The Faddeev-Reshetikhin model from a 4D Chern-Simons theory

We derive the Faddeev-Reshetikhin (FR) model from a four-dimensional Chern- Simons theory with two order surface defects by following the work by Costello and Yamazaki [arXiv:1908.02289]. Then we present a trigonometric deformation of the FR model by employing a boundary condition with an R-operator of Drinfeld-Jimbo type. This is a generalization of the work by Delduc, Lacroix, Magro and Vicedo [arXiv:1909.13824] from the disorder surface defect case to the order one.


Introduction
Searching for a method to describe various integrable models in a unified manner is a significant subject in mathematical physics. A nice idea for such a way is to start from fourdimensional gauge theories by following the works by Costello, Witten and Yamazaki [1][2][3].
In particular, two-dimensional (2D) integrable field theories can be derived from a fourdimensional Chern-Simons (4D CS) theory equipped with a meromorphic one-form ω ω ≡ ϕ(z) dz = dz , (1.2) as proposed by Costello and Yamazaki [4]. The base space is M × C , where M is a 2D manifold and C is a Riemann surface. Introducing 2D defects enables us to consider a dimensionally reduced theory on M . These surface defects are classified into the order defects and the disorder defects. The order defects are defined by introducing new degrees of freedom such as free fermions and free bosons, which are coupled to the 4D bulk gauge theory. For the disorder defects, we allow ω to have zeros on C, and the 2D theories lie on the poles of ω.
In the disorder defect case, ω has been identified with a twist function of the associated integrable system [5]. Then Delduc, Lacroix, Magro and Vicedo has pushed this perspective and elaborated the procedure to derive integrable field theories for disorder defects [6].
Our puporse here is to discuss the order defect case by focusing upon an example.
According to the Hamiltonian analysis in [5], the models in this case should be ultralocal (no δ ′ -term in the Poisson algebra). A famous example of the ultralocal model is the Faddeev-Reshetikhin model [18]. We derive the FR model from a 4D CS theory with two order surface defects. Then we present a trigonometric deformation of the FR model by employing a boundary condition with an R-operator of Drinfeld-Jimbo type [19,20]. This is a generalization of the work [6] from the disorder surface defect case to the order one.
This paper is organized as follows. In section 2, we introduce the basics of the FR model.
In section 3, the FR model is derived from a 4D Chern-Simons theory with two order surface defects. In section 4, we present a trigonometric deformation of the FR model by employing an appropriate boundary condition with the R-operator of Drinfeld-Jimbo type. Section 5 is devoted to conclusion and discussion.
NOTE: Just before completing our draft, we have received an interesting work by Caudrelier, Stoppato and Vicedo [21], where the Zakharov-Mikhailov theory (which is a class of ultralocal models) has been derived with order defects [21] based on the procedure presented in [22]. The FR model is included as a special example. But our derivation is different from theirs and a trigonometric deformation of it has not been discussed there.

The Faddeev-Reshetikhin model
In this section, we shall give a brief review about the Faddeev-Reshetikhin (FR) model [18].

The classical action
The classical action of the FR model is given by where ν is a real parameter and g (±) are group elements of SU (2) . Here M is 2D Minkowski space with the coordinates x α = (x 0 , x 1 ) = (τ, σ) and the metric η αβ = diag(−1, +1) . The light-cone coordinates on M are defined as Here Λ is the Cartan generator of SU(2) taken as where T a (a = 1, 2, 3) are the generators of SU(2) , Here σ a are the Pauli matrices, and the structure constants ε abc are the antisymmetric tensor normalized as ε 123 = 1 . The expression (2.1) of the action is given in [23]. The FR model is closely related to the string sigma model with target space R × S 3 , and the low-energy effective action of (2.1) becomes the Landau-Lifshitz model as explained in [24]. It is easy to generalize the action (2.1) to the SU(N) case as discussed in [23], but we will restrict ourselves to the SU(2) case for simplicity.
The equations of motion obtained from (2.1) are where we have introduced The above equations of motion (2.5) can be rewritten as Therefore, J (α) (α = ±) can be regarded as an on-shell conserved current. While these equations (2.7) have the same forms with the ones derived from the SU(2) PCM, J (±) satisfy additional relations Tr (J (±) ) n : const. (2.8) On the other hand, the conserved current of SU(2) PCM does not satisfy this relation.
As is well known, the FR model (2.1) is classically integrable. Indeed, since the equations (2.7) take the same forms with those for the SU(2) PCM, we can easily construct a Lax where z ∈ CP 1 is a spectral parameter. The flatness condition of the Lax pair (2.9) is equivalent to the equations of motion (2.7): (2.10) As usual, we can obtain infinite (non-local) conserved charges from the monodromy matrix where the symbol P denotes the equal-time path ordering in terms of σ and the spacial component of the Lax pair is defined as

The Poisson structure
The Poisson structure of the FR model is much simpler than that of the SU(2) PCM. In fact, the Poisson brackets of J a (±) (σ) are given by (2.13) These are ultra-local because the term with the derivative of the delta function does not appear in the right hand sides of (2.13), in comparison to the SU(2) PCM. By using the relations in (2.13), the Poisson bracket of the spacial component of the Lax pair can be expressed as where the Poisson bracket in the tensorial notation is defined as and r(z 1 , z 2 ) ∈ g ⊗ g is the classical r-matrix associated with the integrable structure of the system. The resulting classical r-matrix is given by 16) and the twist function ϕ(z) is just one like The classical r-matrix (2.16) satisfies the classical Yang-Baxter equation (CYBE) Here we have introduced the tensorial notation where r ab are the components of the r-matrix The relation (2.14) leads to the Poisson bracket of the monodromy matrices This is the fundamental relation of the FR model.

The FR model from a 4D CS theory
In this section, we shall derive the FR model from a 4D CS theory with two order surface defects. The derivation here is mostly based on a generalization of the method in [6] for the disorder case.

A 4D CS theory with two order surface defects
Let us consider a complexified SU(2) , G C = SU(2) C 1 . The associated complexified Lie algebra is g C ≡ su(2) C . Then, we consider a g C -valued gauge field A defined on M × CP 1 .
The global holomorphic coordinate of CP 1 ≡ C ∪ {∞} is denoted by z . This CP 1 geometry characterizes the rational class of integrable system.
We start from a 4D CS theory coupled with two order surface defects, where the covariant derivatives D ± are defined as The second and third terms of (3.1) describe the two order surface defects sitting at z ± ∈ R , respectively. The first term is the 4D CS action given by where CS(A) is the CS three-form defined as Here, the meromorphic one-form ω is defined in terms of the twist function (2.17) as which has a double pole Note here that since ω is a (1,0)-form, the action (3.3) has an extra gauge symmetry It enables us to take the gauge A z = 0 , i. e.,

Equations of motion
Let us derive the equations of motion of the action (3.1). Taking a variation of (3.1) with respect to A, we obtain where F (A) ≡ dA + A ∧ A is the field strength of A . Here, we have assumed that A vanishes at the boundary of M × CP 1 , and used the relation of the delta function Then, the bulk equations of motion are given by The second and third equations indicate that A has poles at z = z ± . For later discussion, we denote the set of the positions of the order surface defects as 14) It is useful to rewrite the boundary equation of motion as where ξ ∞ ≡ 1/z is the local coordinate around z = ∞ .

Gauge invariance
Let us see here the gauge invariance of the action (3.3) .
In analogy with the disorder defect case [6], it is natural to consider a gauge transfor- where u is a G C -valued function defined on M × CP 1 . Then, at the off-shell level, the action (3.1) is transformed under the transformations (3.16) as where I W Z [u] is the Wess-Zumino (WZ) three-form defined as

Lax form
Let us next introduce a Lax pair associated with the action (3.1).
As in the disorder defect case, a Lax pair is introduced by performing a formal gauge transformation 2 (3.16), whereĝ , g (±) ∈ G C andĝ (±) ≡ĝ| z ± . Here, we take a gauge choice such that Lz = 0 , and hence the one-form L takes the form By substituting (3.21) into (3.12), (3.13), the bulk equations of motion become The currents J (±) are defined as As we will see later, these are going to be identified with the current (2.6). The boundary equations of motion (3.23) and (3.24) indicate that the Lax pair is a g C -valued meromorphic one-form with poles z = z ± .
By substituting (3.20) into (3.1), the 4D action (3.1) can be written as Note that the expression (3.26) is still a 4D action. In the next subsection, we will dimensionally reduce the 4D action (3.26) to the corresponding 2D action by imposing conditions onĝ .

From 4D to 2D via the archipelago conditions
In order to obtain the associated 2D integrable model, it is necessary to impose the archipelago conditions [6] onĝ as in the disorder defect case. Then, the 4D action (3.3) is drastically simplified as follows: where in the second equality we have used the relations The integrand of the first term in (3.27) is apparently a four-form, but as we will see in (3.39) it is localized on the defects at M × {z ± } because dL in the integrand generates delta functions due to the bulk equations of motion (3.23), (3.24).

Reality condition
Let us now discuss the reality condition to ensure that the 4D action (3.1) is real. 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, and then the set of the fixed points under τ defines a real subalgebra g of g C . The involutive automorphism τ satisfies The associated operation on the Lie group G C is denoted byτ : The reality condition is imposed through these involutions as One can see that the action (3.1) is real under the condition (3.30): Note here that µ t (z ± ) = z ± and Λ ∈ g . From the relation (3.20), the reality condition is also expressed asτĝ (±) = µ * tĝ (±) ,τ g (±) = µ * t g (±) , τ L = µ * t L . (3.32)

2D gauge symmetry
The 2D action (3.27) has the "2D gauge symmetry". One can perform the 2D gauge transformations keeping A and G (±) unchanged and preserving the archipelago conditions imposed onĝ . Under the transformation, L ,ĝ (±) and g (±) are transformed as where h is a smooth g-valued function depending on (τ, σ) ∈ M . In contrast to the 4D gauge symmetry (3.16), the 2D gauge symmetry (3.33) is considered as the redundancy in definingĝ without altering A and G (±) .

2D effective action
In order to evaluate (3.27), let us determine the explicit expression of the Lax form.
The first is to solve the boundary equation of motion (3.15) with the following condition: This is a trivial solution to the boundary equation of motion.
As we saw in section 3.2, L ± have poles at z = z ± , respectively. Therefore, it is natural to suppose the following form of L ± : where we have used a formula for delta functions Here U ± are undetermined functions on M and take values in g due to the reality condition (3.32). The 2D gauge symmetry (3.33) allows us to set an archipelago type fieldĝ likê Since the boundary condition (3.34) indicates that U ± = 0 , the Lax form is determined as Finally, let us evaluate the 4D action (3.27). By using the expression (3.38), the first term in (3.27) can be rewritten as Then, the 2D effective action is evaluated as

A trigonometric-deformation of the FR model
In this section, let us consider a trigonometric-deformation of the FR model.

A twist function
For this purpose, we replace the rational classical r-matrix (2.16) with the su(2) trigonometric r-matrix 3 , where we have introduced a deformation parameter η ∈ R and Note that the classical r-matrix (4.1) satisfies the CYBE (2.18). The spectral parameter λ takes a value on a cylinder (rather than CP 1 ) because the classical r-matrix (4.1) is of trigonometric type. Then the fundamental region of λ is represented by By taking a limit η → 0 , the classical r-matrix (4.1) reduces to the rational one (2.16).

(4.4)
Here we have defined the components of the r-matrix as Since the associated twist function is obtained as a measure of the asymmetry of a given classical r-matrix, the (1, 0)-form ω should be taken as

A relationship with Costello and Yamazaki
It is instructive to see that the choice (4.6) of ω is consistent with the expression in [4].
First, let us move from the cylinder C/Z to a plane C × = C\{0} via the map Note that in the z-coordinate system, the trigonometric r-matrix (4.1) becomes This is related to the rational one (2.16) through the relation Then, the (1, 0)-form ω on C × takes the form and has two simple poles The form of ω in (4.10) is the same as in the one in [4].
The reality condition of ω An involution µ t may be defined as follows: In the λ coordinate, the reality condition is trivial: (4.13)

A boundary condition
In the following, we will consider the 4D CS action (3.1) with the (1,0)-form (4.10). We obtain the same bulk equations of motion (3.11), (3.12) and (3.13), but now ω is replaced by the one in (4.10). Note that in this section, the order surface defects lie on z = z ± such that z ± = 1/z ± since the involution µ t is defined as (4.12).
The (1, 0)-form ω has the two simple poles (4.11). Hence, the boundary equations of motion are where the bilinear form is defined as As shown in [7], the boundary condition (4.14) can be solved by assigning the following Drinfeld double to the bilinear form where g δ and g R are defined as  and Tr (R(x)y) = − Tr (xR(y)) , ∀x, y ∈ g . (4.20) Here, let us take the R-operator of the Drinfeld-Jimbo type [19,20] such that R(T ± ) = ∓iT ± , R(T 3 ) = 0 . (4.21) We can easily check that the R-operator satisfies the mCYBE (4.19).
As a result, A α is supposed to satisfy

The associated Lax form and 2D action
Let us next derive the associated Lax form and 2D action.
As in the rational case, we can easily see that the associated Lax form satisfies the equations (3.22), (3.23) and (3.24) though ω is now replaced with (4.10). Hence, an ansatz of L ± is taken as where U ± are undetermined smooth functions M → g C . The reality condition is again realized as in (3.32) .
In order to obtain the expression of U ± , we will take boundary conditions as in (4.22).