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. Avan 1 Work in collaboration with Luc Frappat and ´ Eric Ragoucy 2 1 LPTM 2 LAPTH Cergy-Pontoise 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 : − u + 1 − ψ ¯ �� L ( x , u ) = i ψ � + 2 i ( ψπ − ¯ σ z ψ ¯ π ) 4 u ∂ x ψ − i � � � � 1 − ψ ¯ 1 − ψ ¯ σ + + 2 i 2 ψ ¯ π + ψ ψ 1 − ψ ¯ � u ψ � ∂ x ¯ ψ � + i � � � 1 − ψ ¯ 1 − ψ ¯ ψ ¯ σ − − 2 i 2 ψ π + ψ � 1 − ψ ¯ u ψ (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 − u − 1 − ψ ¯ + i ψ∂ x ¯ ψ − ¯ �� � M ( x , u ) = i ψ ψ∂ x ψ � σ z . 1 − ψ ¯ 4 u ψ ∂ x ψ + i � � � � 1 − ψ ¯ 1 − ψ ¯ σ + + 2 i 2 ψ ¯ π + ψ ψ 1 − ψ ¯ � u ψ � ∂ x ¯ ψ � − i � � � 1 − ψ ¯ 1 − ψ ¯ ψ ¯ σ − − 2 i 2 ψ π + ψ � 1 − ψ ¯ u ψ (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 , u 1 ) ⊗ L ( y , u 2 ) � � � = ∂ x a ( x , u 1 , u 2 ) + a ( x , u 1 , u 2 ) − s ( x , u 1 , u 2 ) , L ( x , u 1 ) ⊗ 1 � �� + a ( x , u 1 , u 2 ) + s ( x , u 1 , u 2 ) , 1 ⊗ L ( x , u 2 ) δ ( x − y ) + 1 � � s ( x , u 1 , u 2 ) + s ( y , u 1 , u 2 ) ( ∂ x − ∂ y ) δ ( x − y ) . (1.3) 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 Complex Sine Gordon 3b: r-matrix where: a ( x , u 1 , u 2 ) = − 1 2 P u 1 + u 2 u 1 − u 2 1 ( ψσ + + ¯ ψσ − ) ⊗ σ z − σ z ⊗ ( ψσ + + ¯ ψσ − ) � � + � 1 − ψ ¯ 8 ψ 1 ( ψσ + + ¯ ψσ − ) ⊗ σ z + σ z ⊗ ( ψσ + + ¯ ψσ − ) � s ( x , u 1 , u 2 ) = � 1 − ψ ¯ 8 ψ (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 � � � � r 12 ( x , u 1 , u 2 ) , r 13 ( x , u 1 , u 3 ) + r 12 ( x , u 1 , u 2 ) , r 23 ( x , u 2 , u 3 ) � � + r 32 ( x , u 3 , u 2 ) , r 13 ( x , u 1 , u 3 ) + K 123 ( x , u 1 , u 2 , u 3 ) − K 132 ( x , u 1 , u 3 , u 2 ) = 0 . (2.1) where: � � r ij ( x , u i , u j ) , L k ( y , u k ) = K ijk ( x , u i , u j , u k ) 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: � ψ ∂ ψ − ψ ∂ � ψ ∂ ψ ∂ � � J z = 2 , J + = 2 ∂ψ , J − = − 2 ¯ 1 − ψ ¯ 1 − ψ ¯ ψ . ∂ ¯ ∂ ¯ ∂ψ (2.3) They realize an sl (2) algebra: J z , J ± � = ± 2 J ± , J + , J − � = J z . � � (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 K 123 ( x , u 1 , u 2 , u 3 ) is given by ab J a r 12 ( x , u 1 , u 2 ) ⊗ σ b , � K − 1 K 123 ( x , u 1 , u 2 , u 3 ) = − 2 (2.5) a , b = z , ± where K ab 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): � � � � � � a 12 , a 13 + a 12 , a 23 + a 13 , a 23 + 1 K ( a ) 123 − K ( a ) 132 + K ( a ) � � = − Ω 123 , (3.1) 231 2 � � � � � � a 12 , s 13 + a 12 , s 23 + s 13 , s 23 + 1 − K ( a ) 123 − K ( s ) 132 + K ( s ) � � = − Ω 123 , (3.2) 231 2 Ω 123 = 1 ǫ ( τ ) σ τ ( z ) ⊗ σ τ (+) ⊗ σ τ ( − ) . � (3.3) 8 τ ∈ S 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 , x 0 = � 1 − ψ ¯ ψ and x − = ψ . Since x ± are complex conjugate, it indicates that the correct deformation parameter manifold is the sphere x + x − + ( x 0 ) 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 + , x 0 , x − ) of the element in the dual Lie algebra su (2) ∗ parametrizing the deformation, and the three Pauli matrices: a ( u 1 , u 2 ) = − 1 2 P u 1 + u 2 u 1 − u 2 1 ( x + σ − + x − σ + + x 0 σ z ) ⊗ σ z − σ z ⊗ ( x + σ − + x − σ + + x 0 σ z ) � � + , 8 x 0 (3.5) s ( u 1 , u 2 ) = − 1 2 P + 1 41 ⊗ 1 1 ( x + σ − + x − σ + + x 0 σ z ) ⊗ σ z + σ z ⊗ ( x + σ − + x − σ + + x 0 σ z ) � � + . 8 x 0 (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
Recommend
More recommend
