×

Norm of a Bethe vector and the Hessian of the master function. (English) Zbl 1072.82012

Summary: We show that the norm of a Bethe vector in the \(sl_{r+1}\) Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct 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.

MSC:

82B23 Exactly solvable models; Bethe ansatz
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
81R12 Groups and algebras in quantum theory and relations with integrable systems
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv