Non-commutativity versus quantization: Painlevé II - a toy modelcomputing.coventry.ac.uk/~mengland/OLD/2012Workshop/... · 2015-05-27 · Applications: M. Bertola - M. Cafasso M

IntroductionPainlevé II equation

Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso

M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference

Non-commutativity versus quantization: Painlevé II- a toy model

Vladimir Roubtsov

LAREMA, U.M.R. 6093 associé au CNRSUniversité d’Angers and Theory Division, ITEP, Moscow

March , 2, 2012 - University of Glasgow, "Matrix models,tau-functions and geometry"

Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,




Based on joint work with Vladimir Retakh (Rutgers University),J. Phys. A: Math. Theor. 43 (2010) 505204,on some results of Mahmood Irfan(Angers University)(J. Geom. Phys. to appear)and on some results of Marco Bertola(CRM, Montreal) and MattiaCafasso (Angers University).(Comm. Math. Phys. 2011)





Plan

1 Introduction

2 Painlevé II equation

3 Non-commutative Painlevé II

4 Applications: M. Bertola - M. Cafasso

5 M. Irfan : Zero-curvature and Lax representations for thealgebraic NC Painlevé II

6 Reference





Quantum Theory and Non-Commutativity-1

String theory in its supersymmetric version - a consistentdescription of quantum gravity.It is still not clear how to merge real (non supersymmetric)fundamental interactions in the strings framework.The unifying theories of strings (or a larger M− theory)contain degrees of freedom which cannot be described byordinary gauge theories.The noncommutative (NC) geometry emerges in relation toparticular string configurations involving branes andfluxes.When non trivial fluxes are turned on, ordinary fieldtheories get deformed by non-commutativity.
























Independently of string theory, NC geometry was initiallyformulated with the hope that it could mild ultravioletdivergences in quantum field theories.Noncommutative relation among space-time coordinates mayalso be interpreted as a possible deformation of geometrybeyond the Planck scale.






Independently of string theory, NC geometry was initiallyformulated with the hope that it could mild ultravioletdivergences in quantum field theories.Noncommutative relation among space-time coordinates mayalso be interpreted as a possible deformation of geometrybeyond the Planck scale.





Motivation to extend to noncommutative spaces

Noncommutative extension of field theories is not just ageneralization of them but a fruitful study direction in both physicsand mathematics.

Noncommutative spaces are characterized by thenoncommutativity of the spatial coordinates xµ

[xµ, xν ] = iθµν

(anti-symmetric tensor θµν -the noncommutative parameter⇔ a real constant closely related to existence of a backgroundfluxResolution of singularities ⇒ U(1)−instantons;





Motivation to extend to noncommutative spaces

Noncommutative extension of field theories is not just ageneralization of them but a fruitful study direction in both physicsand mathematics.

Noncommutative spaces are characterized by thenoncommutativity of the spatial coordinates xµ

[xµ, xν ] = iθµν

(anti-symmetric tensor θµν -the noncommutative parameter⇔ a real constant closely related to existence of a backgroundfluxResolution of singularities ⇒ U(1)−instantons;





"Quantum" = "non-commutative"

Historically the word "quantum" was introduced in relation tothe discretness of the spectrum of operators (Hamiltonian,angular momentum...)Further development of QM had generalized its use to thedescription of non-classical objects(= non-commutingoperators).





"Quantum" = "non-commutative"

Historically the word "quantum" was introduced in relation tothe discretness of the spectrum of operators (Hamiltonian,angular momentum...)Further development of QM had generalized its use to thedescription of non-classical objects(= non-commutingoperators).





Towards noncommutative integrable systems

The integrable equations involving non-commuting variables(i.e.SUSY or fermionic extensions of integrable evolutionequations) are very relevant to modern quantum field Theories.Noncommutative extension of integrable equations such as theKdV/KP equations is also one of the hot topics.Difficulties:These equations imply no gauge field and noncommutativeextension of them perhaps might have no physical picture orno good property on integrability.A commutation rule should be consistent with the evolutionSuch equations should be examined one by one.



































Moyal-product

The Moyal-product is defined for ordinary fields explicitly by

f ? g(x) := exp(i2θµν

∂

∂x ′µ∂

∂x ′′ν)f (x ′)g(x ′′)|x ′=x ′′=x

= f (x)g(x) +i2θµν

∂f∂xµ

∂g∂xν

+O(θ2)

Moyal product has associativity: f ? (g ? h) = (f ? g) ? h,"commutative limit" f ? g → f · g , θµν → 0 and[xµ, xν ]? = xµ ? xν − xν ? xµ = iθµν .





Noncommutative KdV equation

Noncommutative KdV equation in (1 + 1)-dimension, [t, x ]? = iθ,:

∂u∂t

=14∂3u∂x3 +

34{∂u∂x, u}?

where{xµ, xν}? = xµ ? xν + xν ? xµ

.





Noncommutative KP equation

Noncommutative KP equation in (2 + 1)-dimension, [t, x ]? = iθ,:

∂u∂t

=14∂3u∂x3 +

34{∂u∂x, u}? +

34∂−1

x∂2u∂x2 −

34

[u, ∂−1x∂u∂y

]?

where∂−1

x f (x) :=

∫ xf (u)du

.





Painlevé II equation

The Painlevé equations are non-linear ordinary differential equationsof 2nd order, which were discovered by P. Painlevé around 1900 inhis study of algebraic differential equations y ′′ = R(t; y ; y ′) withoutmovable singularities (branching points).

ExamplePainlevé II equation:

PII (u, β) : u′′ = 2u3 − 4ux + 4(β +12

).





Painlevé transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS);They admit a Hamiltonian formulation;They can be expressed as the isomonodromic deformation ofsome linear differential equation with rational coefficients;All Painlevés (except for PI ) admit one-parameter family ofsolutions (in terms of special functions) and for some specialvalues of parameteres they have particular rational solutions;Recently: PII - is non-integrable as a meromorphicHamiltonian system.





























Painlevé II, Hankel matrix and tau-functions

N. Joshi, K. Kajiwara and M.Mazzocco ("Asterisque", 2004): ThePainlevè II (PII ) equation

u′′ = 2u3 − 4xu + 4(β +12

)

admits a unique rational solution for a half-integer value of theparameter β. These solutions can be expressed in terms oflogarithmic derivatives of ratios of Hankel-type determinants: forβ = N + 1

2

u =ddx

logdetAN+1(x)

detAN(x),

where AN(x) = ||ai+j || where i , j = 0, 1, . . . , n− 1. The entries arepolynomials an(x) subjected to the recurrence relations:

a0 = x , a1 = 1, an = a′n−1 +n−1∑i=0

aian−1−i .Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,




"Quantum" Painlevè II

Recently a "quantized" version of PII was proposed (H. Nagoya, B.Grammaticos, A. Ramani)

Three unknown ("operators"): f0, f1, f3 and two "parameters"α0, α1.The commutation rules:

[f0, f2] = [f2, f1] = ~, [f1, f0] = 2~f2.The "Hamiltonian system" ("quantum" PII ):

∂t f0 = f0f2 + f2f0 + α0, ∂t f1 = −f1f2 − f2f1 + α1, ∂t f2 = f1 − f0.

Compatibility of commutation rules and the evolution.f ′′2 = 2f 3

2 − tf2 + α1 − α0











2 − tf2 + α1 − α0











2 − tf2 + α1 − α0











2 − tf2 + α1 − α0











2 − tf2 + α1 − α0





Properties of the quantum PII

It admits the affine Weyl groupA(1)

1 (s2i = 1, π2 = 1, πsi = si+1π, i = 0, 1) action which

preserves the commutation relations;They are the Bäcklund transformations of this equation.The quantum PII Hamiltonian: H = 1

2(f0f1 + f1f0) + α1f2 and"canonical variables";"Commutative time"- t.
































Non-commutative Toda chains

Let R be an associative algebra over a field with a derivation D.Set Df = f ′ for any f ∈ R . Assume that R is a division ring.

Definition(Two-sided NC Toda chains)

(θ′nθ−1n )′ = θn+1θ

−1n − θnθ−1

n−1, n ≥ 1,

assuming that θ1 = φ, θ0 = ψ−1, φ, ψ ∈ R.“Negative" counterpart of it:

(η−1−mη

′−m)′ = η−1

−mη−m−1 − η−1−m+1η−m, m ≥ 1,

where η0 = φ−1, η−1 = ψ.Note that θ′θ−1 and θ−1θ′ are noncommutative analogues of thelogarithmic derivative (log θ)′.Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,




From NC Toda to NC Painlevé

We use the solutions of the Toda equations under a certain ansatzfor constructing solutions of the noncommutative Painlevé IIequation

PII (u, β) : u′′ = 2u3 − 2xu − 2ux + 4(β +12

)

where u, x ∈ R , x ′ = 1 and β is a scalar parameter, β′ = 0.Unlike the quantum Painlevé II we consider here a "purenoncommutative" version of the Painlevé equation without anyadditional assumption for our algebra R .





Quasideterminant

main organizing tool in noncommutative algebra.(Gelfand-Retakh-Wilson)

DefinitionGiven an n × n matrix A over some ring R , the(ij)-quasideterminant |A|ij is defined whenever Aij is invertible, andin that case,|A|ij=

Figure: Quasideterminant |A|ijVladimir Roubtsov Exposé à Glasgow, March 2, 2012,




Example 2x2

ExampleSuppose

A =

(a11 a12a21 a22

);

here are two of its four quasideterminants:|A|11 = a11 − a12a−1

22 a21 and |A|21 = a21 − a22a−112 a11.





Hankel matrix solutions of NC Toda

Assume that θ0 = ψ−1, θ1 = φ and η0 = φ−1, η−1 = ψ.Set a0 = φ, b0 = ψ and

an = a′n−1 +∑

i+j=n−2,i ,j≥0

aiψaj ,

bn = b′n−1 +∑

i+j=n−2,i ,j≥0

biφbj , n ≥ 1.

Construct Hankel matrices An = ||ai+j ||,Bn = ||bi+j ||, i , j = 0, 1, 2 . . . , n.

TheoremSet θp+1 = |Ap|p,p, η−q−1 = |Bq|q,q. The elements θn for n ≥ 1satisfy the Toda system and the elements η−m,m ≥ 1 satisfy the"negative" analogue of the system.





Reduction to Painlevé II

TheoremLet φ and ψ satisfy the following identities:

ψ−1ψ′′ = φ′′φ−1 = 2x − 2φψ,

ψφ′ − ψ′φ = 2β.

Then for n ∈ Nun = θ′nθ

−1n satisfies nc − PII (x , β + n − 1);

u−n = η′−nη−1−n satisfies nc − PII (x , β − n).





Reduction to Painlevé II

TheoremLet φ and ψ satisfy the following identities:

ψ−1ψ′′ = φ′′φ−1 = 2x − 2φψ,

ψφ′ − ψ′φ = 2β.

Then for n ∈ Nun = θ′nθ

−1n satisfies nc − PII (x , β + n − 1);

u−n = η′−nη−1−n satisfies nc − PII (x , β − n).





Remark-1

Our result is a generalization of the following old observation of I.Gelfand and V. Retakh:Let R be a division ring with a derivation ∂. Suppose φ ∈ Rinvertible and all quasideterminants

τn(φ) :=

∣∣∣∣∣∣∣∣∣φ ∂φ . . . ∂n−1φ∂φ ∂2φ . . . ∂nφ. . . . . . . . . . . .

∂n−1φ ∂nφ . . . ∂2n−2φ

∣∣∣∣∣∣∣∣∣are defined and invertible.





Remark-2

If we set φ1 := φ and φn := τn(φ) then

TheoremElements φn, n ≥ 1 satisfy the following identities:

(∂(∂φ1)φ1−1) = φ2φ1

−1,

(∂(∂φn)φ−1n ) = φn+1φ

−1n − φnφ

−1n−1.





NC Painlevé and Fredholm detrminants

Consider the example of the matrix Airy convolution kernel onL2(R+,Cn) defined as:

Ai s(f ) :=

∫R+

Ai(x + y ; s)f (y)dy ,

where

Ai(x ; s) :=

∫γ+

exp θ(µ)C exp θ(µ) exp xµdµ2π

=∣∣∣∣cjkAi(x + sj + sk)

∣∣∣∣θ :=

iµ3

6In + isµ, C ∈ Matn(C)





Lax isomonodromic pair-1

TheoremZero constant matrix NC Painlevé II equation

∂2U = 4(2U3 + {s,U}) (1)

is equivalent to the isomonodromy matrix linear system with2n × 2n matrices

∂jΨ(λ, s) = Sj(λ, s)Ψ(λ, s)

∂λΨ(λ, s) = A(λ, s)Ψ(λ, s).





Lax isomonodromic pair-2

Here σi ∈ Mat2(C), i = 1, 2, 3 -Pauli matrices and

Sj(λ, s) = iλej ⊗ σ3 + i [V , ej ]⊗ I + {U, ej} ⊗ σ1,

A(λ, s) =i2λ2σ̂3 + λU ⊗ σ1 −

12∂2U ⊗ σ2 + i(U2 + s)⊗ σ3,

∂ :=n∑

j=1

∂j , s := diag(s1, . . . , sn), σ̂3 := In ⊗ σ3.





NC Painlevé II and Fredholm determinants for the Airykernel -1

TheoremThere is a unique solution U to (1) with any C ∈ Matn(C) withthe prescribed asymptotics:∣∣∣∣Ujk

∣∣∣∣ = −cjkAi(sj + sk) +O(√S exp−[

43

(2S − 2m)3/2]),

S :=1n

∑j=1

nsj →∞, δj := sj − S are kept fixed, |δj | ≤ m.

If C = C † the solution is pole-free on Rn iff ||C || ≤ 1

Such solutions are called Hastings-McLleod solutions.Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,




NC Painlevé II and Fredholm determinants for the Airykernel -2

TheoremLet U be a Hastings-McLleod solution to (1). Then

det(Id − Ai2s) = exp [−4∫ ∞

S(t − S)TrU2(t + δdt],

where

S :=1n

n∑j=1

sj , t + δ := (t + δ1, . . . , t + δn).





Zero-curvature representation-1

Let A and B be two matrices with non-commutative entries:

A =

(8iλ2 + iv2 − 2iz −ivz + 1

4Cλ−1 − 4λv

ivz + 14Cλ

−1 − 4λv −8iλ2 − iv2 + 2iz

). and

B =

(−2iλ vv 2iλ

)with central λ and constant C .





Zero-curvature representation-2

TheoremThe compatibility (or "zero-curvature") condition Ψzλ = Ψλz forthe linear system

Ψλ = A(z , λ)Ψ, Ψz = B(z , λ)

is equivalent to the following NC Painlevé II equation:

vzz = 2v3 − 2{z , v}+ C .





Lax representation-1

There is another representation of NC Painlevé II which transformsthe equivalent system

v′0 = v2v0 + v0v2 + α0

v′1 = −v2v1 − v1v2 + α1

v′2 = v1 − v0

to the Lax system Lz = [P, L] for the following L− P-pair:

L =

L1 O OO L2 OO O L3

, P =

P1 O OO P2 OO O P3





Lax representation-2

Here

L1 =

(1 0−v0 −1

),L2 =

(1 0−v1 −1

),L3 =

(−1 0−v2 1

)and the elements of the matrix P are given by

P1 =

(ρ1 00 −ρ1

),P2 =

(−ρ2 00 ρ2

),P3 =

(−1 012σ 1

)where

ρ1 = −v2 −12α0v−1

0 , ρ2 = −v2 +12α1v−1

1 , σ = v0 − v1 + 2v2

and

O =

(0 00 0

).









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Non-commutativity versus quantization: Painlevé II - a toy modelcomputing.coventry.ac.uk/~mengland/OLD/2012Workshop/... · 2015-05-27 · Applications: M. Bertola - M. Cafasso M