Duality of Gaudin models

Abstract

We consider actions of the current Lie algebras \gln[t] and \glk[t] on the space Pkn of polynomials in kn anticommuting variables. The actions depend on parameters z¯=(z1,…,zk) and α¯=(α1,…,αn), respectively. We show that the images of the Bethe algebras Bα¯⟨n⟩⊂U(\gln[t]) and Bz¯⟨k⟩⊂U(\glk[t]) under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of Bα¯⟨n⟩ and the spaces of quasi-exponentials describing eigenvectors of Bz¯⟨k⟩. One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians. We also establish the (\glk,\gln)-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.