Integrable sigma model with generalized F structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model

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Introduction
Two-dimensional integrable σ models and their deformations have garnered considerable attention over the past 45 years [1][2][3].The integrable deformation of the principal chiral model on SU (2) was first presented in [4][5][6][7].Additionally, the Yang-Baxter deformation of the chiral model was introduced by Klimcik [8][9][10].The Yang-Baxter sigma model relies on R-operators that satisfy the (modified) classical Yang-Baxter equation ((m)CYBE) [11][12][13][14][15].The integrable sigma model on a Lie group with a complex structure was also studied in [16] as a special case of the Yang-Baxter sigma model [8].Recently, Mohammedi proposed a deformation of W ZW models using two invertible linear operators [17].Utilizing the general method outlined in another work by Mohammedi [18], we have constructed an integrable sigma model on a general manifold, particularly on a Lie group with a complex structure, in our previous work [19].Furthermore, employing this method, we have explored an integrable sigma model on Lie groups equipped with a generalized complex structure [19].
Following the methodology outlined in [17] and [18], and building upon our previous work [19], we aim to construct integrable sigma models on a Lie group with generalized F structure [20] (a generalized Nijenhuis structure J [21,22] with J 3 = −J ).By utilizing the expression of the generalized complex structure on a metric Lie group G in terms of operator relations on its Lie algebra g (as discussed in [19]), we will further develop the Yang-Baxter sigma model with generalized complex structure.Consequently, we will present the multi-Yang-Baxter sigma model.The plane of the paper is outlined as follows: In Section 2, we will review the methods outlined in [17] and [18], as well as the concepts related to generalized complex structure [21,22].Then, in Section 3, we will present the integrable sigma model on a Lie group with generalized F structure.We will illustrate this with two examples: one on the Heisenberg Lie group H 4 ≡ A 4,8 [23], and another on the Lie group GL(2, R).In Section 4, by employing the expression of the generalized complex structure on a metric Lie group G in terms of operator relations on its Lie algebra (as discussed in our previous work [19]), we will construct the Yang-Baxter sigma model with generalized complex structure.An example on the Nappi-Witten Lie group (A 4,10 ) [24] will be presented.Finally, in Section 5, we will introduce the multi-Yang-Baxter sigma model with two and three compatible Nijenhuis structures and provide corresponding examples.
2 Some Review on integrable sigma model and generalized complex structure Before delving into our main discussions, let's provide a brief overview of the two methods proposed by Mohammedi for constructing integrable sigma models [17,18].Additionally, we'll briefly review the fundamental concepts and relations concerning generalized complex structure, as discussed in the literature [21,22].

First Method
Consider the following sigma model action where x µ (z, z) (µ = 1, 2, ..., d) are coordinates of d dimensional manifold M , such that G µν and B µν are invertible metric and anti-symmetric tensor fields on M ; the (z, z) are coordinates of the world-sheet Σ.The equations of motion for this model have the following form [18] ∂∂x λ + Ω λ µν ∂x µ ∂x ν = 0 , where 1 As in [17] and [18] instead of using the world sheet coordinates τ and σ, we will use the complex coordinates (z = σ+iτ, z = σ−iτ ) with ∂ = ∂ ∂z and ∂ = ∂ ∂ z .We also adopt the convention that the Levi-Civita tensor ǫ z z = 1; so we will not have i in front of B field in (1).
such that Γ λ µν is the Christoffel coefficients and components of the torsion are given by As in [18] one can construct a linear system whose consistency conditions are equivalence to the equations of motion (2) as follows: where the matrices α µ and β µ are functions of coordinates x µ .The compatibility condition of this linear system yields the equations of motion, provided that the matrices α µ (x) and β µ (x) must satisfies the following relations [18] such that the second equation of the above set can then be rewritten as where the field strength F µν and covariant derivative corresponding to the matrices α µ are given as follows: Then by splitting symmetric and anti-symmetric parts of ( 9), we have [18] In this manner the integrability condition of the sigma model ( 1) is equivalent to finding the matrices α µ and µ µ such that they satisfy in (9) (or (11) and ( 12)).

Second Method
In [17] Mohammedi give a general integrable deformation of chiral and WZW models.Here we review his results as the following proposition: Proposition [17]: Consider invertible linear operators P ′ and Q ′ (such that P ′ + Q ′ is also invertible) on a metric Lie algebra g (with ad invariant invertible metric2 < , > g ) with Lie group G, such that the operators satisfy the following relations [17]: with , where I is identity operator; λ and ε are parameters.Then the following twodimensional sigma model is classically integrable [17]: where B is a three-dimensional manifold with boundary ∂B.Indeed the equations of motion of the above model are given as conservation of the following currents: i.e.
Such that these currents satisfy the zero curvature condition if P ′ and Q ′ satisfy relation ( 14) [17].

Review on generalized complex structure
The generalized complex structure [21,22] J on a manifold M with even dimension is an endomorphism from J : T M ⊕ T * M → T M ⊕ T * M , such that this structure is invariant with respect to inner product , on and also the generalized Nijenhuis tensor with Courant bracket [25] [ is zero [21,22] .i.e.
Utilizing the generalized complex structure J in the following block form [25] where that by applying the above block form (23) in (20), we have the following relations for tensors J, P and Q: Furthermore, relation (22) (zero of the generalized Nijenhuis tensor) leads to the following relations [26] So the components of a generalized complex structure (23) on a manifold M must be satisfy relations (24)(25)(26)(27)(28)(29)(30)(31).

The model
Now, we will attempt to consider an integrable sigma model on the Lie group G with the aforementioned generalized F structure, subject to conditions (37-45).We propose the action of the sigma model with generalized F structure as follows: where k, k ′ , k ′′ , k 1 , ..., k 4 are real coupling constants.To obtain the condition under which the above model is integrable, one can utilize the first method mentioned in subsection 2.1 by introducing the matrices α µ and µ µ as follows:

Examples
a) Here, we construct the integrable sigma model with generalized F structure (47) on the Heisenberg Lie group H4 ≡ A 4,8 with the following commutation relations for its Lie algebra A 4,8 [23]: To calculate the vierbein we parameterize the corresponding Lie group H 4 with coordinates x µ = {x, y, u, v} and using the g elements as follows g = e vT4 e uT3 e xT1 e yT2 , where we use the generators T α = {P 1 , P 2 , J, T }.Thus the veirbein and ad-invariant metric have the following form [14,28] ( where k 0 ∈ R−{0} is an arbitrary real constant.Now, by utilizing the conditions for the generalized F structure (37-39) and (40-41), along with its integrability conditions (42-45), one can obtain the following results: where b, d ∈ R and a, c ∈ R − {0}.Here, for simplicity, we will choose b = d = 0 and a = c = 1.Then, the action (47) of this model, after omitting surface terms, can be written as: with λ 2 , λ 4 , λ 5 , λ 6 , λ 7 , λ ′ 4 , λ ′ 5 , λ ′ 6 , λ ′ 7 are arbitrary constants and one of them can be considered as spectral parameters.It should be explained that by applying the parameter m from (54) in the model (53), we obtain the following components of the Christoffel symbols and torsion4 To check the integrability of the model (53), one can consider that equation of motion (2) for sigma model ( 53) with Christoffel and torsion components (55) and ( 56), can be written as linear equations ( 5) and ( 6).On the other hand, the W ZW (Witten-Zumino-Witten) model on a Lie group G is defined as follows: so the W ZW model on Heisenberg Lie group H 4 (with K = 4π) is given by the following action [14,28]: by setting m = 0 from (54); the action (53) can be considered as the following integrable deformation of the W ZW action S = S W ZWA48 + k 0 dzdz{− r 2 (∂v ∂x + ∂x ∂v) + r 2 e x (∂y ∂u + ∂u ∂y) Note that, as we expected, this deformation is different from Yang-Baxter deformation of the W ZW model of Heisenberg Lie group H 4 table 1 of ref [14].Indeed, the third, fourth, and fifth terms in the second part of the action (59) are new terms.

b)
As second example, we construct integrable sigma model with generalised F structure on the GL(2, R) Lie group.By considering the following commutation relations between the generators where T 4 is a central generator, and we parameterize the corresponding Lie group GL(2, R) with coordinates x µ = {x, y, u, v} using the g elements as g = e yT2 e xT1 e uT3 e vT4 , then the veirbein and ad-invariant metric are given by [15] (e α µ ) = where k 0 , m ∈ R − {0}.By using the conditions for the integrability of the generalized complex structure (42 -45) and the relations (37-41), then the following components of generalized F structure are obtained where Here for simplicity we will choose b = 0, a = c = d = h = 1, f = 2; then one can construct sigma model (47), as follows: Here, similar to the previous example, we set r = k + k 1 − 2k 2 and s = k ′′ + k 3 .To investigate the integrability of this model such that it satisfies the relation ( 9), we assume that α µ and µ µ are similar to the relation (48).Then, by setting: r = 0, m = −2k 0 (65) such that λ 4 , λ 5 , λ 6 , λ 7 , λ ′ 4 , λ ′ 5 , λ ′ 6 , λ ′ 7 are arbitrary constants (and one of them can be considered as spectral parameter), this model satisfies the integrability condition (9).Indeed by applying the condition (65) the Christoffel and torsion components of the model (64) can be expressed as follows so the conditions (65) and (66) satisfy the integrability condition (9) of the model (64).Consider the WZW model of the GL(2, R) Lie group [15] S W ZW GL(2,R) = dzdz{m∂v ∂v + 2k 0 ∂x ∂x + k 0 e −2x (∂u ∂y + ∂y ∂u) + k 0 e −2x (∂u ∂y − ∂y ∂u)}, if we set r = 0, m = −2k 0 (from (65)), then the action (64) can be written as the perturbed action of the W ZW model as follows (∂u ∂y − ∂y ∂u) + 2k 0 su 2 e −2x (∂x ∂y − ∂y ∂x)

Example c)
As an example we construct the sigma model (84) on Nappi-Witten Lie group A 4,10 , with the following Lie algebra commutators [23,24] [J, We parameterize the corresponding Lie group A 4,10 with coordinates x µ = {a 1 , a 2 , u, v} and consider the following parametrization for the group element [23] g = e ΣaiPi e uJ+vT , with the generators T α = {P 1 , P 2 , J, T }.The veirbein and ad-invariant metric is given by [24]: where k 0 ∈ R. The corresponding components of the generalized complex structure (which satisfy the algebraic form of the relations (24-27), (42-45) and J = −J t ) are so the action (84) for this model is given by (a 2 (∂u ∂a 1 + ∂a 1 ∂u) − a 1 (∂a 2 ∂u + ∂u ∂a 2 ) + 2(∂u ∂v + ∂v ∂u)) this model satisfies the integrability condition (83).Comparing the above model with the results in table 4 of ref [14], we observe that the fourth through eighth terms are new.
5 Multi-Yang-Baxter sigma models The model (84) and its integrability conditions (83) are motivated to consider Yang-Baxter sigma model with two and three (or multiple) compatible Nijenhuis structures 7 .If we consider Q ′ and P ′ in the following forms i.e. a two, three or in general a multi-parametric deformation of principal chiral model then from (15) (with λ = 0) for the following multi-Yang-Baxter sigma model S = dzdz < g −1 ∂g, the condition (13) (by assuming J t i = −J i i = 1, 2, 3) can be written as and integrability condition ( 14) result as the following relations by assuming that i.e. one can construct integrable multi-Yan-Baxter sigma model (91) with three compatible Nijenhuis structures (94) that satisfy in mCYBE (93).One can also write the multi-Yang-Baxter sigma model (91) with two J i such that the conditions (93) and ( 94) are satisfied with i, j = 1, 2 and the conditions (95) is replaced by Indeed the relations (93)-( 94) demonstrate that the two or three Nijenhuis structures are compatible with zero concommitant [29].In this manner one can construct a multi-Yang-Baxter sigma model as multi-parametric Poisson-Lie deformation of principal sigma model with dual Lie algebra g = g ξ1J1+ξ2J2+ξ3J3 [9]; such that g is a Lie algebra with Lie bracket, ∀X, Y . Relations (93-95) can also obtained from satisfying the combination of Nijenhuis structures ξ 1 J 1 + ξ 2 J 2 + ξ 3 J 3 in Yang-Baxter equation.Note that if we have J 2 i = −1 (i.e.complex structure J i ) and have Hermitian structure then we have N = 2 structure and two of J i are independent [29].In the following we investigate two examples on IX + R and G 6,23 Lie groups.

Conclusion
Following the methodology outlined in the previous work [19], we construct a new integrable sigma model with generalized F structure.Additionally, utilizing the expression of the generalized complex structure on the metric Lie group in terms of operator relations on its Lie algebra, we construct the Yang-Baxter sigma model with generalized complex structure.Furthermore, we present the multi-Yang-Baxter sigma model with two or three compatible Nijenhuis structures.Investigating the Poisson-Lie symmetry of these models remains an open problem, as does their conformality up to one loop.