In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N -site GL k Heisenberg chain is dual to the special reduced k + 2-points gl N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.
Motivated by recent progress in the study of supersymmetric gauge theories we propose a very compact formulation of spectral duality between XXZ spin chains. The action of the quantum duality is given by the Fourier transform in the spectral parameter. We investigate the duality in various limits and, in particular, prove it for q → 1, i.e. when it reduces to the XXX/Gaudin duality. We also show that the universal difference operators are given by the normal ordering of the classical spectral curves.Integrable systems provide a key to understanding N = 2 supersymmetric gauge theories. The Seiberg-Witten theory [1] is naturally formulated in terms of integrable systems [2], and ǫ-deformation of the former corresponds to quantization of the latter. Another view on this connection is provided by the AGT correspondence [3], in which a (quantization of) Hitchin integrable system arises in a natural way.Interestingly, it turns out that for a large class of gauge theories there are two different integrable systems, associated with each of them [4]. The equivalence between the two systems is given by the spectral duality. This duality was proposed in [5,6,7] and further explored in different ways in [8, 9, 10] and [4]. On the classical level the duality exchanges the two coordinates in the equation for the spectral curve of the system. The classical Hamiltonians of the two systems, therefore, coincide. On the quantum level the situation is more subtle: the duality comes in several variants. In the weakest version, one claims that some subset of all quantum Hamiltonians is shared by the two systems [8]. In a stronger version [9], the whole Bethe subalgebras, i.e. all the Hamiltonians, of the two systems are identified. In the third version [10] one identifies the orbits of solutions to the Bethe
Bukhvostov and Lipatov have shown that weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model in two dimensions are described by an exactly soluble model containing two coupled Dirac fermions. We propose an exact formula for the vacuum energy of the model for twisted boundary conditions, expressing it through a special solution of the classical sinh-Gordon equation. The formula perfectly matches predictions of the standard renormalized perturbation theory at weak couplings as well as the conformal perturbation theory at short distances. Our results also agree with the Bethe ansatz solution of the model. A complete proof the proposed expression for the vacuum energy based on a combination of the Bethe ansatz techniques and the classical inverse scattering transform method is presented in the second part of this work [40].
The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1+1 dimensions. The model describes weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model. In our previous work [arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.
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