MPP-2022-100

Poisson-Lie T-duality defects
and target space fusion

Saskia Demulder and Thomas Raml

Max-Planck-Institut für Physik (Werner Heisenberg Institut),
Föhringer Ring 6, 80805 München, Germany
E-mail<EMAIL_ADDRESS>
Abstract. Topological defects have long been known to encode symmetries and
dualities between physical systems. In the context of string theory, defects
have been intensively studied at the level of the worldsheet. Although marked
by a number of pioneering milestones, the target space picture of defects is
much less understood. In this paper, we show, at the level of the target
space, that Poisson-Lie T-duality can be encoded as a topological defect. With
this result at hand, we can postulate the kernel capturing the Fourier-Mukai
transform associated to the action of Poisson-Lie T-duality on the RR-sector.
Topological defects have the remarkable property that they can be fused
together or, alternatively, with worldsheet boundary conditions. We study how
fusion of the proposed generalised T-duality topological defect consistently
leads to the known duality transformations for boundary conditions. Finally,
taking a step back from generalised T-duality, we tackle the general problem
of understanding the effect of fusion at the level of the target space. We
propose to use the framework of Dirac geometry and formulate the fusion of
topological defects and D-branes in this language.

###### Contents

  1. 1 Introduction
  2. 2 Poisson-Lie T-duality as a topological defect
    1. 2.1 Sigma-model approach
      1. 2.1.1 Poisson-Lie T-duality topological defect
      2. 2.1.2 Other topological defects
      3. 2.1.3 Including spectators
      4. 2.1.4 Relation to Fourier-Mukai-like transforms
    2. 2.2 Product space and generalised geometry approach
      1. 2.2.1 Poisson-Lie T-duality topological defect – revisited
      2. 2.2.2 Topological defects as Dirac structures
  3. 3 Worldvolume fusion of generalised T-duality defects
    1. 3.1 Lagrangian fusion and gluing conditions
      1. 3.1.1 Gluing matrices and Dirac structures
      2. 3.1.2 Boundaries under Poisson-Lie T-duality via fusion
    2. 3.2 Dirac geometry fusion
  4. 4 Discussion
  5. A Drinfel’d double conventions and the $\omega_{STS}$
  6. B Canonical transformations for Poisson-Lie T-duality
  7. C Reminder of generalised geometry

##  1 Introduction

Topological defects are known to offer a way to encode symmetries and
dualities of a physical system. Studying symmetry and duality-encoding defects
opens an alternative path to explore and understand symmetries in a universal
way, see [1] for many references on the subject. Topological defects have been
extensively studied as objects living on the worldsheet of conformal field
theories, see e.g. [2, 3, 4]. The monicker “topological” refers to their
distinguishing feature: the defect line on the worldsheet can be deformed
continuously. Indeed, the defining property of a topological defect is that
the energy-momentum tensors of the models on both sides continuously connect
at the location of the defect. As a consequence, since the energy-momentum
tensor is the generator of diffeomorphisms, topological defects have the
remarkable property that they can be moved freely on the worldsheet and
superimposed either with another defect or a boundary condition on the
worldsheet. This composition111In fact, in the conformal field theories fusion
at the level of the corresponding operators corresponds to composing the
associated defect operators, see e.g. [5] for additional details. operation of
topological defects (and boundary conditions) is known as “fusion” [6, 7]. The
fusion algebra induced by topological defects is well-understood as the tensor
product of bimodules in the representation category of the chiral symmetry
algebra of the CFT.

In this article, we will however consider a target space picture of
topological defects. We will specialise to topological defects dividing two
theories that are captured by non-linear sigma-model actions. We thus have two
maps $X:\Sigma\rightarrow M$ and
$\tilde{X}:\tilde{\Sigma}\rightarrow\tilde{M}$ from the worldsheets $\Sigma$
and $\tilde{\Sigma}$, that we will take to be separated by a defect line $D$,
to the target spaces $M$ and $\tilde{M}$. As mentioned earlier, topological
defects are objects living on the worldsheets where they can be moved and
fused with other defects or boundary conditions on the worldsheet. Whatever
happens on the worldsheet should translate to an object or operation on the
target space. A line defect on the worldsheet is thus mapped by the product
map $\mathbb{X}=X\times\tilde{X}$ to a submanifold $\mathcal{Y}$ of the
product space $M\times\tilde{M}$. As is the case for D-branes, fields in the
bulk couple to the defect via a line bundle determined by a connection living
on the defect and the associated field strength $\mathcal{F}$. Demanding the
defect line to be topological then translates into constrains on the possible
pairs $(\mathcal{Y},\mathcal{F})$ on $M\times\tilde{M}$.

The target space picture of topological defects was first considered in the
context of WZW models in [8], see also [9], and later in the context of
generalised geometry in [10]. In [8], by computing the scattering of closed
string in a WZW model with D-branes, so-called biconjugacy classes were
identified as the submanifolds $\mathcal{Y}$ realising the target space
picture of the topological line defect on the worldsheet. Due to the
similarity with the D-branes in WZW models, these topological defects were
dubbed “bibranes”. In what follows we will also at times use the term
“bibrane” for any pair $(\mathcal{Y},\mathcal{F})$ defining a topological
defect on the target space of a non-linear sigma-model.

Although it is considered a general fact that dualities and symmetries can be
encoded by topological defects, it is not clear how much the notion of duality
or symmetry can be stretched or weakened before this common lore breaks down.
In string theory, Abelian T-duality is a prime example of a duality that can
be encoded by a topological defect, see [5] for a worldsheet perspective and
[11, 12] for a target space derivation. Although very different in nature,
non-Abelian T-duality [13], was also shown to admit a description as a
topological defect living on the target space of the dual models.

More recently, a more general form of T-duality has gained more and more
attention. This duality takes a step back by no longer hinging on the
existence of isometric directions in the background, instead demanding a
weaker condition known as Poisson-Lie symmetry[14, 15]. Although featuring
many similarities with Abelian T-duality, it still remains unclear if Poisson-
Lie models can be considered as a symmetry between conformal systems. Indeed
Poisson-Lie symmetric models generically do not produce solution to the
conventional supergravity equations but a weaker set of equations, dubbed
modified or generalised supergravity [16]. This new set of equations still
enjoys scale invariance, but is however likely to be plagued by a Weyl
anomaly222A proposal to counter this problem involves considering the
contribution of a modified Fradkin-Tseytlin term, see [17, 18, 19, 20]. Note
also that this problem can be circumvented by constructing the backgrounds
using supergroups, where a non-unimodular r-matrix can always be chosen such
that the background solves the un modified supergravity equations [21, 22]..
In addition, whilst conventional T-duality is a quantum symmetry of string
theory, it is still unclear if Poisson-Lie T-duality can be lifted to a
quantum duality. Some initial steps towards addressing this questions for the
Poisson-Lie case were made in [23, 24]. The quantum aspects of Poisson-Lie
symmetry have also been addressed more recently in [25, 26]. These
observations raise the question of whether Poisson-Lie T-duality is “enough of
a duality” to be described by a topological defect. In what follows, we will
answer this question by the affirmative and we will unravel their properties
under fusion. Let us first summarise the novel results presented in this
article:

#### Summary of the results and overview

  * $\triangleright$

We show, by direct construction, that Poisson-Lie T-duality can be
represented, by using a target space formulation, as a topological defect. The
line-bundle of this defect is controlled by the so-called Semenov-Tian-Shansky
symplectic form on the Drinfel’d double associated to the Poisson-Lie T-dual
models. We establish the existence of a Poisson-Lie T-duality inducing
topological defect when the model includes spectator fields.

  * $\triangleright$

Using the previous result, we provide further evidence that the Fourier-Mukai
transform for Poisson-Lie T-duality has a kernel that is essentially captured
by the Semenov-Tian-Shansky symplectic form of the Drinfel’d double, as was
proposed recently in [27].

  * $\triangleright$

We show how the known transformation rules for the gluing conditions of open
strings under generalised T-duality can be derived by fusing the Poisson-Lie
T-duality defect with a boundary condition.

  * $\triangleright$

In the last section, no longer specialising to Poisson-Lie symmetric
backgrounds, we investigate the result of fusing topological defects and
boundary conditions in terms of the associated target space data. To do so, we
first underline the universal description of D-branes and defects in terms of
Dirac geometry and subsequently exploit this observation to formulate a notion
of fusion at the level of target space.

In section 2, we first show how to construct Poisson-Lie T-duality as a
topological defect. This can be established in two ways, either in terms of
the Lagrangian and its equations of motion or by lifting the discussion to the
associated product space. We will establish the existence of the duality
defect in both formalisms. Whilst the former is the most prevalent approach
for Lagrangian theories, the latter method will act as a stepping stone for
the later discussion of fusion. Section 3 discusses the process of fusion of
topological defects at the level of the target space. With the generalised
T-duality topological defect at hand, in section 3.1, we show how fusing the
generalised T-duality defect with boundary conditions characterised by gluing
conditions yields the known duality transformations. Finally, in section 3.2,
we will take a step back from generalised T-duality. There, we consider the
problem of understanding the well-established notion of worldsheet fusion of
topological defects on the target space. We combine the expectation that the
fusion process should mirror a Fourier-Mukai transformation together with the
insight that Dirac structures are the fitting objects to describe the
worldvolume of D-branes and topological defects. We then propose a fusion
operation that uses the language of generalised and Dirac geometry. In section
4, we draw some final conclusions and future perspectives. We complement the
discussion in the main text with a number of appendices.

##  2 Poisson-Lie T-duality as a topological defect

In this section, we establish Poisson-Lie T-duality as a space-filling
topological defect carrying a line bundle with curvature given by the so-
called Semenov-Tian-Shansky (STS) symplectic structure, both with and without
spectator directions. After showing this first at the Lagrangian level, and
anticipating on our discussion on fusion in the second part of this paper, we
will also show this fact directly at the level of the product space. This will
allow us, following [10], to later formulate the defects in terms of Dirac
structures.

Before introducing the topological defect relevant to Poisson-Lie T-duality,
we first review some necessary facts on this generalised notion of the
T-duality as well its associated algebraic objects. See [15, 14] or the
reviews [28, 29, 30] for a more detailed and thorough discussion of Poisson-
Lie T-duality. In what follows, we will consider sigma-models that are
Poisson-Lie symmetric. This “symmetry” can be seen as a weaker requirement
than demanding to have Abelian or non-Abelian directions in Abelian, resp.
non-Abelian, T-duality. The general form of a sigma-model admitting Poisson-
Lie symmetry is as follows. We have a map $g(X):\Sigma\rightarrow M$, where we
assume a Lie group $\mathcal{G}$ acts on $M$, together with the action [14]

$\displaystyle
S_{\mathrm{PL}}=\int_{\Sigma}L^{a}(E_{0}^{-1}+\Pi[g])^{-1}_{ab}L^{b}\,,$ (2.1)

where we have denoted by $X$ a set of coordinates on a local patch of $M$, the
left-invariant Maurer-Cartan form are denoted by
$L=L^{a}T_{a}=l^{a}_{i}\mathrm{d}X^{i}T_{a}=g^{-1}\mathrm{d}g$ for $g\in M$
and $T_{a}$ are a set of generators for the Lie algebra
$\mathrm{Lie}(\mathcal{G})=\mathfrak{g}$ (and likewise for the tilde
variables) with $a=1,\cdots,\mathrm{dim}\,\mathcal{G}$, $E_{0}$ is a
$\mathrm{dim}\,M\times\mathrm{dim}\,M$-dimensional constant matrix and
$\Pi[g]$ is a Poisson-Lie structure obtained by solving the (modified)
classical Yang-Baxter equation for the Lie algebra
$\mathrm{Lie}(\mathcal{G})=\mathfrak{g}$. The underlying reason to why such
models naturally generalise the conventional form of T-duality is that its
underlying symmetry group is a Poisson-Lie group, that comes with a natural
notion of “dual object” baked in. Indeed, the Poisson-structure $\Pi[g]$
associated to $\mathcal{G}$ does not only define a (multiplicative) Poisson
structure on $\mathcal{G}$ it determines in fact a second Lie bracket on the
algebra $\mathfrak{g}$. Denote by $\tilde{\mathfrak{g}}$ the algebra defined
by this second bracket on the vector space $\mathfrak{g}$. The associated Lie
group $\tilde{\mathcal{G}}$ also admits a Poisson-Lie structure $\tilde{\Pi}$.
The relation is symmetric: the Poisson-Lie dual of $\tilde{\mathfrak{g}}$ is
again ${\mathfrak{g}}$. One can write an action with target space
$\tilde{\mathcal{G}}$; the Poisson-Lie T-dual action,

$\displaystyle\tilde{S}_{\mathrm{PL}}=\int_{\tilde{\Sigma}}\tilde{L}_{a}[(E_{0}+\tilde{\Pi}[\tilde{g}])^{-1}]^{ab}\tilde{L}_{b}\,,$
(2.2)

where $\tilde{\Pi}[\tilde{g}]$ is the Poisson-Lie structure on the Lie group
$\tilde{\mathcal{G}}$ associated to the double
$\mathbb{D}=\mathcal{G}\tilde{\mathcal{G}}$ and $\tilde{l}$ are now the
components of the left-invariant Maurer-Cartan form
$\tilde{L}=\tilde{l}_{ai}\mathrm{d}\tilde{X}^{i}\tilde{T}^{a}$with respect to
an element $\tilde{g}\in\tilde{\mathcal{G}}$. Remarkably, as was shown in [14,
15], the actions in eqs. (2.1) and (2.2) naturally generalise the notion of
T-dual models by taking $\mathbb{D}=U(1)^{2\dim\mathcal{G}}$ for Abelian
T-duality and $\mathbb{D}=\mathcal{G}\times U(1)^{\dim\mathcal{G}}$ for non-
Abelian T-duality.

There is a last but crucial fact left to introduce. The direct sum of the Lie
algebra $\mathfrak{g}$ with its Poisson-Lie T-dual algebra
$\tilde{\mathfrak{g}}$ leads to a so-called Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\oplus\tilde{\mathfrak{g}}$. See appendix A for
definitions and details. An element $\mathbb{g}$ of the exponentiation
$\mathbb{D}$ of this algebra, which is also called the Drinfel’d double of
$(\mathcal{G},\Pi)$, has by construction, locally, two possible decompositions

$\displaystyle\mathbb{g}=\tilde{h}g\quad\text{or}\quad\mathbb{g}=h\tilde{g}\,,\quad\text{with}\;\,\,g,h\in\mathcal{G}\,,\
\tilde{g},\tilde{h}\in\tilde{\mathcal{G}}\,,$ (2.3)

where the Drinfel’d double is assumed to be perfect, such that these
decompositions are unique. In addition, this Lie group $\mathbb{D}$ can be
naturally endowed with a symplectic structure, called the Semenov-Tian-Shansky
(STS) symplectic form [31, 32]

$\displaystyle\omega_{\mathrm{STS}}$
$\displaystyle=\tfrac{1}{2}(r^{a}(g)\wedge\tilde{l}_{a}(\tilde{h})-\tilde{r}_{a}(\tilde{g})\wedge
l^{a}(h))\,,$ (2.4)

where we have used the components of the right-invariant Maurer-Cartan form
$R=r^{a}_{i}\mathrm{d}X^{i}T_{a}=\mathrm{d}gg^{-1}$ for $g\in\mathcal{G}$. The
Drinfel’d double together with the Semenov-Tian-Shansky symplectic form will
play a crucial role in what follows.

###  2.1 Sigma-model approach

At the level of the target space, a topological defect can be described by a
total Lagrangian involving the sigma-model actions of the neighbouring
theories together with a term specifying the coupling of the line bundle on
the defect worldvolume to the bulk fields. We thus consider a pair of target
space maps $X:\Sigma\rightarrow M$ and
$\tilde{X}:\tilde{\Sigma}\rightarrow\tilde{M}$. We glue these two theories
together by combining the worldsheets $\Sigma\cup\tilde{\Sigma}$ along a
defect line $D$, located at $\sigma=0$, so that the target space map
$X(\tau,\sigma)$ is now defined for $\sigma\geq 0$ and
$\tilde{X}(\tau,\sigma)$ for $\sigma\leq 0$. It is thus convenient to define
the product target space map

$\displaystyle\mathbb{X}=X\times\tilde{X}\,:\,\Sigma\cup\tilde{\Sigma}\rightarrow
M\times\tilde{M}:(\tau,\sigma)\mapsto\mathbb{X}^{I}=(X^{i},\tilde{X}^{i})\,.$
(2.5)

The pushforward of the product map $\mathbb{X}$ restricted to the defect line
$D$ will then span a submanifold $\mathcal{Y}$ in product space
$M\times\tilde{M}$. In analogy with D-branes, the bulk fields will couple to a
line bundle whose pullbacked connection $\mathbb{A}$ lives on the worldvolume
of the submanifold $\mathcal{Y}$ corresponding to the defect. The curvature of
the line bundle connection will be denoted by
$\mathcal{F}=\mathrm{d}\mathbb{A}$. That is, from the perspective of the
target space, a defect between two theories with target spaces $M$ and
$\tilde{M}$ is a pair $(\mathcal{Y},\mathcal{F})$ consisting of a submanifold
$\mathcal{Y}$ in the product space $M\times\tilde{M}$ together with a two-form
$\mathcal{F}$. We thus have a total worldsheet action of the form [8, 11]

$\displaystyle\begin{aligned}
S&=\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}\sigma\,\left(G(X)_{ij}\partial_{\mu}X^{i}\partial^{\mu}X^{i}+B(X)_{ij}\epsilon^{\mu\nu}\partial_{\mu}X^{i}\partial_{\nu}X^{j}\right)\\\
&\qquad+\frac{1}{2}\int_{\tilde{\Sigma}}\mathrm{d}^{2}\sigma\,\left(\tilde{G}(X)_{ij}\partial_{\mu}\tilde{X}^{i}\partial^{\mu}\tilde{X}^{i}+\tilde{B}(X)_{ij}\epsilon^{\mu\nu}\partial_{\mu}\tilde{X}^{i}\partial_{\nu}\tilde{X}^{j}\right)+\int_{D}\mathrm{d}\tau\,\mathbb{A}_{I}(\mathbb{X})\partial_{\tau}\mathbb{X}^{I}\,.\end{aligned}$
(2.6)

Locally, the three-form fluxes $H$ and $\tilde{H}$ in each model are
identified with the exterior derivative of the Kalb-Ramond two-forms $B_{ij}$
and $\tilde{B}_{ij}$, i.e. $H=\mathrm{d}B$ and
$\tilde{H}=\mathrm{d}\tilde{B}$. The geometric data associated to the defect
can be encoded into a two-form field $F$ defined on the topological defect by
$F=-(p^{\ast}B-\tilde{p}^{\ast}\tilde{B}+\mathcal{F})$, where the overall
minus sign is chosen for later convenience and where $p$ and $\tilde{p}$ are
the projections from the product space onto $M$, respectively $\tilde{M}$ and
the Kalb-Ramond fields are evaluated at the location of the defect. Note that
by definition we have that the corresponding exterior product is the
difference of the three-form fluxes of the theories on the left and on the
right $\mathrm{d}F=H-\tilde{H}$, when evaluated at the location of the defect.

It is clear that not all pairs of submanifolds and line bundle curvatures
$(\mathcal{Y},\mathcal{F})$ realise the worldvolume of a topological defect.
Indeed the topological property given in eq. (2.8), that requires the energy
momentum tensors of the two neighbouring models to continuously connect along
the defect line, is highly constraining. The energy momentum tensor of a
sigma-model is given by

$\displaystyle
T_{\mu\nu}=G_{ij}(X)\partial_{\mu}X^{i}\partial_{\nu}X^{j}-\frac{1}{2}g_{\mu\nu}g^{\gamma\delta}G_{ij}(X)\partial_{\gamma}X^{i}\partial_{\delta}X^{j}\,,$

where $g_{\mu\nu}$ is the worldsheet metric, or in terms of the coordinates
$z=\tau+\sigma$ and $\bar{z}=\tau-\sigma$ one has

$\displaystyle T=G_{ij}\partial X^{i}\partial
X^{j}\,,\quad\bar{T}=G_{ij}\bar{\partial}X^{i}\bar{\partial}X^{j}\,,$ (2.7)

were the derivatives are given by
$\partial=\frac{1}{2}(\partial_{\tau}+\partial_{\sigma})$ and
$\bar{\partial}=\frac{1}{2}(\partial_{\tau}-\partial_{\sigma})$. A defect is
called topological when it continuously connects holomorphic and anti-
holomorphic components of the left- and right energy-momentum tensors333A
weaker notion of defect is that of conformal defects for which we require the
conservation of the worldsheet energy. This translates into the requirement
that the off-diagonal components of the stress-energy tensors glue
continuously $T^{1}{}_{0}-\hat{T}^{1}{}_{0}=0$, at the location of the defect
$D$. Topological defects are a stronger concept, by additionally demanding
that the worldsheet momentum is also conserved. That is guaranteed when at the
defect the diagonal components of the stress-energy tensor glue continuously
$T^{1}{}_{1}-\hat{T}^{1}{}_{1}=0\,.$ Topological defects in contrast to
conformal defects can be, as long as the defect does not cross the location of
a local operator insertion, deformed smoothly without affecting the values of
correlators. For conformal defects, one has to be more careful as the fusion
operation might lead to singularities and corresponding divergences.

$\displaystyle T=\tilde{T}\,,\qquad\bar{T}=\bar{\tilde{T}}\,,$ (2.8)

at the location of the defect. The constraint in eq. (2.8) has striking
implications. Indeed, the topological defect condition in eq. (2.8) implies
that, since the energy-momentum is the generator of diffeomorphisms, the total
system given by the action in eq. (2.6), remains invariant under continuous
deformations of the defect at values $\sigma\neq 0$. This observation leads to
one of the most striking features of this class of defects: the process of
fusion. Topological defects can be continuously deformed away from their
initial location on the worldsheet. When two defects are superimposed, they
fuse into one or multiple new defects. In a like manner, when the sigma-models
describe open strings, a topological defect can be moved to the boundary of
the worldsheet and fused with the boundary conditions resulting in a new
boundary condition. On a general basis, invertible444A topological defect is
called invertible if there exists a second defect such that their composition
or fusion yield the invisible defect. In conformal field theories, where
defects can be associated to operators, invertibility means a trivial null
space. As a result theories separated by an invertible defect share the same
spectrum. See e.g. [33] for a more rigorous treatment. topological defects are
expected to implement dualities between theories. Indeed, in [11, 13], it was
shown how, at the level of the target space formulation, particular
topological defects can encode T-duality and even non-Abelian T-duality. In
the following section we will prove that one can also construct a topological
defect encoding the most general form of T-duality on group manifolds;
Poisson-Lie T-duality.

####  2.1.1 Poisson-Lie T-duality topological defect

To show that the pair $(\mathbb{D},\omega_{\mathrm{STS}})$ realises the
worldvolume of a topological defect encoding Poisson-Lie T-duality, the
starting point is the total action in eq. (2.6), i.e.

$\displaystyle
S_{\mathrm{tot}}=S+\tilde{S}+S_{\mathrm{defect}}=2\int_{\Sigma}L^{a}E_{ab}L^{b}+2\int_{\tilde{\Sigma}}\tilde{L}_{a}\tilde{E}^{ab}\tilde{L}_{b}+\int_{D}\mathbb{X}^{\ast}\mathbb{A}\,,$
(2.9)

where $E=G+B$ is the background field consisting of metric $G$ and $B$ the
antisymmetric Kalb-Ramond field and likewise for the tilde variables. The
left-invariant Maurer-Cartan form is denoted, as above, by
$L=l^{a}_{i}\mathrm{d}X^{i}T_{a}=g^{-1}\mathrm{d}g$ for $g\in M$ and $T_{a}$
are a set of generators for the Lie algebra
$\mathrm{Lie}(\mathcal{G})=\mathfrak{g}$ (and likewise for the tilde
variables) with $a=1,\cdots,\mathrm{dim}\,\mathcal{G}$. The map
$\mathbb{X}=X\times\tilde{X}|_{D}$ is the product target space map defined in
eq. (2.5). The first action, given by $S$, is taken to be a Poisson-Lie
symmetric model on a Lie group manifold, as given in eq. (2.1) and the second
action $\tilde{S}$ is left unspecified. We will make the Ansatz that the
defect is given by the pair $(\mathbb{D},\omega_{STS})$. That is we take
$\mathrm{d}\mathbb{A}(X,\tilde{X})=\mathcal{F}=\omega_{\mathrm{STS}}(X,\tilde{X})$,
where the Semenov-Tian-Shansky symplectic form was defined in eq. (2.4). We
will see that by picking a particular solution of the equation of motion in
the presence of the defect, the background fields of the target space
$\tilde{\mathcal{G}}$ will be precisely the Poisson-Lie T-dual action
$\tilde{S}_{\mathrm{PL}}$ given in eq. (2.2). In addition, we show that this
defect is topological.

First, it will be more convenient to use a rewriting of this symplectic form

$\displaystyle 2\omega_{STS}=2l^{b}(g)\wedge C_{b}^{\
e}\tilde{l}_{e}(\tilde{g})+l^{b}(g)\wedge C_{b}^{\
m}\tilde{\Pi}_{me}[\tilde{g}]l^{e}(g)-\tilde{l}_{b}(\tilde{g})\wedge\tilde{C}^{b}_{\
m}\Pi^{me}[g]\tilde{l}_{e}(\tilde{g})\,,$ (2.10)

where we have introduced $C=(1-\tilde{\Pi}\Pi)$ and similarly for $\tilde{C}$
with the order of $\Pi,\tilde{\Pi}$ switched. This expression for the STS
symplectic form is derived in the appendix A or see also [27] for an
equivalent but slightly different form. Varying now the action with respect to
$X^{i}$ and keeping only terms localized at $\sigma=0$ we get

$\displaystyle\int_{D}\mathrm{d}\tau\Bigl{.}\Bigl{(}E_{ab}l^{b}_{j}\bar{\partial}X^{j}-E_{ba}l^{b}_{j}\partial
X^{j}\Bigr{)}\Bigr{|}_{\sigma=0}l^{a}_{\ k}\delta
X^{k}=-\left.\delta_{X^{i}}\left(\int_{D}\mathbb{X}^{\ast}\mathbb{A}\right)\right|_{\sigma=0}\,,$
(2.11)

and similarly for $\delta_{\tilde{X}^{i}}$. Taking the variation of the
boundary term

$\displaystyle\delta_{X^{i}}\int_{D}\mathbb{X}^{\ast}\mathbb{A}$
$\displaystyle=\int_{D}\mathrm{d}\tau\left(C_{a}^{\
b}\tilde{\Pi}_{bc}[\tilde{g}]l^{c}_{j}\partial_{\tau}X^{j}+C_{a}^{\
b}\tilde{l}_{bj}\partial_{\tau}\tilde{X}^{j}\right)l^{a}_{i}\delta X^{i}\,.$

From these variations the following equations of motion on the defect line
$\sigma=0$ follow:

$\displaystyle\begin{aligned} E_{ba}l^{b}_{j}\partial
X^{j}-E_{ab}l^{b}_{j}\bar{\partial}X^{j}&=C_{a}^{\
b}\tilde{\Pi}_{bc}[\tilde{g}]l^{c}_{j}\partial_{\tau}X^{j}+C_{a}^{\
b}\tilde{l}_{bj}\partial_{\tau}\tilde{X}^{j}\,,\\\
\tilde{E}^{ba}\tilde{l}_{bj}\partial\tilde{X}^{j}-\tilde{E}^{ab}\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}&=\tilde{C}^{a}_{\
b}\Pi^{bc}[g]\tilde{l}_{cj}\partial_{\tau}\tilde{X}^{j}+l^{b}_{j}C_{b}^{\
a}\partial_{\tau}X^{j}\,.\end{aligned}$ (2.12)

To solve these equations of motion, we pick a particular solution which, as we
will see in the next section, singles out the topological defect encoding
Poisson-Lie T-duality. We set

$\displaystyle\begin{aligned} &\left(E_{ab}+C_{a}^{\
c}\tilde{\Pi}_{cb}[\tilde{g}]\right)l^{b}_{j}\bar{\partial}X^{j}=-C_{a}^{\
b}\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}\,,\quad\left(E_{ba}-C_{a}^{\
c}\tilde{\Pi}_{cb}[\tilde{g}]\right)l^{b}_{j}\partial X^{j}=C_{a}^{\
b}\tilde{l}_{bj}\partial\tilde{X}^{j}\\\
&\left(\tilde{E}^{ab}+\tilde{C}^{a}_{\
c}\Pi^{cb}[g]\right)\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}=-C_{b}^{\
a}l^{b}_{j}\bar{\partial}X^{j}\,,\quad\left(\tilde{E}^{ba}-\tilde{C}^{a}_{\
c}\Pi^{cb}[g]\right)\tilde{l}_{bj}\partial\tilde{X}^{j}=C_{b}^{\
a}l^{b}_{j}\partial X^{j}\,.\end{aligned}$ (2.13)

One can readily check that these relations, if satisfied, solve the equations
of motion at the location of the defect line. Let us now show that the
relations in eq. (LABEL:eq:Eab_on_defect) are nothing but the canonical
relations for Poisson-Lie T-duality [34] which we review in appendix B.
Indeed, one can rewrite the eqs. in eq. (LABEL:eq:Eab_on_defect) as

$\displaystyle\begin{aligned}
\tilde{C}\bar{L}&=-\tilde{C}(E+C\tilde{\Pi})^{-1}C\tilde{L}\,,\quad\tilde{C}L=\tilde{C}(E^{T}-C\tilde{\Pi})^{-1}C\tilde{L}\,,\\\
C\bar{\tilde{L}}&=-C(\tilde{E}+\tilde{C}\Pi)^{-1}\tilde{C}L\,,\quad
C\tilde{L}=C(\tilde{E}^{T}-\tilde{C}\Pi)^{-1}\tilde{C}L\,,\end{aligned}$
(2.14)

where we have suppressed the indices for ease of notation. Consistency of the
equations in eq. (2.14) with respect to those in eq. (LABEL:eq:Eab_on_defect)
forces us to identify

$\displaystyle\tilde{E}^{ab}+\tilde{C}^{a}_{\
c}\Pi^{cb}[g]\equiv\tilde{C}^{a}_{\
d}\left((E+C\tilde{\Pi}[\tilde{g}])^{-1}\right)^{dc}C_{c}^{\ b}\,,$ (2.15)

and similarly for the other equations. These equations are however nothing but
the canonical transformation rules for the background fields of Poisson-Lie
T-dual models [34]

$\displaystyle\begin{aligned}
C^{-1}(E+C\tilde{\Pi})\tilde{C}^{-1}(\tilde{E}+\tilde{C}\Pi)=1\,,\qquad
C^{-1}(E^{T}-C\tilde{\Pi})\tilde{C}^{-1}(\tilde{E}^{T}-\tilde{C}\Pi)=1\,.\end{aligned}$
(2.16)

That is, we have shown that defect
$(\mathcal{Y},\mathcal{F})=(\mathbb{D},\omega_{\mathrm{STS}})$ encodes the
Poisson-Lie T-duality transformation.

Having established that the action in eq. (2.9) with sigma-models given by
eqs. (2.1) and (2.2) and with one-form field defined via
$\mathrm{d}\mathbb{A}=\omega_{\mathrm{STS}}$ describes a defect encoding
Poisson-Lie T-duality, we now show that the defect is in fact topological.
That is, the defect verifies the continuity of the energy-momentum tensors as
give in eq. (2.8) across the defect location. The holomorphic part of the
energy-momentum tensors for the left Poisson-Lie models, after using the
relation in eq. (LABEL:eq:Eab_on_defect) in the expression given in eq. (2.7),
leads to

$\displaystyle T$ $\displaystyle=C_{b}^{\
c}\tilde{\Pi}_{ca}l^{a}_{i}l^{b}_{j}\partial X^{i}\partial X^{j}+C^{\
a}_{b}\tilde{l}_{ai}l^{b}_{j}\partial\tilde{X}^{i}\partial X^{j}\,,$

whilst the energy-momentum tensor for the dual right model takes on the form

$\displaystyle\tilde{T}$ $\displaystyle=\tilde{C}^{b}_{\
c}\Pi^{ca}\tilde{l}_{ai}\tilde{l}_{bj}\partial\tilde{X}^{i}\partial\tilde{X}^{j}+C_{b}^{\
a}l^{b}_{j}\tilde{l}_{ai}\partial X^{j}\partial\tilde{X}^{i}\,.$

Taking their difference we see that the last term from each energy momentum
tensor cancel, leading to

$T-\tilde{T}=\Pi^{ac}C_{c}^{\
b}\tilde{l}_{ai}\tilde{l}_{bj}\partial\tilde{X}^{i}\partial\tilde{X}^{j}-\tilde{\Pi}_{ac}\tilde{C}^{c}_{\
b}l^{a}_{i}l^{b}_{j}\partial X^{i}\partial X^{j}=0\,,$ (2.17)

where the left-over terms vanish identically by virtue of $\Pi C$ and
$\tilde{\Pi}\tilde{C}$ being antisymmetric555This can be shown using the
explicit form of $C,\tilde{C}$ in terms of adjoint actions given in the
appendix A.. One can verify in a very similar fashion that the condition on
the antiholomorphic part of the energy-momentum tensors on both theories is
verified, showing that the defect defined by the total system defined in eq.
(2.15) with field strength $\mathrm{d}\mathbb{A}=\omega_{\mathrm{STS}}$ given
by the Semenov-Tian-Shansky symplectic structure is indeed topological.

####  2.1.2 Other topological defects

In [35] and [1], the relation between the fusion algebra of topological
defects in conformal field theories and the duality group $O(d,d)$ (and its
semi-group extension) was studied. The group $O(d,d)$ is not only generated by
(factorised) T-dualities but also linear transformations and $B$-field
transformations. As later highlighted in [11, 36], one can indeed also realise
the latter two, i.e the $B$-field transformations and linear transformations,
using the target space formulation of defects. $B$-field transformations
between the background field $E=G+B$ on a target space $M$ and the background
field $\tilde{E}$ on $\tilde{M}=\mathcal{G}$, are associated with a “diagonal
defect” determined by the choice $X^{i}=\tilde{X}^{i}$ in the product space
$\mathcal{G}\times\mathcal{G}$ with the two-form field $B$. On can show that
this defect is again topological, we will not repeat the argument in detail,
which can be found in [11, 36]. What is relevant to remember for what is to
come, is that the worldvolume of this defect is realised by a pair
$(\mathcal{G}_{\mathrm{diag}},B)$, where $\mathcal{G}_{\mathrm{diag}}$ is a
copy of $\mathcal{G}$ diagonally embedded in the product space
$\mathcal{G}\times\mathcal{G}$.

Finally, let us remark that for a given Poisson-Lie symmetric target space
$\mathcal{G}$ there are, by construction, as many Poisson-Lie T-duality
defects as there are solutions of the (modified) classical Yang-Baxter
equation for its algebra $\mathfrak{g}$, or equivalently so-called Manin
triples associated with $\mathfrak{g}$. To see this, we need yet another
equivalent expression for the Semenov-Tian-Shansky symplectic form. For each
solution $r$ of the modified classical Yang-Baxter equation for the Lie
algebra $\mathfrak{g}$, we obtain a different dual algebra
$\tilde{\mathfrak{g}}$ or Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\oplus\tilde{\mathfrak{g}}$666Technically this means
that we are looking at “split” type classical modified Yang-Baxter equation.
Indeed, solutions to the non-split classical modified Yang-Baxter equation for
a fixed Lie algebra $\mathfrak{g}$ is essentially unique. We will not discuss
solutions to the homogeneous classical Yang-Baxter equation. More details and
definitions can be found in [31, 37].. It turns out that the Drinfel’d double
$\mathfrak{d}$ is itself also a Poisson-Lie group with r-matrix
$r_{\mathfrak{d}}$ given by the expression

$\displaystyle r_{\mathfrak{d}}=p_{\mathfrak{g}}-p_{\tilde{\mathfrak{g}}}\,,$

where $p_{\mathfrak{g}}$ is the projection into $\mathfrak{g}$ seen as
diagonally embedded into $\mathfrak{d}$ and $p_{\tilde{\mathfrak{g}}}$ is the
projection into the Poisson-Lie dual algebra $\tilde{\mathfrak{g}}$. In terms
of the r-matrix $r_{\mathfrak{d}}$ of the double, the Semenov-Tian-Shansky
symplectic form, or equivalently the corresponding Poisson structure, can be
expressed by, see e.g. [31]:

$\displaystyle\\{\,\cdot\,,\,\cdot\,\\}_{\mathrm{STS}}=\tfrac{1}{2}\langle(r_{\mathfrak{d}}-\mathrm{Ad}_{\mathbb{g}^{-1}}r_{\mathfrak{d}}\mathrm{Ad}_{\mathbb{g}})\nabla\cdot,\nabla\cdot\rangle\,,$
(2.18)

where the adjoint action are taken with respect to an element
$\mathbb{g}\in\mathbb{D}$ of the double and $\nabla$ denotes the left gradient
derivative, i.e. for a function $f\in C^{\infty}(\mathcal{G})$ it takes the
form $\langle\nabla
f(x),X\rangle=\tfrac{\mathrm{d}}{\mathrm{d}t}f(\exp(tX)x)|_{t=0}$, for
$X\in\mathfrak{g}$. That is, for each solution of the (modified) classical
Yang-Baxter associated to the Lie algebra $\mathfrak{g}$ we also have a
different Drinfel’d double and associated Semenov-Tian-Shansky symplectic form
is given in eq. (2.18).

####  2.1.3 Including spectators

When considering spectator fields, the Lie group $\mathcal{G}$ acts freely,
but no longer transitively on the target space $M$. We assume thus that the
target space can be identified with a space of the form $M=\mathcal{G}\times
N$ and is described by coordinates $X^{m}=(X^{i},Y^{\mu})$ with
$i=1,\cdots,\dim\mathcal{G}=n$ and $N$ is the spectator manifold with
coordinates $Y$. In this section, we consider a topological defect lying
between two Poisson-Lie T-dual models but including the freedom of having
spectator directions. We thus start from the same formal action

$\displaystyle S_{\mathrm{tot,spect}}=S+\tilde{S}+S_{\mathrm{defect}}\,,$
(2.19)

where we will first assume that the model on the left of the defect, given by
the action $S$ is Poisson-Lie symmetric with spectator fields. In particular,
this means that the first action is a non-linear sigma-model taking the form

$\displaystyle S$
$\displaystyle=2\int_{\Sigma}\mathrm{d}^{2}\sigma\,\begin{pmatrix}l^{a}_{i}\partial
X^{i}&\partial Y^{\mu}\end{pmatrix}\begin{pmatrix}E_{ab}&E_{a\nu}\\\ E_{\mu
b}&E_{\mu\nu}\end{pmatrix}\begin{pmatrix}l_{i}^{b}\bar{\partial}X^{i}&\bar{\partial}Y^{\nu}\end{pmatrix}\,.$
(2.20)

The form of components of the background field $E_{mn}$ are fixed by imposing
Poisson-Lie symmetry and thus has to satisfy the condition

$\displaystyle
L_{l_{a}}E_{mn}=-\tilde{f}^{bc}{}_{a}E_{mp}l^{p}{}_{b}l^{q}{}_{c}E_{qn}\,,$

where $\tilde{f}^{ab}{}_{c}$ are the structure constants of the algebra of the
Lie group manifold $\tilde{\mathcal{G}}$ and one can determine the total
background field $E_{mn}$ as [14, 15, 34]

$\displaystyle\begin{aligned}
\begin{matrix}[l]E_{a\mu}=E_{ab}((E_{0})^{-1})^{bc}F_{c\mu}\,,&E_{\mu\nu}=F_{\mu\nu}-F_{\mu
a}\Pi^{ab}E_{bc}((E_{0})^{-1})^{cd}F_{d\nu}\,,\\\ \tilde{E}^{a}_{\
\mu}=\tilde{E}^{ab}F_{b\mu}\,,&\tilde{E}_{\mu\nu}=F_{\mu\nu}-F_{\mu
a}\tilde{E}^{ab}F_{b\nu}\,.\end{matrix}\end{aligned}$ (2.21)

Here $F_{\mu\nu},F_{a\mu}$ are arbitrary matrices depending on the choice of
background configuration chosen. These reduce to the conventional Poisson-Lie
symmetric backgrounds upon setting the spectators to zero. Crucially, the
Poisson structure $\Pi$ of this general solution for the background field $E$
in presence of spectator fields, only depends on the coordinates $X$ on
$\mathcal{G}$ and not on the spectator directions $Y$. The matrix
$(E_{0})_{mn}=(E_{0})_{mn}(Y)$ of coupling constant can however have a
$Y$-dependent, but no dependence on the coordinates of the manifold
$\mathcal{G}$ or $\tilde{\mathcal{G}}$.

The last term $S_{\mathrm{defect}}$ in the total action in eq. (2.19) accounts
for the one-form $\mathbb{A}$ on the defect line $D\subset\Sigma$,
contributing with the usual term

$\displaystyle S_{\mathrm{defect}}=\int_{D}\mathbb{X}^{\ast}\mathbb{A}\,,$

where $\mathbb{X}$ is the product map defined in eq. (2.5) from the union of
the worldsheet $\Sigma\cup\tilde{\Sigma}$ to the product space
$\mathcal{G}\times\tilde{\mathcal{G}}\times N$. Note that we do not take two
copies of the spectator manifold $N$. We will show that taking the simplest
Ansatz, that is taking
$\mathrm{d}\mathbb{A}(X,\tilde{X})=\omega_{\mathrm{STS}}(X,\tilde{X})$, will
give a topological defect that encoded Poisson-Lie T-duality with spectator
fields. We will see that taking again the STS symplectic structure to be the
curvature of the line bundle of the topological defect takes care of the
presence of the spectator fields, and will fix the model described by the
action $\tilde{S}$ on the right to be the Poisson-Lie T-dual action, taking on
the same form as the Poisson-Lie action $S$ in eq. (2.20), but defined by a
set of background fields $\tilde{E}$ and (left-invariant) Maurer-Cartan forms
$\tilde{L}=\tilde{l}_{ai}\tilde{T}^{a}\mathrm{d}X^{i}$ depending on the
coordinates $(\tilde{X},Y)$, sharing the same $Y$-direction (the spectator
fields).

Varying the total action given in eq. (2.19) and using again the alternative
expression for the STS form given in eq. (2.10), the equations with spectator
directions become

$\displaystyle E_{ba}l^{b}_{j}\partial X^{j}+E_{\mu a}\partial
Y^{\mu}-E_{ab}l^{b}_{j}\bar{\partial}X^{j}-E_{a\mu}\bar{\partial}Y^{\mu}=C_{a}^{\
b}\tilde{\Pi}_{bc}[\tilde{g}]l^{c}_{j}\partial_{\tau}X^{j}+C_{a}^{\
b}\tilde{l}_{bj}\partial_{\tau}\tilde{X}^{j}\,,$

and similarly for the tilde variables

$\displaystyle\tilde{E}^{ba}\tilde{l}_{bj}\partial\tilde{X}^{j}+\tilde{E}^{\mu
a}\partial
Y^{\mu}-\tilde{E}^{ab}\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}-\tilde{E}^{\mu
b}\bar{\partial}Y^{\mu}=\tilde{C}^{a}_{\
b}\Pi^{bc}[g]\tilde{l}_{cj}\partial_{\tau}\tilde{X}^{j}+l^{b}_{j}C_{b}^{\
a}\partial_{\tau}X^{j}\,.$

Finally varying with respect to the spectators $Y^{\mu}$ we now get a third
equation

$\displaystyle E_{\mu a}l^{a}_{i}\bar{\partial}X^{i}$ $\displaystyle-
E_{a\mu}l^{a}_{i}\partial
X^{i}+E_{\mu\nu}\bar{\partial}Y^{\nu}-E_{\nu\mu}\partial Y^{\nu}$
$\displaystyle=\tilde{E}_{\mu}^{\
a}\tilde{l}_{ai}\bar{\partial}\tilde{X}^{i}-\tilde{E}^{a}_{\
\mu}\tilde{l}_{ai}\partial
X^{i}+\tilde{E}_{\mu\nu}\bar{\partial}Y^{\nu}-\tilde{E}_{\nu\mu}\partial
Y^{\nu}\,.$ (2.22)

The first two equations can be solved by making an Ansatz similar to that of
the non-spectator case but taking now into account a contribution from the
spectator fields. For example the first line in the particular solution given
in eq. (LABEL:eq:Eab_on_defect) now becomes

$\displaystyle E_{a\mu}\bar{\partial}Y^{\mu}+\left(E_{ab}+C_{a}^{\
c}\tilde{\Pi}_{cb}[\tilde{g}]\right)l^{b}_{j}\bar{\partial}X^{j}$
$\displaystyle=-C_{a}^{\ b}\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}\,,$
(2.23)

and mutatis mutandis for the three other equations. Again, by requiring
consistency of these particular solutions leads to the relation

$\displaystyle\tilde{E}^{ab}+\tilde{C}^{a}_{\
c}\Pi^{cb}[g]\equiv\tilde{C}^{a}_{\
d}\left((E+C\tilde{\Pi}[\tilde{g}])^{-1}\right)^{bc}C_{c}^{\
d}\,,\quad\tilde{E}^{a}_{\ \mu}\equiv\tilde{C}^{a}_{\
d}\left((E+C\tilde{\Pi}[\tilde{g}])^{-1}\right)^{bc}E_{c\mu}\,,$

and similarly for the others. We are again lead to conclude that the defect
implements the canonical transformation for the Poisson-Lie T-duality, but
this time including spectator directions.

It remains to show that the third equation of motion coming from the
spectators in eq. (2.1.3) is also satisfied for the specific solution singled
out by the Ansatz. To this purpose we have to use the form of the background
field imposed by the Poisson-Lie symmetry of the model as given in eqs (2.21).
With these expressions at hand, the canonical transformation relating the
left-invariant forms of the Poisson-Lie T-duality backgrounds on $M$ and
$\tilde{M}$ can be written as [38]

$\displaystyle(E_{0}^{\mp})^{-1}E^{\mp}(L_{\pm}\pm\Pi
F^{\mp}\partial_{\pm}Y)=\pm\tilde{E}^{\mp}(\tilde{L}_{\pm}\mp
F^{\mp}\partial_{\pm}Y)\,.$ (2.24)

A little algebra shows that this is equivalent to the relation

$\displaystyle E_{a\mu}l^{a}_{i}\partial X^{i}$
$\displaystyle=\tilde{E}^{a}_{\ \mu}\tilde{l}_{ai}\partial\tilde{X}^{i}-F_{\nu
a}\tilde{E}^{a}_{\ \mu}\partial Y^{\nu}+F_{\nu a}\Pi^{ab}E_{a\mu}\partial
Y^{\nu}\,.$ (2.25)

Using the rewriting of the canonical transformation for Poisson-Lie T-duality
in eq. (2.25) together with the general expressions for the background fields
in eq. (2.21) imposed by Poisson-Lie symmetry, one can then easily show that
the third equation of motion coming from the variation of the spectators in
eq. (2.1.3) is also satisfied. Note finally that in this case the pair
$(\mathcal{Y},\mathcal{F})=(\mathbb{D},\omega_{\mathrm{STS}})$ no longer
defines a space-filling defect since the corresponding worldvolume is again
the Drinfel’d double
$\mathbb{D}\subset\mathcal{G}\times\tilde{\mathcal{G}}\times N$.

Lastly, we turn to the problem of showing that this defect is topological.
Taking into account the spectator directions, the energy momentum tensors
featured in the topological defect condition in eq. (2.8) can be written as
(wlog, for the action on the left)

$\displaystyle T=G_{ab}l^{a}_{i}l^{b}_{j}\partial X^{i}\partial
X^{j}+2G_{a\mu}l^{a}_{i}\partial X^{i}\partial Y^{\mu}+G_{\mu\nu}\partial
Y^{\mu}\partial Y^{\nu}\,.$

We will now show that demanding again that the curvature of the one-form field
$\mathbb{A}$ living on the defect line is the Semenov-Tian-Shansky symplectic
form, guarantees the cancelation of the holomorphic and anti-holomorphic parts
of the energy-momentum tensors. Using the expressions for the background
fields in eq. (2.21), the difference of the holomorphic parts of the energy
momentum tensors on both sides reads

$\displaystyle T-\tilde{T}$ $\displaystyle=E_{a\mu}l^{a}_{i}\partial
X^{i}\partial Y^{\mu}-\tilde{E}^{a}_{\
\mu}\tilde{l}_{ai}\partial\tilde{X}^{i}\partial Y^{\mu}-F_{\mu
a}\Pi^{ab}E_{b\nu}\partial Y^{\mu}\partial Y^{\nu}+F_{\mu a}\tilde{E}^{a}_{\
\nu}\partial Y^{\mu}\partial Y^{\nu}\,,$

where the antisymmetry of the combinations $C\Pi$ and $\tilde{C}\tilde{\Pi}$
was used to cancel a number of terms contributing to the left- and right-
energy momentum tensors. But this last line however is nothing but the
canonical transformation for Poisson-Lie T-duality in presence of spectator
directions as derived in eq. (2.25). An identical computation shows that the
anti-holomorphic components of energy momentum tensors of the theories on the
left and the right of the defect cancel as well, concluding that the defect
does not only encode Poisson-Lie T-duality with spectators but is in fact
topological. In what follows, we will often refer to this topological defect
$(\mathbb{D},\mathcal{F})$ encoding Poisson-Lie T-duality by the shorter
nomicker “Poisson-Lie defects”.

####  2.1.4 Relation to Fourier-Mukai-like transforms

One of our motivations to study whether or not Poisson-Lie T-duality could
also be realised by a topological defect was to better understand if this
generalised form of T-duality can be associated to a Fourier-Mukai-like
transform. It has been known since [39] that the T-duality transformation of
the Ramond-Ramond fields is encoded through a Fourier-Mukai transform. The
latter is an integral transform defining an isomorphism of, possibly
$H$-twisted, cohomologies for the manifolds $M_{1}$ and $M_{2}$ and takes the
formal form

$\displaystyle\Lambda^{\bullet}T^{\ast}M_{1}\rightarrow\Lambda^{\bullet}T^{\ast}M_{2}:\alpha\mapsto(p_{2})_{\ast}(K\cdot
p_{1}^{\ast}\alpha)\,,$ (2.26)

here $p_{i}:M_{1}\times M_{2}\rightarrow M_{i}$ is the projection map onto
$M_{i}$ and $K$ is called the kernel of the integral transform. That is,
differential forms on the manifold $M_{1}$ are first transported to the
product space $M_{1}\times M_{2}$ by the pull-back of the projection map
$p_{1}:M_{1}\times M_{2}\rightarrow M_{1}$. In the product space $M_{1}\times
M_{2}$ the pull-backed object is then “convoluted” with a specific kernel $K$
encoding the duality transformation or isomorphism. The result is then pushed
forward via the projection $p_{2}:M_{1}\times M_{2}\rightarrow M_{2}$ onto the
second factor, to a “dual” differential form defined on the manifold $M_{2}$.
Schematically:

$M_{1}\times M_{2}$$M_{1}$$M_{2}$$p_{1}$$p_{2}$ (2.27)

The question to whether generalisations of T-duality might also admit a
representation as a Fourier-Mukai transform is a challenging task. Foremost,
since Fourier-Mukai transformations between spaces that are not associated to
abelian varieties777 In fact, a Fourier-Mukai transform is an equivalence of
derived category of coherent sheaves. This is realised in topological string
theory, where B-branes are modelled by coherent sheaves and the relevant
kernel to encode Abelian T-duality is given by the curvature of the Poincaré
bundle. Whether such a neat picture could also arise for the non-Abelian or
Poisson-Lie generalisations of T-duality is at present unclear to the authors.
are not well-understood. Remarkably, topological defects associated to
dualities offer a way around this technical and conceptual hurdle.

In [2, 1, 35, 11], it was noticed that the Fourier-Mukai transform naturally
surfaces when studying the worldsheet topological defects encoding Abelian
T-duality. Later, using the target space formulation of topological defects,
the kernels of the Fourier-Mukai transforms for non-Abelian [13] and fermionic
T-duality [36] were shown to again be essentially captured by the data
defining the duality defect. This relation between topological defects and a
Fourier-Mukai like operation is not coincidental and is a direct consequence
of the existence of a fusion operation of T-duality inducing topological
defects. Applying a T-duality to a D-brane should intuitively equate to fusing
the pertinent defect together with the boundary condition. As such, the
objects defining the duality defect have to also contain the necessary
information to understand how the Ramond-Ramond fields, which are charged
under the D-brane, transform under T-duality. More precisely, the authors in
[11, 13] showed how the two-form $\mathcal{F}$ living on the worldvolume of
the topological defect encoding Abelian (and later for non-Abelian) T-duality
determines the kernel $K$ of a Fourier-Mukai-like integral transform acting on
the Ramond-Ramond fields via $\int_{M_{1}}\exp(F_{\mathrm{kernel}}\cdot
p_{1}^{\ast}\alpha)$ where
$F_{\mathrm{kernel}}=-(p^{\ast}B-\tilde{p}^{\ast}\tilde{B}+\mathcal{F})$,
where we recall that $\mathcal{F}$ is the curvature of the line bundle on the
topological defect encoding the duality.

In addition, in [1], see also [35], the authors argued that fusion turns out
to match the composition of the associated geometric integral transformations
in all studied examples. The authors also argue that the fusion of a defect
with two-form curvature $\mathcal{F}$ with any D-brane $\mathcal{Y}_{1}$
determines a transformation that maps D-branes $\mathcal{Y}_{1}$ in $M_{1}$ to
D-branes given by

$\displaystyle(p_{2})_{\ast}(\mathcal{F}\cdot p_{1}^{\ast}\mathcal{Y}_{1})$
(2.28)

on $M_{2}$. This observation, which was first suggested in [2] and later
studied in e.g. [1, 8], will return later when we consider the notion of
fusion.

The discussion in the previous sections suggests, following the same logic,
that one should identify the kernel of the integral transform associated to
Poisson-Lie T-duality with the Semenov-Tian-Shansky symplectic structure of
the double $\mathbb{D}$. This observation nicely supports888Note however that,
at least naively, this form for the kernel of the Fourier-Mukai transform
encoding Poisson-Lie T-duality differs from the one proposed in [40]. the
results of [27], where, by studying the global aspects of Poisson-Lie
T-duality, the Semenov-Tian-Shansky symplectic structure was also proposed as
the relevant kernel of the Fourier-Mukai transform for Poisson-Lie T-duality.
In [27], an expression for the kernel of Poisson-Lie T-duality with spectator
fields was also derived by studying global aspects of the generalised
T-duality and imposing the Bianchi identity on the result of the associated
Fourier-Mukai transformation. This new kernel describing the duality with
spectators displays additional terms that account for the non-trivial
fibration of spectator direction across the manifold. This information can
only be detected when considering the global and topological aspects of
Poisson-Lie T-duality. In the present context however, since the topological
defect encodes a local version of the generalised T-duality transformations,
these terms remain invisible. This explains why, for our purposes, the
additional terms do not play a role in the construction of the associated
defect.

###  2.2 Product space and generalised geometry approach

As was already mentioned in the introduction, topological defects have the
distinguishing feature that they can be deformed continuously from their
initial location. When this defect line is superimposed with another defect
line (or a boundary condition), they “fuse” and one obtains a new
configuration: either a new defect (or new boundary condition) or a
superposition of multiple defects (or boundary conditions). How worldsheet
fusion translate at the level of the target space will be subject of section
3. To prepare the ground to understand how to realise “target space fusion”,
i.e. an operation on the set of submanifolds with two-form field
$(\mathcal{Y},\mathcal{F})$ on the product space $M\times\tilde{M}$ that
defines topological defects, it turns out to be instrumental to reformulate
the notion of bibranes in the setting of generalised geometry.

Indeed, topological defects, using the language of generalised geometry, can
be equally characterised directly in terms of the background fields and the
pair $(\mathcal{Y},\mathcal{F})$ capturing the relevant defect [10]. We will
first review here some of their results useful for our purposes. As was
already stressed in the previous section, on the target space the defect is
described through the corresponding submanifolds in the doubled space
$M\times\tilde{M}$ which carries a two-form $\mathcal{F}$. The condition for
the defect to be topological, see eqs. (2.8), can however be directly
expressed in terms of conditions on the submanifold $\mathcal{Y}$ and its two-
form $\mathcal{F}$. To do so, one first introduces a neutral signature metric
$\mathbb{G}_{AB}$ and B-field $\mathbb{B}_{AB}$ on the product space
$M\times\tilde{M}$ given by

$\displaystyle\mathbb{G}_{AB}=\begin{pmatrix}G_{ab}&0\\\
0&-\tilde{G}_{ab}\end{pmatrix}\,,\qquad\mathbb{B}_{AB}=\begin{pmatrix}B_{ab}&0\\\
0&-\tilde{B}_{ab}\end{pmatrix}\,,$

where $(G,B)$, respectively $(\tilde{G},\tilde{B})$, are the background fields
of the sigma-models on the left, respectively right, of the defect, implicitly
restricted to the boundary $\mathcal{Y}$ where they are well-defined. The
capital index $A$ runs form $1$ to $2n$ with the first $n$ indices belonging
to the target space $\mathcal{G}$ and the latter $n$ to the space
$\tilde{\mathcal{G}}$, for $n=\dim\mathcal{G}=\dim\tilde{\mathcal{G}}$.
Furthermore we define a two-form $\mathbb{F}$ to be the combination

$\displaystyle\mathbb{F}=-\mathrm{d}\mathbb{A}-\iota^{\ast}\mathbb{B}=-\mathcal{F}-\iota^{\ast}\mathbb{B}$
(2.29)

with $\iota:\mathcal{Y}\hookrightarrow\mathcal{G}\times\tilde{\mathcal{G}}$ is
the embedding map and $\mathcal{F}$ is the field strength of the one-form
$\mathbb{A}$ living on the defect worldvolume $\mathcal{Y}$. Kapustin and
Setter showed in [10] that, the condition on the energy momentum tensor can be
translated to a condition on the geometry of the induced target space
worldvolume together with the background fields. In order to define allowed
pairs $(\mathcal{Y},\mathcal{F})$ one has to verify the relation999In [10],
pairs determining defects are chosen to be characterised by the curvature plus
the B-field contribution. Note that in the present paper, we have picked a
slightly different convention having chosen to characterise the pair by
$(\mathcal{Y},\mathcal{F})$ where $\mathcal{F}$ is the curvature of the line
bundle, without the B-fields.

$\displaystyle\left(\left.\mathbb{G}\right|_{S\mathcal{Y}}^{-1}\left.\mathbb{F}\right|_{S\mathcal{Y}}\right)^{2}=\mathds{1}\quad\text{and}\quad\mathrm{ker}\,\mathbb{F}=(T\mathcal{Y})^{\perp}\,,$
(2.30)

where $S\mathcal{Y}$ is the so-called screen distribution since we are working
in a non-Riemannian signature101010See [10, 41]. Since we work in a pseudo-
Riemannian manifold, one can no longer simply split the generalised tangent
space in the usual way
$\iota^{\ast}TM=T\mathcal{Y}\oplus(T\mathcal{Y})^{\perp}$. When $\mathcal{Y}$
is a coisotropic submanifold, i.e.
$T\mathcal{Y}\supset(T\mathcal{Y})^{\perp}$, however, one can show that there
still exists a canonical spitting, albeit by introducing a new object. Indeed,
in this case the tangent space has to be complemented by the so-called screen
distribution $S(T\mathcal{Y})$,
$T\mathcal{Y}=S(T\mathcal{Y})\oplus(T\mathcal{Y})^{\perp}$, where here the
direct sum is between transverse spaces. The (pull-back) of the manifold
$\mathbb{M}$ is then decomposed as [41]
$\displaystyle\iota^{\ast}T\mathbb{M}=T\mathcal{Y}\oplus
N\mathcal{Y}=S(T\mathcal{Y})\oplus(T\mathcal{Y})^{\perp}\oplus
N\mathcal{Y}\,,$ (2.31) where $N\mathcal{Y}$ is a complementary transverse
distribution to $T\mathcal{Y}$ in $\iota^{\ast}TM$.. This equation provides an
equivalent way to characterise topological defects at the level of the product
space $M\times\tilde{M}$ purely in terms of background and target space data.
In what follows we will refer to the topological defect condition given as in
eq. (2.30), as the “target space condition”. When the submanifold
$\mathcal{Y}$ corresponding to the defect is space-filling, i.e
$\mathcal{Y}=\mathcal{G}\times\tilde{\mathcal{G}}$ we have
$(T\mathcal{Y})^{\perp}=0$ and the condition simplifies to $\ker\mathbb{F}=0$
and $(\mathbb{G}^{-1}\mathbb{F})^{2}=\mathds{1}$. Note that in this case, this
also implies that $\mathbb{F}$ is a symplectic structure on the product space
$\mathcal{G}\times\tilde{\mathcal{G}}$. Before reviewing how this condition
can be expressed in generalised geometry, let us show again that the space-
filling defect with two-form field given by the Semenov-Tian-Shansky
symplectic form induces Poisson-Lie T-duality and is topological, however
using this time the target space condition above.

####  2.2.1 Poisson-Lie T-duality topological defect – revisited

Having introduced the target space condition, given in eq. (2.30), which
warrants that a defect pair $(\mathcal{Y},\mathcal{F})$ is topological, we
give an alternative way to show that the pair
$(\mathcal{Y},\mathcal{F})=(\mathbb{D},\omega_{\mathrm{STS}})$ defines in fact
a topological defect inducing Poisson-Lie T-duality. From the Semenov-Tian-
Shansky symplectic structure, using the expression given in eq. (2.10), we can
read off the matrix representation of $\mathrm{d}\mathbb{A}$. The associated
two-form field $\mathbb{F}$ is then given, as in section 2.1, by

$\displaystyle\mathbb{F}=-\mathrm{d}\mathbb{A}-\iota^{\ast}\mathbb{B}=\begin{pmatrix}-C\tilde{\Pi}-B&-C\\\
\tilde{C}&\tilde{C}\Pi+\tilde{B}\end{pmatrix}\,.$ (2.32)

We can now directly compute $(\mathbb{G}^{-1}\mathbb{F})^{2}$ using the
definition for $\mathbb{F}$ in eq. (2.29), yielding the (1,1)- and (1,2)-block
components respectively

$\displaystyle[(\mathbb{G}^{-1}\mathbb{F})^{2}]_{11}$
$\displaystyle=\left(G^{-1}(C\tilde{\Pi}+B)\right)^{2}+G^{-1}C\tilde{G}^{-1}\tilde{C}\,,$
$\displaystyle[(\mathbb{G}^{-1}\mathbb{F})^{2}]_{12}$
$\displaystyle=G^{-1}(C\tilde{\Pi}+B)G^{-1}C+G^{-1}C\tilde{G}^{-1}(\tilde{C}\Pi+\tilde{B})\,.$

It is instructive to first examine the simpler case of Abelian T-duality.
Specialising to Abelian T-duality means that we have that
$\Pi=0,\tilde{\Pi}=0$ and the transformation rules relating the background
fields and their T-dual are known to take on the form [42]

$\displaystyle\tilde{G}=(G-BG^{-1}G)\,,\quad\tilde{G}^{-1}\tilde{B}=-BG^{-1}\,.$
(2.33)

One can readily see that in this case we indeed have that
$(\mathbb{G}^{-1}\mathbb{F})^{2}=\mathds{1}$, and the defect describes Abelian
T-duality and is in addition topological. In a very similar way, using the
eqs. (B.4) derived from the Poisson-Lie canonical transformation in the
Appendix B relating the Poisson-Lie background fields $E=G+B$ and its dual
background field $\tilde{E}=\tilde{G}+\tilde{B}$, one can verify after a
little algebra that the defect $(\mathbb{D},\omega_{\mathrm{STS}})$ that
captures now generalised T-duality is again topological, consistent with what
was established already in section 2.1.1 at the level of the action.

####  2.2.2 Topological defects as Dirac structures

We now turn to the generalised geometry formulation of topological defects for
non-linear sigma-models as was originally given in [10]. Before reviewing the
generalised geometry condition for topological defect, we would like to
anticipate on our discussion about fusion in section 3 and introduce the
notion of Dirac structure in generalised geometry. To do so, we will explain
how D-branes and the worldvolume of topological defects can be described using
the same objects. This can be achieved precisely by promoting their
description to generalised geometry, or rather Dirac geometry. As we will
shortly review, Dirac geometry unifies symplectic structures, Poisson
structures and foliations in the framework of generalised geometry.

The crucial observation is to note that the worldvolume of topological defects
and D-branes are both determined not only by a submanifold but have to be
supplemented by a line-bundle with curvature $\mathcal{F}$ on its worldvolume.
As we will now review, this information is exactly encoded by Dirac
structures. A Dirac structure is a Lagrangian subspace $L$ of a Courant
algebroid111111We refer the reader to appendix C for definitions and a short
introduction to topics in generalised geometry relevant to this section. $E$
over a manifold $M$ that is involutive with respect to the Courant bracket on
$E$. The latter condition guarantees that the projection of the Dirac
structure onto the tangent space at each point of the manifold $M$ defines an
integrable distribution. That is, the Dirac structure $L$ on $M$ defines a
stratification of $M=\sum_{i}\mathcal{Y}_{i}$, where each leaf
$\mathcal{Y}_{i}$ inherit a two-form field $\mathcal{F}_{i}$, see e.g. [43].

A basic example is well-known. A Poisson-structure $\Pi$ on a manifold $M$
determines a foliation of $M$ into symplectic leaves on which the Poisson-
structure $\Pi$ is invertible. Each such leaf $\mathcal{Y}$ of the symplectic
foliation thus carries a two-form field $\omega=(\pi|_{\mathcal{Y}})^{-1}$;
the inverse of the Poisson structure restricted to that leaf. When the Poisson
structure is non-degenerate on $M$, the foliation consists of just one leaf,
the whole manifold $M$ and it is equivalent to a symplectic structure. The
Dirac structure in these examples coincide with the graph of Poisson structure
or the symplectic structure in the generalised tangent space, see appendix C.
A general Dirac structure $L$ can in fact always be identified with a
subbundle that corresponds to the graph of a two-form $\mathcal{F}$ whose
projection to the tangent bundle is an integrable distribution for a
submanifold $\mathcal{Y}$. That is we have that any Dirac structure $L$ can be
identified with a subbundle

$\displaystyle L(\mathcal{Y},\mathcal{F})=\left\\{(X,\xi):X\in
T\mathcal{Y},\xi\in T^{\ast}M\ \text{such that}\
{\mathcal{F}}X=\iota^{\ast}\xi\right\\}\,,$ (2.34)

where $\iota$ again denotes the inclusion map of $\mathcal{Y}$ into $M$.
Conversely, a couple $(\mathcal{Y},\mathcal{F})$ of a submanifold
$\mathcal{Y}\subset M$ and a two-form $\mathcal{F}$ can be shown to define a
Dirac structure $L$ of $E=\mathbb{T}M$.

We can now give some notable examples of Dirac structure describing D-branes.
In WZW models, (twisted) D-branes correspond to so-called Cartan-Dirac
structure carrying the GKHJW-structure [44] and whose leafs are conjugacy
classes. More generally, D-branes described by a particular subset of Dirac
structures have also been studied in [45, 46]. In [47], D-branes in Poisson-
Lie models were identified with the symplectic leaves of the Poisson structure
associated to the Lie group $M=\mathcal{G}$. In [48], by realising that
variational problems associated to two-dimensional sigma-models are associated
to exact (equivariant) Courant algebroids, boundary conditions translate into
the choice of a Dirac structure $L$ in that Courant algebroid $E$ over $M$.

For topological defects, it was shown in [10], although not explicitly coined
as such, that topological defects can be described as Dirac structures $L$ now
living in a Courant algebroid $E$ over the product space $M\times\tilde{M}$,
satisfying an additional compatibility condition. Let us review the argument.
Define a new “generalised metric” $\mathcal{R}_{\mathbb{G}}$ for the product
space $\mathbb{M}=M\times\tilde{M}$ given by

$\displaystyle\mathcal{R}_{\mathbb{G}}=\begin{pmatrix}0&\mathbb{G}^{-1}\\\
\mathbb{G}&0\end{pmatrix}\,,$ (2.35)

defining a generalised almost product structure, i.e an endomorphism
$\mathcal{R}_{\mathbb{G}}:T\mathbb{M}\oplus T^{\ast}\mathbb{M}\to
T\mathbb{M}\oplus T^{\ast}\mathbb{M}$ such that
$\mathcal{R}^{2}_{\mathbb{G}}=\mathrm{id}$. As we remarked earlier, the choice
of a foliation together with the associated two-form is equivalent to having a
Dirac structure of the form given in eq. (2.34). Then the pair
$(\mathcal{Y},\mathcal{F})$, consisting of a two-form $\mathcal{F}$ on
$\mathcal{Y}\subseteq M\times\tilde{M}$ a submanifold, is a topological defect
if and only if121212Its not hard to see that this equivalence: First let
$(\mathcal{Y},\mathcal{F})$ such that
$\mathcal{R}_{\mathbb{G}}(L(\mathcal{Y},\mathcal{F}))=L(\mathcal{Y},\mathcal{F})$
and let $(X,\xi)\in L(\mathcal{Y},\mathcal{F})$ arbitrary. Then
$\mathcal{F}X=\iota^{\ast}\xi$ and there exists some
$Z\in\iota^{\ast}T\mathbb{M}$ such that $\iota^{\ast}\xi=\mathbb{G}Z$.
Therefore
$\mathcal{R}_{\mathbb{G}}(X,\xi)=(\mathbb{G}^{-1}\mathbb{G}Z,\mathbb{G})=(Z,\mathbb{G}X)\in
L(\mathcal{Y},\mathcal{F})$, which implies $\mathcal{F}Z=\mathbb{G}X$ and
$Z\in T\mathcal{Y}$ and therefore $(\mathcal{Y},\mathcal{F})$ is a topological
bibrane. The converse direction requires bit more work, see [10] for details.
the following holds

$\mathcal{R}_{\mathbb{G}}(L(\mathcal{Y},\mathcal{F}))=L(\mathcal{Y},\mathcal{F})\,.$

That is topological defects correspond to Dirac structures of a Courant
algebroid over $M\times\tilde{M}$ that are stable under the product structure
$\mathcal{R}_{\mathbb{G}}$. This formulation is indeed reminiscent of how
D-brane compatible with extended supersymmetry [49], D-branes in para-
Hermitian geometry [50] or complex submanifolds in generalised complex
geometry [43] are characterised.

##  3 Worldvolume fusion of generalised T-duality defects

In this section we turn to the question of how the fusion of topological
defects can be understood as an operation on the target space. In contrast to
worldsheet fusion, very little is known on how to consistently define fusion
in target space. First insights came from studying the particular case of
fusing topological defects (or also called bibranes) in WZW models [8]. As can
be expected from the study of worldsheet fusion, target space fusion of two
topological defects or bibranes $(\mathcal{Y}_{i},\mathcal{F}_{i})_{i=1,2}$
results in single or possibly a superposition of multiple defects,
schematically

$\displaystyle(\mathcal{Y}_{1},\mathcal{F}_{1})\star(\mathcal{Y}_{2},\mathcal{F}_{2})=\sum_{\alpha}N_{12}{}^{\alpha}(\mathcal{Y}_{\alpha},\mathcal{F}_{\alpha})\,,$
(3.1)

where $N_{12}{}^{\alpha}$ are some constant coefficients for a collection of
bibranes $(\mathcal{Y}_{\alpha},\mathcal{F}_{\alpha})$. The transparent or
invisible defect, which separates $M$ with itself, acts as the unit element of
the fusion operation. Then a defect is invertible for the operation of fusion
when it can be fused with a second to yield the invisible defect. Remarkably,
it turns out that for WZW models the fusion operation was suggested to yield
the Verlinde algebra [8]. In general however, the fusion operation $\star$
will not define a group operation as some elements might fail to have an
inverse or the algebra might not even close.

In what follows, we first set out to understand the fusion properties of the
Poisson-Lie T-duality defect derived earlier. We provide a fusion realisation
of the generalised T-duality transformations of boundary conditions
characterised in terms of a gluing matrix. Subsequently, we will address the
effect of fusion at the level of the target space. Combining a Fourier-Mukai
like operation and the observation that Dirac structures both encode the
worldvolume data of defects and D-brane, we write down a target space fusion
operation encoding the D-brane resulting from the fusion of defect and
D-brane.

###  3.1 Lagrangian fusion and gluing conditions

In this section, we first examine the result of fusing a generalised T-duality
defect with a boundary condition at the Lagrangian level. We first remind the
reader of how boundary conditions can be encoded using a gluing matrix. We
then show how fusing the Poisson-Lie topological defect in this way correctly
predicts the transformation rules under Poisson-Lie T-duality for the boundary
conditions of open strings in terms of a gluing matrix.

####  3.1.1 Gluing matrices and Dirac structures

A convenient way to encode boundary conditions for open strings is by using
so-called gluing matrices [51]. The boundary condition for open strings are
then imposed by demanding that the left- and right-moving worldsheet fields
are related on the boundary by a gluing operator $R$:

$\displaystyle\partial_{-}X|_{\sigma=0,\pi}=R|_{\sigma=0,\pi}\cdot\partial_{+}X|_{\sigma=0,\pi}\,.$
(3.2)

The matrix $R$ cannot be arbitrary and has to satisfy a number of conditions
reflecting the consistency of the underlying boundary conditions. In
particular, conformal invariance requires the matrix $R$ to preserve the
metric. The Dirichlet directions, those normal to the D-brane, correspond to
the $-1$ eigenspace of the matrix $R$. These can be singled out by the
projector $RP_{\parallel}=P_{\parallel}R=-P_{\parallel}$. In addition, to
warrant for the existence of a smooth underlying submanifold modelling the
D-brane on the target space, the direction along the orthogonal directions
obtained from the projector $P_{\perp}=\mathds{1}-P_{\parallel}$, encoding the
Neumann directions, should generate an integrable distribution and a
corresponding foliation of the manifold $M$. On each such leaf
$\mathcal{Y}\subset M$ of the foliation, there exists a two-form $\mathcal{F}$
such that $\mathrm{d}\mathcal{F}=H|_{\mathcal{Y}}$.

Under a T-duality transformation, the gluing matrix $R$ capturing the specific
boundary conditions is mapped to a new, dual, gluing matrix $\tilde{R}$ for
the boundary conditions in the dual space. How the gluing matrix transforms
under the action of Poisson-Lie T-duality was derived in [52, 53]. We will
show in this section that the fusion operation of the Poisson-Lie T-duality
defect $(\mathbb{D},\omega_{\mathrm{STS}})$ with a boundary condition encoded
by a gluing matrix $R$ consistently yields the expected Poisson-Lie T-dual
gluing matrix.

For later use, let us however first review how boundary conditions arising
from this type of gluing conditions can be formulated within generalised
geometry. As pointed out in [49], the gluing condition can be reformulated in
an $O(d,d)$ covariant way. Introduce a “doubled” gluing matrix on the
generalised tangent space $\mathcal{R}:\mathbb{T}M\rightarrow\mathbb{T}M$
which is taken to be an $O(d,d)$ element. This $O(d,d)$-valued gluing matrix
$\mathcal{R}$ is defined in terms of the projectors induced by the gluing
matrix $R$ acting on the tangent space via

$\displaystyle\mathcal{R}=\begin{pmatrix}P_{\Delta}&0\\\
\mathcal{F}P_{\Delta}+P_{\Delta}^{T}\mathcal{F}&-P_{\Delta}^{T}\end{pmatrix}\,,$

where $P_{\Delta}=(P_{\parallel}-P_{\perp})$ and the two-form
$\mathrm{d}\mathcal{F}=H|_{\mathcal{Y}}$ on a leaf $\mathcal{Y}$ of the
foliation. This gluing matrix is unipotent $\mathcal{R}^{2}=1$ and has to be
compatible with the generalised metric. From the consistency condition of the
conventional gluing matrix $R$ above, one should have that the $+1$-eigenspace
generate the integrable distribution of the worldvolume of the D-brane when
restricted to the tangent space. In particular, the projector
$\tfrac{1}{2}(1-\mathcal{R})$ can be used to define a Dirac structure
$L(\mathcal{Y},\mathcal{F})\subset\mathbb{T}M$ given by the expression

$\displaystyle
L(\mathcal{Y},\mathcal{F})=\\{\tfrac{1}{2}(1-\mathcal{R})(X+\xi)=(X+\xi)\mid
X+\xi\in T\mathcal{D}\oplus
T^{\ast}M|_{\mathcal{D}}\,,\;\xi|_{\mathcal{D}}=\iota_{X}\mathcal{F}\\}\,.$
(3.3)

That is, the boundary condition characterised by the gluing matrix in eq.
(3.2) can be, on the target space, equivalently be described in terms of the
Dirac structure in eq. (3.3).

####  3.1.2 Boundaries under Poisson-Lie T-duality via fusion

We will now show how to obtain the Poisson-Lie T-dual boundary conditions by
fusing boundary condition of the form given in eq. (3.2) with the topological
defect encoding Poisson-Lie T-duality that was put forward in the first
section. The set-up is the same as in section 2.1.1, with the additional
assumption that the system on, say the left, is subject to boundary condition
described by the gluing condition as in eq. (3.2) for a given gluing matrix
$R$131313Note that this $R$ should not be confused with the right-invariant
Maurer-Cartan of $\mathcal{G}$. Since in this section, we will solely work
with the left-invariant form $L=l_{i}dX^{i}$, there should be no source of
confusion.. Since the Poisson-Lie T-duality defect is topological we can
(formally) move the defect line from its initial position at $\sigma=0$ on the
worldsheet to the boundary of the left system. At the boundary we now both
have to impose the boundary conditions and the defect equations. Remembering
the equations of motion taking into account the defect contribution given in
eq. (2.12), these relations however have to be supplemented with the boundary
conditions encoded by the gluing matrix, i.e. $L_{-}=RL_{+}$. Plugging this
gluing condition into the first equation of motion in eq. (2.12), leads to

$\displaystyle\tilde{l}_{bj}\partial_{\tau}\tilde{X}^{j}=(C^{-1})_{b}^{\
a}E^{R}_{ac}l^{c}_{j}\partial
X^{j}-\tilde{\Pi}_{bc}[\tilde{g}]l^{c}_{j}\partial_{\tau}X^{j}\,,$

where we have defined the combination $E^{R}\equiv E^{T}-ER$. Substituting
this last relation into the equation of motion of the dual model in eq. (2.12)
and taking into account the relation $\Pi C^{-1}=\tilde{C}^{-1}\tilde{\Pi}$,
leads to

$\displaystyle l^{a}_{j}\partial_{\tau}X^{j}$
$\displaystyle=\tilde{E}^{ba}\tilde{l}_{bj}\partial\tilde{X}^{j}-\tilde{E}^{ab}\tilde{l}_{bj}\bar{\partial}\tilde{X}^{j}-\Pi^{ab}E^{R}_{bc}l^{c}_{j}\partial
X^{j}\,.$ (3.4)

Using the expression $l^{a}_{j}\partial_{\tau}X^{j}=L_{+}+L_{-}$ and enforcing
that the Maurer-Cartan forms on $\mathcal{G}$ and $\tilde{\mathcal{G}}$ are
related by the canonical transformation for Poisson-Lie T-duality given in eq.
(B.1) reviewed in appendix B, one obtains that the dual boundary conditions
are described by the “dual” gluing matrix $\tilde{R}$ given by

$\displaystyle\tilde{L}_{-}=\tilde{R}\tilde{L}_{+}\,,\quad\tilde{R}=-\tilde{E}^{-1}E_{-}^{-1}ER(E^{T})^{-1}E_{0}^{T}\tilde{E}^{T}\,.$
(3.5)

This expression for the dual gluing matrix is precisely the expression
obtained in [52].

###  3.2 Dirac geometry fusion

As was already briefly mentioned in section 2.1.4, it is natural to postulate
that the fusion at the level of the target space emulates that of the integral
transformation of the defect in the worldsheet picture. The idea that a
Fourier-Mukai like operation controls the fusion of target space defects was
already put forward for the fusion of topological defects (or bibranes) and
boundary conditions in WZW models by [8]. The authors proposed that the
submanifold that results from the fusion operation between a D-brane and a
topological defect separating two WZW models on a Lie group $\mathcal{G}$ is
given by

$\displaystyle\mathcal{Y}_{B}\star\mathcal{Y}_{D}\equiv
p_{1}\left(\mathcal{Y}_{B}\cap p_{2}^{-1}(\mathcal{Y}_{D})\right)\,,$ (3.6)

where $\mathcal{Y}_{B}$ is the worldvolume of the bibrane or topological
defect in the product space $M_{1}\times M_{2}=\mathcal{G}\times\mathcal{G}$,
$\mathcal{Y}_{D}$ is the worldvolume of the D-brane in $M_{2}=\mathcal{G}$ and
$p_{i}$ are the usual projections $p_{i}:M_{1}\times M_{2}\rightarrow M_{i}$.
Note that this time the role of the kernel for this Fourier-Mukai inspired
operation is played by the worldvolume of the topological defect. In general
however, the object resulting from the formula in eq. (3.6) is ill-defined:
the operation in eq. (2.28) is in general just a subset and not a submanifold.
A second drawback is that this proposal for the fusion operation does not
include an explicit expression for the two-form $\mathcal{F}$ living on the
fused boundary condition.

In the present article we will take, besides the Fourier-Mukai operation, a
second guiding principle to propose an Ansatz for the target space fusion. In
previous sections, we stressed that the target space data
$(\mathcal{Y},\mathcal{F})$ capturing both D-branes and defects in some
manifold $M$ can be encoded in terms of Dirac structures
$L=L(\mathcal{Y},\mathcal{F})\subset M$. This observation strongly suggest
that generalised geometry is the right setting to define the operation of
fusion at the level of the target space. Using the Fourier-Mukai
transformation as our blue print, we will vindicate in this section a fusion
operation of a topological defect with boundary condition using the framework
of generalised geometry.

In section 2.1.4, we reviewed the relation between the Fourier-Mukai type
transformation and the fusion of topological defects with boundary conditions.
We would like to translate this transformation as a pull-back of the boundary
condition to the product space, its “convolution” with the “kernel” of the
topological defect and subsequence reduction to the second space. In terms of
generalised geometry that means we have to transport Dirac structures from
Courant algebroids over the different manifolds. In the present context it
will be relevant to restrict to a particular class of Courant algebroids to
obtain a well-defined notion of fusion. The following discussion relies on the
results of [48, 54, 55] on Courant algebroids and the related reductions.

We will first assume that all three Courant algebroids $E_{i}$ over $M_{i}$
for $i=1,2$ and $\mathbb{E}$ over $\mathbb{M}=M_{1}\times M_{2}$ are so-called
exact Courant algebroids. This class was discovered and classified by Ševera
in [56, 57]. These Courant algebroids have the distinguishing property that
they are locally isomorphic to the standard Courant algebroid $TM\oplus
T^{\ast}M$, if one allows for the Courant bracket to be twisted by a closed
three-form $H$. Exact Courant algebroids are then uniquely determined by a
class in $H^{3}_{\mathrm{d}R}(M,\mathbb{R})$. Making this choice essentially
specialises the theories on both sides of the defect to be determined by two-
dimensional sigma-models. Indeed one can show that an exact $H$-twisted
Courant algebroid $E$ over a manifold $M$ is equivalent to specifying the
phase space of a two-dimensional sigma-model where the Kalb-Ramond field has
curvature $H$, see e.g. [58]. Schematically we now have a Fourier-Mukai-like
diagram transporting D-branes and defects as Dirac structure from one Courant
algebroid to the other

$\mathbb{E}\cong\mathbb{T}(M_{1}\times
M_{2})_{H_{1}-H_{2}}$$E_{1}\cong(\mathbb{T}M_{1})_{H_{1}}$$E_{2}\cong(\mathbb{T}M_{2})_{H_{2}}$
(3.7)

where the right, respectively left, arrows are understood to be the pull-back,
resp. the reduction of the Dirac structure via the associated pull-
back/reduction of the Courant algebroids. To warrant the reduction process to
be well-defined, we will in addition demand that these exact Courant
algebroids admit the action of a Lie group $\mathcal{G}$ that is free and
transitive such that the associated 1st Pontraygin class vanishes. Such
Courant algebroids are known as $\mathcal{G}$-equivariant (exact) Courant
algebroids. In addition, these assumptions guarantee that any Dirac structure
in the respective Courant algebroids remain Dirac in the new pull-
backed/reduced Courant algebroids [48].

We are left with the task of specifying the “convolution” operation, and the
relevant “kernel”, taking place in between the pull-back to the product space
and the reduction in eq. (3.7). A simple Ansatz is a natural product on the
space of Dirac structures $\mathrm{Dir}(\mathbb{D})$, which we will denote by
$\circledast$ to distinguish it from the fusion operation $\star$. As we will
see that the product $\circledast$ on the space of Dirac structure over $M$,
automatically reproduces the desired submanifold resulting from fusion
proposed by [8] and given in eq. (3.6). Indeed, take two Dirac structures
$L_{1}\equiv L(\mathcal{Y}_{1},\omega_{1}),L_{2}\equiv
L(\mathcal{Y}_{2},\omega_{2})\subset\mathbb{T}M$, one can then define a
pointwise product141414Dually, we could represent a Dirac structure
$L=L(\mathcal{Y},\pi)$ where
$L(\mathcal{Y},\pi)=\\{X+\pi^{\sharp}(\eta)+\eta:\eta\in
T\mathcal{Y}^{\circ}\,,\;X\in T\mathcal{Y}\\}$, where the annihilator space if
denoted by a $\circ$. For which we can in turn define a product
$L_{1}\circledast L_{2}=\\{X_{1}+X_{2}+\eta:X_{1}+\eta\in
L_{1}\,,X_{2}+\eta\in L_{2}\\}\,.$

$\displaystyle(L_{1}\circledast L_{2})_{\mathbb{g}}\equiv
L_{1,\mathbb{g}}\circledast
L_{2,\mathbb{g}}\subset\mathbb{T}_{p}\mathbb{M}\,,\quad\forall\mathbb{g}\in\mathbb{M}\,,$

given explicitly by

$\displaystyle L(\mathcal{Y}_{1},\omega_{1})\circledast
L(\mathcal{Y}_{2},\omega_{2})$ $\displaystyle=\\{X+\eta+\nu:X+\eta\in
L_{1},X+\nu\in L_{2}\\}$
$\displaystyle=L(\mathcal{Y}\subset\mathcal{Y}_{1}\cap\mathcal{Y}_{2},\omega_{1}|_{\mathcal{Y}}+\omega_{2}|_{\mathcal{Y}})\,.$

If the resulting subbundle is smooth, the product $L_{1}\circledast L_{2}$ is
also a Dirac structure. See e.g. [59] for more details. In addition, when
$\mathcal{Y}_{1}\cap\mathcal{Y}_{2}$ is a connected submanifold, we have that
$\mathcal{Y}=\mathcal{Y}_{1}\cap\mathcal{Y}_{2}$. In general this intersection
will have different connected components, possibly of different dimensions.
Note finally that this Dirac structure product $\circledast$ also naturally
realises B-field shifts, since for a closed two-form $B$, we have that
$L(\mathbb{D}_{\mathrm{diag}},B)\circledast
L(\mathcal{Y}_{1},\mathcal{F})=L(\mathcal{Y}_{1},\mathcal{F}+B)$.

Summarising, we have the following diagram

$\bar{L}(\mathcal{Y}_{B},\mathcal{F}_{B})\circledast\iota_{\ast}L(\mathcal{Y}_{1},\mathcal{F}_{1})$$L(\mathcal{Y}_{\mathrm{fus}},\mathcal{F}_{\mathrm{fus}})$$L(\mathcal{Y}_{1},\mathcal{F}_{1})$
(3.8)

This proposed fusion operation really contains two pieces of data: on the one
hand the resulting submanifold, and on the other the expression for the
induced two-form field

$\displaystyle\mathcal{Y}_{\mathrm{fus}}\equiv\mathcal{Y}_{B}\star\mathcal{Y}_{1}=p(\mathcal{Y}_{B}\cap{\iota}(\mathcal{Y}_{1}))\subset
M_{2}\,,\quad\mathcal{F}_{\mathrm{fus}}\equiv(p)_{\ast}\left(\mathcal{F}_{B}-{\iota}^{\ast}(\mathcal{F}_{1})\right)\in\wedge^{2}M_{2}\,,$

where $(\mathcal{Y}_{B},\mathcal{F}_{B})$ is the target space data
characterising the topological defect in $M_{1}\times M_{2}$ and
$(\mathcal{Y}_{1},\mathcal{F}_{1})$ the world-volume data of the D-brane in
$M_{1}$. The notation $\bar{L}$, denoted the Dirac structure with minus the
two-form, i.e. $\bar{L}(\mathcal{Y}_{B},\mathcal{F}_{B})\equiv
L(\mathcal{Y}_{B},-\mathcal{F}_{B})$. Looking now at the two-form we have that
$\mathrm{d}\mathcal{F}_{B}=H_{1}|_{\mathrm{def.}}-H_{2}|_{\mathrm{def.}}$,
where we have restricted the three-forms to the defect location, for the two-
form on the worldvolume of the topological defect. First note that this
relation reproduces the fused submanifold given in eq. (3.6) of [8]. In
addition, by construction we have that
$\mathrm{d}\mathcal{F}_{\mathrm{fus}}=H_{2}$.

Let us conclude this section with connecting back to the Poisson-Lie T-duality
topological defect. Defining the fusion in this way incidentally matches also
exactly the procedure described in [60] of how Dirac structure transform under
T-duality. To see this in more detail, we have the Poisson-Lie T-duality
defects $(\mathcal{Y}_{B},\mathcal{F}_{B})=(\mathbb{D},\omega_{\mathrm{STS}})$
and we take a D-brane that is a symplectic leaf associated to the Lie group
$\mathcal{G}$ [47]. To embed a symplectic leaf $\mathcal{Y}\subset\mathcal{G}$
in the Drinfel’d double, we can simply multiply every point
$g\in\mathcal{Y}\subset\mathcal{G}$ with a constant element
$\mathbb{g}\in\mathbb{D}$ of the Drinfel’d double. The intersection for this
particular case is trivial:
$\mathbb{D}\cap\iota\mathcal{Y}=\iota\mathcal{Y}\subset\mathbb{D}$. The last
step, the projection onto $\tilde{\mathcal{G}}$, requires one to decompose
every point on the embedded surface $\iota\mathcal{Y}$

$\displaystyle
p_{1}(\iota\mathcal{Y})=\\{\tilde{h}\mid\mathbb{g}g=\tilde{h}h\;\text{for}\;g\in\mathcal{Y}\;\text{and}\;h\in\mathcal{G},\tilde{h}\in\tilde{\mathcal{G}}\\}\,,$

yielding $\mathcal{G}$-dressing orbits or equivalently the symplectic leaves
in the dual Lie group $\tilde{\mathcal{G}}$ for the Poisson structure
$\tilde{\Pi}$ associated to the double $\mathbb{D}$, in accordance with [47].
In the special case where element $\mathbb{g}=e$ is the identity element, the
leaf is dual to a single point. Finally, one can easily check that
$\mathcal{F}_{\mathrm{fus}}=(\tilde{\Pi}|_{p_{1}(\iota\mathcal{Y})})^{-1}$
which is well-defined since $p_{1}(\iota\mathcal{Y})$ is precisely a
symplectic leaf of the Poisson-Lie structure $\tilde{\Pi}$ on
$\tilde{\mathcal{G}}$.

##  4 Discussion

In this article, we showed how the most general notion of T-duality, despite
its potential disparities with more conventional notions of T-duality, can in
fact be encoded as a defect separating Poisson-Lie T-dual models. Using a
target space formulation, we have shown, by direct construction, that Poisson-
Lie T-duality can be understood as a space-filling topological defect. This
Poisson-Lie T-duality defect is characterised by the natural symplectic
structure on the Drinfel’d double associated to the Poisson-Lie T-duality
pair, the so-called Semenov-Tian-Shanshy symplectic structure. In addition, we
argued, along the same line as [11], that the Fourier-Mukai transform
associated to Poisson-Lie T-duality should have the Semenov-Tian-Shansky
symplectic structure as kernel. This nicely agrees with the same result
obtained by [27] from a very different perspective, that of QP-manifolds. We
subsequently demonstrated how the Poisson-Lie T-duality defect consistently
yields the correct transformation rules for the boundary condition given by a
gluing matrix under generalised T-duality.

The final part of this paper took a more general look at the target space
realisation of fusion, without specialising to the above constructed
generalised T-duality defect. Although very well-understood at the level of
the worldsheet, the target space realisation of fusion is mired by a veil of
mystery and remains largely ill-understood. A crucial step forwards was made
in [8, 35], by realising that the fusion process should be similar in spirit
to the Fourier-Mukai transform. In the present article, we put forward that a
second crucial ingredient is that the worldvolumes of both defects and
D-branes can be described within generalised geometry via Dirac structures.
Exploiting both insights, we proposed a well-defined operation for the
worldvolume fusion using Dirac geometry. The resulting object is always a
submanifold carrying a well-defined two-form field whose exterior derivative
matches the three-form flux.

Let us conclude with some future directions and open questions. It is tempting
to believe that, in analogy with topological defects WZW models [8], the
worldvolume of topological defects separating Poisson-Lie symmetry models
could be related, not as for WZW models to conjugacy classes, but to dressing
orbits of the associated Drinfel’d double. Dressing orbits can be seen as a
non-linear generalisation of conjugacy classes associated to a natural action
of Poisson-Lie dual pairs. The worldvolume of these bibranes would then be
some sort of “bi-dressing orbits”.

Integrable defects form a special subclass of topological defects. This then
begs the question of whether the integrable defects in Yang-Baxter models
constructed in [61] admit a notion of fusion. The fusion of integrable defects
with boundary conditions was e.g. already considered in [62] for Sinh-Gordon
and Lee-Yang models. On the other hand, it would be interesting to
specialising the duality defects and the associated fusion to particular
instances of models which are related by Poisson-Lie T-duality, such as the
$\eta$-or $\lambda$-deformations [63, 64, 65] and their boundary conditions
[66].

In view of the recent formulation of Poisson-Lie T-duality in terms of QP
manifolds, let us comment that Dirac structures on generalised geometry can
straightforwardly be lifted to QP geometry. In the supergeometric setting the
Courant algebroid becomes a degree two symplectic N-manifold with Courant
bracket and anchor encoded by an Hamiltonian vector field $\Theta$, i.e.
$\\{\Theta,\Theta\\}=0$, and Dirac structures $L$ of $E$ become Lagrangian
submanifolds $\mathcal{L}$ along which $\Theta|_{\mathcal{L}}$ is constant
[67].

Figure 1: When an algebra admits two Drinfel’d double one can fuse the
associated Poisson-Lie T-duality defects $\mathrm{PL}_{i}$ separating the
sigma-models $\Sigma\rightarrow\mathcal{G}$ with
$\tilde{\Sigma}_{i}\rightarrow\tilde{\mathcal{G}}_{i}$, for $i=1,2$.

Finally, in the present work, the resulting operation of fusion at the level
of the target space was restricted to fusion processes between a topological
defect and a boundary condition or D-brane. In fact, the fusion of two
topological defects associated to generalised T-dualities appear to be
somewhat more subtle and has to be handled with care. To see this, remember
that a given Lie group $\mathcal{G}$ might admit two different Poisson-Lie
structures $\mathrm{PL}_{i}$, $i=1,2$, and thus two Poisson-Lie dual groups
$\tilde{\mathcal{G}}_{i}$ with corresponding doubles $\mathbb{D}_{i}$, see
fig. 1. Assume that we place the worldsheets of Poisson-Lie symmetry on
$\mathcal{G}$ between those with target spaces $\tilde{\mathcal{G}}_{1}$ (on
the left) and $\tilde{\mathcal{G}}_{2}$ (on the right), with the worldsheets
divided by the respective generalised T-duality defects. Being topological, we
can fuse these two topological defects together. More often than not however,
the resulting product space
$\tilde{\mathcal{G}}_{1}\times\tilde{\mathcal{G}}_{2}$ will not be a Drinfel’d
double and this defect will not encode a (single) Poisson-Lie T-duality. One
could however imagine that the fused defect encodes a complicated solution
generating transformation that results from applying the two Poisson-Lie
T-duality after one another. It has been anticipated in [6] (see also [5])
that fusion of conformal defects could lead to a solution generating algebra,
similar to the role of Ehlers-Geroch transformations in general relativity.
This simple reasoning seems to suggest that studying the fusion algebra
generated by the Poisson-Lie defect on the target space can potentially enable
us to pinpoint crucial differences between Abelian and Poisson-Lie T-duality.

#### Acknowledgments

We would like to thank Alex Arvanitakis, Chris Blair, Falk Hassler, Dan
Thompson, Fridrich Valach, Ilka Brunner, David Osten and Pavol Ševera for
interesting discussions and Dieter Lüst and Alessandra Gnecchi for support and
encouragement. We would also like to thank the organisers and the participants
of the “Integrability, dualities and deformations” conference at Humboldt
Universität zu Berlin for the many interesting discussions.

##  Appendix A Drinfel’d double conventions and the $\omega_{STS}$

In this appendix, we set our notation used in the main text as well as sketch
the derivation of an alternative expression for the Semenov-Tian-Shansky
symplectic form.

A Drinfel’d double $\mathbb{D}$ is an even-dimensional Lie group with Lie
algebra $\mathfrak{d}$ (also referred to as the Drinfel’d double) equipped
with a non-degenerate symmetric bilinear form
$\langle\cdot,\cdot\rangle_{\mathfrak{d}}$ such that the algebra can be
written as $\mathfrak{d}=\mathfrak{g}\oplus\tilde{\mathfrak{g}}$, with
$\mathfrak{g}$ and $\tilde{\mathfrak{g}}$ maximally isotropic subalgebras with
respect to $\langle\cdot,\cdot\rangle_{\mathfrak{d}}$. The generators of the
Drinfel’d double $\mathfrak{d}$ are denoted by
$\mathbb{T}_{A}=(T_{a},\tilde{T}^{a})$ with $T_{a},\tilde{T}^{a}$ the
generators of the maximally isotropic subalgebras
$\mathfrak{g},\tilde{\mathfrak{g}}$. The triple
$(\mathfrak{d},\mathfrak{g},\tilde{\mathfrak{g}})$ corresponding to this three
quantities is called a Manin triple. The isotropy condition means that

$\displaystyle\langle
T_{a},T_{b}\rangle_{\mathfrak{d}}=0\,,\quad\langle\tilde{T}^{a},\tilde{T}^{b}\rangle_{\mathfrak{d}}=0\,,\quad\langle
T_{a},\tilde{T}^{b}\rangle_{\mathfrak{d}}=\delta_{a}^{b}\,,$

and the maximality conditions requires the Lie algebra to be half of the
dimension of the corresponding Drinfel’d double, i.e.
$\mathrm{dim}\,\mathfrak{d}=2\,\mathrm{dim}\,\mathfrak{g}=2\,\mathrm{dim}\,\tilde{\mathfrak{g}}$.
The Lie groups $\mathcal{G}$ and $\tilde{\mathcal{G}}$ obtained after
exponentiating the Lie algebras $\mathfrak{g}$ and $\tilde{\mathfrak{g}}$
carry a natural Poisson-Lie structure $\Pi$ and $\tilde{\Pi}$ compatible with
the group multiplication. For group elements
$g\in\mathcal{G}=\exp\mathfrak{g},\tilde{g}\in\tilde{\mathcal{G}}=\exp\tilde{\mathfrak{g}}$,
we define the adjoint action matrices by

$\displaystyle g^{-1}T_{a}g$ $\displaystyle=a_{a}^{\ b}[g]T_{b}\,,\quad
g^{-1}\tilde{T}^{b}g=b^{ab}[g]T_{b}+(a[g]^{-1})_{b}^{\ a}\tilde{T}^{b}\,,$
$\displaystyle\tilde{g}^{-1}\tilde{T}^{a}\tilde{g}$
$\displaystyle=\tilde{a}^{a}_{\
b}[\tilde{g}]\tilde{T}^{b}\,,\quad\tilde{g}^{-1}T_{a}\tilde{g}=\tilde{b}_{ab}[\tilde{g}]\tilde{T}^{b}+(\tilde{a}[\tilde{g}]^{-1})^{b}_{\
a}T_{b}\,.$

In terms of these adjoint action matrices, the Poisson-Lie structures can
easily be computed using

$\displaystyle\Pi^{ab}[g]=b^{ca}[g]a_{c}^{\
b}[g]\,,\quad\tilde{\Pi}_{ab}[\tilde{g}]=\tilde{b}_{ca}[\tilde{g}]\tilde{a}^{c}_{\
b}[\tilde{g}]\,.$ (A.1)

One can show by direct computation that, as should be for Poisson structures,
these bivectors are antisymmetric $\Pi^{T}[g]=-\Pi[g]$ and idem for the tilde
object.

We show now how one can get the particular form of the Semenov-Tian-Shansky
symplectic form $\omega_{\mathrm{STS}}$ we use in order to define the defect
from its standard expression given in [68, 32]

$\displaystyle
2\omega_{\mathrm{STS}}=r^{a}(g)\wedge\tilde{l}_{a}(\tilde{h})-\tilde{r}_{a}(\tilde{g})\wedge
l^{a}(h)\,.$ (A.2)

The expression for the Semenov-Tian-Shansky we need in the main text closely
resembles an expression first derived in [27]. In order to do so we first need
to collect a number of identities relating the different decompositions of the
group element on the Drinfel’d double
$\mathbb{D}=\mathcal{G}\tilde{\mathcal{G}}$. Recall that an element
$\mathbb{g}$ on the double $\mathbb{D}$ can be written in two different ways

$\displaystyle\mathbb{g}=\tilde{h}g\quad\text{or}\quad\mathbb{g}=h\tilde{g}\,,\quad\text{with}\;\,\,g,h\in\mathcal{G}\,,\
\tilde{g},\tilde{h}\in\tilde{\mathcal{G}}\,.$

First we compare the adjoint action on the double in these two different
parameterisations

$\displaystyle\mathbb{a}[\mathbb{g}=\tilde{h}g]T_{a}$
$\displaystyle=(\tilde{h}g)^{-1}T_{a}\tilde{h}g=g^{-1}\left(\tilde{b}_{ab}[\tilde{h}]\tilde{T}^{b}+(\tilde{a}^{-1})^{b}_{\
a}[\tilde{h}]T_{b}\right)g$
$\displaystyle=\tilde{b}_{ab}[\tilde{h}]\left(b^{bc}[g]T_{c}+(a^{-1})_{c}^{\
b}[g]\tilde{T}^{c}\right)+(\tilde{a}^{-1})^{b}_{\ a}[g]a_{b}^{\ c}[g]T_{c}\,,$

and for the alternative decomposition of the Drinfel’d double element

$\displaystyle\mathbb{a}[\mathbb{g}=h\tilde{g}]T_{a}$ $\displaystyle=a_{a}^{\
b}[h]\left(\tilde{b}_{bc}[\tilde{g}]\tilde{T}^{c}+(\tilde{a}^{-1})^{c}_{\
}[\tilde{g}]T_{c}\right)\,.$

Collecting the terms in valued in $\mathfrak{g}$, we get the identifications

$\displaystyle a_{a}^{\ b}[h](\tilde{a}^{-1})^{c}_{\ b}[\tilde{g}]$
$\displaystyle=\tilde{b}_{ab}[\tilde{h}]b^{bc}[g]+(\tilde{a}^{-1})^{b}_{\
a}[\tilde{h}]a_{b}^{\ c}[g]$ (A.3)
$\displaystyle\tilde{b}_{ab}[\tilde{h}](a^{-1})_{c}^{\ b}[g]$
$\displaystyle=a_{a}^{\ b}[h]\tilde{b}_{bc}[\tilde{g}]\,.$ (A.4)

Doing the same for the terms in valued in the dual algebra $\mathfrak{g}$,
leads to

$\displaystyle\tilde{a}^{a}_{\ b}[\tilde{h}](a^{-1})^{b}_{\ c}[g]$
$\displaystyle=b^{ab}[h]\tilde{b}_{bc}[\tilde{g}]+(a^{-1})_{b}^{\
a}[h]\tilde{a}_{c}^{\ b}[\tilde{g}]$ (A.5) $\displaystyle\tilde{a}^{a}_{\
b}[\tilde{h}]b^{bc}[g]$ $\displaystyle=b^{ab}[h](\tilde{a}^{-1})^{c}_{\
b}[\tilde{g}]\,.$ (A.6)

We proceed analogously with the invariant forms
$\mathbb{L}=\mathbb{g}^{-1}\mathrm{d}\mathbb{g}$ and
$\mathbb{R}=\mathrm{d}\mathbb{g}\mathbb{g}^{-1}$.

$\displaystyle\mathbb{L}[\mathbb{g}=\tilde{h}g]$
$\displaystyle=g^{-1}\tilde{l}(\tilde{h})g+l(g)=\tilde{l}_{a}(\tilde{h})(b^{ab}[g]T_{b}+(a^{-1})_{b}^{\
a}[g]\tilde{T}^{b})+l^{b}(g)T_{b}$
$\displaystyle\mathbb{L}[\mathbb{g}=h\tilde{g}]$
$\displaystyle=\tilde{g}^{-1}l(h)\tilde{g}+\tilde{l}(\tilde{g})=l^{a}(h)(\tilde{b}_{ab}[\tilde{g}]\tilde{T}^{b}+(a^{-1})^{b}_{\
a}[\tilde{g}]T_{b}+\tilde{l}_{b}(\tilde{g})\tilde{T}^{b}$

Therefore we have

$\displaystyle\tilde{l}_{a}(\tilde{h})b^{ab}[g]+l^{a}(g)$
$\displaystyle=l^{a}(h)(a^{-1})[\tilde{g}]\,\quad\tilde{l}_{a}(\tilde{h})(a^{-1})_{b}^{\
a}[g]=l^{a}(h)\tilde{b}_{ab}[\tilde{g}]+\tilde{l}_{a}(\tilde{g})$

as well as from $\mathbb{R}$

$\displaystyle r^{b}(h)$ $\displaystyle=r^{a}(g)\tilde{a}^{b}_{\
a}[\tilde{h}]-\tilde{r}_{a}(\tilde{g})b^{ab}[h^{-1}]\,,\quad\tilde{r}_{b}(\tilde{h})=\tilde{r}_{a}(\tilde{g})a_{b}^{\
a}[h]-r^{a}(g)\tilde{b}_{ab}[\tilde{h}^{-1}]\,,$

which implies the following relation between left- and right-invariant Maurer-
Cartan fields

$\displaystyle l^{a}(h)$ $\displaystyle=r^{d}(g)\tilde{a}^{b}_{\
d}[\tilde{h}]a_{b}^{\ a}[h]-\tilde{r}_{d}(\tilde{g})b^{db}[h^{-1}]a_{b}^{\
a}[h]\,,$ (A.7) $\displaystyle\tilde{l}_{a}(\tilde{h})$
$\displaystyle=\tilde{r}_{d}(\tilde{g})a_{b}^{\ d}[h]\tilde{a}^{b}_{\
a}[\tilde{h}]-r^{d}(g)\tilde{b}_{db}[\tilde{h}^{-1}]\tilde{a}^{b}_{\
a}[\tilde{h}]\,.$ (A.8)

Lastly we also need to find an expression for the matrix $C$ featured in the
Poisson-Lie T-duality transformation rules. In order to do so, we start from
equation (A.3) and apply eq. (A.4) as well as $\Pi^{ab}[g]=b^{ca}[g]a_{c}^{\
b}[g],\tilde{\Pi}_{ab}[\tilde{g}]=\tilde{b}_{ca}[\tilde{g}]\tilde{a}^{c}_{\
b}[\tilde{g}]$ to get

$\displaystyle a_{a}^{\ b}[h](\tilde{a}^{-1})^{c}_{\ b}[\tilde{g}]$
$\displaystyle=\tilde{b}_{ab}[\tilde{h}]b^{bc}[g]+(\tilde{a}^{-1})^{b}_{\
a}[\tilde{h}]a_{b}^{\ c}[g]\,.$

This relation then allows one to write the following expression for the matrix
$C$ in terms of the adjoint maps

$\displaystyle C_{c}^{\
h}\equiv(\delta^{c}_{h}-\tilde{\Pi}_{hl}[\tilde{g}]\Pi^{lc}[g])^{-1}$
$\displaystyle=(a^{-1})_{c}^{\ b}[g](\tilde{a})^{a}_{\ b}[\tilde{h}]a_{a}^{\
e}[h](\tilde{a}^{-1})^{h}_{\ e}[\tilde{g}]\,.$ (A.9)

Having collected all the necessary expressions at hand, we can finally compute
the relevant expression for the Semenov-Tian-Shansky symplectic form. Starting
from the defining expression for the symplectic form in equation (A.2) and
recalling that $\tilde{r}_{a}=\tilde{l}_{b}(\tilde{a}^{-1})^{b}_{\
a}[\tilde{g}],r^{a}=l^{b}(a^{-1})_{b}^{\ a}$, we can consecutively rewrite the
symplectic structure as

$\displaystyle 2\omega_{STS}$ $\displaystyle=l^{b}(g)(a^{-1})_{b}^{\
a}[g]\wedge\left(\tilde{r}_{d}(\tilde{g})a_{c}^{\ d}[h]\tilde{a}^{c}_{\
a}[\tilde{h}]-r^{d}(g)\tilde{b}^{dc}[\tilde{h}^{-1}]\tilde{a}^{c}_{\
a}[\tilde{h}]\right)$
$\displaystyle\quad-\tilde{l}_{b}(\tilde{g})(\tilde{a}^{-1})^{b}_{\
a}[\tilde{g}]\wedge\left(r^{d}(g)\tilde{a}^{c}_{\ d}[\tilde{h}]a_{c}^{\
a}[h]-\tilde{r}_{d}(\tilde{g})b^{dc}[h^{-1}]a_{c}^{\ a}[h]\right)$
$\displaystyle=l^{b}(g)\wedge\left((a^{-1})_{b}^{\ a}[g]\tilde{a}^{c}_{\
a}[\tilde{h}]a_{c}^{\ d}[h](\tilde{a}^{-1})^{e}_{\
d}[\tilde{g}]\right)\tilde{l}_{e}(\tilde{g})$
$\displaystyle\quad-l^{b}(g)\wedge\left((a^{-1})_{b}^{\ a}[g]\tilde{a}^{c}_{\
a}[\tilde{h}]a_{c}^{\ d}[h]\tilde{b}_{de}[\tilde{g}]\right)l^{e}(g)$
$\displaystyle\quad-\tilde{l}_{b}(\tilde{g})\wedge\left((a^{-1})_{e}^{\
d}[g]\tilde{a}^{c}_{\ d}[\tilde{h}]a_{c}^{\ a}[h](\tilde{a}^{-1})^{b}_{\
a}[\tilde{g}]\right)l^{e}(g)$
$\displaystyle\quad+\tilde{l}_{b}(\tilde{g})\wedge\left(\tilde{a}^{c}_{\
d}[\tilde{h}]b^{de}[g]a_{c}^{\ a}[h](a^{-1})^{b}_{\
a}[\tilde{g}]\right)\tilde{l}_{e}(\tilde{g})\,,$

Here we used equations (A.7) and (A.8) to obtain the first line. The second
equality results from using (A.4) and (A.6). This last line, then finally
leads, looking back at eq. (A.9), to the expression relevant for the
symplectic form on the double used in section 2.2.1:

$\displaystyle 2\omega_{STS}=2l^{b}(g)\wedge C_{b}^{\
e}\tilde{l}_{e}(\tilde{g})+l^{b}(g)\wedge C_{b}^{\
m}\tilde{\Pi}_{me}[\tilde{g}]l^{e}(g)-\tilde{l}_{b}(\tilde{g})\wedge\tilde{C}^{b}_{\
m}\Pi^{me}[g]\tilde{l}_{e}(\tilde{g})\,.$ (A.10)

Finally, note that here we have used a different decompositions of elements in
the double as compared to [27], where $\mathbb{g}=\tilde{g}g$ or
$\mathbb{g}=g^{\prime}\tilde{g}^{\prime}$. Substituting
$\tilde{g}\to\tilde{h},g^{\prime}\to h,\tilde{g}^{\prime}\to\tilde{g}$ and
using some of the above identities one can recover their parametrization,
which they propose as a kernel for the Fourier-Mukai transform.

##  Appendix B Canonical transformations for Poisson-Lie T-duality

Poisson-Lie T-duality was identified as being a canonical transformations in
[14, 15, 38, 34] and when relating the left-invariant Maurer-Cartan form and
its dual counterpart it takes the form

$\displaystyle(E_{0}^{\mp})^{-1}E^{\mp}L_{\pm}=\pm\tilde{E}^{\mp}\tilde{L}_{\pm}$
(B.1)

with $L_{+}=L^{a}_{+}T_{a}=l^{a}_{i}\partial X^{i}T_{a}$ and
$L_{-}=L^{a}_{-}T_{a}=l^{a}_{i}\bar{\partial}X^{i}T_{a}$ (mutatis mutandis for
the tilde variables $\tilde{L}_{\pm}$), and the background fields are given by
the usual

$\displaystyle E^{\pm}$
$\displaystyle=((E_{0}^{\pm})^{-1}\pm\Pi)^{-1}\,,\quad\tilde{E}^{\pm}=(E_{0}^{\pm}\pm\tilde{\Pi})^{-1}\,.$
(B.2)

Note that in this notation we have $(E_{0}^{+})^{T}=E_{0}^{-}$.

Here we rewrite the canonical transformation for Poisson-Lie T-duality into a
form amenable to show the topological nature of the Poisson-Lie defect in
section 2.1.1. The canonical relations can be written as

$\displaystyle\begin{aligned}
C^{-1}(E+C\tilde{\Pi})\tilde{C}^{-1}(\tilde{E}+\tilde{C}\Pi)&=1\,,\\\
C^{-1}(E^{T}-C\tilde{\Pi})\tilde{C}^{-1}(\tilde{E}^{T}-\tilde{C}\Pi)&=1\,.\end{aligned}$
(B.3)

From these relations we want to extract the dual background fields
$\tilde{G},\tilde{B}$ in terms of the original fields $G,B$. Take for example
the first relation in eq. (2.16) which is equivalent to

$\displaystyle\left(\tilde{C}^{-1}\tilde{G}C^{-1}+\tilde{C}^{-1}(\tilde{B}+\tilde{C}\Pi)C^{-1}\right)^{-1}$
$\displaystyle=G+(B+C\tilde{\Pi})\,.$

Using that the first, respectively second, term (inside the brackets on the
left-hand side) on each side is symmetric, respectively antisymmetric, one
obtains the relations151515For $S,\tilde{S}$ symmetric invertible matrices and
$A,\tilde{A}$ antisymmetric matrices $(\tilde{S}+\tilde{A})^{-1}=(S+A)$ then
$\displaystyle\tilde{S}^{-1}\tilde{A}=-AS^{-1}\,,\quad\tilde{S}=(S+AS^{-1}A)^{-1}\,,$
when $A$ is also invertible one has also that $\tilde{A}=(A+SA^{-1}S)^{-1}$.
We will however not require the B-field $B$ to be invertible, since only the
first two relations will be needed.

$\displaystyle\begin{aligned}
\tilde{C}^{-1}\tilde{G}C^{-1}&=\left(G-(B+C\tilde{\Pi})G^{-1}(B+C\tilde{\Pi})\right)^{-1}\\\
C\tilde{G}^{-1}(B+C\tilde{\Pi})C^{-1}&=-(B+C\tilde{\Pi})G^{-1}\,.\end{aligned}$
(B.4)

When including spectator variables $Y^{i}$, the canonical transformations read
[34]

$(E_{0}^{\mp})^{-1}E^{\mp}(L_{\pm}\pm\Pi
F^{\mp}\partial_{\pm}Y)=\pm\tilde{E}^{\mp}(\tilde{L}_{\pm}\mp
F^{\mp}\partial_{\pm}Y)\,.$ (B.5)

where again

$\displaystyle E^{\pm}$
$\displaystyle=((E_{0}^{\pm})^{-1}\pm\Pi)^{-1}\,,\quad\tilde{E}^{\pm}=(E_{0}^{\pm}\pm\tilde{\Pi})^{-1}\,.$
(B.6)

and the background fields with spectator directions are restricted to be of
the form [34] (setting $E\equiv E^{+}$ for the sake of clarity)

$\displaystyle\begin{matrix}[l]E_{a\mu}=E_{ab}((E_{0})^{-1})^{bc}F_{c\mu}\,,&E_{\mu\nu}=F_{\mu\nu}-F_{\mu
a}\Pi^{ab}E_{bc}((E_{0})^{-1})^{cd}F_{d\nu}\,,\\\ \tilde{E}^{a}_{\
\mu}=\tilde{E}^{ab}F_{b\mu}\,,&\tilde{E}_{\mu\nu}=F_{\mu\nu}-F_{\mu
a}\tilde{E}^{ab}F_{b\nu}\,,\end{matrix}$ (B.7)

with $F_{\mu\nu},F_{a\mu}$ arbitrary matrices depending on the choice of
background configuration chosen. The canonical transformation for fields not
involving any spectator direction are still given by (2.16) while for mixed
coordinates we have [34]

$\displaystyle\begin{aligned} \tilde{E}^{d}_{\
\mu}&=(\tilde{C}(E+C\tilde{\Pi})^{-1})^{dc}E_{c\mu}\,,\quad\tilde{E}^{\
d}_{\mu}=(\tilde{C}(E^{T}-C\tilde{\Pi})^{-1})^{dc}E_{c\mu}\,.\end{aligned}$
(B.8)

##  Appendix C Reminder of generalised geometry

In this appendix, we collect a number of relevant definitions and results in
generalised geometry and Courant algebroid theory. See e.g. [43, 48] for a
pedagogical or more complete exposition.

##### Isotropic and Lagrangian subspaces.

For $V$ a real vector space equipped with a real bilinear form
$\langle\cdot,\cdot\rangle$, a subspace $L$ of $V\otimes V^{\ast}$, where
$V^{\ast}$ is linear vector space dual to $V$, is called isotropic if $\langle
X,Y\rangle=0$ for all $X,Y\in L$. It will become clear in the following that
so-called maximal isotropic subspace play a special role. Since the signature
of the bilinear form is $(\dim V,\dim V)$, a maximal isotropic subspace (or
Lagrangian subspace) of $V$ is of dimension $\dim V$. Lagrangian subspaces
$L\subset V\otimes V^{\ast}$ for a linear vector space $V$ are equivalent to
picking a subspace $\Delta\subset V$ together with a two-form
$F\in\wedge^{E}\Delta^{\ast}$ via the identification

$\displaystyle L=L(\Delta,\epsilon)=\\{X+\eta\in\Delta\oplus
V^{\ast}\;:\;\iota^{\ast}\eta=\imath_{X}F\\}\,,$

where $\iota:\Delta\hookrightarrow V$ is the inclusion map. An important
property of Lagrangian subspace of linear vector spaces is that $B$-field
transformations of such Lagrangian subspace $L$, i.e.

$\displaystyle e^{B}L(\Delta,F)=L(\Delta,F+\iota^{\ast}B)\,,\quad
B\in\wedge^{2}V\,,$

does not change the projection $\Delta$ onto the vector space $V$. Note that
using $B$-field transformation any Lagrangian subspace $L$ can be brought to
the form $L(\Delta,0)$. Acting with a $\beta$-transformation will in general
modify $\Delta$ and its dimension.

##### Courant algebroids.

A Courant algebroid is a bundle $E$ over an $m$-dimensional smooth manifold
$M$ equipped with a bilinear form $\langle\cdot,\cdot\rangle$ of signature
$(m,m)$. In addition it comes with an anchor map $a:E\rightarrow TM$ , that
allows to map any section on $E$ to the more familiar notion of vector fields
on the tangent bundle $TM$ of $M$ and a bracket called the Courant bracket
$\llbracket\cdot,\cdot\rrbracket_{C}$. Contrary to a Lie bracket, this bracket
need not be skew-symmetric. Instead one requires the following axioms

$\displaystyle\llbracket u,\llbracket v,w\rrbracket\rrbracket$
$\displaystyle=\llbracket\llbracket u,v\rrbracket,w\rrbracket+\llbracket
v,\llbracket u,w\rrbracket\rrbracket\,,$ $\displaystyle[u,fv]$
$\displaystyle=f[u,v]+(\rho(u)f)v\,,$ $\displaystyle\rho(u)(v,w)$
$\displaystyle=\langle[u,v],w\rangle+\langle v,[u,w]\rangle\,,$
$\displaystyle[u,v]+[v,u]$ $\displaystyle=\rho^{T}\mathrm{d}\langle
u,v\rangle\,,$

for section $u,v,w$ of $E$, $f\in C^{\infty}(M)$ a continuous function and
$\rho^{T}:T^{\ast}M\rightarrow E$ is the transpose of the anchor map, using
the non-degenerate innerproduct $\langle\cdot,\cdot\rangle$ to identify $E$
with $E^{\star}$. The quadruple
$(E,\langle\cdot,\cdot\rangle,a,\llbracket\cdot,\cdot\rrbracket_{C})$ is
called a Courant algebroid over $M$. We will now consider a smooth manifold
$M$ together with its generalised tangent bundle $TM\oplus T^{\ast}M$.

##### Exact Courant algebroids.

Take $E$ a Courant algebroid over a manifold $M$ with anchor map $a$. One then
has automatically that $a\circ a^{T}=0$ leading to the chain complex

$\displaystyle 0\rightarrow
T^{\ast}M\xrightarrow[]{a^{T}}E\xrightarrow[]{a}TM\rightarrow 0\,.$

When this sequence is exact, then $E$ is called an exact Courant algebroid.
The reason why this forms a distinguished subclass amongst Courant algebroids
is that exact Courant algebroids are classified by closed three-forms
$H\in\Omega^{3}(M)$, which in the sigma-model setting coincides with the
three-form flux $\mathrm{d}B=H$ of the Kalb-Ramond two-form $B$. To extract
such three-from for a given exact Courant algebroid one uses that since the
above sequence is exact, for any Lagrangian subbundle $L$ of $E$ the map
$a|_{L}:L\rightarrow TM$ is an isomorphism. This allows us to make a choice
for the embedding $\sigma:TM\rightarrow E$ in the Courant algebroid by
demanding that its image is the Lagrangian subbundle $L$. Then

$\displaystyle H(X,Y,Z)\equiv\langle[\sigma(X),\sigma(Y)],\sigma(Z)\rangle\,,$

defines a closed three-form on $M$. Exact Courant algebroids can be identified
with the canonical generalised tangent bundle $TM\oplus T^{\ast}M$ via the map
$\sigma\oplus a^{T}$ at the cost of twisting the Courant bracket by the three
form field $H$:

$\displaystyle\llbracket(X,\xi),(Y,\eta)\rrbracket_{H}=([X,Y],L_{X}\eta-\imath_{Y}\mathrm{d}\xi+H(X,Y,\cdot))\,.$

Conversely, for any closed three-form $H\in\Omega^{3}(M)$, the above bracket
turns the canonical generalised tangent bundle $TM\oplus T^{\ast}M$ into an
exact Courant algebroid. Since changing the choice of splitting map $\sigma$
by a two-form $\epsilon\in\Omega^{2}(M)$, the three-form $H$ is modified by
the shift $H+\mathrm{d}\epsilon$, we have the exact Courant algebroids over a
manifold $M$ are uniquely determined by $H^{3}(M,\mathbb{R})$.

#####  $\mathcal{G}$-equivariant Courant algebroids.

Take $\mathfrak{g}$ a Lie algebra equipped with an invariant bilinear
symmetric form and denote by $\mathcal{G}$ the corresponding Lie group. A
$\mathfrak{g}$-equivariant Courant algebroid over a manifold $M$ is a Courant
algebroid $E$ together with an injective linear map
$\chi:\mathfrak{g}\rightarrow\Gamma(E)$ which verifies

$\displaystyle[\chi(X),\chi(Y)]=[X,Y]\,,\qquad\langle\chi(X),\chi(Y)\rangle=\langle
X,Y\rangle\,,$

for all $X,Y$ in $\mathfrak{g}$. This definition guarantees that the Courant
algebroid $E$ admits the action of the algebra $\mathfrak{g}$, in turn
inducing an action of $\mathfrak{g}$ on the manifold $M$. When the action of
the Courant algebroid $E$ integrates to an action of the associated connected
Lie group $\mathcal{G}$, the algebroid $E$ is called
$\mathcal{G}$-equivariant.

One can show that when a Lie group $\mathcal{G}$ acts freely and transitively
over $M$, there exists a $\mathcal{G}$-equivariant exact Courant algebroid
over $M$ if and only if the first Pontryagin class $[\langle
F,F\rangle_{\mathfrak{g}}]$ of the bundle $M\rightarrow M/\mathcal{G}$
vanishes.

##### Dirac structures.

Dirac structures generalise (or unify) the notion of Poisson structures,
symplectic structures and foliations using the framework of generalised
geometry. The crucial observation is that these three examples can be
equivalently described by a Lagrangian subbundle that is closed with respect
to the Courant bracket. Indeed, to each symplectic structure $\omega$ or
Poisson structure $\pi$ define the associate bundle maps

$\displaystyle\omega^{\sharp}$ $\displaystyle:TM\rightarrow
T^{\ast}M:\;X\mapsto\imath_{X}\omega\,\quad\text{or}\quad\Pi^{\sharp}:T^{\ast}M\rightarrow
TM:\;\alpha\mapsto\imath_{\alpha}\Pi\,,$ (C.1)

then their respective graphs

$\displaystyle L_{\Pi}$
$\displaystyle=\text{graph}(\pi^{\sharp})=\\{(\pi^{\sharp}(\xi),\xi)\mid\xi\in\mathbb{T}^{\ast}M\\}\,,$
$\displaystyle L_{\omega}$
$\displaystyle=\text{graph}(\omega^{\sharp})=\\{(X,\omega^{\sharp}(X))\mid
X\in\mathbb{T}M\\}\,,$

generate Lagrangian subbundles of $\mathbb{T}M=TM\oplus T^{\ast}M$. These
subbundles are in addition closed under the Courant bracket by virtue of $\Pi$
satisfying the Schouten bracket, respectively $\omega$ being closed.

More generally, a Dirac structure $L$ on $M$ is a subbundle
$L\subset\mathbb{T}M$ such that

  * (i)

$L=L^{\perp}$ with respect to $\langle\cdot,\cdot\rangle$ ,

  * ii)

$L$ is involutive with respect to the Courant bracket:
$\llbracket\Gamma(L),\Gamma(L)\rrbracket\subset\Gamma(L)$.

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