x − α(t))N ∂m1+...+mr Y(x, t) ∂m1t1 . . . ∂mr tr = 0.

Parametrized monodromy

Consider an analytic family

(S) ∂xY = A(x, t)Y

where A(x, t), analytic in Ω × U, is a rational function of x,

◮ U ⊂ Cr a neighbourhood of t = 0 in the parameter space ◮ Ω ⊂ C, 0 ∈ Ω, an open subset of P1(C) such that P1(C) \ Ω

is the union of m disjoint disks Di

◮ for each t ∈ U, Equation (S) has exactly one singularity

αi(t) in each Di

◮ ∂x = d dx , ∂ti = d dti , with t = (t1, . . . , tr) multiparameter ◮ ∆ = {∂x, ∂t1, . . . , ∂tr }, ∆t = {∂t1, . . . , ∂tr }

Parametrized monodromy (2)

Fix a fundamental solution Y(x, t) of (S) in a neighbourhood of x = 0. For each fixed t ∈ U , let Mi(t) be the monodromy matrices Mi(t) of Y with respect to the singularities αi(t). Mi(t)= the parametrized monodromy matrices of (S) with base-point 0 around the moving singularities αi(t). Theorem (M. - Singer) : Given (S) as before,

◮ assume A ∈ gln(C0(x)) for some differentially closed

∆t-field C0 containing C,

◮ let C1 be a differentially closed ∆t-field containing

C0 and the entries of the parametrized monodromy matrices Mi(t).

◮ Then the Mi(t) belong to G(C1), where G is the PPV-group

• f (S) over the ∆-field C0(x).

◮ Moreover, if all the singularities αi(t) are regular singular

(in the parametrized sense) we get an analogue of the Schlesinger theorem. Theorem (M. - Singer) : With the same notation,

◮ assume that (S) has regular singularities only, near each

αi(0),

◮ let k be a differentially closed ∆t-field containing the

x-coefficients of the entries of A(x, t), the singular points αi(t) and the entries of the parametrized monodromy matrices Mi(t).

◮ Then the Mi(t) generate a Kolchin-dense subgroup of the

PPV-group of S) over k(x).

PPV and (iso-)monodromy

Consider a regular singular parametrized equation (S) ∂x = A(x, t)Y with A(x, t) rational in x, ∆ = {∂x, ∂t1, . . . , ∂tr }. Base field : ∆-field k = k0(x), where k0 = C∂x

k is the ∆-closure

• f the field generated over C by the x-coefficients of A.

Theorem (Cassidy-Singer): (S) is isomonodromic iff Gal∆(S) is conjugate in GL(n, k0) to a linear algebraic subgroup of GL(n, C).

The Darboux-Halphen example

The Darboux-Halphen fifth order non-linear equation (in the time variable t) (DH V)            ω′

1

= ω2ω3 − ω1(ω2 + ω3) + φ2 ω′

2

= ω3ω1 − ω2(ω3 + ω1) + θ2 ω′

3

= ω1ω2 − ω3(ω1 + ω2) − θφ φ′ = ω1(θ − φ) − ω3(θ + φ) θ′ = − ω2(θ − φ) − ω3(θ + φ),

• ccurs in the Bianchi IX cosmological model, as a reduction of

SDYM (Self-Dual Yang-Mills equation).

• Y. Ohyama has shown that DH-V is equivalent to Halphen’s

second equation H-II x′

i = Q(xi), i = 1, 2, 3,

where Q(x) = x2 + a(x1 − x2)2 + b(x2 − x3)2 + c(x3 − x1)2 (a, b, c, constants)

DH-V (2)

References

• S. Chakravarty, M. J. Ablowitz, Reductions of Self-Dual

Yang-Mills Fields and Classical Systems, Physical Review Letters 65, 9 (1990), 1085–1087

• S. Chakravarty, M. J. Ablowitz, Integrability, monodromy

evolving deformations, and self-dual Bianchi IX systems, Physical Review Letters 76, 6 (1996), 857–860

• Y. Ohyama, Monodromy evolving deformations and

Halphen’s equation in Groups and Symmetries, CRM Proc. Lecture Notes 47 (2009), Amer. Math. Soc. (2009) Chakravarty and Ablowitz show that these equations describe a special type of Fuchsian ‘evolving monodromy’, in the same way as the Schlesinger equation describes the Schlesinger isomonodromy.

DH-V (3)

◮ Contrary to other SDYM reductions (like Painlevé

equations) DH-V does not satisfy the Painlevé property (there is a whole boundary of movable essential singularities) ⇒ DH-V is not likely to rule isomonodromy !

◮ H-II is the integrability condition of a Lax pair of order 2

systems ∂Y ∂x =

• µ

(x − x1)(x − x2)(x − x3) I +

3

• i=1

λi x − xi K

• Y

(1) ∂Y ∂t =

• νI +

3

• i=1

λixiK

• Y − Q(x)∂Y

∂x (2) with moving poles xi = xi(t)

DH-V (4)

◮ K constant matrix, traceless ◮ µ, λi constants with µ = 0, λ1 + λ2 + λ3 = 0, ◮ ν solution of

∂ν ∂x = − x + x1 + x2 + x3 (x − x1)(x − x2)(x − x3)µ.

◮ Note that ν is not rational in x. ◮ The parametrized equation (1) is non isomonodromic! (by

Sibuya’s criterion).

Darboux-Halphen (monodromy)

◮ Equation (1) of the Lax pair is a parametrized family (S) as

above.

◮ With notation from previous slides, fix neighbourhoods Di

• f xi(0), and x0 ∈ Di, i = 1, 2, 3.

◮ Fix Y, a fundamental solution of the Lax pair at x0. ◮ For each i, analytic continuation of Y near xi(t) writes

Y(x, t) = Yi(x, t) · (x − xi(t))Li.

◮ The parametrized monodromy matrix Mi(t) around xi(t) is

Mi = e2π

√ −1Li

.

◮ Mi(t) is actually here of the form

Mi(t) = ci(t) Gi,

◮ Gi is constant, Gi = e2π √ −1Li(t0) ◮ ci(t) is analytic,

ci(t) = e

−2π √ −1µ t

t0 αi(t)dt

with x + 3

i=1 xi(t)

3

i=1(x − xi(t))

=

3

• i=1

αi(t) x − xi(t)

Projective isomonodromy

Given a parametrized system

(S) ∂xY = A(x, t)Y

A(x, t) analytic in Ω × D (Ω and D as before) with π1(Ω; x0) generated by [γ1], . . . , [γm] (fixed x0 ∈ Ω) Definition: (S) is projectively isomonodromic if, for all i, there are

• constant matrices Gi ∈ GL(n, C)
• analytic functions ci : D → C∗ such that, for each fixed t ∈ D,

some fundamental solution Yt(x) of (S) has monodromy Mi(t) = ci(t)Gi . Remark: Yt(x) need not be analytic in t...

Projectively isomonodromic solutions

Proposition If (S) is projectively isomonodromic, then (with notations of the definition ) there is a fundamental solution Yt(x) analytic in Ω × D such that Mi(t) = ci(t)Gi. The proof follows Bolibrukh’s method for the Fuchsian case : If (S) is projectively isomonodromic, let ˜ Y any analytic solution in Ω × D, with monodromy Gi(t) = Γ(t)−1ci(t)GiΓ(t) then there is an analytic Γ(t) conjugating the monodromy of ˜ Y to the "projective monodromy" ci(t)Gi of the definition. Bolibrukh’s proof uses in particular the fact that for fiber bundles

• ver a Stein variety, topologically trivial

= ⇒ analytically trivial.

Fuchsian projective isomonodromy

For Fuchsian equations, we get the following criterion Proposition Fuchsian (analytic) parametrized system (F) ∂xY =

m

• i=1

Ai(t) x − xi(t) is projectively isomonodromic if and only if for all i Ai = Bi + bi I, where bi : D → C and Bi : D → gl(n, C) are analytic functions such that the family of equations ∂xY = (

m

• i=1

Bi(t) x − xi(t))Y is isomonodromic.

DH-V example : Equation (1) of the Lax pair meets this condition ∂xY =

• µ

(x − x1)(x − x2)(x − x3) I +

3

• i=1

λi x − xi K

• Y.

Here bi = µ (x − x1)(x − x2)(x − x3), Bi = λi x − xi and ∂xY = 3

• i=1

λi x − xi K

• Y

is clearly isomonodromic since K is constant.

Galois conditions of projective isomonodromy

◮ Given (S), parametrized system over the ∆-field k = k0(x),

with differentially closed ∆- field of ∂x-constants k0, assume (S) has parametrized regular singularities only.

◮ Proposition: (S) is projectively isomonodromic if and only

if its PPV-group G over k is conjugate in GL(n, k0) to a subgroup of GL(n, C) · Scal(n, k0).

◮ Corollary: If (S) is projectively isomonodromic, then

(G, G) is conjugate in GL(n, k0) to a subgroup of GL(n, C).

◮ The converse of Corollary only holds under the following

asumption.

PPV characterization of projective isomonodromy

Theorem: If (S) is absolutely irreducible over k, then (S) is projectively isomonodromic iff (G, G) is conjugate in GL(n, k0) to a subgroup of GL(n, C).

absolutely irreducible: irreducible over any algebraic extension The proof uses Schur’s Lemma and Kolchin/Zariski topological arguments in the following lemmas

Lemma 1 If g ∈ GL(n, k0) normalizes an irreducible constant subgp H ⊂ GL(n, C), then g ∈ GL(n, C) · Scal(n, k0). Lemma 2 Let H ⊂ GL(n, k0) be a linear differential algebraic

• group. Assume H is Kolchin-connected and its Zariski-closure

H is irreducible. Then H ⊂ (H, H)∆ · Scal(n, k0) where (H, H)∆ is the Kolchin-closure of (H, H). Lemma 3 If G ⊂ GL(n, k0) is a linear differential algebraic group such that

• (G, G) ⊂ GL(n, C)
• G

0 is irreducible

then G ⊂ GL(n, C) · Scal(n, k0).

Thank you for your attention