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Register a new userEigenvalues of Bethe vectors in the Gaudin model
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A theorem of Feigin, Frenkel and Reshetikhin provides expressions for the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in terms of elements of the center of the affine vertex algebra at the critical level. In our recent work, explicit Harish-Chandra images of generators of the center were calculated in all classical types. We combine these results to calculate the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in an explicit form. The Harish-Chandra images can be interpreted as elements of classical $W$-algebras. We provide a direct connection between the rings of $q$-characters and classical $W$-algebras by calculating classical limits of the corresponding screening operators.
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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, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.
We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.
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 derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G$_2$. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. 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 show that the points of the spectrum of the Gaudin model in type G$_2$ are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces - the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.
The Jaynes-Cummings-Gaudin model describes a collection of $n$ spins coupled to an harmonic oscillator. It is known to be integrable, so one can define a moment map which associates to each point in phase-space the list of values of the $n+1$ conserved Hamiltonians. We identify all the critical points of this map and we compute the corresponding quadratic normal forms, using the Lax matrix representation of the model. The normal coordinates are constructed by a procedure which appears as a classical version of the Bethe Ansatz used to solve the quantum model. We show that only elliptic or focus-focus singularities are present in this model, which provides an interesting example of a symplectic toric action with singularities. To explore these, we study in detail the degeneracies of the spectral curves for the $n=1$ and $n=2$ cases. This gives a complete picture for the image of the momentum map (IMM) and the associated bifurcation diagram. For $n=2$ we found in particular some lines of rank 1 which lie, for one part, on the boundary of the IMM, where they behave like an edge separating two faces, and which go, for another part, inside the IMM.
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