Affine Gaudin models

Transcription

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

2 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

3 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

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

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

6 Symmetrisable Kac-Moody algebras Gaudin models: associated with symmetrisable Kac-Moody algebras Kac-Moody algebra g Lie bracket: [ I a, I b] = fab ab c I c 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

7 Symmetrisable Kac-Moody algebras Gaudin models: associated with symmetrisable Kac-Moody algebras Kac-Moody algebra g Lie bracket: [ I a, I b] = fab ab c I c 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

8 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 (s)]a = δ rs f ab ab c I c (r) Commuting quadratic Hamiltonians: Ĥ r = s r κ ab I a (r) I b (s) λ r λ s, [Ĥr, Ĥs] A = 0 λ r C position of the site V r Sylvain Lacroix Affine Gaudin models RAQIS 18 6 / 20

9 Lax matrix and integrable structure Lax matrix: L(λ) = N κ ab I a I(r) b r=1 λ λ r g A Linear Sklyanin bracket: C 12 = κ ab I a I b [ [ ] L1 (λ), L 2 (µ)] = C12 A µ λ, L 1 (λ) + L 2 (µ) Spectral dependent quadratic Hamiltonian: Ĥ(λ) = appropriate ordering of 1 2 κ( ) L(λ), L(λ) Ad-invariance of κ [ Ĥ(λ), Ĥ(µ)] = 0 for all λ, µ C A Partial fraction decomposition N 1 Ĥ(λ) = 2 r=1 Ω (r) (λ λ r ) 2 + Ĥr λ λ r Sylvain Lacroix Affine Gaudin models RAQIS 18 7 / 20

10 Lax matrix and integrable structure Lax matrix: L(λ) = N κ ab I a I(r) b r=1 λ λ r g A Linear Sklyanin bracket: C 12 = κ ab I a I b [ [ ] L1 (λ), L 2 (µ)] = C12 A µ λ, L 1 (λ) + L 2 (µ) Spectral dependent quadratic Hamiltonian: Ĥ(λ) = appropriate ordering of 1 2 κ( ) L(λ), L(λ) Ad-invariance of κ [ Ĥ(λ), Ĥ(µ)] = 0 for all λ, µ C A Partial fraction decomposition N 1 Ĥ(λ) = 2 r=1 Ω (r) (λ λ r ) 2 + Ĥr λ λ r Sylvain Lacroix Affine Gaudin models RAQIS 18 7 / 20

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

12 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: [ QΨ (λ), Q Ξ (µ)] A = 0, Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models (g affine algebra) κ invariant quadratic polynomial Ĥ(λ) No higher degree invariant polynomials additional commuting charges? λ, µ C Sylvain Lacroix Affine Gaudin models RAQIS 18 9 / 20

13 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: [ QΨ (λ), Q Ξ (µ)] A = 0, Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models (g affine algebra) κ invariant quadratic polynomial Ĥ(λ) No higher degree invariant polynomials additional commuting charges? λ, µ C Sylvain Lacroix Affine Gaudin models RAQIS 18 9 / 20

14 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: [ QΨ (λ), Q Ξ (µ)] A = 0, Quadratic Hamiltonians with Ψ = 1 2 κ Large number of conserved commuting charges Affine Gaudin models (g affine algebra) κ invariant quadratic polynomial Ĥ(λ) λ, µ C No higher degree invariant polynomials [Chari Ilangovan 84] additional commuting charges? Sylvain Lacroix Affine Gaudin models RAQIS 18 9 / 20

15 Affine Gaudin models Sylvain Lacroix Affine Gaudin models RAQIS / 20

16 Affine Gaudin models g affine Kac-Moody algebra (infinite dimensional) Lax matrix: A n (λ) g g = g[t, t 1 ] CD CK L(λ) = A n (λ)t n + i ϕ(λ) D + D(λ) K n Z }{{} L(λ) g[t,t 1 ] Coordinate on the circle : t e ix, x [0, 2π[ L(λ) n Z A n (λ)e inx g-valued field field theory on the circle [Vicedo 17] Twist function ϕ(λ): rational function characteristic of the model Sylvain Lacroix Affine Gaudin models RAQIS / 20

17 Classical hierarchy for Affine Gaudin models Sklyanin bracket: [ L1 (λ), L 2 (µ) ] A = [ ] C12 µ λ, L 1 (λ) + L 2 (µ) [SL Magro Vicedo ] Appropriate choice of polynomials Φ n on g of degree n: S n (λ) = Φ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { Qn,i, Q m,i } A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

18 Classical hierarchy for Affine Gaudin models Sklyanin bracket: [ L1 (λ), L 2 (µ) ] A depends on ϕ(λ) and ϕ(µ) [ C1 µ ] [SL Magro Vicedo ] Appropriate choice of polynomials Φ n on g of degree n: S n (λ) = Φ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { Qn,i, Q m,i } A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

19 Classical hierarchy for Affine Gaudin models Sklyanin bracket: { L1 (λ), L 2 (µ) } [ ] A depends on ϕ(λ) and ϕ(µ) C1 µ [SL Magro Vicedo ] Appropriate choice of polynomials Φ n on g of degree n: S n (λ) = Φ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { Qn,i, Q m,i } A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

20 Classical hierarchy for Affine Gaudin models Sklyanin bracket: { L1 (λ), L 2 (µ) } [ ] A depends on ϕ(λ) and ϕ(µ) C1 [SL Magro Vicedo ] Appropriate choice of polynomials Ψ n on g of degree n: S n (λ) = Ψ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = non-zero as Ψ n s non-invariant µ Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { Qn,i, Q m,i } A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

21 Classical hierarchy for Affine Gaudin models Sklyanin bracket: { L1 (λ), L 2 (µ) } [ ] A depends on ϕ(λ) and ϕ(µ) C1 [SL Magro Vicedo ] Appropriate choice of polynomials Ψ n on g of degree n: S n (λ) = Ψ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = ϕ(λ)( ) + ϕ(µ) ( ) µ Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { Qn,i, Q m,i } A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

22 Classical hierarchy for Affine Gaudin models Sklyanin bracket: { L1 (λ), L 2 (µ) } [ ] A depends on ϕ(λ) and ϕ(µ) C1 [SL Magro Vicedo ] Appropriate choice of polynomials Ψ n on g of degree n: S n (λ) = Ψ n ( L(λ) ) Poisson bracket { Sn (λ), S m (µ) } A = ϕ(λ)( ) + ϕ(µ) ( ) µ Zeros ζ i s of the twist function: ϕ(ζ i ) = 0 { } Q n,i = S n (ζ i ), Qn,i, Q m,j A = 0 classical hierarchy of conserved charges in involution Sylvain Lacroix Affine Gaudin models RAQIS / 20

23 Quantum hierarchy for affine Gaudin models? Classical hierarchy { Q n,i, Q m,j }A = 0 Quantum hierarchy [ Qn,i, Q m,j ]A = 0? Naive guess ( ) Ŝ n (λ) = Ψ n L(λ) + quantum corrections and Qn,i = Ŝn(ζ i ) does not work [SL Vicedo Young ] Conjecture 1: Let P(λ) be such that λ log P(λ) = ϕ(λ). Then Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ γ i for some closed contour γ i (h dual Coxeter number) Conjecture originates from the study of affine opers Sylvain Lacroix Affine Gaudin models RAQIS / 20

24 Quantum hierarchy for affine Gaudin models? Classical hierarchy { Q n,i, Q m,j }A = 0 Quantum hierarchy [ Qn,i, Q m,j ]A = 0? Naive guess ( ) Ŝ n (λ) = Ψ n L(λ) + quantum corrections and Qn,i = Ŝn(ζ i ) does not work [SL Vicedo Young ] Conjecture 1: Let P(λ) be such that λ log P(λ) = ϕ(λ). Then Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ γ i for some closed contour γ i (h dual Coxeter number) Conjecture originates from the study of affine opers Sylvain Lacroix Affine Gaudin models RAQIS / 20

25 Hypergeometric integrals on Pochhammer contour Twist function with simple poles: N k r N ϕ(λ) =, P(λ) = (λ λ r ) kr λ λ r r=1 r=1 Hypergeometric integrals: Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ γ i P(λ) multi-valued contour γ i on which P(λ) is single-valued Typical examples: Pochhammer contours γ λ r λ s Sylvain Lacroix Affine Gaudin models RAQIS / 20

26 Classical limit of the quantum hierarchy Reintroduce : k r kr Q n,i = P(λ) n 1 h Ŝ n (λ) dλ γ i Classical limit 0: saddle point approximation localisation at extrema of P(λ) localisation at zeros ζ i of ϕ(λ) Coherent with the construction of the classical hierarchy Counting: # independent Pochhammer contours = # zeros of ϕ Sylvain Lacroix Affine Gaudin models RAQIS / 20

27 Classical limit of the quantum hierarchy Reintroduce : k r kr Q n,i = P(λ) n 1 h Ŝ n (λ) dλ γ i Classical limit 0: saddle point approximation localisation at extrema of P(λ) localisation at zeros ζ i of ϕ(λ) Coherent with the construction of the classical hierarchy Counting: # independent Pochhammer contours = # zeros of ϕ Sylvain Lacroix Affine Gaudin models RAQIS / 20

28 Commutation of the quantum charges Classical case: involution of the Q n,i s from { Sn (λ), S m (µ) } A = ϕ(λ)( ) + ϕ(µ) ( ) Quantum case: how to get commutation of the Q n,i s? Twisted derivatives: n,ϕ λ f (λ) = λf (λ) γ i P(λ) (n 1)/h = A n,ϕ λ (n 1)ϕ(λ) h f (λ) n,ϕ λ f (λ) dλ = 0 Conjecture 2: There exist Ân,m(λ, µ) and B n,m (λ, µ) such that [Ŝn Ŝm(µ)] (λ), Â n,m (λ, µ) + µ m,ϕ B n,m (λ, µ) Commutation of Q n,i s: Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ = γ i [ Qn,i, Q m,j ] A = 0 Sylvain Lacroix Affine Gaudin models RAQIS / 20

29 Commutation of the quantum charges Classical case: involution of the Q n,i s from { Sn (λ), S m (µ) } A = ϕ(λ)( ) + ϕ(µ) ( ) Quantum case: how to get commutation of the Q n,i s? Twisted derivatives: n,ϕ λ f (λ) = λf (λ) γ i P(λ) (n 1)/h = A n,ϕ λ (n 1)ϕ(λ) h f (λ) n,ϕ λ f (λ) dλ = 0 Conjecture 2: There exist Ân,m(λ, µ) and B n,m (λ, µ) such that [Ŝn Ŝm(µ)] (λ), Â n,m (λ, µ) + µ m,ϕ B n,m (λ, µ) Commutation of Q n,i s: Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ = γ i [ Qn,i, Q m,j ] A = 0 Sylvain Lacroix Affine Gaudin models RAQIS / 20

30 Commutation of the quantum charges Classical case: involution of the Q n,i s from { Sn (λ), S m (µ) } A = ϕ(λ)( ) + ϕ(µ) ( ) Quantum case: how to get commutation of the Q n,i s? Twisted derivatives: n,ϕ λ f (λ) = λf (λ) γ i P(λ) (n 1)/h = A n,ϕ λ (n 1)ϕ(λ) h f (λ) n,ϕ λ f (λ) dλ = 0 Conjecture 2: There exist Ân,m(λ, µ) and B n,m (λ, µ) such that [Ŝn Ŝm(µ)] (λ), Â n,m (λ, µ) + µ m,ϕ B n,m (λ, µ) Commutation of Q n,i s: Q n,i = P(λ) (n 1)/h Ŝ n (λ) dλ = γ i [ Qn,i, Q m,j ] A = 0 Sylvain Lacroix Affine Gaudin models RAQIS / 20

31 Cubic charge [SL Vicedo Young ] Untwisted affine algebras of type A: ŝl N for N 3, classic cubic charges Q 3,i s Construction of the cubic operator Ŝ3(λ) and the cubic quantum charges Q 3,i satisfying the conjectures 1 and 2 (vertex algebras techniques) Eigenvalues of Q 3,i s by the Bethe ansatz with zero and one excitation Sylvain Lacroix Affine Gaudin models RAQIS / 20

32 Conclusion and perspectives Sylvain Lacroix Affine Gaudin models RAQIS / 20

33 Conclusion Additional results: [SL Vicedo Young ] relations with affine opers (affine version of the approach started in [Feigin Frenkel Reshetikhin 94] for finite Gaudin models) applications to quantum Boussinesq equation and relations with [Bazhanov Hibberd Khoroshkin 02] First results towards the quantisation of hierarchy in affine Gaudin models Perspectives: achieve the construction of the whole hierarchy and the description of its spectrum various generalisations (cyclotomy, reality conditions,...) application to integrable quantum field theories (KdV, σ-models, Toda field theories,...) link with the ODE/IM correspondence Sylvain Lacroix Affine Gaudin models RAQIS / 20

34 Thank you for your attention! Sylvain Lacroix Affine Gaudin models RAQIS / 20