Remarks on the Gaudin model modulo p

Journal of Singularities
volume 18 (2018), 486-499

Received: 14 September 2017. Accepted: 22 February 2018.

DOI: 10.5427/jsing.2018.18aa

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Abstract:

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,...,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,...,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.

Author(s) information:

Alexander Varchenko
Department of Mathematics
University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA
email: anv@email.unc.edu