Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group $H_4$

We show that the WZW model on the Heisenberg Lie group $H_4$ has Poisson-Lie symmetry only when the dual Lie group is ${ A}_2 \oplus 2{ A}_1$. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group $H_4$ and its dual pair, ${ A}_2 \oplus 2{ A}_1$, as the target space in such a way that the original model is the same as the $H_4$ WZW model. Furthermore, we show that the dual model is conformal up to two loops order. Finally, we discuss $D$-branes and the worldsheet boundary conditions defined by a gluing matrix on the $H_4$ WZW model. Using the duality map obtained from the canonical transformation description of the Poisson-Lie T-duality transformations for the gluing matrix which locally defines the properties of the $D$-brane, we find two different cases of the gluing matrices for the WZW model based on the Heisenberg Lie group $H_4$ and its dual model.


Introduction
The WZW models on non-semi-simple Lie groups [1], [2], [3], and [4] play an important role in string theory, since some of them provide exact string backgrounds that have a target space dimension equal to the integer and irrational Virasoro central charge of the affine non-semi-simple algebra [3] and [5]. The first of these models was based on the group E c 2 , a central extension of the twodimensional Euclidean group; and the corresponding sigma model describes string propagation on a four-dimensional space-time in the background of a gravitational plane wave [1]. This construction was subsequently extended to other non-semisimple Lie groups [3] and [4] in such a way that the WZW model on the Heisenberg group with arbitrary dimension was, for the first time, introduced by Kehagias and Meessen [4].
On the other hand, the T-duality is a very important symmetry of string theories, or more generally, two-dimensional sigma models [6] and the Poisson-Lie T-duality [7], [8] is a generalization of Abelian and non-Abelian target space duality (T-duality). So far, there is one example for conformal sigma models related by Poisson-Lie T-duality [9] in a way that the duality relates the standard SL(2, R) WZW model to a constrained sigma model defined on the SL(2, R) group space. Moreover, we have recently shown that the WZW models on the Lie supergroups GL(1|1) [10] and (C 3 +A) [11] contain super Poisson-Lie symmetry such that in this process the dual Lie supergroups are the respective B ⊕ A ⊕ A 1,1 | .i and C 3 ⊕ A 1,1 | .i. In this paper we show that the WZW model on the Heisenberg Lie group H 4 has Poisson-Lie symmetry only when the dual Lie group is A 2 ⊕ 2A 1 .
Furthermore, we show that the dual model is conformal up to two-loop order and in this manner we obtain the general form of the dilaton field of the dual model. We also study the worldsheet boundary conditions for our model and its dual.
The outline of the paper is as follows. In section 2 we show that the WZW model on the Heisenberg Lie group H 4 has Poisson-Lie symmetry only when the dual Lie group is A 2 ⊕ 2A 1   In this section, based on our previous works [10] and [11], we will describe a new example of a WZW model containing Poisson-Lie symmetry. The model is constructed on the Heisenberg Lie group H 4 , a non-semi-simple Lie group of dimension four. As mentioned in introduction section, the WZW model based on the Heisenberg group H 4 was, for the first time, constructed by Kehagias and Meessen [4]. Here, we first obtain the WZW model on the Heisenberg group H 4 with a new background. Then, we will show that the model has Poisson-Lie symmetry. Before proceeding to construct model, let us first introduce the Lie algebra h 4 of the Lie group H 4 (the oscillator Lie algebra). The Lie algebra h 4 is generated by the generators {N, A + , A − , M } with the following non-zero Lie brackets One can show that the Lie algebra h 4 is isomorphic to the Drinfel'd double of a two-dimensional Lie bialgebra, i.e., (A 2 , I 2 ) [12], [13] where A 2 and I 2 are two-dimensional non-Abelian and Abelian Lie algebras, respectively. The isomorphic transformation between the Lie algebras h 4 and (A 2 , I 2 ) is given by where {T 1 , · · · , T 4 } are generators of the Lie algebra of the Drinfel'd double (A 2 , I 2 ) and α 0 ∈ ℜ; β 0 , γ 0 ∈ ℜ − {0}.
Let us now turn into the construction of our model. In general, given a Lie algebra with generators X a and structure constants f c ab , to define a WZW model, one needs a non-degenerate ad-invariant symmetric bilinear form Ω ab =< X a , X b > on Lie algebra G such that it satisfies the following relation [1] Using the commutation relations (2.1), one can obtain the non-degenerate ad-invariant bilinear form Ω ab on the Lie algebra h 4 as where κ 0 is a non-zero real constant. In general, we know that the WZW model based on a Lie group G is defined on a Riemannian surface Σ as a worldsheet by the following action [1] where the components of the left-invariant one-forms L a α 's are defined via g −1 ∂ α g = L a α X a , in which g : Σ → G is an element of Lie group G. Here B is a three-manifold bounded by worldsheet Σ and σ ± = 1 √ 2 (τ ± σ) are the standard light-cone variables on the worldsheet. To calculate the L a α 's we parameterise the corresponding Lie group H 4 with coordinates x µ = {x, y, u, v} so that its elements can be written as Hence using relations (2.1), (2.3) and (2.6) and some algebraic calculations, the WZW action on the H 4 Lie group is worked out to be of the form As we know the non-linear sigma model for a bosonic string propagating in a d-dimensional spacetime with the metric g µν , the anti-symmetric tensor field b µν and the dilaton fileld Φ is given Here the space-time geometry is described by a Lorentz signature metric g µν and anti-symmetric tensor field b µν , both of which may depend on the space-time coordinates x µ . For the action (2.7), the corresponding space-time metric and the anti-symmetric tensor field are, respectively, given by Thus, by identifying the action (2.7) with the sigma model of the form (2.9) we can read off the background matrix E µν = g µν + b µν in the coordinate base {dx, dy, du, dv} as The sigma model (2.9) has Poisson-Lie symmetry with respect to the dual Lie groupG (with the same dimension G) when background matrix satisfies in the following relation [7], [8] Now, by substituting relations (2.11) and (2.13) on the right hand side of (2.12) and, then, by direct calculation of Lie derivative of E µν with respect to V a , one can obtain the structure constants of the dual Lie algebra to the Lie algebra h 4 in such a way that only non-zero commutation relation of the dual pair is (2.14) The Lie algebra deduced in this process is a four-dimensional decomposable Lie algebra. In the classification of four-dimensional Lie algebras [14] it has denoted by Nevertheless, the (h 4 , A 2 ⊕ 2A 1 ) as a Lie bialgebra satisfies mixed Jacobi identities. Having a Drinfel'd double which is simply a Lie group D, we can construct the Poisson-Lie symmetric sigma models on it. So we will, first, form the Drinfel'd double generated by the Lie algebra h 4 and its dual pair .., X 4 ;X 1 , ...,X 4 } so that they obey the following set of non-zero commutation relations In the next section, we shall construct a pair of Poisson-Lie T-dual sigma models which is associated with the Drinfel'd double (H 4 , A 2 ⊕ 2A 1 ) and will show that the original sigma model on the H 4 is the same as the WZW model obtained in (2.7).
3 Poisson-Lie T-dual sigma models built on the Drinfel'd double (H 4 , A 2 ⊕2A 1 ) As mentioned above, having Drinfel'd doubles we can construct the Poisson-Lie T-dual sigma models on them. The construction of the models has been described in [7] and [8]. The models have target spaces as the Lie groups G andG and are, respectively, given by the actions where R a ± and (R ± ) a are the components of the right-invariant one-forms on the Lie groups G and G, respectively, and are defined by furthermore, the background fields E + (g) andẼ + (g) are defined by in which E + 0 (e) andẼ + 0 (ẽ) are the sigma model constant matrices at the unit element of G andG, respectively, and are related to each other in the following way [8] Here Π(g) is a bivector field on the Lie group manifold which gives a Poisson-Lie bracket on G and is defined as where a(g) and b(g) are sub-matrices of the adjoint representation of G on the Lie algebra of the Drinfel'd double [8]. AnalogouslyΠ(g) is a Poisson-Lie bracket on the dual Lie group manifoldG.
To construct the mutual T-dual sigma models with the Drinfel'd double (H 4 , A 2 ⊕ 2A 1 ) whose Lie algebra defined in (2.15), we use the same parameterisation (2.5) for both the original and dual models 5 . Using relation (2.5) and (3.3), one can calculate R a ± 's as and considering relation (2.5) for the corresponding tilted symbols, we obtain Then, choosing the constant matrix at the unit element of H 4 as T-dual sigma models (3.1) and (3.2) are found to be given by The parametersx,ỹ,ũ andṽ are applied for the dual model. Now by rescaling κ 0 to −2π K in action (2.7) one can conclude that action (3.10) is nothing but the WZW action based on the Lie group H 4 . Thus, we showed that the Poisson-Lie T-duality relates the H 4 WZW model to a sigma model defined on the dual Lie group of H 4 , i.e., A 2 ⊕ 2A 1 . It is seen that in this case, the Poisson-Lie T-duality transforms rather extensive and complicated action (3.11) to much simpler form such as (3.10).

Conformal invariance of the dual sigma model up to the two-loop B-functions
Consistency of the string theory requires that the action (2.8) defines a conformally invariant quantum field theory, and the conditions for conformal invariance can be interpreted as effective field equations for g µν , b µν and Φ of the string effective action [15,16]. The conditions for conformal invariance of the sigma model to the order α ′2 (at the two-loop level) have been obtained in [17] (see also Ref. [18]). It has shown that these conditions can be derived from any one of a family of space-time effective actions.
When the β-functions are trivial, i.e., they vanish up to the ambiguities inherent in their definition, then the theory is rigid scale invariant, i.e., the integrated trace anomaly vanishes. The local scale or conformal invariance needed here requires that the trace anomaly vanishes locally, which requires the vanishing of certain B-functions [17]. The vanishing of two-loop B-function gives us the conformal invariance conditions of the sigma model (2.8) to the order α ′2 [17]. In the following, we use these conditions to show that the dual sigma model (3.11) is conformally invariant up to two-loop order.
where R µν is the Ricci tensor of the metric g µν , is the torsion of the anti-symmetric field b µν , (H 2 ) µν = H µρσ H ρσ ν and H 2 = H µνρ H µνρ . Notice that the original model (the model described by action (3.10)) as a WZW model should be conformally invariant. One can find that the only non-zero components of R µν and H are R xx = − 1 2 and H xyu = − e x 2 , respectively. Thus, it is straightforward to get R = 0 = H 2 and verify that the only non-zero component of (H 2 ) µν is 2H xyu H yu x = − 1 2 . Consequently, the metric of this model is flat in the sense that its scalar curvature vanishes. Employing the above results in the vanishing of equations (3.12) we conclude that dilaton is constant. Nevertheless, by solving the vanishing of equations (3.12) one can also find a non-constant dilaton as where Φ 0 and C are integration constants.
Let us now turn into the dual model. With regard to action (3.11) the line element of the dual model is Analogously, for the dual model we find that the only non-zero component ofR µν isRṽṽ = − 1 2(e −ṽ −2) 2 e −2ṽ − 4e −ṽ + 16 ; asgṽṽ = 0,R = 0. Therefore, the metric of dual model is also flat in the sense that its scalar curvature vanishes. As shown in the above, the dual metric has an apparent singularity. This singularity is the coordinate singularity in the metric. In general relativity [19], to investigate the types of singularities one has to study the invariant characteristics of space-time. To detect the singularities it is sufficient to study only three of them, the Ricci scalar R, R µν R µν and the so-called Kretschmann scalar R µνσρ R µνσρ .
whereΦ 0 andΦ 1 are the constants of integration.

Conformal invariance of the dual sigma model up to the two-loop B-functions
The two-loop B-functions found by Hull and Townsend [17] (see also Ref. [18]) are given by where Φ ′ = Φ + α ′ qH 2 , (H 2 ) µν = H µρσ H ν ρσ and R µρσλ is the Riemann tensor field. We note that round brackets denote the symmetric part on the indicated indices whereas square brackets denote the anti-symmetric part. For the line element (3.15) of the dual model one finds that the only non-zero components ofR µνρσ andR µνρσ areRỹṽũṽ = − e −2ṽ −4e −ṽ +16 , respectively. Furthermore, we find that the only non-zero components of (H 2 ) µν and ∇ λHµνρ are and ∇ṽHỹũṽ = e −ṽ (e −ṽ −2) 3 . We must also note that sinceH 2 = 0, hence, Φ ′ =Φ. Putting all these together into equations (3.18) we conclude that the vanishing of the two-loop B-functions satisfy. Therefore the dual model is conformally invariant up to two-loop order.

Worldsheet boundary conditions under the Poisson-Lie T-duality
In this section we generally consider the worldsheet boundary conditions and their transformation under the Poisson-Lie T-duality. Then, we study the worldsheet boundary conditions specially for the H 4 WZW model. Consider a d-dimensional target space with Dp-branes, i.e., there are d − (p + 1) Dirichlet directions along which the field x i (i = p + 1, ..., d − 1) is frozen. At any given point on a Dp-brane we can choose local coordinates such that x i are the directions normal to the brane and x m (m = 0, ..., p) as label Neumann directions are coordinates on the brane. Such a coordinate system is called adapted to the brane [22]. With this choice the Dirichlet condition takes the familiar form The worldsheet boundary is by definition confined to a D-brane. Since the boundary relates leftmoving fields ∂ + x µ to the right-moving fields ∂ − x µ , one can make a general ansatz for this relation.
The goal is then to find the restrictions on this ansatz arising from varying the action (2.8). The most general local boundary condition may be expressed as [23] ∂ where R ρ ν is a locally defined object which is called the gluing matrix. This matrix encodes the information about the Neumann and Dirichlet directions in its the eigenvalues and eigenvectors.
We assume that R ρ ν is in the form of a 2 × 2 block matrix as where the submatrices R m n and R i j are Neumann-Neumann and Dirichlet-Dirichlet parts, respectively. By going to adapted coordinates at a point and by using equations (4.1) and (4.2) we get R i j = −δ i j . The Neumann condition R m n still remains very general. The boundary conditions mentioned above preserve conformal invariance at the boundary. We know that each symmetry corresponds to a conserved current, obtained by varying the action with respect to the appropriate field. For the case of conformal invariance, the corresponding current is the energy-momentum tensor and is derived by varying the action (2.8) with respect to the metric h αβ . Its components in lightcone coordinates are [22] The T ++ component is called the left-moving current, whereas T −− is referred to as right-moving current. In the conformally invariant case, energy-momentum conservation requires that the T ++ and T −− components depend only on σ + and σ − , respectively. To ensure conformal symmetry on the boundary, we need to impose boundary conditions on the currents (4.4). In general, one can find the boundary condition for a given current by using its associated charge. Applied to the energy-momentum tensor, the result is Now, using the equations (4.2), (4.4) and (4.5) we find This condition states that the gluing matrix R µ ν preserves the metric g µν . In the next, we investigate structures on D-branes. We begin by defining a Dirichlet projector Q µ ν on the worldsheet boundary, which projects vectors onto the space normal to the brane. These vectors (Dirichlet vectors) are eigenvectors of R µ ν with eigenvalue −1. Hence we can use it to write the Dirichlet condition (4.1) on the desired covariant form By contracting equations (4.2) and (4.7), we then obtain Similarly, we may define a Neumann projector N µ ν which projects vectors onto the tangent space of the brane (vectors tangent to the brane are eigenvectors of R µ ν with eigenvalue 1) and is defined as complementary to Q µ ν , i.e., The Neumann projector satisfies the following conditions [24], [25] The condition (4.10) is a condition on the Neumann-Neumann part of R µ ν . In fact it states the definition of the b-field. In adapted coordinates, for a spacefilling brane (along the Neumann directions) equation (4.10) implies, schematically, 7 R = E −1 E T . The condition (4.11) implies the diagonalisation of the metric with respect to the D-brane and the latter condition is the integrability 7 The superscript "T" means transposition of the matrix. condition for N µ ν [25]. Before starting the main result of this section, let us first to write down the boundary conditions (4.2), (4.6), (4.8) and (4.10)- (4.12) in the Lie algebra frame (related to the model on the Lie group).
To this end, we use relations Ω ab = ( and similarly for Q µ ν and N µ ν . Then, we get = Ω ab , (4.14) From the relation (4.13) it is seen that the object R a b as a gluing map between currents at the worldsheet boundary maps R a + to R a − , which are elements of the Lie algebra. As explained above in adapted coordinates, the Dirichlet-Dirichlet block of R is −δ i j . To obtain the non-zero Neumann-Neumann block of R one must use the condition (4.16). Then R takes, schematically, the following To continue, we obtain the transformation of the gluing matrix which defines how the Poisson- Lie T-duality acts on the sigma model boundary conditions. In this way, we use the canonical transformation of the Poisson-Lie T-duality transformations found by Sfetsos [26] and [27]. The canonical transformation between the pairs of variables (R a σ , P a ) and (R σ ) a ,P a ) is given by [27] where and (4.21) as a transformation from R ± toR ± . The resulting map is Ultimately, by substituting equation (4.13) into (4.25) and then by using (4.24) we obtain is the duality transformation of the gluing matrix. Note that one can immediately get detR = det(−R). We use relation (4.27) for analyzing the dual branes of the H 4 WZW model in the following subsection.

Example
In this subsection we study the worldsheet boundary conditions and D-branes in conformal Poisson-Lie symmetric sigma models generated by the Heisenberg Lie group H 4 and its dual pair, i.e., the Lie group A 2 ⊕ 2A 1 . In this example we analyze the consequences of the gluing matrix duality transformation (4.27). The conformal Poisson-Lie T-dual sigma models built by the Drinfel'd double H 4 , A 2 ⊕ 2A 1 have been given by relation (3.10) and (3.11). For these models the background fields E + ab (g) andẼ +ab (g) can be obtained from equations (3.4)-(3.6) and (3.9) as follows: (4.28) Thus, inserting E + ab (g) andẼ +ab (g) from (4.28) together with the E + 0 (e) of relation (3.9) into equation (4.27) one can get the dual gluing matrixR for any given original matrix R. From the form of the matrix E + ab (g) (relation (4.28)) it is seen that 1× 1, 2× 2 and 3× 3 submatrices of E + ab (g) are not invertible. Therefore according to relation (4.19) the brane does not include one, two and or three Neumann directions. Consequently, we can have two different types of D-branes: D(−1) and D3; that is, all directions are either Dirichlet or Neumann. In the following we compute the dual gluing matrix for each of these cases.
Case (1): In this case all directions are Dirichlet; that is, Q a b = δ a b and N a b = 0. Since the numbers of Neumann directions are p + 1, so we have, in this case, a D(−1)-brane. From relation (4.19) the corresponding gluing matrix is given by Then, equation (4.27) yields the dual gluing matrix It has determinant detR = det(−R) = 1, so the the dual brane may include the following directions: (1.i) Four Dirichlet directions, i.e., the dual brane is also a D(−1)-brane. The dual D(−1)-brane is nontrivially embedded in the dual manifold, and the embedding can be found by diagonalizing R. Then, in this case, the only solution isR = −1I, which happens only for backgrounds E + (g) (1.ii) Two Dirichlet directions and two Neumann directions, i.e., it is a D1-brane.
(1.iii) Zero Dirichlet directions. This is a D3-brane whose embedding inG is given byR.
Since it is spacefilling it should satisfy the dual version of equation ( Case (2): In this case we have a spacefilling brane, i.e., a D3-brane. The corresponding gluing matrix according to equation (4.16) is given by Its determinant is detR = 1, so the dual brane may include the following directions: (2.i) Four Dirichlet directions. In this case we obtain exactly the reverse situation of case (1.iii), that is, we haveR = −1 and the D3-brane is dual to a D(-1)-brane provided the detΠ oñ G vanishes.
We note that since detR = 1, the dual brane may include zero Dirichlet directions, i.e., a D3brane. But, on the other hand, since it is spacefilling it should satisfy the dual version of (4.16), R =Ẽ + −1Ẽ + T . In this situation, equation (4.27) would require E + 0 + E + 0 T = 0 and hence a vanishing metric. Thus, we conclude that D3-branes are dual either to D1-branes or D(-1)-branes provided by detΠ = 0, but that D3-branes are never dual to D3-branes.

Concluding remarks
In  (2) we have a spacefilling brane, i.e., a D3-brane; in this case we have shown that D3-branes are dual either to D1-branes or D(-1)-branes provided by detΠ = 0, but that D3-branes are never dual to D3-branes.