Yang-Baxter deformations of the AdS5×S5 supercoset sigma model from 4D Chern-Simons theory

We present homogeneous Yang-Baxter deformations of the AdS5×S5 supercoset sigma model as boundary conditions of a 4D Chern-Simons theory. We first generalize the procedure for the 2D principal chiral model developed by Delduc et al. [5] so as to reproduce the 2D symmetric coset sigma model, and specify boundary conditions governing homogeneous Yang-Baxter deformations. Then the conditions are applicable for the AdS5×S5 supercoset sigma model case as well. In addition, homogeneous bi-Yang-Baxter deformation is also discussed.


Introduction
A significant subject in mathematical physics is to establish a unified picture to describe integrable systems [1,2]. By focusing upon 2D classical integrable systems including nonlinear sigma models (NLSMs), such a nice way was originally proposed by Costello and Yamazaki [3] based on a 4D Chern-Simons (CS) theory with a meromorphic 1-form ω . Notably, this 1-form ω is identified with a twist function characterizing the Poisson structure of the integrable system by Vicedo [4]. Recently, this procedure has been elaborated by Delduc, Laxcroix, Magro and Vicedo [5] so as to describe systematic ways to perform integrable deformations of 2D principal chiral model (PCM) including the Yang-Baxter (YB) deformation [6][7][8][9][10][11][12] and the λ-deformation [13,14]. For other recent works on this subject, see [15,16].
Our aim here is to generalize the preceding result on the PCM [5] to symmetric coset sigma models. By starting from a twist function in the rational description (with a slightly different parametrization of the spectral parameter), we specify a boundary condition associated with a symmetric coset. Then, the boundary condition is generalized so as to describe homogeneous YB deformations. It is straightforward to carry out the same analysis for the AdS 5 ×S 5 supercoset sigma model. As a result, the homogeneous YB deformations of the AdS 5 ×S 5 supercoset sigma model have been derived as specific boundary conditions of the 4D CS theory. This paper is organized as follows. Section 2 explains how to derive 2D NLSMs from 4D CS theory. In section 3, we derive 2D symmetric coset sigma models as boundary conditions of the 4D CS theory and then specify boundary conditions which describes homogeneous JHEP09(2020)100 Yang-Baxter deformation. In section 4, the results obtained in section 3 are generalized to the AdS 5 ×S 5 supercoset sigma model case. Section 5 is devoted to conclusion and discussion. Appendix A explains the computation concerned with a dressed R-operator in detail. In appendix B, we present homogeneous bi-Yang-Baxter deformed sigma models as boundary conditions of the 4D CS theory.
Note. Just before submitting this manuscript to the arXiv, we have found an interesting work [17]. The content of [17] has some overlap with us on the integrability of the AdS 5 ×S 5 superstring.

2D NLSM from 4D CS theory
This section explains how to derive 2D NLSMs from a 4D CS theory by following [3,5].
Let us begin with a 4D CS action [3], 1 where A is a g C -valued 1-form and CS(A) is the CS 3-form defined as Then ω is a meromorphic 1-form defined as and ϕ is a meromorphic function on CP 1 . This function is identified with a twist function characterizing the Poisson structure of the underlying integrable field theory [4]. Note that the z-component of A can always be gauged away like A = A σ dσ + A τ dτ + Az dz , (2.4) because ϕ(z) depends only on z and hence the action (2.1) has an extra gauge symmetry A → A + χ dz . (2.5) The pole and zero structure of ϕ will be important in the following discussion. The set of poles is denoted as p and that of zeros is z . At each point of z, the 1-form A cannot be regular because otherwise the action (2.1) is degenerate and hence the equations of motion at z cannot be determined.
By taking a variation of the classical action (2.1) , we obtain the bulk equation of motion and the boundary equation of motion dω ∧ A, δA = 0 . (2.7) 1 For the notation and convention here, see [18].

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Note that the boundary equation of motion (2.7) has the support only on M × p ⊂ M × and only the pole of ϕ can contribute as a distribution. The boundary conditions satisfying (2.7) are crucial to describe integrable deformations [3,5]. The bulk equation of motion (2.6) can be expressed in terms of the component fields: The factor ω is kept in order to cover the case ∂zA σ and ∂zA τ are distributions on CP 1 supported by z . It is also helpful to rewrite the boundary equation of motion (2.7) into the form x∈p p≥0 where ij is the antisymmetric tensor. Here the local holomorphic coordinates ξ x is defined as ξ x ≡ z − x for x ∈ p\{∞} and ξ ∞ ≡ 1/z if p includes the point at infinity. The relation (2.11) manifestly shows that the boundary equation of motion does not vanish only on M × p .
Lax form. By taking a formal gauge transformation with a smooth functionĝ : M × CP 1 → G C , the following gauge is realized Hence the 1-form L takes the form and we call L the Lax form. This will be specified as a Lax pair for 2D theory later. In terms of the Lax form L, the bulk equations of motion are expressed as It follows that L is a meromorphic 1-form with poles at the zeros of ω , namely z is regarded as the set of poles of L .

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Reality condition. It is natural to suppose some condition for the form of ω and its boundary condition on A so as to ensure the reality of the 4D action (2.1) and the resulting action (2.21) [5]. For a complex coordinate z, complex conjugation z → z defines an involution µ t : Let τ : g C → g C be an anti-linear involution. Then the set of the fixed point under τ is a real Lie subalgebra g of g C . The anti-linear involution τ satisfies The associated operation to the Lie group G is denoted byτ : Introducing the involutions, one can see that the reality of the action (2.21) is ensured by the conditions Recalling the relation (2.12), we suppose that so as to satisfy (2.19).
From 4D to 2D via the archipelago conditions. Whenĝ satisfies the archipelago conditions [5], the 4D action (2.1) is reduced to a 2D action with the WZ term by performing an integral over CP 1 as follows: Here R x is the radius of the open disk U x . The action (2.21) is invariant under a gauge transformation with a local function h : M → G C . This gauge symmetry can be seen as the remnant after taking the gauge (2.13) . Note here that we have not imposed the reality condition by following [18], in comparison to [5]. The reality condition will be introduced later when fixing a boundary condition ofĝ .

YB deformations of the symmetric coset sigma model
In this section, we will reproduce the action of a symmetric coset sigma model and homogeneous Yang-Baxter deformations of it from the 4D CS theory (2.1) by generalizing the work [5,18]. The symmetric coset case has been discussed in [3] in a slightly different way.

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Symmetric coset. Let G and H be a Lie group and its subgroup, and the Lie algebras for G and H are denoted as g and h , respectively. We assume that the Lie algebra g enjoys a Z 2 -grading, namely, g is decomposed like g = h ⊕ m as the vector space and the following relations are satisifed Twist function. The twist function for a symmetric coset sigma model is given by 2 where we have followed the notation in [19]. The meromorphic 1-form ω indeed satisfies the reality condition (2.18). The poles and zeros of ϕ c (z) are listed as where these poles are double poles, and each zero is a single zero. As we will see later, the twist function (3.3) is applicable not only to symmetric cosets, but also to homogeneous YB deformed sigma models.
Boundary condition. In order to specify a 2D integrable model, we need to choose a solution to the boundary equations of motion, Here the double bracket is defined as (x, y), (x , y ) p ≡ (res p ω) x, x + (res p ξ p ω) x, y + x , y ) = 4p K x, y + x , y . The boundary equations of motion (3.5) take the same form as in the PCM case. In the following, we will consider two classes of solutions. Note that A| z=±1 and ∂ z A| z=±1 take values in the real Lie algebra g, supposing the reality conditions (2.18) and (2.19) and the points z = ±1 are fixed points of the involution µ t .
The first class is where {0} g ab is an abelian copy of g defined as The twist function (3.3) is the same as the one for PCM, and they are related by a transformation where z is the spectral parameter for PCM.

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This configuration obviously solves the boundary equations of motion and lead to a symmetric coset sigma model as we will see later.
The second class is where g R and gR are defined as Here the linear R-operator R : g → g satisfies the homogeneous classical Yang-Baxter equation (hCYBE), The other R-operatorR : g → g is defined as where f : g → g is a Z 2 -grading automorphism of g . An explicit represetation will be given in (3.24). For f in (3.24) , we can show thatR also solves the hCYBE (3.11) if the Roperator R is a solution to the equation (3.11). Furthermore, thanks to the hCYBE (3.11), we can check that the second configuration in (3.9) solves the boundary equations of motion (3.5). The choice of the boundary conditions (3.9) is motivated by the one of homogeneous bi-YB deformations (For the details, see appendix B). Note that the first and the second solutions are related by a β-transformation at the Lie algebra level (For the details, see appendix A of [18]).
Lax form. Before deriving sigma model actions, we shall summarize our notation used in the following. We will takeĝ at each pole of the twist function (3.3) aŝ where g ,g ∈ G . Here g andg take values in G (not G C ) due to the reality condition (2.20) . The reality condition has been implicitly imposed at this moment. The associated leftinvariant currents are defined as Then, the relation between the gauge field and the Lax pair at each pole becomes From the zeros of the twist function (3.3), we suppose an ansatz for the Lax pair as where U ± , V ± ∈ g are undetermined functions of σ , τ , and the light-cone coordinates are defined as As we will see, the ansatz (3.16) of the Lax pair works well for the two classes of boundary conditions.

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i) symmetric coset sigma model. Let us first see the class i) that describes a symmetric coset sigma model. Under the boundary condition (3.7), the relations in (3.15) are rewritten as By solving these equations with respect to U ± and V ± , we obtain As a result, the Lax pair is expressed as Then, the residues of ϕ c L at z = ±1 are evaluated as By substituting (3.21) into (2.21), the 2D action is given by Ifg is independent of g, then by using the gauge symmetry of the 4D CS theory, we can rewrite the 2D action (3.22) to that of PCM with Lie group G [3,5].
Here we would like to impose a relation between j andj . Note that the resulting action (3.22) is invariant under the exchange of j andj . This invariance should be respected in a relationj = f (j) and hence the automorphism f : g → g should satisfy the following conditions: In order to obtain the known result, we will take f satisfying the following relations: Here we have introduced the generators of the decomposed vector space g = h ⊕ m as By employing the automorphism (3.24),j is evaluated as where the projection operators P (0) and P (2) are defined as, respectively,

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Then, by using the expression ofj in (3.26), the 2D action can be further rewritten as (3.28) and the Lax pair (3.20) becomes These are the standard expressions of the classical action and the associated Lax pair for a symmetric coset sigma model.
ii) homogeneous YB deformations. The next one we will discuss is the class ii) in (3.9) that describes homogeneous YB deformations of a symmetric coset sigma model. The condition gives a constraint on the gauge field A at each pole of the twist function, We again suppose the same ansatz (3.16) for the Lax pair. Then, the constraints in (3.30) lead to By solving these equations with respect to U ± and V ± , we obtain The residues of ϕ c L at z = ±1 take the same forms as (3.21), but V ± are given by (3.32). Thus the 2D action is given by Note that in the present case, the resulting action is invariant under the exchange of g and g, not j andj. The exchange symmetry of the action (3.33) at the level of group element leads to a slight change in the previous case: for group elements, we impose an additional conditioñ where an automorphism F : G → G has the Z 2 -grading property F • F (g) = g . To specify an explicit representation of F , let us take a parameterization of an element g ∈ G as where Xǎ and Xâ are functions of τ and σ . Then, in a neighborhood of the identity, F (g) can be written by using the automorphism f : g → g as follows, JHEP09(2020)100 or equivalently, Now let us rewrite the 2D action (3.33) by requiring (3.34). As shown in appendix A.1, we can show that the dressed R-operators R g andRg satisfy the following relation: (3.38) The relation (3.38) indicates Furthermore, by using (3.39), U ± can be rewritten as As a result, we obtain the 2D action and the Lax pair These are the standard expressions of the classical action and the Lax pair for a homogeneous YB deformed symmetric coset sigma model [12].

YB deformations of the AdS 5 × S 5 supercoset sigma model
In this section, we shall reproduce the Green-Schwarz (GS) action of the AdS 5 × S 5 supercoset sigma model [20] and homogeneous YB deformations of it [11] from the 4D CS theory.

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Twist function. The Poisson structure of the AdS 5 ×S 5 superstring has been considered in [21,22], and the twist function of the AdS 5 × S 5 supercoset sigma model is given by 3 Here ω is invariant under the involution µ t since ϕ str satisfies ϕ str (z) = ϕ str (z) . The poles and zeros of the twist function (4.2) are listed as where the poles are double poles and the zeros are triple zeros.
Boundary condition. The associated boundary equations of motion are where the double bracket is defined as As in the symmetric coset case, one may consider two classes of solutions to the boundary equations of motion (4.5). Now by considering the reality condition (2.19) leads to Due to the reality condition (4.8) , the relation holds for the following two boundary conditions. For the AdS 5 × S 5 supercoset sigma model, we take the following solution: where su(2, 2|4) p,ab is an abelian copy of su(2, 2|4) and su(2, 2|4) C : JHEP09(2020)100 The second choice for a homogeneous YB deformed AdS 5 × S 5 supercoset sigma model is given by ii) (A| z=p , ∂ z A| z=p ) ∈ su(2, 2|4) p,Rn p (p ∈ p) . (4.12) The subscript n p of R denotes the label of the poles as {n 1 , n i , n −1 , n −i } ≡ {1, 2, 3, 4} , and su(2, 2|4) p,Rn p is defined as Here the linear operators R k : g C → g C (k = 1, 2, 3, 4) are where the linear R-operator R : su(2, 2|4) C → su(2, 2|4) C is a solution to the hCYBE for su(2, 2|4) C , and f s : su(2, 2|4) C → su(2, 2|4) C is a Z 4 -grading automorphism of g C . An explicit representation is given in (4.27). For this representation, one can show that the R-operator R k also satisfies the hCYBE (3.11) for su(2, 2|4) C if R is a solution to the equation (3.11). Therefore, the boundary conditions (4.12) can be taken as solutions to the boundary equations of motion (4.5).
Lax form. Similarly to the symmetric coset sigma model case, let us takeĝ at each pole of the twist function (4.2) aŝ where g k ∈ SU(2, 2|4) (k = 1, 3) , g k ∈ SU(2, 2|4) C (k = 2, 4) , such that τ g 2 = g 4 . The associated left-invariant currents are defined as 17) and the relations between the gauge field A and the Lax pair L at each pole are written as From the zero structure of the twist function (4.2), we suppose the following ansatz for the Lax pair as where V [n] ± (n = −1, 0, 1) , V [±2] ± : M → su(2, 2|4) are smooth functions such that τ L = µ * t L. As we will see later, the ansatz (4.19) works well for both solutions to the boundary equations of motion. Note that the above ansatz (4.19) is not the only possible choice. One may consider other ansatz corresponding to the pure spinor formalism by following [3].

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i) the AdS 5 × S 5 supercoset sigma model. Let us reproduce the GS action of the AdS 5 × S 5 supercoset sigma model from the 4D CS action (2.1).

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where g (0) ⊕g (2) and g (1) ⊕g (3) are the bosonic and fermionic parts of su(2, 2|4) , respectively, and g (0) is identified with a bosonic subgroup so(1, 4) × so (5) . The commutation relations of g (m) satisfy [g (m) , g (n) ] ⊂ g (k) (m + n = k mod 4) . (4.26) In order to obtain the GS action, let us take the Z 4 -grading automorphism f s such that each subspace g (k) (k = 0, 1, 2, 3) is the eigenspace of f s satisfying Note that after taking a supermatrix realization of su(2, 2|4) , we can write down the explicit expression of f s (For the details, see [23]). The additional condition (4.24) enables us to express the functions V [n] in terms of the . In fact, by using (4.24) and (4.27), the left-invariant currents in (4.24) are rewritten as (4.28) Then, by substituting (4.28) into (4.21), the functions V [n] are given by From this result, we immediately obtain the Lax pair The expression (4.30) is precisely the same as the Lax pair constructed in [24]. Next, let us evaluate the 2D action (4.23). By using (4.29), we can see that the contribution to the 2D action from each pole is identical, namely, where d ± are the linear combinations of the projection operators P (i) like This fact comes from the cyclic symmetry of the 2D action (4.23). As a result, we obtain This is nothing but the Metsaev-Tseytlin action of the AdS 5 × S 5 supercoset sigma model [20].

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ii) homogeneous YB deformations. Let us next discuss homogeneous YB deformations of the AdS 5 × S 5 supercoset sigma model [11]. We consider the boundary condition (4.12). To avoid confusion of notations, we will replace the functions V [n] ± appeared in the Lax pair (4.19) with V [n] ± (∈ su(2, 2|4)). Then, from the boundary condition (4.12), we obtain the relations where the dressed R-operator R g k (k = 1 , . . . , 4) is defined as By introducing the linear operator the equations (4.34) are rewritten as ± take the expressions (4.21). Since the operator R (p) g is skewsymmetric, the equations (4.37) for V [n] ± can be uniquely solved and the associated 2D action can also be written down. However, the resulting 2D action has a rather complex form, and so instead of giving its explicit expression, we will only show that the associated 2D action is invariant under the cyclic permutation of g k (k = 1, . . . , 4) .
For this purpose, let us define the map Under this transformation, the linear operator R ± in (4.21) are transformed as P R and (4.40)

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From the transformation rules in (4.39) and (4.40), and the equations (4.37), the functions V [n] ± follow the same transformation rules as the functions V [n] (4.41) This fact indicates that the residues res p (ϕ str L) (p ∈ p) satisfy P (res p (ϕ str L)) = res p+1 (ϕ str L) , (4.42) where res p (ϕ str L) (p ∈ p) is given by Therefore, the associated 2D action (2.21) is invariant under the permutation of g k (k = 1, . . . , 4) . Thanks to the cyclic symmetry of the 2D action, we can require an additional condition where g ∈ SU(2, 2|4) , and the map F s : SU(2, 2|4) → SU(2, 2|4) is an automorphism of SU(2, 2|4) satisfying F 4 s = 1 . As in the symmetric coset case, let us take F s so as to be induced by f s defined in (4.27). More concretely, when a parameterization of an element g ∈ SU(2, 2|4) is taken as the automorphism F s is defined as Here, X A k are functions of τ and σ . By definition, F s is an automorphism of SU(2, 2|4) with the Z 4 -grading property. Then, as shown in appendix A.2, the dressed R-operator R g k that act on the generators of su(2, 2|4) should satisfy (4.48) This relation indicates . (4.49)

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By using the relation (4.49), the equations in (4.37) can be solved as where the deformed current J ± is defined as Thus the Lax pair is given by This is nothing but the Lax pair of homogeneous YB deformations of the AdS 5 × S 5 supercoset sigma model [11].
Next, let us derive the associated 2D action. By using (4.50), we find that the contribution to the 2D action from each pole is identical, namely, As a result, we obtain This action (4.54) is precisely the same as that of homogeneous YB deformations of the AdS 5 × S 5 supercoset sigma model [11].

Conclusion and discussion
In this paper, we have generalized the preceding result on the PCM to the case of the symmetric coset sigma model. By employing the same twist function in the rational description, we have specified boundary conditions which lead to the symmetric coset sigma model and the homogeneous YB-deformed relatives. The same analysis is applicable for the AdS 5 ×S 5 supercoset sigma model. As a result, homogeneous YB-deformations of the AdS 5 ×S 5 supercoset sigma model have been derived from the 4D CS theory as boundary conditions. In order to discuss the AdS 5 ×S 5 superstring beyond the sigma model, we have to take the Virasoro conditions into account by following the seminal work [17] in the present formulation. This is one of the most significant issues and the result will be reported in another place [25].
There are some open questions. It is well known that homogeneous YB deformations with abelian classical r-matrices can be seen as twisted boundary conditions [26][27][28][29][30] via non-local gauge transformations. It is interesting to consider the interpretation of this fact from the viewpoint of the 4D CS theory. It is also significant to understand how to realize the sine-Gordon model from the 4D CS theory. The sine-Gordon model can be reproduced JHEP09(2020)100 from the O(3) NLSM via the Pohlmeyer reduction at the classical level. Hence it would be nice to study how the Pohlmeyer reduction works in the context of the 4D CS theory.
It is also interesting to study the η-deformation based on the modified classical YB equation as well, though we have discussed only the homogeneous YB-deformations. We will report the result in another place [25].

A Relations for dressed R-operators
Here we shall prove the relations (3.38) and (4.48) that dressed R-operators should satisfy.

A.1 Z 2 -grading case
Let us first give a proof of the relation (3.38) for a dressed R-operator.
To begin with, we examine how a dressed R-operator R g acts on the generators. The adjoint operation with a group element g on the generators Pǎ and Jâ is expressed as Next, let us see the adjoint actions ofg , which is related to g through the Z 2 -grading automorphism (3.37). By using the Campbell-Baker-Hausdorff formula and the Z 2 -grading JHEP09(2020)100 Then, the action ofRg on Pǎ defined in (3.12) is given bỹ

A.2 SU(2, 2|4) case
Next, let us show that the action of the dressed R-operator R g k (k = 1, . . . , 4) on the su(2, 2|4) generators satisfies the relation (4.48). As in the previous case, we can see that the adjoint action with g k on the generators of su(2, 2|4) is written as

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By using these relations and the definition (4.14) of R g k , the projected dressed R-operator P (m) • R g k • P (n) can be expressed as Thus the relation (4.48) has been shown.

B Homogeneous bi-YB deformed sigma model
In this appendix, let us derive the action of a homogeneous bi-YB deformed principal chiral model, which is a two-parameter generalization of homogeneous YB deformation. In this case, we use the twist function (3.3) which is the same as in the symmetric coset case.
Boundary condition. A solution to the boundary equations of motion (3.5) is given by (A| z=1 , ∂ z A| z=1 ) ∈ g C R R , (A| z=−1 , ∂ z A| z=−1 ) ∈ g C R L , (B.1) where g C R and g C L are defined as Here η R and η L are the deformation parameters, and R R and R L are linear R-operators satisfying the hCYBE (3.11).

(B.4)
Since we use the same twist function (3.3) with the symmetric coset case, we suppose the same ansatz for the Lax pair:

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The solution (B.1) leads to A| z=1 = 2η R R R (∂ z A| z=1 ) , A| z=−1 = −2η L R L (∂ z A| z=−1 ) . (B.6) By using (B.4), (B.5) and (B.6), we obtain By solving these equations and removing U ± from the Lax pair, we obtain the following expression: where V ± contains both g R and g L like . (B.10) Deformed action. Now, we can obtain the action of the homogeneous bi-YB deformed sigma model. By using the expression of the Lax pair (B.9) , the residues of ϕ c L at z = ±1 are evaluated as Then the 2D action becomes This is an unusual form of the action of the homogeneous bi-YB deformed sigma model. In order to see the standard expression, let us use a complexified 2D gauge invariance g x → g x h (h ∈ G C ) . Then, we can realize the following configuration: g R = g , g L = 1 , (B.14) where g ∈ G . With this gauge, the action (B.13) reduces to This is the standard expression of the homogeneous bi-YB deformed sigma model action.
Then the Lax pair (B.9) is also simplified as

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