Doubling, T-Duality and Generalized Geometry: a Simple Model

A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated as a 0+1 field theory, aiming at further investigating the relation between Generalized/Double Geometry on the one hand and Doubled World-Sheet Formalism/Double Field Theory, on the other hand. The model is defined over the group manifold of SU(2) and a dual model is introduced having the Poisson-Lie dual of SU(2) as configuration space. A generalized action with configuration space SL(2,C), i.e. the Drinfel'd double of the group SU(2), is then defined: it reduces to the original action of the rotator or to its dual, once constraints are implemented. The new action contains twice as many variables as the original. Moreover, its geometric structures can be understood in terms of Generalized Geometry. keywords: Generalized Geometry, Double Field Theory, T-Duality, Poisson-Lie symmetry.


Introduction
Generalized Geometry (GG) was first introduced by N. J. Hitchin in Ref. [1]. As Hitchin himself states in his pedagogical lectures [2], it is based on two premises: the first consists in replacing the tangent bundle T of a manifold M with T ⊕ T * , a bundle with the same base space M but fibers given by the direct sum of tangent and cotangent spaces. The second consists in replacing the Lie bracket on sections of T , which are vector fields, with the Courant bracket which involves vector fields and one-forms. The construction is then extended to general vector bundles E over M so to have E ⊕ E * and a suitable bracket for the sections of the new bundle.
The framework where Double Field Theory (DFT) [3] can be inserted aims at understanding the behavior under duality transformations of physical models, including strings, described by field theories exhibiting such transformations as a symmetry of the dynamics; the symmetry however is not manifest at the level of the action functional. More specifically, DFT is a proposal to incorporate T-duality, a peculiar symmetry of a toroidally compactified string, as a manifest symmetry of the string effective field theory. In order to achieve this goal, the action of such field theory has to be defined on a configuration space that is doubled with respect to the original. What makes T-duality a distinctive symmetry of strings is that it refers to extended objects that, just like strings and differently from particles, can wrap non-contractible cycles. Such a wrapping implies the presence of winding modes that have to be added to the ordinary momentum modes which take integer values along compact dimensions. T-duality is an O(D, D) symmetry of the dynamics of a closed string under, roughly speaking, the exchange of winding and momentum modes: it establishes, in this way, a connection between the physics of strings defined on different geometric backgrounds. Then, a T-duality symmetric field theory must contain information about windings. A possible way to implement such information consists in introducing a new set of dual coordinates, canonically conjugate to the winding modes. Hence, to each compact space coordinate characterized by a certain winding, one can associate a new coordinate conjugate to the winding. It is then possible to describe the dynamics in the Hamiltonian setting by means of Poisson brackets which generalize the canonical ones to such an extended setting. In this sense DFT is a double theory: it doubles the coordinates of the compact space. But actually the doubling can involve the whole D-dimensional target space, since one can also formally associate to each non-compact dimension its own dual coordinate, on which no physical quantities really depend.
DFT is formulated in terms of the background fields G ij (the target-space metric tensor) and B ij (the Kalb-Ramond field), with i, j = 1, . . . , D, of closed strings in a D-dimensional target manifold M , in addition to a dilaton scalar field φ. These fields depend, in that framework, on doubled coordinates x i andx i even if there is no doubling of their tensor indices. They are double fields. The gauge symmetry parameters for DFT are the vector fields ξ i (x,x), which parametrize diffeomorphisms and live in the tangent bundle of the doubled manifold, together with the oneformsξ i (x,x), which describe gauge transformations of B ij and live in the cotangent bundle of the doubled manifold. In Ref. [4] it has been shown that the algebra of the gauge symmetries of DFT is determined by the so-called C-brackets, first introduced, together with other relevant aspects of DFT, in Refs. [5]. C-brackets provide an O(D, D) covariant, DFT generalization of the Courant bracket. More precisely, it can be shown that they reduce to Courant brackets if one drops the dependence of the doubled fields on the coordinatesx i . This establishes a relation between GG and DFT that needs to be more deeply analyzed.
Models whose carrier manifold of the dynamics is a Lie group G can be very helpful in understanding such a relation on a rigorous basis, because the notion of dual of a Lie group is well established together with that of double Lie group and the so called Poisson-Lie symmetries [6,7]. The idea of investigating such geometric structures in relation to duality in field theory has already been applied to topological sigma models by Klimčík andŠevera in [8] (also see [9], [10]) where the authors first introduced the notion of Poisson-Lie T-duality. On the other hand, in Ref. [11], the phase space T * G was already proposed as a toy model for discussing conformal symmetries of chiral models, in a mathematical framework which is very similar to the one adopted here. Double Field Theory on group manifolds, including its relation with Poisson-Lie symmetries, has been analyzed in [12]. In the present paper, we propose a fresh look at the subject in relation to the recent developments in GG and DFT. This is the first of a series of two papers. Detailed here within lies the analysis of the isotropic rigid rotator (IRR) having as configuration space the group SU (2) and of its dual model on the dual group SB(2, C). Their Poisson-Lie symmetries are considered and an extended model on the double group, the so-called classical Drinfel'd double SL(2, C), is formulated. An alternative description of the IRR model on the Drinfel'd double was already proposed in [13], although there the dual model was not analyzed, the accent being not on duality, but on the possibility of describing the same dynamics with a different phase space, the group manifold SL(2, C), by relying on the fact that the latter is symplectomorphic to to the cotangent bundle of SU (2) [14].
Let us stress that the model considered here is too simple to exhibit symmetry under duality transformation, but it is readily generalizable. We may look at it as a 0 + 1 field theory, thus paving the way for a genuine 1 + 1 field theory, the SU (2) principal chiral model, which, while being modeled on the IRR system, will result to be duality invariant, with all the richness of DFT and generalized geometry. This will be the subject of a forthcoming paper [15].
The paper is organized as follows. In section 2 the dynamics of the IRR on the group manifold of the group SU (2) is reviewed. In section 3 an account of the mathematical framework that is going to be used is given, with Poisson-Lie groups and their Drinfel'd doubles discussed in some detail. In 4 the dual model on the dual group of SU (2), the group SB(2, C), is introduced and its dynamics analyzed. Finally, in section 5 a dynamical model on the Drinfel'd double of SU (2) is proposed: it has doubled configuration variables with respect to the original IRR coordinates, and doubled generalized velocities, (I i ,Ĩ i ) which are dual to each other. The latter can be interpreted as tangent and cotangent space coordinates on the generalized T ⊕ T * bundle over SU (2) (or, dually, over the group SB(2, C)), with Poisson brackets yielding a Poisson realization of C-brackets. Their Hamiltonian vector fields can be derived, yielding, in turn, an algebra which is closed under Lie brackets, which can be seen as derived C-brackets [16,17].
While completing the article we have become aware of the work in Refs. [18,19]. In the first one non-Abelian T-duality is analyzed within the same mathematical framework, whereas the latter studies an interesting mechanical model, the electron-monopole system, within the DFT context. Their relation with the present work should be further investigated; we plan to come back to this issue in the future.

The Isotropic Rigid Rotator
The Isotropic Rigid Rotator (IRR) provides a classical example of dynamics on a Lie group, the group being in this case SU (2) with its cotangent bundle T * SU (2), the carrier space of the Hamiltonian formalism, carrying the group structure of a semi-direct product. In this section, the Lagrangian and Hamiltonian formulations of the model on the group manifold are reviewed. Although being simple, the model captures relevant characteristics of the dynamics of many interesting physical systems, both in mechanics and in field theory, such as Keplerian systems, gravity in 2+1 dimensions in its first order formulation, with and without cosmological constant [20], Palatini action with Holst term [21], and principal chiral models [22].

The Lagrangian and Hamiltonian Formalisms
As carrier space for the dynamics of the three dimensional rigid rotator in the Lagrangian [Hamiltonian] formulation we can choose the tangent [cotangent] bundle of the group SU (2). We follow Ref. [23] for the formulation of the dynamics over Lie groups.
A suitable action for the system is the following with g : t ∈ R → SU (2), the group-valued target space coordinates, so that is the Maurer-Cartan left-invariant one-form, which is Lie algebra-valued, σ k are the Pauli matrices, α k are the basic left-invariant one-forms, * denotes the Hodge star operator on the source space R, such that * dt = 1, and Tr the trace over the Lie algebra. Moreover, g −1ġ is the contraction of the Maurer-Cartan one-form with the dynamical vector field Γ = d/dt, g −1ġ ≡ (g −1 dg)(Γ). Let us remind here that the Lagrangian is written in terms of the nondegenerate invariant scalar product defined on the SU (2) manifold and given by a|b = Tr(ab) for any two group elements. The model can be regarded as a (0 + 1)-dimensional field theory which is group-valued.
The group manifold can be parametrized with R 4 coordinates, so that g ∈ SU (2) can be read as g = 2(y 0 e 0 + iy i e i ), with (y 0 ) 2 + i (y i ) 2 = 1, e 0 = I/2, e i = σ i /2 the SU (2) generators. One has then: By observing that we define the left generalized velocitiesQ i aṡ . . , 3 are therefore tangent bundle coordinates, with Q i implicitly defined. Starting with a Lagrangian written in terms of right-invariant one-forms, one could define right generalized velocities in an analogous way. They give an alternative set of coordinates over the tangent bundle.
The Lagrangian L 0 in Eq. (2.1) can be rewritten as: In the intrinsic formulation, which is especially relevant in the presence of non-invariant Lagrangians, the Euler-Lagrange equations of motion are represented by: the Lagrangian one-form and L Γ the Lie derivative with respect to Γ. By projecting along the basic left-invariant vector fields X i dual to α i , one obtains: Since L Γ and i X i commute over the Lagrangian one-form, one gets: because of the rotation invariance of the product and the antisymmetry of the structure constants of SU (2) as a manifestation of the invariance of the Lagrangian under rotation.
Equivalently, the equations of motion can be rewritten as: being, from Eq. (2.2): Cotangent bundle coordinates can be chosen to be (Q i , I i ) with the I i 's denoting the left momenta: An alternative set of fiber coordinates is represented by the right momenta, which are defined in terms of the right generalized velocities.
The Legendre transform from T SU (2) to T * SU (2) yields the Hamiltonian function: By introducing a dual basis {e i * } in the cotangent space, such that e i * |e j = δ i j , one can consider the linear combination: The dynamics of the IRR is thus obtained from the Hamiltonian (2.7) and the following Poisson brackets which are derived from the first-order formulation of the action functional with θ the canonical one-form. Indeed the symplectic form ω is readily obtained as Upon inverting ω one finds the Poisson algebra (2.9)-(2.11). The fiber coordinates I i are associated to the angular momentum components and the base space coordinates (y 0 , y i ) to the orientation of the rotator. The resulting system is rotationally The Hamilton equations of motion for the system are: Thus the angular momentum I i is a constant of motion, while g undergoes a uniform precession. Since the Lagrangian and the Hamiltonian are invariant under right and left SU (2) action, as well-known right momenta are conserved as well, being the model super-integrable.
Let us remark here that, while the fibers of the tangent bundle T SU (2) can be identified, as a vector space, with the Lie algebra of SU (2), su(2) ≃ R 3 , withQ i denoting vector fields components, the fibers of the cotangent bundle T * SU (2) are isomorphic to the dual Lie algebra su(2) * . As a vector space this is again R 3 , but the I i 's are now components of one-forms. This remark is relevant in the next section, when the Abelian structure of su(2) * is deformed.
As a group, T * SU (2) is the semi-direct product of SU (2) and the Abelian group R 3 , with the corresponding Lie algebra given by: (2.14) Then, the non-trivial Poisson bracket on the fibers of the bundle, (2.10), can be understood in terms of the coadjoint action of the group SU (2) on its dual algebra su(2) * ≃ R 3 and it reflects the non-triviality of the Lie bracket (2.12). In this picture the Lie algebra generators L i 's are identified with the linear functions on the dual algebra. 1 .
Before concluding this short review of the canonical formulation of the dynamics of the rigid rotator, let us stress the main points which we are going to elaborate further: • The carrier space of the Hamiltonian dynamics is represented by the semi-direct product of a non-Abelian Lie group, SU (2), and the Abelian group R 3 which is nothing but the dual of its Lie algebra.
• The Poisson brackets governing the dynamics are the Kirillov-Poisson-Souriau brackets induced by the coadjoint action.
It has been shown in Ref. [13] that the carrier space of the dynamics of the rigid rotator can be generalized to the semisimple group SL(2, C), which is obtained by replacing the Abelian subgroup R 3 of the semidirect product above, with a non-Abelian group. The generalization is obtained by considering the double Lie group of SU (2). In this paper such generalization will be further pursued giving rise to a doubled dynamical model, the simplest instance of a DFT, together with its double geometry.
The underlying mathematical construction of Drinfel'd double Lie groups and their relation with the structures of Generalized Geometry is the subject of the next section.
3 Poisson-Lie Groups and the Double Lie Algebra sl(2, C) In this section we shortly review the mathematical setting of Poisson-Lie groups and Drinfel'd doubles, see [7,31,32] for details, with the aim of introducing, in the forthcoming sections, new Lagrangian and Hamiltonian formulations of the IRR with a manifest symmetry under duality transformation. More precisely, in Section 4, a model which is dual to the one described in Section 2 is introduced, while in Section 5 built a new model is built with doubled dynamical variables and with a manifest symmetry under duality transformation. The dynamics derived from the new action describes two models, dual to each other, being one the ordinary rigid rotator, the other a"rotator-like" system, with the rotation group SU (2) replaced by its Poisson-Lie dual, the group SB(2, C) of Borel 2 × 2 complex matrices.
A Poisson-Lie group is a Lie group equipped with a Poisson structure which makes the product µ : G × G → G a Poisson map if G × G is equipped with the product Poisson structure. Linearization of the Poisson structure at the unit e of G provides a Lie algebra structure over the dual algebra g * = T * e (G) by the relation The compatibility condition between the Poisson and Lie structures of G yields the relation: with v, w ∈ g * , X, Y ∈ g and ad * X , ad * v the coadjoint actions of the Lie algebras g, g * on each other. This allows one to define a Lie bracket in g ⊕ g * through the formula: If G is connected and simply connected, (3.2) is enough to integrate [ , ] * to a Poisson structure on G that makes it Poisson-Lie and the Poisson structure is unique. The symmetry between g and g * in (3.2) implies that one has also a Poisson-Lie group G * with Lie algebra (g * , [ , ] * ) and a Poisson structure whose linearization at e ∈ G * gives the bracket [ , ]. G * is the dual Poisson-Lie group of G. The two Poisson brackets on G, G * , which are dually related to the Lie algebra structure on g * , g, respectively, when evaluated at the identity of the group are nothing but the Kirillov-Souriau-Konstant Poisson brackets on coadjoint orbits of Lie groups. The Lie group D, associated to the Lie algebra d = g ⊕ g * is the Drinfel'd double group of G (or G * , being the construction symmetric). 2 There is a dual algebraic approach to the picture above, mainly due to Drinfel'd [6], which starts from a deformation of the semi-direct sum g ⊕ R n , with R n ≃ g * , into a fully non-Abelian Lie algebra, which coincides with d. The latter construction is reviewed below.
To be specific to our problem, we focus on the group SU (2) whose Drinfel'd double can be seen to be the group SL(2, C) [6]. An action can be shown to be written on the tangent bundle of D, in such a way that the usual Lagrangian description of the rotator can be recovered by reducing the carrier manifold to the tangent bundle of SU (2).
The structure of d = sl(2, C) as a double algebra is shortly reviewed here.
With this purpose, we start recalling that the complex Lie algebra sl(2) is completely defined by the Lie brackets of its generators: By considering complex linear combinations of the basis elements of sl (2), say e i , b i , i = 1, 2, 3, respectively given by: the real algebra sl(2, C) can be easily obtained with its Lie brackets: In a similar way, one can introduce the combinations: which are the dual basis of the generators (3.6), with respect to the scalar product naturally defined on sl(2, C) as: Indeed, it is easy to show that Hence, {ẽ j } is the dual basis of {e i } in the dual vector space su(2) * . Such a vector space is in turn a Lie algebra, the special Borel subalgebra sb(2, C) with the following Lie brackets: In a more compact form, the generators (3.11) can be written as: (3.15) and the corresponding the Lie brackets can be derived: For future convenience we also note that: The following relations can be easily checked: so that both su(2) and sb(2, C) are maximal isotropic subspaces of sl(2, C) with respect to the scalar product (3.12). 3 Therefore, the Lie algebra sl(2, C) can be split into two maximally isotropic dual Lie subalgebras with respect to a bilinear, symmetric, non degenerate form defined on it. The couple (su (2), sb(2, C)), with the dual structure described above, is a Lie bialgebra.
Since the role of su(2) and its dual algebra can be interchanged, (sb(2, C), su (2)) is a Lie bialgebra as well. The triple (sl(2, C), su (2), sb(2, C)) is called a Manin triple [6]. The total algebra d = g ⊲⊳ g * which is the Lie algebra defined by the Lie brackets (3.8), (3.16), (3.17), with its dual d * is also a Lie bialgebra. 4 The couple (d, d * ) is called the double of (g, g * ) [7]. The double group D is meant to be the Lie group of d endowed with some additional structures such as a Poisson structure on the group manifold compatible with the group structure; more details are given in the next section. The two partner groups, SU (2) and SB(2, C) with suitable Poisson brackets, are named dual groups and sometimes indicated by G, G * . Their role can be interchanged, so that they share the same double group D.
The splitting of sl(2, C) is realized with respect to the scalar product (3.12). This is given by the Cartan-Killing metric g ij = 1 2 c ip q c jq p , induced by the structure constants c ij k of sl(2, C) in its adjoint representation.
But this is not the only decomposition of sl(2, C) one can give. There is another nondegenerate, invariant scalar product, represented by In this case, for the basis elements, one gets: giving rise to a metric which is not positive-definite. With respect to the scalar product defined in Eq. (3.20), new maximal isotropic subspaces can be defined in terms of: It turns out that: Let us notice that neither of them spans a Lie subalgebra. By denoting by C + and C − the two subspaces spanned by {e i } and {b i } respectively, one can notice [33] that the splitting d = C + ⊕ C − defines a positive definite metric G on d via: As in Ref. [33], the inner product is here used to identify d with its dual, so that the metric G may be viewed as an automorphism of d which is symmetric and which squares to the identity, i.e. G 2 = 1. Let us indicate the Riemannian metric with double round brackets. One has then: In order to come back to the main subject of the paper, namely the relation between GG and DFT, introducing the following notation for the sl(2, C) generators reveals to be very helpful: (2), e i ∈ sb(2, C), ((ẽ i ,ẽ j )) = δ ip δ jq ((b p + e l ǫ l p3 ))((b q + e k ǫ k j3 )) = δ ij + ǫ i l3 δ lk ǫ j k3 ; (3.29) Therefore, one has: This metric satisfies the relation: It is interesting to see how the metric L in Eq. (3.28) and the metric G in Eq. (3.31) naturally emerge out in the framework here in exam. They correspond, in the usual context of DFT, to the O(d, d) invariant metric and to the so-called generalized metric [34,35], respectively (see also [?]). In particular, in the latter, the role of the graviton field is played by the Kronecker delta δ ij while the role of the Kalb-Ramond field is played by the three-dimensional Levi-Civita symbol ǫ ij3 with one of the indices being fixed.

Para-Hermitian Geometry of SL(2, C)
The two non-degenerate scalar products of SL(2, C), discussed above, have been widely applied in many physical contexts where the Lorentz group and its universal covering SL(2, C) play a role, starting from the pioneering work by E. Witten [20]. While the first scalar product, i.e. the one defined in Eq. (3.12), is nothing but the Cartan-Killing metric of the algebra, the Riemannian structure G can be mathematically formalized in a way which clarifies its role in the context of Generalized Complex Geometry [33,36]. Let us shortly review the derivation.
The splitting of sl(2, C) in su(2) and sb(2, C) implies the existence of a (1, 1)-tensor: such that R 2 = 1 and of eigenspaces given by su (2), with eigenvalue +1, and sb(2, C), with eigenvalue −1. This can be seen as the local expression of a (1, 1)-tensor on SL(2, C) called product structure, since it has integrable eigenbundles T SU (2) and T SB(2, C), that, at every point of SL(2, C), are given by su (2) and sb(2, C) and are such that T SL(2, C) = T SU (2) ⊕ T SB(2, C). These two eigenbundles are maximal isotropic with respect to the scalar product (3.12) and, being integrable, they give rise to two transversal foliations of SL(2, C), the one with SU (2) as leaves, the other with SB(2, C).
Moreover, the tensor R is compatible with the scalar product (3.12) meaning that the following equation holds: where Γ(T SL(2, C)) denotes the vector fields over the group manifold. In a more compact form, one has: R T LR = −L, with L(X, Y ) = X, Y , ∀X, Y ∈ Γ(T SL(2, C)). This condition implies that a 2-form field can be defined as: In other words, a para-Kähler structure (also called bi-Lagrangian) [36] can be defined on the manifold SL(2, C), where R is the product structure, (3.12) is the scalar product compatible with the Lorentzian signature and ω is the closed two-form. In this sense, the scalar product (3.31) can be read as a metric with Riemannian signature considering the bases (3.22).
In fact, expressing the bases (3.22) as linear combinations of {e i } and {ẽ i } yields the following: and generating, respectively, the subspaces V + and V − of sl(2, C), which are maximal isotropic with respect to (3.20).
Then, given the splitting sl(2, C) = V + ⊕ V − , there exists the (1, 1)-tensor H such that and implying that V + and V − are eigenspaces of H with eigenvalues +1 and −1 respectively.
One can immediately notice that the Lie bracket of any two elements in {f + i } and {f − i } is not involutive, hence V + and V − are not Lie subalgebras of sl(2, C).
As described at the beginning of this section, Eq. (3.35) can be read as the definition, at any point, of a (1, 1)-tensor field H as an almost product structure on SL(2, C), since the eigenbundles V + and V − , obtained as distributions that, at any point, are V + and V − respectively, are not integrable. The local change of splitting implies that T SL(2, C) = V + ⊕ V − .
In order to write down the dual bases {f j * + } and {f j * − } of {f + j } and {f − j } respectively, by using the duality relation between {e i } and {ẽ i }, the duality conditions have to be imposed: Therefore, the almost product structure H turns out to be the following: which, in the bases {e i }, {ẽ i }, becomes: The metric (3.28) can be used on H for raising and lowering indices. In fact, in the doubled formalism, one can write the matrix: which is exactly the generalized metric (3.31), i.e. G = LH. Moreover, it is easy to verify that the metric with Lorentzian signature arising from (3.20) is given by: which takes the form and is compatible with H, i.e. H T KH = −K. The compatibility condition of K and H gives a closed two-form that, in the bases {e i } and {ẽ i }, gets the following expression: This can be read as an explicit form of the product structure R and the closed two-form Ω.
In conclusion, it has been here shown that the natural scalar product (3.20) and the almost product structure H define an almost para-Kähler structure on the manifold SL(2, C).
Finally, let us describe the structure arising from (3.26). In the previous section, a positive definite metric G (3.25) on SL(2, C) has been defined by the splitting sl(2, C) = C + ⊕ C − . In order to explicitly write down the metric tensor G, dual bases of {e i } and {b i } have been introduced. After noticing that b i = δ ijẽ j + ǫ k3 i e k and by imposing the conditions It is worth to stress that changing the splitting also changes the dual bases. Thus, the metric tensor of Eq. (3.25) can be retrieved: It takes the form (3.36) in the bases ({e i }, {ẽ i }), is symmetric and squares to the identity. Moreover, the metric G can be seen to be given by the composition of two generalized complex structures [33], I J and I ω , respectively defined by an almost complex structure J and a symplectic structure ω. Therefore, one has a pair (I J , I ω ) of commuting generalized complex structures on T SL(2, C) inducing a positive definite metric G. This is usually called a generalized Kähler structure. Consequently, it has been shown how the almost para-Kähler structure and the generalized metric G on the manifold SL(2, C) are related.
The discussion that has been just completed shows the existence of two non-degenerate, invariant scalar products on the total algebra d. In the forthcoming sections both products will be considered in order to define action functionals for the dynamical systems in exam.

The Dual Model
In the previous section the dual group of SU (2), the group SB(2, C), has been introduced as the partner of SU (2) in a kind of Iwasawa decomposition of the group SL(2, C). The latter has been regarded as a deformation of the cotangent bundle of SU (2) with fibers F ≃ R 3 replaced by the group SB(2, C). It is then legitimate to reverse the paradigma and regard SL(2, C) as a deformation of the cotangent bundle T * SB(2, C), with fibersF ≃ R 3 now replaced by SU (2). In this section a dynamical model on the configuration space SB(2, C) is proposed with an action functional that is formally analogous and indeed dual to (2.1). The model is described with its symmetries in the Lagrangian and Hamiltonian formalisms.
In sect. 5 a generalized action containing both the models is finally introduced on the whole group SL(2, C). The Poisson algebra encoding the dynamics, as well as the algebra of generalized vector fields describing infinitesimal symmetries, turns out to be related to the so-called C-bracket of DFT.

The Lagrangian and Hamiltonian Formalisms
As carrier space for the dynamics of the dual model in the Lagrangian (respectively Hamiltonian) formulation one can choose the tangent (respectively cotangent) bundle of the group SB(2, C). A suitable action for the system is the following: withg : t ∈ R → SB(2, C), the group-valued target space coordinates, so that is the Maurer-Cartan left invariant one-form on the group manifold, with β k the left-invariant basic one-forms, * the Hodge star operator on the source space R, such that * dt = 1. The symbol T r is here used to represent a suitable scalar product in the Lie algebra sb(2, C). Indeed, since the algebra is not semi-simple, there is no scalar product which is both non-degenerate and invariant. Therefore, one has two possible different choices: the scalar product defined by the real and/or imaginary part of the trace, given by Eqs. (3.12) and (3.20) which is SU (2) and SB(2, C) invariant, but degenerate; or one could use the scalar product induced by the Riemannian metric G, which, on the algebra sb(2, C) takes the form (3.29) which is positive definite and non-degenerate, but only invariant under left SB(2, C) action and SU (2) invariant. Indeed, by observing that the generatorsẽ i are not Hermitian, (3.29) can be verified to be equivalent to: The associated dynamical models are obviously different. The non-degenerate scalar product defined in Eq. (3.29) is used here, therefore the Lagrangian (4.1) is only left/right SU (2) and left-SB(2, C) invariant, differently from the Lagrangian of the rigid rotator (2.1) which is invariant under both left and right actions of both groups. As in the previous case, the model can be regarded as a (0 + 1)-dimensional field theory which is group-valued.
The group manifold can be parametrized with R 4 coordinates, so thatg ∈ SB(2, C) reads g = 2(u 0ẽ 0 + iu iẽ i ), with u 2 0 − u 2 3 = 1 andẽ 0 = I/2. One has then: where the last product is defined as twice the real part of the trace, in order to be consistent with the others. By observing that the Lagrangian in (4.1) can be rewritten as: the left generalized velocities and the metric defined by the scalar product. By repeating the analysis already performed for the IRR, one finds the equations of motion: withX j being the left invariant vector fields associated to SB(2, C). Differently from the IRR case, the Lagrangian now is not invariant under right action, therefore, being the left invariant vector fields the generators of the right action, the l.h.s. in Eq. (4.4) is not expected to be zero and through a straightforward calculation it results to be: It has to be noticed here that, analogously to the IRR case, one could define the right generalized velocities on the fibers starting from right invariant one-forms, but, differently from that case, the right invariant Lagrangian is not equivalent to the left invariant one, as already stressed.
The cotangent bundle coordinates are (Q i ,Ĩ i ) withĨ i the conjugate left momentã The latter is in turn invertible, yielding: so that the Legendre transform from T SB(2, C) to T * SB(2, C) leads to the Hamiltonian function:H the inverse of Eq. (4.3). Similarly to Eq. (2.8), the linear combination over the dual basis is introduced:Ĩ = iĨ jẽ * j (4.8) with e j * |ẽ i = δ i j . Then, the first order dynamics can be obtained from the Hamiltonian (4.6) and the following Poisson brackets: which are derived from the first order formulation of the action functional. Since the results are slightly different from the IRR case, let us present the derivation in some detail.
The first-order action functional reads in this case as: Observing that: the symplectic formω reads as: where the relation d[g −1 dg] = i 2 β i ∧ β j f ij kẽ k has been used. By invertingω, one finally finds the Poisson algebra (4.9)-(4.11).
Hamilton equations are readily obtained from the Poisson brackets. In particular one gets: which is consistent with Eq. (4.5) and is different from zero, expressing the non-invariance of the Hamiltonian under right action. Vice-versa, by introducing the right momentaJ i as the Hamiltonian functions of right-invariant vector fields, which in turn generate the left action, and observing that left and right invariant vector fields commute, one readily obtains: namely, right momenta are constants of the motion and the Hamiltonian is invariant under left action, as we expected.
By using (4.11) it is possible to find: consistently with Eq. (4.2). Right momenta are therefore conserved, as for the rigid rotator, while left momenta are not.
Let us remark here that, while the fibers of the tangent bundle T SB(2, C) can be identified, as a vector space, with the Lie algebra of SB(2, C), sb(2, C) ≃ R 3 , withQ i denoting vector fields components, the fibers of the cotangent bundle T * SB(2, C) are isomorphic to the dual Lie algebra sb(2, C) * . As a vector space this is again R 3 , butĨ j are now components of one-forms. This remark will be relevant in the next section where the Abelian structure of sb(2, C) * is deformed.
As a group, T * SB(2, C) is the semi-direct product of SB(2, C) and the Abelian group R 3 , with Lie algebra the semi-direct sum represented by Then, as before, the non-trivial Poisson bracket on the fibers of the bundle, (4.10), can be understood in terms of the coadjoint action of the group SB(2, C) on sb(2, C) * ≃ R 3 , i.e. its dual algebra, and it reflects the non-triviality of the Lie bracket (4.16) with the Lie algebra generators B i identified with linear functions on the dual algebra.
To summarize the results of this section, the model that has been introduced is dual to the Isotropic Rigid Rotator in the sense that the configuration space SB(2, C) is dual, as a group, to SU (2).
In the next section, a generalized action is constructed on the Drinfel'd double group and it encodes the duality relation between the two models and the global symmetries that have been discussed.

A New Formalism for the Isotropic Rotator: the Double Field Theory Formulation
In the previous sections, two dynamical models have been introduced with configuration spaces being Lie groups which are dually related. The Poisson algebras for the respective cotangent bundles, T * SU (2), T * SB(2, C), which we restate for convenience in the form: have both the structure of a semi-direct sum dualizing the semi-direct structure of the Lie algebras su(2)⊕R 3 and sb(2, C)⊕R 3 . Upon identifying the dual algebras R 3 , in both cases, with an Abelian Lie algebra, we have that each semi-direct sum has the the form (3.3), with R 3 generators satisfying trivial brackets and with a trivial ad * action: To this, it is sufficient to expand the group variables, g,g and compute the related Poisson brackets in Eqs. (5.1), (5.2) to first order in the parameters. One gets: The new dynamical variables J i ,J i will be identified in the forthcoming section with I i ,Ĩ i upon unifying the cotangent bundles T * SU (2), T * SB(2, C) into the Drinfel'd double SL(2, C). The brackets (5.4), (5.5) will then emerge naturally as appropriate limits of the Poisson-Lie brackets on the dual groups G, G * , when evaluated at the identity of the respective groups as in Eq. (3.3).

The Lagrangian Formalism
We are now ready to introduce the new action for the Isotropic Rigid Rotator using the Lagrangian formalism on T SL(2, C). As in the conventional formulation described above, its description can be read as a (0 + 1)-dimensional field theory which is group-valued, with g(t) ∈ SU (2) now replaced by γ : t ∈ R → γ(t) ∈ SL(2, C). The left invariant one-form on the group manifold is then: where, however, both components are tangent bundle coordinates for SL(2, C) 5 . By using the scalar product (3.12) the components of the generalized velocity can be explicitly obtained: with the Hodge operator defined as previously, namely * dt = 1, the proposed action is the following: where k 1 , k 2 are real parameters, and γ −1 dγ ∧ , * γ −1 dγ is defined in terms of the scalar product in Eq. (3.28) while (γ −1 dγ ∧ , * γ −1 dγ) is defined in terms of the scalar product in Eq. (3.31), namely: Explicitly, in terms of the chosen splitting of the Drinfel'd double sl(2, C) = su(2) ⊲⊳ sb(2, C), one has, up to an overall constant: δ lk and where the position k 1 /k 2 ≡ k has been made. This leads to: The Lagrangian one-form is therefore: and the equations of motion read as: where C K IP are the structure constants of sl(2, C). The matrix k L IJ + G IJ is non-singular, provided k 2 = 1, which will be assumed from now on.

Recovering the Standard Description
The standard dynamics of the isotropic rigid rotator is now shown to be recovered from the new Lagrangian.
In order to be definite, let us fix a local decomposition for the elements of the double group SL(2, C): γ =gg, with g ∈ SU (2) andg ∈ SB(2, C). From Eq. (5.7), one can see that L is invariant under left and right action of the group SU (2), but only under left action of the group SB(2, C), given by SB(2, C) L : γ →hγ =hgg, ∀h ∈ SB(2, C).
In order to recover the usual description of the rotator, the SB(2, C) L invariance has to be promoted to a local gauge symmetry. One has then: the gauge connection. The following split can be performed: Then, Eq. (3.13) implies: so that, using (3.12) leads to: (5.14) Analogously, one can compute: so that, from (3.12) 2Im Tr (γ −1ẽj γe i ) = a j k h i s δ k s one derives: After replacing the Lagrangian (5.10) with the gauged Lagrangian one gets: Let us introduce now the combination: 16) allowing to rewrite the Lagrangian L C as follows: This can be used for writing the partition function of the system under analysis as: and integrate over the gauge potential. Therefore, the integration with respect to C i can be traded for the integration with respect to U i . The functional integral (5.17) can be performed by changing the integration variable. Therefore, by inverting the relation (5.14), one can calculate det δC i δU j and see that it is a constant, because the matrices involved in the definition of U i are all invertible. Consequently, the functional integral in the partition function becomes: where It is worth to notice that, in (5.16), the tensor T ij = kδ ij − ǫ ij3 defines, for k = 0, a constant invertible map T : sb(2, C) → su(2), so one can introduce the endomorphism E of d = su(2) ⊕ sb(2, C) which preserves the splitting, defined by the constant matrix: This acts on any element of d in the following way: where W i is given by (5.16) and We can write down the left invariant forms The constant endomorphism (5.20) induces a map exp(E) : SL(2, C) → SL(2, C) such that γ =gg → γ ′ =g ′ g ′ . Then, one can see that the path integral measure can be transformed giving DgDg = Dg ′ Dg ′ up to a constant factor, i.e. the determinant of the constant map exp(E).
Thus the path integral (5.18) can be written, up to constant factors, as: where the path integral overg ′ gives a constant and the other integral is exactly the partition function of the action of the IRR defined up to a constant factor.

The Hamiltonian Formalism
In the doubled description introduced above, the left generalized momenta are represented by: The Hamiltonian reads then as: From (5.21) one can explicitly write the generalized momenta P I in terms of the components oḟ Q I ≡ (A i , B j ), finding: In terms of the components I i ,Ĩ j , it turns out that: which can be rewritten as In order to obtain the Hamilton equations for the generalized model on the Drinfel'd double, one can proceed as in the previous section with the determination of Poisson brackets from the first-order action functional: We stress once again that P I , α J are respectively generalized momenta and basis one-forms on the doubled configuration space SL(2, C). The symplectic form on T * SL(2, C) ≃ SL(2, C) × sl(2, C) * is therefore: which yields for the generalized momenta the Poisson brackets: while the Poisson brackets between momenta and configuration space variables g,g are unchanged with respect to T * SU (2), T * SB(2, C). We shall come back to the Poisson algebra (5.22) in the next subsection.
In order to derive Hamilton equations, it is sufficient to write in compact form: {P I , P J } = C K IJ P K with C K IJ the structure constants specified above. We have then:
Let us verify that we actually recover Eqs. (2.9)-(2.11). In order to obtain the PB on the fibers of the cotangent bundle, the matrix r is rescaled by a real parameter λ and the elements of G * are made dependent on the same parameter. By expanding up to first order, one gets: g(λ) = e iλI i e i = 1 + iλI i e i + O(λ 2 ). (5.31) Substituting this in (5.28) yields, for the left-hand side:  By equating the two sides, in the limit λ → 0, one obtains the Poisson bracket: {I i , I j } = ǫ k ij I k . (5.33) Let us consider the second Poisson bracket, Eq. (5.29). In order to compute its l.h.s. we use for g the parametrization g = y 0 σ 0 + iy i σ i . We have, up to first order in λ {g 1 , g 2 } = 2iλ {I i , y 0 }e i ⊗ e 0 + i{I i , y j }e i ⊗ e j + O(λ 2 ) (5.34) while for the r.h.s.
−g 1 rg 2 ≃ −2 (1 + iλI i e i ) ⊗ 1 (λe k ⊗ e k ) 1 ⊗ (y 0 e 0 + iy j e j ) = −2λe k ⊗ e k 1 ⊗ (y 0 e 0 + iy j e j ) + O(λ 2 ) = −2λ( 1 2 y 0 e k ⊗ e k + iy j e k ⊗ e k e j ) + O(λ 2 ) = −λ(y 0 e k ⊗ e k + iy j e k ⊗ (δ kj e 0 + iǫ i kj e i ) (5.35) By equating (5.34) with (5.35) one finally gets at order λ: where the first one is compatible with the second one, upon using (y 0 ) 2 = 1 − k y k y k . Finally, let us consider (5.30). The l.h.s. yields: In order to underline the symmetric role played by the group SU (2) and its dual, one can perform a slightly different analysis by considering r * as an independent solution of the Yang-Baxter equation ρ = µe k ⊗ e k (5.39) and expanding g ∈ SU (2) as a function of the parameter µ: By repeating the same analysis as above, it is straightforward to prove that the Poisson structure induced by ρ is the one that correctly reproduces the canonical Poisson brackets on the cotangent bundle of G * = SB(2, C) derived in Eqs. (4.9)-(4.11).
Last but not least, it is possible to consider a different Poisson structure on the double, given by [7] :

Conclusions and Outlook
Starting from an existing description of the dynamics of the Isotropic Rigid Rotator on Heisenberg doubles [13], we have introduced a new dynamical model which is dual to the standard IRR.
To this, we have used the notion of Poisson-Lie groups and Drinfel'd double for understanding the duality between the carrier spaces of the two models. Specifically, we have used the Drinfel'd double of the group SU (2) as the target configuration space for the dynamics of a generalized model, with doubled degrees of freedom. This model exhibits non-Abelian duality and is an ideal arena to analyze in a simple context the meaning to physics of generalized and double geometry structures. Moreover, we have shown that, from the generalized action, the usual description with half the degrees of freedom, can be recovered by gauging one of its symmetries.
The simple model of the IRR is especially interesting as a toy model for field theories with non-trivial target spaces such as Principal Chiral Models. In their original formulation [22] these are nonlinear sigma models with the principal homogeneous space of the Lie group SU (N ) as its target manifold, where N is the number of quark flavors. Therefore, the dynamical fields of the model, so called currents take value in the cotangent bundle of the Lie group, while the canonical formalism is described by a Poisson algebra which takes the form of a semi-direct sum. The analogy with the IRR is thus very strict: the analysis we have performed can be readily generalized, starting from an alternative description of Principal Chiral Models given in ref.s [24], [38,39,40] (also see [41] where Principal Chiral Models are analyzed in the DFT context). The work is under preparation and the results will be presented in a forthcoming paper [15].