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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,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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)
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
Quantum Theory and Non-Commutativity-2
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
Quantum Theory and Non-Commutativity-2
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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;
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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;
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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 .
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
"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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
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-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,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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 .
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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)
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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.
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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 .
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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 .
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
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
).
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,
IntroductionPainlevé II equation
Non-commutative Painlevé IIApplications: M. Bertola - M. Cafasso
M. Irfan : Zero-curvature and Lax representations for the algebraic NC Painlevé IIReference
THANK YOU FOR YOUR ATTENTION!
Vladimir Roubtsov Exposé à Glasgow, March 2, 2012,