Some analytical aspects of the Kontsevich matrix model Mattia - - PowerPoint PPT Presentation

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Some analytical aspects of the Kontsevich matrix model

Mattia Cafasso

Laboratoire Angevin de REcherche en MAthématiques (LAREMA), Angers.

Geometry of Integrable Systems. SISSA - International School of Advanced Studies, 04-06-2017.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Plan of the talk

• The Witten’s conjecture and the Witten–Kontsevich tau function.
• The Painlevé I hierarchy and the string equation.
• Kontsevich’s model.
• The convergence of the Kontsevich model to (some) solutions of the

Painlevé I hierarchy.

Collaboration with M. Bertola, arXiv : 1603.06420 (Comm. in Math. Phys. , 2017).

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

The Deligne-Mumford moduli space of Riemann surfaces

Mg,n :=

• Riemann surfaces with n marked points
• ∼

A point in Mg,n is a (possibly singular) Riemann surface with n marked points (modulo isomorphisms).

Mg,n is a complex orbifold of dimension 3g − 3 + n. We denote with Lj the tautological line bundle, their fibers over [C] are given by T⋆

pjC, ψj will denote the

corresponding Chern classes.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Intersection numbers

Intersection numbers are given by the integrals

• τ k0

0 τ k1 1 . . .

• g,n :=
• Mg,n

ψℓ1

1 ∧ · · · ∧ ψℓn n ,

where kj = ♯ occurences of j as an exponent. Example : < τ 2

0 τ 3 1 τ2 >=

• ψ1 ∧ ψ2 ∧ ψ3 ∧ ψ2

4 ∧ ψ0 5 ∧ ψ0 6.

The numbers ki satisfy 3g − 3 + n =

∞

• k=0

jkj, n =

• kj.

Let’s define F(T0, T1, . . . , ) :=

• τ k0

0 τ k1 1 . . . τ kℓ ℓ . . .

T

kj j

kj!

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Witten’ conjecture (Kontsevich theorem) :

Let ˜ Rn[U] be the Lenard polynomials defined by the recursion ˜ R0[U] = U, ∂˜ Rn+1 ∂T0 = 1 2n + 1 ∂U ∂T0 + 2U ∂ ∂T0 + 1 4 ∂3 ∂T3

• ˜

Rn. Theorem : The formal series F(T0, T1, . . . , ) :=

• τ k0

0 τ k1 1 . . . τ kℓ ℓ . . .

T

kj j

kj! is uniquely determined by the following conditions : 1) U := ∂2F ∂T2 is a solution of the Korteweg de-Vries hierarchy ∂U ∂Ti = ∂ ∂T0 ˜ Ri[U], i ≥ 0. 2) F satisfies the string equation ∂F ∂T0 =

• i≥0

Ti+1 ∂F ∂Ti + T2 2 . In other words, eF = τ is a tau function for the KdV hierarchy, uniquely determined by the Virasoro constraints or -equivalently- by its initial value ln τ = T3

6 .

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

String equation and the Painlevé I hierarchy

             ∂2F ∂T2 =

• i≥0

Ti+1 ∂2F ∂Ti∂T0 + T0 ∂2F ∂T0∂Ti = ˜ Ri

• ∂2F

∂T2

• =

⇒ U − T0 =

• i≥0

Ti+1˜ Ri+1[U]. Putting Ti = 0 for all i = 0, N we get the collection of equations TN˜ RN[U] = U − T0, N ≥ 1 known as Painlevé I hierarchy. Remark : The same equations can be written as

• L, M
• = 1,

L := ∂2 ∂T2 − U. (Douglas, “String in less than one dimension”).

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

The Kontsevich matrix model

Zn(x; Y) :=

• Hn

dMe

Tr

• i M3

3 −YM2+ixM

• Hn

dMe−Tr(YM2) , (Matrix Airy function) Hn :=

• M = M† ∈ Matn×n(C)
• Y := diag(y1, . . . , yn)

(x is added for later convenience). Theorem (Kontsevich, 1992) : When n → ∞, the following formal identity holds F( T) = lim

n→∞ ln Zn(0; Y),

i.e. τWK( T) = lim

n→∞Zn(0; Y),

under the identification (Miwa’s variables) Tj = Tj(Y) := −(2j − 1)!!

n

• ℓ=1

1 y2j+1

ℓ

, for |Y| → ∞.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

A natural question How do we choose Y in such a way that Zn(0; Y) converges to a solution of the PI hierarchy ? What are the properties of such solutions ?

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

The main idea

It’s easy to prove that Zn(x; Y) can be written as a “wronskian” - type determinant Zn(x; Y) = 2nπ

n 2 e 2 3 TrY3+xTrY

det

• Ai(j−1)(y2

k + x)

• k,j≤n

n

j=1(yj)

1 2

• j<k(yj − yk)

Reyj > 0, and this suggest a link with Darboux transformations... Let’s consider the system          ∂λΨ0(x; λ) =

• −i

i(λ + x)

• Ψ0(x; λ),

∂xΨ0(x; λ) =

• −i

i(λ + x)

• Ψ0(x; λ),

and let’s add poles on the points {λ1, . . . , λn} , λk = y2

k to get the new system

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

The main idea II

                     ∂λΨn = AΨn, A = iσ+ − i

• λ + x

2 − ∂xa(n) σ− −

n

• j=1

Aj λ − λj , ∂xΨn = UΨn, U = iσ+ − i

• λ − 2∂xa(n)

σ−, ∂λkΨn = − Ak λ − λk Ψn, k = 1, . . . , n. (1) The isomonodromic (Jimbo-Miwa-Ueno) tau function associated to the system above is defined by the equations ∂λk ln τn = resλkTrA2dλ, ∂x ln τn = a(n) and we will prove that τn(x, {λk}) = e

x3 12 Zn(x, Y).

Once this equality is established, one can study the large n limit of the Riemann–Hilbert problem associated to the system (1)...

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

An extension of the Kontsevich matrix model I :

Remark : Ai(λ) = e− 2

3 λ 3 2

2√πλ

1 4

• 1 + O(λ− 3

2 )

• ,

λ → ∞, = ⇒ e

2 3 y3+xyAi(y2 + x) =

           1 2√π√y(1 + O(y−3)) for Rey > 0, e

4 3 y3+2xy

2√π√y(1 + O(y−3)) for Rey < 0, = ⇒ Zn(x; Y) = 2nπ

n 2 e 2 3 TrY3+xTrY det

• Ai(j−1)(y2

k + x)

• k,j≤n

n

j=1(yj)

1 2

• j<k(yj − yk)

, this expression admits a “regular” expansion if Re(yi) > 0 for all i.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

An extension of the Kontsevich matrix model II :

Y − → Y(0) ⊔ Y(1) ⊔ Y(2); Zn(x; Y(0), Y(1), Y(2)) = Cn e

2 3 TrY3+xTrY

• j<k(yj − yk) det

       

• Ai(k−1)

(y2

j + x)

• yj∈Y(0)

1≤k≤n

• Ai(k−1)

1

(y2

j + x)

• yj∈Y(1)

1≤k≤n

• Ai(k−1)

2

(y2

j + x)

• yj∈Y(2)

1≤k≤n

        . Ais(λ) := Ai(ωsλ) , ω := e

2iπ 3 .

This determinant have a regular expansion if Y(a) ∋ yj → ∞, yj ∈ Sa. S0 : ; S1 : S2 : . For what follows let’s introduce the parameters {λi, µj} such that yi = √λi if Re(yi) > 0 et yj = −√µj if Re(yj) ≤ 0. d( λ, µ) :=

• j=1

√µj + √ λ √µj − √ λ

• j=1
• λj −

√ λ

• λj +

√ λ .

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

A Riemann-Hilbert problem for Zn

Theorem (M. Bertola, M.C.) : Zn(x; Y(0), Y(1), Y(2)) = e

x3 12 τn(x,

λ, µ), τn tau function of the Riemann–Hilbert problem with asymptotics Γn(λ) ∼ λ− σ3

4 1 + iσ1

√ 2

• 1 + a(n)(x;

λ, µ) √ λ σ3 + O(λ−1)

• ,
• 1

d( λ, µ)e

−

4 3

λ

3 2

− 2 x √ λ

1

• 1

d−1( λ, µ)e

4 3 λ 3 2 +2x

√ λ

1

• 1

d−1( λ, µ)e

4 3

λ

3 2

+ 2 x √ λ

1

• 1

−1

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

The matrix Ψn(λ; x, λ, µ) := Γne(− 2

3 λ 3 2 −x

√ λ)σ3D−1(λ)

D(λ) :=      

n2

• j=1

(

• λj +

√ λ)

n1

• j=1

(√µj − √ λ)

n2

• j=1

(

• λj −

√ λ)

n1

• j=1

(√µj + √ λ)       is a solution of the isomonodromic system ∂ ∂λΨn(λ; x, λ, µ) = A(λ; x, λ, µ)Ψn(λ; x, λ, µ) ∂ ∂xΨn(λ; x, λ, µ) = U(λ; x, λ, µ)Ψn(λ; x, λ, µ) ∂ ∂λk Ψn(λ; x, λ, µ) = −Ak(x, λ, µ) λ − λk Ψn(λ; x, λ, µ), ∂ ∂µk Ψn(λ; x, λ, µ) = −Bk(x, λ, µ) λ − µk Ψn(λ; x, λ, µ).

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Riemann-Hilbert for the tronquées solutions of PIN−1

2arg(t) 2N+1

̟0 ̟0 ̟+ ̟− ← −(Example N = 11)

ϑ(λ; t, x) := tλ

2N+1 2

+ 2 3λ

3 2 + xλ 1 2 ,

jumps are given by : 1 + e−2ϑ(λ;t,x)σ+, λ ∈ ̟0 1 + e2ϑ(λ;t,x)σ−, λ ∈ ̟± iσ2, λ ∈ R−. Γ(λ) = λ− σ3

4 1 + iσ1

√ 2

• 1 + a σ3

√ λ + O(λ−1)

• ,

λ → ∞. (The matrix Ψ(λ) = Γ(λ)e−ϑ(λ) is the solution of the corresponding RH problem)

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Riemann-Hilbert for the tronquées solutions of PIN−1

Let’s choose three integers k+, k0, k− ∈

• −
• N−1

2

• , . . . ,
• N−1

2

• with

k+ > k−, k+ ≥ k0 ≥ k−, and θ0 ∈

• −

π 2N + 1, π 2N + 1

• +

4k0π 2N + 1 − 2arg(t) 2N + 1 , θ± ∈

• −

π 2N + 1, π 2N + 1

• + (4k± ± 2)π

2N + 1 − 2arg(t) 2N + 1 M(λ) =          1 + e−2ϑ(λ;t,x)σ+ λ ∈ ̟0 := eiθ0R+ 1 + e2ϑ(λ;t,x)σ− λ ∈ ̟± := eiθ±R+ iσ2 λ ∈ R− ϑ(λ; t, x) := tλ

2N+1 2

+ 2 3λ

3 2 + xλ 1 2

Find a matrix Γ(λ) such that Γ+(λ) = Γ−(λ)M(λ), λ ∈ ̟−,0,+ , Γ+(λ) = Γ−(λ)iσ2, λ ∈ R− with asymptotics Γ(λ; t) = λ− σ3

4 1 + iσ1

√ 2

• 1 + a(t) σ3

√ λ + O(λ−1)

• , λ → ∞.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Which solutions are they ?

• For N = 2 there is just one solution, the one studied by Boutroux.
• For N = 3 there 4 solutions. One is the tritronquée solution U0 of PI2

related to the Dubrovin’s conjecture on universality Dubrovin (with t fixed) (for t = 0 it had been studied by Brezin-Marinari-Parisi and Moore).

• The other three belong to the set of “two parameters solutions” studied

by Grava-Kapaev-Klein.

• For N générique, the analog of U0 had been used by

Claeys-Its-Krasovsky to describe “higher order” Tracy–Widom distributions.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

How to go from Zn to PIN−1 ?

Jumps for Zn : Mn(λ) =              1 + d(λ)e− 4

3 λ 3 2 −2xλ 1 2 σ+

λ ∈ ̟0 := eiθ0R+ 1 +

1 d(λ) e

4 3 λ 3 2 +2xλ 1 2 σ−

λ ∈ ̟± := eiθ±R+ iσ2 λ ∈ R− Jumps for PIN−1 : M(λ) =          1 + e−2ϑ(λ;t,x)σ+ λ ∈ ̟0 := eiθ0R+ 1 + e2ϑ(λ;t,x)σ− λ ∈ ̟± := eiθ±R+ iσ2 λ ∈ R− ϑ(λ; t, x) = tλ

2N+1 2

+ 2 3λ

3 2 + xλ 1 2

So we need to approximate e−tλ

2N+1 2

using the rational function d(λ)...

Padé approximants !

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Padé’s Approximants

Let Pr be the r-th Padé approximant for e−z : e−z = Pr(z) Pr(−z) + O(z2r+1), z → 0. The distribution of its zeros is known [SaffVarga78], they are all contained in the region Rez > 0. µe1+µ = 1, µ ≃ 0, 278...

µ 1 + 2

3n

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Let’s fix N ∈ N and Y(0) ⊔ Y(1) ⊔ Y(2) := Y =

• y :

Pr(2ty2N+1) = 0

• ,

n = r(2N + 1). IMPORTANT : There’s an ambiguity on the choice of the location of the zeroes...

• k0 = ( #{Yκ} in the second quadrant assigned to Y(2)) - (#{Yκ} in the third

quadrant assigned to Y(1))

• k− = −

N

2

• + ( #{Yκ} in the first quadrant assigned to Y(1) ) ;
• k+ =

N

2

• ( #{Yκ} in the fourth quadrant assigned to Y(2) ) ;

Y(0) Yκ Y(1) Y(2) FIGURE: Exemple avec N = 3, k+ = 1, k0 = 0, k− = −1. Y(0) Yκ Y(1) Y(2) FIGURE: Example with N = 4, k+ = 1, k0 = 0, k− = −1.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Theorem (M. Bertola, M.C.) : Let’s fix N ∈ N and choose Y = {y1, . . . , yn} as above. Zn(x; Y) converges, for n → ∞, to the tau function PIN identified by the corresponding k0, k±. In particular, u(x, t) := 2∂2

x τ(x, t) satisfies the equation

(2N + 1)tRN[u(x; t)] + u(x; t) + x = 0, with Rj defined by the Lenard’s recursion ∂ ∂x RN+1[u] = 1 4 ∂3 ∂x3 + u(x) ∂ ∂x + 1 2ux(x)

• RN[u],

R0[u] = 1. Example : N = 2; 5 8t

• u′′ + 3u2

+ u + x = 0, N = 3; 7 32t

• u(4) + 10uu′′ + 5(u′)2 + 10u3

+ u + x = 0 .

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité

Thanks !