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Aspects of the ODE/IM correspondence

Panagiota Adamopoulou

University of Kent

3rd South East Mathematical Physics Seminar

Panagiota Adamopoulou (UKC) Aspects of the ODE/IM correspondence 3rd SEMPS 1 / 35

Plan

• Introduction to the ODE/IM correspondence

• Generalisations of the correspondence

Third-order ODE

PDEs and massive quantum field theories


The ODE/IM Correspondence

What is the ODE/IM correspondence?

(Dorey, Dunning, Tateo, Bazhanov, Lukyanov, Zamolodchikov, Suzuki, . . .)

−→ Linear ordinary differential equations (ODEs) defined in the complexplane

Spectral determinants related to certain eigenvalue problems for the ODEs.

−→ Quantum integrable models (IMs)/quantum field theories (QFTs) in twodimensions

Baxter’s TQ relation, T and Q operators of Bazhanov, Lukyanov, Zamolodchikov for

quantum field theory.

? Correspondence based mainly on functional relations (and BetheAnsatz-type equations) that appear on both sides



Eigenvalues of ODEs involved satisfy Bethe Ansatz-type equations (BAE)

Real eigenvalues −→ mapped to BAE with real solutions: ground state ofquantum integrable model

Some results on excited states 1

1V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Higher-level eigenvalues ofQ-operators and Schrodinger equation, Adv Theor and Math Phys 7 (2004), 711-725



ODEs successfully mapped to Bethe Ansatz systems

An−1 :(

(−1)n+1Dn(g) + p(x,E))ψ(x) = 0

Dn :(Dn(g†)

(d

dx

)−1

Dn(g)−√p(x,E)

(d

dx

)√p(x,E)

)ψ(x) = 0

Bn :(Dn(g†)Dn(g) +

√p(x,E)

(d

dx

)√p(x,E)

)ψ(x) = 0

Cn :(Dn(g†)

(d

dx

)Dn(g)− p(x,E)

(d

dx

)−1

p(x,E))ψ(x) = 0

• p(x,E) = xnM − E• Dn(g) = D (gn−1 − (n− 1))D (gn−2 − (n− 2)) . . . D (g0), withD(g) = (d/dx− g/x)

• g = gn−1, . . . , g1, g0 and g† = n− 1− g0, n− 1− g1, . . . , n− 1− gn−1Panagiota Adamopoulou (UKC) Aspects of the ODE/IM correspondence 3rd SEMPS 5 / 35


Applications/Connections:

PT -symmetric Quantum Mechanics

Quasi-exact solvability of differential equations

Spectral equivalences between differential equations

and others. . .



Goals:

Further explore connections

Enlarge sets of Differential Equations that enter the correspondence

Generalise to other quantum integrable models


Third-order ODE

One direction of extension:

Consider the third-order ODE:(d3

dx3+A

x2

d

dx+B

x3+ x3M + αx2M−1 − E

)ψ(x) = 0

with A = −2 + g0g1 + g0g2 + g1g2 , B = −g0g1g2 and gi,M, α ∈ R

Find E for which there exists a solution decaying exponentially asx→ +∞ and algebraically as x→ 0


Third-order ODE

The ODE admits the following asymptotic solution

y ∼ x−M−α3 exp

(− x

M+1

M + 1

)as |x| → ∞ in the sector | arg x| < 4π/(3M + 3)

We define rotated solutions that decay in certain sectors of the complexplane

yk(x,E, α,g) ≡ ωk−αk3 ω−k(M+1)

y(ω−kx, ω−3kME,ω−k(M+1)α,g)

with ω = exp(

2πi3(M+1)

) For k ∈ N the yk solve the ODE. Also define

α0 ≡ α , α1 ≡ ω−(M+1)α , α2 ≡ ω−2(M+1)α


Third-order ODE

The functions yk, yk+1, yk+2 are linearly independent

y = C(1)(E,α) y1 + C(2)(E,α) y2 + C(3)(E,α) y3

The C(i) are called Stokes multipliers and can be expressed in terms ofWronskians of rotated solutions yk

Expand the solution y in basis of solutions to the ODE at the origin

y = D0(E,αi)χ0 +D1(E,αi)χ1 +D2(E,αi)χ2

with χi ∼ xgi , i = 1, 2, 3


Third-order ODE

Combining the relations for solutions at the origin and at infinity

C(1,i)(E)D(1,i+1)(ω−3ME)D(2,i+1)(ω−3ME) =

ωg0−1+αi+1

3 D(1,i)(E)D(2,i+1)(ω−3ME)

+ ωg1−1+ 13 (2αi+1−αi+2)D(1,i+2)(ω−6ME)D(2,i)(E)

+ ω2−g0−g1− 13 (2αi+αi+2−αi+1)D(1,i+1)(ω−3ME)D(2,i+2)(ω−6ME)

with

C(1,i)(E) = C(1)(E,αi) , D(1,i)(E) = D0(E,αi) , D(2,i) ∼W [y, y1]


Third-order ODE

Three sets (one for each value of α) of Bethe Ansatz-type equations forthe eigenvalues E

−→ Map to known quantum integrable model

The third-order ODE is quasi-exactly solvable, for certain values of α(subset of eigenvalues and eigenfunctions can be constructed)


Massive ODE/IM Correspondence

Another direction of extension: include other Differential Equations andmassive QFTs

• The correspondence concerned the mapping of certain ODEs to masslessquantum field theories

• Lukyanov & Zamolodchikov showed how to include massive quantum fieldtheories

• They had as a starting point the classical sinh-Gordon equation

• Here a correspondence between classical A(1)n−1 Toda field theories

(classical integrable PDEs) and An−1 Bethe Ansatz systems will bepresented 2

2P. Adamopoulou and C. Dunning, Bethe ansatz equations for the classical A(1)n affine

Toda field theories, J.Phys.A 47 (2014), 205205


Toda equations

Describe the motion of n particles on a closed chain with exponentialinteractions 3

d2ηidt2

= 2e2ηi+1−2ηi − 2eηi−2ηi−1 , i = 1, . . . , n

where ηi is the displacement of the i-th particle from its equilibriumposition, with ηn+1 = η1

A two-dimensional continuous generalisation of this model reads 4

∂2t ηi − ∂2

xηi = 2e2ηi+1−2ηi − 2eηi−2ηi−1 , i = 1, . . . , n

with ηi ≡ ηi(x, t) ∈ R and x , t ∈ R

3M. Toda, Vibration of a Chain with Nonlinear Interaction, Journal of the PhysicalSociety of Japan 22 (1967), 431-436

4A. V. Mikhailov, Integrability of the two-dimensional generalization of toda chain,JETP Letters 30 (1979), 414-418Panagiota Adamopoulou (UKC) Aspects of the ODE/IM correspondence 3rd SEMPS 14 / 35

A(1)n−1 Toda field equations

The two-dimensional A(1)n−1 Toda field theories are described by the

Lagrangian

L =1

2

n∑i=1

(∂tηi)2 − (∂xηi)

2 −n∑i=1

exp(2ηi+1 − 2ηi)

with ηi ≡ ηi(x, t), periodic boundary conditions, and∑ni=1 ηi = 0

Using coordinates w = x+ t and w = x− t, the corresponding equationsof motion are

2 ∂w∂wηi = exp(2ηi − 2ηi−1)− exp(2ηi+1 − 2ηi) , i = 1, . . . , n ,

with ηi ≡ ηi(w, w) and (w, w) ∈ C2



We are interested in a particular class of real-valued solutions to the A(1)n−1

Toda equations

To this end, we first apply asymptotic analysis in certain asymptotic

limits to a reduction of the A(1)n−1 Toda equations



Asymptotic Analysis

The combination ww remains invariant under a scaling of the variables

−→ perform a symmetry reduction:

t =√

2ww , ηi(w, w) = yi(t)

The A(1)n−1 Toda equations become a system of n coupled nonlinear

ordinary differential equations

d2

dt2yi +

1

t

d

dtyi + e2yi+1−2yi − e2yi−2yi−1 = 0 , i = 1, . . . , n ,

which is of Painleve III-form



Asymptotic Analysis

Asymptotic analysis to the system of equations provides the followingleading order behaviours for yi(t)

• As t→ 0

yi(t) ∼ (n− i− gn−i) ln t+ bi + power series in t

with gi, bi ∈ R

∑n−1i=0 gi = n(n− 1)/2

g0 < g1 < · · · < gn−1 and g0 + n > gn−1

• As t→∞yi(t) = O(1)

→ The constants gi: related to certain parameters which enter (the ODEs) ofthe ODE/IM correspondence

→ The asymptotic analysis provides for free certain relations which wereimposed to the gi


Lax pair A(1)n−1

The A(1)n−1 Toda field equations are integrable

The associated linear problem (Lax pair) is(∂w + U(w, w, λ)

)Φ = 0 ,

(∂w + V (w, w, λ)

)Φ = 0

• U, V are functions from (z, z) ∈ C2 to A(1)n−1[λ, λ−1]

• λ = eθ ∈ C is the spectral parameterwe have that

U(w, w, λ) = ∂wηi δij + λC , V (w, w, λ) = −∂w ηi δij + λ−1C ,

(C)ij = exp(ηj+1 − ηj) δi−1,j j = 1, . . . , n

The compatibility condition of the linear system of equations

∂wV − ∂wU + [U , V ] = 0

(zero-curvature condition) is equivalent to the A(1)n−1 Toda field equations


How can we make a connection to quantum integrability?


Modified A(1)n−1 Toda field equations

−→ We consider a modified version of the A(1)n−1 Toda equations

• Change of variables: (w, w)→ (z, z)

dw = p(z)1/ndz , dw = p(z)1/ndz ,

withp(t) = tnM − snM and M, s ∈ R+

• Transformation of the fields

ηi(z, z)→ ηi(z, z)− 14n

(2i− n− 1) ln(p(z)p(z))



The A(1)n−1 Toda equations transform to the modified A

(1)n−1 Toda equations

2∂z∂zηi = e2ηi−2ηi−1 − e2ηi+1−2ηi for i = 2, . . . , n− 1 ,

2∂z∂zη1 = p(z)p(z) e2η1−2ηn − e2η2−2η1 ,

2∂z∂zηn = e2ηn−1−2ηn − p(z)p(z) e2η1−2ηn .

with ηi ≡ ηi(z, z) and (z, z) ∈ C2



Solutions

A particular family of asymptotic solutions to the modified A(1)n−1

equations is obtained form the asymptotic solutions to the original A(1)n−1

equations:

• As zz → 0

ηi(z, z) ∼ 12(n− i− gn−i) ln(zz) + bi +

∞∑k=1

γik(znkM + znkM

)+ p.s. in zz

• As zz →∞ηi(z, z) = 1

4(2i− n− 1)M ln(zz) + o(1)


Lax Pair Modified A(1)n−1

The Lax pair for the A(1)n−1 Toda equations is associated to that for the

modified A(1)n−1 Toda equations by a gauge transformation

(∂w + U(w, w, λ)

)Φ = 0(

∂w + V (w, w, λ))Φ = 0

(w,w)→(z,z)−−−−−−−−−→transformηi

(∂z + U(z, z, λ)

)Φ = 0(

∂z + V (z, z, λ))Φ = 0

gauge−−−−−→transf.

(∂z + U(z, z, λ)

)Ψ = 0 ,

(∂z + V (z, z, λ)

)Ψ = 0 ,

with

A(z, z, λ) = g−1gz + g−1A(z, z, λ)g , Φ = gΨ

and

(g)ij =

(p(z)

p(z)

)n− 2i−14n

δij



Symmetries

It is convenient to introduce z = ρ eiφ, z = ρ e−iφ with (ρ, φ) ∈ R2

We define the transformations

• Ω : φ→ φ+ 2πnM

, θ → θ − 2πinM

• S : A(λ)→ S A(σ−1λ

)S−1 or A(θ)→ S A(θ − 2πi

n)S−1

for any n× n matrix A(λ), with σ = exp(

2πin

)and (S)ij = σi δi,j

Such groups of transformations are known as reduction groups 5

5A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica DNonlinear Phenomena 3 (1981), 73-117



Symmetries

The matrices U , V of the linear problem are invariant under the action ofthe transformations Ω, S

Ω (U(ρ, φ, θ)) = U(ρ, φ, θ) , Ω (V (ρ, φ, θ)) = V (ρ, φ, θ)

S (U(ρ, φ, θ)) = U(ρ, φ, θ) , S (V (ρ, φ, θ)) = V (ρ, φ, θ)

The symmetries of U , V affect the auxiliary solution Ψ of the Lax pair



Solution

Consider a vector Ψ = (Ψ1, . . . ,Ψn)T

A general solution to the modified A(1)n−1 linear problem reads

Ψi(z, z, λ) =

−λ−1eηi−ηi+1

(∂zΨi+1 + ∂zηi+1Ψi+1

)for i = n− 1, . . . , 1

eηnψ for i = n

Where ψ ≡ ψ(z, z, λ) satisfies the nth-order differential equation((−1)n+1Dn(η) + λnp(z)

)ψ = 0

with Dn(η) the nth-order operator

Dn(η) = (∂z + 2 ∂zη1) (∂z + 2 ∂zη2) . . . (∂z + 2 ∂zηn)



Solution

−→ Finding an asymptotic solution for ψ allows for a solution Ψ to the Laxpair to be determined

−→ Different asymptotic solutions ηi provide different potentials to the ODEfor ψ



Solution

• As zz → 0: solutions to the ODE for ψ behave as zgj , giving rise to nsolutions to the Lax pair

Ξj ∼(

0, . . . , 0, e(gj−j)(θ+iφ), 0 . . . , 0)T

where the (n−j)th component is non-zero

• As zz →∞: a WKB analysis of the ODE provides a solution ψ whichleads to the asymptotic solution to the Lax pair

Ψ(ρ, φ, θ,g) ∼ (Ψ1, . . . ,Ψn)T exp(−2 ρM+1

M+1 cosh(θ + iφ(M + 1)))

where Ψj = exp (iφM(n− (2j − 1))/2), g = g0, . . . , gn−1

Expand Ψ in the basis of solutions Ξj

Ψ(ρ, φ, θ,g) =

n−1∑j=0

Qj(θ,g) Ξj(ρ, φ, θ,g)


Q-functions

The Qj-functions are characterised by certain properties:

• they are expressed in terms of determinants of the solutions Ξj , Ψ

• they are quasi-periodic functions of θ:

Qj(θ,g) = exp(− 2πi

n (gj − 1))Qj(θ − 2πi

n(M+1)M ,g

)which follows from the symmetries Ω, S of the Lax pair

• they satisfy certain functional relations from which the An−1 Betheansatz equations arise


Massless limit

−→ Consider the limit:

Take z → 0 with z finite and small

Then set x = z eθ

M+1 , E = snMenMθM+1 and take the limit z ∼ s→ 0, θ →∞

in the ODE for ψ

This way the nth-order ODE appearing in the massless An−1 ODE/IMcorrespondence is recovered

−→ In this limit the coefficients Qi coincide with those of the masslessquantum field theory related to the An−1 Lie algebra


ODE/IM correspondence

linear ODEsODE/IM correspondence−−−−−−−−−−−−−−−−→ BAE (massless QFTs)

nonlinear integrable PDEsODE/IM correspondence−−−−−−−−−−−−−−−−→ BAE (massive QFTs)


Current Work

• Systems of nonlinear ODEs (second order) have emerged in the analysis

of the classical A(1)n−1 Toda equations

• It is conjectured (ARS conjecture6) that the above ODEs possess veryinteresting analytic properties (Painleve property)

• Confirm this conjecture and examine the integrability properties (Laxpair, Backlund transformations) (reduction to PIII)

Also

• Study analytic properties of Q-functions: asymptotic behaviour, patternof zeros

6Ablowitz, M. J., Ramani, A, Segur, A., Nonlinear evolution equations and ordinarydifferential equations of Painleve type, Lett. Nuovo Cimento 23 (1978), 333-338


Thank you for your attention


References

• P. Dorey, T.C. Dunning, R. Tateo, The ODE/IM correspondence, Journal of Physics AMathematical General 40 (2007)

• S. L. Lukyanov and A. B. Zamolodchikov, Quantum sine(h)-Gordon model andclassical integrable equations, Journal of High Energy Physics 07 (2010)

• P. Dorey, S. Faldella, S. Negro, and R. Tateo, The Bethe Ansatz and theTzitzeica-Bullough-Dodd equation, Phil.Trans.R.Soc.A. 371 (2013), 20120052

• K. Ito and C. Locke, ODE/IM correspondence and modified affine Toda fieldequations, arXiv:1312.6759 [hep-th] (2013)


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