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A QP perspective on topology change in Poisson-Lie T-duality

We describe topological T-duality and Poisson-Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QP-manifold on doubled non-abelian"correspondence"space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of"bibundle"structure on correspondence space realises changes in the global fibration structure under Poisson-Lie duality. Our approach can be directly translated to the worldsheet to derive dual string current algebras. Finally, the canonical transformations appearing in our reduction procedure naturally suggest a Fourier-Mukai integral transformation for Poisson-Lie T-duality.


Introduction
Topological T-duality The attraction of studying T-duality is that it is one of the simplest examples of the interesting new features that appear when passing from theories of particles to theories of strings. Physicists understand T-duality in practice as a set of transformation rules that take string theory in one spacetime, M , admitting d (compact) abelian isometries, to string theory in another spacetime, M , also admitting d abelian isometries. These transformation rules express how the components of the metric and other fields on M are determined in terms of those on M . This is therefore at least naively a very local perspective.
A perspective which captures some essential global features of T-duality is provided by topological T-duality [1,2]. In the simplest version of this approach, the manifold M is the total space of Topological T-duality has proven to be of mathematical interest in its own right, while the correspondence space can be seen as a precise mathematical realisation of a 'doubled' space, an idea often invoked by physicists in other (non-topological) approaches to T-duality.
Topological T-duality also interacts with the Courant algebroid [3,4] structures that describe the gauge algebra of the NS-NS sector of type II supergravity. For example it is known that the exact Courant algebroids atop M and M can be quotiented to provide isomorphic "invariant" Courant algebroids atop the base B [5] which can be seen as another manifestation of topological T-duality.
Generalisations of T-duality Here, we will study an extension of the framework of topological T-duality to treat generalised versions of T-duality. The simplest generalisation focuses on manifolds M admitting non-abelian isometries [11], but a more general version of T-duality surprisingly applies in spacetimes without isometries but which admit a group action of a Poisson-Lie group. This is known as Poisson-Lie T-duality [12,13]. Poisson-Lie T-duality is known to be an equivalence of sigma models in the phase space formulation [12][13][14], and it contains non-abelian and abelian Tduality as particular cases. The difficulty here is that questions of topology are far less tractable.
As the dual manifolds in these settings do not involve circle bundles, there is no Gysin sequence to exploit as was done in [1,2] for the abelian case. The challenge is further increased by the fact that the dual fibres (roughly speaking the spaces on which the duality acts) are not even diffeomorphic.
Despite these challenges, there are many structural similarities between abelian and Poisson-Lie duality.
A crucial feature of Poisson-Lie T-duality is that it admits a naturally doubled interpretation.
This is best understood first in the simplest case B = {point}. Here the target space is simply a group manifold G equipped with the action of a second Lie group G, and the duality transform serves to exchange the role of G and G. Consistency dictates that given a G, G can not be chosen arbitrarily. Instead, G and G are required to be subgroups of a larger group manifold D whose algebra d is equipped with an ad-invariant pairing under-which g = Alg(G) andg = Alg( G) are maximal isotropic sub-algebras. The Lie group D, known as a classical Drinfel'd double, thus has double the dimension of G and has a similar significance as the doubled T 2 fibration in the abelian example above.
In this paper We exhibit changes in the global fibration structure under Poisson-Lie T-duality with spectators. To this end, we extend the diamond diagram (1.1) from abelian to Poisson-Lie T-duality. At the top of the diamond, we consider the principal D-bundle D ֒→ M ։ B. We then study reductions under the left group actions G\M ≡ M or G\M ≡ M which take us to the middle rung of the diagram. In the presence of spectators, i.e. a non-trivial base B, we will find that the bundle D ֒→ M ։ B can not be arbitrary -instead, to enable the reduction to M or M to take place, at least within the approach we are taking here, further conditions need to be imposed.
We show a sufficient condition for reduction is that M be a bibundle (a right bundle equipped with a right-equivariant structure map) [15][16][17] that further more obeys a topological factorisation condition. This factorisation condition requires that M admits a reduction of its structure group to Topological T-duality via QP-manifolds Our approach here will be to situate the discussion in the context of QP-manifolds. These are graded symplectic manifolds equipped with cohomological vector field (the Q-structure). This formulation is very practical in the present setting, due to the fact that a Courant algebroid can be very compactly realised as a degree 2 QP-manifold [18]. in which the arrows correspond to symplectic reductions that preserve a Q-structure. The point of view that duality can be studied in terms of reductions of QP -manifolds was articulated in the paper [19] and in the final line of a proceeding, both by Ševera [20], and in [10,21] a Courant algebroid perspective was developed. Here we will flesh out the idea that Courant algebroid reduction in the graded setup is just a special case of symplectic reduction 2 .
Our ultimate motivation for this QP-manifold picture is however the possibility of a generalisation to M-theory and U-duality. In that context the algebroid structure of target space is no longer that of a Courant algebroid, but a QP-manifold picture is known [22,23] and symplectic reduction works in the same way.
Canonical transformations and a Fourier-Mukai transformation An important feature of our approach is that the natural Q-structure appearing at the top of the diamond is not immediately compatible with the symplectic reduction. Instead one must first perform a symplectomorphism of M (i.e. a canonical transformation which preserves the P -structure but which transforms the Q-structure). Roughly speaking the Q-structure can be associated with a three-form on M and the canonical transformation serves to amend this by the exterior derivative of a certain two-form, Φ ( Φ), to produce aG-basic (G-basic) three-form that defines a Q-structure In the abelian case, we will see that the difference of these canonical transformation ̟ = Φ − Φ plays a vital role in topological T-duality; it provides the kernel for a Fourier-Mukai integral transform [24] that acts on the Ramond-Ramond (RR) sector of the superstring and eventually lifts to the K-theory perspective of T-duality. In the Poisson-Lie situation we show the same ̟ has an elegant interpretation, in the absence of spectators, as that of the Semenov-Tian-Shansky symplectic form on D [12,13] 3 . With spectators, our techniques give a natural candidate for the 1 The graded vector bundle T [1]M is the reversed-parity tangent bundle, where the fibre coordinates are declared to be anticommuting and carry degree 1. The bundle T ⋆ [P ]V for some graded vector bundle V first forms the cotangent bundle T ⋆ V and then shifts degrees on fibre coordinates of T ⋆ V by +P . The notation is standard in the context of supergeometry and AKSZ constructions. 2 We thank Fridrich Valach for private communication on this point. 3 An additional role played by ̟ is as the 'fundamental 2-form' appearing as part of a para-Hermitian structure Fourier kernel that determines the duality transformation of the RR forms, and we conjecture that it should play a similarly vital role in future developments on the topological nature of PL duality in the context of D-branes.
From QP-manifolds to the worldsheet A final benefit of the QP-perspective that we employ is that it can be directly linked to the associated worldsheet theory. This is a generalisation of the link between Courant algebroids and worldsheet current algebras established in the seminal work of Alekseev-Strobl [28]. Here this is achieved through a transgression into the space of maps into M from a supermanifold associated to the worldsheet, as in the AKSZ construction [29]. Recent work by one of us [30] has shown that a zero-locus reduction procedure can be employed to define a canonical algebra of currents on the world-volume of any brane starting with a Q-structure on any QP-manifold M. Specialised to the present situation we show how this recovers the current algebra on LD, the loop-group of D, that defines the PL sigma models of Klimčík and Ševera [12], and also the PL dual pair of current algebras on LG, L G.
Outline of this paper We structure the paper as follows. and discuss how the general framework of section 2 applies to them and furnishes PL duality with spectators. We end with some general remarks and speculations.
2 A "master" QP-manifold on non-abelian doubled space

The QP non-abelian set-up
At the top of the diamond diagram, we consider the principal D-bundle D ֒→ M ։ B endowed with a connection one-form To obtain a QP perspective we move to the graded supermanifold M ≡ T ⋆ [2]T [1]M which is equipped with a natural Poisson structure in the following way. We can choose local coordinates, x µ in the Born geometry of correspondence space [25], which has also been studied in the Poisson-Lie setting in [26,27].
on the base and Y I on the fibre both of degree zero, ψ µ , θ A degree 1 coordinates on the corresponding shifted tangent spaces, and finally χ µ , ξ A (degree 1) and p µ , Ô A (degree 2) coordinates on the shifted cotangent bundle to T [1]M. With these the symplectic structure on M is given by This gives rise to the non-zero Poisson brackets In the above, L A and V A are left-invariant Maurer-Cartan forms and fundamental vector fields on D. We can recover a conventional momentum variable by Ô I = L A I Ô A ; this has {Y I , Ô J } = δ I J . We can endow M with a standard Q-structure Q 0 = {Θ 0 , •} defined by a (degree 3) Hamiltonian corresponding to the exterior derivative on M which in this trivialisation becomes a sum of the exterior derivative on the base and that on the group manifold fibre. The Jacobi identity ensures that {Θ 0 , Θ 0 } = 0 and hence that Q 0 is a cohomological vector field. From this perspective the degree 1 coordinates ξ A generate a global right action on We then augment the Q-structure with an H-flux H M ∈ Λ 3 T * M given by in which H ∈ Λ 3 T * B is basic and ω CS (α) = α ∧ , dα + 2 3 α ∧ α is the Chern-Simons three-form. We then define the Hamiltonian which is evaluated explicitly using the decomposition of eq. (A.8) (with the replacement L A → θ A and A A → A A µ ψ µ ) as (2.7) A short calculation shows that {Θ, Θ} = 0 when dH M = 0, or Everything we have described is so far completely general as long as we have an adjoint invariant inner product , . Henceforth we specialise to the Drinfel'd double case relevant to Poisson-Lie T-duality. We therefore assume that D is even-dimensional and its Lie algebra d admits a splitting into two sub-algebras g andg which are maximally isotropic with respect to a split signature inner product. We denote this inner product by η, such that η AB ≡ η(T A , T B ) ≡ T A , T B has components η ab = 0 = η ab , η a b = η b a = δ b a , with generators T a for g and T a forg. We also make the technical assumption that this is a "perfect" double as defined in appendix A.4, which essentially means that one can globally factorise group elements ∈ D as =gg with g ∈ G andg ∈ G.

Conditions for symplectic reduction
To proceed down to the middle level of the diamond, we now wish to implement a symplectic reduction of the QP-manifold (M, Q) by a G action. We will see however that this can not be done immediately, instead we are required to first perform a canonical transformation described by a two-form Φ which we shall characterise in general.
If G is a subgroup of D, we can consider the quotient under the left group action G\M ≡ M which will be the base of a principal G-bundle G ֒→ M ։ G\M if G ֒→ D ։ G\D is another principal bundle [31, Proposition 2.1], which we assume. The left action has a cotangent lift to an action on M ≡ T ⋆ [2]T [1]M infinitesimally generated by a subset of the right-invariant variableŝ ξ A ,Ô A that we will denoteξ a ,Ô a . Since the constraintsξ a ,Ô a form a closed algebra when the generators T a ∈ g form a Lie subalgebra in d (which is the case). The Q-structure Q -or more accurately its hamiltonian -descends 4 to the symplectic reduction when Qξ a = QÔ a = 0 modξ a ,Ô a . However, for Q = {Θ, •} of (2.6) we calculate Qξ a = −Ô a + 1 2 dα a . (2.10) Note that here we make explicit use of the Drinfel'd double inner product used in the Chern-Simons term. The presence of the dα a term in (2.10) clearly obstructs the reduction of the Q-structure.
To fix this, we need to consider an equivalent Q-structureQ related to Q by the canonical where we identified the contraction ι U a against the right-invariant vector fields U a on M that correspond to g with the Poisson bracket {ξ a , •}. Therefore the condition for the reduction of the Q-structureQ is simply given by the existence of Φ such that ι U a dΦ = 1 2 dα a . (2.14) The reduction condition (2.13) has a natural implication for this three-form namely that it be G-basic (i.e. G invariant and horizontal). Horizontality follows from which together with dH M = 0 gives invariance L U aH M = 0. HenceH M can be thought of as the pull back of a conventional three-form of Λ 3 T ⋆ (M ) that we shall denote simply as H.
It remains to produce an explicit Φ which obeys (2.13). A sufficient condition that ensures this is for Φ to be G invariant and obey This motivates an ansatz of the form From the second condition of (2.16), we immediately have that Since L U a α B = −F a C B α C the first condition of (2.16) requires that, using the form of the Drinfel'd double structure constants (see appendix A) At this point, the form of Ω ab and Φ may be suggestive of familiar structures in generalised Tduality. In the subsequent sections of this paper, we will make this precise, and obtain the conditions defining Ω ab , by considering in succession the cases where D is abelian (corresponding to abelian T-duality), where D is non-abelian and we assume we are dealing with a trivial bundle over a point (corresponding to Poisson-Lie T-duality without spectators), and finally the full non-abelian case with a non-trivial bibundle structure (corresponding to Poisson-Lie T-duality with spectators).

Topological T-duality
We first consider the case where D = T d × T d is abelian and has dimension 2d. The original QP Hamiltonian (2.7) in this case takes the simple form where we have explicitly reinstated the inner product η AB . Here there is an obvious action of and so on) which is a canonical transformation of the symplectic structure (2.2), and in particular is a diffeomorphism of the total space M of the doubled torus bundle. This gives a morphism O ⋆ : , and an isomorphism of QP-manifolds where the new Q-structure Q ′ takes the form (3.1) in terms of the transformed variables, with We now complete the symplectic reduction described in section 2.2. We assume the principal D = T d × T d bundle is a principal T d bundle with respect to the action of both the torus T d and the dual torus T d inside T 2d . Our symplectic reduction leaves M/ T d ∼ = M as a principal T d -bundle over the base B. We identify α a = A a with the connection one-form on M , with curvature F a = dA a .
We also denote α a = A a and let F a = d A a . 5 We need to solve equation (2.19) for the object Ω ab , which in the absence of structure constants becomes simply ι U a dΩ bc = 0, and we can take the trivial solution Ω ab = 0. Thus we have Φ = 1 2 α a α a = 1 2 A a A a . This leads to the QP Hamiltonian and H-flux Alternatively, we could have carried out the 'dual' symplectic reduction leading to M/T d ∼ = M . This is straightforwardly done directly, or can be viewed as arising from the O(d, d; Z) transformations above by first using the transformation that exchanges Y a ↔ Y a and then reducing as before. Either way one finds that now A a is the connection one-form on M , and The exchange of the roles of A a and A a , and hence of F a and F a , realises topological T-duality [1].
We can continue the reduction procedure to the bottom of the diamond diagram. Starting with (3.3) and reducing by the action of p a leads to a new Hamiltonian Θ = −p µ ψ µ +H + F a (θ a +A µ a ψ µ ). This is not manifestly invariant under the exchange of A a and A a . This can be fixed by instead first performing a canonical transformation generated by Φ = ξ a A µ a ψ µ which replaces (3.3) with after quotienting by the action of p a .
We have now arrived at a transitive Courant algebroid whose associated QP-manifold is M/ /T d = . This is manifestly self-T-dual under the canonical transformation that swaps ξ a ↔ θ a . This is the QP-manifold version of the equivalence of 'invariant' Courant algebroids under T-duality [5] which was originally noticed by Barmaz [6].

(Topological) Poisson-Lie T-duality
A quotient of the Drinfel'd double D by its maximal isotropic subgroup G produces the group manifold G (due to the parameterisation =gg). We will upgrade this to a symplectic reduction of the associated QP-manifolds using the general machinery of subsection 2.2. This will give rise to Poisson-Lie T-duality. We summarise some essential conventions and notations in appendix A. In particular associated to the decomposition =gg we introduce the adjoint actions a of g andã ofg acting on the double. Now, the absence of spectators, the canonical transformation described by Φ takes the form involving the 0-form Ω ab antisymmetric in its indices. To solve the condition (2.13) ensuring the symplectic reduction, we can choose forΠ ab the Poisson bivector on G that satisfies (A.22) and endows it with Poisson-Lie structure. 6 The effect of the canonical transformation generated by this Φ is to eliminate the term On the other hand, by identifying F ABC θ A θ B θ C ≡ F ABC L A ∧ L B ∧ L C and exploiting the adjoint invariance of F, we can again use (3.10) to write from which we calculateΘ = Θ − dΦ: The general machinery of subsection 2.2 realises the symplectic reduction G where both QP-manifolds have the standard Q-structure (2.4) (that arises from the de Rham differential) up to canonical transformations. To display this explicitly, we exploit (3.10) again to write the Q-structure and symplectic form on D in terms of the ones on each isotropic subgroup G, G. For the fermionic variables we writeθ,ξ for the fermions associated to left-invariant forms and vector fields on G, andθ,ξ for the fermions associated to the right-invariant ones on G.
Then we have (3.14) From this we can obtain a symplectomorphism let ω D be the symplectic form that produces the Poisson brackets (2.3). Similarly we can introduce analogous symplectic forms : By requiring that ω D = ω G + ω G we can then solve for the momentap a , p R a on G, G respectively.
This determines the momenta Ô A in terms of an affine transformation of the momentap a , p R a . 7 In these new momenta variables we havē In the same new variables, the hamiltoniansξ a ,Ô a = −Qξ a of section 2.2 that generate the left Symplectic reduction effectively sets these to zero in the Q-structure and symplectic form. Using also (3.17) we see the reduced symplectic form and Q-structure are the canonical ones for T ⋆ [2]T [1]G, 7 We have been consistently using hats to denote left-invariant variables that are dressed by an adjoint action. The left-invariant fermions are transformed by the hat into right-invariant ones. However the momenta transform affinely instead of linearly when we demand that the symplectic form ωĜ in right-invariant variables pulls back to the symplectic form ωG in left-invariant ones, so schematically p R aT a = a(paT a + θξ) =p + θξ .
namely the standard cotangent bundle form and the Q-structure arising from the exterior derivative: we could similarly perform a reduction with respect to the left action of G. This procedure invokes a canonical transformation generated by This leads to a (primed) dual version of (3.19), with the dual structure constantsf ab c appearing in place of the original f ab c , realising the symplectic reduction We will sometimes write G ′ and G ′ instead of G and G to clarify when the associated factorisation is = g ′g′ or the original =gg.
Further reduction and a target-space avatar of the Klimčík-Ševera current algebra.
In fact by replacing Φ → λΦ for real λ this calculation shows that on any Poisson-Lie group (G, Π) there is a pair of anticommuting Q-structures on T ⋆ [2]T [1]G: one given by (3.22), and the other by Such Q-structures appeared as an example of a "bi-QP" manifold in [35]. We group the fermionic variables into A T A = θ a T a + ξ aT a , which obey the following suggestive identities given the Qstructure (3.23): The A will later give rise to the string sigma-model currents that exhibit Poisson-Lie T-duality, that were originally constructed by Klimčík and Ševera [12]. Replacing G with G and repeating the above steps leads to another set of 2d fermionic variables A that obey the same suggestive identities, as long as the Lie algebras g,g form a Manin triple d = g ⊕g with Lie algebra structure constants F AB C . We perform the symplectic reduction of T ⋆ [2]T [1]G with Q-structure (3.23) modulo the left G action, which has generators −p a ≡ a a b p b . Since Qp a = 0 modp, the Q-structure descends to the quotient whose Q and P-structures are obtained from (3.23) and (3.22) by settingp a = 0: For this QP-manifold Q 2 = 0 is equivalent to the property that d = g ⊕g is a Manin triple [35,36].
Evidently, had we started with G instead of G we would have arrived at an isomorphic QP-manifold, with the replacements (θ, ξ) → ( ξ, θ).
Poisson-Lie duality, canonical transformations, and a lagrangian correspondence. The upshot of this subsection is the following diamond diagram of symplectic reductions:

Bibundles and Poisson-Lie duality with spectators
We now reinstate the spectator connection field, with the first goal of carrying out the symplectic reduction from the top to the middle rung of the diamond. The challenge here is to understand the global construction that allows us to solve the general conditions, c.f. eq. (2.16), required of the canonical transformation that enables this reduction. In the absence of spectators the key to doing so was to parameterise the element of D as =gg and to then exploit the adjoint action ofg on D as in eq. (3.10). In the case of spectators, in which D is potentially non-trivially fibred over B we can proceed on a local patch in much the same fashion, although even here we shall encounter some surprising features. However to give such manipulations a coherent global meaning we impose additional structure on D ֒→ M ։ B, namely that it be a bibundle.

A local perspective
To get an intuition of how to proceed, let us fix a local section : We now want to employ the factorisation =gg and work out the accompanying decomposition of the connection. A subtle but important point is that the projection of A into g does not define a g-connection. Instead, we find that it is the combination A a = A a + A b Π ba g that transforms like a connection under local right G action, i.e. if g → gh then A a T a → h −1 dh + h −1 A a T a h. As a result, we find that the following quantities: are invariant under the local right G action. However, in contrast to the abelian case, one cannot directly interpret r ∇ as a connection in a G-bundle. The first way to see this is that the explicit appearance of Π ba g means A a is not a pull back of a one-form on the base. More sharply, r ∇ fails to meet the equivariance condition required of a connection under the global left G action. Instead, under the global G left action generated by the vector fields u a (dual to the right invariant forms r a ) we have that 8 such that the appearance of A in the first equality is the deviation away from r ∇ being a connection.
Regardless of these issues, using the decomposition =gg and the definitions (4.29) we have α =g l + A + r ∇ g −1 . (4.5) We can next evaluate the Chern-Simons form to give (where contractions involving η AB are implicit) This provides a local picture of how spectators are to be included but raises critical questions: 1. In defining the decomposition of eq. (4.5) and in solving the conditions for the canonical transformation Φ, we have assumed three things; first that the fibre coordinates can be split as =gg, second that the adjoint action [ ] exists on a bundle, and third the adjoint action can factorise to give a well-defined mapsã[g] and the bi-vectorΠ. What structure needs to be imposed on D ֒→ M ։ B for these to exist?
2. What is the global status of r ∇ and A, and how should they be better understood?
To address both of these points we now introduce the aforementioned bibundle structure on D ֒→ M ։ B.

A class of bibundles with Drinfel'd double fibre
We begin by presenting the definition(s) of a bibundle (in which D need not be a Drinfel'd double) [16,17]. which is right-equivariant in the sense M is also a principal D-bibundle for the left action ⊲, which is defined by 11) and in particular the structure map is equivariant under the opposite group action: We now specialise to the case where D is a Drinfel'd double. In addition to allowing for the extra structure provided by a bibundle, we want to allow the factorisation =gg to be applied.
For this we need to introduce a topological condition: where provide the central elements in G (resp.G) which are also central in D. 9 To be absolutely clear, by Type 1 bibundles we mean the (Aut(D), D) bibundles of [17] whose type map M → Out(D) equals 1 everywhere. This implies the existence of a principal Z(D)-subbundle explicitly given by equation (4.39); we refer to that paper for more details. Going the other way around, the existence of the reduction of the structure group to Z(D) implies the existence of a Z(D) subbundle M1, which can be used to construct a left D action on M (if we are initially given a right one) such that (4.39) is satisfied. that respectively fix the subgroups G, G inside D, and which yield a factorisation of : Moreover a,ã enjoy the following equivariance properties involving the inclusions ι,ι of G, G in D and the actions ⊲, ⊳ of the latter on M: (4.17) Note that this proposition depends on the parameterisation =gg on the Drinfel'd double.
The same argument goes through with the opposite choice = g ′g′ (the 'primed' parameterisation) of (A.29) and yields 'primed' structure maps a ′ ,ã ′ . As a consequence of (4.17), a,ã restrict to bibundle structure maps for groups G, G (with the same total space M): Corollary 1. The following expression defines a left G action ⊲ on M, alongside the right G action ⊳ induced by the restriction of the right D action:

(4.19)
Similarly, the G bibundle structure is given by the group actions Furthermore these bibundle structures are relevant for the reduced spaces: the quotient G\M inherits the G-bibundle structure of M. This is in fact a type 1 bibundle G ֒→ G\M ։ B by Definition 3, as its structure group Z(G, D) (4.13) is a subgroup of the centre Z(G).
We postpone the proof of the proposition and corollary to section 4.5 and now turn instead to its consequences.

Bi-vectors from bibundles
The above construction leads to well-defined mapsã and Π that were invoked in the local construction of r ∇ , A and the canonical transformation Φ (and, of course, equivalents without tildes).
Since the structure map , and its factorisations a andã, take values in Inn(D) ⊆ Aut(D) we can take a derivative to define maps, that we shall also call, , a,ã that take values in Aut(d). In

terms of the generators T A we have for instanceã[Ñ](T A ) ≡ã[Ñ]
A B T B . Under the flow generated by the leftG action on M, (where the minus is conventional) the equivariance of the (factorised) structure maps given by eq. (4.17) ensures

Connections from projections
In this section we extract the relevant objects r ∇ and A from the connection α. To invoke the split of d = g ⊕g into its maximal isotropic subalgebras, we introduce the commuting projectors P : d ։ g and P : d ։g. By composing these projections with the factorised structure maps we obtain new projectors, defined point-wise in M, which equals the Poisson bivector on G in the absence of spectators in the convention of (A.9). We then see that Similarly we have (4.28) Using these we can extract the globally well-defined object in which ≈ corresponds to the evaluation in a local trivialisation of the kind described in (4.40), in which we can identify (see section 4.5) We can now identify the combination Turning to A, let us first define the 1-form A on M viã where again ≈ corresponds to the evaluation in a local trivialisation. We can interpret A as a global left connection one-form for the G action generated byũ a ≡ U a i.e. it obeys the two conditions The algebraic condition on the connection A follows immediately from eq. (4.27) since ιũa P ã α = P ã ιũaα = P ã T a . The differential follows easily as for x ∈ d, a constant element of the algebra, we have Thus It is important to stress that A is a connection for aG-bundle over M =G\M (rather than over the base B). The aforementioned A can thus be viewed as the local gauge potential for this connection.
We are left then with a small puzzle in that the reduced H-flux derived from (4.6) (η contractions remain implicit) depends on A and so appears at first sight to be only a local construction. However, the combination In this diagram M 1 is the principal bundle with structure group the centre Z(D) of D whose existence is equivalent to the bibundle M being of type 1 (Definition 3). This has a concrete realisation as the locus of points Ñ 1 ∈ M where left and right actions are equal [17]: Denote the associated local section ofM 1 byÑ 1 , soÑ 1 [b] ∈M 1 for all points b ∈ U . Then, for all Ñ ∈ M in the fibre above b ∈ U we can write (omitting the inclusion maps ι 1 •ι, ι,ι) Well-definedness of the factorisation. We can directly calculate the bibundle structure map in terms of the parameterisation (4.40): To confirm a,ã are well-defined we choose another local sectionÑ ′ 1 : U →M 1 , which leads to the new parameterisation (where the primes are unrelated to the 'primed' parameterisation of (A.29) where G and G appear in the other order) (4.43) The new section should be related to the old one by a gauge transformation taking values in , which leads tog ′z =g, zg ′ = g once we factorise Þ =zz inz ∈ G and z ∈ G. Since z is central in G and ιz is central in D we find where we took note of the factÑ 1 [b] ∈ M 1 to employ (4.39). Therefore we can read off The proof forã works similarly. Here we can exhibit the structures that appear in the dual reduced fluxes of the form (4.36) (associated to the reduction modulo the left action of G) and its 'primed' analogue (involving the left action of G). Consider firstly the reduction to the left side of the diamond diagram after quotienting by G. The local quantities appearing are

Examples
Hence r ∇ is in this case a global G-connection 1-form, denoted A. The reduced H-flux is The global definitions of the objects appearing here are simply A = P α and d A = F. Now consider the reduction to the right side, after quotienting by G ′ . The local quantities and now One sees that A ′ = A and so the final term in the H-flux has exchanged F for the field strength

Duality on the worldsheet
A direct application of the QP-manifold approach we have taken is to immediately obtain a string worldsheet description. A classic result of Alekseev and Strobl [28] shows that the symmetries of a non-linear sigma model are realised through currents specified by a pair of a (pull back of a target space M ) vector and a one-form, and that these currents form an algebra under the Poisson bracket whose closure invokes the Courant bracket of T M + T ⋆ M . The QP perspective was recently found [30] to give a route to this worldsheet result, using a mechanism known as Zero Locus Reduction (ZLR) [35] alongside the AKSZ construction. As orientation, we first illustrate this in the abelian set-up before proceeding to the case where D is a Drinfel'd double for which we show the ZLR procedure recovers the Klimčík-Ševera current algebra. We define the zero-locus or phase space, Z ∆ , to be the quotient of C ∞ (Maps) by the ideal I ∆ generated by ∆. As was shown in [30] Z ∆ is equipped with a degree zero Poisson bracket under which the currents, so constructed, form an algebra where the signs are determined by the degrees of f , g and ǫ and p.
In the present context, we will specify N = T [1]S 1 where the S 1 is to be thought of as a spatial section of the string worldsheet parameterised by a coordinate σ, and D be the de Rham differential d such that superfields of degree n are expanded as f = f (n) (σ) + f (n−1) (σ)dσ and the integration N simply picks out the top-form and integrates.
Abelian currents We first display this ZLR procedure in the abelian case starting at the bottom of the diamond. The Hamiltonian was derived in equation (3.6), and the corresponding QP-manifold can be seen to be T ⋆ [2]T [1]B × R 2d [1] with the P and Q structure specified by ω = dp µ dx µ − dψ µ dχ µ − dξ a dθ a , Θ = −p µ ψ µ + H + F a θ a + F a ξ a . (5. 3) The corresponding target-space algebroid is the "transitive" Courant algebroid of T d -invariant sec- Since this is not an exact Courant algebroid the Alekseev-Strobl result [28] does not apply, and the ZLR technology is needed.
The degree 1 coordinates ξ and θ correspond to sections of the generalised tangent bundle that point along fibre directions. These give rise to corresponding currents e.g.

The bracket formula (5.2) then gives for any background
This motivates considering the currents A |ǫ in which A T A = θ a T a + ξ aT a are previously introduced combinations but now viewed as superfields. The one-form component of A , which we also call A , then obey worldsheet Poisson bracket relations The non-zero righthand side is the classical precursor of Schwinger term in a current algebra. For zero background H = F =F = 0 the full ZLR and nonvanishing Poisson brackets are Solving the ZLR condition for ξ a , θ a as ξ a = dỹ a , θ a = dy a (5.8) exposes the familiar [40] non-commutativity between positions/dual positions, The spectator coordinates x µ and their superfields x µ give rise to an Alekseev-Strobl type current (sub)algebra for zero background, but for nonzero fluxes the fibre and spectator currents will have nonvanishing brackets determined via (5.2).
Non-abelian currents We may extend this to the full non-abelian set up, taking for expediency the base B = {point} such that the relevant QP-manifold is g[1] ⊕ g ⋆ [1] with Q and P-structures given as in eq. (3.27) namely Then for the currents A |ǫ with ǫ, η set as 0-forms we see very rapidly using the suggestive identities Hence, the one-form component of A , which we also call A , then obey on the worldsheet the Klimčík-Ševera [12] Poisson bracket relations Gauging worldsheet currents vs. symplectic reduction The procedure of section 4.5 that yields (topological) PL duality via symplectic reductions of QP-manifolds naturally suggests how to implement duality on the worldsheet. We outline briefly how this works.
Consider firstly gauging a current J f (σ, t) corresponding via (5.1) to some arbitrary f ∈ C ∞ (M).
In the hamiltonian formalism we would simply introduce a lagrange multiplier λ(σ, t) and add dt dσ J f (σ, t)λ(σ, t) (5.13) to the action. Upon switching to the notation actually used in (5.1), we would rewrite this as dt f |λ (t) where now the test (poly)form λ also depends on time t, and f | does as well via the superfield f (t) = ϕ(t) ⋆ f , for ϕ : R → Maps. Therefore the closure of the gauge algebra is controlled by the bracket formula (5.2). Moreover we see that if we gauge a current f | associated to f , we are simulaneously gauging the current Qf | associated to Qf : if λ = dµ (for d the worldspace de Rham) then by integration by parts and zero locus reduction f |λ = (−1) f +1 df |µ = (−1) f +1 Qf |µ . to the action. It is easy to see that this resolves into an expression linear in momenta along G. The gauge algebra indeed closes due to (2.9).
Clearly the same reasoning works mutatis mutandis also for the case with spectators, and the case where we want to obtain the model with target M instead of M .

The Fourier-Mukai kernel
The idea here is that we first implement a pull back to the doubled space with p ⋆ and then a push forwardp ⋆ by integrating over the group G ′ in the "dual" decomposition D = G ′ × G ′ . Note that as submanifolds of D, G and G ′ are identical (at least in the case of perfect doubles, see formulas (A.27), (A.29) and associated discussion in the appendix). While this integral transform appears naturally, further development is required e.g. to provide a specification of the integral measure taking into account that G (′) need not be compact.
In the abelian case, this integral transformation is known to describe the transformation under T-duality of the Ramond-Ramond field strengths of Type II string theory. In that case, the results of section 3.1 immediately give A a ), and this indeed reproduces the Fourier-Mukai kernel discussed in the context of topological T-duality in [1,2].
We therefore suggest the same holds for Poisson-Lie duality. The RR fields are G ∈ Λ G is compact (which can be arranged, for instance, in the non-abelian duality example discussed in subsection 4.6).
For the full Poisson-Lie cases, we now make the expressions for ̟ more explicit and make some further comments.  (5.23) in which Π g is the PL bivector of G, Πg that of G 11 and O ≡ 1 −ΠgΠ g −1 .
For applications as a Fourier-Mukai kernel it is useful to further recast ̟ in terms of the oneforms r andr ′ which would enter the definition of the two T-dual theories. In this case one obtains in which now a g is the adjoint action of g on g andã ′g ′ that ofg ′ ong, and N ≡ 1 − Π g Π ′g ′ . We can specialise this to the case of semi-abelian Drinfel'd doubles, the context for non-abelian T-duality for which a Fourier-Mukai kernel was proposed in [42]. In this case,g = u(1) d . With which agrees (modulo conventions) with the expression in [42]. In this case the relationships with the primed variables are Π = a Π ′ a t ,l ′ = a −1 gl −Π ′g ′ l and l ′ = l, such that we could alternatively write 2̟ = ra g Πg′a t g r + 2ra gr ′ , (5.27) manifestly depending only on y andỹ ′ .
There is another proposal for an FM transform in the literature [43] which was motivated by the expected transformation of RR fields under PL T-duality [44]; however, the precise relation of that to the present result remains unclear.
Proposed Fourier-Mukai kernel with spectators In terms of the objects introduced in section 4.1, we would have If M has bibundle structure satisfying the conditions of Proposition 1, we can rewrite this using This is globally well-defined, and of course reduces to the simpler spectator-free and abelian cases described above. What is notable is that it involves both the genuine G, G connections A and A of (4.32) as well as the connection-like 1-forms r ∇ ,r ′∇ of (4.29).

Discussion
The main result of this paper has been to show how certain "type 1" bibundles (Definition 3) provide examples of PL duality with spectators. Although we were able to exhibit topology change in the fibration structure (trivial vs. non-trivial bundle), the type 1 restriction is a rather severe topological constraint that essentially excludes e.g. simply connected spectator manifolds B in the sense that they cannot accommodate non-trivial type 1 bibundles. It would be interesting to alleviate this type restriction. This leads to non-trivial bibundle structures even without spectators [17], with underlying manifold D, right action the canonical one of D on itself, and a left action twisted by an outer automorphism. It is likely that pushing forward in this "bibundle" direction might elucidate some puzzles that remain even in the type 1 case that we treated, such as the geometric character (2-connection [16]?) of the form r ∇ of subsection (4.4) that arose in the reduction of the (ordinary) principal D-bundle connection α.
We also obtained a formula for the Fourier-Mukai kernel that enters into the transformation of RR fluxes, that automatically respects the Bianchi identities in the sense that RR polyforms with d ± H∧ = 0 are mapped into dual ones that have d ± H∧ = 0. In the abelian case, one can prove that the FM transform descends to an isomorphism in cohomology. In the PL case that we are instead interested in, this statement will need to be qualified on account of how at least one of G, G are non-compact in known examples. Our proposal for the FM transform agrees with that of [42] in the non-abelian case (where one group is abelian), but the relation with the FM transform proposed in [43] is currently unclear.
One virtue of the the QP-manifold approach that we have taken, beyond its expediency for calculation, is that it is amenable to generalisation to D-branes and M-theory [22,30,45]. This provokes the exciting notion of studying topological U-duality, including the recently proposed Poisson-Lie U-duality [46,47] where the Drinfel'd double Lie algebra is replaced by a Leibniz algebra known as the exceptional Drinfel'd algebra. We hope to report on progress in these directions in the future.

A.1 QP-manifolds
A QP-manifold of degree n is a graded manifold M equipped with a symplectic structure ω, and a compatible cohomological vector field Q, a vector field of degree +1 that obeys Q 2 = 0 and The second of these, the classical master equation, enforces nilpotency of Q.

A.2 General group and bundle conventions
The constructions described in this paper use the notation of tables 1, 2 and 3 below.
Group and algebra conventions For a Lie algebra d corresponding to Lie groups D, we let {T A } be a basis such that [T A , T B ] = F AB C T C . We define the left and right invariant Maurer-Cartan one-forms as Inner product η κ κ    Bundle conventions We work with connections for principal left bundles (i.e. there is a global left action but transition functions use local right action). We define such that Under a local right transformation, α and ̥ are invariant while The Chern-Simons three-form is (A.8) The three-form ω CS (α) is a globally-defined section of Λ 3 T ⋆ M; ω CS (A) is the conventional Chern-Simons three-form, locally defined on B.

A.3 Poisson-Lie conventions
Poisson-Lie group A Lie group G is said to be Poisson-Lie (PL) when it is further equipped with a g ∧ g valued one-cocycle Π = Π ab T a ⊗ T b that vanishes at the identity, is skew, compatible with right multiplication, and obeys in particular Moreover Π ab defines a Poisson structure on G. Here f ab c are can be viewed as structure constants for an algebra g, or as defining a g ∧ g valued one-cocycle, δ, on g via δ(T a ) = f bc a T b ⊗ T c . In this way (g, δ) forms a Lie bi-algebra.
Drinfel'd double A Drinfel'd Double algebra d is an even-dimensional Lie algebra equipped with an ad-invariant split sign-signature pairing η, that admits at least one decomposition d = g + g in which g and g are sub-algebras maximally isotropic with respect to η. The triple of algebras {d, g, g} is called a Manin triple. We denote the generators of each of the algebras contained in a Manin triple as T A , T a , T a respectively. We then have that η(T a , T b ) = η( T a , T b ) = 0 , η(T a , T b ) = η( T b , T a ) = δ b a . (A.10) Adjoint invariance determines that We can specify the adjoint action of G = exp g via The left-invariant forms l a and dual vector fields v a obey as above the differential identities L va v b = −f ab c v c , dl a = + 1 2 f bc a l b ∧ l c , L va l b = f ac b l c . In addition Π ab obeys (A.9). It is useful to also define π a ≡ Π ab v b , such that We could also repeat the above identities starting with the adjoint action of G = exp g on d and defining the Poisson-Lie bivector of G via

A.4 Parametrisation of Drinfel'd Double
Locally, at least, we can always parameterise ∈ D in terms of group elements g ∈ G andg ∈G by 2. ιG ∩ι G = 1 ∈ D.
The map ϕ is not a group homomorphism, unless D is abelian. (These are called "double Lie groups" in [48], without the second condition.) This map ϕ realises the parameterisation of (A.23) globally.
Given ϕ we can also define a "primed" factorisation where G and G appear in the other order. This is explicitly given by the diffeomorphism ϕ ′ : G × G → D defined via its inverse (where I G is the inversion map g → g −1 of G, resp. for other groups, and σ permutes G × G → G × G):