ODE/IQFT correspondence for the generalized affine $\mathfrak{ sl}(2)$ Gaudin model

Abstract

An integrable system is introduced, which is a generalization of the sl(2) quantum affine Gaudin model. Among other things, the Hamiltonians are constructed and their spectrum is calculated within the ODE/IQFT approach. The model fits within the framework of Yang-Baxter integrability. This opens a way for the systematic quantization of a large class of integrable non-linear sigma models. There may also be some interest in terms of Condensed Matter applications, as the theory can be thought of as a multiparametric generalization of the Kondo model.

Four-dimensional Chern–Simons theory and integrable field theories

“…In the context of the Kondo problems and their relation to four-dimensional Chern-Simons theory, these ODE take particular form which depends on the one-form ω underlying the four-dimensional setup. This is in agreement with the fact, mentioned in subsection 5.4, that the four-dimensional Chern-Simons theory is related to the so-called affine Gaudin models: indeed, it was proposed in [5] that these models can serve as a natural framework for studying the ODE/IQFT correspondence (see also [40,98,100,101] for further developments on the quantisation of affine Gaudin models and their relation to the ODE/IQFT correspondence).…”

“…Let us finally mention that in the work [29], it was shown that certain integrable systems called Kondo problems can also be related to the four-dimensional Chern-Simons theory. Quantum aspects of these systems are discussed in [29], in particular in the framework of the so-called ordinary differential equations (ODE)/integrable quantum field theory (IQFT) correspondence [93][94][95][96] (see also [97] for previous results on the ODE/IQFT correspondence for Kondo problems and [40,98,99] for further recent developments). This correspondence relates certain observables of an IQFT, in particular the spectrum of its commuting operators, to the properties of some well-chosen ODE.…”

Anisotropic Kondo line defect and ODE/IM correspondence

“…Anisotropic vs coset: from the ODE (4.1), it is interesting to perform a change of coordinate y = e − x and we find e 2θ −2 y −(2+ 2 ) i (1 − g i y) k i + t(y) (4.3) where t(y) ≡ −2 y −2 t(x) − 1 4y 2 . This is precisely the ODE proposed in equation (7.7) of [53]. On the other hand, according to [16], it describes a Kondo defect in a coset su(2) −(2+ 2 ) ⊕ i su (2)…”

“…Note added: While the manuscript is close to completion, we became aware of [53] which has some overlap with our results. In particular, our proposal of the ODE (3.17) and (4.1) is equivalent to (7.7) in [53] via a suitable change of coordinate. We thank Gleb A. Kotousov and Sergei L. Lukyanov for sharing their work with us before publishing.…”

On Bethe equations of 2d conformal field theory

“…In the related quantum toroidal setting such relation is understood thanks to so-called Miki automorphism [63] which exchanges the mode and spin spaces (which is geometrically related to fiber-base duality [64]) and the fact that there can be different systems of Bethe ansatz equations associated to the same physical system is an example of more general phenomenon called spectral duality [65][66][67][68]. There is also a close connection to (affine) Gaudin models, starting from [24] and recently discussed for example in [69][70][71][72][73].…”