Integrable Deformations from Twistor Space

Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons in 6 dimensions. We provide the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. Starting from 6d holomorphic Chern-Simons theory on twistor space with a particular meromorphic 3-form $\Omega$, we construct the defect theory to find a novel 4d integrable field theory, whose equations of motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry reducing, we find a multi-parameter 2d integrable model, which specialises to the $\lambda$-deformation at a certain point in parameter space. The same model is recovered by first symmetry reducing, to give 4d Chern-Simons with generalised boundary conditions, and then constructing the defect theory.


Introduction
Developing a systematic understanding of the landscape of integrable 2-dimensional field theories [IFT 2 ] has been a longstanding challenge.In the case of non-linear sigma-models with target space a group manifold G, with associated Lie algebra g, a significant advance was the introduction of integrable deformations of both the principal chiral model [PCM] and the Wess-Zumino-Witten [WZW] model, known respectively as η-, or Yang-Baxter [YB] [1], and λ-models [2].These discoveries have paved the way to wider classes of integrable deformations with applications to worldsheet string theory and holography (for a recent survey see, e.g., [3]).
At the classical level, the integrability of these models can be captured by a flat g C -valued Lax connection D = d + L which depends meromorphically on a spectral parameter ζ ∈ CP 1 .The Poisson algebra of the spatial component L σ can be entirely characterised by a single meromorphic function φ(ζ) known as the twist function.The form of this algebra implies that an infinite tower of higher-spin conserved charges in involution can be constructed [4].
In a remarkable sequence of works [5,6,7,8,9] it was shown that, by geometrising the spectral parameter plane and considering it as part of space-time, such 2-dimensional integrable models have a 4-dimensional origin as a Chern-Simons type (CS 4 ) theory.
The action of CS 4 for a g C -valued gauge field A is defined as in which ω is a meromorphic differential on CP 1 .To recover the integrable 2-dimensional theories above, this meromorphic differential should be specified in terms of the relevant twist function as To fully define the theory, the action should be complemented with a choice of boundary conditions on the gauge field at the location of the poles of ω.With suitable boundary conditions in place this 4-dimensional theory localises to a 2-dimensional integrable theory defined on the worldsheet Σ.For the case of η-and λ-models, the relevant boundary conditions were constructed in [9].See also [10] for a recent review.
Four-dimensional Chern-Simons theory can in turn be understood [11] as coming from a reduction of 6-dimensional holomorphic Chern-Simons [hCS 6 ] on Euclidean twistor space, An action of the form (1.3) was first considered in [12] as the cubic open string field theory action for the type B topological string.In the context of type B topological string theory, the target space-time is necessarily Calabi-Yau which ensures it is complemented with a trivial canonical bundle, admitting a globally holomorphic top form Ω. Twistor space however is not Calabi-Yau and as such does not possess a trivial canonical bundle.Therefore, to study (1.3) on PT we instead require that Ω is a meromorphic (3, 0)-form on PT which, in this work, is assumed to be nowhere vanishing.Schematically, this process can be understood as defining a non-compact Calabi-Yau 3-fold by excising the poles of Ω from PT, which we can now take to be a consistent target space of our type B topological string [13].This action has also been studied in [14] with a focus on dimensionally reduced gravity and supergravity.
The connection between hCS 6 and CS 4 can be immediately anticipated since PT is diffeomorphic to E 4 × CP 1 (see appendix A.1 for twistor space conventions).Identifying the CP 1 with the spectral parameter plane, we specify a reduction ansatz, known as symmetry reduction, which identifies the worldsheet Σ → E 4 .The details, however, are subtle.While ω has both zeroes and poles, Ω only has poles (the zeroes arise from the data that specifies the symmetry reduction).Moreover, suitable boundary conditions on A are less well understood and currently only explicitly known for a limited class of theories, primarily Dirichlet-type boundary conditions that give, e.g., the principal chiral model with Wess-Zumino term.The generalisation away from Dirichlet boundary conditions necessary to recover lines of continuous integrable deformations, including the aforementioned η-and λ-models, is not known.Indeed, obstacles to the construction of integrable deformations from 6 dimensions were highlighted in [15].
Starting in 6 dimensions, an appealing prospect is to swap the order of symmetry reduction and localisation.Indeed performing the integration over CP 1 (localising to the poles of Ω) results in a 4-dimensional theory.Here the avatar of integrability is that the equations of motions can be recast as an anti-self-dual Yang-Mills [ASDYM] equation.The connection between integrable equations and ASDYM has long been known (see [16]), and ASDYM has been shown to provide a 4-dimensional analogue of 2-dimensional rational CFTs (the WZW model in particular) [17,18,19].We then anticipate a return to the same 2-dimensional integrable model by performing symmetry reduction.This gives rise to a diamond correspondence of theories hCS 6 in which the wavy arrows indicate symmetry reduction and straight arrows indicate localisation (integration over CP 1 ).In the context of integrable deformations of IFT 2 , the right-hand side of this diamond is less well understood.As such, a key goal of the present work is to develop this side of the diamond for deformed models, and in particular for λ-type deformations of the WZW and coupled WZW models.
Briefly, the key results of this work are: 1 We establish the consequence of a new class of boundary conditions for hCS 6 .These reduce to a wider class of boundary conditions in CS 4 than have previously been considered (relaxing the assumption of an isotropic subalgebra of the defect algebra).
2 Integrating over CP 1 results in a novel multi-parametric IFT 4 whose equations of motion can be recast in terms of an anti-self-dual Yang-Mills connection.This new IFT 4 exhibits two semi-local symmetries, which can be understood as the residual symmetries preserving the boundary conditions.
For each of these two semi-local symmetries, the Noether currents can be used to construct two inequivalent Lax formulations of the dynamics.
3 Upon symmetry reduction, this IFT 4 descends to the 2-field λ-type IFT 2 of [20] providing a new multi-parametric sigma-model example of the Ward conjecture [21].Generically the semi-local symmetries of the IFT 4 reduce to global symmetries of the IFT 2 and the two Lax formulations of the IFT 4 give rise to two Lax connections of IFT 2 .

Holomorphic 6-Dimensional Chern-Simons Theory
Our primary interest in this work will be the hCS 6 diamond containing the λ-deformed IFT 2 originally constructed in [2].By proposing a carefully chosen set of boundary conditions, we will be able to find a diamond of theories that arrives at a multi-parametric class of integrable λ-deformations between coupled WZW models.
To this end we restrict our study of hCS 6 , defined by the action to the case where the (3, 0)-form is given by, in the basis of (1, 0)-forms defined in appendix A.2, (2.2) Here, we view PT as diffeomorphic to E 4 × CP 1 and adopt the standard coordinates x AA ′ on E 4 and homogeneous coordinates π A ′ on CP 1 .See appendix A.1 for further details of twistor notation and conventions.The gauge field is similarly written in the basis of (0, 1)-forms as and the action is invariant under shifts of A by any (1, 0)-form, i.e.A → A + ρ where ρ ∈ Ω (1,0) (PT).
The first step in studying the 6-dimensional theory is to impose conditions ensuring the vanishing of the 'boundary' term that appears in the variation of the action Since Ω is meromorphic, as opposed to holomorphic, this receives contributions from the poles at α, α, and β.We assume that Dirichlet conditions A A | π=β = 0 are imposed at the second-order pole.At the first-order poles, we can then evaluate the integral over CP 1 to obtain 6 the condition (2.5) For reasons that will shortly become apparent, let us introduce a unit norm spinor µ A about which we can expand any spinor X A as Expanding the gauge field components in terms of the basis µ A and μA , and solving locally pointwise on E 4 , this condition may be written as (2.7) The boundary conditions we are led to consider are

.8)
6 To compute the boundary variation of the action, we have used the identities e C ∧ e C ∧ ēA ∧ ēB = −2 vol 4 ε AB (where where we have introduced the free parameter σ, which will play the role of the deformation parameter in the IFT 4 .
Let us note that these boundary conditions are invariant under the following discrete transformations (2.10) These will descend to transformations that leave the IFT 4 invariant.

Residual Symmetries and Edge Modes
A general feature of Chern-Simons theory with a boundary is the emergence of propagating edge modes as a consequence of the violation of gauge symmetry by boundary conditions.A similar effect underpins the emergence of the dynamical field content of the lower dimensional theories that descend from hCS 6 .
Generally, group-valued degrees of freedom, here denoted by h and h, would be sourced at the locations of the poles of Ω.If however, the boundary conditions (2.8) admit residual symmetries, then these will result in symmetries of the IFT 4 potentially mixing the h and h degrees of freedom.These may be global symmetries, gauge symmetries, or semi-local symmetries depending on the constraints imposed by the boundary conditions.It is thus important to understand the nature of any residual symmetry preserved by the boundary conditions (2.8).
Gauge transformations act on the hCS 6 gauge field as ĝ : In the bulk, i.e. away from the poles of Ω, these are unconstrained, but at the poles they will only leave the action invariant if they preserve the boundary conditions.For later convenience, we will denote the values of the gauge transformation parameters at the poles by (2.12) Firstly, the transformation acting at β must preserve the constraint A A | β = 0. Initially, one might suppose that only constant ℓ would preserve this boundary condition, but in fact it is sufficient for ℓ to be holomorphic with respect to the complex structure defined by β These differential constraints arise from the fact that the anti-holomorphic vector fields ∂A = π A ′ ∂ AA ′ are valued in O(1).In other words, they depend explicitly on the CP 1 coordinate (see appendix A.1 for more details).
Secondly, the transformations acting at α and α must preserve the boundary conditions (2.8), implying the constraints r = r , (2.14) These residual symmetries are neither constant (i.e.global symmetries) nor fully local (i.e.gauge symmetries).Instead, we expect that our IFT 4 should exhibit two semi-local symmetries subject to the above differential constraints, akin to the semi-local symmetries of the 4d WZW model first identified in [17,18] 7 .

Symmetry reduction
As we progress around the diamond, we will perform 'symmetry reduction' (see §4 and §5 for details).In essence, this will mean we restrict to fields and gauge parameters that are independent of two directions, i.e. they obey the further differential constraints (where γ A ′ is some constant spinor) We can then predict some special points in the lower dimensional theories by considering how these differential constraints interact with those imposed by the boundary conditions.Generically, these four differential constraints (two from the boundary conditions and two from symmetry reduction) will span a copy of E 4 at each pole, meaning that only constant transformations (i.e.global symmetries) will survive.
However, if the symmetry reduction is carefully chosen, the two sets of constraints may partially or entirely coincide.In the case that they entirely coincide, the lower dimensional symmetry parameter will be totally unconstrained, meaning that the IFT 2 will possess a gauge symmetry.Alternatively, if the constraints partially coincide then the lower dimensional theory will have a symmetry with free dependence on half the coordinates, e.g. the chiral symmetries of the 2d WZW model.Let us now proceed in navigating the top right-hand side of the diamond.By integrating over CP 1 we will 'localise' hCS 6 on PT to an effective theory defined on E 4 .This resulting theory is 'integrable' in the sense that its equations of motion can be encoded in an anti-self-dual connection.
The localisation analysis is naturally presented in terms of new variables A ′ and ĥ, which are related to the fundamental field by A = ĥ−1 A ′ ĥ + ĥ−1 ∂ĥ .
However, there is some redundancy in this new parametrisation.There are internal gauge transformations (leaving A invariant) given by ĝ : These allow us to impose the constraint A ′ 0 = 0, i.e. it has no leg in the CP 1 -direction.This is done so that A ′ may be interpreted as an anti-self-dual Yang-Mills connection on E 4 .
There are still internal gauge transformations that are CP 1 -independent, and we can use these to fix the value of ĥ at one pole.We will therefore impose the additional constraint ĥ| β = id so that we have resolved this internal redundancy.The values of ĥ at the remaining poles will be dynamical edge modes as a consequence of the violation of gauge symmetry by boundary conditions.
As we will now see, the entire action localises to a theory on E 4 depending only on these edge modes.
The hCS 6 action is written in these new variables as PT Ω ∧ Tr ĥ−1 ∂ĥ ∧ ĥ−1 ∂ĥ ∧ ĥ−1 ∂ĥ . (3.4) The cubic term in A ′ has dropped out since we have imposed A ′ 0 = 0. Inspecting the terms in our action involving ĥ, we see that the second term localises to the poles due to the anti-holomorphic derivative acting on Ω.The third term similarly localises to the poles.For this, we consider a manifold whose boundary is PT. 8 We take the 7-manifold PT × [0, 1] and extend our field ĥ over this interval.Denoting this extension with the same symbol, we see that the third term in our action may be equivalently written as d Ω ∧ Tr ĥ−1 d ĥ ∧ ĥ−1 d ĥ ∧ ĥ−1 d ĥ . (3.5) Then, using the closure of the Wess-Zumino 3-form and the fact that all of the holomorphic legs on PT are saturated by Ω, this is equal to ∂Ω ∧ Tr ĥ−1 d ĥ ∧ ĥ−1 d ĥ ∧ ĥ−1 d ĥ . (3.6) Therefore, this contribution also localises, meaning that the only information contained in the field ĥ : PT → G are its values9 at the poles of Ω. Explicitly, this contribution is given by where are the (2, 0)-forms with respect to the complex structure on E 4 defined by α A ′ and αA ′ respectively. 10nowing that the latter two terms in the action (3.4) localise to the poles, we are one step closer to deriving the IFT 4 .There are two unresolved problems: the first term is still a genuine bulk term; and the second term contains A ′ , rather than being written exclusively in terms of the fields h and h.Both of these problems will be resolved by invoking the bulk equations of motion for A ′ .This will completely specify its CP 1 -dependence, and, combined with the boundary conditions, we will then be able to solve for A ′ in terms of the edge modes h and h.
Varying the first term in the action, which is the only bulk term, we find the equation of motion ∂0 A ′ A = 0, which implies that these components are holomorphic.Combined with the knowledge that A ′ A has homogeneous weight 1, we deduce that the CP 1 -dependence is given by Turning our attention to the boundary conditions, we first consider the double pole where we have imposed A A | β = 0. Recalling that ĥ| β = id, this simply translates to A ′ A | β = 0.This tells us that A ′ A = ⟨πβ⟩B A for some B A .Therefore, we have that The solution for B A found by solving the remaining two boundary conditions (2.8) is written more concisely if we introduce some notation.We will make extensive use of the operators which enjoy the useful identities where transposition is understood to be with respect to the ad-invariant inner product on g.In terms of the components of ĥ−1 ∂ AA ′ ĥ, defined with useful normalisation factors, we find that the solutions to the remaining boundary conditions may be written as Recovering the IFT 4 is then simple.The first term in the action (3.4) vanishes identically on shell, and we can substitute in our solution for A ′ in terms of h and h to get a 4d theory only depending on these edge modes.This results in the action where Observe that the 4d IFT (3.16) with (3.17) is mapped into itself under the formal transformation interchanging the positions of the two poles.This directly follows from the invariance (2.9) of the hCS 6 boundary conditions.On the other hand, looking at how the transformation (2.10) descends to the IFT 4 , we find It is then straightforward to check that the action (3.16) with (3.17) is invariant under this transformation.
Let us emphasise that, to our knowledge, the IFT 4 described by the action (3.16) with (3.17) has not been considered in the literature before.In the following subsections we will study some properties of this model starting with its symmetries, and moving onto its equations of motion and their relation to the 4d ASDYM equations.

Symmetries of the IFT 4
Having derived the action functional for the IFT 4 , we will now examine those symmetries that leave this action invariant.While they may not be obvious from simply looking at the action, in §2.1 we leveraged the hCS 6 description to predict the symmetries of the IFT 4 .These may then be verified by explicit computation.
To this end, we recall that the hCS 6 gauge transformations act as ĝ : and we denoted the value of this transformation parameter at the poles by Tracing through the derivation above, we find that these result in an induced action on the IFT 4 fields, (ℓ, r, r) : h → ℓhr , h → ℓ hr , (3.22) where ℓ, r and r obey the constraints (2.13) and (2.14) respectively.One can explicitly verify that the IFT 4 is indeed invariant under these transformations.Key to this is exploiting a Polyakov-Wiegmann identity such the variation of S WZ4 in eq.(3.17) produces a total derivative.This gives a contribution on E 4 that cancels the variation of the remainder of eq.(3.16).Useful intermediate results to this end are in which the constraints (2.13) and (2.14) have been used.
The only contribution to the variation localises to β and is given by

.25)
Since A A | β ∝ ⟨πβ⟩ (recall that we fix ĥ| β = id) the only way the integrand will be non-vanishing is for the , and hence the variation becomes If we think of the ℓ-symmetry as a global transformation, then this would provide the conservation law associated to the Noether current, i.e.
and indeed we will see later that this conservation law follows from the equations of motion of the IFT 4 .
As the parameter ϵ (ℓ) is allowed to holomorphic with respect to the complex structure defined by β, the interpretation is more akin to a Kac-Moody current.
For the case corresponding to the r-symmetry we have In this case the variation receives two contributions with an opposite sign Integrating by parts gives Introducing new currents defined by the conservation law associated to the r-symmetry is given by Recalling from eq. (3.9) that A A = ⟨πβ⟩Ad −1 ĥ B A + π A ′ ĥ−1 ∂ AA ′ ĥ, we can relate the B current to the C and C currents as follows where we have used the identities (3.15).The transformation of these currents under the (ℓ, r)-symmetries is given by (ℓ, r) : As a consequence notice that the 4d (CP 1 -independent) gauge field, the right action, whereas the left action acts as a conventional gauge transformation (ℓ, r) : albeit semi-local rather than fully local.The transformation of these currents and the 4d ASD connection also follows from the hCS 6 description.While the original gauge transformations act on A, we observe that r and r become right-actions on ĥ, leaving A ′ and A AA ′ invariant.By comparison, after fixing ĥ| β = id, it is only a combination of the 'internal' transformations and the original gauge transformations that preserve this constraint.In particular, ℓ has an induced action on h, h and A ′ , thus leading to the above transformations of B A and A AA ′ .
As we will show momentarily, the equations of motion of the theory correspond to anti-self duality of the field strength of the connection A AA ′ , hence it immediately follows that the equations of motion are preserved by the symmetry transformations (3.35).To close the section we note that the action is concisely given in terms of the currents as

Equations of Motion, 4d ASDYM and Lax Formulation
The equations of motion of the IFT 4 can be obtained in a brute force fashion by performing a variation of the action (3.16).This calculation requires the variation of the operators U but is otherwise straightforward.The outcome is that the equations of motion can be written as Taking a weighted sum and difference equations gives the first of which is the anticipated conservation equation for the ℓ-symmetry.Making use of the definitions of C and C in (3.32), the equations of motion are equivalently expressed as we can take the difference of these equations to obtain which is the anticipated conservation law for the r-symmetry.
ASDYM.We will now justify the claim that this theory is integrable by constructing explicit Lax pair formulations of the dynamics in two different fashions.First we will show the equations of motion (3.39) can be recast as the anti-self-dual equation for a Yang-Mills connection.Before demonstrating that this holds for our particular model, let us highlight that this follows from the construction of hCS 6 by briefly reviewing the Penrose-Ward correspondence [23].Recalling that we have resolved one of the hCS 6 equations of motion it follows that the remaining system of equations should be equivalent to the vanishing of the other components of the field strength F ′ AB = 0. We may express this in terms of the anti-holomorphic covariant derivative D′ A = ∂A + A ′ A as [ D′ A , D′ B ] = 0, which may also be written as This is equivalent to the vanishing of π A ′ π B ′ F AA ′ BB ′ where F is the field strength of the 4d connection To make contact with the anti-self-dual Yang-Mills equation, note that an arbitrary tensor that is anti-symmetric in Lorentz indices, e.g.F µν , can be expanded in spinor indices as Here, ϕ and φ are both symmetric and correspond to the self-dual and anti-self-dual components of the field strength respectively.Explicitly computing the contraction (3.43), we find that the remaining equation is simply ϕ = 0 which is indeed the anti-self-dual Yang-Mills equation.In effect, this argument demonstrates that a holomorphic gauge field on twistor space (which is gauge-trivial in CP 1 ) is in bijection with a solution to the 4-dimensional anti-self-dual Yang-Mills equation -this statement is the Penrose-Ward correspondence.
Now, returning to the case at hand, recall that our connection is of the form A AA ′ = B A β A ′ , so the anti-self-dual Yang-Mills equation specialises to This should hold for any π A ′ ∈ CP 1 and we can extract the key information by expanding π A ′ in the basis formed by α A ′ and αA ′ , that is Substituting into (3.45),we find two independent equations with CP 1 -dependent coefficients ⟨πβ⟩⟨π α⟩ and ⟨πβ⟩⟨πα⟩ respectively.These are explicitly given by the two equations in eq.(3.39).Therefore, as expected, the equations of motion for this IFT 4 are equivalent to the anti-self-dual Yang-Mills equation for A AA ′ .

B-Lax.
The anti-self-dual Yang-Mills equation is also 'integrable' in the sense that it admits a Lax formalism.Using the basis of spinors µ A and μA , we define the Lax pair L and M by in which the normalisations are for later convenience.It is important to emphasise that here π A ′ is not just an ad hoc spectral parameter.It is introduced directly as a result of the hCS 6 equations of motion and is the coordinate on CP 1 → PT.The vanishing of [L (B) , M (B) ] = 0 for any π A ′ ∈ CP 1 is equivalent to the anti-self-dual Yang-Mills equation.

C-Lax.
Let us now turn to the equations of motion cast in terms of the C-currents in eq.(3.41).
Evidently, looking at the definition of these currents eq.(3.32), we see that when σ = 1 we have C = C and the equations of motion have the same form as the B-current equations (3.39).Therefore, when σ = 1, we can package the C-equations in terms of a ASDYM connection A (C) Away from this point, when C ̸ = C it is not immediately evident if these equations follow from an ASDYM connection.
Regardless, we can still give these equations a Lax pair presentation as follows.
Letting ϱ ∈ C denote a spectral parameter we define Unlike the spectral parameter π A ′ entering the B-Lax, there is no clear way to associate the spectral parameter of the C-Lax, ϱ, with the CP 1 coordinate alone.Indeed, under a natural assumption, we will see that ϱ has origins from both the CP 1 geometry and the parameters that enter the boundary conditions.
The existence of a second Lax formulation of the theory, distinct from the ASDYM equations encoded via the B-Lax, is an unexpected feature.We will see shortly that, upon symmetry reduction, this twin Lax formulation is inherited by the IFT 2 .

Reality Conditions and Parameters
To understand how the action of the IFT 4 (3.16) with (3.17) depends on the parameters K, σ, α A ′ , αA ′ , β A ′ and µ A , let us denote our coordinates such that and Substituting into the action (3.16) with (3.17), we see that the IFT 4 now only depends explicitly on two parameters which we see by noting that U † ± = U T ∓ .With symmetry reductions to Lorentz-invariant theories in mind, it is also interesting to analytically where σ ± and τ ± are real coordinates.The action again only depends on the parameters K ′ and σ, and is real when these parameters are real.The space-time symmetry of the action is now SO(1, 1) acting on the light-cone coordinates in the usual way, σ ± → e ±ϑ σ ± and τ ± → e ±ϑ τ ± .
In §5, we will be interested in symmetry reducing while preserving the space-time symmetry, recovering an action on R 2 or R 1,1 that is invariant under the Euclidean or Poincaré groups respectively.We have freedom in how we do this since the action is not invariant under SO(2) rotations acting on (z, w) and (ẑ, ŵ), or equivalently (σ + , τ + ) and (σ − , τ − ).Therefore, we can choose different directions along which to symmetry reduce in each of these planes, in principle introducing an additional two parameters.We should note that in the Euclidean case, since the two planes are related by conjugation, we will break the reality properties of the action unless the two symmetry reduction directions are also related by conjugation, reducing the number of parameters by one for a real action.This is not an issue in the Lorentzian case since the coordinates are real, hence we expect to find a four-parameter real Lorentz-invariant IFT 2 .We will indeed find that this is the case in §5.
A final interesting real slice to consider is If we also assume that h and h are related by conjugation, then the action is real for real K ′ and σ, since Again, the action is invariant under the space-time symmetry (3.54).Let us note that we can reach these reality conditions by tuning α = −α and ⟨α β⟩ = ⟨αβ⟩.One way to understand this, is that after complexification, we can use an SL(2, C) transformation on CP 1 to fix the three points α, α and β, such that when we take the real slice we recover a real action.This is then equivalent to first taking the real slice and then modifying the conjugation properties of the coordinates.

Equivalent Forms of the Action and its Limits
In this section we describe alternative, but equivalent ways of writing the action of the 4d IFT (3.16) with (3.17), and consider two interesting limits of the theory.These constructions are motivated by analogous ones that are important in the context of the 2d λ-deformed WZW model.
First, let us note that the IFT 4 (3.16) with (3.17) can be written in the following two equivalent forms where Written in this way, it is straightforward to see that the symmetries of the 4d IFT are given by transformations of the form (3.22) with which, as expected, coincide with (2.13) and (2.14) upon using the definitions (3.50).
We can also introduce auxiliary fields B A , C A and C A to rewrite the action as (3.61) Here we take the auxiliary fields B A , C A and C A to be undetermined.Varying the action and solving their equations of motion, we find that on-shell, they are given by the expressions introduced above in eqs.(3.13) and (3.33).Moreover, substituting their on-shell values back into (3.61)we recover the 4d IFT.Using the symmetry (2.9), we note that the action can also be written in a similar equivalent form, in which tilded and untilded quantities are swapped, σ → σ −1 , K ′ → −K ′ and B = B.This can also be seen by making the off-shell replacements all of which are compatible with the on-shell values of the auxiliary fields.
The first limit we consider is σ → 0, in which the action becomes This has the form of a current-current coupling between two building blocks that could be described as 'holomorphic WZW 4 ' of the form This somewhat unusual theory has derivatives only in the holomorphic two-plane singled out by the complex structure on E 4 defined by α (i.e.only ∂ w and ∂ ŵ enter), although the field depends on all coordinates of E 4 .This structure is quite different (both in the kinetic term and Wess-Zumino term) from the conventional WZW 4 [24] for which the action 13 is The Kähler point of the theory is achieved when β = α such that ϖ α,β is the Kähler form associated to the complex structure defined by α.In fact, the WZ term of our holomorphic WZW 4 is of this general form with β = α such that ϖ α,β defines a (2, 0)-form.However even in the β = α case, the kinetic term does not match that of the holomorphic WZW 4 action.
Returning to the holomorphic WZW 4 , we can establish that the theory is invariant under a rather large set of symmetries.Since only w and ŵ derivatives enter, it is immediate that the transformation h → l(ẑ, z)hr(ẑ, z) leaves the action eq.(3.63) invariant.These are further enhanced, as in a WZW 2 , to From this perspective holomorphic WZW 4 can be considered the embedding of a WZW 2 in 4 dimensions.
Similarly, we have a symmetry for the holomorphic WZW 4 for h ( l, r) : h → l(ẑ, ŵ, w) h r(z, ŵ, w) . (3.66) The interaction term, j ȷ , in the action (3.62) preserves the right actions, but breaks the enhanced independent ℓ, l left actions.Instead a new 'diagonal' left action emerges It is important to emphasise that here the right actions on h and h are independent (r and r are not the same).This stems from the enlargement of the residual symmetries of the 6-dimensional boundary conditions.The constraints of eq.(2.14) are relaxed such that gauge parameters at different poles are unrelated but are chiral.
In this limit the currents associated to the left and right action become (3.68) The conservation laws become To compute the O(σ) correction to the action (3.62) we first note that and that the combination Then from the expression of the IFT 4 action in terms of currents (3.36), we see that the leading correction to SIFT4 is given by i.e. the perturbing operator is the product of two currents associated to the right-acting symmetries.
The second limit we consider is σ → 1. Recall that in this limit, we have that C = C from eqs. (3.32) and (3.33), and a symmetry emerges interchanging B and C, as well as h and h−1 .This is also evident if we set σ = 1 in (3.61).An alternative way to take σ → 1 is to first set h = exp( ν K ′ ) and h = exp( ν K ′ ), along with σ = e 1 K ′ and take K ′ → ∞.In this limit the 4d IFT becomes which is reminiscent of a 4d version of the non-abelian T-dual of the principal chiral model, albeit with two fields instead of one.If instead we take the limit in the action with auxiliary fields (3.61), also setting , we find after integrating by parts.Integrating out the auxiliary field B A , we recover the action (3.72).It would be interesting to instead integrate out the fields ν and ν to give a 4d analogue of 2d non-abelian T-duality.
However, note that, unlike in 2 dimensions, ν and ν do not enforce the flatness of a 4d connection, hence there is no straightforward way to parametrise the general solution to their equations.Returning to hCS 6 , we now descend via the top left-hand side of the diamond by performing a symmetry reduction of the action.Doing so, we find the resulting theory is CS 4 .
The idea of symmetry reduction is to take a truncation of a d-dimensional theory specified by a d-form Lagrangian L d depending on a set of fields {Φ} to obtain a lower dimensional theory.We assume here that we are reducing along two directions singled out by vector fields V i , i = 1, 2. The reduction procedure imposes that all fields are invariant, L Vi Φ = 0, with dynamics now specified by the d − 2-form Lagrangian While similar in spirit to a dimensional reduction, there is no requirement that V i span a compact space, hence there is no scale separation in this truncation.
In order to perform the symmetry reduction, we will introduce a unit norm spinor γ A ′ about which we can expand any spinor X A ′ as The spinor γ A ′ defines another complex structure on E 4 which coincides with the complex structure on It coincides with the opposite complex structure -swapping holomorphic and anti-holomorphic -at the antipodal point π A ′ = γA ′ .Using the spinor µ A , we can define a basis of one-forms adapted to this complex structure, We will perform symmetry reduction along the vector fields dual to dz and dz, The symmetry reduction along the ∂ z and ∂ z directions takes us from a theory on PT to a theory on Σ × CP 1 in which w, w are coordinates on the worldsheet Σ.
To perform this reduction, it is expedient [25] to make use of the invariance of the action (2.1) under the addition of any (1, 0)-form to A → Â = A + ρ 0 e 0 + ρ A e A .By choosing ρ A appropriately, we can ensure that Â has no dz or dz legs and is given by where these components are related to those of A by An important feature to note is that Â necessarily has singularities at γ and γ.While at the 6-dimensional level this is a mere gauge-choice artefact, it plays a crucial role in the construction of the CS 4 theory.
In these variables, the boundary variation and boundary condition of hC6 6 are restated as where we have introduced the combinations Upon symmetry reduction to CS 4 , r ± will correspond to the residues of the 1-form ω.
Since the shifted gauge field Â manifestly has no dz or dz legs, and we impose L z Â = L z Â = 0, the contraction by ∂ z and ∂ z only hits Ω.It then follows that the symmetry reduction yields in which the meromorphic 1-form on CP 1 is given by (4.10) To compare with the literature, we will also translate to inhomogeneous coordinates on CP 1 .The CP 1 coordinate itself will be given by ζ = π 2 ′ /π 1 ′ on the patch π 1 ′ ̸ = 0. We also specify representatives for the various other spinors in our theory.Without loss of generality we can choose such that We also denote the inhomogeneous coordinates for γ A ′ and γA ′ by Then, the meromorphic 1-form ω is written in inhomogeneous coordinates as To complete the specification of the theory we simply note that the 6d boundary conditions immediately Before we discuss the residual symmetries of the CS 4 models, let us make two related observations.First, fixing the shift symmetry to ensure Â is horizontal with respect to the symmetry reduction introduces poles into our gauge field Â at γ and γ.Thus, despite starting with potentially smooth field configurations in 6 dimensions we are forced to consider singular ones in 4 dimensions.We can understand the origin of these singularities by recalling the holomorphic coordinates on E 4 with respect to the complex structure on PT = CP 3 \CP 1 .Described in detail in appendix A.1, PT is only diffeomorphic to E 4 × CP 1 , and the complex structure is more involved in these coordinates.The holomorphic coordinates on E 4 with respect to this complex structure are given by v A = π A ′ x AA ′ , which align with our coordinates {z, w} at π ∼ γ and {z, w} at π ∼ γ.It is precisely at these points that we are forced to introduce poles by the symmetry reduction procedure.
Second, in line with the singular behaviour in the gauge field, we have also introduced zeroes in ω at π ∼ γ and π ∼ γ whereas Ω in 6 dimensions was nowhere vanishing.Of course, given the pole structure of Ω, the introduction of two zeroes was inevitable given the Riemann-Roch theorem.

Residual Symmetries and the Defect Algebra
Let us take a moment to consider the residual symmetries of these CS 4 models.Here the residual symmetry preserving the boundary condition (4.15) is generically constrained to only include constant modes, At the special 'diagonal' point in parameter space where t = s = 1, notice these differential equations are identically solved and we find a local gauge symmetry.This enhancement of residual gauge freedom matches with previous considerations in the context of CS 4 , where diagonal boundary conditions of the form A| α = A| α are known to give rise to the λ-deformed WZW as an IFT 2 , a theory that depends on a single field h.The residual gauge symmetries are those satisfying ĝ| α = ĝ| α and can be used to reduce the number of fields appearing in the resulting IFT 2 to one (see §5.4 in [9]).
Another interesting point occurs when we take t = s or t = s −1 in which case we retain a chiral residual symmetry.When t = 0 the boundary conditions admit an enlarged residual symmetry as there is no requirement that r = ĝ| α and r = ĝ| α match.Instead they must be chiral and of opposite chiralities i.e. ∂ w r = ∂ w r = 0.As mentioned earlier, for more generic values of t and s the residual symmetries will be constrained, preventing them from being used to eliminate any degrees of freedom.While these boundary conditions have not been yet considered for t, s ̸ = 1 and t ̸ = 0, we will see that they give rise to the multi-parametric class of λ-deformations between coupled WZW models introduced in [20].
To make further contact with the literature, it is helpful to rephrase the boundary conditions (4.15) in terms of a defect algebra, which in the case at hand is simply the Lie algebra d = g + g equipped with an ad-invariant pairing The boundary conditions (4.15) read Since ⟨⟨l t , l t −1 ⟩⟩ = 0 the boundary conditions are suitable, however it should be noted that l t is itself neither a subalgebra nor an isotropic subspace of d.This is more general than boundary conditions previously considered 14 in the context of 4-dimensional Chern Simons theory.In particular, we might expect that generalising [27,28,29] to boundary conditions defined by subspaces that are neither a subalgebra nor an isotropic subspace of d will lead to novel families of 2-dimensional integrable field theories.
It is worth highlighting that these boundary conditions still define maximal isotropic subspaces, but now inside the space of algebra-valued 1-forms, rather than just the defect algebra.Consider the space of g-valued 1-forms on Σ × CP 1 , equipped with the symplectic structure15 The boundary conditions above define maximal isotropic subspaces with respect to this symplectic structure, that is half-dimensional subspaces Y ⊂ Ω 1 (Σ × CP 1 ) ⊗ g such that W(X, Y ) = 0 for all X, Y ∈ Y. Indeed, this is required for them to be 'good' boundary conditions.The isotropic subspaces of the defect algebra described earlier are then special cases of these subspaces.In this section we will apply the same symmetry reduction previously applied to hCS 6 to the IFT 4 .In doing so we derive the IFT 2 corresponding to the CS 4 model described above.
Recalling that the reduction requires that the fields h and h depend only on w, w and not on z, z, we can simply set ∂ z = ∂ z = 0 in the action eq.(3.16).To compare with the literature, when discussing 2-dimensional theories we will define ∂ + ≡ ∂ w and ∂ − ≡ ∂ w (implicitly rotating to 2d Minkowski space where the action is rendered real for real parameters) and denote To evaluate the symmetry reduction, denoted by ⇝, of the IFT 4 action we first note that 2) The resulting 2-dimensional Lagrangian is given by (trace implicit) This theory, depending on two G-valued fields h and h, exactly matches a theory introduced in [20] as a multi-field generalisation of the λ-deformed WZW model [2].To make a precise match with [20] we relate their fields (g 1 , g 2 ) to our fields (h, h−1 ).The model in [20] is defined by two WZW levels k 1,2 and by two deformation matrices which we take to be proportional to the identity with constants of proportionality λ 1,2 .The mapping of parameters is then (5.4) In 2d Minkowski space, the Lagrangian (5.3) is real if the parameters r ± , s t and σ are all real.Assuming K and σ are real, this is the case if r + and r − have the opposite sign and the parameters α ± and γ ± lie on the same line in C, which we can take to be the real line without loss of generality.This follows since r ± , s and t are all expressed as ratios of differences of α ± and γ ± , hence are invariant under translations and scalings. 16

Limits
The four-parameter model has a number of interesting limits, many of which are discussed in [20].Here, we briefly summarise some key ones.First, let us take t → 0. In order to have a well-defined limit we keep σt −1 = λ finite, implying σ → 0 as well. 17The resulting 2d Lagrangian is given by This current-current deformation preserves half the chiral symmetry of the G r+ × G −r− WZW model, which corresponds to the UV fixed point λ = 0. Indeed, this model can be found by taking chiral Dirichlet boundary conditions in 4d CS [8,26], corresponding to the special case t = 0 in the boundary conditions we find from symmetry reduction (4.19).Assuming −r − > r + , in the IR we have that λ = s −1 .At this point the Lagrangian can be written as (5.6) Redefining h → hh, we find the G r+ × G −r−−r+ WZW model.In the case of equal levels r − = −r + this reduces to the G r+ WZW model.
The equal-level, r − = −r + , version of (5.5), whose classical integrability was first shown in [30], is canonically equivalent [31] and related by a path integral transformation [32] to the λ-deformed WZW model.Indeed, from the point of view of 4d CS, these two models have the same twist function.To recover (5.5) with equal levels, we take chiral Dirichlet boundary conditions, t = 0, s = 1 in (4.19), while to recover the λ-deformed WZW model we take diagonal boundary conditions t = s = 1.
It follows that if we take t = s = 1 in eq. ( 5.3), we expect to recover the λ-deformed WZW model.
Indeed, setting r − = −r + and t = 1, the Lagrangian (5.3) becomes As explained in subsection 2.1, at this point in parameter space the symmetry reduction directions are aligned such that the constrained symmetry transformations (2.14) become a gauge symmetry of the IFT 2 .This allows us to fix h = 1, recovering the standard form of the λ-deformed WZW model [2] with σ playing the role of λ.Further taking σ → 0, we recover the G r+ WZW model.
Another point in parameter space where we expect a gauge symmetry to emerge is when the symmetry reduction preserves the left-acting symmetry.This corresponds to setting s = 1, i.e. r − = −r + and t = σ.
Doing so we find where we recall that U ± are defined in eq.(3.59).This Lagrangian is invariant under a left-acting gauge symmetry as expected, which can be used to fix h = 1.We again recover the standard form of the λ-deformed WZW model with σ playing the role of λ.The CS 4 description of this limit is analysed in appendix C.
Before we move onto the integrability of the 2d IFT and its origin from the 4d IFT, let us briefly note the symmetry reduction implications of the formal transformations (3.18) and (3.19), which in turn descended from the discrete invariances of the hCS 6 boundary conditions (2.9) and (2.10).The first (3.18)implies that the 2d IFT is invariant under (5.9) 17 An analogous limit is to take σ → 0 and keep t finite.
recovering the 'duality' transformation of [20].Since the second involves interchanging w and z, it tells us the parameters are transformed if we symmetry reduce requiring that the fields h and h only depend on z, z, instead of w, w.We find that σ → σ −1 and t → tσ −2 .

Integrability and Lax Formulation
The analysis of [20] shows that the equations of motion (5.3) are best cast in terms of auxiliary fields18 B ± , C ± which are related to the fundamental fields by (5.10) The equations of motion for h and h, together with the Bianchi identities obeyed by their associated Maurer-Cartan forms, can be repackaged into the flatness of two Lax connections with components (5.11) Here ζ GS is the spectral parameter used in [20].Taken together, the flatness of this pair of Lax connections contains implies both the Bianchi identities and the equations of motions.However, if one is prepared to enforce the definition (5.10) of auxiliary fields in terms of fundamental fields (such that the Bianchi equations are automatically satisfied) then either Lax will generically (i.e.away from special points in parameter space such as λ i = 1) imply the equations of motion 19 of the theory.
We can relate this discussion to the construction above by symmetry reducing the 4d Lax operators (3.47) and (3.48).First we note that the currents corresponding to the (ℓ, r)-symmetries reduce to simple combinations of the auxiliary fields introduced in eq. ( 5.10) (5.12) Notice that all explicit appearances of the operators U ± have dropped out such that these currents reduce exactly to the 2-dimensional auxiliary gauge fields.
Using the complex coordinates adapted for symmetry reduction defined in eq.(4.2), and introducing a specialised inhomogeneous coordinate on CP 1 given by ς = ⟨πγ⟩/⟨πγ⟩, the 4d B-Lax pair (3.47) may be written as We can symmetry reduce the 4d Lax pairs, L (B/C) , M (B/C) of eqs.(3.47) and (3.48) to obtain (5.14) Now using the inhomogeneous coordinates introduced in eqs.(4.11) and (4.13), and the relations between parameters (5.4), the 4d Lax operators immediately descend upon symmetry reduction to the 2d Lax connections (5.11), provided the 4d and 2d spectral parameters are related as The relation between ζ GS and ς can be recast in the standard CP 1 homogeneous coordinate π ∼ (1, ζ) as such that if we choose to fix γ ± = ±1 then ζ GS = ζ −1 .If we make the assumption that the ζ GS entering in the two different Lax formulations have the same origin then we can map between between ϱ and the CP 1 homogeneous coordinate (5.17) Therefore, under this assumption, we see that ϱ depends on the parameter t, which is part of the specification of boundary conditions and not just geometric data of CP 1 .Indeed ϱ becomes constant when t → 1, hence there is no spectral parameter dependence left.In contrast, when t → 0, we have ϱ → −ζ.
6 Localisation of CS 4 to IFT 2 Finally, we localise the CS 4 theory obtained by symmetry reduction of hCS 6 in §4.This will result in a 2-dimensional theory on Σ, which matches the IFT 2 derived from symmetry reduction of the IFT 4 in §5.
In the following discussion we will make use of the E-model formulation of CS 4 [27,28,29].In this approach we accomplish localisation via algebraic means, constructing from the data of our CS 4 a defect algebra and projectors.The choice of boundary conditions in CS 4 corresponds to a choice of isotropic subspace of our defect algebra, from which we can then write down the action and Lax connection for the corresponding 2d IFT.To obtain the IFT 2 we could also follow an analogous route to that taken §3, namely integrating out the CP 1 directions directly.Details of this approach are presented in appendix B.
The gauge field Â is related to the 2-dimensional Lax connection by a change of variables that takes the form of a gauge transformation, which importantly does not preserve the boundary conditions, Â = ĥ−1 d ĥ + ĥ−1 L ĥ .( As before, we fix the redundancy in this parametrisation by demanding that L has no legs in the CP 1 direction, though may of course depend on it functionally, and that ĥ| β = id.The bulk equations of motion, ω ∧ F [ Â] = 0, ensures that L is flat and meromorphic in ζ with analytic structure mirroring ω.
The key idea is that the field ĥ evaluated at the poles serve as edge modes that become the degrees of freedom of the IFT 2 , and the boundary conditions will determine the form of the Lax connection L in terms of these fields.However, the complete construction requires a more careful treatment, especially when Ω has higher order poles [27].
Let us start by recalling the boundary conditions eq. (4.19) phrased in terms of A w and A w valued in the defect algebra d = g + g.These are that A w ∈ l t and A w ∈ l t −1 , where these subspaces are mutually orthogonal ⟨⟨l t , l t −1 ⟩⟩ = 0. Given these boundary conditions for the gauge field at the simple poles, we introduce a group valued field h ∈ D = exp d and an algebra element L ∈ d such that L can be understood as the 2d Lax connection lifted to the double by evaluating it at the poles of the spectral parameter.From this, and the known singularity structure of Â, we will recover the full Lax connection.
It is important to emphasise that most previous treatments have assumed that A w and A w lie in the same subspace, and moreover that this space is a maximal isotropic subalgebra l ⊂ d.The only exception that we know of are the chiral Dirichlet boundary conditions of [8,26], which are a special case of our boundary conditions.Taking l to be an isotropic subalgebra of d ensures that the resulting IFT 2 has a residual gauge symmetry given by the left action of exp (l) on h.This can be fixed by setting h ∈ D/ exp(l).For example, if we take r + = −r − , then l = g diag , the diagonally embedded g in d = g + g, is a suitable isotropic subalgebra and letting 20 h = (h −1 , h−1 ) the residual symmetry can be fixed by setting h = id.Here, however, we do not have any such residual gauge symmetry in general and the 2-dimensional theory will depend on the entire field content in h.
As above, we switch notation in 2 dimensions to The projectors W ± h : d → d are defined via their kernel and image where we recall that ζ = γ ± are the zeroes of ω.Care is required to correlate the zeroes of omega with the pole structure of A, which has been determined by our symmetry reduction data.In the case at hand for instance, the zeroes at π = γ and π = γ are associated to poles in A w and in A w respectively.
In order to unpack the IFT 2 action, let us outline the calculation of W + h (h −1 ∂ + h).Defining the useful we can parameterise the kernel and image of W + h as We decompose h −1 ∂ + h into the kernel and image by solving This yields in which we see the reappearance of the auxiliary combinations encountered earlier in eq.(5.10).It follows that ) 20 Including inverses in this parametrisation makes it easier to compare to results obtained elsewhere. 21In terms of these parameters, the relations (4.8) become from which we can evaluate In a similar fashion we find that and Taking the difference of eq. ( 6.10) and eq.( 6.12) we find that the Lagrangian of the 2-dimensional action eq.( 6.3) exactly matches the IFT 2 obtained previously in eq. ( 5.3) by descent on the other side of the diamond.This explicitly verifies our diamond of theories.
Let us note that this IFT 2 has also been constructed from CS 4 in a two-step process in [34].First, a more general 2-field model based on a twist function with additional poles and zeroes, and the familiar isotropic subalgebra boundary conditions, is constructed.Second, a special decoupling limit is taken, where a subset of these poles and zeroes collide.It remains to understand how to recover our boundary conditions (4.15) from those considered in [34].
To complete the circle of ideas we can also directly obtain a Lax formulation from CS 4 .This is essentially achieved by undoing the map into the defect algebra as follows.Given an element X = (x, y) ∈ d, we determine a, b ∈ g such that We introduce a map ℘ into the space of g-valued meromorphic functions in terms of which the components of the Lax connection are given by In this way we recover the Lax L 1 that we obtained from symmetry reduction of the 4d ASDYM Lax pair with the identification (5.16).There does not appear an algebraic derivation in this spirit of the other Lax L 2 .This in contrast to the derivation of this model from CS 4 in [34] where the extra data associated to the additional poles means that both Lax in (5.11) can be directly constructed.More generally, this highlights an interesting question that we leave for the future about the integrability and the counting of conserved charges, beyond the existence of a Lax connection, when we consider boundary conditions not based on isotropic subalgebras.

RG Flow
Let us recall the RG equations given in [20] λi where dot indicates the derivative with respect to RG 'time' d d log µ .The levels k 1 and k 2 and λ 0 = k 1 /k 2 are RG invariants.In this section we will interpret this flow in terms of the data that is more natural from the perspective of 4d CS, namely the poles and zeroes of the differential and the boundary conditions of the theory.
Using the map between parameters given in eq. ( 5.4) we can infer from eq. ( 6.16) a flow on the parameters {t, α ± , γ ± , K}.Let us first consider the parameter t = λ 2 .As discussed in [20], there is a flow from t = 0 in the UV to t = λ 0 in the IR (assuming that λ 0 < 1).Explicitly the flow equation has the solution The interesting observation is that the boundary conditions display algebraic enhancements at the fixed points.In the UV, t = 0 limit, these boundary conditions become chiral, A w ∈ g R ⊂ d and A w ∈ g L ⊂ d.While g L,R are now subalgebras, neither are isotropic with respect to the inner product (4.17).In non-doubled notation the UV limit becomes Âw | α = 0 , Â w| α = 0 .( On the other hand in the IR limit, t = λ 0 , we see that A w ∈ g diag ⊂ d, again a subalgebra, but only an isotropic one for k 1 = k 2 , i.e. r + = −r − .In non-doubled notation the IR limit becomes 22   Âw While in general, there are no residual gauge transformations preserving the boundary conditions, in the UV and IR limits we notice chiral boundary symmetries emerging.For example, in the IR these are those satisfying g −1 ∂ wg = 0, which corresponds to t = s −1 in eq.(4.16).
Let us now turn to the action of RG on the differential ω.An immediate observation is that the RG invariant WZW levels are given by monodromies about simple poles 23 exactly in line with the conjecture of Costello (reported and supported by Derryberry [35]).While there are more parameters in ω than there are RG equations, we can form the ratios of poles and zeroes .24) in terms of which the RG system of [20] translates to . (6.25) 22 The seemingly more democratic boundary condition of t = 1, which does define an isotropic space of d (not a subalgebra however) is not attained along this flow. 23The monodromy about the double pole at infinity is trivially RG invariant since the sum of all the residues vanishes.
The RG invariants are given by which allows us to retain either of q ± as independent variables.We can directly solve these equations and a remarkable feature, also conjectured by Costello, is that this quantity is precisely the contour integral between zeroes d d log µ To best understand the action of the RG flow on the locations of the poles directly, we replace K with the RG invariant k 2 (or k 1 ), and fix the zeroes to be located at γ ± = ±1.This yields the RG invariant and a flow equation the solution of which is As illustrated in fig.6.1, this system displays a finite RG trajectory linking fixed points.In the UV limit the poles accumulate to different zeroes, and in the IR the poles accumulate to the same zero.Let us consider the upper red trajectory of fig.6.1 in which we choose λ 0 < 1 and pick the positive branch of the With this choice we see that there are finite fixed points 24 such that the right hand side of eq.(6.30) vanishes at UV : (α − , α + ) = (−1, 1) , λ 1 = 0 , IR : (α − , α + ) = (1, 1) , λ 1 = λ 0 , (6.32) in which we recall the map . (6.33) One of the appealing features of the IFT 2 (5.3) is that it provides a classical Lagrangian interpolation that includes its own UV and IR limits [20].That is to say these CFTs can be obtained directly from the Lagrangian eq. ( 5.3) by tuning the parameters of the theory to their values at the end points of the RG flow.Given the interpretation of these RG flows as describing poles colliding with zeroes it is natural to expect that a similar interpolation can be obtained directly in 4d by taking limits of the differential ω in eq.(6.17).
Here we will explore how this works for the IFT 2 (5.3) in the IR.The limit we will consider is to collide the poles at α ± with the zero at γ + , following the upper red trajectory in fig.6.1.This corresponds to taking q ± → 0. In order to be consistent with the RG invariants (6.26), we take this limit as .34)Taking this limit in (6.17), and redefining the spectral parameter such that the remaining pole and zero are fixed to 1 and −1 respectively, yields  Let us consider the implication of this limit from the CS 4 perspective.Given that the pole structure of ω is modified in this limit, so will the double d, and thus we should be careful in our interpretation of the boundary conditions.If we take ω to be given by (6.35) and consider the boundary conditions (6.20) with t = 0, the condition A w ∈ l t becomes Âw | α = 0. From eq. (4.5) we know that Âw has a pole at γ + .
In other words, we can write Âw = with Ξ(ζ) regular as α + → γ + .Hence, in the IR there is no boundary condition for Âw at ζ = 1.On the other hand, the boundary condition A w ∈ l t −1 for t = 0 is Â w| ᾱ = 0 which in the limit α + , γ + → 1 becomes a chiral boundary condition for the w component Â w| ζ=1 = 0 .(6.37) For this choice of boundary condition one can localise the CS 4 action following the procedure described in either §4 or appendix B, and the resulting two dimensional IFT is the WZW model at level k 2 − k 1 .
In contrast, from the 2-dimensional perspective it is known that the full result at this IR fixed point is actually a product WZW model on G k2 × G k1−k2 [20].This indicates that there is some delicacy in taking the IR limit directly as a Lagrangian interpolation in 4d even when it possible in 2d.One reason for this is there is also the freedom to perform redefinitions of the spectral parameter, which can, in general, produce non-equivalent limits of ω.Such limits are known as decoupling limits [36,34] in the literature, and have been investigated for the UV fixed point of the bi-Yang-Baxter model in [37].

Discussion and Outlook
In this work we have constructed a diamond of integrable models related by localisation and symmetry reduction.Starting from holomorphic Chern-Simons theory with the meromorphic (3,0)-form (2.2), we have found a new choice of admissible boundary conditions, which leads to a well-defined 6-dimensional 24 There are also fixed points to the RG flow at α + = 0 with α however by assumption k 2 > k 1 , and so these do not correspond to real values of α − and consequently λ 1 is imaginary.We do not consider such complex limits here.
theory.This generalises the analysis carried out in [11,14] to a new class of boundary conditions not of Dirichlet type.
By first viewing twistor space, PT, as a CP 1 bundle over E 4 , we solved the equations of motion along the CP 1 fibres.In doing so we fully specified the dependence of the integrand on the CP 1 fibre and thus could perform a fibrewise integration along those directions.Consequently our 6-dimensional theory then localised to the poles of Ω, leading to a new 4-dimensional theory on E 4 given by the action (3.16).Indeed, this 4-dimensional theory is 'integrable' in the sense that its equations of motion can be encoded in an anti-self-dual connection, as expected from the Penrose-Ward correspondence.Moreover, this new IFT 4 exhibits two semi-local symmetries, which can be understood as the residual symmetries preserving the boundary conditions of hCS 6 .For each of these semi-local symmetries, the Noether currents can be used to construct inequivalent Lax formulations of the dynamics.
On the other hand, symmetry reducing hCS 6 along two directions of the E 4 in PT ∼ = CP 1 × E 4 , leads to an effective CS 4 theory on CP 1 × E 2 .Under this procedure, the meromorphic (3, 0)-form reduces to the meromorphic (1, 0)-form used in [9] to construct the λ-model, whereas the 6-dimensional boundary conditions reduce to a class of boundary conditions in CS 4 that have not been previously considered.
Specifically, they relax the assumption of an isotropic subalgebra of the defect algebra.By performing the standard localisation procedure of CS 4 we obtain the 2-field λ-type IFT 2 introduced in [20].
Notably, this same multi-parametric class of integrable λ-deformations between coupled WZW models can be obtained by symmetry reduction (along the same directions) of the novel IFT 4 mentioned above.From the perspective of the IFT 2 , there is a close relationship between the notions of Poisson-Lie symmetry, duality and integrability [38,39,40,41].This poses an interesting question as to the implications of such dualities for both the IFT 4 and hCS 6 .For the model considered here, we might seek to understand the semi-local symmetries of IFT 4 in the context of the q-deformed symmetries expected to underpin the IFT 2 .
In this work, the integrable models we have studied can be viewed as descending from the open string sector of a type B topological string.An interesting direction for future work is to consider the closed string sector [42] and its possible integrable descendants.A tantalising prospect is to understand the closed string counterparts of the integrable deformations we have considered in the context of the non-linear graviton construction for self-dual space-times [43,44,45].
By coupling the open and closed string sectors [46,47,48,49] one can find an anomaly free quantization to all loop orders in perturbation theory of the coupled hCS 6 -BCOV action.This mechanism has already proven a powerful tool in the context of the top-down approach to celestial holography [13,50].This could This basis of 1-forms is split into the (1, 0)-forms and (0, 1)-forms with respect to the complex structure inherited from CP 3 .They should also be understood as being valued in line bundles, since although they are normalised to have weight zero under the rescaling πA ′ ∼ r πA ′ , they carry weight under π A ′ ∼ rπ A ′ .
The dual basis of vector fields is given by

A.3 Hyper-Kähler Structure
Recall that E 4 can be given a hyper-Kähler structure consisting of a triplet J i , i = 1, 2, 3 of complex structures obeying J 2 i = −id and J 1 J 2 J 3 = id.For each J i the corresponding Kähler forms ϖ i are self-dual with respect to the metric ds 2 = 3 µ=0 dx µ ⊗ dx µ and may be given explicitly as Taking combinations of these J i , we find a space of complex structures parameterised by π We can also define this CP 1 -dependent complex structure in spinor notation with π A ′ ∼ (1, ζ) as Thinking of PT E as the bundle of complex structures over E 4 , the coordinates v A = π A ′ x AA ′ are holomorphic coordinates with respect to the complex structure J π .We may denote these coordinates by v A = (z π , w π ).The Kähler form corresponding to J π can equally be expressed as Holomorphic self-dual (2, 0)-and (0, 2)-forms are given by

B Alternative Localisation of CS 4 to IFT 2
In §6 the defect algebra approach to 4d CS (for a complete discussion see [28,29]) was utilised in the passage to the 2d IFT.In this appendix we will perform the localisation of 4d CS to the 2d IFT via a more standard route, mirroring the localisation approach from 6d hCS to the 4d IFT, showing that we also land on (5.3).
We recall the action (4.9) with the meromorphic one form of eq.(4.10) and boundary conditions as per eq.(4.7).

C Alternative CS 4 Setup for the λ-Model
In this section, we will consider an alternative symmetry reduction of our hCS 6 setup which also recovers the λ-deformed IFT 2 .In order to recover a 1-field IFT 2 , we need one of the semi-local residual symmetries of the IFT 4 to become a gauge symmetry under symmetry reduction.Let us denote the symmetry reduction vector fields by V 1 and V 2 .Taking the example of the residual left-action parameterised by ℓ, this must obey the constraint β A ′ ∂ AA ′ ℓ = 0.In order for this to become a gauge symmetry of the IFT 2 , the symmetry reduction constraints L V1 ℓ and L V2 ℓ = 0 must coincide with the pre-existing constraints on ℓ.This means that we must choose to symmetry reduce along the vector fields Following the recipe described elsewhere in this paper, we deduce that the CS 4 1-form is given by26  In the 4d CS description, we can already see that we have eliminated one degree of freedom relative to other symmetry reductions.The symmetry reduction zeroes have eliminated the double pole at β, effectively removing one field from the IFT 2 .
Furthermore, if we denote the surviving coordinates on Σ by y 1 = μA βA ′ ∂ AA ′ and y 2 = −µ A βA ′ ∂ AA ′ , the boundary conditions reduce to Since the localisation from CS 4 to the IFT 2 has been described in detail elsewhere, we will be brief in this section.In the parametrisation Â = ĥ−1 L ĥ + ĥ−1 d ĥ , (C.4) we fix the constraints Lζ = 0 and denote the values of ĥ at the poles by ĥ| α = h and ĥ| α = id.We can then use the bulk equations of motion and the boundary conditions to solve for L 1 and L 2 in terms of h.
We find the solutions Finally, the action localises to 2d and is given, up to an overall factor of K/(α + − α − ), by This can be recognised as the λ-deformed IFT 2 .
, and that b, b, b and b are related as b

. 38 )
in which we invoke the definitions of b, b, b and b above in eqs.(3.13) and (3.14).These equations can be written in terms of the current B A as

Furthermore, the
semi-local symmetries of the IFT 4 reduce to global symmetries of the IFT 2 and the two Lax formulations of the IFT 4 give rise to two Lax connections for the IFT 2 .When the directions of the symmetry reduction are aligned to these semi-local symmetries, the IFT 2 symmetries are enhanced to either affine or fully local (gauge) symmetries.In the latter case, the IFT 2 becomes the standard (1-field) λ-model.This work opens up a range of interesting further directions.There are a selection of direct generalisations that can be made to incorporate the wide variety of integrable deformations known in the literature.Perhaps the most interesting outcome of this would be the construction of swathes of new four-dimensional integrable field theories.Our work focused on the case where Ω was nowhere vanishing; it would be interesting to explore the relaxation of this condition together with its possible boundary conditions, and how the ASDYM equations are modified.Moreover, one might hope that the study of boundary conditions in hCS 6 could lead to a full classification of the landscape of integrable sigma-models in 2d, and perhaps result in theories not yet encountered in the literature.