Integrable Kondo problems

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.

Kondo line defects and affine Gaudin models

“…However, the residues of (4.19) do correspond to a shifted Weyl reflection of this highest weight. Indeed, the simple roots of sl 2 act by shifted Weyl reflections on the residues at the origin as 19) describes a generalised Miura λ-oper in the sense of [20,49]. We conjecture that the state J + −2 |0, −1 is described by this generalised Miura λ-oper.…”

“…The couplings g i [z] need to be renormalized and acquire a scale dependence. It was conjectured in [19] that the RG flow factors through a flow of the spectral parameter, so that the spectral parameter may be identified with the (complexified) renormalization scale e θ via dimensional transmutation. In other words, the RG flow defines a one-parameter family of commuting line defects.…”

“…Following [19], we adopt the convention that the physical RG flows start from asymptotically free defects and the couplings grow in the positive real direction approaching the infrared. Therefore the UV expansion is concerned with small positive g i .…”

“…1. Integrable Kondo defects in products of chiral SU(2) WZW models i sl(2) k i [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

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

“…The ODE (7.83) with r > 1 and arbitrary k a was introduced in the recent work [43]. Note that in the case A (vac,±) = 0 it corresponds to the chiral primary states for the ⊗ r a=1 sl ka (2) WZW model.…”