Kondo line defects and affine Gaudin models

Abstract

We describe the relation between integrable Kondo problems in products of chiral SU(2) WZW models and affine SU(2) Gaudin models. We propose a full ODE/IM solution of the spectral problem for these models.

4-dimensional Chern-Simons theory and integrable field theories

“…In the context of the Kondo problems and their relation to 4-dimensional Chern-Simons theory, these ODE take particular form which depends on the 1-form ω underlying the 4-dimensional setup. This is in agreement with the fact, mentioned in Subsection 5.4, that the 4-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 [92,93,95,96] 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 [27], it was shown that certain integrable systems called Kondo problems can also be related to the 4-dimensional Chern-Simons theory. Quantum aspects of these systems are discussed in [27], in particular in the framework of the so-called ODE/IQFT correspondence [87][88][89][90] (see also [91] for previous results on the ODE/IQFT correspondence for Kondo problems and [92][93][94] for further recent developments). This correspondence relates certain observables of an Integrable Quantum Field Theory (IQFT), in particular the spectrum of its commuting operators, to the properties of some well-chosen Ordinary Differential Equations (ODE).…”

“…Instead we will focus on the second approach proposed in [4] to generate integrable 2-dimensional models from 4-dimensional Chern-Simons theory, which is based on so-called disorder defects and which occurs in the case where the 1-form ω has zeroes. This approach was further developed in [23,[25][26][27][28][30][31][32][34][35][36][38][39][40][44][45][46][47][48]92]: in these lectures, we will mainly review the results of [4] and [23]. Overview.…”

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

“…Renewed interest in the Kondo problem came from refs. [43,44], where some previously obtained facts concerning the isotropic case were rediscovered. Here we explain the relation between the differential equation (7.7) and the one that was considered in those works.…”

Integrable Kondo problems

“…The IM/DE for excited states will be discussed in a companion paper. [46] 4 The su(2) Kondo line defects…”