Integrable deformed $T^{1,1}$ sigma models from 4D Chern-Simons theory

Recently, a variety of deformed $T^{1,1}$ manifolds, with which 2D non-linear sigma models (NLSMs) are classically integrable, have been presented by Arutyunov, Bassi and Lacroix (ABL) [arXiv:2010.05573]. We refer to the NLSMs with the integrable deformed $T^{1,1}$ as the ABL model for brevity. Motivated by this progress, we consider deriving the ABL model from a 4D Chern-Simons (CS) theory with a meromorphic one-form with four double poles and six simple zeros. We specify boundary conditions in the CS theory that give rise to the ABL model and derive the sigma-model background with target-space metric and anti-symmetric two-form. Finally, we present two simple examples 1) an anisotropic $T^{1,1}$ model and 2) a $G/H$ $\lambda$-model. The latter one can be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev model.


Introduction
A significant subject in String Theory is the integrability in the AdS/CFT correspondence [1][2][3] (For a comprehensive review, see [4]). Although there are a lot of research directions, we are interested in the sigma-model classical integrability here. In the typical case of AdS/CFT, the string-theory side is basically described by a 2D non-linear sigma model (NLSM) 1 with target space AdS 5 ×S 5 together with the Virasoro constraints after fixing 2D diffeomorphism. Then the classical integrability is ensured by the fact that AdS 5 × S 5 is described as a symmetric coset which exhibits the Z 2 -grading.

2D NLSM from 4D CS theory
In this section, we give a derivation of 2D NLSMs from a 4D CS theory by following [20,21].
Let G be a Lie group with the Lie algebra g, and g C denotes the complexification of g .
We now consider a 4-dimensional space M × CP 1 , where M and CP 1 are parametrized by coordinates (τ, σ) and (z,z), respectively. A 4D CS action is defined as [20] 3 , The discussion in [43] is based on an affine Gaudin model [44][45][46] and covers more general cases. This family of T 1,1 models is a special case of it. For the off-critical value of the B-field, classical chaos appears for some initial conditions [47,48]. 3 For the notation and convention here, see [22].
where A is a g C -valued one-form and CS(A) is the CS three-form defined as ·, · is a non-degenerate adjoint-invariant bilinear form g C × g C → C . Then ω is a meromorphic one-form defined as and ϕ is a meromorphic function on CP 1 . This function is found to be a twist function characterizing the Poisson structure of the underlying integrable field theory [46]. The pole and zero structure of ϕ will be important in the following discussion. We denote the sets of poles and zeros by p and z , respectively.
Note that an extra gauge symmetry can alway gauge away the z-component of A like By taking a variation of the action (2.1) , we obtain the bulk equation of motion where ǫ ij is the antisymmetric tensor. Here the local holomorphic coordinates ξ x are defined as ξ x ≡ z − x for x ∈ p\{∞} and ξ ∞ ≡ 1/z if p includes the point at infinity. The expression (2.9) manifestly shows that the boundary equation of motion has the support only on M×p .
In terms of the components, the bulk equation of motion (2.6) reads The factor ω is kept since ∂zA σ and ∂zA τ are in general distributions on CP 1 supported by z .

Lax form
By performing a formal gauge transformation with a smooth functionĝ : M × CP 1 → G C , thez-components of L can be taken to zero: (2.14) Hence the one-form L takes the form The one-form will be specified as a Lax pair for 2D theory later, and so we refer to L as the Lax form.
The bulk equations of motion (2.6) in terms of the Lax form L are expressed as These equations means that L is a meromorphic one-form with poles at the zeros of ω , namely z can be regarded as the set of poles of L .

Reality condition
To ensure the reality of the 4D action (2.1) and the resulting action (2.22), we suppose some condition for the form of ω and the configuration of A [21].
For a complex coordinate z, an involution µ t : CP 1 → CP 1 is defined by complex conjugation z →z . Let τ : g C → g C be an anti-linear involution which satisfies Then a real Lie subalgebra g of g C is given as the set of the fixed points under τ . The associated operation to the Lie group G is denoted byτ : One can see that the action (2.1) is real if ω and A satisfy Recalling the relation (2.13), the conditioñ leads to (2.20).

From 4D to 2D via the archipelago conditions
The 4D action (2.1) can be reduced to a 2D action with the WZ term whenĝ satisfies the archipelago condition [21]. By performing an integral over CP 1 , we obtain Here I W Z [u] is the Wess-Zumino (WZ) three-form defined as where R x is the radius of the open disk on CP 1 .
The action (2.22) is invariant under a gauge transformation with a local function h : M → G . One can seen this as the residual gauge symmetry after taking the gauge (2.14).

5
In this section, we shall consider 2D NLSMs with a family of deformed T 1,1 manifolds, which have been presented by Arutyunov-Bassi-Lacroix [43]. We will refer to them as the ABL model for brevity as explained in Introduction.
Here, let us reproduce the ABL model from 4D CS theory. In the ABL model, the 2D surface M is embedded into the Lie group G × G . By defining a subgroup H ⊂ G as fixed points of an involutive automorphism, this model exhibits a gauge H diag -symmetry, where H diag is the diagonal subgroup of G × G . Then the phase space is reduced to a coset

Twist function
Let us start with the following meromorphic one-form, where ϕ ABL (z) is a twist function with ζ ± ∈ CP 1 and z 1 , z 2 ∈ R . This ω has the four double poles and the six simple zeros The twist function in (3.1) corresponds to the case with N = 2 and T = 2 in (3.14) in [43].

Boundary condition
In specifying a 2D integrable model associated with ω , we need to choose a solution to the boundary equations of motion, Here the double bracket is defined as where the overall constants c p (p ∈ p) are given by To derive the ABL model, we take the following solution: where g ab is an abelian copy of g .

Lax form
Before deriving the sigma model action, we shall introduce our notation used in the following.
We takeĝ at each pole of the twist function (3.1) aŝ where g k ∈ G (k = 1, . . . , 4) . Note that g k take values in G (not G C ) due to the reality condition (2.21) . The associated left-invariant currents are defined as 8) and the relations between the gauge field A and the Lax form L at each pole are written as where the adjoint action Ad g : g C → g C is defined as Ad g L = gLg −1 . Taking account of the configuration of the zeros (3.1) , we suppose an ansatz for the Lax form as where U [k] ± (k = 1, . . . , 4) are undetermined smooth functions of τ and σ , and the light-cone coordinates are defined as

2D action
Let us derive the 2D action by employing the boundary condition (3.6) .
Under the boundary condition (3.6), the relations in (3.9) are rewritten as By solving these equations with respect to U [k] Then the Lax pair can be rewritten as and the coefficients η , η , η (1) (3.22) Next, let us evaluate the residues of ϕ ABL L at z = ±z 1 , ±z 2 . By using the expression (3.20) of the Lax form, we obtain 2,+ c where the constants ρ (k) rs,± (r, s = 1, 2 , k = 0, 1) are defined as and ρ (1) Furthermore, the constants c s,± (s = 1, 2) are (3.26) Note that the above constants satisfy the relations 2,+ + c The residues of ω at each pole are Then, the 2D action is given by Here we would like to impose a relation between j s andj s (s = 1, 2) . Note that the re- The vector subspace h is also a subalgebra of g , and thus there exists the associated Lie subgroup H . Then the projection operators into h, m are defined as and thenj s and J By using the commutation relation of the Lie algebra for the symmetric coset, we can see Hence, we obtain (3.40) Then, by using the expressions ofj s in (3.36), the 2D action can be further rewritten as (3.41) The Lax form (3.20) becomes The expressions (3.41) and (3.42) are the same as the classical action and the associated Lax pair given in [43].

Gauge invariance
The action (3.41) exhibits a local H diag -symmetry, which is regarded as a gauge symmetry.
The diagonal subgroup H diag = {(h, h) ∈ G × G | h ∈ H} acts on G × G as

EX. 1) Anisotropic T 1,1 model
For simplicity, let us first impose the following condition: This condition (3.54) is solved in terms of z 1 , z 2 as By introducing a new quantity r defined as 11 . (3.57) Then the metric, B-field and the twist function are given by, respectively, dθ 2 1 + sin 2 θ 1 dφ 2 1 + r dθ 2 2 + sin 2 θ 2 dφ 2 2 + r 1 + r dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 2 , (3.58) (3.60) There remain three independent parameters ρ 11 , r and z 1 /z 2 now. Note here that the original T 1,1 case is not included in this example because the vanishing B-field means that r = 0 . The GMM model is also not included. The parameter r may be rather seen as an anisotropic parameter. In the isotropic case with r = 1 , the coefficient of the U(1)-fiber is fixed as 1/2 and the B-field remains complicated.
Finally, the Lax pair is given by (3.61)

EX. 2) G/H λ-model
As the second example, let us suppose the following conditions: Then the parameters in (3.24) and (3.25) are expressed as (3.63) (3.1) coincide. In this case, the resulting action and Lax pair are given by, respectively, This is nothing but the Lagrangian and Lax pair of a G × G/H sigma model related to the tripled G/H λ-model formulation 4 [51,52] . Notably, in the limit z 1 /z 2 → 0 , the action (3.64) reduces to that of the GMM model [49].
The flatness condition for this Lax pair is obtained as and this is equivalent to the equations of motion of the model. Note that as we take the limit z 1 /z 2 → 0 , the term with 1/z is lost, and thus we cannot reproduce all of the equations of motions. A possivle way to care this point would be to prepare another Lax pair by scaling the spectral parameter as z ′ = zz 2 /z 1 [43].

(3.68)
This is a deformed GMM background with a parameter z 1 /z 2 . When z 1 /z 2 = 0 , the target space of the original GMM model is reproduced.
In this paper, we have derived the ABL model from a 4D CS theory with a meromorphic one-form (3.1) with four double poles and six simple zeros by specifying a boundary condition. Then we have explicitly derived the sigma-model background with metric and anti-symmetric two-form (i.e., the ABL background). As its special cases, we have presented an anisotropic T 1,1 model and a G/H λ-model. The latter can be regarded as a one-parameter integrable deformation of the GMM model.
It would be very interesting to consider the ABL background in the context of AdS/CFT.
The first task is to find out a possible string-theory embedding of the ABL background. It is nice to try to identify the remaining components of type IIB supergravity for the ABL background. A possibility is to consider a variant of the ABL model for G = SL (2) and . This is a natural extension of the work [53], which considered the GMM model for G = SU(2) × SL(2) and H = U(1) × U(1) so as to be a supergravity background. Once the ABL background has been embedded into string theory, it would be nice to explore the dual gauge theory.
Another future direction is to consider a relation between the present result and 6D holomorphic CS theory [54,55]. Along this line, it may be possible to derive a new family of 4D integrable systems. We would like to report some results in the near future in another place.
We hope that the ABL background we have derived would open up a new arena in the study of the integrability in AdS/CFT.
The other equations of motion comes from the flatness condition of another Lax pair which can be obtained by taking a different scaling limit. To see this, let us redefine z as z/α , and then take the same limit (A.1). Then, the twist function (3.1) becomes ϕ 2 (z) = − 2(λ 2 − λ 2 2 z 2 ) z(z 2 − 1) 2 . (A.7) The double poles and simple zeros of ϕ 2 (z) are listed as