Iterative construction of eigenfunctions of the matrix elements of - - PowerPoint PPT Presentation

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

• f the monodromy matrix
• S. Derkachov

PDMI, St.Petersburg

RAQIS’16 Recent Advances in Quantum Integrable Systems 22-26 August 2016

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

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 on

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

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

• perator 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

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

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 are more simple in comparison with finite-dimensional representations."

• the model describes high-energy behaviour in Yang-Mills theory
• L. N. Lipatov, High-energy asymptotics of multicolor QCD and

two-dimensional conformal field theories, Phys. Lett. B 309 (1993) 394. High-energy asymptotics of multicolor QCD and exactly solvable lattice models,hep-th/9311037.

• L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completely

integrable model, Phys. Lett. B 342 (1995) 311.

• diagonalization of B(u) ↔ Sklyanin SOV representation
• E. 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 magnet Yang-Baxter equation R-operators,Q-operators and Baxter equation Construction of eigenfunctions

SL(2, C) spin magnet

The quantum SL(2, C) spin magnet is a straightforward generalization of the standard 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 acting nontrivially 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 principal series representations of the SL(2, C) group Vk = C2s+1 → Vk = L2(C)

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

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 − ¯ b¯ z−2¯

s φ

• −c + az

d − bz , −¯ c + ¯ a¯ z ¯ d − ¯ b¯ z

• 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 + 2¯

s ¯ z commutation relations: [S+, S−] = 2S , [S, S±] = ±S± , [¯ S+, ¯ S−] = 2¯ S , [¯ S, ¯ S±] = ±¯ S± conjugation: S†

± = −¯

S± , S† = −¯ S

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

Intertwining operator

There exists an operator W which intertwines a pair of principal series representations 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)2−2¯

s

• The operator W is well-defined at generic s, ¯

s and the problems emerge for the discrete set of points 2s = −n , 2¯ s = −¯ n at n, ¯ n ∈ Z≥0

• At these special values of spins an (n + 1)(¯

n + 1)-dimensional representation decouples from the general infinite-dimensional case

• T (− n

2 ,− ¯ n 2 )(g) Φ

• (z, ¯

z) = (d − bz)n ¯ d − ¯ b¯ z¯

n Φ

• −c + az

d − bz , −¯ c + ¯ a¯ z ¯ d − ¯ b¯ z

• 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.

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

Normalized intertwining operator

[z]α ≡ zα ¯ z ¯

α – single valued function in the complex plane provided that

α − ¯ α ∈ Z Fourier transformation I

• d2z ei(pz+¯

p¯ z)

[z]α = π iα− ¯

α a(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¯

p¯ x

x 1+α¯ x 1+ ¯

α = pα¯

p ¯

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−2¯

s

πa(2 − 2s)

• d2x [z − x]2s−2 Φ(x, ¯

x) W (s) = [i∂z]1−2s

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

SL(2, C) spin magnet

The Hilbert space of the SL(2, C) spin magnet is given by the tensor product of the 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) and ¯ TN(¯ 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) =

• AN(u)

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

• ,

¯ T(¯ u) =

¯

AN(¯ u) ¯ BN(¯ u) ¯ CN(¯ u) ¯ DN(¯ u)

• The operators AN(u) , BN(u) are differential operators of N−th order in the

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

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

Eigenfunctions of AN(u) and ¯ AN(¯ u)

AN(u) , ¯ AN(¯ u) are polynomials of degree N in u and ¯ u correspondingly AN(u) = uN + iuN−1 S +

N

• k=2

uN−kak , S =

N

• k=1

S(k) ¯ AN(¯ u) = ¯ uN + i¯ uN−1 ¯ S +

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

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

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 polynomial eigenvalues 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

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

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 ¯ z ¯

α – single valued function in the complex plane provided that

α− ¯ 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∂) zix−s¯

zi¯

x−¯ s = (u − x) zix−s¯

zi¯

x−¯ s

¯ A1(¯ u) zix−s¯ zi¯

x−¯ s = i

¯ s − i¯ u + ¯ z ¯ ∂ zix−s¯ zi¯

x−¯ s = (¯

u − ¯ x) zix−s¯ zi¯

x−¯ s

• rthogonality x1 = − in1

2 + ν1 , ¯

x1 = in1

2 + ν1 , x2 = − in2 2 + ν2 , ¯

x2 = in2

2 + ν2

• d2z ΨA(x1|z)ΨA(x2|z) = 2π2δ2(x1 − x2) , δ2(x1 − x2) = δn1n2 δ(ν1 − ν2)

completeness x = − in

2 + ν ,

¯ x = in

2 + ν

• Dx ΨA(x|z1)ΨA(x|z2) = 2π2δ2(z1 − z2) ,
• Dx =
• n∈Z

+∞

−∞

dν

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

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−kbk , 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− operators and the roots xk, ¯ xk, k = 1, . . . , N − 1 x = {x1, . . . , xN−1, xN = (p, ¯ p)}

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

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¯

p¯ z

B1 eipz+i¯

p¯ z = −i∂ eipz+i¯ p¯ z = p eipz+i¯ p¯ z

¯ B1 eipz+i¯

p¯ z = −i ¯

∂ eipz+i¯

p¯ z = ¯

p eipz+i¯

p¯ z

• rthogonality
• d2z ΨB(p1|z)ΨB(p2|z) = π2δ2(p1 − p2)

completeness

• d2p ΨB(p|z1)ΨB(p|z2) = π2δ2(z1 − z2)

everything is based on well known formula

+∞

−∞

dp eipx = 2πδ(x)

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

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 = − 1

2 ,¯

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 = − 1

2: invariant subspace with basis {1, ¯

z3} R12(u − v, ¯ u − ¯ v) ¯ L1(¯ u) ¯ L2(¯ v) = ¯ L2(¯ v) ¯ L1(¯ u) R12(u − v, ¯ u − ¯ v)

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

L-operator

L(u) = i

• z∂ + s − iu

−∂ z2∂ + 2s z −z∂ − s − iu

• =

= i

• 1

z 1 s − iu − 1 −∂ −s − iu 1 −z 1

• restrictions
• s3 = − 1

2 ,¯

s3 = 0 R13(u, ¯ u) → L1(u) = L1(u1, u2) =

• 1

z1 1 u1 −∂1 u2 1 −z1 1

• R23(v, ¯

v) → L2(v) = L2(v1, v2) =

• 1

z2 1 v1 −∂2 v2 1 −z2 1

• s3 = 0 ,¯

s3 = − 1

2

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

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

RLL-relations

RLL-relation in a matrix form R12

• z1∂1 + s1 − iu

−∂1 z2

1∂1 + 2s1z1

−z1∂1 − s1 − iu z2∂2 + s2 − iv −∂2 z2

2∂2 + 2s2z2

−z2∂2 − s2 − iv

• =
• z2∂2 + s2 − iv

−∂2 z2

2 ∂2 + 2s2z2

−z2∂2 − s2 − iv z1∂1 + s1 − iu −∂1 z2

1 ∂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)

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

R-operator

generators 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)

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

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

z1 u2 u1 −∂z1 1 1 z21 1 1 −∂z2 v2

• v1

−z2 1

• =

=

• 1

z1 u2 v2 −∂z1 1 1 z21 1 1 −∂z2 u1 v1 −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

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

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)

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

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

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

R-operator

factorization of L-operator L(u1, u2) = i

• u1 + 1 + z∂

−∂ z2∂ + (u1 − u2 + 1) z v2 − z∂

• = iZ
• u1

−∂ u2

• Z −1

Z =

• 1

z 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−u1 Φ(w1, z2) The main building block is the operator R12(x) which interchanges the second parameters 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)

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

R-operator

[R12(x)Φ] (z1, z2) =

• d2w1

[z1 − z2]1−2s [w1 − z1]1−s−ix[w1 − z2]1−s+ix Φ(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 following relations

• 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)N R12...N0(u + i, ¯ u)

¯

AN(¯ u) + ¯ z0 ¯ BN(¯ u) R12...N0(u, ¯ u) = (¯ u + i¯ s)N R12...N0(u, ¯ u + i)

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

Commutativity

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

• =

= AN(u) + z0BN(u) BN(u) s − iu − 1 −∂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.

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

Baxter equation

R12(x) L1(u1, u2) L2(u1, ix − iu) = L1(u1, ix − iu) L2(u1, u2) R12(x) ⇓ Z −1

1

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 −1

1

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

• (u + is) R12(u + i)

−iR12(u)∂1 (u − is) R12(u − i)

• · · ·
• (u + is) RN0(u + i)

. . . . . .

• As a consequence of this matrix relation we derive Baxter equation

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

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

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 β γ 1 − α 1 − β 1 − γ Figure: The chain and star– triangle relations, α + β + γ = 2.

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

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 following expression 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 where α = 1 − s − ix , β = 1 − s + ix , γ = 2s − 1

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

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 − β2 γ 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)

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

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−iu QB(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)N QA(u + i, ¯ u) ¯ AN(¯ u) QA(u, ¯ u) = (¯ u + i¯ s)N QA(u, ¯ u + i) QB-operator [QB(v) , QB(u)] = [QB(v) , BN(u)] = 0 BN(u) QB(u, ¯ u) = (u + is)N QB(u + i, ¯ u) ¯ BN(¯ u) QB(u, ¯ u) = (¯ u + i¯ s)N QB(u, ¯ u + i)

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

Iterative construction of eigenfunctions

The operator R12...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 to the 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 z2

N∂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)

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

Iterative construction of eigenfunctions

Next 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 last

• perator B1(u) is eipz1+i¯

p¯ z1

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

p¯ z1

A-operator action on the function Ψ(z1 . . . zN−1)z−s+ix

N

with special dependence on zN AN(u) R12...N (x) Ψ(z1 . . . zN−1)z−s+ix

N

= (u − x) R12...N (x) z−s+ix

N

AN−1(u)Ψ(z1 . . . zN−1) Iterative procedure AN(u) R12...N (x1) z−s+ix1

N

· · · R12 (xN−1) z

−s+ixN−1 2

Ψ(z1) = = (u − x1) · · · (u − xN−1) R12...N (x1) z−s+ix1

N

· · · 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+ixN

1

ΨA(x|z) ∼ R12...N (x1) · · · R12(xN−1) [zN]−s+ix1[zN−1]−s+ix2 · · · [z1]−s+ixN

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

Eigenfunctions and Q-operators

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

p¯ z1 =

=

N−1

• k=1
• πa(αk, ¯

βk, γ + 1)k R12...N (x1) · · · R12(xN−1) eipz1+i¯

p¯ z1

• symmetry with respect any permutations of parameters xk = (xk, ¯

xk) due to 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)

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

Eigenfunctions and Q-operators

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

N

• k=1
• πa(αk, ¯

βk, γ + 1)k R12...N (x1) · · · R12(xN−1) [zN]−s+ix1 · · · [z1]−s+ixN

• symmetry with respect any permutations of parameters xk = (xk, ¯

xk) due to 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)

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

Orthogonality

• rthogonality
• d2Nz ΨA(x′|z)ΨA(x|z) = µA

−1(x) δN(x − x′)

• d2Nz 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) ≡ δnkn′

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

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

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′) = 2NπN2+N

N

• k=1

δ2( zk − z′

k)

• d2p DN−1x [p]N−1

k<j

[xk−xj] ΨB(x|z) ΨB(x|z′) = 2N−1πN2+N

N

• k=1

δ2( zk− z′

k)

The symbol DNx stands for DNx =

N

• k=1
• ∞
• nk=−∞

∞

−∞

dνk

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

Open problems

• appropriate proof of the completeness
• generalization of Sklyanin SOV representation to the higher rank SL(N, C)
• SOV for the modular magnet

Andrei 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