Affine Gaudin models Sylvain Lacroix Laboratoire de Physique, ENS - PowerPoint PPT Presentation

Affine Gaudin models Sylvain Lacroix Laboratoire de Physique, ENS de Lyon RAQIS’18, Annecy September 10th, 2018 [SL, Magro, Vicedo, 1703.01951] [SL, Vicedo, Young, 1804.01480, 1804.06751]

Introduction Finite Gaudin models: quantum integrable spin chains [Gaudin ’76 ’83] Associated with finite dimensional semi-simple algebras Large number of conserved commuting charges → integrability Spectrum described by Bethe ansatz Affine Gaudin models: associated with affine Kac-Moody algebras (infinite dimensional) Classical level: integrable two-dimensional field theories (integrable σ -models, ...) [Vicedo ’17] infinite number of local charges in involution [SL Magro Vicedo ’17] Quantum commuting charges ? Spectrum ? Sylvain Lacroix Affine Gaudin models RAQIS’18 2 / 20

Introduction Finite Gaudin models: quantum integrable spin chains [Gaudin ’76 ’83] Associated with finite dimensional semi-simple algebras Large number of conserved commuting charges → integrability Spectrum described by Bethe ansatz Affine Gaudin models: associated with affine Kac-Moody algebras (infinite dimensional) Classical level: integrable two-dimensional field theories (integrable σ -models, ...) [Vicedo ’17] infinite number of local charges in involution [SL Magro Vicedo ’17] Quantum commuting charges ? Spectrum ? Sylvain Lacroix Affine Gaudin models RAQIS’18 2 / 20

Contents Gaudin models 1 Affine Gaudin models 2 Conclusion and perspectives 3 Sylvain Lacroix Affine Gaudin models RAQIS’18 3 / 20

Gaudin models Sylvain Lacroix Affine Gaudin models RAQIS’18 4 / 20

Symmetrisable Kac-Moody algebras Gaudin models: associated with symmetrisable Kac-Moody algebras Kac-Moody algebra g � I a , I b � = f ab ab c I c Lie bracket: Finite type: semi-simple finite dimensional Lie algebras Affine and indefinite type: infinite dimensional Symmetrisable Cartan matrix → non-degenerate ad-invariant bilinear form κ on g Quadratic Casimir Ω: appropriate ordering of κ ab I a I b in U ( g ) Sylvain Lacroix Affine Gaudin models RAQIS’18 5 / 20

Symmetrisable Kac-Moody algebras Gaudin models: associated with symmetrisable Kac-Moody algebras Kac-Moody algebra g � I a , I b � = f ab ab c I c Lie bracket: Finite type: semi-simple finite dimensional Lie algebras Affine and indefinite type: infinite dimensional Symmetrisable Cartan matrix → non-degenerate ad-invariant bilinear form κ on g Quadratic Casimir Ω: appropriate ordering of κ ab I a I b in U ( g ) Sylvain Lacroix Affine Gaudin models RAQIS’18 5 / 20

Gaudin models Gaudin model: quantum integrable model associated with g Hilbert space: H = V 1 ⊗ · · · ⊗ V N , with V r ’s representations of g Algebra of observables: A = U ( g ) ⊗ N , generated by I a ( r ) ’s � � I a ( r ) , I b A = δ rs f ab ab c I c ( s ) ( r ) Commuting quadratic Hamiltonians: κ ab I a ( r ) I b � � � � ( s ) � H r , � H r = , H s A = 0 λ r − λ s s � = r λ r ∈ C position of the site V r Sylvain Lacroix Affine Gaudin models RAQIS’18 6 / 20

Lax matrix and integrable structure Lax matrix: κ ab I a ⊗ I b N � ( r ) � L ( λ ) = ∈ g ⊗ A λ − λ r r =1 Linear Sklyanin bracket: C 12 = κ ab I a I b � C 12 � � � L 1 ( λ ) , � � µ − λ, � L 1 ( λ ) + � L 2 ( µ ) A = L 2 ( µ ) Spectral dependent quadratic Hamiltonian: � � � H ( λ ) = appropriate ordering of 1 � L ( λ ) , � L ( λ ) 2 κ � � � H ( λ ) , � Ad-invariance of κ → H ( µ ) A = 0 for all λ, µ ∈ C Partial fraction decomposition N � � Ω ( r ) 1 H r � H ( λ ) = ( λ − λ r ) 2 + 2 λ − λ r r =1 Sylvain Lacroix Affine Gaudin models RAQIS’18 7 / 20

Lax matrix and integrable structure Lax matrix: κ ab I a ⊗ I b N � ( r ) � L ( λ ) = ∈ g ⊗ A λ − λ r r =1 Linear Sklyanin bracket: C 12 = κ ab I a I b � C 12 � � � L 1 ( λ ) , � � µ − λ, � L 1 ( λ ) + � L 2 ( µ ) A = L 2 ( µ ) Spectral dependent quadratic Hamiltonian: � � � H ( λ ) = appropriate ordering of 1 � L ( λ ) , � L ( λ ) 2 κ � � � H ( λ ) , � Ad-invariance of κ → H ( µ ) A = 0 for all λ, µ ∈ C Partial fraction decomposition N � � Ω ( r ) 1 H r � H ( λ ) = ( λ − λ r ) 2 + 2 λ − λ r r =1 Sylvain Lacroix Affine Gaudin models RAQIS’18 7 / 20

Bethe ansatz � � � H ( λ ) , � H ( µ ) A = 0 for all λ, µ ∈ C → eigenvectors basis of � H ( λ ) ? eigenvalues ? Bethe ansatz [Schechtman Varchenko ’91] Sylvain Lacroix Affine Gaudin models RAQIS’18 8 / 20

Additional commuting charges Finite Gaudin models ( g finite algebra) Ψ ad-invariant polynomial on g : � � � � Q Ψ ( λ ) = Ψ L ( λ ) + quantum corrections Sklyanin bracket + ad-invariance of Ψ , Ξ: � � Q Ψ ( λ ) , � � Q Ξ ( µ ) A = 0 , ∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models ( g affine algebra) κ invariant quadratic polynomial → � H ( λ ) No higher degree invariant polynomials → additional commuting charges ? Sylvain Lacroix Affine Gaudin models RAQIS’18 9 / 20

Additional commuting charges Finite Gaudin models ( g finite algebra) Ψ ad-invariant polynomial on g : � � � � Q Ψ ( λ ) = Ψ L ( λ ) + quantum corrections Sklyanin bracket + ad-invariance of Ψ , Ξ: � � Q Ψ ( λ ) , � � Q Ξ ( µ ) A = 0 , ∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models ( g affine algebra) κ invariant quadratic polynomial → � H ( λ ) No higher degree invariant polynomials → additional commuting charges ? Sylvain Lacroix Affine Gaudin models RAQIS’18 9 / 20

Additional commuting charges Finite Gaudin models ( g finite algebra) Ψ ad-invariant polynomial on g : � � � � Q Ψ ( λ ) = Ψ L ( λ ) + quantum corrections Sklyanin bracket + ad-invariance of Ψ , Ξ: � � Q Ψ ( λ ) , � � Q Ξ ( µ ) A = 0 , ∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models ( g affine algebra) κ invariant quadratic polynomial → � H ( λ ) No higher degree invariant polynomials [Chari Ilangovan ’84] → additional commuting charges ? Sylvain Lacroix Affine Gaudin models RAQIS’18 9 / 20

Affine Gaudin models Sylvain Lacroix Affine Gaudin models RAQIS’18 10 / 20

Affine Gaudin models g affine Kac-Moody algebra (infinite dimensional) g [ t , t − 1 ] ⊕ C D ⊕ C K g = ˚ Lax matrix: A n ( λ ) ∈ ˚ g � � A n ( λ ) t n L ( λ ) = + i ϕ ( λ ) D + D ( λ ) K n ∈ Z � �� � � g [ t , t − 1 ] L ( λ ) ∈ ˚ Coordinate on the circle : t �→ e ix , x ∈ [0 , 2 π [ � � A n ( λ ) e inx ˚ L ( λ ) �− → g -valued field n ∈ Z → field theory on the circle [Vicedo ’17] Twist function ϕ ( λ ): rational function characteristic of the model Sylvain Lacroix Affine Gaudin models RAQIS’18 11 / 20

Classical hierarchy for Affine Gaudin models Sklyanin bracket: � C 12 � � � L 1 ( λ ) , � � µ − λ, � L 1 ( λ ) + � L 2 ( µ ) A = L 2 ( µ ) [SL Magro Vicedo 1703.01951] Appropriate choice of polynomials Φ n on g of degree n : � � S n ( λ ) = Φ n L ( λ ) Poisson bracket � � S n ( λ ) , S m ( µ ) A = Zeros ζ i ’s of the twist function: ϕ ( ζ i ) = 0 � � Q n , i , Q m , i A = 0 → classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS’18 12 / 20

Classical hierarchy for Affine Gaudin models Sklyanin bracket: � C 1 � � � L 1 ( λ ) , � � L 2 ( µ ) A depends on ϕ ( λ ) and ϕ ( µ ) µ [SL Magro Vicedo 1703.01951] Appropriate choice of polynomials Φ n on g of degree n : � � S n ( λ ) = Φ n L ( λ ) Poisson bracket � � S n ( λ ) , S m ( µ ) A = Zeros ζ i ’s of the twist function: ϕ ( ζ i ) = 0 � � Q n , i , Q m , i A = 0 → classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS’18 12 / 20

Classical hierarchy for Affine Gaudin models Sklyanin bracket: � C 1 � � � L 1 ( λ ) , L 2 ( µ ) A depends on ϕ ( λ ) and ϕ ( µ ) µ [SL Magro Vicedo 1703.01951] Appropriate choice of polynomials Φ n on g of degree n : � � S n ( λ ) = Φ n L ( λ ) Poisson bracket � � S n ( λ ) , S m ( µ ) A = Zeros ζ i ’s of the twist function: ϕ ( ζ i ) = 0 � � Q n , i , Q m , i A = 0 → classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS’18 12 / 20

Classical hierarchy for Affine Gaudin models Sklyanin bracket: � C 1 � � � L 1 ( λ ) , L 2 ( µ ) A depends on ϕ ( λ ) and ϕ ( µ ) µ [SL Magro Vicedo 1703.01951] Appropriate choice of polynomials Ψ n on g of degree n : � � S n ( λ ) = Ψ n L ( λ ) Poisson bracket � � S n ( λ ) , S m ( µ ) A = non-zero as Ψ n ’s non-invariant Zeros ζ i ’s of the twist function: ϕ ( ζ i ) = 0 � � Q n , i , Q m , i A = 0 → classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS’18 12 / 20