Gaudin Hamiltonians Research Articles

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

An identity in the Bethe subalgebra of C[Sn]$\mathbb {C}[\mathfrak {S}_n

AbstractAs part of the proof of the Bethe ansatz conjecture for the Gaudin model for , Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro–Shapiro conjecture. In this paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.

Completeness of Bethe Ansatz for Gaudin Models with 𝔤𝔩(1|1) Symmetry and Diagonal Twists

We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and showed that a bijection exists 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. We also gave dimensions of the generalized eigenspaces.

Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.

Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation

We continue the exploration of multipoint scalar comb channel blocks for conformal field theories in 3D and 4D. The central goal here is to construct novel comb channel cross ratios that are well adapted to perform projections onto all intermediate primary fields. More concretely, our new set of cross ratios includes three for each intermediate mixed symmetry tensor exchange. These variables are designed such that the associated power series expansion coincides with the sum over descendants. The leading term of this expansion is argued to factorise into a product of lower point blocks. We establish this remarkable factorisation property by studying the limiting behaviour of the Gaudin Hamiltonians that are used to characterise multipoint conformal blocks. For six points we can map the eigenvalue equations for the limiting Gaudin differential operators to Casimir equations of spinning four-point blocks.

Frobenius-like structure in Gaudin model

We introduce a Frobenius-like structure for the [Formula: see text] Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.

Completeness of Bethe ansatz for Gaudin models associated with [formula omitted

We study the Gaudin models associated with gl(1|1). We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation 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 confirm [7, Conjecture 8.3]. We also give dimensions of the generalized eigenspaces. Moreover, we express the generating pseudo-differential operator of Gaudin transfer matrices associated to antisymmetrizers in terms of the quadratic Gaudin transfer matrix and the center of U(gl(1|1)[t]).

Rational so(3) Gaudin model with general boundary terms

We study the so(3) Gaudin model with general boundary K-matrix in the framework of the algebraic Bethe ansatz. The off-shell action of the generating function of the so(3) Gaudin Hamiltonians is determined. The proof based on the mathematical induction is presented on the algebraic level without any restriction whatsoever on the boundary parameters. The so(3) Gaudin Hamiltonians with general boundary terms are given explicitly as well as their off-shell action on the Bethe states. The correspondence between the Bethe states and the solutions to the generalized so(3) Knizhnik-Zamolodchikov equations is established. In this context, the on-shell norm of the Bethe states is determined as well as their off-shell scalar product.

Bethe states and Knizhnik-Zamolodchikov equations of the trigonometric Gaudin model with triangular boundary

We present a comprehensive treatment of the non-periodic trigonometric sℓ(2) Gaudin model with triangular boundary, with an emphasis on specific freedom found in the local realization of the generators, as well as in the creation operators used in the algebraic Bethe ansatz. First, we give Bethe vectors of the non-periodic trigonometric sℓ(2) Gaudin model both through a recurrence relation and in a closed form. Next, the off-shell action of the generating function of the trigonometric Gaudin Hamiltonians with general boundary terms on an arbitrary Bethe vector is shown, together with the corresponding proof based on mathematical induction. The action of the Gaudin Hamiltonians is given explicitly. Furthermore, by careful choice of the arbitrary functions appearing in our more general formulation, we additionally obtain: i) the solutions to the Knizhnik-Zamolodchikov equations (each corresponding to one of the Bethe states); ii) compact formulas for the on-shell norms of Bethe states; and iii) closed-form expressions for the off-shell scalar products of Bethe states.

From Gaudin Integrable Models to d-Dimensional Multipoint Conformal Blocks.

In this work, we initiate an integrability-based approach to multipoint conformal blocks for higher-dimensional conformal field theories. Our main observation is that conformal blocks for N-point functions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides us with a complete set of differential equations that can be used to evaluate multipoint blocks.

Algebraic Bethe Ansatz for the Trigonometric sℓ(2) Gaudin Model with Triangular Boundary

In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model.

Quantum-classical duality for Gaudin magnets with boundary

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.

The solutions of $${\mathfrak {g}}{\mathfrak {l}}_{M|N}$$ Bethe ansatz equation and rational pseudodifferential operators

We describe a reproduction procedure which, given a solution of the glM|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 rescaling) of all glM|N Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.

Generalized sℓ(2) Gaudin algebra and corresponding Knizhnik–Zamolodchikov equation

The Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.

A Proof of the Gaudin Bethe Ansatz Conjecture

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 that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.

Classical N-reflection equation and Gaudin models

We introduce the notion of N-reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the N=2 case. The basic theory is established and illustrated with several examples of solutions of the N-reflection equation associated with the rational and trigonometric r-matrices. A central result is the construction of a Poisson algebra associated with a non-skew-symmetric r-matrix whose form is specified by a solution of the N-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of BC_L-type.

Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus

The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.

Remarks on the Gaudin model modulo p

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 that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation. If $k=1$ and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map.

Eigenvalues of Bethe vectors in the Gaudin model

According to the Feigin–Frenkel–Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical W-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of q-characters and classical W-algebras.

Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.