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SL(2, C) spin magnet Yang-Baxter equation R-operators,Q-operators and Baxter equation Construction of eigenfunctions

Iterative construction of eigenfunctions of the matrix elements

of the monodromy matrix

S. Derkachov

PDMI, St.Petersburg

RAQIS’16 Recent Advances in Quantum Integrable Systems22-26 August 2016


Plan

• SL(2,C) spin magnet

• principal series representations of SL(2,C)I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin (1966)

Generalized functions. Vol. 5: Integral geometry and representation theory

• monodromy matrix

• operators A(u) and B(u)

• Yang-Baxter equation

• Construction of eigenfunctions

• R-matrices and Q-operators

• Iterative construction

• Eigenfunctions and Q-operators

based onS. Derkachov, G. Korchemsky, A. Manashov Nucl.Phys. B617 (2001) 375-440

Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q

operator and separation of variables

S. Derkachov, A.Manashov J.Phys. A47 (2014) 305204 Iterative construction of

eigenfunctions of the monodromy matrix for SL(2,C) magnet


Motivation

• SL(2,C) spin magnet• principal series representations of SL(2,C) I. M. Gelfand, M. A. Naimark,

Unitary representations of the classical groups,Trudy Mat. Inst. Steklov., vol. 36, Izdat. Nauk SSSR, Moscow - Leningrad,1950; German transl.: Academie - Verlag, Berlin, 1957.

"In some sense infinite-dimensional representations in many respects aremore simple in comparison with finite-dimensional representations."

• the model describes high-energy behaviour in Yang-Mills theoryL. N. Lipatov, High-energy asymptotics of multicolor QCD andtwo-dimensional conformal field theories, Phys. Lett. B 309 (1993) 394.High-energy asymptotics of multicolor QCD and exactly solvable latticemodels,hep-th/9311037.

L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completelyintegrable model, Phys. Lett. B 342 (1995) 311.

• diagonalization of B(u) ↔ Sklyanin SOV representationE. K. Sklyanin, Quantum Inverse Scattering Method.Selected Topics, in

Quantum Groups and Quantum Integrable Systems, (Nankai lectures), ed.Mo-Lin Ge, pp. 63-97, World Scientific Publ., Singapore 1992, [hep-th/9211111]

Separation of variables - new trends, Prog.Theor.Phys.Suppl. 118 (1995) 35-60,

[solv-int/9504001]

• diagonalization of A(u)L. N. Lipatov, Integrability of scattering amplitudes in N=4 SUSY, J. Phys. A 42

(2009) 304020.


SL(2,C) spin magnetThe quantum SL(2,C) spin magnet is a straightforward generalization of thestandard XXXs spin chain.XXXs spin chain

The Hilbert space of the XXXs model is given by the tensor product of the(2s + 1)-dimensional representations of the SU(2) group

HN = V1 ⊗ V2 ⊗ · · · ⊗ VN , Vk = C2s+1

, k = 1, . . . ,N.

To each site k we associate the quantum L-operators with subscript k actingnontrivially on the k−th space in the tensor product

Lk(u) =

(u + iS (k) iS

(k)−

iS(k)+ u − iS (k)

)

; [S(k)+ ,S

(k)− ] = 2S

(k), [S (k)

,S(k)± ] = ±S

(k)±

The monodromy matrix is defined as a product of L−operators

T (u) = L1(u)L2(u) . . . LN(u) =

(AN(u) BN(u)CN(u) DN(u)

)

SL(2,C) spin magnet

(2s + 1)-dimensional representations of the SU(2) group −→ unitary principalseries representations of the SL(2,C) group

Vk = C2s+1 → Vk = L2(C)


unitary principal series representations of SL(2,C)SL(2,C) spin magnet

g =

(a b

c d

)

∈ SL(2,C) ; g → T(s,s)(g) : L2(C)→ L2(C)

[T

(s,s)(g) φ](z, z) = (d − bz)−2s

(d − bz

)−2sφ

(−c + az

d − bz,−c + az

d − bz

)

〈φ |ψ〉 =

∫

d2z φ(z, z)ψ(z, z) ; 〈T (s,s)(g)φ |T (s,s)(g)ψ 〉 = 〈φ |ψ〉

The spins s and s are parameterized as follows (ns ∈ Z , νs ∈ R)

s =1 + ns

2+ iνs ; s =

1− ns

2+ iνs

generators:

S− = −∂z , S = z∂z + s , S+ = z2∂z + 2s z

S− = −∂z , S = z∂z + s , S+ = z2∂z + 2s z

commutation relations:

[S+,S−] = 2S , [S,S±] = ±S± , [S+, S−] = 2S , [S, S±] = ±S±

conjugation:

S†± = −S± , S

† = −S


Intertwining operator

There exists an operator W which intertwines a pair of principal seriesrepresentations T (s,s) and T (1−s,1−s) at generic complex s and s

W (s, s)T(s,s)(g) = T

(1−s,1−s)(g)W (s, s)

Integral operator

[W (s, s)Φ ] (z, z) = const

∫

C

d2x

Φ(x , x)

(z − x)2−2s(z − x)2−2s

• The operator W is well-defined at generic s, s and the problems emergefor the discrete set of points 2s = −n , 2s = −n at n, n ∈ Z≥0

• At these special values of spins an (n + 1)(n + 1)-dimensionalrepresentation decouples from the general infinite-dimensional case

[

T(− n

2,− n

2)(g) Φ

]

(z, z) = (d − bz)n(d − bz

)nΦ

(−c + az

d − bz,−c + az

d − bz

)

The space of polynomials spanned by (n + 1)(n + 1) basis vectors zk z k ,where k = 0, 1, · · · , n and k = 0, 1, · · · , n is invariant subspace.


Normalized intertwining operator[z]α ≡ zα zα – single valued function in the complex plane provided thatα− α ∈ Z

Fourier transformation I∫

d2z

e i(pz+pz)

[z]α= π i

α−αa(α)

1

[p]1−α

a(α) ≡ a(α, α) =Γ(1− α)

Γ(α), a(α,β, γ, . . .) = a(α)a(β)a(γ) . . .

Fourier transformation II

1

πiα−αa(1 + α)

∫

d2x

eipx+i px

x1+αx1+α= p

αp

α

p → i∂z , p → i∂z

[i∂z ]α Φ(z, z) =

1

πiα−αa(1 + α)

∫

d2x [z − x ]−1−α Φ(x , x)

normalized intertwining operator

[W (s, s)Φ ] (z, z) =i2s−2s

πa(2− 2s)

∫

d2x [z − x ]2s−2 Φ(x , x)

W (s) = [i∂z ]1−2s


SL(2,C) spin magnet

The Hilbert space of the SL(2,C) spin magnet is given by the tensor product ofthe unitary principal series representations of the SL(2,C) group

HN = V1 ⊗ V2 ⊗ · · · ⊗ VN , Vk = L2(C) , k = 1, . . . ,N.

The space HN is the space of functions Ψ(z1, z1 . . . , zN , zN).To each site k we associate the pair of quantum L-operators with subscript k

acting nontrivially on the k−th space in the tensor product

Lk(u) =

(u + iS (k) iS

(k)−

iS(k)+ u − iS (k)

)

, Lk(u) =

(u + i S (k) i S

(k)−

i S(k)+ u − i S (k)

)

,

where u is complex conjugate to u. The monodromy matrices TN(u) andTN(u) are defined as a product of L operators

T (u) = L1(u)L2(u) . . . LN(u) , T (u) = L1(u)L2(u) . . . LN(u)

T (u) =


)

, T (u) =


)

The operators AN(u) ,BN(u) are differential operators of N−th order in thevariables z1, . . . , zN and AN(u) , BN(u) are differential operators of N−th orderin the variables z1, . . . , zN .


Eigenfunctions of AN(u) and AN(u)

AN(u) , AN(u) are polynomials of degree N in u and u correspondingly

AN(u) = uN + iu

N−1S +

N∑

k=2

uN−k

ak , S =

N∑

k=1

S(k)

AN(u) = uN + i u

N−1S +

N∑

k=2

uN−k

ak , S =

N∑

k=1

S(k)

by construction [AN(u), AN(v)] = 0→ [S, S] = [ak , aj ] = 0

RTT → [AN(u),AN(v)] = 0 , [AN(u), AN(v)] = 0→ [ak , aj ] = [ak , aj ] = 0

conjugation rules for generators → AN(u) = (AN(u))† → a

†k = ak

AN(u) , AN(u) generate the set of commuting self-adjoint operators

{

i(S + S), S − S, ak + ak , i(ak − ak), k = 2, . . .N}

and can be diagonalized simultaneously


Eigenfunctions of AN(u) and AN(u)

AN(u)ΨA(x |z) = (u − x1) · · · (u − xN)ΨA(x|z)

AN(u)ΨA(x |z) = (u − x1) · · · (u − xN)ΨA(x|z)

The eigenfunctions ΨA(z1, z1 . . . , zN , zN) are labeled by zeroes of polynomialeigenvalues

x = {x1, . . . , xN}, xk = (xk , xk)

z = {z1, . . . , zN}, zk = (zk , zk )

Further, the operator AN(u) is a hermitian adjoint of AN(u), AN(u) = (AN(u))†

and it results in the following relation for the eigenvalues(

N∏

k=1

(u − xk )

)∗

=

N∏

k=1

(u∗ − xk)

that, in its turn, implies that x∗k = xk . Together with the condition

(s − ixk )− (s − i xk) = −i(xk − xk) + ns ∈ Z

it results in the following parametrization: nk ∈ Z , νk ∈ R

xk = −ink

2+ νk , xk =

ink

2+ νk


Simplest example N = 1

L(u) = i

(s − iu + z∂ −∂z2∂ + 2sz −s − iu − z∂

)

=

(A1(u) B1(u)C1(u) D1(u)

)

eigenfunctions

ΨA(x|z) = [z]ix−s ≡ zix−s

zi x−s

[z]α ≡ zα zα – single valued function in the complex plane provided thatα− α ∈ Z→ (s− ix)−(s− i x) = −i(x− x)+ns ∈ Z→ x = − in

2+ν , x = in

2+ν

A1(u) zix−s

zi x−s = i (s − iu + z∂) z

ix−sz

i x−s = (u − x) zix−s

zi x−s

A1(u) zix−s

zi x−s = i

(s − i u + z∂

)z

ix−sz

i x−s = (u − x) zix−s

zi x−s

orthogonality x1 = − in12+ ν1 , x1 = in1

2+ ν1 , x2 = − in2

2+ ν2 , x2 = in2

2+ ν2

∫

d2z ΨA(x1|z)ΨA(x2|z) = 2π2

δ2(x1 − x2) , δ

2(x1 − x2) = δn1n2 δ(ν1 − ν2)

completeness x = − in2+ ν , x = in

2+ ν

∫

Dx ΨA(x|z1)ΨA(x|z2) = 2π2δ

2(z1 − z2) ,

∫

Dx =∑

n∈Z

∫ +∞

−∞

dν


Eigenfunctions of BN(u) and BN(u)

BN(u) , BN(u) are polynomials of degree N − 1

BN(u) = iuN−1

S− +

N∑

k=2

uN−k

bk , S− =

N∑

k=1

S(k)−

BN(u) = i uN−1

S− +

N∑

k=2

uN−k

bk , S− =

N∑

k=1

S(k)−

Eigenfunctions ΨB(x|z)

BN(u)ΨB(x|z) = p(u − x1) · · · (u − xN−1)ΨB(x|z)

BN(u)ΨB(x|z) = p(u − x1) · · · (u − xN−1)ΨB(x|z)

are parameterized by the momenta p, p – eigenvalues of the S−, S− operatorsand the roots xk , xk , k = 1, . . . ,N − 1

x = {x1, . . . , xN−1, xN = (p, p)}


Simplest example N = 1

L(u) = i

(s − iu + z∂ −∂z2∂ + 2sz −s − iu − z∂

)

=

(A1(u) B1(u)C1(u) D1(u)

)

eigenfunctions

ΨB(p|z) = eipz+i pz

B1 eipz+i pz = −i∂ e

ipz+i pz = p eipz+i pz

B1 eipz+i pz = −i ∂ e

ipz+i pz = p eipz+i pz

orthogonality∫

d2z ΨB(p1|z)ΨB(p2|z) = π

2δ

2(p1 − p2)

completeness∫

d2p ΨB(p|z1)ΨB(p|z2) = π

2δ

2(z1 − z2)

everything is based on well known formula∫ +∞

−∞

dp eipx = 2πδ(x)


R-matrix and Yang-Baxter equation

R(u, u) is SL(2,C)-invariant solution of Yang-Baxter equation

Yang-Baxter equation V1 ⊗ V2 ⊗ V3

R12(u − v , u − v)R13(u, u)R23(v , v) = R23(v , v)R13(u, u)R12(u − v , u − v)

SL(2,C)-invariance

Rij(u, u) : Vi ⊗ Vj → Vi ⊗ Vj ,

T(si ,si )(g) ⊗ T

(sj ,sj )(g)Rij(u, u) = Rij(u, u)T(si ,si )(g) ⊗T

(sj ,sj )(g)

RLL-relation

V1 ,V2 generic and V3 - two-dimensional: there are two variants

• s3 = − 12, s3 = 0: invariant subspace with basis {1, z3}

R12(u − v , u − v)L1(u)L2(v) = L2(v) L1(u)R12(u − v , u − v)

• s3 = 0 , s3 = − 12: invariant subspace with basis {1, z3}

R12(u − v , u − v) L1(u) L2(v) = L2(v) L1(u)R12(u − v , u − v)


L-operator

L(u) = i

(z∂ + s − iu −∂z2∂ + 2s z −z∂ − s − iu

)

=

= i

(1 0z 1

)(s − iu − 1 −∂

0 −s − iu

)(1 0−z 1

)

restrictions

• s3 = − 12, s3 = 0

R13(u, u)→ L1(u) = L1(u1, u2) =

(1 0z1 1

)(u1 −∂1

0 u2

)(1 0−z1 1

)

R23(v , v)→ L2(v) = L2(v1, v2) =

(1 0z2 1

)(v1 −∂2

0 v2

)(1 0−z2 1

)

• s3 = 0 , s3 = − 12

R13(u, u)→ L1(u) = L1(u1, u2) ; R23(v , v)→ L2(v) = L2(v1, v2)

u, s1 −→ u1 = s1 − iu − 1 , u2 = −s1 − iu ; v , s2 −→ v1 = s2 − iv − 1 , v2 = −s2 − iv


RLL-relations

RLL-relation in a matrix form

R12

(z1∂1 + s1 − iu −∂1

z21∂1 + 2s1z1 −z1∂1 − s1 − iu

)(z2∂2 + s2 − iv −∂2

z22∂2 + 2s2z2 −z2∂2 − s2 − iv

)

=

(z2∂2 + s2 − iv −∂2

z22∂2 + 2s2z2 −z2∂2 − s2 − iv

)(z1∂1 + s1 − iu −∂1

z21∂1 + 2s1z1 −z1∂1 − s1 − iu

)

R12

RLL-relation – compact notations

R12(u − v) L1(u)L2(v) = L2(v) L1(u)R12(u − v)

RLL-relation – explicit notations

R12(u1, u2|v1, v2) L1(u1, u2)L2(v1, v2) = L2(v1, v2)L1(u1, u2)R12(u1, u2|v1, v2)

R-operator The full-fledged notation would be R12(u − v , u − v |s1, s1, s2, s2)


R-operatorgenerators are first order differential operators with respect to the variables z1

and z2 acting on the function ψ(z1, z2)It is useful to extract operator of permutation from R-operator

P12 : P12 ψ(z1, z2) = ψ(z2, z1)

R12 = P12 R12 ,

R12(u1, u2|v1, v2) L1(u1, u2)L2(v1, v2) = L1(v1, v2)L2(u1, u2)R12(u1, u2|v1, v2)

u ≡ (v1, v2, u1, u2)←→ group of permutations S4

R12(u)↔ σ = σ2σ1σ3σ2 ; σ(v1, v2, u1, u2) = (u1, u2, v1, v2)

elementary transpositions σ1, σ2 and σ3 – generators in S4

σ1u = (v2, v1, u1, u2) ; σ2u = (v1, u1, v2, u2) ; σ3u = (v1, v2, u2, u1)

(

S1︷︸︸︷v1 , v2, u1, u2) : S1(u) L2(v1, v2) = L2(v2, v1) S1(u)

(v1

S2︷︸︸︷v2, u1, u2) : S2(u) L1(u1, u2)L2(v1, v2) = L1(v2, u2) L2(v1, u1) S2(u)

(v1, v2,

S3︷︸︸︷u1 , u2) : S3(u)L1(u1, u2) = L1(u2, u1)S3(u)


R-operator

S1 and S3 – intertwining operators

u1 ⇄ u2 ←→ s ⇄ 1− s

W L(u1, u2) = L(u2, u1)W←→W = [i∂]1−2s = [i∂]u2−u1

S1(u) = [i∂2]v2−v1 ; S3(u) = [i∂1]

u2−u1

Defining equation for the last operator S2(u) in explicit form (zij = zi − zj )

S2

(1 0z1 u2

)(u1 −∂z1

0 1

)(1 0

z21 1

)(1 −∂z2

0 v2

)(v1 0−z2 1

)

=

=

(1 0z1 u2

)(v2 −∂z1

0 1

)(1 0

z21 1

)(1 −∂z2

0 u1

)(v1 0−z2 1

)

S2

[S2, z1] = [S2, z2] = 0←→ S2(u) = [z12]u1−v2

σ = σ2σ1σ3σ2 → R12(u) = S2(σ1σ3σ2u) S1(σ3σ2u) S3(σ2u)S2(u) =

= [z12]u2−v1 [i∂2]

u1−v1 [i∂1]u2−v2 [z12]

u1−v2


R-operator

R-operator

R12(u − v , u − v) = [z12]u2−v1 [i∂2]

u1−v1 [i∂1]u2−v2 [z12]

u1−v2

RLL-relation – explicit check

[z12]u2−v1 [i∂2]

u1−v1 [i∂1]u2−v2 [z12]

u1−v2 L1(u1, u2)L2(v1, v2) =

[z12]u2−v1 [i∂2]

u1−v1 [i∂1]u2−v2 L1(v2, u2)L2(v1, u1) [z12]

u1−v2 =

[z12]u2−v1 [i∂2]

u1−v1 L1(u2, v2)L2(v1, u1) [i∂1]u2−v2 [z12]

u1−v2 =

[z12]u2−v1 L1(u2, v2)L2(u1, v1) [i∂2]

u1−v1 [i∂1]u2−v2 [z12]

u1−v2 =

L1(v1, v2)L2(u1, u2) [z12]u2−v1 [i∂2]

u1−v1 [i∂1]u2−v2 [z12]

u1−v2

Yang-Baxter relation: braided form ←→ star-triangle relation

R23(u − v , u − v)R12(u, u)R23(v , v) = R12(v , v)R23(u, u)R12(u − v , u − v)


Star-triangle relation

[i∂1]a [z12]

a+b [i∂1]b = [z12]

b [i∂1]a+b [z12]

a,

[i∂2]a [z12]

a+b [i∂2]b = [z12]

b [i∂2]a+b [z12]

a

These identities are equivalent to the star-triangle relation

[i∂z ]a [z]a+b [i∂z ]

b = [z]b [i∂z ]a+b [z]b

A.P. Isaev Nucl.Phys. B662 (2003) 461-475 Multiloop Feynman integrals and

conformal quantum mechanics

equivalent form – integral identity

∫

d2w

1

[z − w ]α [w − x ]β [w − y ]γ=

πa(α,β, γ)

[z − x ]1−γ [z − y ]1−β [y − x ]1−α

α+ β + γ = α+ β + γ = 2


R-operator

factorization of L-operator

L(u1, u2) = i

(u1 + 1 + z∂ −∂

z2∂ + (u1 − u2 + 1) z v2 − z∂

)

= iZ

(u1 −∂0 u2

)

Z−1

Z =

(1 0z 1

)

, u1 = s − 1− iu , u2 = −s − iu .

R-operator interchange u2 ↔ v2 in the product of L−operators

R12 L1(u1, u2)L2(v1, v2) = L1(u1, v2)L2(v1, u2)R12 ,

R12 L1(u1, u2)L2(v1, v2) = L1(u1, v2)L2(v1, u2)R12

explicit expression

[R12 Φ](z1, z2) =

∫

d2w1

[z1 − z2]u2−u1

[w1 − z1]1+u2−v2 [w1 − z2]v2−u1Φ(w1, z2)

The main building block is the operator R12(x) which interchanges the secondparameters u2 = −s − iu and v2 = ix − iu in the product of two L-operators

R12(x) L1(u1, u2) L2(u1, ix − iu) = L1(u1, ix − iu) L2(u1, u2)R12(x)


R-operator

[R12(x)Φ] (z1, z2) =

∫

d2w1

[z1 − z2]1−2s

[w1 − z1]1−s−ix [w1 − z2]1−s+ixΦ(w1, z2)

Let us define operator

R12...N0(x) = R12(x)R23(x) · · ·RN−1 N(x)RN0(x)

Using the defining relation for R12(x) it is possible to prove the followingrelations

• commutation relation with operator AN(u) + z0BN(u)

R12...N0(x) (AN(u) + z0BN(u)) = (AN(u) + z0BN(u)) R12...N0(x)

• mutual commutativity

R12...N0(u)R12...N0(v) = R12...N0(v)R12...N0(u)

• Baxter relations

(AN(u) + z0BN(u)) R12...N0(u, u) = (u + is)NR12...N0(u + i , u)

(AN(u) + z0BN(u)

)R12...N0(u, u) = (u + i s)N

R12...N0(u, u + i)


Commutativity


⇓

R12...N0(x) L1 (u1, u2) · · · LN (u1, u2) L0 (u1, ix − iu) =

= L1 (u1, ix − iu) L2 (u1, u2) · · · LN (u1, u2)L0 (u1, u2) R12...N0(x)

v2 = ix − iu enters the second row of the matrix L1 (u1, ix − iu) only

R12...N0 (x)(

AN(u) + z0BN(u) BN(u))(

s − iu − 1 −∂0

0 ix − iu

)

=

=(

AN(u) + z0BN(u) BN(u))(

s − iu − 1 −∂0

0 −s − iu

)

R12...N0 (x)

as a consequence we obtain the commutation relation

R12...N0 (x) (AN(u) + z0BN(u)) = (AN(u) + z0BN(u)) R12...N0 (x)

The commutativity

R12...N0(u)R12...N0(v) = R12...N0(v)R12...N0(u)

can be proved diagrammatically.


Baxter equation


⇓

Z−11 R12(x) L1(u1, u2)Z2 =

((x + is)R12(x + i) + (u − x)R12(x) −iR12(x)∂1

−(u − x) z12 R12(x) (x − is)R12(x − i) + (u − x)R12(x)

)

Note the important property: for x = u we obtain triangular matrix

Z−11 R12(u)R23(u) · · ·RN0(u) L1(u)L2(u) · · · LN(u)Z0 =

(

(u + is)R12(u + i) −iR12(u)∂1

0 (u − is)R12(u − i)

)

· · ·

(

(u + is)RN0(u + i) . . .

0 . . .

)

As a consequence of this matrix relation we derive Baxter equation

R12...N0(u, u) (AN(u) + z0BN(u)) = (u + is)NR12...N0(u + i , u) .


Diagram technique: main rules

w z

α

= [z − w]−α

Figure: The diagrammatic representation of the propagator.

= π(−1)γ−γa(α, β, γ)α β

= π a(α, β, γ)

α + β − 1

α

βγ

1− α

1− β 1− γ

Figure: The chain and star– triangle relations, α + β + γ = 2.


Diagram technique: cross relation and commutativity

=

α

1− α′

β

1− β ′

α′−α

1− α

α′ β ′

1− β

β−β′

a(α, β)a(α′, β′)

Figure: The cross relation, α + β = α′ + β′.

The kernel of the integral operator R12...N0(x) is given by the followingexpression

R12...N0(z1, . . . , zN , z0|w1, . . . ,wN) =

N−1∏

k=1

[zk − zk+1]−γ [wk − zk ]

−α[wk − zk+1]−β

[zN − z0]−γ [wN − zN ]

−α[wN − z0]−β

whereα = 1− s − ix , β = 1− s + ix , γ = 2s − 1


Diagram technique: cross relation and commutativity

z1

z2

z3

z0

w1

w2

w3

γ

α

β

α1

α2

β1

β2

α1

1− α1

β2 − β1

β1 1− β2

α1 1− α2

β1

1− α2

1− β2γ

γ 1− γ

1− γ

γ

z0

z0

z0

z0

Figure: Diagrammatic proof of commutativity

commutativity

R12...N0(u)R12...N0(v) = R12...N0(v)R12...N0(u)


Q-operators

In two special cases z0 → 0 and z0 →∞ we obtain two operators:

R12...N0(u)z0→0−→ QA(u)

R12...N0(u)z0→∞−→ [z0]

−s−iuQB(u)

with all characteristic properties of Q-operators

QA-operator

[QA(v) ,QA(u)] = [QA(v) ,AN(u)] = 0

AN(u)QA(u, u) = (u + is)NQA(u + i , u)

AN(u)QA(u, u) = (u + i s)NQA(u, u + i)

QB-operator

[QB(v) ,QB(u)] = [QB(v) ,BN(u)] = 0

BN(u)QB(u, u) = (u + is)NQB(u + i , u)

BN(u)QB(u, u) = (u + i s)NQB(u, u + i)


Iterative construction of eigenfunctions

The operatorR12...N(x) = R12(x)R23(x) · · ·RN−1 N(x)

moves v2 = ix − iu from the right hand side of the product of L-operators tothe left hand side

R12...N(x)L1 (u1, u2) · · · LN−1 (u1, u2)LN (u1, ix − iu) =

= L1 (u1, ix − iu) L2 (u1, u2) · · · LN (u1, u2)R12...N(x)

Relation for the first row

iR12...N (x) (AN−1(u) BN−1(u))

(s − iu + zN∂N −∂N

z2N∂N + (s − ix) zN ix − iu − zN∂N

)

=

= (AN(u) BN(u)) R12...N (x)

B-operator

action on the function Ψ(z1 . . . zN−1) which does not depend on zN

BN(u)R12...N (x)Ψ(z1 . . . zN−1) = (u − x)R12...N (x)BN−1(u)Ψ(z1 . . . zN−1)


Iterative construction of eigenfunctionsNext we continue up to the last step

BN(u)R12...N (x1) · · ·R12...k(xk−1) · · ·R12(xN−1)Ψ(z1) =

= (u − x1) · · · (u − xN−1)R12...N (x1) · · ·R12...k(xk−1) · · ·R12(xN−1)B1(u)Ψ(z1)

where R12...k(x) = R12(x)R23(x) · · ·Rk−1,k(x). The eigenfunction of the lastoperator B1(u) is e

ipz1+i pz1

ΨB(x|z) ∼ R12...N (x1) · · ·R12(xN−1) eipz1+i pz1

A-operator

action on the function Ψ(z1 . . . zN−1)z−s+ixN with special dependence on zN

AN(u)R12...N (x)Ψ(z1 . . . zN−1)z−s+ixN =

(u − x)R12...N (x) z−s+ixN AN−1(u)Ψ(z1 . . . zN−1)

Iterative procedure

AN(u)R12...N (x1) z−s+ix1N · · ·R12 (xN−1) z

−s+ixN−1

2 Ψ(z1) =

= (u − x1) · · · (u − xN−1)R12...N (x1) z−s+ix1N · · ·R12 (xN−1) z

−s+ixN−1

2 A1(u)Ψ(z1)

The eigenfunction of the last operator A1(u) = u + i(s + z1∂1) is z−s+ixN1

ΨA(x|z) ∼ R12...N (x1) · · ·R12(xN−1) [zN ]−s+ix1 [zN−1]

−s+ix2 · · · [z1]−s+ixN


Eigenfunctions and Q-operators

ΨB(x|z) = QA (x1) · · ·QA(xN−1) eipz1+i pz1 =

=

N−1∏

k=1

(πa(αk , βk , γ + 1)

)kR12...N (x1) · · ·R12(xN−1) e

ipz1+i pz1

• symmetry with respect any permutations of parameters xk = (xk , xk) dueto commutativity of Q-operators

• Baxter equation for Q-operators → AN(xk) is a shift operator

AN(xk)ΨB(. . . xk , xk . . . |z) = (xk + is)N ΨB(. . . xk + i , xk . . . |z)

AN(xk)ΨB(. . . xk , xk . . . |z) = (xk + i s)N ΨB(. . . xk , xk + i . . . |z)

ΨB(x|z) is eigenfunction of the operator QB(u)

QB(u, u)ΨB(x|z) = qB(u, x)ΨB(x|z) ,

qB(u, x) = πa(1− s − i u)[p]−s−iu

N−1∏

k=1

πa(1− s − iu, s + ixk , iu − ixk ) .

exchange relation

Q(N)B (u)R12...N (x) = πa(1− s − iu, s + ix , iu − ix)R12...N (x) Q

(N−1)B (u)


Eigenfunctions and Q-operators

ΨA(x |z) = QB (x1) · · ·QB(xN) [zN ]−2s · · · [z1]

−2s =

=

N∏

k=1

(πa(αk , βk , γ + 1)

)kR12...N (x1) · · ·R12(xN−1) [zN ]

−s+ix1 · · · [z1]−s+ixN

• symmetry with respect any permutations of parameters xk = (xk , xk) dueto commutativity of Q-operators

• Baxter equation for Q-operators → BN(xk) is a shift operator

BN(xk)ΨB(. . . xk , xk . . . |z) = (xk + is)N ΨB(. . . xk + i , xk . . . |z)

BN(xk)ΨB(. . . xk , xk . . . |z) = (xk + i s)N ΨB(. . . xk , xk + i . . . |z)

QA(u)ΨA(x|z) = q(u, x)ΨA(x|z) ,

q(u, x) =

N∏

k=1

πa(1− s − iu, s + ixk , ixk − iu) .

exchange relation

Q(N)A (u)R12...N (x) [zN ]

−s+ix =

πa(1− s − iu, s + ix , iu − ix)R12...N (x) [zN ]−s+ix

Q(N−1)A (u)


Orthogonalityorthogonality

∫

d2N

z ΨA(x ′|z)ΨA(x|z) = µA−1(x) δN(x − x

′)

∫

d2N

z ΨB(x ′|z)ΨB(x|z) = µB−1(x) δ2(~p − ~p′) δN−1(x − x

′)

Here the delta function δN(x − x ′) is defined as follows:

δN(x − x′) =

1

N!

∑

SN

δ(x1 − x′k1) . . . δ(xN − x

′kN) ,

where summation goes over all permutations of N elements and

δ(xk − x′m) ≡ δnk n′

mδ(νk − ν

′m)

Sklyanin measure

µA(x) = (2π)−Nπ

−N2 ∏

k<j≤N

[xk − xj ]

µB(x) = 2(2π)−Nπ

−N2

[p]N−1∏

k<j≤N−1

[xk − xj ]

∏

[xk − xj ] =∏(

(νk − νj)2 +

1

4(nk − nj)

2)


Completeness

completeness

∫

Dx µ(x)Ψ(x|z)Ψ(x|z ′) =

N∏

k=1

δ2(~zk − ~z

′k)

⇓

∫

DNx∏

k<j

[xk − xj ] ΨA(x|z)ΨA(x|z ′) = 2Nπ

N2+N

N∏

k=1

δ2(~zk − ~z

′k )

∫

d2pDN−1x [p]N−1

∏

k<j

[xk−xj ] ΨB(x|z)ΨB(x|z ′) = 2N−1π

N2+N

N∏

k=1

δ2(~zk−~z

′k)

The symbol DNx stands for

DNx =

N∏

k=1

(∞∑

nk=−∞

∫ ∞

−∞

dνk

)


Open problems

• appropriate proof of the completeness

• generalization of Sklyanin SOV representation to the higher rank SL(N,C)

• SOV for the modular magnetAndrei G. Bytsko, Jorg Teschner Quantization of models with non-compact

quantum group symmetry: Modular XXZ magnet and lattice sinh-Gordon model

J.Phys. A39 (2006) 12927-12981

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