Dual Baxter equations and quantization of the affine Jacobian

Abstract

Abstract.A quantum integrable model is considered which describes a quantization of affine hyper-elliptic Jacobian. This model is shown to possess the property of duality: a dual model with inverse Planck constant exists such that the eigen-functions of its Hamiltonians coincide with the eigen-functions of Hamiltonians of the original model. We explain that this duality can be considered as duality between homologies and cohomologies of quantized affine hyper-elliptic Jacobian.0 Membre du CNRS 1 Laboratoire associé au CNRS.

Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain

“…Note that the Baxter equation (3.63a) has been obtained in [37] in the framework of separation of variables [32]. A similar system of Baxter equations (3.63) appeared for the first time in [11], [38] in the models different from ours, but with the same type of duality property.…”

The Baxter equation for quantum discrete Boussinesq equation

“…We remark that this duality of the Baxter equation also appeared in studies of the discrete KdV equation [31], where given was the interpretation from the viewpoint of the algebraic geometry.…”

Dual Baxter's equation and quantum algebraic geometry

“…The main conjecture concerning these dual integrable models is that their spectrum is defined by different solutions of the equations (5.1,5.2) with Q(ζ) being an entire function of its argument with certain asymptotic at the infinity [3]. The above construction proves, in particular, existence of solutions to these equations.…”

“…Following [4,3] one introduces the weight of integration in the space of functions of ζ j consistent with the scalar product in H. For the matrix elements we find:…”