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

1

Gaudin models

2

Affine Gaudin models

3

Conclusion and perspectives

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

Lie bracket:

• I a, I b

= f ab

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 κabI aI 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

Lie bracket:

• I a, I b

= f ab

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 κabI aI 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 = V1 ⊗ · · · ⊗ VN, with Vr’s representations of g Algebra of observables: A = U(g)⊗N, generated by I a

(r)’s

• I a

(r), I b (s)

• A = δrs f ab

ab c I c (r)

Commuting quadratic Hamiltonians:

• Hr =
• s=r

κab I a

(r)I b (s)

λr − λs , Hr, Hs

• A = 0

λr ∈ C position of the site Vr

Sylvain Lacroix Affine Gaudin models RAQIS’18 6 / 20

Lax matrix and integrable structure

Lax matrix:

• L(λ) =

N

• r=1

κab I a ⊗ I b

(r)

λ − λr ∈ g ⊗ A Linear Sklyanin bracket: C12 = κabI aI b

• L1(λ),

L2(µ)

• A =

C12 µ − λ, L1(λ) + L2(µ)

• Spectral dependent quadratic Hamiltonian:
• H(λ) = appropriate ordering of 1

2κ L(λ), L(λ)

• Ad-invariance of κ →

H(λ), H(µ)

• A = 0 for all λ, µ ∈ C

Partial fraction decomposition

• H(λ) =

N

• r=1

1 2 Ω(r) (λ − λr)2 +

• Hr

λ − λr

Sylvain Lacroix Affine Gaudin models RAQIS’18 7 / 20

Lax matrix and integrable structure

Lax matrix:

• L(λ) =

N

• r=1

κab I a ⊗ I b

(r)

λ − λr ∈ g ⊗ A Linear Sklyanin bracket: C12 = κabI aI b

• L1(λ),

L2(µ)

• A =

C12 µ − λ, L1(λ) + L2(µ)

• Spectral dependent quadratic Hamiltonian:
• H(λ) = appropriate ordering of 1

2κ L(λ), L(λ)

• Ad-invariance of κ →

H(λ), H(µ)

• A = 0 for all λ, µ ∈ C

Partial fraction decomposition

• H(λ) =

N

• r=1

1 2 Ω(r) (λ − λr)2 +

• Hr

λ − λr

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:

• QΨ(λ) = Ψ

L(λ)

• + quantum corrections

Sklyanin bracket + ad-invariance of Ψ, Ξ:

• QΨ(λ),

QΞ(µ)

• A = 0,

∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1

2κ

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:

• QΨ(λ) = Ψ

L(λ)

• + quantum corrections

Sklyanin bracket + ad-invariance of Ψ, Ξ:

• QΨ(λ),

QΞ(µ)

• A = 0,

∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1

2κ

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:

• QΨ(λ) = Ψ

L(λ)

• + quantum corrections

Sklyanin bracket + ad-invariance of Ψ, Ξ:

• QΨ(λ),

QΞ(µ)

• A = 0,

∀ λ, µ ∈ C Quadratic Hamiltonians with Ψ = 1

2κ

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 = ˚ g[t, t−1] ⊕ CD ⊕ CK Lax matrix: An(λ) ∈ ˚ g

• L(λ) =
• n∈Z

An(λ)tn

• L(λ) ∈˚

g[t,t−1]

+ i ϕ(λ) D + D(λ) K Coordinate on the circle : t → eix, x ∈ [0, 2π[

• L(λ) −

→

• n∈Z

An(λ)einx ˚ 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’18 11 / 20

Classical hierarchy for Affine Gaudin models

Sklyanin bracket:

• L1(λ),

L2(µ)

• A =

C12 µ − λ, L1(λ) + L2(µ)

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Φn on g of degree n: Sn(λ) = Φn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A =

Zeros ζi’s of the twist function: ϕ(ζi) = 0

• Qn,i, Qm,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:

• L1(λ),

L2(µ)

• A depends on ϕ(λ) and ϕ(µ)

C1 µ

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Φn on g of degree n: Sn(λ) = Φn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A =

Zeros ζi’s of the twist function: ϕ(ζi) = 0

• Qn,i, Qm,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:

• L1(λ), L2(µ)
• A depends on ϕ(λ) and ϕ(µ)

C1 µ

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Φn on g of degree n: Sn(λ) = Φn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A =

Zeros ζi’s of the twist function: ϕ(ζi) = 0

• Qn,i, Qm,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:

• L1(λ), L2(µ)
• A depends on ϕ(λ) and ϕ(µ)

C1 µ

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Ψn on g of degree n: Sn(λ) = Ψn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A = non-zero as Ψn’s non-invariant

Zeros ζi’s of the twist function: ϕ(ζi) = 0

• Qn,i, Qm,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:

• L1(λ), L2(µ)
• A depends on ϕ(λ) and ϕ(µ)

C1 µ

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Ψn on g of degree n: Sn(λ) = Ψn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A = ϕ(λ)
• · · ·
• + ϕ(µ)
• · · ·
• Zeros ζi’s of the twist function: ϕ(ζi) = 0
• Qn,i, Qm,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:

• L1(λ), L2(µ)
• A depends on ϕ(λ) and ϕ(µ)

C1 µ

• [SL Magro Vicedo 1703.01951]

Appropriate choice of polynomials Ψn on g of degree n: Sn(λ) = Ψn

• L(λ)
• Poisson bracket
• Sn(λ), Sm(µ)
• A = ϕ(λ)
• · · ·
• + ϕ(µ)
• · · ·
• Zeros ζi’s of the twist function: ϕ(ζi) = 0

Qn,i = Sn(ζi),

• Qn,i, Qm,j
• A = 0

→ classical hierarchy of conserved charges in involution

Sylvain Lacroix Affine Gaudin models RAQIS’18 12 / 20

Quantum hierarchy for affine Gaudin models ?

Classical hierarchy

• Qn,i, Qm,j
• A = 0

Quantum hierarchy Qn,i, Qm,j

• A = 0 ?

Naive guess

• Sn(λ) = Ψn

L(λ)

• + quantum corrections

and

• Qn,i =

Sn(ζi) does not work [SL Vicedo Young 1804.01480] Conjecture 1: Let P(λ) be such that ∂λ log P(λ) = ϕ(λ). Then

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ for some closed contour γi (h∨ dual Coxeter number) Conjecture originates from the study of affine opers

Sylvain Lacroix Affine Gaudin models RAQIS’18 13 / 20

Quantum hierarchy for affine Gaudin models ?

Classical hierarchy

• Qn,i, Qm,j
• A = 0

Quantum hierarchy Qn,i, Qm,j

• A = 0 ?

Naive guess

• Sn(λ) = Ψn

L(λ)

• + quantum corrections

and

• Qn,i =

Sn(ζi) does not work [SL Vicedo Young 1804.01480] Conjecture 1: Let P(λ) be such that ∂λ log P(λ) = ϕ(λ). Then

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ for some closed contour γi (h∨ dual Coxeter number) Conjecture originates from the study of affine opers

Sylvain Lacroix Affine Gaudin models RAQIS’18 13 / 20

Hypergeometric integrals on Pochhammer contour

Twist function with simple poles: ϕ(λ) =

N

• r=1

kr λ − λr , P(λ) =

N

• r=1

(λ − λr)kr Hypergeometric integrals:

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ P(λ) multi-valued → contour γi on which P(λ) is single-valued Typical examples: Pochhammer contours

λr λs

γ

Sylvain Lacroix Affine Gaudin models RAQIS’18 14 / 20

Classical limit of the quantum hierarchy

Reintroduce : kr − → kr

• Qn,i =
• γi

P(λ)− n−1

h∨

Sn(λ) dλ Classical limit → 0: saddle point approximation → localisation at extrema of P(λ) → localisation at zeros ζi of ϕ(λ) Coherent with the construction of the classical hierarchy Counting: # independent Pochhammer contours = # zeros of ϕ

Sylvain Lacroix Affine Gaudin models RAQIS’18 15 / 20

Classical limit of the quantum hierarchy

Reintroduce : kr − → kr

• Qn,i =
• γi

P(λ)− n−1

h∨

Sn(λ) dλ Classical limit → 0: saddle point approximation → localisation at extrema of P(λ) → localisation at zeros ζi of ϕ(λ) Coherent with the construction of the classical hierarchy Counting: # independent Pochhammer contours = # zeros of ϕ

Sylvain Lacroix Affine Gaudin models RAQIS’18 15 / 20

Commutation of the quantum charges

Classical case: involution of the Qn,i’s from

• Sn(λ), Sm(µ)
• A = ϕ(λ)
• · · ·
• + ϕ(µ)
• · · ·
• Quantum case: how to get commutation of the

Qn,i’s ? Twisted derivatives: ∂n,ϕ

λ f (λ) = ∂λf (λ) − (n − 1)ϕ(λ)

h∨ f (λ)

• γi

P(λ)−(n−1)/h∨∂n,ϕ

λ f (λ) dλ = 0

Conjecture 2: There exist An,m(λ, µ) and Bn,m(λ, µ) such that

• Sn(λ),

Sm(µ)

• A = ∂n,ϕ

λ

• An,m(λ, µ) + ∂m,ϕ

µ

• Bn,m(λ, µ)

Commutation of Qn,i’s:

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ = ⇒

• Qn,i,

Qm,j

• A = 0

Sylvain Lacroix Affine Gaudin models RAQIS’18 16 / 20

Commutation of the quantum charges

Classical case: involution of the Qn,i’s from

• Sn(λ), Sm(µ)
• A = ϕ(λ)
• · · ·
• + ϕ(µ)
• · · ·
• Quantum case: how to get commutation of the

Qn,i’s ? Twisted derivatives: ∂n,ϕ

λ f (λ) = ∂λf (λ) − (n − 1)ϕ(λ)

h∨ f (λ)

• γi

P(λ)−(n−1)/h∨∂n,ϕ

λ f (λ) dλ = 0

Conjecture 2: There exist An,m(λ, µ) and Bn,m(λ, µ) such that

• Sn(λ),

Sm(µ)

• A = ∂n,ϕ

λ

• An,m(λ, µ) + ∂m,ϕ

µ

• Bn,m(λ, µ)

Commutation of Qn,i’s:

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ = ⇒

• Qn,i,

Qm,j

• A = 0

Sylvain Lacroix Affine Gaudin models RAQIS’18 16 / 20

Commutation of the quantum charges

Classical case: involution of the Qn,i’s from

• Sn(λ), Sm(µ)
• A = ϕ(λ)
• · · ·
• + ϕ(µ)
• · · ·
• Quantum case: how to get commutation of the

Qn,i’s ? Twisted derivatives: ∂n,ϕ

λ f (λ) = ∂λf (λ) − (n − 1)ϕ(λ)

h∨ f (λ)

• γi

P(λ)−(n−1)/h∨∂n,ϕ

λ f (λ) dλ = 0

Conjecture 2: There exist An,m(λ, µ) and Bn,m(λ, µ) such that

• Sn(λ),

Sm(µ)

• A = ∂n,ϕ

λ

• An,m(λ, µ) + ∂m,ϕ

µ

• Bn,m(λ, µ)

Commutation of Qn,i’s:

• Qn,i =
• γi

P(λ)−(n−1)/h∨ Sn(λ) dλ = ⇒

• Qn,i,

Qm,j

• A = 0

Sylvain Lacroix Affine Gaudin models RAQIS’18 16 / 20

Cubic charge

[SL Vicedo Young 1804.06751] Untwisted affine algebras of type A: slN → for N ≥ 3, classic cubic charges Q3,i’s Construction of the cubic operator S3(λ) and the cubic quantum charges Q3,i satisfying the conjectures 1 and 2 (vertex algebras techniques) Eigenvalues of Q3,i’s by the Bethe ansatz with zero and one excitation

Sylvain Lacroix Affine Gaudin models RAQIS’18 17 / 20

Conclusion and perspectives

Sylvain Lacroix Affine Gaudin models RAQIS’18 18 / 20

Conclusion

Additional results: [SL Vicedo Young 1804.01480 1804.06751]

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’18 19 / 20

Thank you for your attention !

Sylvain Lacroix Affine Gaudin models RAQIS’18 20 / 20