Integrable Kondo problems

TL;DR

This work develops an integrability framework for Kondo-type defects, where a 1d impurity couples to a 2d chiral CFT to drive a defect RG flow. It unifies 2d CFT techniques, ODE/IM correspondence, and 4d Chern-Simons theory to derive commuting transfer matrices and Hirota-type relations across several Kondo variants, including the basic $\mathfrak{su}(2)_1$ case and multichannel generalizations. The paper provides perturbative analyses, nonperturbative RG insights, and explicit ODE/IM mappings for transfer matrices $T_n[\theta]$, $T_{n;l}[\theta]$, and their IR asymptotics via WKB networks, with Ising-model and WZW examples illustrating the approach. It also outlines broad generalizations to cosets, higher groups, and affine Gaudin models, suggesting a universal ODE/IM architecture governing defect integrability in chiral CFTs and their 4d CS realizations.

Abstract

We discuss the integrability and wall-crossing properties of Kondo problems, where an 1d impurity is coupled to a 2d chiral CFT and triggers a defect RG flow. We review several new and old examples inspired by constructions in four-dimensional Chern-Simons theory and by affine Gaudin models.

Figures (6)

• Figure 1: IR fate of the deformed line defect $L_S[\theta]$ for different $\operatorname{Im}\theta$.
• Figure 2: WKB diagram for the differential equation \ref{['eq:IsingODE']} defined in \ref{['eq:Isingquadraticdiff']}. Generic flow lines and WKB lines are colored blue and red respectively.
• Figure 3: The RG flow pattern over the complex $g$ plane and the complex $1/g$ plane. The top two and he bottom two are plotted using the beta function \ref{['eq:betafunctiongeff']} and \ref{['eq:betafunctiongeffwithzero']} respectively. Note that the lower left figure is the same as Fig. 1 in Nakagawa_2018
• Figure 4: WKB diagram for $k=1$ (left) and $k=2$ (right). Generic flow lines and WKB lines are colored blue and red respectively. $\theta$ is chosen to be $0$ and $-i\frac{\pi}{2}$ respectively. We number the WKB lines on large positve $x$ side increasingly from top. There are $k+2$ WKB lines that are connected to the zero, numbered from $n_0$ to $n_0+k+1$.
• Figure 5: WKB diagram for the differential equation \ref{['eq:app:secondorderODE']}. Generic flow lines and WKB lines are colored blue and red respectively.