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Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Algebraic structure of classical integrability for complex sine-Gordon

• J. Avan1

Work in collaboration with Luc Frappat and ´ Eric Ragoucy2

1LPTM

Cergy-Pontoise

2LAPTH

Annecy arXiv 1911.06720, SciPost Phys. 8, 033 (2020)

September 2020

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Complex Sine Gordon 1

Complex Sine Gordon is 1+1 classically integrable field theory, with Lax representation

• ∂x + L(x, u), ∂t + M(x, u)
• = 0 :

L(x, u) = i 4

• − u + 1 − ψ ¯

ψ u + 2i(ψπ − ¯ ψ¯ π)

• σz

+ 2i

• 2
• 1 − ψ ¯

ψ ¯ π + ∂xψ

• 1 − ψ ¯

ψ − i u

• 1 − ψ ¯

ψ ψ

• σ+

−2i

• 2
• 1 − ψ ¯

ψ π + ∂x ¯ ψ

• 1 − ψ ¯

ψ + i u

• 1 − ψ ¯

ψ ¯ ψ

• σ−
• (1.1)
• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Complex Sine Gordon 2

and M(x, u) = i 4

• − u − 1 − ψ ¯

ψ u + i ψ∂x ¯ ψ − ¯ ψ∂xψ 1 − ψ ¯ ψ

• σz
• .

+ 2i

• 2
• 1 − ψ ¯

ψ ¯ π + ∂xψ

• 1 − ψ ¯

ψ + i u

• 1 − ψ ¯

ψ ψ

• σ+

−2i

• 2
• 1 − ψ ¯

ψ π + ∂x ¯ ψ

• 1 − ψ ¯

ψ − i u

• 1 − ψ ¯

ψ ¯ ψ

• σ−
• (1.2)
• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Problem of integrability

existence of r-matrix structure (Maillet ’86) of Lax matrix guarantees classical integrability Quantum integrability has anomalies (Maillet-De Vega ’83) Classical Yang Baxter equation written (Maillet ’86) but lacks full algebraic formulation

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Complex Sine Gordon 3a: r-matrix

r-matrix: Ultralocal Poisson structure endows Lax matrix L with non-ultralocal r-matrix Poisson structure given by

• L(x, u1) ⊗ L(y, u2)
• =
• ∂xa(x, u1, u2) +
• a(x, u1, u2) − s(x, u1, u2), L(x, u1) ⊗ 1
• +
• a(x, u1, u2) + s(x, u1, u2), 1 ⊗ L(x, u2)
• δ(x − y)

+ 1 2

• s(x, u1, u2) + s(y, u1, u2)
• (∂x − ∂y)δ(x − y) .

(1.3)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Complex Sine Gordon 3b: r-matrix

where: a(x, u1, u2) = −1 2P u1 + u2 u1 − u2 + 1 8

• 1 − ψ ¯

ψ

• (ψσ+ + ¯

ψσ−) ⊗ σz − σz ⊗ (ψσ+ + ¯ ψσ−)

• s(x, u1, u2) =

1 8

• 1 − ψ ¯

ψ

• (ψσ+ + ¯

ψσ−) ⊗ σz + σz ⊗ (ψσ+ + ¯ ψσ−) (1.4) Non skew symmetric (terms a, s) Dynamical (contains fields) and Yang Baxter equation for a, s is not of Gervais-Neveu (’84) Felder (’95) type!

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

The Yang Baxter equation

Problem: Find complete algebraic interpretation of classical Yang Baxter equations for r-matrix components a, s; r = a + s

• r12(x, u1, u2), r13(x, u1, u3)
• +
• r12(x, u1, u2), r23(x, u2, u3)
• +
• r32(x, u3, u2), r13(x, u1, u3)
• + K123(x, u1, u2, u3) − K132(x, u1, u3, u2)

= 0. (2.1) where:

• rij(x, ui, uj), Lk(y, uk)
• = Kijk(x, ui, uj, uk) δ(x − y).

(2.2)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Poisson brackets of L: the differential operators

Poisson bracket with L takes explicit algebraic form: Introduce differential operators: Jz = 2

• ¯

ψ ∂ ∂ ¯ ψ − ψ ∂ ∂ψ

• , J+ = 2
• 1 − ψ ¯

ψ ∂ ∂ψ , J− = −2

• 1 − ψ ¯

ψ ∂ ∂ ¯ ψ . (2.3) They realize an sl(2) algebra:

• Jz , J±

= ±2J± ,

• J+ , J−

= Jz . (2.4)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Result 1: PB’s of L as su(2) derivation

The Poisson bracket of the matrix r with the Lax matrix L now takes an algebraic form: Proposition The kernel K123(x, u1, u2, u3) is given by K123(x, u1, u2, u3) = −2

• a,b=z,±

K −1

ab Ja r12(x, u1, u2) ⊗ σb ,

(2.5) where Kab is the Killing form of su(2). Hence non-abelian dynamical algebra (Ping Xu ’02, not GNF)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

The MCYB equations

The matrices a and s satisfy the following modified classical Yang–Baxter equations (MCYBE):

• a12, a13
• +
• a12, a23
• +
• a13, a23
• + 1

2

• K (a)

123 − K (a) 132 + K (a) 231

• = −Ω123 ,

(3.1)

• a12, s13
• +
• a12, s23
• +
• s13, s23
• + 1

2

• − K (a)

123 − K (s) 132 + K (s) 231

• = −Ω123 ,

(3.2) Ω123 = 1 8

• τ∈S3

ǫ(τ) στ(z) ⊗ στ(+) ⊗ στ(−) . (3.3)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

MCYB for a,s

The modified classical Yang–Baxter equation (with or without its added dynamical shift) is a well-known object described in e.g. Semenov-Tjan-Shanskii ’83 (without dynamics) or Etingof-Varchenko ’98, Ping ’02 (with dynamics). The adjoint modified classical Yang–Baxter set however, to the best of our knowledge, has not been identified in a given system or defined a priori before.

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Dynamical manifold

Differential operators realize a linear representation (spin 1 su(2)) when acting on the three functions x+ = ¯ ψ, x0 =

• 1 − ψ ¯

ψ and x− = ψ. Since x± are complex conjugate, it indicates that the correct deformation parameter manifold is the sphere x+x− + (x0)2 = 1. (3.4) The classical Poisson algebra is therefore identified with a non-abelian su(2)∗ dynamical reflection a/s structure, realized on a moduli space of deformations = submanifold (3.4) of the full dual space su(2)∗. Cf Ping ’02. Cf: proposed construction of CSG as a (deformed) WZWN model on SU(2)/U(1) by Bakas ’94.

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Canonical form

Dynamical dependance of a and s is now a canonical product of the projective components (x+, x0, x−) of the element in the dual Lie algebra su(2)∗ parametrizing the deformation, and the three Pauli matrices: a(u1, u2) = −1 2P u1 + u2 u1 − u2 + 1 8x0

• (x+σ− + x−σ+ + x0σz) ⊗ σz − σz ⊗ (x+σ− + x−σ+ + x0σz)
• ,

(3.5) s(u1, u2) = −1 2P + 1 41 ⊗ 1 + 1 8x0

• (x+σ− + x−σ+ + x0σz) ⊗ σz + σz ⊗ (x+σ− + x−σ+ + x0σz)
• .

(3.6)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

General features

Identified structure: pair of matrices a,s; hence reflection algebra Dynamical properties of non abelian type, on submfd of su(2). Hence dynamical RA. Modified classical YB equations: quasi-associator at quantum level?

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Which dynamical reflection algebra ?

Answer: dynamical reflection algebra of the type“twisted dynamical” (JA and Eric Ragoucy, ’12). 3 dynamical extensions known for the general abelian quadratic algebra A12K1B12K2 = K2C12K1D12, parametrized as A12K1(−ǫRh(2))B12K2(+ǫLh(1)) = K2(−ǫRh(1))C12K1(+ǫLh(2))D12 . (4.1) ǫR = −ǫL defines the boundary dynamical algebra (original one, Behrend et al ’96, Fan et al.’97), ǫR = ǫL defines the twisted boundary dynamical algebra (JA, ER ’12), ǫR = 0, ǫL = 1 defines semi-dynamical algebras (JA et al. ’04, Arutyunov et al. ’98). Here: signs suggest non-abelian version of the twisted boundary algebra.

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Non abelian shift?

Non-abelian case: how to formulate exponentiated dynamical shifts? Solved formally by Ping ’02: introducing a star-product ∗ entailing a formal, ordered, Taylor series expansion, both to define R ∗ T ∗ T and R(q“ + ”h) Dynamical deformation = Drinfel’d twist, twisted coboundary equation expressed in terms of the star-product. Manipulate explicitly such objects? coproduct, comodule or trace structures explicitly moreover here non-abelian dynamical quantum REFLECTION algebra

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Modified YB implies quasi-associator ?

Associative Ω123 transforms the classical YBE into a modified classical YBE for a and modified adjoint classical YBE for the matrices a/s Non-equivariance of a Hence an obstruction to associativity in the quantum case. Quasi-associativity? Ω123 = classical limit of the quasi-associator Φ123 in the quantum Yang–Baxter associator equation (Drinfeld ’90): R12 Φ312 R13 Φ−1

132 R23 Φ123 = Φ321 R23 Φ−1 231 R13 Φ213 R12 .

(4.2)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

The quasi-associator

Without dynamical deformation, Φ123 = 1123 + 2 Ω123 + o(2): known (non-dynamical) modified classical YBE for an a matrix A Drinfel’d twist (non-abelian) of the structure then should yield a dynamical deformation with above equation as classical limit. Ω123 obeys coproduct relations for Φ123 at first leading order

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Gauging/quantizing

Complex Sine Gordon = gauged deformed SU(2)/U(1) WZNW Hence here : gauge THEN quantize Delduc et al ’13. Quantize deformed WZNW (with a,s non dynamical pair) THEN gauge to get QCSG. here a, s are dynamized by gauging/factorization procedure (see how dynamical r-matrices arise in Calogero Moser)

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Space-labeled algebras

Classical algebra here= dynamically deformed structure parametrized by the field variables ψ, ¯ ψ identified as coordinates on the target manifold SU(2)/U(1) of the related WZWN model. For each value of the space variable x, a deformation algebra Uψ, ¯

ψ(su(2)) is thus to be defined.

The classical a, s Yang–Baxter structure is ultralocal (contrary to the L-Poisson structure which is not ultralocal, an issue which we will also address in this respect).

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Monodromy along deformation indices ?

Monodromy matrix for the Lax matrix at the classical level (Maillet’86) = evolution along the x-axis. Quantum monodromy matrix = usually iteration of the coproduct (for an RTT structure) or a comodule (for an RKRK structure) of given underlying quantum algebra = a tensor product over the “neighboring” quantum spaces. Here monodromy is iteration over the index x i.e. over the index labeling deformation manifold of the algebra. As far as we know, such a “monodromy matrix” as tensor product of algebras with distinct but isomorphic deformation moduli spaces, has never been defined.

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon

Introduction: The problem of CSG r-matrix structure Algebraic structures of Poisson brackets of L The YB equations for a,s On the

Intertwining objects

Introduction of intertwining structures “between” the different x-labeled deformed algebras Then generate the Lax matrix and the terms ∂xa δ(x − y) and (s + s) (∂x − ∂y)δ(x − y) by a classical limit. Lax matrix may arise as a combination of a K-matrix in a reflection algebra structure at fixed x, ultralocal, yielding the δ(x − y) commutator terms of the Poisson bracket, with matrices A and/or S, yielding the ∂xa δ(x − y) and (s + s) (∂x − ∂y)δ(x − y) terms.

• J. Avan, L.Frappat, ´
• E. Ragoucy

RAQIS 2020 Complex Sine Gordon