Anisotropic Kondo line defect and ODE/IM correspondence

We study the anisotropic Kondo line defects in products of chiral $SU(2)$ WZW models. We propose an ODE/IM correspondence for the anisotropic Kondo problems by considering the four-dimensional Chern Simons theory in the trigonometric case. We verify the claim both by explicit perturbative calculations in the ultraviolet and by exact WKB analysis in the infrared. In doing so, we derived the infrared dynamics of a large class of anisotropic line defects.


Introduction
Kondo model was invented to describe a single magnetic impurity in a condensed matter system and now has become a prototypical example of quantum impurity [1][2][3][4][5][6][7].See e.g.[8,9] for a detailed account of the historical developments and further references.In the language of quantum field theory (e.g.[10][11][12][13][14]), a local impurity coupled to the bulk gives rise to a line defect, which is generically nonconformal.In particular, the Kondo line defect that we will consider in this paper is a large class of (nonconformal) chiral line defects where the impurity is only coupled to the chiral half of a bulk conformal field theory.Therefore throughout this paper, we will only be concerned with the chiral half of a CFT since the Kondo defect is transparent to the anti-chiral sector.We will assume a basic familiarity with [15,16] and only review some necessary background to set up the notations.
One of the basic examples is when the bulk CFT is chiral SU (2) k WZW model with the defect Hamiltonian H = g t a J a (t, 0) , where t a is n dimensional irreducible matrix representation of SU( 2) and J a is the chiral SU(2) k WZW currents.It produces a line defect, with the trace taken in the n dimensional irrep of SU (2).To be well-defined as a quantum operator, one has to carry out a careful renormalization procedure.One regularization scheme is given by [17] where the renormalized operator is explicitly computed in the first few orders of g and1 k .(The result for finite k can be found in [15]).Their regularization scheme is particularly nice because it preserves • translation invariance along the defect so that aside from the renormalization of the coupling g, the only local counterterm is the identity • rigid translation invariance perpendicular to the defect • global symmetry SU (2) It turns out that the Kondo defect is asymptotically free and dynamically generates a scale µ ≡ e θ .Therefore we will denote the Kondo defect below as Tn [θ ].Kondo line defects are not topological.Rather, it is believed to carry the integrable structure of the bulk CFT [18], which, among other things, claims the commutativity for any n, n ′ and θ , θ ′ Tn and the Hirota fusion relation As operator equations, they have been verified explicitly in [15].It is one of the surprising incarnations of the integrability that (quantum) Kondo line defect corresponds to a (classical) ordinary differential equation in the spirit of ODE/IM correspondence 1 [21]: the expectation value of the Kondo line defect in a state |ℓ〉 is identified with the Stokes data of an ODE [15,16]. 2 The most basic example is the Kondo defect in the chiral SU (2) k WZW model which corresponds to the ODE 3 ∂ 2 x ψ(x) = e 2θ e 2x (1 + g x) k + t(x) ψ(x) , (5) where t(x) = 0 if |ℓ〉 is the vacuum state and for other generic states, t(x) is determined by a set of Bethe equation [19,20,25,26,28].See also [29][30][31] for a more recent discussion.A special case of the ODE (5) in vacuum state at level k = 1 has appeared in [32] in the early days of the ODE/IM correspondence.We think of (5) as the most basic example of ODE/IM correspondence, based on which, one can easily generalize ODE to other chiral algebras associated to su(2) k , including the multichannel i SU(2) k i and the coset i SU (2) k i /SU (2) k i [15,16].Generalizations to higher rank Lie algebras similar to [33] should also be straightforward (e.g.see a discussion for sl 3 in [16]).
In general, it is very hard to determine the strongly coupled physics in the infrared where the perturbation theory breaks down.Thanks to the ODE/IM correspondence, one can easily derive the nonperturbative infrared dynamics of a large class of Kondo line defects by utilizing the techniques of exact WKB analysis.See [15,16] and references therein for more details.Notably, we discovered an interesting pattern of the wall-crossing phenomenon in the complex θ plane.In particular, as a special case, we reproduced the well-known conjecture of underscreening-overscreening transition from [34,35].
Let us remark that it is not yet understood if ODE/IM correspondence can be obtained from a direct relationship between two physical systems.We expect one should be able to find its string theory embedding 4 in the same vein as in [36].As a step towards this end [15], we embed the Kondo problems in the recently proposed four-dimensional Chern Simons theory [37][38][39][40][41] and conjecture an identification of the meromorphic one form with the logarithmic derivative of the potential in (5).
On a different front, in this article, we would like to ask the following natural question: what happens if the coupling between the bulk and impurity spin is anisotropic, i.e. when the defect Hamiltonian is not SU(2) symmetric with g ⊥ = g z being isotropic.This is a well-studied quantum impurity problem in condensed matter literature starting from [42] and further developed by many others including [43][44][45][46][47].
One can also find extended discussions in reviews like [48][49][50][51] and references therein.The perturbation theory yields the renormalization group equation in the following form which is symmetric under g ⊥ ↔ −g ⊥ .Without loss of generality, let's take g ⊥ > 0 and the RG flow diagram is shown in 1 Importantly, we have a line of UV fixed point labelled by finite g z at g ⊥ = 0 and renormalization will bring us to strong coupling in the infrared.We then expect to have a family of Kondo line defects labelled by the UV fixed point.In section 2, we will properly define and renormalize the line defect produced by the anisotropic Kondo model.In particular, the quantum line defect we define also make sense for finite deformation parameter.We will give explicit results in the leading nontrivial order.In section 3, we will discuss its embedding in the trigonometric 4d Chern Simons theory and propose an ODE (62) that corresponds to the SU(2) k anisotropic Kondo line defect in the fashion of ODE/IM correspondence.The proposal will be verified both in the UV and IR.In doing so, we derive the infrared dynamics in a variety of classes of line defects.In the simplest case, i.e. physical RG flow of SU(2) k WZW we reproduce the known result in the literature.In section 4, we generalize the construction to multichannel i SU(2) k i .Note added: While the manuscript was close to completion, we became aware of [52] which has some overlap with our results.In particular, our proposal of the ODE (62) and ( 101) is equivalent to (7.7) in [52] via a suitable change of coordinate.As remarked in section 3.1, the perspective developed in this article has pleasantly many different features manifested.We thank Gleb A. Kotousov and Sergei L. Lukyanov for sharing their work with us before publishing.

Twist fields and defect changing operators
We are interested in (anisotropic) Kondo defects associated with the SU(2) k WZW model.Since (anisotropic) Kondo defect will be defined only using chiral currents of the bulk chiral algebra, we can embed su (2) , which is the chiral half of k compact bosons at the self-dual radius.
Let us start by reviewing some necessary facts about a single compact boson.We will work with the normalization such that5 The vertex operator V α =: e iαϕ : has conformal dimension h = α 2 4 and has the OPE with the current In particular, consider the vertex operator J ± ≡ V ±2 , they have conformal dimension 1 and the OPE with the current i∂ ϕ as follows Therefore the current algebra from i∂ ϕ can be extended by J ± to give us the chiral algebra su(2) 1 .
The current J 0 ≡ i∂ ϕ alone generates a U(1) symmetry with the topological line operator (with anticlockwise orientation) Its actions can be determined from the OPE to be so that L 1 is identified with the identity line L 0 .The topological line L δ can end on twist fields, which are by definition elements of the defect Hilbert space.For example, the ground state of the defect Hilbert space can be written in terms of vertex operator V ±δ .If we bring a bulk local operator J(z) around the twist field, it undergoes a group action, as shown pictorially in Fig. 2.
As a result, the current J 0 is single-valued so the mode expansion is as usual whereas Equivalently, the current J ± have the following mode expansion where J ± n±α and J 0 n generate the twisted affine algebra, related to untwisted su(2) 1 by the spectral flow transformation U α .Please refer to appendix B for more details.Importantly the Figure 2: Red dots denote twist fields living at the end of the topological line defect and red crosses denote the defect-changing operators living at the junction of two topological line defects.
Hilbert space after the spectral flow transformation is isomorphic to the untwisted one with a shift of the L 0 eigenvalue and J 0 0 eigenvalue [53] In particular, vacuum operator, under the spectral flow U α , maps to an operator with dimension 4 living at the end of the twist defect L α , namely Defect fields are local operators living on the topological line L δ .More generally, there are fields, referred to as defect-changing operators, living on the junction of two topological lines, denoted as the red cross in Fig. 2. Similar to the space of twist fields living at the end of the line, the space of defect-changing operators is also isomorphic to the space of bulk local operators related to the spectral flow.To see this, take a look at figure 2. If one brings a local operator J(z) around the defect-changing operator, it gets acted on twice Therefore, the space of defect changing operator H L δ →L −δ living at the junction from a line L δ to a line L −δ is simply U 2δ H bulk .In particular, the image of bulk local current J ± under the spectral flow is In particular we are interested in

Ultraviolet analysis of the anisotropic Kondo defects
We are now ready to give a precise definition of the anisotropic Kondo defects.In particular, the definition makes sense even for finite deformation parameter δ.We will consider the case of spin 1 2 , or equivalently n = 2. Consideration of higher spin will be left to future work, as we will explain in section 4.
Consider k copies of the chiral algebra su(2) 1 discussed in section 2.1 which contains the chiral algebra su(2) k generated by the total currents such that Just like k = 1, J 0 generates a U(1) global symmetry whose topological symmetry line is The UV fixed point of the spin 1 2 anisotropic Kondo defect line of anisotropicity δ is defined to be the direct sum of two such lines L δ ⊕ L −δ , given explicitly by exp δσ z dz i i∂ ϕ i = exp 2δ dz t 0 J 0 (z) , (30) with t 0 ≡ 1 2 σ z is half of the Pauli matrix σ Z .In other words, it is a deformation of a direct sum of two identity lines L 0 ⊕ L 0 by the exactly marginal operator t 0 J 0 .As we reviewed in the previous section, the space of the defect-changing operators is related to the space of bulk local operators via the spectral flow operation.In particular, we are interested in where The operator t ± J ∓ δ has dimension which is smaller than 1 when δ ∈ [0, 2  k ].Since they are slightly relevant, it makes sense to turn them on and formally write them as Tr Pexp This expression is only classical since it subjects to renormalization once λ is nonzero.We will refer to the line defect after an appropriate quantization as the anisotropic Kondo defect T2 (λ, δ), where the subscript 2 comes from the fact the trace is taken in a two-dimensional representation of su (2).It is then one of the tasks of this section to quantize this line defect.When δ = λ, we are reduced to the isotropic Kondo defect, where a nice recipe of quantization has been given in [17] and further studied in [15].We are interested in the expectation value of T2 (λ, δ).More generally, given twist parameter α, we can also introduce a twist defect line L α with twist fields V ±α ≡: e i ⃗ α• ⃗ ϕ : living at the ends, where ⃗ α is a k-component vector α1 = (α, α, . . ., α).This is shown pictorially in figure 3. Then the correlation function 〈V −α (∞) T2 (λ, δ)V α (0)〉 can be loosely written as interpretted as the expectation value of the Kondo defect T2 (λ, δ) in the state |α〉.
We will work in the renormalization scheme where δ is not being renormalized At the leading order, we can turn off the coupling g and Kondo defect reads where we insert a twist inside the trace, which is necessary to make sure the integrand makes sense, i.e. single-valued.To figure out what is the correct choice of α, recall that the defect operators t ∓ J ± δ are not single-valued around the circle if there is no such insertion.More specifically, as depicted in figure 3 whenever they go cross the line L α , they pick up a phase exp iπ ⃗ α • ⃗ β a .To cancel this phase, one can think of inserting e 2πi αt 0 at the intersection of the Kondo defect and L α line, shown as the pink triangle in figure 3. Due to the identity we can set α = 1 2 ⃗ α • ⃗ β a so that the phase from crossing the line L α cancels the phase from commuting with the insertion e 2πi αt 0 .Therefore at the leading order, we find O(λ) order has to vanish since one cannot insert a single defect-changing operator on the Kondo line defect.At the order of O(λ 2 ), we have the integrand is simply a correlation function of four vertex operators, which equals where Figure 3: The correlation function 〈V −α (∞) T2 (λ, δ)V α (0)〉 involving twist lines L α and the anisotropic Kondo lines T2 (λ, δ).The top/bottom left diagrams are for the computations at the order g 0 and g 1 respectively.On the right is a pictorial derivation of the twisting factor inserted on the pink triangle.
The contour integral is over the configuration space Conf 2 (S 1 ) of two ordered points on the circle of radius R. Explicitly, if we parametrize z a = e iζ a for a = 1, 2, then the integration is a two dimensional integral of ζ 1 and ζ 2 over 0 < ζ 1 < ζ 2 < 2π while identifying 0 and 2π.We will review some basic facts about configuration space in appendix A, where we explain Conf 2 (S 1 ) is a cylinder The integral is greatly simplified if we choose a coordinate system that is rotationally invariant along S 1 .The result yields 〈α|T (δ, g)|α〉 = 2 cos π α where we renamed the circumference of the cylinder to be R ≡ e θ .Note that this makes sense since λ has a positive dimension and we have a nice expansion in terms of a dimensionless parameter Some remarks are in order.Obviously, the renormalization of the anisotropic Kondo line defect described in this section is much easier than the isotropic case given in [15,17].In fact, this is a generic feature of conformal perturbation theory with relevant couplings, where the renormalization is made easier by an analytical continuation of the parameter, i.e. δ in this case.
Due to the commutativity of the Kondo operators (3), the correlation function (44) are essentially nonlocal integral of motions [54].In fact, the Kondo defects are expected to coincide with the transfer matrix in [54] after an appropriate identification.We discuss more of this in section 4.

Proposal
What should be the corresponding ODE for the anisotropic Kondo defect?Since the ODE for the isotropic Kondo defect is obtained from the four-dimensional Chern Simons theory in the rational setting, it is natural to expect the anisotropic ODE can be found in 4d CS in the trigonometric setting.Let us first recall some key steps in the rational construction [15].
The action for the 4d CS theory [37][38][39] is where CS(A) ≡ Tr AdA + 2 3 A ∧ A ∧ A is the Chern Simons three form built out of the partial connection A = A x d x + A y d y + A z dz.Classically, rational case just corresponds to considering the spacetime to be 2 × and ω = 1.It is shown in [15] that if we couple to the 4d CS a 2d chiral WZW model living on a surface defect wrapping 2 × {z 0 }, we obtain an isotropic Kondo problem after integrating out the transverse z direction.The Wilson line wrapping a line L ⊂ 2 ×{z} will become a Kondo defect in the two-dimensional system with z −z 0 playing the role of the spectral parameter.The commutativity (3) and Hirota relation (4) of Kondo line defect then follow automatically from the properties of the Wilson lines.
There is one important subtlety to the story above we would like to stress.The coupling between the chiral WZW on the surface defect and the bulk gauge field will induce a gauge anomaly.To cancel this anomaly, we need to correct the one form ω by so that the spectral parameter θ is identitied to be the primitive dθ = ω(z)dz.The conjecture proposed in [15] is then the identification between the (twisted) meromorphic one form ω(z)dz and the logarithmic derivative of the quadratic differential P(x)d x 2 from the ODE ω ↔ 1 2 from which the ODE takes the form By renaming the coupling ħ h z 0 → −g, we find the isotropic ODE given in [15]: From rational case to trigonometric case, classically one just needs to replace the one form dz by dz z and work with * instead of .On the other hand, since the anomaly is a UV effect which is not sensitive to the global structure, we can therefore, without doing any computations, find the correction due to the gauge anomaly of the same form where z = e εx , ε = ħ h, and z 0 = g.In x coordinate, we find the correct residue k 2 at the pole x 0 = log g, as expected from (47), which leads to the following proposal In fact, the ODE for the single-channel anisotropic Kondo problem for the vacuum state in chiral SU(2) k WZW has been proposed 6
Compared to (53), the form of ( 52) has many new features manifested.Importantly, (52) takes a universal form as (50) since they both have 4d Chern Simons origin.Therefore, as we will show below, we can translate effortlessly most of the techniques we understand well in the isotropic case, including the Hirota equation, WKB analysis and a straightforward generalization to the excited states and the multichannel i SU(2) k i .Noticeably, as shown in sections below, (52) enables a clear match of physical observables with defect RG flows, which have otherwise remained elusive.
We will refer to P(x) ≡ e 2x 1 − g e −εx k as the potential and consider With this choice, the Stokes data at infinity is still defined in terms of small solutions along Stokes lines at large positive real part of x, spaced by iπ in the x plane and we can avoid the appearance of Stokes sectors at negative infinity.We will discuss the structure of the WKB diagram in detail later in section 3.3.
The construction we use to extract Stokes data and make contact with the Kondo defect is the same as in the isotropic case [15], which we now review.When the real part of x + θ is large, the potential is dominated by the part e 2θ e 2x , where we define small solutions.Let's start by defining a small solution ψ 0 (x; θ ) to be the unique solution (up to normalization) that decreases asymptotically fast along the line8 of large real positive x + θ .We fix the normalization of ψ 0 so that it agrees with the WKB asymptotics for large positive real x + θ ψ 0 (x; θ ) ∼ 1 Then we define an infinite sequence of small solutions which can be easily seen to have the asymptotics (55) at large positive real x + θ + nπi.This normalization ensures that all the Wronskians9 between neighboring solutions equals −i identically, i.e. i(ψ n , ψ n+1 ) = 1.We further define T -functions to be The collection of the T -functions encodes the Stokes data of the ODE (52).
Compared to the isotropic case in [15], a new feature is that the equation ( 52) is invariant under x → x − 2πi ε accompanied by θ → θ + 2πi ε .As we defined above in (55), the small solutions have a behaviour at infinity controlled by a hypergeometric function which is invariant under these translations.We thus have and thus the T functions are periodic under θ → θ + 2πi ε , i.e. to be functions of the spectral parameter z ≡ e εθ .
Hirota relation [60][61][62] automatically follow from the construction which takes the same form as the isotropic SU(2) Hirota relation [16].However, it is customary to write it in terms of the spectral parameter z, where the Hirota relations become multiplicative, involving multiplicative shifts of z by powers of q = e iπε .
T n q − 1 2 z T n q From the seminal work of [19,20], given the ODE for the vacuum state, one can find the one for excited states by introducing singularities of trivial monodromy.Since the ODE (52) for the vacuum state takes the same form as the isotropic one (50), the recipe to write down the ODE for excited states is straightforward.One just need to add t(x) = a 2 + + ∂ x a + (x) to the potential, so that with and the requirement that all the singularities s, u i and u ′ i have trivial monodromy.This is done by defining a − (x) ≡ −a + (x) − 1 2 Note that, close to each pole we have the same behaviour as in the isotropic case [16].For example, close to s a , we have Then by the same reasoning given in [16,25,26,[28][29][30]63], the trivial monodromy condition is just realized by the Bethe equation In the fashion of ODE/IM correspondence, we claim the following identification [15,16] ℓ Tn ℓ = i ψ 0 On the left-hand side is the expectation value of the anisotropic Kondo line defect defined in section 2 in the state |ℓ〉, which, by state-operator correspondence, can be either genuine bulk local operators or twist field living at the end of a twist topological line, as we have discussed in section 2. Before we conclude this section, let us comment on the isotropic limit.We expect to find the isotropic ODE by taking the limit ε → 0 in an appropriate way.For example, if we expand the potential in small ε Upon a shift of coordinate, we find e 2θ (εg) k e −2 1−g εg e 2x x k .On the other hand, a coordinate shift of e 2 θ e 2x (1 + ĝ x) k yields e 2 θ ĝk e −2/ĝ e 2x x k , we therefore find the identification The same is true for the Miura part t(x) = a(x) 2 + ∂ x a(x) as well.For example, if Expand in the limit ε → 0 as expected.
In the following sections, we perform explicit computations both in the UV and in the IR in the fashion of [15] to verify the claim.As the fact that excited states are controlled by the addition of t(x) in (62) follows immediately from the same reasoning in the isotropic case, we will not replicate the same computation in the most general case, rather just to consider a + (x) = − α and a − (x) = 1 + α, leading to the following ODE

Ultraviolet analysis
In this section, we perform the explicit UV analysis, i.e. small g expansion, to verify the claim (68).The first observation is that if we perform a shift of the coordinate x → x + x 0 in (52), with g ≡ e εx 0 , the potential then becomes e 2θ +2x 0 e 2x (1 − e −εx ) k , which depends on three independent parameters k, ε and g eff (θ ) ≡ e 2θ +2x 0 ε/2 = g e εθ . (74) We work with the convention that the physical RG flow corresponds to increasing θ along the real axis while keeping ε and k constant.Then it is not hard to see we have all the ingredients to match the UV physics of the anisotropic Kondo problem: 1. k should correspond to the level of the bulk chiral SU(2) k WZW model.
2. The RG flows of the Kondo defect are labelled by UV fixed points, defined by a δt 3 J 3 deformation.This role is played by parameter ε.
3. e θ corresponds to the RG scale in the Kondo problem, so g eff (θ ) ≡ g e εθ signals the deformation from an operator of dimension 1 − ε 2 .On the other hand, at the UV fixed points, σ + J + and σ − J − have deformed dimensions h = kδ 2 − 2δ + 1. We, therefore, conjecture the following identification 4. The RG flow is produced by a λ(σ + J + + σ − J − ) deformation of the UV fixed point.We then expect λ should just be g, upon an appropriate change of coordinate.
We will now solve the ODE perturbatively in g, which will verify the conjectured correspondence above.The recipe for such perturbative calculation is given in [15], which we briefly outline here.We start by writing down the perturbative solution in small g, more precisely in the region g e −εx < 1. Plug in ψ I I = g i ψ (i) .At the leading order of g, we have ∂ 2 x ψ (1) − e 2θ +2x + α 2 ψ (1) = −e −xε e 2θ +2x kψ (0) .(78) The solutions are for arbitrary constant c 0 and c 1 .Note that the lower limit of the first integration is arbitrary since c 1 is arbitrary.We need to match with the asymptotic behaviours of the Bessel functions in the other region g e −εx > 1, or roughly speaking large negative x.With the assumption ε ∈ (0, 2 k ), the equation takes the simple form whose solutions are parametrized by 10 10 Since to compare the asymptotics, without loss of generality here in this section we assume α ≥ 0. The other sign works the same way and the final result is invariant under α → − α since the original ODE (76) is.
We can then consider the solution ψ I I in the limit x → −∞, and use the asymptotics of the Bessel functions (84) we find ψ II ≈ ψ (0) + gψ (1) , where and D is arbitrary due to the arbitrary constant c 1 in the solution (80).One natural choice is to fix T 1 (θ , ε, α) = 1 identically by choosing which give us So upon identifying this α with the one defined in (43) in the previous section and we find an exact match with (44)!A remark is in order.In writing down the ODE (52), we required ε ∈ (0, 2 k ) to have desired behaviours around the negative infinity.What does such a requirement mean in the Kondo problem?Since the dimension of the defect changing operators and we require them to be relevant, i.e. h < 1, we might naively conclude a different range ε ∈ (0, 2).The discrepancy is resolved by noticing h = kδ 2 − 2δ + 1 actually has a minimum 1 − 1 k at δ = 1 k .So in terms of ε, we indeed have ε ∈ (0, 2 k ), which serves as an amusing check.

IR physics
Let's first recall the key aspects of the isotropic Kondo lines.The global symmetry 11 of the isotropic Kondo model is SU (2), The analysis of the IR physics is then two-folded.Firstly, we 11 Technically we cannot discuss the symmetry group without specifying the information of the anti-chiral part of the bulk CFT, which we try to avoid.Therefore, the symmetry we consider here is only about the operator algebra, which might have 't Hooft anomaly or other global issues when we realize the symmetry on the Hilbert space.would like to know the properties of the defect lines at the IR fixed point.Secondly, we would like to know what are the corrections to the physical observables, e.g.expectation value of the defect lines away from the IR fixed point.This is done by looking for SU(2) invariant irrelevant operators in the IR.It has been known for a long time [34,35] that the answer depends on how the dimension n = 2 j + 1 of the representation of the spin is compared to the level k.
• j < k 2 : the Kondo defect line flowing to the IR fixed point becomes the Verlinde line with label j and the corrections come from the descendant of the spin 1 primary J a −1 φ a with dimension 2 k+2 + 1.
• j ≥ k 2 : the Kondo defect line flowing to the IR fixed point becomes the tensor product between the invertible Verlinde line of label k/2 and an IR-free Kondo defect line with an impurity of spin j − k 2 .The corrections come from the operator J a J a of dimension 2.
Let's now discuss the global symmetry in the anisotropic Kondo problem.Since we break the reletive coefficient between [J + t − + J − t + ] δ and J 0 t 0 , we are left with automorphism group whereas 2 acts by One can indeed verify together they form the group 12 O(2).
In the case of spin j = 1 2 or n = 2, below are some examples of possible IR scenarios with the leading O(2)-preserving irrelevant deformations.
The IR defect could be the same as the isotropic case, namely a Verlinde line L 1 2 of spin j = 1 2 for the current algebra su(2) k • At k = 1, the only operators living on the line L 1 2 are current algebra descendant of the identity operator, starting from J 0 J 0 , J + J − and J − J + with dimension 2.
• At k ≥ 2, line L 1 2 support a spin 1 primary operator denoted as φ a .So the leading , which is small than that of J 0 J 0 , J + J − and J − J + .
The IR line defect might be Verlinde lines for the chiral algebra 13 u(1) k , which is the current algebra from i∂ ϕ and extended by the vertex operator : e i2 kϕ : of dimension k.There are k Verlinde lines L u(1) s , labelled by s = 0, . . .k − 1, all of which have quantum dimension 1.Another natural possibility is having an IR-free Kondo defect line of spin j = 1 2 , which becomes a direct sum of two identity lines with a possible twist δ IR J 0 t 0 in the far IR.This mirrors what happens in the UV: There are operators of the form Recall in the UV, as discussed in section 2, we use them to initiate a relevant RG flow with δ UV ∈ [0, 2  k ].They will be natural irrelevant operators in the IR with δ IR taken outside this region.and k = 3 on the right.

WKB analysis in the IR
In this section, we perform the exact WKB analysis on the ODE (52) in the limit θ → ∞, which can tell us the infrared behaviours of the line defect.Here we are using a more refined WKB analysis developed in [15,16] than the Voros/GMN-style one since the latter is only applicable to meromorphic potentials with simple zeroes [64][65][66][67][68].We will refer the readers to the appendices of [15,16] and references therein for more details.Physical RG flow with g > 0: this corresponds to considering WKB analysis in large real θ limit for the ODE (52) with g > 0.
An example of the WKB diagrams is given in the left panel of figure 4.There is an infinite sequence of zeros of order k.Local behaviours around each zero are exactly the same as the order k zero considered in the isotropic case.Since we are only interested in the case of j = 1 2 , i.e. n = 2 whose associated Stokes lines are connected at the zero on the real axis, we will only need to analyze the local behaviour around this zero.So we have the same behaviour as in the isotropic case: the leading order of T -functions is the quantum dimension of the su(2) k current algebra.And the corrections come in powers of γ = e −θ 2 k+2 .This means that in this region of the parameter, the anisotropic Kondo lines flow to Verlinde lines L 1 2 as in the isotropic case.But this makes sense since there is no O(2)preserving operators that have dimension smaller than 1 + 2 k+2 .This provides a direct derivation of the conjectured result from an infinitesimal analysis in small ε given in [43], whereas for us ε can be finite.'Unphysical' RG flow with g < 0: this corresponds to considering WKB analysis in large real θ limit for the ODE (52) with g < 0. Given (89), this RG flow might look 'unphysical' since it will correspond to the coupling λ being purely imaginary in the Kondo line defect (34).However, surprisingly it is physically meaningful to consider such an analytically continued line defect including complex scale θ and complex coupling g in both formal theory and condensed matter applications [15,69].
The WKB diagram is given in the right panel of figure 4. The structure is very similar to the case above with g > 0, except that two Stokes lines closest to the real axis (corresponding to ψ −1 and ψ 1 respectively) are connected to the negative infinity instead of at a zero.
To analyze the local behaviour around the negative infinity, suppose y = x − x −∞ is the local coordinate around x −∞ , whose real part is very large negative.In this coordinate, the potential is We might want to choose then the potential takes the form of with again g eff (θ ) ≡ g e εθ .Since we assume 0 < ε < 2 k , the exponent of g eff (θ ) is necessarily negative.In the infrared g eff (θ ) → +∞, the leading potential is then e (2−kε) y + α 2 and with appropriate normalization, we have the usual Wronskians The corrections come in integer powers of g eff (θ ) −1− εk 2−kε , which, by dimensional analysis, corresponds to an irrelevant operator with the dimension It must be a boundary-changing operator associated with a direct sum of two identity lines with deformation δ IR t 0 J 0 .In section 2, we have derived its scaling dimension to be h IR = kδ 2 IR −2δ IR +1.
Equating with (97), we find where δ UV is what we call δ in section 2, i.e. the deformation parameter labelling the UV fixed point.We add this subscript to emphasize the difference from δ IR .

When ε becomes larger
When ε is small, zeros are far apart.Each zero source k + 2 Stokes lines and we are approximately dealing with isotropic case close to each zero.This has been made precise in section 3.1.When ε becomes large, zeros come closer but the span in the imaginary direction of the k+2 lines stays the same.We might worry that the span of k+2 lines between neighbouring zeros would overlap.Indeed, zeros of 1 − g e εx = 0 for g > 0 are given by So zeros are separated by 2πi ε in the imaginary direction.Since each zero sources k + 2 WKB lines that span (k + 1)π in the imaginary direction, they will overlap when 2 k+1 < ε < 2 k .This is particularly relevant when k is small.We plot some WKB diagrams for k = 3 and g > 0 in figure 5.When j < k 2 , the behaviour of physical RG flow (real θ ) stays the same, i.e. it flows to the Verlinde line L j of su (2) k as in the isotropic case.However, the physical RG flow for j > k 2 and the complex RG flow exhibit an increasing complexity as the WKB diagram gets more complex when ε is larger.None of this can be deduced from the Kondo defect picture.This perfectly exemplifies the advantage of ODE/IM correspondence and exact WKB analysis in deriving the IR behaviours of the defect RG flows.When ε > 2 k , there will also be small solutions associated with the negative infinity, rather than just positive infinity, which is outside the scope of this paper.

Generalizations and future directions 4.1 Multichannel generalization
We have verified that the Stokes data of the proposed ODE (52) matches the expectation value of the anisotropic Kondo defect line in the chiral SU(2) k WZW model.In the construction of 4d Chern Simons, the generalization to the multichannel case i SU(2) k i is obvious.We just need to have multiple insertions of surface defect wrapping 2 ×{z 1 , z 2 . . .}.The meromorphic one-form is 1 where z = e εx , ε = ħ h, and z i = g i .Therefore the ODE is given by where t(x) ≡ a 2 + + ∂ a + with the straightforward generalization of a + (x) where u a and u ′ b are fixed by the trivial monodromy condition as in ( 66) and (67).As for the Kondo defect, we just replace λ t ± J ∓ δ with i λ i t ± J ∓ i δ , which can be checked with the ODE (101) above by a straightforward but a bit tedious calculation in the fashion of the appendix of [16].Since the structure is the same, we will not repeat it here.Note that one might be tempted to turn on different δ i for different factors of SU (2), which corresponds to having ε i different from each other.However, there is no obvious way to generalize (102) or to generalize the 4d Chern Simons theory to have multiple ħ h i .Therefore we conjecture such line defects are not integrable.It would be interesting to explore this further.

Higher spin
In this article, one of the pieces of evidence for the ODE/IM correspondence is that we have verified explicitly (68) in the UV for n = 2 or spin 1  2 .While the construction of T n [θ ] as Stokes data from the ODE side works for any n, we only discussed the anisotropic Kondo defect for n = 2.In the case with impurities of higher spin n > 2, the main idea is the same: the UV fixed point we start from is a certain marginal deformation of a direct sum of n identity defect; then the RG flow is initiated by turning on defect-changing operators.However, both the space of marginal deformations and the space of defect-changing operators are much larger and the unbroken U(1) symmetry does not give us enough constraints as it does for n = 2.A k , there will also be small solutions associated with the negative infinity rather than just positive infinity, which is outside the scope of this paper.Whenever needed (except ε = 2 3 ), a small imaginary part is given to the potential P(x) to avoid the appearance of the saddle connection, i.e.Stokes line connecting two zeros.See the discussion of ϑ-WKB diagram in [68] for more details.priori, we don't know which RG flow in this large space of couplings is integrable.Based on our experience with 4d Chern Simons theory in the trigonometric setting [37,38], we expect to find the finite-dimensional matrix representations 14 of U q (sl 2 ), which in principle can be verified by enforcing the commutativity relation (3) and the Hirota fusion relation (4).
In fact, due to the simplicity 15 of the renormalization as we discussed at the end of the section 2, we can explicitly show the equivalence of the anisotropic Kondo defect with the transfer matrix [18,54,70] for U q ( sl( 2)).Using the notation used in this article, the transfer matrix for the level k = 1 reads 16,17   T ℓ (λ) ≡ Tr ℓ e i2πβ t 0 J 0 0 Pexp λ 2π 0 du V −β q t 0 t + + V β q −t 0 t − (103) = Tr ℓ e i2πβ t 0 J 0 0 ∞ m=0 λ m σ 1 ,...,σ m =± q σ 1 t 0 t σ 1 . . .q σ m t 0 t σ m (104) where t 0 , t ± are generators of U q (sl 2 ) and the trace is taken in the 2l + 1 dimensional matrix representation.The deformation parameter q = e iπβ 2 /4 Let's first look at the ℓ = 1 2 case.Note that for the two-dimensional representation, we have q ±t 0 t ± 2 = q −1 q 2t 0 t ± 2 = 0 , (106) q t 0 t + q −t 0 t − = qt + t − , (107) q −t 0 t − q t 0 t + = qt − t + , (108) we then find Using the identity used in (37) and the argument in figure 3, we can show that the twist e 4πbt 0 J 0 0 in the trace enables us to cyclically permute operators without introducing monodromies.Therefore two terms are actually the same.Recall from (20) that J 0 acts on a state 18 in the defect Hilbert space by we then find an exact match with the Kondo defect given in section 2 with the following identification λ ↔ q as before we can define α ≡ 1 2 βα.The equivalence we just derived between the spin 1 2 anisotropic Kondo defect and the transfer matrix for U q (sl 2 ) confirms the common lore: the anisotropic Kondo line defect is integrable when the matrix t 0 , t ± are in the representations of U q (sl 2 ) instead of sl 2 .We will leave it as a future direction to explicitly verify this in the higher spin.

Anisotropic vs coset
From the ODE (101), it is interesting to perform a change of coordinate y = e −εx and we find where t( y) ≡ ε −2 y −2 t(x) − 1 4 y 2 .This is precisely the ODE proposed in equation (7.7) of [52].On the other hand, according to [15], it describes a Kondo defect in a coset where the excited states are identified with those of the (101) after a spectral flow due to the Schwarzian contribution − 1 4 y 2 .The existence of the branch cut makes it less preferred for the analysis than (101).However, abstractly it would be really interesting to understand the relationship between two interpretations.This curiosity already exists in the isotropic ODE (5), but it seems to be a bit more intriguing in the anisotropic case.For example, we can try to lift them to 4D Chern Simons theory.They both have a 4d Chern Simons construction in a trigonometric setting on * but with different boundary conditions at the origin.
There is also an involutive automorphism induced by the Weyl group W(sl 2 ) = 2 We therefore have Aut( sl 2 ) = ⋊ 2 .In particular, the even part is inner and corresponds to the affine Weyl group W( sl 2 ) = (2 ) ⋊ 2 .Consequently, the induced action by U 2 maps each integral highest weight representation into itself, whereas more general U α maps between the (twisted) modules.For example,

Figure 1 :
Figure 1: A schematic RG flow diagram for the anisotropic Kondo model (6).The green dashed line is the isotropic flow.

Figure 5 :
Figure 5: WKB diagrams for k = 3, g = 1 5 and a sequence of increasing ε as labelled inside the figures.The zeros cease to be isolated when the span of k +2 lines overlap, i.e. ε ∈ [ 2 k+1 , 2 k ].When ε > 2k , there will also be small solutions associated with the negative infinity rather than just positive infinity, which is outside the scope of this paper.Whenever needed (except ε = 23 ), a small imaginary part is given to the potential P(x) to avoid the appearance of the saddle connection, i.e.Stokes line connecting two zeros.See the discussion of ϑ-WKB diagram in[68] for more details.