Perfect integrability and Gaudin models

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.


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antum spin chains are important models in integrable system. ese models have numerous deep connections with other areas of mathematics and physics. In this article, we would like to suggest the notion of perfect integrability for quantum spin chains.
We deal with Gaudin models and XXX spin chains. Let g be a simple (or reductive) Lie (super)algebra and G the corresponding Lie group. Let A g be an affinization of g where U(g) can be identified as a Hopf subalgebra of A g . In this paper, A g is either the universal enveloping algebra of the current algebra U(g [t]) which describes the symmetry for Gaudin models, or Yangian Y(g) associated to g for XXX spin chains. In both cases the algebra A g has a remarkable commutative subalgebra called the Bethe algebra. We denote the Bethe algebra by B g . e Bethe algebra B g commutes with U(g). Take any finite-dimensional irreducible representation M of A g , then B g acts naturally on the space of singular vectors M sing . Let B g (M) be the image of B g in End(M sing ). e problem is to study the spectrum of B g (M) acting on M sing 1 . In this case, we say that the corresponding spin chain has periodic boundary condition.
With the agreement with the philosophy of geometric Langlands correspondence, it is important to understand and describe the finite-dimensional algebra B g (M) and the corresponding scheme spec(B g (M)). Or more generally, find a geometric object parameterizing the eigenspaces of B g when M runs over all finite-dimensional irreducible representations (up to isomorphism). In Gaudin models, the underlying geometric objects are described by the sets of monodromy-free L gopers with regular singularities of prescribed residues at evaluation points, see [FFRy10,Ryb18], where L g is the Langlands dual of g. Moreover, when g = gl N , the Bethe algebra B g (M) is interpreted as the space of functions on the intersection of suitable Schubert cycles in a Grassmannian variety, see [MTV09]. is interpretation gives a relation between representation theory and Schubert calculus useful in both directions which has important applications in real algebraic geometry, see [MTV09,MT16].
Any finite-dimensional unital commutative algebra B is a module over itself induced by le multiplication. We call this module the regular representation of B. e dual space B * is naturally a B-module which is called the coregular representation. A Frobenius algebra is a finite-dimensional unital commutative algebra whose regular and coregular representations are isomorphic, see Section 2.5.
Based on the extensive study of quantum spin chains, see the evidence from [MTV08, MTV09, FFRy10, MTV14, Ryb18, LM19b, CLV20], the following conjecture is expected to hold.
Conjecture 1.1. e Bethe algebra B g (M) is a Frobenius algebra and B g (M) acts on M sing cyclically.
When Conjecture 1.1 holds, we say that the corresponding quantum spin chain is perfectly integrable or the B g (M)-module M sing is perfectly integrable. Note that in this case, the B g (M)module M sing is isomorphic to the regular and coregular representations of B g (M).
In fact there is a family of commutative Bethe algebras B µ g depending on an element µ ∈ g * (or an element µ in the Lie group G for XXX spin chains). We have B 0 g = B g . If µ ∈ g * is a regular semi-simple element, we say that the corresponding spin chain has regular quasi-periodic boundary condition.
For regular quasi-periodic spin chains the Bethe algebra does not commute with U(g) and one replaces M sing with M. For more general µ ∈ g * , one has to replace M sing with an appropriate subspace of M depending on µ, see Section 2.7. e perfect integrability was shown for • Gaudin models of gl N in [MTV08,MTV09] with periodic and regular quasi-periodic boundary conditions; • XXX (resp. XXZ) spin chains of gl N associated to irreducible tensor products of vector representations in [MTV14] (resp. [RTV15]) with periodic and regular quasi-periodic boundary conditions; • XXX spin chains of gl 1|1 associated to cyclic tensor products of polynomial representations in [LM19b] with periodic and regular quasi-periodic boundary conditions; • XXX spin chains of gl m|n associated to irreducible tensor products of vector representations in [CLV20] with periodic boundary condition. Our suggestion to call the situation in Conjecture 1.1 "perfect integrability" is motivated by Lemma 1.2 below.
Let B be a finite-dimensional unital commutative algebra. Let V be a B-module and E : B → C a character, then the B-eigenspace and generalized B-eigenspace associated to E in V is defined by a∈B ker(a| V − E(a)) and Lemma 1.2. If the B V -module V is perfectly integrable, then every B-eigenspace in V has dimension one, and there exists a bijection between B-eigenspaces in V and closed points in spec(B V ). Moreover, each generalized B-eigenspace is a cyclic B-module, and the algebra B V is a maximal commutative subalgebra in End(V ) of dimension dim V .
is lemma easily follows from general well-known facts about regular and coregular representations of a finite-dimensional unital commutative algebra, see e.g. [MTV09, Section 3.3].
Note that we expect that the dimensions of eigenspaces are one from the general philosophy of Bethe ansatz conjecture. e integrabilty in any sense always asserts that the algebra of Hamiltonians is maximal commutative. And we also expect that the Bethe algebra has geometric nature based on the geometric Langlands correspondence [Fre07].
It is proved in [Ryb18, eorem 3.2] (resp. [FFRy10, Corollary 5]) that B g (resp. B µ g with regular µ) acts cyclically on M sing (resp. M). For generic values of evaluation parameters (in the periodic case or in the case of generic regular µ ∈ h * ) the action of Bethe algebra is diagonalizable and we immediately obtain that eigenspaces have dimension one. However, we cannot make such a conclusion for arbitrary parameters. Indeed, if a linear operator acts cyclically on a vector space then all its eigenspaces have dimension one. But the same result fails if we replace a single operator by a set of commuting linear operators, as the following simple example shows.
Consider the regular representation A. en the eigenspace corresponding to zero character is spanned by x 1 and x 2 which is two-dimensional.
We supplement the results of [FFRy10] and [Ryb18] with the nondegenerate symmetric bilinear form on M sing which makes B µ g (M) Frobenius which allows us to use Lemma 1.2. e bilinear form comes from the tensor product of Shapovalov forms on M, we show that all elements of Bethe algebra B µ g (M) with µ ∈ h * are symmetric with respect to this form, see Lemma 2.6. We expect the conjecture with proper modification also holds for XXZ and XYZ spin chains. e author is grateful to E. Mukhin and V. Tarasov for interesting discussions and helpful suggestions. is work was partially supported by a grant from the Simons Foundation #353831.

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2.1. Feigin-Frenkel center. In this section, we recall the definition of Feigin-Frenkel center and its properties. Let g be a complex simple Lie algebra of rank r. Consider the affine Kac-Moody algebra g, where C[t, t −1 ] is the algebra of Laurent polynomials in t. For X ∈ g and s ∈ Z, we simply write Let h ∨ be the dual Coxeter number of g. Define the module V −h ∨ (g) as the quotient of U( g) by the le ideal generated by g[t] and K + h ∨ . We call the module V −h ∨ (g) the Vaccum module at the critical level over g. e vacuum module V −h ∨ (g) has a vertex algebra structure.
Define the subspace z( g) of V −h ∨ (g) by Using the PBW theorem, it is clear that V −h ∨ (g) is isomorphic to U(g − ) as vector spaces. ere is an injective homomorphism from z( g) to U(g − ). Hence z( g) is identified as a commutative subalgebra of U(g − ). We call z( g) the Feigin-Frenkel center. We remark that Feigin-Frenkel center is slightly different from the Bethe algebra in the introduction. We refer the reader to e.g. [Mol12, Section 5] for more detail about obtaining the Bethe algebras from Feigin-Frenkel center. ere is a distinguished element S 1 ∈ z( g) given by where {X a } is an orthonormal basis of g with respect to the Killing form. e element S 1 is called a Segal-Sugawara vector.

Corollary 2.2. e Feigin-Frenkel center z( g) is invariant under the Cartan anti-involution ̟.
Proof. Since by Proposition 2.1, z( g) is the centralizer of S 1 in U(g − ), the statement follows from the fact that ̟(S 1 ) = S 1 .

2.2.
Affine Harish-Chandra homomorphism. Let n + be the nilpotent Lie subalgebra generated by e 1 , . . . , e r . Let n − be the nilpotent Lie subalgebra generated by f 1 , . . . , f r . Let h be the Cartan subalgebra generated by h 1 , . . . , h r . One has the triangular decomposition g = n + ⊕h⊕n − . e Lie algebra g is considered as a subalgebra of g via identifying X ∈ g with X[0] ∈ g. e Lie subalgebra h acts on g adjointly and hence acts on U(g − ). Let U(g − ) h be the centralizer of h in U(g − ).
Let J be the le ideal of U(g − ) generated by t −1 n − [t −1 ]. en we have the direct sum of vector spaces, Hence we have the projection It is clear that f is a homomorphism of algebras. We call f the affine Harish-Chandra homomorphism. We use the same le er f for the restriction map f : Proof. Now take S ∈ z( g) and write the decomposition of S as in (2.1), en ̟(S) = ̟(S h ) + ̟(S j ). Note that ̟ fix elements in and also the intersection of U(g − ) h with the right ideal of U(g − ) generated by t −1 n + [t −1 ]. It follows that Note that by Corollary 2.2 both S and ̟(S) are elements in z( g). Since by Proposition 2.3 the homomorphism f : z( g) → U(t −1 h[t −1 ]) is injective, we conclude that S = ̟(S), completing the proof.
Let λ = (λ 1 , . . . , λ ℓ ) be a sequence of dominant integral weights. Denote by V λ i the finitedimensional irreducible g-module of highest weight λ i . We set Here we identify h * with the subspace of g * consisting of all elements annihilating n + ⊕ n − . By the construction of A z,µ , M λ,µ is an A z,µ -module. e image of the Gaudin algebra A z,µ acting on V λ coincides with that of Bethe algebra B µ g acting on tensor product of evaluation modules V λ with evaluation points at z = (z 1 , . . . , z ℓ ), see [FFRe94,Ryb06,FFT10]. Note that in this case, all finite-dimensional irreducible U(g[t])-modules are tensor products of evaluation modules with pairwise distinct evaluation parameters.
Let A z,µ be the algebra of Hamiltonians and M λ,µ the phase space. We call the corresponding integrable system the Gaudin model. We say that the Gaudin model has periodic boundary condition if µ = 0 and regular quasi-periodic boundary condition if µ ∈ h * is regular. We would like to study the spectrum of A z,µ acting on M λ,µ . e following theorem is obtained in [FFRy10, Corollary 5] for any regular µ ∈ g * and in [Ryb18, eorem 3.2] for µ = 0. eorem 2.5. If µ ∈ h * is regular or if µ = 0, then the space M λ,µ is cyclic as an A z,µ -module.
2.4. Shapovalov form. For a dominant integral weight λ, there is a unique nondegenerate symmetric bilinear form S λ on V λ such that where v λ is a highest weight vector of V λ and v, w ∈ V λ . We call S λ the Shapovalov form on V λ . e Shapovalov form S λ is positive definite on the real part of V λ . e Shapovalov forms S λ i induce a nondegenerate symmetric bilinear form S λ = ⊗ ℓ i=1 S λ i on V λ . e restriction of S λ on the singular subspace (V λ ) sing is also nondegenerate.
Suppose µ ∈ h * , then it is clear that for all v, w ∈ V λ and X ∈ g. Let ρ λ,z,µ : A z,µ → End(M λ,µ ) be the representation of the natural action of A z,µ on M λ,µ . Let A λ,z,µ be the image of A z,µ under ρ λ,z,µ .
2.5. Frobenius algebra. Let A be a finite-dimensional commutative unital algebra. If there exists a nondegenerate symmetric bilinear form (·, ·) on A such that (ab, c) = (a, bc) for all a, b, c ∈ A, then it is clear that the regular and coregular representations of A are isomorphic. us A is a Frobenius algebra.
We prepare the following lemma for the proof of the main theorem. Suppose A is a unital commutative algebra acting on a finite-dimensional space V , ρ : A → End(V ). Let A be the image of A under ρ in End(V ). Clearly, A is a finite-dimensional unital commutative algebra.
Lemma 2.7. Suppose A acts on V cyclically. If there is a nondegenerate symmetric bilinear form (·|·) on V such that (av|w) = (v|aw), for all a ∈ A, v, w ∈ V, then the algebra A is a Frobenius algebra. In particular, the A-module V is perfectly integrable.
Proof. Let v + be a cyclic vector of the action of A on V . Define a linear map ξ by Clearly, ξ is surjective.
We claim that ξ is injective. Indeed, suppose that a ∈ ker ξ, then a ∈ End(V ) and av + = 0. Hence aa ′ v + = a ′ av + = 0 for all a ′ ∈ A, namely a ξ(A) = 0. Since ξ(A) = V , we conclude that aV = 0. erefore a = 0, which implies ξ is injective and hence a bijection. en it is clear that ξ defines an A-module isomorphism between the regular representation of A and the A-module V .
Hence A is a Frobenius algebra.
2.6. Perfect integrability of Gaudin models. e following is our main theorem which asserts Gaudin models with periodic and regular quasi-periodic boundary conditions are perfectly integrable. eorem 2.8. If µ ∈ h * is regular or if µ = 0, then the A λ,z,µ -module M λ,µ is perfectly integrable.
By Lemma 2.6, we can apply Lemma 2.7 for the case A = A λ,z,µ , V = M λ,µ , and (·|·) = S λ (·, ·). erefore we conclude that the algebra A λ,z,µ is a Frobenius algebra. eorem 2.8 gives the following important facts. By eorem 2.8, Lemma 1.2, and [Ryb18, Corollary 3.3], we see that the joint eigenvectors (up to proportionality) of the Gaudin algebra in V sing λ are in one-to-one correspondence with monodromy-free L g-opers on the projective line with regular singularities at the points z 1 , . . . , z ℓ , ∞ and the prescribed residues at the singular points. Here z 1 , . . . , z ℓ are arbitrary pairwise distinct complex numbers. Similarly, when g is of type B or C (resp. G 2 ), one deduces from [LMV17, eorem 4.5] (resp. [LM19a, eorem 5.8]) that there exists a bijection between joint eigenvectors (up to proportionality) of the Gaudin algebra in V sing λ and self-dual (resp. self-self-dual) spaces of polynomials in a suitable intersection of Schubert cells in Grassmannian. 2.7. Conjecture for general µ ∈ g * . For an arbitrary µ ∈ g * , there exists an element g ∈ G such that gµg −1 is in the negative Borel part b − = n − ⊕ h. us, without loss of generality, we can assume that µ ∈ b − . Let z µ (g) be the centralizer of µ in g. It is known that A z,µ commutes with the diagonal action of z µ (g), see [Ryb06,Proposition 4].