We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two.
A new object is the Bethe vector averaging maps.
We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of regular functions on the Calogero-Moser space.
The generating function for elements of the Bethe subalgebra of Hecke algebra is constructed as Sklyanin's transfer-matrix operator for Hecke chain. We show that in a special classical limit q -> 1 the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of Hecke chain. We consruct a non-local analogue of the Gaudin Hamiltonians for the case of Hecke algebras.
Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master …
We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we conf…
We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In …
We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.
We show that the Gaudin Hamiltonians H_1,...,H_n generate the Bethe algebra of the n-fold tensor power of the vector representation of gl_N. Surprisingly the formula for the generators of the Bethe algebra in terms of the Gaudin Hamiltonians does not depend on N. Moreover, this formula coincides with Wilson's formula for the stationary Baker-Akhiezer function on the adelic Grassmannian.
We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes.
In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the gl_N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [MTV4] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic.…
In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.
We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space of G-bundles on a complex curve X and flat G^L-bundles on X. Beilinson and Drinfeld construct it by applying a localization functor to representations of affine algebras of critical level. We show that in genus zero the corresponding D-modules are closely related to the diagonalization problem in the Gaudin model associated to G. This allows us to give a ne…
The eigenvectors of the Hamiltionians of the XYZ Gaudin model are constructed by means of the algebraic Bethe Ansatz. The construction is based on the quasi-classical limit of the corresponding results for the inhomogeneous higher spin eight vertex model.
We consider the trigonometric classical $r$-matrix for $\mathfrak{gl}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to 1$ to elements of the Bethe subalgebra for the $XXZ$ model.
The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl_2-invariant Gaudin model. As opposed…
The eigenvectors of the osp(1|2) invariant Gaudin hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov equation yields the norm of the eigenvectors.
To any simple Lie algebra $\mathfrak g$ and automorphism $σ:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $σ= \text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenke…
Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl_{r+1} irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight. We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Piñeiro mult…
We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.
The XYZ spin chain is considered in the framework of the generalized algebraic Bethe ansatz developed by Takhtajan and Faddeev. The sum of norms of the Bethe vectors is computed and expressed in the form of a Jacobian. This result corresponds to the Gaudin hypothesis for the XYZ spin chain.
Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states…
We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family $P$ of other solutions called the population. To a population we associate a rational pseudodifferential operator $R$ and a superspace $W$ of rational functions. We show that if at least one module is typical then the population $P$ is canonically identified with the set of minimal factorizations of $R$ and with the space of full superflags in $W$. We conjecture that the singular eigenvectors (up to rescal…
This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g…
The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundament…
We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions). To a solution of the Bethe Ansatz equation, one assigns a common eigenvector (called the Bethe vector) of the trigonometric Gaudin op…
Generalizing the construction of the cyclotomic Gaudin algebra from arXiv:1409.6937, we define the universal cyclotomic Gaudin algebra. It is a cyclotomic generalization of the Gaudin models with irregular singularities defined in arXiv:math/0612798. We go on to solve, by Bethe ansatz, the special case in which the Lax matrix has simple poles at the origin and arbitrarily many finite points, and a double pole at infinity.
We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $gl_N$. The models are defined on a tensor product irreducible $gl_N$-modules. For each model, there exist $N$ one-parameter families of commuting operators on the tensor product, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.
We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra should be encoded by affine opers associated to the Langlands dual affine algebra. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of th…
This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without monodromy in general gl(n) case modulo a hypothesys on the analytic properties of the solution of a KZ-type equation. Firstly we explore the quantum analog of the characteristic polynomial which is a differential operator in a variable $u$ with the coefficients in U(gl(n))^{\otimes N}. We will call it "universal G-oper". It is constructed by the formula "Det"(L(u)-\partial_u) where L(u) is the quantum L…
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain "extended" master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an "extended" non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin…
We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This in particular implies the following result: If all ramification points of a parametrized rational curve $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linea…
In this article we exploit the known commutative family in Y(gl(n)) - the Bethe subalgebra - and its special limit to construct quantization of the Gaudin integrable system. We give explicit expressions for quantum hamiltonians QI_k(u), k=1,..., n. At small order k=1,...,3 they coincide with the quasiclassic ones, even in the case k=4 we obtain quantum correction.
A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quant…
We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of …
We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct ex…
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bet…
Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, whic…
We conjecture that quantum Gaudin models in affine types admit families of local higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further con…
Asymptotic solutions to the quantized Knizhnik-Zamolodchikov equation associated with $\frak{gl}_{N+1}$ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is an analogue of the Gaudin-Korepin formula for the norm of the Bethe vector. It is shown that, generically, the Bethe vectors form a base for the $\frak{gl}_2$ case.
We show that the Bethe vectors are non-zero vectors in the sl_{r+1} Gaudin model. Moreover, we show that the norm of a Bethe vector is equal to the Hessian of the corresponding master function at the corresponding non-degenerate critical point. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As other byproducts of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental sl_{r+1} modules and we show transversality of some Schubert cycles in the Grassmannian of r+1-dimensional plane…
Recently a new class of quantum integrable models, the cyclotomic Gaudin models, were described in arXiv:1409.6937, arXiv:1410.7664. Motivated by these, we identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac-Moody type Lie algebras with diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero.
We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explici…
Let $(\otimes_{j=1}^nV_j)[0]$ be the zero weight subspace of a tensor product of finite-dimensional irreducible $\frak{sl}_2$-modules. The dynamical elliptic Bethe algebra is a commutative algebra of differential operators acting on $(\otimes_{j=1}^nV_j)[0]$-valued functions on the Cartan subalgebra of $\frak{sl}_2$. The algebra is generated by values of the coefficient $S_2(x)$ of a certain differential operator $D=(d/dx)^2+S_2(x)$, defined to V. Rubtsov, A. Silantyev, D. Talalaev in 2009. We express $S_2(x)$ in terms of the KZB operators introduced by G. Felder and C. Wieszerkowski in 1994. …
Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra $\mathfrak{g}$. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the…
We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking t…
