further supposing that the rank of $\rho$ is constant, i.e. the foliation
${\mathcal{F}}$ is regular, as our main interest in this paper concerns
gaugings with Lie algebroids characterized by injective anchor maps, and we
will be particularly interested in this case for our applications to Born
sigma-models later on.

In the case of a regular foliation ${\mathcal{F}}$ of $M,$ there is a short
exact sequence of vector bundles

$0\longrightarrow
T{\mathcal{F}}\xlongrightarrow{i}TM\xlongrightarrow{\bar{q}}N{\mathcal{F}}\longrightarrow
0$

where $N{\mathcal{F}}=TM/T{\mathcal{F}}$ is the normal bundle of
${\mathcal{F}}$ and $\bar{q}:TM\rightarrow N{\mathcal{F}}$ is the quotient
map. We can always choose an orthogonal splitting

$s_{\perp}:N{\mathcal{F}}\longrightarrow TM$

of this exact sequence with respect to ${\mathcal{H}}$, so that ${\rm
Im}(s_{\perp})=T{\mathcal{F}}^{\perp}$ and

$TM=T{\mathcal{F}}\oplus T{\mathcal{F}}^{\perp}\ ,$

where $T{\mathcal{F}}^{\perp}\simeq N{\mathcal{F}}$ is the orthogonal
complement of $T{\mathcal{F}}$ in the metric ${\mathcal{H}}.$ This also
induces a splitting of the cotangent bundle
$T^{*}M=T^{*}{\mathcal{F}}\oplus(T{\mathcal{F}}^{\perp})^{*},$ where
$(T{\mathcal{F}}^{\perp})^{*}\simeq N^{*}{\mathcal{F}}$ with
$N^{*}{\mathcal{F}}$ the conormal bundle of ${\mathcal{F}}.$ Then we can
decompose the metric ${\mathcal{H}}$ as

${\mathcal{H}}=g_{\parallel}+g_{\perp}\ ,$

where $g_{\parallel}$ is a fiberwise metric on the sub-bundle $T{\mathcal{F}}$
and $g_{\perp}$ is a fiberwise metric on $T{\mathcal{F}}^{\perp}.$ This allows
us to rephrase the condition (4.13) by saying that the Lie derivative
$\pounds_{X_{\parallel}}{\mathcal{H}},$ for every vector field
$X_{\parallel}\in\mathsf{\Gamma}(T{\mathcal{F}}),$ can only have components in
$\mathsf{\Gamma}(T^{*}{\mathcal{F}}\otimes T^{*}{\mathcal{F}}),$
$\mathsf{\Gamma}(T^{*}{\mathcal{F}}\otimes(T{\mathcal{F}}^{\perp})^{*})$ and
$\mathsf{\Gamma}((T{\mathcal{F}}^{\perp})^{*}\otimes T^{*}{\mathcal{F}}),$
i.e. $(\pounds_{X_{\parallel}}{\mathcal{H}})_{\perp}=0$ in
$\mathsf{\Gamma}((T{\mathcal{F}}^{\perp})^{*}\otimes(T{\mathcal{F}}^{\perp})^{*})$.
It is easy to see that this constraint is equivalent to

$\pounds_{X_{\parallel}}g_{\perp}=0\ ,$ (4.14)

for all $X_{\parallel}\in\mathsf{\Gamma}(T{\mathcal{F}}).$ This implies that
the component $g_{\perp}$ of the metric ${\mathcal{H}}$ is transverse
invariant. In other words, the fiberwise metric $g_{\perp}$ induced by
${\mathcal{H}}$ satisfies ${\rm Ker}(g_{\perp})=T{\mathcal{F}}$ and also
(4.14) whenever the gauging is possible. These gauging constraints are simply
the defining conditions for $(M,g_{\perp},{\mathcal{F}})$ to be a Riemannian
foliation [81].

We will now show that this statement about the gauging of sigma-models is
equivalent to saying that $(M,{\mathcal{H}},{\mathcal{F}})$ is a foliated
manifold with a bundle-like metric [82, 83] whenever the foliation is regular.

######  Definition 4.15.

A Riemannian metric ${\mathcal{H}}$ on a foliated manifold $(M,{\mathcal{F}})$
is (_totally geodesics_) _bundle-like_ if

$\pounds_{X_{\parallel}}{\mathcal{H}}(Y_{\perp},Z_{\perp})=0\ ,$

for all $X_{\parallel}\in\mathsf{\Gamma}(T{\mathcal{F}})$ and
$Y_{\perp},Z_{\perp}\in\mathsf{\Gamma}(T{\mathcal{F}}^{\perp}).$

The leaf holonomy invariance of bundle-like metrics is discussed in [83]. The
motivating example for this structure is given by the compatible generalized
metrics of Example 2.39.

######  Theorem 4.16.

Let $M$ be a manifold endowed with a regular foliation ${\mathcal{F}}$ and a
Riemannian metric ${\mathcal{H}}.$ Then the gauging condition (4.12) holds if
and only if ${\mathcal{H}}$ is bundle-like.

###### Proof.

Suppose that ${\mathcal{H}}$ is a bundle-like metric. Let ${\sf
A}=T{\mathcal{F}}$ be the Lie algebroid with anchor map given by the inclusion
$i:T{\mathcal{F}}\hookrightarrow TM.$ We choose the Lie algebroid connection
$\nabla$ to be the unique Bott connection141414See [84, 85] for background on
Bott connections on foliated Riemannian manifolds. In our case we use the Bott
connection on the tangent bundle rather than on the normal bundle.
$\nabla^{\tt B}$ defined by $(M,{\mathcal{H}},{\mathcal{F}})$ restricted to
$T{\mathcal{F}}$:

$\nabla^{\tt
B}:\mathsf{\Gamma}(TM)\times\mathsf{\Gamma}(T{\mathcal{F}})\longrightarrow\mathsf{\Gamma}(T{\mathcal{F}})$

given by $\nabla^{\tt B}_{X}Y_{\parallel}\in\mathsf{\Gamma}(T{\mathcal{F}}),$
for all $X\in\mathsf{\Gamma}(TM)$ and
$X_{\parallel}\in\mathsf{\Gamma}(T{\mathcal{F}})$. The restriction is always
well-defined as a Lie algebroid connection since this reflects one of the
properties of the Bott connection. To show that (4.12) holds, we give the
representation $\prescript{\tau}{}{\nabla}^{\tt B}$ for this restriction of
the Bott connection by using (4.10):

$\prescript{\tau}{}{\nabla}^{\tt
B}_{Y_{\parallel}}X=[Y_{\parallel},X]+\nabla^{\tt B}_{X}Y_{\parallel}\ .$
(4.17)

The only non-zero component of the torsion tensor $T^{\tt B}$ of a Bott
connection is $T^{\tt
B}(X_{\perp},Y_{\perp})\in\mathsf{\Gamma}(T{\mathcal{F}}),$ i.e. $T^{\tt
B}(X_{\perp},Y_{\parallel})=T^{\tt B}(X_{\parallel},Y_{\parallel})=0.$ This
implies

$\nabla^{\tt B}_{X}Y_{\parallel}=\nabla^{\tt
B}_{Y_{\parallel}}X+[X,Y_{\parallel}]\ ,$

which when substituted in (4.17) gives

$\prescript{\tau}{}{\nabla}^{\tt B}_{Y_{\parallel}}X=\nabla^{\tt
B}_{Y_{\parallel}}X\ .$

Since $\nabla^{\tt B}$ is a metric connection for the bundle-like metric
${\mathcal{H}},$ the compatibility condition (4.12) follows.

Conversely, we have already shown above that (4.12) is equivalent to the
condition (4.14), which is the defining condition of a bundle-like metric. ∎

To understand when the condition (4.12) allows us to recover a sigma-model for
the quotient space of a foliation, we need an additional concept.

######  Definition 4.18.

Let $(M,{\mathcal{H}})$ and $({\mathcal{Q}},g)$ be Riemannian manifolds. Let
${\mathit{\Pi}}:M\rightarrow{\mathcal{Q}}$ be a surjective submersion, so that
the orthogonal complement ${\rm Ker}(\mathrm{d}{\mathit{\Pi}})^{\perp}\subset
TM$ defines the horizontal sub-bundle complementary to ${\rm
Ker}(\mathrm{d}{\mathit{\Pi}}).$ Then ${\mathit{\Pi}}$ is a _Riemannian
submersion_ if the isomorphism $\mathrm{d}{\mathit{\Pi}}:{\rm
Ker}(\mathrm{d}{\mathit{\Pi}})^{\perp}\rightarrow T{\mathcal{Q}}$ is an
isometry:

${\mathcal{H}}(X_{\tt h},Y_{\tt h})=g\big{(}\mathrm{d}{\mathit{\Pi}}(X_{\tt
h}),\mathrm{d}{\mathit{\Pi}}(Y_{\tt h})\big{)}\ ,$

for all $X_{\tt h},Y_{\tt h}\in\mathsf{\Gamma}({\rm
Ker}(\mathrm{d}\phi)^{\perp})$.

Whenever the quotient ${\mathcal{Q}}=M/{\mathcal{F}}$ of a foliated manifold
$(M,{\mathcal{F}})$ equipped with a bundle-like metric ${\mathcal{H}}$ is
smooth, we can identify $(T{\mathcal{F}}^{\perp})^{*}\simeq
N^{*}{\mathcal{F}}$ with $T^{*}{\mathcal{Q}}$ and find that $g_{\perp}$
induces a metric $g$ on the quotient manifold ${\mathcal{Q}}.$ This describes
a Riemannian submersion of $(M,{\mathcal{H}})$ onto $({\mathcal{Q}},g).$ We
interpret the Riemannian submersion
$(M,{\mathcal{H}})\rightarrow({\mathcal{Q}},g)$ as a way to relate a sigma-
model ${\mathcal{S}}({\mathcal{H}})$ for a foliated Riemannian manifold
$(M,{\mathcal{H}})$ to a sigma-model $S(g)$ for the leaf space ${\mathcal{Q}}$
endowed with a Riemannian metric $g$ obtained from ${\mathcal{H}}$ through the
gauging. From this point of view, the constraint (4.14) is simply a condition
for the metric $g_{\perp}$ to be well-defined on the leaf space
${\mathcal{Q}}.$

Under these conditions, the gauged sigma-model thus constructed is invariant
under the Lie algebra of sections of the pullback to $\Sigma$ of the Lie
algebroid $(T{\mathcal{F}},i)$ on $M$, which is a Lie subalgebroid of the
tangent algebroid $(TM,\mathds{1}_{TM})$. An integrating Lie groupoid for $TM$
is given by the pair groupoid $M\times M\rightrightarrows M$ of the manifold
$M$, with orbit space $M$, while $T{\mathcal{F}}$ is integrated by the Lie
subgroupoid of $M\times M$ given by the graph of the equivalence relation on
$M$ defined by the surjective submersion $M\to{\mathcal{Q}}$ [86], with orbit
space $M/{\mathcal{F}}={\mathcal{Q}}$. The pullback of this to $\Sigma$
determines the groupoid of gauge transformations which leaves the resulting
gauged sigma-model invariant.

####  4.2.3. Incorporating the $B$-Field

Finally, we wish to include a topological term in the action functional of the
sigma-model, i.e. a $B$-field. This extension of the present formalism is also
discussed in [44].

######  Definition 4.19.

Let $({\mathsf{A}},\rho)$ be a Lie algebroid over $(M,\Phi),$ where
$\Phi\in\mathsf{\Gamma}(T^{*}M\otimes T^{*}M)$ is a non-degenerate bilinear
form. Endow ${\mathsf{A}}$ with a connection $\nabla$ and let
$\psi\in\mathsf{\Gamma}(T^{*}M\otimes{\sf End}({\mathsf{A}})).$ Then
$({\mathsf{A}},\nabla,\psi)$ and $(M,\Phi)$ are _compatible_ if

$(\prescript{\tau}{}{\nabla}^{+}\otimes\mathds{1}+\mathds{1}\otimes\prescript{\tau}{}{\nabla}^{-})(\Phi)=0\
,$ (4.20)

where

$\nabla^{\pm}=\nabla\pm\psi$

and $\prescript{\tau}{}{\nabla}$ is given in (4.10).

To apply this to the case at hand, we also describe the local expression of
(4.20). Let us write ${\sf Sym}\\{\Phi\\}={\mathcal{H}}$ and ${\sf
Alt}\\{\Phi\\}=\omega$ for the symmetric and skew-symmetric parts of the
$(0,2)$-tensor $\Phi$, and let $\\{a_{i}\\}$ be a local basis of
$\mathsf{\Gamma}({\mathsf{A}})$ such that

$\nabla a_{i}=\Omega^{j}_{i}\otimes a_{j}\qquad\mbox{and}\qquad\psi
a_{i}=\psi^{j}_{i}\otimes a_{j}\ ,$

where $\Omega^{j}_{i}$ and $\psi^{j}_{i}$ are 1-forms on $M$. Then (4.20) can
be cast in the form

$\displaystyle\pounds_{\rho(a_{i})}{\mathcal{H}}$
$\displaystyle=\Omega^{j}_{i}\odot\iota_{\rho(a_{j})}{\mathcal{H}}+\psi^{j}_{i}\odot\iota_{\rho(a_{j})}\omega\
,$ (4.21) $\displaystyle\pounds_{\rho(a_{i})}\omega$
$\displaystyle=\Omega^{j}_{i}\wedge\iota_{\rho(a_{j})}\omega+\psi^{j}_{i}\wedge\iota_{\rho(a_{j})}{\mathcal{H}}\
.$ (4.22)

These equations represent the generalization of (4.13), thus the condition
expressed by (4.20) is usually referred to as a _generalized isometry_ or as
the condition for a _Lie algebroid gauging_.

A relevant example for our purposes is given by a sigma-model on a foliated
manifold. Let $(M,\Phi)$ be a manifold foliated by ${\mathcal{F}}.$ We then
naturally consider as Lie algebroid ${\mathsf{A}}=T{\mathcal{F}}.$ The gauging
of the corresponding sigma-model is possible if (4.20) is satisfied and the
bilinear form $\Phi$ induces a bilinear form $\Phi^{{\mathcal{Q}}}$ on
$N^{*}{\mathcal{F}}.$ Whenever the quotient ${\mathcal{Q}}=M/{\mathcal{F}}$ is
smooth, the bilinear form $\Phi^{\mathcal{Q}}$ defines a section of
$T^{*}{\mathcal{Q}}\otimes T^{*}{\mathcal{Q}}.$

In this paper we are mostly concerned with the conditions that the geometry of
the target space must satisfy in order to admit a gauging of the generalized
isometry. For a discussion involving the structure of the pullback gauge Lie
algebroid on the worldsheet $\Sigma$, see [87].

###  4.3. Gauging the Born Sigma-Model

The discussion of Section 4.2 can be applied to a Born manifold
$(M,K,\omega,{\mathcal{H}})$ simply by setting

$\Phi={\mathcal{H}}+\omega$

in $\mathsf{\Gamma}(T^{*}M\otimes T^{*}M),$ which is non-degenerate by
construction. By assuming that one of the eigenbundles $L_{\pm}$ of $K$ is
integrable, we obtain a foliation ${\mathcal{F}}_{\pm}$ of $M.$ Then we can
regard the map $(M,\Phi)\rightarrow({\mathcal{Q}},\Phi^{{\mathcal{Q}}}),$
where ${\mathcal{Q}}=M/{\mathcal{F}}_{\pm}$ and

$\Phi^{\mathcal{Q}}=g+b\ ,$

as a reduction of a Born sigma-model ${\mathcal{S}}({\mathcal{H}},\omega)$ to
a sigma-model $S(g,b)$ for the leaf space ${\mathcal{Q}}.$ The specialization
of the general formalism for the gauging for a foliated manifold comes from
the compatibility conditions that the Riemannian metric ${\mathcal{H}}$ now
satisfies with the para-Hermitian structure.

Here we shall focus on the local construction of the gauging of a Born sigma-
model. From now on ${\mathcal{H}}$ will be the compatible generalized metric
on a Born manifold $M$ and $\omega$ will be its fundamental 2-form. To apply
the theory of gauged sigma-models with generalized isometry reviewed in
Section 4.2, we need a foliated target space $M.$ Since the target space of a
Born sigma-model is an almost para-Hermitian manifold $(M,K,\eta),$ the
tangent bundle is given by the splitting $TM=L_{+}\oplus L_{-}$ as described
in Section 2. We require one of the eigenbundles $L_{\pm}$ of $K$ to be
integrable; let us take it to be $L_{-}$ for definiteness. Then $M$ is
foliated by the leaves of ${\mathcal{F}}_{-}$ such that
$T{\mathcal{F}}_{-}=L_{-}.$ From a local perspective, this amounts to the
assumption that any frame for $TM$ closes a Lie algebra of the form (2.42)
with $R^{mnk}=0$; we shall return to the case when $R^{mnk}\neq 0$ in Section
4.4. The bundle $L_{-}$ is a Lie subalgebroid of the tangent algebroid $TM$
with the anchor map given by the inclusion $i:L_{-}\hookrightarrow TM$. When
the space of leaves is a smooth manifold, $L_{-}$ is integrated by the Lie
subgroupoid of the pair groupoid $M\times M\rightrightarrows M$ given by the
graph of the equivalence relation on $M$ defined by the surjective submersion
$M\to{\mathcal{Q}}=M/{\mathcal{F}}_{-}$; we shall discuss the case when the
leaf space is not smooth in Section 4.4. It is natural to use this Lie
algebroid in the application of the formalism of Section 4.2, and investigate
under which geometric constraints the generalized isometry conditions (4.21)
and (4.22) are satisfied for a Born sigma-model with a foliated para-Hermitian
target space. For this, we require a Lie algebroid connection on $L_{-}$; as
discussed in the proof of Theorem 4.16, a natural candidate is the Bott
connection $\nabla^{\tt B}$.

The traditional approach to the gauging of a generalized isometry involves
applying a non-standard variation of the dynamical fields $\phi$ and $A$ of
the gauged Born sigma-model, where $A$ is the $L_{-}$-valued connection 1-form
on $M$. Since the anchor map is the inclusion $i$ of $L_{-}$ in $TM,$ the Lie
algebroid bracket is locally given by

$[\tilde{Z}^{m},\tilde{Z}^{n}]=Q_{k}{}^{mn}\,\tilde{Z}^{k}\ .$ (4.23)

The map $\mathrm{d}\phi$ is covariantized upon introducing

$\mathrm{D}^{A}\phi^{I}=\mathrm{d}\phi^{I}-\bar{i}^{Ij}\,\bar{A}_{j}\ ,$
(4.24)

where $\bar{i}$ is the pullback of $i$ along $\phi.$ The gauged sigma-model
with topological term is thus written as

${\mathcal{S}}[\phi,A]=\frac{1}{4}\,\int_{\Sigma}\,\bar{\mathcal{H}}_{IJ}\,\mathrm{D}^{A}\phi^{I}\wedge\star\,\mathrm{D}^{A}\phi^{J}+\frac{1}{4}\,\int_{\Sigma}\,\bar{\omega}_{IJ}\,\mathrm{D}^{A}\phi^{I}\wedge\mathrm{D}^{A}\phi^{J}\
,$ (4.25)

and the variations of the fields $\phi$ and $A$ under the infinitesimal gauge
transformations generated by the vector fields $\tilde{Z}^{i}$ read

$\displaystyle\delta_{\epsilon}\phi^{I}$
$\displaystyle=\bar{i}^{Ij}\,\epsilon_{j}\ ,$ (4.26)
$\displaystyle\delta_{\epsilon}\bar{A}_{i}$
$\displaystyle=\mathrm{d}\epsilon_{i}+\bar{Q}_{i}{}^{jk}\,\epsilon_{k}\,\bar{A}_{j}+\bar{\Omega}_{iJ}^{k}\,\epsilon_{k}\,\mathrm{D}^{A}\phi^{J}+\bar{\psi}_{iJ}^{k}\,\epsilon_{k}\,\star\,\mathrm{D}^{A}\phi^{J}\
,$ (4.27)

where $\epsilon_{i}\in C^{\infty}(\Sigma).$ It can be verified that the gauged
action functional (4.25) is invariant under these transformations:

$\delta_{\epsilon}{\mathcal{S}}[\phi,A]=0\ ,$

if and only if the conditions (4.21) and (4.22) are satisfied [88].

We also give the gauging of the Born sigma-model in local coordinates

$\mathbb{X}^{I}=(x^{i},\tilde{x}_{i})$

on $M,$ where the foliation ${\mathcal{F}}_{-}$ has leaves whose adapted
coordinates are given by $\tilde{x}_{i}$ for $i=1,\dots,d$. Then the gauged
Born sigma-model coming from the local expression (4.7) is obtained by
replacing the pulled back 1-forms $\mathrm{d}\bar{x}^{i}$ and
$\mathrm{d}\bar{\tilde{x}}{}_{i}$ by the covariantized maps151515Here and in
the following we suppress the pullback of the inclusion $i.$

$\mathrm{D}^{A}\bar{x}^{i}=\mathrm{d}\bar{x}^{i}\qquad\mbox{and}\qquad\mathrm{D}^{A}\bar{\tilde{x}}{}_{i}=\mathrm{d}\bar{\tilde{x}}{}_{i}-\bar{A}_{i}\
.$ (4.28)

In the quantum theory, this minimal coupling allows
$\mathrm{d}\bar{\tilde{x}}{}_{i}$ to be absorbed into a shift of the
worldsheet gauge fields $\bar{A}_{i}$. The gauged Born sigma-model in local
coordinates thus reads

${\mathcal{S}}[\phi,A]=\frac{1}{4}\,\int_{\Sigma}\,\bar{\mathcal{H}}_{IJ}\,\mathrm{D}^{A}\bar{\mathbb{X}}^{I}\wedge\star\,\mathrm{D}^{A}\bar{\mathbb{X}}^{J}+\frac{1}{4}\,\int_{\Sigma}\,\bar{\omega}_{IJ}\,\mathrm{D}^{A}\bar{\mathbb{X}}^{I}\wedge\mathrm{D}^{A}\bar{\mathbb{X}}^{J}\
,$ (4.29)

and the variations (4.26) and (4.27) under which the gauged sigma-model is
invariant are written as

$\displaystyle\delta_{\epsilon}\bar{x}^{i}$ $\displaystyle=0\ ,$
$\displaystyle\delta_{\epsilon}\bar{\tilde{x}}{}_{i}$
$\displaystyle=\epsilon_{i}\ ,$ $\displaystyle\delta_{\epsilon}\bar{A}_{i}$
$\displaystyle=\mathrm{d}\epsilon_{i}+\bar{Q}_{i}{}^{jk}\,\epsilon_{k}\,\bar{A}_{j}+\bar{\Omega}_{iJ}^{k}\,\epsilon_{k}\,\mathrm{D}^{A}\bar{\mathbb{X}}^{J}+\bar{\psi}_{iJ}^{k}\,\epsilon_{k}\,\star\,\mathrm{D}^{A}\bar{\mathbb{X}}^{J}\
.$

In the Born sigma-model, the almost para-Hermitian structure appears solely in
the topological term. In the sigma-model without topological term, the
generalized isometry condition for ${\mathcal{H}}$ only involves the
assumption that $M$ must be foliated and does not capture deeper information
about the almost para-Hermitian manifold; in [20] the topological term was
introduced for doubled torus bundles in order to maintain invariance under
large gauge transformations in the corresponding gauged sigma-model and found
to play an important role in the quantum theory [47]. To understand how the
generalized isometry conditions (4.21) and (4.22) specialise to a Born sigma-
model, let us discuss the local reduction of a Born sigma-model
${\mathcal{S}}({\mathcal{H}},\omega)$ to a sigma-model $S(g,b)$ for the leaf
space.

In the coordinates $\mathbb{X}^{I}=(x^{i},\tilde{x}_{i})$ adapted to the
foliation ${\mathcal{F}}_{-},$ where $\tilde{x}_{i}$ are the leaf coordinates,
the local frame and dual coframe which respectively span $\mathsf{\Gamma}(TM)$
and $\mathsf{\Gamma}(T^{*}M)$, and which diagonalize the almost para-complex
structure $K$, are given in the form

$Z_{i}=\frac{\partial}{\partial
x^{i}}+N_{ij}\,\frac{\partial}{\partial\tilde{x}_{j}}\qquad\mbox{and}\qquad\tilde{Z}^{i}=\frac{\partial}{\partial\tilde{x}_{i}}\
,$

for local functions $N_{ij}$ on $M$, since there always exists a local
completion $\\{\tilde{Z}^{i}\\}$ of the holonomic frame for
$\mathsf{\Gamma}(T{\mathcal{F}}_{-}),$ and

$\Theta^{i}=\mathrm{d}x^{i}\qquad\mbox{and}\qquad\tilde{\Theta}_{i}=\mathrm{d}\tilde{x}_{i}-N_{ji}\,\mathrm{d}x^{j}\
,$

which form a local basis for $L_{+}^{*}$ and $L_{-}^{*}$ respectively. The
split signature metric $\eta$ assumes the form

$\eta=\eta^{i}_{j}\,\bigl{(}(\mathrm{d}\tilde{x}_{i}-N_{ki}\,\mathrm{d}x^{k})\otimes\mathrm{d}x^{j}+\mathrm{d}x^{j}\otimes(\mathrm{d}\tilde{x}_{j}-N_{kj}\,\mathrm{d}x^{k})\bigl{)}\
,$ (4.30)

while the fundamental 2-form $\omega$ reads

$\omega=\eta^{j}_{i}\,\mathrm{d}x^{i}\wedge\mathrm{d}\tilde{x}_{j}-\eta^{k}_{j}\,N_{ik}\,\mathrm{d}x^{i}\wedge\mathrm{d}x^{j}\
.$

Finally, a compatible generalized metric ${\mathcal{H}}$ on $(M,K,\eta)$ is
equivalent to specifying a fiberwise metric $g_{+}$ on $L_{+},$ which locally
reads

$g_{+}=(g_{+})_{ij}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}\ .$

Then the complete local expression for ${\mathcal{H}}$ is given by

$\displaystyle{\mathcal{H}}$
$\displaystyle=\bigl{(}(g_{+})_{ij}+(g_{-})^{kl}\,N_{ik}\,N_{jl}\bigr{)}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}+(g_{-})^{ij}\,\mathrm{d}\tilde{x}_{i}\otimes\mathrm{d}\tilde{x}_{j}$
$\displaystyle\qquad-(g_{-})^{jk}\,N_{ik}\,\mathrm{d}x^{i}\otimes\mathrm{d}\tilde{x}_{j}-(g_{-})^{ik}\,N_{jk}\,\mathrm{d}\tilde{x}_{i}\otimes\mathrm{d}x^{j}\
,$ (4.31)

where $g_{-}$ is defined in (2.29).

The Born sigma-model ${\mathcal{S}}({\mathcal{H}},\omega)$ in the coordinates
adapted to the foliation is written as in (4.7) and its gauging is obtained
upon introducing the covariant derivatives (4.28) to get the action functional
(4.29). Then the gauged Born sigma-model reads

$\displaystyle{\mathcal{S}}[\phi,A]$
$\displaystyle=\frac{1}{4}\,\int_{\Sigma}\,\Big{(}\bigl{(}(\bar{g}_{+})_{ij}+\bar{N}_{ik}\,(\bar{g}_{-})^{kl}\,\bar{N}_{jl}\bigr{)}\,\mathrm{d}\bar{x}^{i}\wedge\star\,\mathrm{d}\bar{x}^{j}$
$\displaystyle\hskip
56.9055pt-2\,(\bar{g}_{-})^{ik}\,\bar{N}_{jk}\,\mathrm{D}^{A}\bar{\tilde{x}}{}_{i}\wedge\star\,\mathrm{d}\bar{x}^{j}+(\bar{g}_{-})^{ij}\,\mathrm{D}^{A}\bar{\tilde{x}}{}_{i}\wedge\star\,\mathrm{D}^{A}\bar{\tilde{x}}{}_{j}\Big{)}$
$\displaystyle\quad\,+\frac{1}{2}\,\int_{\Sigma}\,\Big{(}\bar{\eta}^{j}_{i}\,\mathrm{d}\bar{x}^{i}\wedge\mathrm{D}^{A}\bar{\tilde{x}}{}_{j}-\bar{\eta}^{k}_{j}\,\bar{N}_{ik}\,\mathrm{d}\bar{x}^{i}\wedge\mathrm{d}\bar{x}^{j}\Big{)}\
.$ (4.32)

To recover a reduced sigma-model on the leaf space, we impose the constraint
obtained from the equation of motion for the worldsheet gauge field $A,$
$\delta{\mathcal{S}}[\phi,A]/\delta A_{i}=0,$ which appears quadratically as
an auxiliary field. It is given explicitly by

$\mathrm{D}^{A}\bar{\tilde{x}}{}_{i}=\bar{N}_{ki}\,\mathrm{d}\bar{x}^{k}+(\bar{g}_{-}^{-1})_{il}\,\bar{\eta}^{l}_{k}\,\star\,\mathrm{d}\bar{x}^{k}\
.$ (4.33)

By using (4.30) and (4.31) we can write (4.33) in a more covariant form as

$\mathrm{D}^{A}\bar{\mathbb{X}}^{I}=\bar{\eta}^{IJ}\,\bar{\mathcal{H}}_{JK}\,\star\,\mathrm{d}\bar{\mathbb{X}}^{K}\
,$

which for $A=0$ is the immediate generalization of the self-duality constraint
written in [20, 47, 24]. By substituting the constraint (4.33) in (4.32), we
obtain the local expression

$S[\phi]=\frac{1}{2}\,\int_{\Sigma}\,(\bar{g}_{+})_{ij}\,\mathrm{d}\bar{x}^{i}\wedge\star\,\mathrm{d}\bar{x}^{j}+\int_{\Sigma}\,\bar{N}_{ik}\,\bar{\eta}^{k}_{j}\,\mathrm{d}\bar{x}^{i}\wedge\mathrm{d}\bar{x}^{j}\
.$ (4.34)

In the quantum theory, integrating out $A_{i}$ in the functional integral
formally generates a determinant involving
$\det(\bar{g}_{-})=\det(\bar{g}_{+})^{-1}$ which contributes a Fradkin-
Tseytlin term with dilaton field

$\Psi=-\tfrac{1}{2}\log\det(g_{+})$

in the sigma-model action functional; this gives the required generalized
T-duality invariant correction to the dilaton.

It follows that the reduced sigma-model (4.34) is well-defined on the leaf
space if the condition

$\pounds_{X_{-}}g_{+}=0$

holds for all $X_{-}\in\mathsf{\Gamma}(L_{-}),$ so that ${\mathcal{H}}$ is a
bundle-like metric; in other words,
$\pounds_{X_{-}}{\mathcal{H}}\in\mathsf{\Gamma}(T^{*}M\otimes L_{-}^{*}),$ and
$(M,g_{+},{\mathcal{F}}_{-})$ is a Riemannian foliation. If
$\psi_{i}^{j}\in\mathsf{\Gamma}(L_{-}^{*}),$ then the Lie derivative of
${\mathcal{H}}$ along any vector field in $\mathsf{\Gamma}(L_{-})$ is still an
element of $\mathsf{\Gamma}(T^{*}M\otimes L_{-}^{*})$ and hence the condition
(4.21) still holds, since $\iota_{X_{-}}\omega\in\mathsf{\Gamma}(L_{+}^{*}),$
for all $X_{-}\in\mathsf{\Gamma}(L_{-}).$

Let us focus now on the topological term of the reduced sigma-model (4.34). We
would like to find conditions under which the 2-form

$b=N_{ik}\,\eta^{k}_{j}\,\mathrm{d}x^{i}\wedge\mathrm{d}x^{j}$

is at least locally well-defined on the leaf space
${\mathcal{Q}}=M/{\mathcal{F}}_{-}.$ This 2-form arises from a local frame
spanning the sub-bundle $L_{+},$ so it involves the locally defined functions
$N_{ik}$ which characterize the frame for $L_{+}.$ To obtain a condition
involving these functions, we will consider the transverse components to the
foliation of the Lie derivative of $\omega$ along vector fields parallel to
the foliation. Following [44], we require the condition

${}^{\tau}\nabla\omega=0\ ,$

which implies transversal invariance of the fundamental 2-form, i.e. for all
$X_{-}\in\mathsf{\Gamma}(L_{-})$:

$\pounds_{X_{-}}\omega(Y_{+},Z_{+})=0\ ,$ (4.35)

for all $Y_{+},Z_{+}\in\mathsf{\Gamma}(L_{+}).$ This implies that the split
signature metric satisfies the invariance condition $\pounds_{X_{-}}\eta=0$,
and the frame spanning $L_{+}$ which diagonalizes $K$ is composed of (local)
projectable vector fields. If we write any section of $L_{+}$ in the local
basis diagonalizing $K$, then this condition imposes the local constraint

$\pounds_{X_{-}}N_{ik}=0\ .$

Then the metric (4.30) coincides with the pp-wave type split signature metrics
proposed by [34], proving here that these exhaust (locally) the allowed non-
constant split signature metrics for double field theory. It follows that then
the local 2-form
$b=N_{ik}\,\eta^{k}_{j}\,\mathrm{d}x^{i}\wedge\mathrm{d}x^{j}$ is well-defined
on the leaf space.

Because of the isotropy of $L_{-}$ with respect to $\omega$ and its
involutivity, it is also easy to show that

$\pounds_{X_{-}}\omega(Y_{-},Z_{-})=0\ ,$

for all $Y_{-},Z_{-}\in\mathsf{\Gamma}(L_{-}).$ Together with (4.35) this
implies that $\pounds_{X_{-}}\omega$, like $\omega$, is an element of
$\mathsf{\Gamma}(L_{+}^{*}\wedge L_{-}^{*}).$ It follows that the Lie
derivative of $\omega$ along any vector field from $\mathsf{\Gamma}(L_{-})$
satisfies (4.22) if the connection coefficients of $\nabla$ satisfy
$\Omega_{i}^{j}\in\mathsf{\Gamma}(L_{-}^{*}).$ Combining this constraint with
the constraint obtained from the generalized isometry condition for
${\mathcal{H}}$, we may still consider the Bott connection $\nabla^{\tt B}$ as
the Lie algebroid connection on $T{\mathcal{F}}_{-}$ and restrict it further
to get a map

$\nabla^{\tt
B}:\mathsf{\Gamma}(T{\mathcal{F}}_{-})\times\mathsf{\Gamma}(T{\mathcal{F}}_{-})\longrightarrow\mathsf{\Gamma}(T{\mathcal{F}}_{-})\
,$

so that $\nabla^{\tt B}\pm\psi$ with
$\psi\in\mathsf{\Gamma}(L_{-}^{*}\otimes{\sf End}(T{\mathcal{F}}_{-}))$ give
well-defined connections on $T{\mathcal{F}}_{-}.$ This restriction of
$\nabla^{\tt B}$ is well-defined because of the properties of Bott
connections.

The 2-form $b$ on the leaf space emerging from this description is not always
globally defined; a global construction involving the 3-form
${\mathcal{K}}=\mathrm{d}\omega$ should, in principle, resemble the
construction implemented in [89]. A first step towards extending this gauging
procedure to further include open string sigma-models can be performed
following the formalism of Hamiltonian Lie algebroids [90].

###  4.4. Generalized T-Duality and Non-Geometric Backgrounds

We will now discuss the role of ${\sf O}(d,d)(M)$-transformations for Born
sigma-models and how they relate to their gauging. We saw in Section 3 that,
given an almost para-Hermitian manifold $(M,\eta,K)$ with compatible
generalized metric ${\mathcal{H}},$ the Born structure
$(\eta,K,{\mathcal{H}})$ is mapped into another Born structure
$(\eta,K_{\vartheta},{\mathcal{H}}_{\vartheta})$ on $M$ by
$\vartheta\in{\mathsf{O}}(d,d)(M).$ Thus starting from a Born geometry which
defines a Born sigma-model ${\mathcal{S}}({\mathcal{H}},\omega)$ with the
target space $M,$ an ${\mathsf{O}}(d,d)(M)$-transformation gives another Born
sigma-model ${\mathcal{S}}({\mathcal{H}}_{\vartheta},\omega_{\vartheta})$ with
the new Born structure on the same target space. This is a generalized
T-duality transformation which relates two Born sigma-models.

To interpret generalized T-duality in the context of gauged Born sigma-models,
we first focus on ${\mathsf{O}}(d,d)(M)$-transformations which relate Born
structures $(\eta,K,\omega,{\mathcal{H}})$ and
$(\eta,K_{\vartheta},\omega_{\vartheta},{\mathcal{H}}_{\vartheta})$ for which
both $K$ and $K_{\vartheta}$ have at least one integrable eigenbundle, which
without loss of generality we may assume corresponds to the same eigenvalue
$-1$. We write $L_{-}$ for the integrable sub-bundle of $K$ and
$L^{\vartheta}_{-}$ for the integrable sub-bundle of $K_{\vartheta}.$ Then the
Born manifold $(M,\eta,K,{\mathcal{H}})$ is foliated by ${\mathcal{F}}_{-}$
such that $L_{-}=T{\mathcal{F}}_{-}$ and
$(M,\eta,K_{\vartheta},{\mathcal{H}}_{\vartheta})$ is foliated by
${\mathcal{F}}^{\vartheta}_{-}$ such that
$L^{\vartheta}_{-}=T{\mathcal{F}}^{\vartheta}_{-}.$ Whenever the backgrounds
defining both Born sigma-models ${\mathcal{S}}({\mathcal{H}},\omega)$ and
${\mathcal{S}}({\mathcal{H}}_{\vartheta},\omega_{\vartheta})$ for $M$ satisfy
the generalized isometry conditions, we can gauge both sigma-models to reduce
them to two distinct non-linear sigma-models $S(g,b)$ and
$S(g_{\vartheta},b_{\vartheta})$ for the leaf spaces $M/{\mathcal{F}}_{-}$ and
$M/{\mathcal{F}}^{\vartheta}_{-}$ respectively. The reduced sigma-models are
defined, respectively, by the Riemannian metric $g$ and the 2-form $b$ induced
by $({\mathcal{H}},\omega)$ on $M/{\mathcal{F}}_{-}$, and the metric
$g_{\vartheta}$ and 2-form $b_{\vartheta}$ induced by
$({\mathcal{H}}_{\vartheta},\omega_{\vartheta})$ on
$M/{\mathcal{F}}^{\vartheta}_{-}.$ We say that the non-linear sigma-models
$S(g,b)$ and $S(g_{\vartheta},b_{\vartheta})$ recovered in this way are _T-
dual_ to each other.

We may picture this prescription through the diagram

${(M,{\mathcal{H}},\omega)}$${(M,{\mathcal{H}}_{\vartheta},\omega_{\vartheta})}$${(M/{\mathcal{F}}_{-},g,b)}$${(M/{\mathcal{F}}_{-}^{\vartheta},g_{\vartheta},b_{\vartheta})}$$\scriptstyle{\vartheta}$$\scriptstyle{{\mathit{\Pi}}}$$\scriptstyle{{\mathit{\Pi}}_{\vartheta}}$$\scriptstyle{{\mathscr{T}}}$

where $\vartheta\in{\mathsf{O}}(d,d)(M)$, and the dashed arrow ${\mathscr{T}}$
is not a map but rather the generalized T-duality relation between the sigma-
models $S(g,b)$ and $S(g_{\vartheta},b_{\vartheta})$ defined by the
backgrounds on the respective leaf spaces. The vertical arrows are the
Riemannian submersions ${\mathit{\Pi}}$ of $(M,{\mathcal{H}},\omega)$ onto
$(M/{\mathcal{F}}_{-},g,b),$ which is physically defined by imposing the
dynamical self-duality constraint
$\delta{\mathcal{S}}({\mathcal{H}},\omega)/\delta A_{i}=0$, and
${\mathit{\Pi}}_{\vartheta}$ of
$(M,{\mathcal{H}}_{\vartheta},\omega_{\vartheta})$ onto
$(M/{\mathcal{F}}^{\vartheta}_{-},g_{\vartheta},b_{\vartheta})$ which is
similarly defined by the self-duality constraint
$\delta{\mathcal{S}}({\mathcal{H}}_{\vartheta},\omega_{\vartheta})/\delta
A_{i}=0.$ These constraints relate derivatives of the pullback of the leaf
coordinates $\tilde{x}_{i}$ to derivatives of the pullback of the physical
coordinates $x^{i}$ on the space of leaves as in (4.33), and together with the
generalized isometry conditions they constitute the generalization of the
strong constraint of double field theory.

The reduced sigma-models may also be used to geometrically characterize the
choice of polarization. When the leaf space
${\mathcal{Q}}=M/{\mathcal{F}}_{-}$ is a smooth manifold and the reduced
sigma-model $S(g,b)$ involves well-defined background fields on
${\mathcal{Q}},$ it corresponds to a _geometric background_ and the
corresponding polarization is called a _geometric polarization_. Otherwise, if
the leaf space ${\mathcal{Q}}$ does not admit a smooth structure but the
background fields $(g,b)$ are still well-defined on ${\mathcal{Q}}$, i.e. they
come from a background $({\mathcal{H}},\omega)$ satisfying the generalized
isometry conditions for the Born sigma-model, we call it a _locally geometric
background_. Using the terminology of [20], we will refer to such leaf spaces
as _T-folds_ , and the corresponding polarization defining the Born sigma-
model which leads to a T-fold will be called a _T-fold polarization_. In
contrast to common lore, initiated by [69], it is possible for both geometric
and non-geometric backgrounds in this sense to have non-vanishing ‘$Q$-flux’,
as (4.23) and the general analysis of Section 4.3 shows. That $Q$-flux is not
necessarily an obstruction to global geometry was also highlighted by [25].

In a T-fold polarization, the foliation ${\mathcal{F}}_{-}$ defines a Lie
subalgebroid of the tangent algebroid $(TM,\mathds{1}_{TM})$ that is naturally
integrated by the holonomy groupoid of ${\mathcal{F}}_{-}$ presenting the
space of leaves, which however is no longer a Lie subgroupoid of the pair
groupoid $M\times M\rightrightarrows M$ [86]. The (singular) quotient
${\mathcal{Q}}=M/{\mathcal{F}}_{-}$ can also be presented in a more invariant
way as a _smooth_ stack, even for singular foliations ${\mathcal{F}}_{-}$, and
generalized T-duality can be realized as a morphism of stacks. This
perspective was developed by [91] in the more general context of stratified
spaces, which include the orbifolds and symmetric spaces that appear in the
following, while topological T-duality and T-folds are described in such a
geometric framework by [92] for the polarizations obtained from torus bundles
with NS–NS $H$-flux. An interesting special class of T-folds which admit a
precise geometric description are the foliated Born manifolds whose leaves are
compact. Since the generalized isometry condition for the compatible
generalized metric ${\mathcal{H}}=g_{+}+g_{-}$ implies that
$(M,g_{+},{\mathcal{F}}_{-})$ is a Riemannian foliation, in these cases the
leaf space admits the structure of an orbifold with isotropy group given by
the leaf holonomy group, and ${\mathit{\Pi}}:M\to{\mathcal{Q}}$ is an orbifold
submersion, see [81, 71, 93] for further details; in this case, a more
invariant description of the orbifold ${\mathcal{Q}}$ is as a smooth real
Deligne-Mumford stack.

The other scenario which can arise is when a generalized T-duality
${\vartheta}\in{\sf O}(d,d)(M)$ maps a Born manifold
$(M,\eta,K,{\mathcal{H}})$ with an integrable eigenbundle $L_{-}\subset TM$ of
$K$ into another Born manifold
$(M,\eta,K_{{\vartheta}},{\mathcal{H}}_{{\vartheta}})$ with eigenbundle
$L^{{\vartheta}}_{-}\subset TM$ of $K_{\vartheta}$ which is no longer
integrable; this is the case of non-vanishing ‘$R$-flux’ $R^{mnk}\neq 0$ in
(2.42). In this case, there is still a well-defined Born sigma-model
${\mathcal{S}}({\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})$ which is
related to the Born sigma-model ${\mathcal{S}}({\mathcal{H}},\omega)$ by an
${\sf O}(d,d)(M)$-transformation. However, even if a frame spanning
$\mathsf{\Gamma}(L^{{\vartheta}}_{-})$ and
${\mathcal{S}}({\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})$ satisfy the
generalized isometry conditions, a gauging of
${\mathcal{S}}({\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})$ which
recovers a conventional spacetime description is not possible since $M$ is no
longer a foliated manifold: there is no submersion from
$(M,{\mathcal{H}}_{{\vartheta}},\omega_{\vartheta})$ because there is no leaf
space in this case. This situation is summarized by the diagram

${(M,{\mathcal{H}},\omega)}$${(M,{\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})}$${(M/{\mathcal{F}}_{-},g,b)}$${(\
\cdot\ ,\ \cdot\ ,\ \cdot\
)}$$\scriptstyle{\vartheta}$$\scriptstyle{{\mathit{\Pi}}}$$\scriptstyle{{\mathscr{T}}}$

The vertical dashed arrow here indicates the impossibility of recovering any
conventional background, even locally.

In this instance one could try to implement a similar version of the gauging
of the Born sigma-model
${\mathcal{S}}({\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})$ upon
introducing the analogue of a covariantized map $\mathrm{D}^{\mathcal{A}}\phi$
which is defined by the bundle maps

$T\Sigma\xlongrightarrow{\bar{{\mathcal{A}}}}L^{{\vartheta}}_{-}\xlongrightarrow{i}TM$

where $\bar{{\mathcal{A}}}:T\Sigma\rightarrow L^{{\vartheta}}_{-}$ is a bundle
map which is _not_ generally induced by the pullback of a Lie algebroid
connection, since $L^{{\vartheta}}_{-}$ is not naturally a Lie algebroid in
this case. There is again the pullback bundle $\phi^{*}{\rm
Im}(i\,\circ\,\bar{{\mathcal{A}}}\,)\subset\phi^{*}TM$ and the induced map
$\overline{i({\mathcal{A}})}:T\Sigma\rightarrow\phi^{*}TM$ which permits us to
write the analogue of the covariant derivative

$\mathrm{D}^{\mathcal{A}}\phi=\mathrm{d}\phi-\overline{i({\mathcal{A}})}\ .$

We can associate to this map the tensor
$\underline{\mathrm{D}^{\mathcal{A}}\phi}\in\mathsf{\Gamma}(T^{*}\Sigma\otimes\phi^{*}TM).$
A “gauged” sigma-model can still be defined with this map since the norm
$\|\,\underline{\mathrm{D}^{\mathcal{A}}\phi}\,\|_{h,{\mathcal{H}}}$ is well-
defined on the vector space of sections
$\mathsf{\Gamma}(T^{*}\Sigma\otimes\phi^{*}TM).$ This construction depends on
the choice of bundle map $\bar{\mathcal{A}}$.

To give a physical meaning to this construction, we pass to the local picture
and introduce the analogue of a covariant derivative for only half of the
coordinates; however these coordinates now have no particular geometric
significance. We can write down a self-duality constraint
$\delta{\mathcal{S}}({\mathcal{H}}_{{\vartheta}},\omega_{{\vartheta}})/\delta{\mathcal{A}}_{i}=0,$
but the solution to this constraint does not eliminate the dependence of the
background fields on the “gauged coordinates”, and is moreover expected to
involve a non-local expression. This means that it is not possible to find
even a locally defined conventional background on some open subset of $M.$
Thus there is no reduced sigma-model that can be recovered, since there is no
well-defined quotient and hence no physical spacetime to serve as a target for
a reduced sigma-model. The polarization in which this happens can thus only be
described in the full doubled formalism based on the Born manifold
$(M,{\mathcal{H}}_{\vartheta},\omega_{\vartheta})$; using the terminology of
[60], we say that this polarization is associated with an _essentially doubled
space_ , and call the corresponding polarization an _essentially doubled
polarization_.

###  4.5. Weakly versus Strongly T-Dual Sigma-Models

We will now discuss how to distinguish T-dual sigma-models based on the
geometry of the underlying foliations. For this, we stress a further
distinction amongst generalized T-duality transformations. We may apply an
${\sf O}(d,d)(M)$-transformation $\vartheta$ preserving the foliation induced
by the almost para-complex structure and preserving the transverse metric
$g_{+}$ to the foliation, i.e. $(M,{\mathcal{F}}_{-},{\mathcal{H}},\omega)$
and $(M,{\mathcal{F}}_{-},{\mathcal{H}}_{\vartheta},\omega_{\vartheta})$ are
both Riemannian foliations with respect to the same metric $g_{+}$ and have
the same leaf space ${\mathcal{Q}}=M/{\mathcal{F}}_{-}.$ In this case, the
only difference between the reduced sigma-models is given by the topological
term for the leaf space ${\mathcal{Q}},$ i.e. the T-dual sigma-models are
given by the backgrounds $({\mathcal{Q}},g_{+},b)$ and
$({\mathcal{Q}},g_{+},b_{\vartheta}).$ These sigma-models thus have the same
dynamical content, since the same transverse metric $g_{+}$ appears in the
background of each. In this sense, such sigma-models are _weakly_ T-dual to
each other; weak T-duality acts on an exact Courant algebroid, and is
physically a manifest symmetry of the low-energy effective target space
supergravity theory on ${\mathcal{Q}}$. In contrast, _strongly_ T-dual sigma-
models arise when applying ${\mathsf{O}}(d,d)(M)$-transformations which map a
Riemannian foliation into a different Riemannian foliation, each associated
with a different Born structure.

It follows from Remark 3.18 that a weak generalized T-duality transformation
is exactly a $B$-transformation. Let
$(M,{\mathcal{F}}_{-},\eta,\omega,{\mathcal{H}})$ be a foliated Born manifold
with splitting of its tangent bundle

$TM=T{\mathcal{F}}_{-}\oplus L_{+}$

induced by the almost para-complex structure $K.$ The compatible generalized
metric is

${\mathcal{H}}=g_{+}+g_{-}$

in this splitting, where $g_{+}$ and $g_{-}$ are fiberwise metrics on $L_{+}$
and $T{\mathcal{F}}_{-}$, respectively, which are related by (2.29). We may
think of this splitting as the bundle map

$s:N{\mathcal{F}}_{-}\longrightarrow TM$

that splits the short exact sequence of vector bundles

$0\longrightarrow T{\mathcal{F}}_{-}\longrightarrow TM\longrightarrow
N{\mathcal{F}}_{-}\longrightarrow 0$ (4.36)

which is maximally isotropic with respect to the split signature metric $\eta$
and orthogonal to $T{\mathcal{F}}_{-}$ in the compatible generalized metric
${\mathcal{H}}.$ We assume that $(M,g_{+},{\mathcal{F}}_{-})$ is a Riemannian
foliation, therefore the associated Born sigma-model
${\mathcal{S}}({\mathcal{H}},\omega)$ can be reduced to a conventional non-
linear sigma-model $S(g_{+},\omega\rvert_{L_{+}})$ for the leaf space
${\mathcal{Q}},$ i.e. there exists a Riemannian submersion

${\mathit{\Pi}}:(M,{\mathcal{H}},\omega)\longrightarrow({\mathcal{Q}},g_{+},\omega\rvert_{L_{+}})\
.$

A weakly T-dual sigma-model is obtained by applying a $B_{+}$-transformation,
which preserves $T{\mathcal{F}}_{-}$ and is generated by a basic 2-form
$b_{+}\in\mathsf{\Gamma}(\text{\Large$\wedge$}^{2}L_{+}^{*})$, as discussed in
Section 3.3. The $B_{+}$-transformed Born structure is given by
$(K^{B_{+}},\eta,{\mathcal{H}}^{B_{+}},\omega^{B_{+}}),$ where
$\omega^{B_{+}}=\omega+2\,b_{+}$ and $K^{B_{+}}=K+2\,B_{+}$ with $B_{+}$ a
bundle map from $L_{+}$ to $T{\mathcal{F}}_{-}.$ It induces the splitting

$TM=T{\mathcal{F}}_{-}\oplus L_{+}^{B_{+}}$

that can be regarded as a different choice of splitting

$s^{B_{+}}:N{\mathcal{F}}_{-}\longrightarrow TM$

of the short exact sequence (4.36) such that ${\rm Im}(s^{B_{+}})$ is still
maximally isotropic with respect to $\eta,$ which naturally follows from the
fact that $s$ changes by a $B_{+}$-transformation.

Let us concentrate on the action of the foliation-preserving
$B_{+}$-transformation on the compatible generalized metric. We can easily
show that an ${\mathcal{F}}_{-}$-preserving ${B_{+}}$-transformation is an
isometry of the fiberwise metric $g_{+}$ on $L_{+}$: any section
$X_{+}\in\mathsf{\Gamma}(L_{+})$ transforms as

$X_{+}^{B_{+}}=e^{B_{+}}(X_{+})=X_{+}+{B_{+}}(X_{+})\ ,$

so that

$g_{+}\big{(}X^{B_{+}}_{+},Y^{B_{+}}_{+}\big{)}=g_{+}(X_{+},Y_{+})\ .$

Thus the eigenbundle $L_{+}^{B_{+}}$ of $K^{B_{+}}$, whose sections are of the
form $X^{B_{+}}_{+}$, inherits the fiberwise Riemannian metric $g_{+},$ and
the structure of $(M,g_{+},{\mathcal{F}}_{-})$ as a Riemannian foliation is
preserved. We can also show that the only effect of the
${B_{+}}$-transformation on the compatible generalized metric ${\mathcal{H}}$
is to introduce a new fiberwise metric on $T{\mathcal{F}}_{-}.$ In other
words, the fiberwise metric $g_{-}$ on $T{\mathcal{F}}_{-}$ does not have
$L_{+}^{B_{+}}$ as its kernel:

$g_{-}\big{(}X_{+}^{B_{+}},Y_{+}^{B_{+}}\big{)}=g_{-}\big{(}{B_{+}}(X_{+}),{B_{+}}(Y_{+})\big{)}\
,$

and so the change of sub-bundle $L_{+}^{B_{+}}$ is associated with a change of
fiberwise metric on $T{\mathcal{F}}_{-},$ now given by $g_{-}^{B_{+}}$ such
that ${\rm Ker}(g_{-}^{B_{+}})=L_{+}^{B_{+}}.$ The ${B_{+}}$-transformed
compatible generalized metric thus reads

${\mathcal{H}}^{B_{+}}=g_{+}+g_{-}^{B_{+}}$

in the polarization $TM=T{\mathcal{F}}_{-}\oplus L_{+}^{B_{+}}$ defined by
$K^{B_{+}},$ i.e. by the splitting $s^{B_{+}}$ of (4.36) which is now
orthogonal to $T{\mathcal{F}}_{-}$ with respect to ${\mathcal{H}}^{B_{+}}.$

Since the ${B_{+}}$-transformation preserves the Riemannian foliation and the
2-form $b_{+}$ is basic, we can still obtain a well-defined Riemannian
submersion from $M$ to the leaf space ${\mathcal{Q}}$ given by

${\mathit{\Pi}}^{B_{+}}:(M,{\mathcal{H}}^{B_{+}},\omega^{B_{+}})\longrightarrow({\mathcal{Q}},g_{+},\omega\rvert_{L_{+}}+2\,b_{+})\
.$

Then the sigma-models $S(g_{+},\omega\rvert_{L_{+}})$ and
$S(g_{+},\omega\rvert_{L_{+}}+2\,b_{+}),$ each defined with the same leaf
space ${\mathcal{Q}}$ as target space, are T-dual. They have the same
dynamical content, as they are defined by the same metric $g_{+}$, whereas the
topological term changes. In this sense they are weakly T-dual to each other:
Classically they have the same local degrees of freedom and differ only in
their global structure (which can lead to differences in the quantum theory).
It follows that weakly T-dual sigma-models are classified by the cohomology of
basic 2-forms.

On the other hand, a strongly T-dual sigma-model of
$S(g_{+},\omega\rvert_{L_{+}})$ may be thought of as induced by a maximally
isotropic splitting

$s^{\vartheta}:N{\mathcal{F}}^{\vartheta}_{-}\longrightarrow TM$

with respect to $\eta$ of the short exact sequence

$0\longrightarrow T{\mathcal{F}}^{\vartheta}_{-}\longrightarrow
TM\longrightarrow N{\mathcal{F}}^{\vartheta}_{-}\longrightarrow 0$

for $\vartheta\in{\sf O}(d,d)(M)$, which corresponds to an almost para-
Hermitian structure $(K^{\vartheta},\eta,\omega^{\vartheta})$ on $M$. The
compatible generalized metric ${\mathcal{H}}^{\vartheta}$ decomposes as

${\mathcal{H}}^{\vartheta}=g^{\vartheta}_{+}+g^{\vartheta}_{-}\ ,$

where $g^{\vartheta}_{+}$ is a fiberwise metric on $L^{\vartheta}_{+}$ such
that $(M,g^{\vartheta}_{+},{\mathcal{F}}^{\vartheta}_{-})$ is a Riemannian
foliation.

In the remainder of this paper we will illustrate the constructions of this
section through several explicit examples.

##  5\. Born Sigma-Models for Phase Spaces

A large class of examples which are well-suited to explicit realization of the
formalism of Section 4 come from Born structures on fiber bundles. These
examples naturally supply Riemannian submersions from the total space $M$ to
the base space ${\mathcal{Q}}$, which can be regarded as the smooth quotient
$M/{\mathcal{F}}$ of the total space with respect to the foliation of the
bundle given by the fibers ${\mathcal{F}}.$ We will also consider a different
quotient, which in Section 7 will be used to give a geometric interpretation
of the prototypical T-folds in this framework. Our working example will be the
cotangent bundle $T^{*}{\mathcal{Q}}$, which can be thought of as the phase
space for a closed string with target space ${\mathcal{Q}}$. This nicely ties
our worldsheet formalism from Section 4 with the old sigma-models for duality-
symmetric string theory based on phase space targets [46] and with more recent
discussions of phase spaces as instances of doubled geometry [61, 94, 95, 28,
96, 97, 98, 99, 41, 65].

###  5.1. Para-Kähler Structure on the Cotangent Bundle

We first recall how to define a Born structure on the cotangent bundle of any
smooth manifold by specializing the general discussion of Examples 2.10 and
2.39. Let ${\mathcal{Q}}$ be a smooth manifold with
$\mathrm{dim}({\mathcal{Q}})=d.$ Its cotangent bundle is the vector bundle

$\pi:T^{*}{\mathcal{Q}}\longrightarrow{\mathcal{Q}}\ ,$ (5.1)

where $\pi$ is the canonical projection and the typical fiber ${\mathcal{F}}$
is diffeomorphic to $\mathbb{R}^{d}.$ Since $\pi$ is a surjective submersion,
there is a short exact sequence of vector bundles

$0\longrightarrow L_{\tt
v}(T^{*}{\mathcal{Q}})\xlongrightarrow{i}T(T^{*}{\mathcal{Q}})\xlongrightarrow{\hat{\pi}}\pi^{*}(T{\mathcal{Q}})\longrightarrow
0$ (5.2)

where $L_{\tt
v}(T^{*}{\mathcal{Q}})=\mathrm{Ker}(\pi_{*})=\mathrm{Ker}(\hat{\pi})$ is the
vertical sub-bundle defined by the differential of the projection and
$\pi^{*}(T{\mathcal{Q}})$ is the pullback bundle of $T{\mathcal{Q}}$ over
$T^{*}{\mathcal{Q}}$ along $\pi.$ The map $i:L_{\tt
v}(T^{*}{\mathcal{Q}})\rightarrow T(T^{*}{\mathcal{Q}})$ is the canonical
inclusion of the vertical vector sub-bundle into $T(T^{*}{\mathcal{Q}}).$ The
vertical sub-bundle is integrable and can be regarded as a tangent bundle
$L_{\tt v}(T^{*}{\mathcal{Q}})\simeq T\mathbb{R}^{d}.$ The bundle map
$\hat{\pi}:T(T^{*}{\mathcal{Q}})\rightarrow\pi^{*}(T{\mathcal{Q}})$ is
surjective and covers $\pi,$ since there is also a surjective submersion of
$\pi^{*}(T{\mathcal{Q}})$ onto ${\mathcal{Q}}.$

We define an almost para-complex structure on $T^{*}{\mathcal{Q}}$ by choosing
a splitting of the short exact sequence (5.2), i.e. we fix a right inverse $C$
of $\hat{\pi}$:

${T(T^{*}{\mathcal{Q}})}$${\pi^{*}(T{\mathcal{Q}})}$$\scriptstyle{\hat{\pi}}$$\scriptstyle{C}$
(5.3)

so that $T(T^{*}{\mathcal{Q}})=\mathrm{Im}(C)\oplus\mathrm{Ker}(\hat{\pi}).$
The sub-bundle $\mathrm{Im}(C)=L^{C}_{\tt h}(T^{*}{\mathcal{Q}})$ is one of
the possible choices of horizontal distribution: the map $C$ is usually
understood as a horizontal lift of sections of $T{\mathcal{Q}}$ to
$T(T^{*}{\mathcal{Q}}).$ This defines an Ehresmann connection on
$T^{*}{\mathcal{Q}},$ with

$T(T^{*}{\mathcal{Q}})=L^{C}_{\tt h}(T^{*}{\mathcal{Q}})\oplus L_{\tt
v}(T^{*}{\mathcal{Q}})\ .$

The horizontal lift $C$ thus defines a vector sub-bundle $L^{C}_{\tt
h}(T^{*}{\mathcal{Q}})$ of $T(T^{*}{\mathcal{Q}}),$ which is generally not
involutive. A splitting (5.3) of $T(T^{*}{\mathcal{Q}})$ is equivalent to a
choice of an almost para-complex structure on $T^{*}{\mathcal{Q}}$: we define
the almost para-complex structure $K_{C}\in{\sf
Aut}_{\mathds{1}}(T(T^{*}{\mathcal{Q}}))$ by

$K_{C}\big{|}_{L^{C}_{\tt h}(T^{*}{\mathcal{Q}})}=\mathds{1}_{L^{C}_{\tt
h}(T^{*}{\mathcal{Q}})}\qquad\mbox{and}\qquad K_{C}\big{|}_{L_{\tt
v}(T^{*}{\mathcal{Q}})}=-\mathds{1}_{L_{\tt v}(T^{*}{\mathcal{Q}})}\ .$

The phase space $T^{*}{\mathcal{Q}}$ is endowed with a canonical symplectic
2-form $\omega_{0}$ with respect to which $L_{\tt v}(T^{*}{\mathcal{Q}})$ is
maximally isotropic. We may then ask whether the almost para-complex structure
$K_{C}$ and the canonical symplectic 2-form $\omega_{0}$ satisfy a
compatibility condition such that they induce a split signature metric on
$T^{*}{\mathcal{Q}}.$ In other words, we may ask for conditions ensuring
existence of a split signature metric compatible with $K_{C}$, in the sense of
almost para-Hermitian structures, such that $\omega_{0}$ is the corresponding
fundamental 2-form. Recall that the requisite compatibility condition is

$\omega_{0}\big{(}K_{C}(X),Y\big{)}+\omega_{0}\big{(}X,K_{C}(Y)\big{)}=0\ ,$
(5.4)

for all $X,Y\in\mathsf{\Gamma}(T(T^{*}{\mathcal{Q}})).$ It is straightforward
to check that (5.4) holds if and only if the chosen splitting is isotropic
with respect to $\omega_{0}$. Since $L_{\tt v}(T^{*}{\mathcal{Q}})$ is
maximally isotropic because it is in the kernel of the tautological 1-form on
$T^{*}{\mathcal{Q}},$ this means we have to choose $C$ such that $L^{C}_{\tt
h}(T^{*}{\mathcal{Q}})$ is isotropic with respect to $\omega_{0}.$ We denote
the split signature metric given by such a choice by $\eta_{C}$:

$\eta_{C}(X,Y)=\omega_{0}\big{(}K_{C}(X),Y\big{)}\ ,$

for all $X,Y\in\mathsf{\Gamma}(T(T^{*}{\mathcal{Q}})).$ We then obtain an
almost para-Kähler structure on $T^{*}{\mathcal{Q}}$ given by
$(K_{C},\eta_{C},\omega_{0}).$

######  Remark 5.5.

This construction generalizes to any fiber bundle
$\pi:M\rightarrow{\mathcal{Q}},$ with ${\rm dim}(M)=2\,{\rm
dim}({\mathcal{Q}}),$ which is endowed with a Liouville 1-form [100]. For
this, consider again the short exact sequence of vector bundles (2.11) from
Example 2.10. A _Liouville 1-form_ $\alpha\in\mathsf{\Gamma}(T^{*}M)$ is a
horizontal 1-form on $M,$ i.e. $\iota_{X_{\tt v}}\alpha=0,$ for all $X_{\tt
v}\in\mathsf{\Gamma}(L_{\tt v}(M)).$ Then the foliation given by the fibers of
$M$ is Lagrangian with respect to the symplectic 2-form
$\omega=\mathrm{d}\alpha$ associated with $\alpha,$ since ${\rm
Ker}(\alpha)=L_{\tt v}(M).$ Any choice of an isotropic splitting
$s:\pi^{*}(T{\mathcal{Q}})\rightarrow TM$ of (2.11) with respect to $\omega$
defines an almost para-Kähler structure on $M.$ These structures are all
diffeomorphic to those defined on the cotangent bundle of the base manifold
${\mathcal{Q}}$ by considering the tautological 1-form as a Liouville 1-form.

######  Example 5.6.

Let ${\mathcal{Q}}$ be the configuration space of a dynamical system and
consider the tangent bundle $\pi:T{\mathcal{Q}}\rightarrow{\mathcal{Q}}$ as a
carrier space of the dynamics. The equations of motion of the system is thus
defined by a second order vector field
${\mathit{\Sigma}}\in\mathsf{\Gamma}(T(T{\mathcal{Q}}))$. A regular Lagrangian
${\mathscr{L}}\in C^{\infty}(T{\mathcal{Q}})$ for the dynamical system, when
it exists, defines a Liouville 1-form in the following way.

The _vertical lift_ $X_{\tt v}\in\mathsf{\Gamma}(L_{\tt v}(T{\mathcal{Q}}))$
of a vector field $X\in\mathsf{\Gamma}(T{\mathcal{Q}})$ to
$\mathsf{\Gamma}(T(T{\mathcal{Q}}))$ is the infinitesimal generator of
translations along the fibers, i.e. of the one-parameter group of
diffeomorphisms defined by $\mathbb{R}\ni t\mapsto(q,t\,X\rvert_{q})\in
T_{q}{\mathcal{Q}}$; this induces a map $\bar{{\tt
v}}:T{\mathcal{Q}}\rightarrow L_{\tt v}(T{\mathcal{Q}}).$ The _vertical
endomorphism_ $\upsilon\in\mathsf{End}(T(T{\mathcal{Q}}))$ is the bundle map
given by the composition of the vertical lift and the tangent projection:
$\upsilon=\bar{{\tt v}}\,\circ\,\pi_{*},$ or equivalently the endomorphism of
$T(T{\mathcal{Q}})$ which makes the diagram

${T(T{\mathcal{Q}})}$${T{\mathcal{Q}}}$${T(T{\mathcal{Q}})}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\upsilon}$$\scriptstyle{\bar{{\tt
v}}}$

commute. Then the Liouville 1-form of the Lagrangian dynamics is given
by161616In this context, $\alpha_{\mathscr{L}}$ is usually called the ‘Cartan
1-form’. $\alpha_{\mathscr{L}}=\upsilon(\mathrm{d}{\mathscr{L}})$, since ${\rm
Ker}(\alpha_{\mathscr{L}})={\rm Ker}(\upsilon)=L_{\tt v}(T{\mathcal{Q}}).$
This gives the Lagrangian symplectic 2-form
$\omega_{\mathscr{L}}=\mathrm{d}\alpha_{\mathscr{L}}.$

The dynamical vector field
${\mathit{\Sigma}}\in\mathsf{\Gamma}(T(T{\mathcal{Q}}))$ induces an isotropic
splitting, with respect to $\omega_{\mathscr{L}}$, of the canonical short
exact sequence (2.11) of vector bundles from Example 2.10 with
$M=T{\mathcal{Q}}.$ Hence $(T{\mathcal{Q}},{\mathscr{L}})$ admits a para-
Kähler structure. One can also show that the Lagrangian ${\mathscr{L}}$
induces a compatible generalized metric on $T{\mathcal{Q}}$ [39]. The
symplectomorphism induced by a regular Lagrangian, given by the Legendre
transform from $T{\mathcal{Q}}$ to $T^{*}{\mathcal{Q}},$ is a vector bundle
isomorphism covering the identity which induces a map from the dynamical para-
Kähler structure to an isotropic splitting of (5.2); see [39, 101] for further
details.

As discussed in [39], in a local description we may describe the horizontal
lift of a holonomic frame $\big{\\{}\frac{\partial}{\partial q^{i}}\big{\\}}$
of $T{\mathcal{Q}},$ where $q^{i}$ are local coordinates on ${\mathcal{Q}}$
(pulled back from ${\mathcal{Q}}$ to $T^{*}{\mathcal{Q}}$ by the projection
$\pi$), by the vector fields

${\tt h}_{i}=C\Big{(}\frac{\partial}{\partial
q^{i}}\Big{)}=\frac{\partial}{\partial q^{i}}+C_{ij}\,\frac{\partial}{\partial
p_{j}}\ \in\ \mathsf{\Gamma}\big{(}L^{C}_{\tt h}(T^{*}{\mathcal{Q}})\big{)}$

where $(q^{i},p_{i})$ are local Darboux coordinates on $T^{*}{\mathcal{Q}},$
and $C_{ij}$ are smooth functions on the chosen open subset of
$T^{*}{\mathcal{Q}}$ defining the Darboux chart. This gives a local basis of
sections of the horizontal sub-bundle. Then it is straightforward to see that
$L^{C}_{\tt h}(T^{*}{\mathcal{Q}})$ is maximally isotropic with respect to

$\omega_{0}=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$

if and only if $C_{ij}=C_{ji}$ is symmetric.

To define a Born sigma-model for $T^{*}{\mathcal{Q}}$ we need to define a
compatible generalized metric ${\mathcal{H}}_{C}$ for the almost para-Kähler
structure $(K_{C},\eta_{C},\omega_{0}).$ Following Example 2.39, we endow the
base manifold ${\mathcal{Q}}$ with a Riemannian metric $g$. Then a fiberwise
metric $g_{+}$ on $L^{C}_{\tt h}(T^{*}{\mathcal{Q}})$ is given by the pullback
$g_{+}=\pi^{*}g$:

$g_{+}\big{(}X_{\tt h},Y_{\tt h}\big{)}=g(X,Y)\ ,$ (5.7)

where $X_{\tt h}=C(X)$ and $Y_{\tt h}=C(Y),$ for all
$X,Y\in\mathsf{\Gamma}(T{\mathcal{Q}}),$ i.e. $X_{\tt h}$ and $Y_{\tt h}$ are
arbitrary horizontal lifts. This gives a compatible generalized metric for the
almost para-Kähler structure $(K_{C},\eta_{C},\omega_{0})$ which takes the
form

${\mathcal{H}}_{C}=g_{+}+g_{-}$

in the splitting induced by $K_{C},$ with

$g_{-}(X_{\tt v},Y_{\tt v})=g_{+}^{-1}\big{(}\eta_{C}{}^{\flat}(X_{\tt
v}),\eta_{C}{}^{\flat}(Y_{\tt v})\big{)}$

for all $X_{\tt v},Y_{\tt v}\in\mathsf{\Gamma}(L_{\tt
v}(T^{*}{\mathcal{Q}}))$.

###  5.2. Phase Space Born Sigma-Model and its Gauging

We now have all of the ingredients needed to write down a Born sigma-model for
$T^{*}{\mathcal{Q}}.$ The cotangent bundle Born sigma-model
${\mathcal{S}}({\mathcal{H}}_{C},\omega_{0})$ is given by

${\mathcal{S}}_{0}[\phi]=\frac{1}{4}\,\int_{\Sigma}\,\bigl{(}(\bar{g}_{+})_{ij}\,\mathrm{d}\bar{q}^{i}\wedge\star\,\mathrm{d}\bar{q}^{j}+(\bar{g}_{-})^{ij}\,\bar{\zeta}_{i}\wedge\star\,\bar{\zeta}_{j}\bigr{)}$

and

${\mathcal{S}}_{\rm
top}[\phi]=\frac{1}{2}\,\int_{\Sigma}\,\mathrm{d}\bar{q}^{i}\wedge\mathrm{d}\bar{p}_{i}\
,$

where $\phi$ is a map from the closed string worldsheet $\Sigma$ to the phase
space $T^{*}{\mathcal{Q}}$. Here we wrote the compatible generalized metric
${\mathcal{H}}_{C}$ as

${\mathcal{H}}_{C}=(g_{+})_{ij}\,\mathrm{d}q^{i}\otimes\mathrm{d}q^{j}+(g_{-})^{ij}\,\zeta_{i}\otimes\zeta_{j}\
,$

with $\mathrm{d}q^{i}$ and

$\zeta_{i}=\mathrm{d}p_{i}-C_{ij}\,\mathrm{d}q^{j}$

dual 1-forms to ${\tt h}_{i}$ and $\frac{\partial}{\partial p_{i}}$
respectively, and $(g_{+})_{ij}=g_{ij}$; we also used
$\mathrm{d}q^{i}\wedge\zeta_{i}=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$ in
writing the topological term. The topological term is defined by the
symplectic 2-form $\omega_{0}=-\mathrm{d}\alpha,$ where $\alpha$ is the
tautological 1-form on $T^{*}{\mathcal{Q}}$ (in a Darboux chart,
$\alpha=p_{i}\,\mathrm{d}q^{i}$). Since $\omega_{0}$ is exact and we assume
that $\Sigma$ is closed, the topological term vanishes. However, even in the
case when $\Sigma$ has a non-empty boundary, since $\omega_{0}$ does not have
a component in $\mathsf{\Gamma}(\text{\Large$\wedge$}^{2}L_{{\tt
v}}(T^{*}{\mathcal{Q}})^{*})$ we do not expect any topological term to arise
in the reduced sigma-model. We will keep the topological term explicit in the
following to show that this is indeed the case.

This sigma-model can be gauged, as discussed in Section 4, by considering the
vertical distribution $L_{\tt v}(T^{*}{\mathcal{Q}})$ as a Lie algebroid. In
this case $L_{\tt v}(T^{*}{\mathcal{Q}})$ is the Lie algebroid of symmetries
of the Born sigma-model ${\mathcal{S}}({\mathcal{H}}_{C},\omega_{0}),$ since

$\pounds_{Z_{\tt v}}\,g_{+}=0\ ,$

for all $Z_{\tt v}\in\mathsf{\Gamma}(L_{\tt v}(T^{*}{\mathcal{Q}})),$ because
$\pounds_{Z_{\tt v}}X_{\tt h}\in\mathsf{\Gamma}(L_{\tt
h}^{C}(T^{*}{\mathcal{Q}}))$ with $X_{\tt h}$ the horizontal lift of a vector
field $X\in\mathsf{\Gamma}(T{\mathcal{Q}}),$ and because (5.7) holds. The
gauging is also possible since $\pounds_{Z_{\tt v}}\,\omega_{0}$ has vanishing
component in $\mathsf{\Gamma}(\text{\Large$\wedge$}^{2}L_{\tt
h}^{C}(T^{*}{\mathcal{Q}})^{*}).$ We introduce the connection 1-form $A$ on
$T^{*}{\mathcal{Q}}$ obtained from the Lie algebroid of generalized isometries
of ${\mathcal{H}}_{C}$ and $\omega_{0}.$ As discussed in Section 4, we define
the covariant derivatives

$\mathrm{D}^{A}\bar{q}^{i}=\mathrm{d}\bar{q}^{i}\qquad\mbox{and}\qquad\mathrm{D}^{A}\bar{p}_{i}=\mathrm{d}\bar{p}_{i}-\bar{A}_{i}\
,$

where we covariantize only the pullback of the differential of the leaf
coordinates $p_{i}.$ Here we work with a Darboux chart $(q^{i},p_{i})$, so
that $(p_{i})$ are coordinates adapted to the leaves of $T^{*}{\mathcal{Q}},$
as discussed in [43, 102].

The action functional ${\mathcal{S}}[\phi,A]$ of the resulting gauged Born
sigma-model has two terms and is given by

$\displaystyle{\mathcal{S}}[\phi,A]$
$\displaystyle=\frac{1}{4}\,\int_{\Sigma}\,\Big{(}\big{(}(\bar{g}_{+})_{ij}+\bar{C}_{im}\,(\bar{g}_{-})^{mn}\,\bar{C}_{jn}\big{)}\,\mathrm{d}\bar{q}^{i}\wedge\star\,\mathrm{d}\bar{q}^{j}$
$\displaystyle\hskip
56.9055pt+(\bar{g}_{-})^{ij}\,\mathrm{D}^{A}\bar{p}_{i}\wedge\star\,\mathrm{D}^{A}\bar{p}_{j}-2\,(\bar{g}_{-})^{ik}\,\bar{C}_{jk}\,\mathrm{D}^{A}\bar{p}_{i}\wedge\star\,\mathrm{d}\bar{q}^{j}\Big{)}$
$\displaystyle\quad\,+\frac{1}{2}\,\int_{\Sigma}\,\mathrm{d}\bar{q}^{i}\wedge\mathrm{D}^{A}\bar{p}_{i}\
.$

Then the self-duality constraints $\delta{\mathcal{S}}[\phi,A]/\delta A_{i}=0$
are given by

$\star\,(\bar{g}_{-})^{ij}\,\mathrm{D}^{A}\bar{p}_{j}-(\bar{g}_{-})^{ij}\,\bar{C}_{kj}\,\star\,\mathrm{d}\bar{q}^{k}-\mathrm{d}\bar{q}^{i}=0\
.$

By imposing this constraint we obtain a sigma-model for the quotient
$T^{*}{\mathcal{Q}}/\mathbb{R}^{d}\simeq{\mathcal{Q}}$ with background given
by the Riemannian metric $g$ in which the holonomic basis of $T{\mathcal{Q}}$
is orthonormal, i.e. the reduced sigma-model $S(g,b)$ is given by

$S[\phi]=\frac{1}{2}\,\int_{\Sigma}\,(\bar{g}_{+})_{ij}\,\mathrm{d}\bar{q}^{i}\wedge\star\,\mathrm{d}\bar{q}^{j}\
,$

where here $\phi$ is the harmonic map with its image projected to the leaf
space ${\mathcal{Q}}$ and $(g_{+})_{ij}=g_{ij}$. Thus the sigma-model $S(g,b)$
for ${\mathcal{Q}}$ is characterized by the metric
$g=g_{ij}\,\mathrm{d}q^{i}\otimes\mathrm{d}q^{j}$ on ${\mathcal{Q}},$ which is
not surprising since the compatible generalized metric ${\mathcal{H}}_{C}$ is
defined as a horizontal lift of $g$ to $T^{*}{\mathcal{Q}}$. However, what was
not obvious from the start is that the reduced sigma-model has vanishing Kalb-
Ramond field $b=0,$ i.e. even starting with a topological term, in the almost
para-Kähler case the reduced sigma-models involve a background with vanishing
$B$-field on ${\mathcal{Q}}$. This also means that the Riemannian submersion
$(T^{*}{\mathcal{Q}},{\mathcal{H}}_{C},\omega_{0})\rightarrow({\mathcal{Q}},g,b=0)$
is simply given by the bundle projection
$\pi:T^{*}{\mathcal{Q}}\rightarrow{\mathcal{Q}},$ as expected.

The properties of the class of examples described in this section extend more
generally to arbitrary choices of compatible generalized metric, given by a
fiberwise metric $g_{+}\in\mathsf{\Gamma}(\text{\Large$\odot$}^{2}L_{\tt
h}^{C}(T^{*}{\mathcal{Q}})^{*})$ such that $\pounds_{Z_{\tt v}}\,g_{+}=0,$ for
all $Z_{\tt v}\in\mathsf{\Gamma}(L_{\tt v}(T^{*}{\mathcal{Q}})).$ Although
this simple class of examples gives the obvious result, it aids in
understanding how to deal with the gaugings in general. Furthermore, we can
still use it to understand how to obtain a background for the reduced sigma-
model in this case with a non-trivial $B$-field.

###  5.3. Weakly T-Dual Sigma-Model with $B$-Field

In order to describe T-dual sigma-models for the background
$({\mathcal{Q}},g,b=0),$ we give the first simple example of the construction
discussed in Section 4.5 by considering a $B_{+}$-transformation of the Born
structure introduced in Section 5.1, which is a pushforward of
$(K_{C},\eta_{C},{\mathcal{H}}_{C})$ by $e^{B_{+}}.$ For this, we recall that
$e^{B_{+}}\in{\mathsf{O}}(d,d)(T^{*}{\mathcal{Q}})$ is generated by a skew map
$B_{+}:L^{C}_{\tt h}(T^{*}{\mathcal{Q}})\rightarrow L_{\tt
v}(T^{*}{\mathcal{Q}})$ such that the new almost para-complex structure
$K_{C}^{B_{+}}$ has $L_{\tt v}(T^{*}{\mathcal{Q}})$ and
$L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})$ as its eigenbundles, where
$L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})$ is the sub-bundle obtained from
$L^{C}_{\tt h}(T^{*}{\mathcal{Q}})$ after the $B_{+}$-transformation as
discussed in Section 3.3:

$L_{B_{+}}^{C}(T^{*}{\mathcal{Q}})=\big{\\{}X_{\tt h}+B_{+}(X_{\tt h})\
\big{|}\ X_{\tt h}\in L^{C}_{\tt h}(T^{*}{\mathcal{Q}})\big{\\}}\ .$

Since a $B_{+}$-transformation preserves the vertical sub-bundle, we may think
of the sub-bundle $L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})$ as the horizontal
distribution defining a new splitting (5.3) of the short exact sequence of
vector bundles (5.2):

$T(T^{*}{\mathcal{Q}})=L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})\oplus L_{\tt
v}(T^{*}{\mathcal{Q}})\ ,$

which represents a different Ehresmann connection

$C_{B_{+}}:\pi^{*}(T{\mathcal{Q}})\longrightarrow T(T^{*}{\mathcal{Q}})\ .$

Thus given an isotropic splitting of the short exact sequence (5.2) with
respect to $\omega_{0}$, we can obtain non-isotropic splittings by acting with
$B_{+}$-transformations, which preserve the vertical distribution. The
splittings obtained in this way are maximally isotropic with respect to the
split signature metric $\eta_{C}.$ The associated almost para-Hermitian
structure $(K^{B_{+}}_{C},\eta_{C},\omega^{B_{+}}_{0})$ is obtained as a
$B_{+}$-transformation of the almost para-Kähler structure
$(K_{C},\eta_{C},\omega_{0}),$ whose fundamental 2-form is no longer the
canonical symplectic 2-form on $T^{*}{\mathcal{Q}}$.

It is important to stress that $L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})$ is not
isotropic with respect to $\omega_{0},$ therefore the splitting $C_{B_{+}}$
does not induce an almost para-complex structure which is compatible with
$\omega_{0}.$ In fact, the fundamental 2-form of the $B_{+}$-transformed
structure is not symplectic in general. Recall that, since
$e^{B_{+}}\in{\mathsf{O}}(d,d)(T^{*}{\mathcal{Q}}),$ the
$B_{+}$-transformation preserves the metric $\eta_{C},$ but not the
fundamental 2-form which transforms to

$\omega^{B_{+}}_{0}=\omega_{0}+2\,b_{+}\ ,$

where $b_{+}$ is a horizontal 2-form,171717$b_{+}$ is the pullback of a 2-form
in $\Omega^{2}({\mathcal{Q}})$ since the map $B_{+}$ is constant along the
fibers. i.e. $\iota_{X_{\tt v}}\,b_{+}=0,$ for all $X_{\tt
v}\in\mathsf{\Gamma}(L_{\tt v}(T^{*}{\mathcal{Q}}))$; hence
$\omega^{B_{+}}_{0}$ is closed if and only if $b_{+}$ is closed. Thus
$(T^{*}{\mathcal{Q}},K_{C}^{B_{+}},\eta_{C})$ is generally only an almost
para-Hermitian manifold.

In a local description, we may regard $L^{C}_{B_{+}}(T^{*}{\mathcal{Q}})$ as
the sub-bundle locally spanned by sections

${\tt h}^{B_{+}}_{i}=C_{B_{+}}\Big{(}\frac{\partial}{\partial
q^{i}}\Big{)}=\frac{\partial}{\partial
q^{i}}+\big{(}C_{ij}+(b_{+})_{ij}\big{)}\,\frac{\partial}{\partial p_{j}}\ ,$

where $C_{ij}+(b_{+})_{ij}$ is not symmetric. The map $B_{+}$ can be regarded
as a tensor $\underline{B_{+}}\in\mathsf{\Gamma}(L^{C}_{\tt
h}(T^{*}{\mathcal{Q}})^{*}\otimes L_{\tt v}(T^{*}{\mathcal{Q}}))$ such that

$\underline{B_{+}}=(b_{+})_{ij}\,\mathrm{d}q^{i}\otimes\frac{\partial}{\partial
p_{j}}\ ,$

where $\\{\mathrm{d}q^{i}\\}$ is the local coframe that spans
$\mathsf{\Gamma}(L^{C}_{\tt h}(T^{*}{\mathcal{Q}})^{*}),$ and so is dual to
$\big{\\{}C(\frac{\partial}{\partial q^{i}})\big{\\}}.$ Thus the horizontal
2-form $b_{+}$ reads

$b_{+}=\tfrac{1}{2}\,(b_{+})_{ij}\,\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}\ .$

In summary, we consider a $B_{+}$-transformation, which preserves the vertical
sub-bundle so that it can still be regarded as the Lie algebroid of
generalized isometries of the new structure.

A $B_{+}$-transformation of the compatible generalized metric
${\mathcal{H}}_{C}$ gives rise to a new compatible generalized metric
${\mathcal{H}}_{C}^{B_{+}}$ as in (2.30). In the setting of Section 5.1, the
horizontal lift $g_{+}$ of a Riemannian metric $g$ on ${\mathcal{Q}}$ to
$T^{*}{\mathcal{Q}}$ is mapped into another horizontal lift $g_{+}^{B_{+}}$ of
the same metric $g$ given by

$g_{+}^{B_{+}}\big{(}C_{B_{+}}(X),C_{B_{+}}(Y)\big{)}=g(X,Y)\ ,$

for all $X,Y\in\mathsf{\Gamma}(T{\mathcal{Q}}).$ In this case we can write

${\mathcal{H}}_{C}^{B_{+}}=\big{(}g_{+}^{B_{+}}\big{)}_{ij}\,\mathrm{d}q^{i}\otimes\mathrm{d}q^{j}+\big{(}g_{-}^{B_{+}}\big{)}^{ij}\,\zeta^{B_{+}}_{i}\otimes\zeta^{B_{+}}_{j}\,,$

where $(g_{+}^{B_{+}})_{ij}=g_{ij}$ and
$g_{-}^{B_{+}}=\eta_{C}\,\circ\,(g_{+}^{B_{+}})^{-1}\,\circ\,\eta_{C},$ while

$\zeta^{B_{+}}_{i}=\mathrm{d}p_{i}-\big{(}C_{ij}-(b_{+})_{ij}\big{)}\,\mathrm{d}q^{j}$

is the local coframe spanning $\mathsf{\Gamma}(L_{\tt
v}(T^{*}{\mathcal{Q}})^{*}),$ i.e. it is dual to $\frac{\partial}{\partial
p_{i}}$ in the $B_{+}$-transformed polarization.

We thus obtain a new Born sigma-model
${\mathcal{S}}\big{(}{\mathcal{H}}_{C}^{B_{+}},\omega^{B_{+}}_{0}\big{)}$ for
$T^{*}{\mathcal{Q}}$ given by

${\mathcal{S}}^{B_{+}}[\phi]=\frac{1}{4}\,\int_{\Sigma}\,\Big{(}\big{(}\bar{g}_{+}^{B_{+}}\big{)}_{ij}\,\mathrm{d}\bar{q}^{i}\wedge\star\,\mathrm{d}\bar{q}^{j}+\big{(}\bar{g}_{-}^{B_{+}}\big{)}^{ij}\,\bar{\zeta}^{B_{+}}_{i}\wedge\star\,\bar{\zeta}^{B_{+}}_{j}\Big{)}+\frac{1}{2}\,\int_{\Sigma}\,\mathrm{d}\bar{q}^{i}\wedge\bar{\zeta}^{B_{+}}_{i}$

and it can still be gauged with respect to the generalized isometries
generated by the vertical distribution. Following the same steps as in Section
5.2, the generalized isometry condition

$\pounds_{Z_{\tt v}}\,g^{B_{+}}_{+}=0$

holds for all $Z_{\tt v}\in\mathsf{\Gamma}(L_{\tt v}(T^{*}{\mathcal{Q}}))$
since $g_{+}^{B_{+}}$ is the horizontal lift of a Riemannian metric $g$ on
${\mathcal{Q}}$, and the component $b_{+}$ of the fundamental 2-form
$\omega^{B_{+}}_{0}$ must satisfy the condition

$\pounds_{Z_{\tt v}}\,b_{+}=0\ ,$

for all $Z_{\tt v}\in\mathsf{\Gamma}(L_{\tt v}(T^{*}{\mathcal{Q}})),$ which
follows here since $b_{+}$ is the pullback of a 2-form on ${\mathcal{Q}}.$ In
the case of a Born structure arising from a $B_{+}$-transformation one obtains
a global expression for this component of the fundamental 2-form. This is not
always the case, since generally it has a local characterization in terms of
the local expression of a splitting and therefore the induced 2-form on the
leaf space, even when it is smooth, is not necessarily globally defined.

The gauging is analogous to the gauging for the Born sigma-model
${\mathcal{S}}({\mathcal{H}}_{C},\omega_{0})$ described in Section 5.2. In the
present case, the self-duality constraint
$\delta{\mathcal{S}}^{B_{+}}[\phi,A]/\delta A_{i}=0$ from gauging reads

$\star\,\big{(}\bar{g}_{-}^{B_{+}}\big{)}^{ij}\,\mathrm{D}^{A}\bar{p}_{j}-\star\,\big{(}\bar{g}_{-}^{B_{+}}\big{)}^{ij}\,\big{(}\bar{C}_{kj}+(\bar{b}_{+})_{kj}\big{)}\,\mathrm{d}\bar{q}^{k}-\mathrm{d}\bar{q}^{i}=0\
.$

Then the sigma-model for ${\mathcal{Q}}$ obtained by imposing these self-
duality constraints is

$S^{B_{+}}[\phi]=\frac{1}{2}\,\int_{\Sigma}\,\big{(}\bar{g}_{+}^{B_{+}}\big{)}_{ij}\,\mathrm{d}\bar{q}^{i}\wedge\star\,\mathrm{d}\bar{q}^{j}+\int_{\Sigma}\,\bar{b}_{+}\
.$

Hence the reduced sigma-model $S(g,b_{+})$ for the quotient
$T^{*}{\mathcal{Q}}/\mathbb{R}^{d}\simeq{\mathcal{Q}}$ is defined by the same
Riemannian metric $g=g_{ij}\,\mathrm{d}q^{i}\otimes\mathrm{d}q^{j}$ as in the
previous gauging, and Kalb-Ramond field given by
$b_{+}=\frac{1}{2}\,(b_{+})_{ij}\,\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}.$ It
follows that the only effect of a $B_{+}$-transformation, which leaves
unchanged the integrable sub-bundle generating the generalized isometries, is
to give a new topological term for this class of sigma-models. In summary, we
have shown that the sigma-models $S(g,0)$ and $S(g,b_{+})$ for ${\mathcal{Q}}$
can be considered as weakly T-dual sigma-models in the sense of Section 4.5,
i.e. they are related by a weak generalized T-duality transformation. This is
our working example of weakly T-dual sigma-models.

###  5.4. Strongly T-Dual Sigma-Models

An example of how a strongly T-dual sigma-model can be constructed is obtained
by considering a foliation ${\mathcal{F}}^{\vartheta}$ of $T^{*}{\mathcal{Q}}$
of codimension ${\rm dim}({\mathcal{Q}})$ such that
$T{\mathcal{F}}^{\vartheta}$ is maximally isotropic with respect to
$\eta_{C},$ for $\vartheta\in{\mathsf{O}}(d,d)(T^{*}{\mathcal{Q}})$. We then
obtain the short exact sequence

$0\longrightarrow T{\mathcal{F}}^{\vartheta}\longrightarrow
T(T^{*}{\mathcal{Q}})\longrightarrow N{\mathcal{F}}^{\vartheta}\longrightarrow
0$ (5.8)

and an isotropic splitting

$s^{\vartheta}:N{\mathcal{F}}^{\vartheta}\longrightarrow
T(T^{*}{\mathcal{Q}})$

with respect to $\eta_{C}$ defines an almost para-Hermitian structure with
different fundamental 2-form $\omega_{0}^{\vartheta}$ which is not the
canonical symplectic 2-form. Whenever the compatible generalized metric
${\mathcal{H}}_{C}^{\vartheta}$ induces a Riemannian foliation, we then obtain
a reduced sigma-model for the leaf space
$T^{*}{\mathcal{Q}}/{\mathcal{F}}^{\vartheta}$ which is strongly T-dual to the
natural sigma-model constructed in Section 5.2.

This discussion is a natural prelude to describing Born geometries associated
with Lagrangian foliations of $T^{*}{\mathcal{Q}}$ with respect to
$\omega_{0}.$ Consider a foliation ${\mathcal{F}}^{\vartheta}$ of
$T^{*}{\mathcal{Q}}$ such that $T{\mathcal{F}}^{\vartheta}$ is maximally
isotropic with respect to $\omega_{0}.$ A maximally isotropic splitting of the
short exact sequence (5.8) with respect to $\omega_{0}$ gives an almost para-
Kähler structure $(K^{\vartheta},\eta^{\vartheta},\omega_{0}).$ Such a
splitting has split signature metric $\eta^{\vartheta}$ which is in general
different from $\eta_{C}.$ Therefore this structure cannot be obtained via an
${\mathsf{O}}(d,d)(T^{*}{\mathcal{Q}})$-transformation of the canonical almost
para-Kähler structure discussed in Section 5.1. Moreover, any compatible
generalized metric ${\mathcal{H}}^{\vartheta}$ induced by a fiberwise metric
$g^{\vartheta}_{+}$ giving rise to a Riemannian foliation defines a Born
sigma-model which is not T-dual to the canonical Born sigma-model of Section
5.2. In this sense, the distinct T-duality orbits of phase space Born sigma-
models, giving rise to distinct T-duality chains, are classified by Lagrangian
foliations and their allowed Riemannian foliation structures.

###  5.5. Worldsheet Description of Essentially Doubled Backgrounds

We shall now discuss the case in which the gauging is not possible, i.e. it
does not lead to any submersion from $T^{*}{\mathcal{Q}}$ to any orbit space.
For a phase space Born sigma-model, this happens whenever we apply a
$B_{-}$-transformation. For this, let $(K_{C},\eta_{C})$ be an almost para-
Kähler structure on $T^{*}{\mathcal{Q}}$ which is compatible with the
canonical symplectic 2-form $\omega_{0},$ obtained as discussed in Section
5.1. Consider the automorphism $e^{B_{-}}\in{\sf O}(d,d)(T^{*}{\mathcal{Q}})$
covering the identity that is generated by a skew map $B_{-}:L_{\tt
v}(T^{*}{\mathcal{Q}})\rightarrow L^{C}_{\tt h}(T^{*}{\mathcal{Q}}),$ which
can be regarded as a tensor $\underline{B_{-}}\in\mathsf{\Gamma}(L_{\tt
v}(T^{*}{\mathcal{Q}})^{*}\otimes L^{C}_{\tt h}(T^{*}{\mathcal{Q}})),$ as
discussed in Section 3.3. As a section of this tensor bundle, in a local
coordinate chart $\underline{B_{-}}$ takes the form

$\underline{B_{-}}=(\beta_{-})^{ij}\,\zeta_{i}\otimes
C\Big{(}\frac{\partial}{\partial q^{j}}\Big{)}\ ,$

where $\\{\zeta_{i}\\}$ is a local basis of vertical 1-forms and
$\big{\\{}C(\frac{\partial}{\partial q^{i}})\big{\\}}$ is a local basis of
horizontal vector fields, as described in Sections 5.1 and 5.2. The
$B_{-}$-transformation determines a horizontal bivector $\beta_{-}$ which in
local coordinate form reads

$\beta_{-}=\frac{1}{2}\,(\beta_{-})^{ij}\,C\Big{(}\frac{\partial}{\partial
q^{i}}\Big{)}\wedge C\Big{(}\frac{\partial}{\partial q^{j}}\Big{)}\ .$

We then obtain a new almost para-Hermitian structure by pushing forward
$(K_{C},\eta_{C})$ by $e^{B_{-}}.$ The new almost para-complex structure is
given by

$K_{C}^{B_{-}}=e^{B_{-}}\circ K_{C}\circ e^{-B_{-}}=K_{C}+2\,B_{-}\ ,$

while the split signature metric $\eta_{C}$ is preserved. The fundamental
2-form is no longer closed in general and becomes the 2-form
$\omega_{0}^{B_{-}}$, where

$\displaystyle\omega_{0}^{B_{-}}(X,Y)=\omega_{0}(X,Y)+2\,\beta_{-}\big{(}\eta_{C}^{\flat}(X),\eta_{C}^{\flat}(Y)\big{)}$
(5.9)

for all $X,Y\in\mathsf{\Gamma}(T(T^{*}{\mathcal{Q}}))$. The eigenbundles of
$K_{C}^{B_{-}}$ are given by $L^{C}_{\tt h}(T^{*}{\mathcal{Q}}),$ i.e. the
horizontal sub-bundle remains unchanged, and $L_{B_{-}}(T^{*}{\mathcal{Q}})$
as a deformation of the vertical distribution, which is no longer a vertical
sub-bundle. In fact, we may regard (local) sections of
$L_{B_{-}}(T^{*}{\mathcal{Q}})$ as spanned by vector fields of the form

$P^{i}=\frac{\partial}{\partial
p_{i}}+(\beta_{-})^{ij}\,C_{jk}\,\frac{\partial}{\partial
p_{k}}+(\beta_{-})^{ij}\,\frac{\partial}{\partial q^{j}}\ ,$

obtained from the local span of the vertical distribution $L_{\tt
v}(T^{*}{\mathcal{Q}})$ via $e^{B_{-}}.$ The transformation $e^{B_{-}}$
induces a new splitting of $T(T^{*}{\mathcal{Q}}),$ but it is no longer given
by a choice of an Ehresmann connection on $T^{*}{\mathcal{Q}},$ i.e. by a
splitting of the short exact sequence (5.2).

Despite the fact that the eigenbundle $L^{C}_{\tt h}(T^{*}{\mathcal{Q}})$
remains the same, the new complementary sub-bundle (which is maximally
isotropic with respect to $\eta_{C}$) is no longer vertical. This can be
easily seen by considering the pullback $\pi^{*}f\in
C^{\infty}(T^{*}{\mathcal{Q}})$ of any function $f\in
C^{\infty}({\mathcal{Q}}),$ and noticing that

$\pounds_{X}\,\pi^{*}f\neq 0\ ,$

in general for $X\in\mathsf{\Gamma}(L_{B_{-}}(T^{*}{\mathcal{Q}})).$ Therefore
there is still an eigenbundle decomposition

$T(T^{*}{\mathcal{Q}})=L^{C}_{\tt h}(T^{*}{\mathcal{Q}})\oplus
L_{B_{-}}(T^{*}{\mathcal{Q}})\ ,$

but this splitting does not arise from the fiber bundle structure of
$T^{*}{\mathcal{Q}}$; in other words, the transformation $e^{B_{-}}$ does not
map a horizontal distribution into another horizontal distribution, so it does
not preserve the vertical distribution. Hence in this case it no longer makes
sense to distinguish the eigenbundles of $K^{B_{-}}_{C}$ as vertical and
horizontal. Furthermore, neither of the eigenbundles is integrable in general,
so $T^{*}{\mathcal{Q}}$ does not generally admit any foliation associated with
the almost para-Hermitian structure $(K_{C}^{B_{-}},\eta_{C}).$ This is the
reason for refering to this polarization as an ‘essentially doubled
polarization’: It exhibits an unavoidable obstruction to obtaining a reduced
sigma-model associated with the Born sigma-model in this polarization.

To describe the Born sigma-model associated with this polarization, we
consider the generalized metric ${\mathcal{H}}_{C}^{B_{-}}$ which is
compatible with the pullback almost para-Hermitian structure
$(K_{C}^{B_{-}},\eta_{C})$ obtained via $e^{B_{-}}$ from the generalized
metric ${\mathcal{H}}_{C}.$ We then obtain

${\mathcal{H}}_{C}^{B_{-}}=\big{(}g_{+}^{B_{-}}\big{)}_{ij}\,\theta^{i}\otimes\theta^{j}+\big{(}g_{-}^{B_{-}}\big{)}^{ij}\,\lambda_{i}\otimes\lambda_{j}\
,$

where

$\lambda_{i}=\mathrm{d}p_{i}-C_{ik}\,\mathrm{d}q^{k}$

is the coframe dual to $\\{P^{i}\\}$ and

$\theta^{i}=(\beta_{-})^{ij}\,\mathrm{d}p_{j}+\big{(}\delta^{i}{}_{k}-(\beta_{-})^{ij}\,C_{jk}\big{)}\,\mathrm{d}q^{k}$

is the dual coframe to $\big{\\{}C(\frac{\partial}{\partial q^{i}})\big{\\}}$
in the splitting given by $K_{C}^{B_{-}}.$ In this coframe, the fundamental
2-form (5.9) can be written as

$\omega_{0}^{B_{-}}=\theta^{i}\wedge\lambda_{i}\ .$

We now have all the data needed to write down the Born sigma-model action
functional in the essentially doubled polarization:

${\mathcal{S}}^{B_{-}}[\phi]=\frac{1}{4}\,\int_{\Sigma}\,\Big{(}\big{(}\bar{g}_{+}^{B_{-}}\big{)}_{ij}\,\bar{\theta}^{i}\wedge\star\,\bar{\theta}^{j}+\big{(}\bar{g}_{-}^{B_{-}}\big{)}^{ij}\,\bar{\lambda}_{i}\wedge\star\,\bar{\lambda}_{j}\Big{)}+\frac{1}{2}\,\int_{\Sigma}\,\bar{\theta}^{i}\wedge\bar{\lambda}_{i}\
,$ (5.10)

where $\phi$ is a map from the closed string worldsheet $\Sigma$ to the phase
space $T^{*}{\mathcal{Q}}.$ It is clear that we cannot obtain from the sigma-
model ${\mathcal{S}}({\mathcal{H}}_{C}^{B_{-}},\omega_{0}^{B_{-}})$ any
reduced sigma-model associated with the almost para-Hermitian structure
$(K_{C}^{B_{-}},\eta_{C})$, since this structure does not yield any foliation.

Neverthess, let us still attempt to follow the discussion of Section 4, and in
particular our interpretation of generalized T-duality from Section 4.4. We
may try to force the gauging with respect to the $-1$-eigenbundle
$L_{B_{-}}(T^{*}{\mathcal{Q}})$ of $K_{C}^{B_{-}},$ which cannot be regarded
as a Lie algebroid. Despite the fact that the Lie derivatives
$\pounds_{P^{k}}\,{\mathcal{H}}^{B_{-}}_{C}$ and
$\pounds_{P^{k}}\,\omega_{0}^{B_{-}}$ can still be cast in the form of the
conditions for a generalized isometry, there are no leaf coordinates whose
pullback differentials can be covariantized. Thus in order to introduce some
kind of gauging, we might try to covariantize $\mathrm{d}\bar{p}_{i}$ by
minimal coupling to a 1-form ${\mathcal{A}}$ on $\Sigma$ which is neither
valued in a Lie algebra nor in a Lie algebroid. In principle, ${\mathcal{A}}$
should be valued in the vector sub-bundle $L_{B_{-}}(T^{*}{\mathcal{Q}})$ and
might be interpreted as induced by a vector bundle morphism
$\bar{{\mathcal{A}}}:T\Sigma\rightarrow L_{B_{-}}(T^{*}{\mathcal{Q}})$ which
covers $\phi:\Sigma\to T^{*}{\mathcal{Q}}$ giving the pullback bundle
$\phi^{*}{\rm Im}(i\,\circ\,\bar{{\mathcal{A}}})$ on $\Sigma,$ a vector sub-
bundle of $\phi^{*}T(T^{*}{\mathcal{Q}}),$ and an associated tensor
$\underline{\bar{\mathcal{A}}}\in\mathsf{\Gamma}(T^{*}\Sigma\,\otimes\,\phi^{*}T(T^{*}{\mathcal{Q}})).$
Here the choice of $p_{i}$ as “gauged” coordinates is arbitrary, since they do
not have a geometric meaning in this polarization as leaf coordinates. Despite
this, we introduce the “covariant derivatives”

$\mathrm{D}^{\mathcal{A}}\bar{q}^{i}=\mathrm{d}\bar{q}^{i}\qquad\mbox{and}\qquad\mathrm{D}^{\mathcal{A}}\bar{p}_{i}=\mathrm{d}\bar{p}_{i}-\bar{\mathcal{A}}_{i}\
,$ (5.11)

and write down the “gauged” sigma-model
${\mathcal{S}}^{B_{-}}[\phi,{\mathcal{A}}]$ in the usual way by replacing
$\mathrm{d}\bar{p}_{i}$ and $\mathrm{d}\bar{q}^{i}$ with the maps in (5.11).

We obtain the self-duality constraint as the equation of motion for
${\mathcal{A}}$ by imposing
$\delta{\mathcal{S}}^{B_{-}}[\phi,{\mathcal{A}}]/\delta{\mathcal{A}}_{i}=0,$
which reads

$\displaystyle\Big{(}\big{(}\bar{g}_{+}^{B_{-}}\big{)}_{ij}\,(\bar{\beta}_{-})^{im}\,(\bar{\beta}_{-})^{jn}+\big{(}\bar{g}_{-}^{B_{-}}\big{)}^{mn}\Big{)}\,\star\,\mathrm{D}^{\mathcal{A}}\bar{p}_{n}$
$\displaystyle\hskip
85.35826pt+\Big{(}\big{(}\bar{g}_{+}^{B_{-}}\big{)}_{ij}\,(\bar{\beta}_{-})^{im}\,\big{(}\delta^{j}{}_{l}-(\bar{\beta}_{-})^{jk}\,\bar{C}_{kl}\big{)}-\big{(}\bar{g}_{-}^{B_{-}}\big{)}^{mj}\,\bar{C}_{jl}\Big{)}\,\star\,\mathrm{d}\bar{q}^{l}$
$\displaystyle\hskip
170.71652pt-(\bar{\beta}_{-})^{mn}\,\mathrm{D}^{\mathcal{A}}\bar{p}_{n}+\big{(}2\,(\bar{\beta}_{-})^{mi}\,\bar{C}_{il}-\delta^{m}{}_{l}\big{)}\,\mathrm{d}\bar{q}^{l}=0\
.$

These equations are solved formally by the non-local expression

$\displaystyle\mathrm{D}^{\mathcal{A}}\bar{p}_{k}=\bigg{(}\frac{1}{\bar{g}_{-}^{B_{-}}-\bar{\beta}_{-}\,\bar{g}_{+}^{B_{-}}\,\bar{\beta}_{-}-\bar{\beta}_{-}\,\star}\bigg{)}_{km}\,\Big{(}$
$\displaystyle\big{(}\bar{g}_{+}^{B_{-}}\big{)}_{ij}\,(\bar{\beta}_{-})^{im}\,\big{(}\delta^{j}{}_{l}-(\bar{\beta}_{-})^{jk}\,\bar{C}_{kl}\big{)}\,\mathrm{d}\bar{q}^{l}$
$\displaystyle-\big{(}\bar{g}_{-}^{B_{-}}\big{)}^{mj}\,\bar{C}_{jl}\,\mathrm{d}\bar{q}^{l}+\big{(}2\,(\bar{\beta}_{-})^{mi}\,\bar{C}_{il}-\delta^{m}{}_{l}\big{)}\,\star\,\mathrm{d}\bar{q}^{l}\Big{)}$

and substitution into the gauged extension of (5.10) gives an action
functional with the $\mathrm{d}\bar{p}_{n}$ dependence removed. However, even
in the simplest instances where $\beta_{-}$ and $C$ guarantee that the sub-
bundle $L_{B_{-}}(T^{*}{\mathcal{Q}})$ is non-integrable, the “reduced” local
action functional still involves both sets of coordinates $q^{i}$ and $p_{i}.$
In other words, the worldsheet formulation for an essentially doubled
polarization does not permit the writing of any reduced sigma-model, not even
in a local form, which has dependence on only half of the coordinates. In this
sense the Born sigma-model itself is needed to describe string theory on the
essentially doubled background.

##  6\. Born Sigma-Models for Doubled Groups

Another broad class of examples of Born geometries come from Lie groups which
can be endowed with an almost para-Hermitian structure, and their discrete
quotients. In particular, cotangent bundles of Lie groups (and their discrete
quotients) furnish natural examples of doubled groups which can be nicely
combined with the phase space formalism of Section 5. Worldsheet theories for
these types of doubled geometries are discussed in [48, 103, 24, 49, 104, 64,
25, 105, 106, 73, 107, 108]. In particular, doubled groups provide concrete
examples where both the exact Courant algebroid and doubled geometry
descriptions of the string background are understood, and the connections
between them were described by [64] in a similar spirit to the framework of
the present paper. These doubled sigma-models are defined using the natural
left-invariant metric and 3-form on the group manifold, so that quotients by
left-acting discrete subgroups of the doubled group can be treated using the
standard isometric gauging techniques reviewed in Section 4.2.1. Particular
changes of polarization of doubled groups are related to non-abelian T-duality
and some aspects of Poisson-Lie T-duality; double field theory on these sorts
of extended spacetimes was formulated in [109, 110], and in this context
Poisson-Lie T-duality for Drinfel’d double groups was studied in [111, 67,
112]. In this section we shall re-examine gauged sigma-models for doubled
groups (and their discrete quotients) from the perspective of the Lie
algebroid gauging of Born sigma-models developed in Section 4, and hence
provide a more intrinsic geometric description of them.

###  6.1. Invariant Para-Hermitian Structures on Lie Groups

We begin by describing the para-Hermitian geometry on a Lie group which is
invariant under the action of the group on itself. We refer to such groups as
‘doubled groups’.

######  Definition 6.1.

A Lie group ${\mathsf{D}}$ of even dimension $2d$ is a _doubled group_ if it
is endowed with a left-invariant almost para-Hermitian structure181818One can
also define a doubled group with a right-invariant almost para-Hermitian
structure. $(K^{\textrm{\tiny L}},\eta^{\textrm{\tiny L}}),$ so that

${\tt L}_{\gamma}^{*}\eta^{\textrm{\tiny L}}=\eta^{\textrm{\tiny
L}}\qquad\mbox{and}\qquad{{\tt L}_{\gamma}}_{*}\circ K^{\textrm{\tiny
L}}=K^{\textrm{\tiny L}}\circ{{\tt L}_{\gamma}}_{*}$

for all $\gamma\in{\mathsf{D}}$, where ${\tt
L}_{\gamma}:{\mathsf{D}}\to{\mathsf{D}}$ is the map induced by left
multiplication of elements of ${\mathsf{D}}$ with $\gamma$.

Let $T_{M}$, with $M=1,\dots,2d$, be generators of the Lie algebra
$\mathfrak{d}={\sf Lie}({\mathsf{D}})$, with the brackets

$[T_{M},T_{N}]=t_{MN}{}^{P}\,T_{P}\ .$ (6.2)

This preserves a constant ${\sf O}(d,d)$-invariant metric defined by
$\eta^{\textrm{\tiny L}}$, and so a doubled group is a $2d$-dimensional
subgroup of ${\sf O}(d,d)$. The polarization defined by the left-invariant
almost para-complex structure $K^{\textrm{\tiny L}}$ splits the generators
$T_{M}$ into two sets $T_{m}$ and $\tilde{T}^{m}$, with $m=1,\dots,d$, such
that the Lie algebra (6.2) takes the form

$\displaystyle[T_{m},T_{n}]$
$\displaystyle=f_{mn}{}^{p}\,T_{p}+H_{mnp}\,\tilde{T}^{p}\ ,$
$\displaystyle[\tilde{T}^{m},T_{n}]$
$\displaystyle=f_{np}{}^{m}\,\tilde{T}^{p}-Q_{n}{}^{mp}\,T_{p}\ ,$
$\displaystyle[\tilde{T}^{m},\tilde{T}^{n}]$
$\displaystyle=Q_{p}{}^{mn}\,\tilde{T}^{p}+R^{mnp}\,T_{p}\ ,$

with constant fluxes $H_{mnp}$, $f_{mn}{}^{p}$, $Q_{p}{}^{mn}$ and $R^{mnp}$.
The Jacobi identity for the Lie brackets (6.2) implies a set of algebraic
Bianchi identities for the generalized fluxes which can be found in e.g. [39].

Corresponding to $T_{M}$ there is a global frame of left-invariant vector
fields $Z_{M}$ on ${\mathsf{D}}$ which trivialize the tangent bundle
$T{\mathsf{D}}\simeq{\mathsf{D}}\times\mathbb{R}^{2d}$ and generate the right
action of ${\mathsf{D}}$ on itself; they generate the Lie algebra (6.2) with
respect to the Lie bracket of vector fields. The left-invariant Maurer-Cartan
one-forms $\Theta^{M}$, dual to the left-invariant vector fields $Z_{M}$, form
a global coframe trivializing the cotangent bundle $T^{*}{\mathsf{D}}$ which
satisfy the Maurer-Cartan equations

$\mathrm{d}\Theta^{M}+\tfrac{1}{2}\,t_{NP}{}^{M}\,\Theta^{N}\wedge\Theta^{P}=0\
.$

The polarization selects a splitting of these bases as
$Z_{M}=(Z_{m},\tilde{Z}^{m})$ and
$\Theta^{M}=(\Theta^{m},\tilde{\Theta}_{m})$. The left-invariant almost para-
Hermitian structure $(K^{\textrm{\tiny L}},\eta^{\textrm{\tiny L}})$ can then
be expressed in terms of this global frame and coframe as

$\underline{K^{\textrm{\tiny L}}}=\Theta^{m}\otimes
Z_{m}-\tilde{\Theta}_{m}\otimes\tilde{Z}^{n}\qquad\mbox{and}\qquad\eta^{\textrm{\tiny
L}}=\Theta^{m}\otimes\tilde{\Theta}_{m}+\tilde{\Theta}_{m}\otimes\Theta^{m}\
,$

and the corresponding fundamental 2-form is

$\omega^{\textrm{\tiny L}}=\Theta^{m}\wedge\tilde{\Theta}_{m}\ ,$ (6.3)

with field strength

$\displaystyle{\mathcal{K}}^{\textrm{\tiny L}}=\mathrm{d}\omega^{\textrm{\tiny
L}}=\tfrac{1}{2}\,\big{(}$ $\displaystyle
H_{mnp}\,\Theta^{m}\wedge\Theta^{n}\wedge\Theta^{p}+f_{mn}{}^{p}\,\Theta^{m}\wedge\Theta^{n}\wedge\tilde{\Theta}_{p}$
$\displaystyle-
Q_{m}{}^{np}\,\Theta^{m}\wedge\tilde{\Theta}_{n}\wedge\tilde{\Theta}_{p}+R^{mnp}\,\tilde{\Theta}_{m}\wedge\tilde{\Theta}_{n}\wedge\tilde{\Theta}_{p}\big{)}\
.$ (6.4)

We will now introduce a suitable notion of left-invariant generalized metric.

######  Definition 6.5.

A _generalized metric_ on a doubled group ${\mathsf{D}}$ is an automorphism
$I^{\textrm{\tiny L}}\in{\sf Aut}_{\mathds{1}}(T{\mathsf{D}})$ such that
$(I^{\textrm{\tiny L}})^{2}=\mathds{1}$, $I^{\textrm{\tiny
L}}\neq\pm\,\mathds{1}$ and $I^{\textrm{\tiny L}}\circ{{\tt
L}_{\gamma}}_{*}={{\tt L}_{\gamma}}_{*}\circ I^{\textrm{\tiny L}}$ for all
$\gamma\in{\mathsf{D}}$, which defines a left-invariant Riemannian metric
${\mathcal{H}}^{\textrm{\tiny L}}$ by

${\mathcal{H}}^{\textrm{\tiny L}}(X,Y)=\eta^{\textrm{\tiny
L}}\big{(}I^{\textrm{\tiny L}}(X),Y\big{)}$

for all $X,Y\in\mathsf{\Gamma}(T{\mathsf{D}})$.

An example is provided by the generalized metric (2.32) on ${\mathsf{D}}={\sf
SL}(2,\mathbb{C})$ from Example 2.31; see also [107] for a two-parameter
family of almost para-Hermitian structures in this case which arise in the
context of integrable deformations of the principal chiral model.

A left-invariant Born geometry is a compatible generalized metric on
${\mathsf{D}}$ which is determined in the usual way by choosing a left-
invariant fiberwise metric $g_{+}$ on $L_{+}$ (or $g_{-}$ on $L_{-}$), where
$T{\mathsf{D}}=L_{+}\oplus L_{-}$ is the splitting induced by
$K^{\textrm{\tiny L}}$. We will often work with the simplest example of a Born
metric on ${\mathsf{D}}$ which is constructed from the left-invariant 1-forms
as

${\mathcal{H}}^{\textrm{\tiny
L}}=\delta_{MN}\,\Theta^{M}\otimes\Theta^{N}=\delta_{mn}\,\Theta^{m}\otimes\Theta^{n}+\delta^{mn}\,\tilde{\Theta}_{m}\otimes\tilde{\Theta}_{n}\
.$ (6.6)

This is the unique left-invariant Riemannian metric on ${\mathsf{D}}$ in which
the selected frame $\\{Z_{M}\\}$ is orthonormal.

####  6.1.1. Matrix Lie Groups

To make contact between our framework and previous treatments of the geometry
of doubled groups in the literature, as well as to work out some explicit
examples, we will now specialise to the case that ${\mathsf{D}}$ is a matrix
group. Then the Maurer-Cartan 1-forms are given by

$\Theta=\gamma^{-1}\,\mathrm{d}\gamma=\Theta^{M}\,T_{M}$

where $\gamma\in{\mathsf{D}}$. In a neighbourhood of the identity, we can
introduce local coordinates

$\mathbb{X}^{M}=(x^{m},\tilde{x}_{m})$

on the group manifold of ${\mathsf{D}}$ by using the polarization to write a
general group element $\gamma\in{\mathsf{D}}$ through the exponential
parameterization

$\gamma(\mathbb{X})=\tilde{\sigma}(\tilde{x})\,\sigma(x)$

where

$\sigma(x)=\exp\big{(}x^{m}\,T_{m}\big{)}\qquad\mbox{and}\qquad\tilde{\sigma}(\tilde{x})=\exp\big{(}\tilde{x}_{m}\,\tilde{T}^{m}\big{)}\
.$

So far we have not said anything about the integrability of the eigenbundles
of $K^{\textrm{\tiny L}}$, and this splitting of coordinates can always
naturally be made for a doubled group. With it we can express the global
Maurer-Cartan 1-forms $\Theta^{M}$ on ${\mathsf{D}}$ as local
$C^{\infty}({\mathsf{D}})$-linear combinations of the holonomic basis
$\mathrm{d}\mathbb{X}^{M}$.

The Born geometry of the doubled group ${\mathsf{D}}$ may then be expressed in
this parameterization by following [24, 25] to introduce the
$\mathfrak{d}$-valued 1-form

$\Xi=\sigma\,\Theta\,\sigma^{-1}=\tilde{\sigma}^{-1}\,\mathrm{d}(\tilde{\sigma}\,\sigma)\,\sigma^{-1}=\mathrm{d}\sigma\,\sigma^{-1}+\tilde{\sigma}^{-1}\,\mathrm{d}\tilde{\sigma}\
.$ (6.7)

Using the polarization we can expand the $\mathfrak{d}$-valued 1-forms on the
right-hand side of (6.7) as

$\mathrm{d}\sigma\,\sigma^{-1}=\varrho^{m}\,T_{m}+\varrho_{m}\,\tilde{T}^{m}\qquad\mbox{and}\qquad\tilde{\sigma}^{-1}\,\mathrm{d}\tilde{\sigma}=\tilde{\ell}^{m}\,T_{m}+\tilde{\ell}_{m}\,\tilde{T}^{m}\
.$

The component form $\Xi=\Xi^{M}\,T_{M}$ is thus given by

$\Xi^{M}=(p^{m},\tilde{q}_{m})$

where

$p^{m}=\varrho^{m}+\tilde{\ell}^{m}\qquad\mbox{and}\qquad\tilde{q}_{m}=\varrho_{m}+\tilde{\ell}_{m}\
.$

The inverse of this change of coframe, $\Theta=\sigma^{-1}\,\Xi\,\sigma$, is
given by

$\Theta^{M}={\mathcal{E}}_{N}{}^{M}(x)\,\Xi^{N}\ ,$

where ${\mathcal{E}}_{N}{}^{M}(x)$ depends only on the local coordinates
$x^{m}$ and is given by the adjoint action of $\sigma^{-1}(x)$ on the Lie
algebra $\mathfrak{d}$. The adjoint action preserves the split signature
metric $\eta^{\textrm{\tiny L}}$ and so ${\mathcal{E}}(x)\in{\mathsf{O}}(d,d)$
for each $x$. Similarly to the discussion in Remark 3.11, we may parameterize
it with respect to the splitting of $T^{*}{\mathsf{D}}$ associated to the
almost para-complex structure $K^{\textrm{\tiny L}}$ as

${\mathcal{E}}=\begin{pmatrix}e&e\,\beta\\\ e^{-1}\,b&e^{-1}\end{pmatrix}$

where $e(x)\in{\sf GL}(d,\mathbb{R})$ while $b(x)$ and $\beta(x)$ are skew-
symmetric $d\times d$ matrices which depend only on the local coordinates
$x^{m}$.

The fundamental 2-form (6.3) can then expressed in the coframe $\Xi^{M}$ as

$\omega^{\textrm{\tiny
L}}=\tfrac{1}{2}\,\hat{\omega}_{MN}(x)\,\Xi^{M}\wedge\Xi^{N}$

where

$\hat{\omega}=\begin{pmatrix}2\,b&\mathds{1}+b\,\beta\\\
-\mathds{1}-b\,\beta&-2\,\beta\end{pmatrix}\ ,$ (6.8)

while the compatible generalized metric (6.6) can be written as

${\mathcal{H}}^{\textrm{\tiny
L}}=\hat{\mathcal{H}}_{MN}(x)\,\Xi^{M}\otimes\Xi^{N}$

where

$\hat{\mathcal{H}}=\begin{pmatrix}g-b\,g^{-1}\,b&g\,\beta-b\,g^{-1}\\\
-\beta\,g+g^{-1}\,b&g^{-1}-\beta\,g\,\beta\end{pmatrix}\ .$ (6.9)

Here $g(x)=e(x)^{\rm t}\,e(x)$ is a symmetric non-degenerate $d\times d$
matrix depending only on the local coordinates $x^{m}$.

If the $R$-flux $R^{mnp}$ vanishes, then $\tilde{T}^{m}$ generate a
$d$-dimensional subgroup ${\mathsf{G}}\subset{\mathsf{D}}$ with the Lie
algebra

$[\tilde{T}^{m},\tilde{T}^{n}]=Q_{p}{}^{mn}\,\tilde{T}^{p}\ .$

In this case $\tilde{\ell}^{m}=0$, so that $p^{m}=\varrho^{m}$, and
$\tilde{\sigma}^{-1}\,\mathrm{d}\tilde{\sigma}$ gives the left-invariant
Maurer-Cartan 1-forms $\tilde{\ell}_{m}$ on ${\mathsf{G}}$. The Lie group
${\mathsf{G}}$ gives a maximally isotropic foliation of the doubled group
${\mathsf{D}}$ and we can analyse the generalized isometry conditions which
enable the gauging of the corresponding Born sigma-model. This reduces to a
non-linear sigma-model for a conventional geometric background
${\mathsf{D}}/{\mathsf{G}}$ with local coordinates $x^{m}$, and will be
studied in detail in Sections 6.2 and 6.3.

If the $R$-flux is non-zero, then the generators $\tilde{T}^{m}$ do not close
a Lie subalgebra of $\mathfrak{d}$. In this instance there is no foliation and
any “gauging” of the Born sigma-model will yield a reduced sigma-model
description that depends explicitly on both sets of local coordinates $x^{m}$
and $\tilde{x}_{m}$, so that there is no interpretation in terms of a
conventional $d$-dimensional spacetime. The resulting background is therefore
essentially doubled.

####  6.1.2. Quotienting by a Discrete Group

When a Lie group ${\mathsf{G}}$ is non-compact, a Scherk-Schwarz dimensional
reduction [113] on ${\mathsf{G}}$ does not give a proper compactification. In
order to lift it to string theory one should introduce a discrete cocompact
subgroup ${\mathsf{G}}(\mathbb{Z})\subset{\mathsf{G}}$ and consider instead
compactification on the compact space ${\mathsf{G}}/{\mathsf{G}}(\mathbb{Z})$
[114]. If ${\mathsf{G}}$ foliates a doubled group ${\mathsf{D}}$, then taking
the quotient by a discrete cocompact subgroup
${\mathsf{D}}(\mathbb{Z})\subset{\mathsf{D}}$ gives a compact manifold
$M={\mathsf{D}}/{\mathsf{D}}(\mathbb{Z})$, where
${\mathsf{G}}(\mathbb{Z})\subset{\mathsf{D}}(\mathbb{Z})$ acts only on
${\mathsf{G}}$ and leaves the leaf space ${\mathsf{D}}/{\mathsf{G}}$
invariant. Thus the doubled group construction in string theory is restricted
to Lie groups which admit a discrete cocompact subgroup. A widely studied
class of examples are the nilpotent Lie groups which can be defined over the
rationals, and taking the quotient by a discrete cocompact subgroup gives a
compact nilmanifold; we will study in detail an example from this class in
Section 7. Generally, writing the left-invariant 1-forms in the holonomic
basis as

$\Theta^{I}=E^{I}{}_{J}\,\mathrm{d}\mathbb{X}^{J}$

identifies the Scherk-Schwarz twist matrix $E=\big{(}E^{I}{}_{J}\big{)}\in{\sf
GL}(2d,\mathbb{R})$ in this formalism.

Taking the subgroup ${\mathsf{D}}(\mathbb{Z})$ to have a left action
$L_{\xi}:{\mathsf{D}}\to{\mathsf{D}}$ for all
$\xi\in{\mathsf{D}}(\mathbb{Z})$, then the left-invariant almost para-
Hermitian structure $(K^{\textrm{\tiny L}},\eta^{\textrm{\tiny L}})$ and
compatible generalized metric ${\mathcal{H}}^{\textrm{\tiny L}}$ descend to a
well-defined almost para-Hermitian structure $(K,\eta)$ and compatible
generalized metric ${\mathcal{H}}$ on the quotient
$M={\mathsf{D}}/{\mathsf{D}}(\mathbb{Z})$. The group of large diffeomorphisms
${\sf Diff}(M;\mathbb{Z})$ is the automorphism group ${\sf
Aut}\big{(}{\mathsf{D}}(\mathbb{Z})\big{)}$ of the lattice
${\mathsf{D}}(\mathbb{Z})$, and in the quantum theory physical T-duality
transformations will then live in a subgroup of the discrete group

${\mathsf{O}}(d,d)({\mathsf{D}})\,\cap\,{\sf
Aut}\big{(}{\mathsf{D}}(\mathbb{Z})\big{)}$ (6.10)

of automorphisms of the doubled group ${\mathsf{D}}$ that preserve
${\mathsf{D}}(\mathbb{Z})$ and the split signature metric $\eta^{\textrm{\tiny
L}}$. For example, when ${\mathsf{D}}=\mathbb{R}^{2d}$ then
${\mathsf{D}}(\mathbb{Z})=\mathbb{Z}^{2d}$ with
$M={\mathsf{D}}/{\mathsf{D}}(\mathbb{Z})=\mathbb{T}^{2d}$ and ${\sf
Aut}(\mathbb{Z}^{2d})={\sf GL}(2d,\mathbb{Z})$, so that (6.10) is the
T-duality group ${\mathsf{O}}(d,d)\cap{\sf
GL}(2d,\mathbb{Z})={\mathsf{O}}(d,d;\mathbb{Z})$ of string theory on a
$d$-dimensional toroidal compactification.

Gauging the generalized isometry generated by the vector fields
$\tilde{Z}^{m}$ in the corresponding Born sigma-model then gives a
conventional reduced sigma-model for the quotient space $M/{\mathsf{G}}$.
However, in contrast to the Born structure on the doubled group manifold
${\mathsf{D}}$, where the quotient ${\mathsf{D}}/{\mathsf{G}}$ always yields a
geometric background, the geometric nature of the quotient $M/{\mathsf{G}}$
depends on the way in which the subgroups ${\mathsf{G}}$ and
${\mathsf{D}}(\mathbb{Z})$ are embedded into ${\mathsf{D}}$ [24, 49]. If the
subgroup ${\mathsf{G}}\subset{\mathsf{D}}$ commutes with the action of
${\mathsf{D}}(\mathbb{Z})$, so that

$\xi\,{\mathsf{G}}\,\subseteq\,{\mathsf{G}}\,\xi\ ,$

for $\xi\in{\mathsf{D}}(\mathbb{Z})$, then the quotient space $M/{\mathsf{G}}$
is smooth and describes a conventional geometric background. On the other
hand, if the subgroup ${\mathsf{G}}$ does not commute with
${\mathsf{D}}(\mathbb{Z})$, then the quotient space $M/{\mathsf{G}}$ is not
smooth and the resulting background is a T-fold. In Section 7 we shall study a
concrete example which illustrates all of these general features explicitly
from a different geometric point of view.

###  6.2. Manin Pairs and Drinfel’d Doubles

We will now consider some general Lie algebraic structures that naturally lead
to doubled groups. We demonstrate how doubled groups arise from Manin pairs,
and also Manin triples such as Drinfel’d doubles which generalize Example
2.31. Let us start by providing some definitions which will be central to the
rest of this paper, following [115].

######  Definition 6.11.

A _Manin pair_ $({\mathfrak{d}},{\mathfrak{g}})$ is a $2d$-dimensional Lie
algebra ${\mathfrak{d}}$ endowed with an invariant symmetric non-degenerate
pairing $\braket{\,\cdot\,,\,\cdot\,}$ of signature $(d,d)$, together with a
Lie subalgebra ${\mathfrak{g}}\subset\mathfrak{d}$ which is maximally
isotropic with respect to $\braket{\,\cdot\,,\,\cdot\,}.$

A short exact sequence of vector spaces is naturally associated with any Manin
pair:

$0\longrightarrow\mathfrak{g}\xlongrightarrow{i}\mathfrak{d}\xlongrightarrow{i^{*}}\mathfrak{g}^{*}\longrightarrow
0$ (6.12)

where $i:\mathfrak{g}\hookrightarrow\mathfrak{d}$ is the inclusion map,
$\mathfrak{g}^{*}$ is the dual vector space of $\mathfrak{g}$, and the map
$i^{*}$ is defined by

$\braket{\,i({\mathsf{x}})\,,\,{\mathsf{w}}\,}=\braket{\,{\mathsf{x}}\,,\,i^{*}({\mathsf{w}})\,}\
,$

for all ${\mathsf{x}}\in\mathfrak{g}$ and ${\mathsf{w}}\in\mathfrak{d}.$ We
can always choose an isotropic splitting of the short exact sequence (6.12),
which is an injective map

$j:\mathfrak{g}^{*}\longrightarrow\mathfrak{d}\qquad\mbox{with}\quad
i^{*}\circ j=\mathds{1}_{\mathfrak{g}^{*}}\ .$

In this case

$\mathfrak{d}=\mathfrak{m}\oplus\mathfrak{g}\ ,$ (6.13)

where $\mathfrak{m}={\rm Im}(j)$ is a maximally isotropic subspace with
respect to the pairing $\braket{\,\cdot\,,\,\cdot\,},$ but not generally a Lie
subalgebra of $\mathfrak{d}$. We call $(\mathfrak{d},\mathfrak{g};j)$ a _split
Manin pair_.

The choice of an isotropic splitting of the short exact sequence (6.12)
defines an almost para-Hermitian structure on the Lie algebra $\mathfrak{d}$.
It is given by an almost para-complex structure $\kappa\in{\sf
Aut}(\mathfrak{d})$ such that

$\kappa\big{(}j({\mathsf{x}})+i({\tilde{\mathsf{x}}})\big{)}=j({\mathsf{x}})-i({\tilde{\mathsf{x}}})\
,$ (6.14)

for all ${\mathsf{x}}\in{\mathsf{g}}^{*}$ and
$\tilde{{\mathsf{x}}}\in{\mathsf{g}},$ and the symmetric non-degenerate
pairing $\braket{\,\cdot\,,\,\cdot\,}.$ The almost para-complex structure
$\kappa$ is compatible with the pairing $\braket{\,\cdot\,,\,\cdot\,}$ by
construction. Then the fundamental 2-form
${\mathcal{W}}\in\text{\Large$\wedge$}^{2}\,\mathfrak{d}^{*}$ induced by
$\kappa$ and the pairing is

${\mathcal{W}}({\mathsf{w}},{\mathsf{z}})=\braket{\,\kappa({\mathsf{w}})\,,\,{\mathsf{z}}\,}\
,$

for all ${\mathsf{w}},{\mathsf{z}}\in\mathfrak{d},$ which by using isotropy of
$\mathfrak{m}$ and $\mathfrak{g}$ with respect to the pairing reads

${\mathcal{W}}\big{(}j({\mathsf{x}})+i(\tilde{{\mathsf{x}}})\,,\,j({\mathsf{y}})+i(\tilde{{\mathsf{y}}})\big{)}=\braket{\,j({{\mathsf{x}}})\,,\,i(\tilde{\mathsf{y}})\,}-\braket{\,i(\tilde{\mathsf{x}})\,,\,j({{\mathsf{y}}})\,}\
,$

for all ${{\mathsf{x}}},{{\mathsf{y}}}\in\mathfrak{m}$ and
$\tilde{\mathsf{x}},\tilde{\mathsf{y}}\in\mathfrak{g}.$ Thus the subspaces
$\mathfrak{m}$ and $\mathfrak{g}$ are also maximally isotropic with respect to
${\mathcal{W}},$ so that
${\mathcal{W}}\in\mathfrak{m}^{*}\wedge\mathfrak{g}^{*}.$

There is a notion of $B$-transformations in this case preserving the Lie
subalgebra $\mathfrak{g}$ which are generated by bivectors
$\Lambda\in\text{\Large$\wedge$}^{2}\,\mathfrak{g}.$ Once a splitting $j$ is
fixed, we may then obtain a new subspace $\mathfrak{m}_{\Lambda}={\rm
Im}(j_{\Lambda})$, where

$j_{\Lambda}({{\mathsf{x}}})=j({{\mathsf{x}}})+i\big{(}\Lambda({\mathsf{x}})\big{)}\
,$

for all ${{\mathsf{x}}}\in\mathfrak{g}^{*}.$ The subspace
$\mathfrak{m}_{\Lambda}$ is again isotropic, so this gives a transformation
that maps an isotropic splitting $j$ into another isotropic splitting
$j_{\Lambda}.$ The difference between these two splittings is given by the
associated almost para-complex structure, which we formally write as

$\kappa_{\Lambda}=\kappa+2\,\Lambda\ .$

Correspondingly, the fundamental 2-form for $j_{\Lambda}$ reads

${\mathcal{W}}_{\Lambda}={\mathcal{W}}+2\,i(\Lambda)\ .$

Generally, changes of polarization
$\vartheta\in{\mathsf{O}}(d,d)(\mathfrak{d})$ which map a split Manin pair
$(\mathfrak{d},\mathfrak{g};j)$ into another split Manin pair
$(\mathfrak{d},\mathfrak{g}_{\vartheta};j_{\vartheta})$ are called _non-
abelian T-duality transformations_ [106].

Suppose now that ${\mathsf{D}}$ is a Lie group which integrates the Lie
algebra $\mathfrak{d}$, i.e. $\mathfrak{d}={\sf Lie}({\mathsf{D}})$. The
corresponding tangent Lie group is the semi-direct product

$T{\mathsf{D}}\simeq{\mathsf{D}}\ltimes\mathfrak{d}$

by the adjoint action of ${\mathsf{D}}$ on $\mathfrak{d}\simeq\mathbb{R}^{2d}$
regarded as an abelian Lie group. Thus ${\mathsf{D}}$ inherits an almost para-
Hermitian structure $(K^{\textrm{\tiny L}},\eta^{\textrm{\tiny
L}},\omega^{\textrm{\tiny L}})$ from
$(\kappa,\braket{\,\cdot\,,\,\cdot\,},{\mathcal{W}})$ by using the isomorphism
between $\mathfrak{d}$ and the left-invariant vector fields on ${\mathsf{D}},$
which by construction is left-invariant with respect to the left action of
${\mathsf{D}}$ on itself. Hence ${\mathsf{D}}$ is a doubled group. As a vector
bundle, the tangent bundle admits a splitting into left-invariant
distributions

$T{\mathsf{D}}=L_{\mathfrak{m}}({\mathsf{D}})\oplus
L_{\mathfrak{g}}({\mathsf{D}})\ ,$

which corresponds fiberwise to the vector space splitting (6.13). Here
$L_{\mathfrak{m}}({\mathsf{D}})$ is the sub-bundle of $T{\mathsf{D}}$
associated with the subspace $\mathfrak{m},$ and
$L_{\mathfrak{g}}({\mathsf{D}})\simeq T{\mathsf{G}}$ with ${\mathsf{G}}$ the
Lie subgroup of ${\mathsf{D}}$ whose Lie algebra is $\mathfrak{g},$ hence
sections of $L_{\mathfrak{g}}({\mathsf{D}})$ are given by left-invariant
vector fields on ${\mathsf{G}}$. Clearly $L_{\mathfrak{m}}({\mathsf{D}})$ is
not generally integrable, whereas ${\mathsf{G}}$ defines a foliation of
${\mathsf{D}}.$

It follows that a compatible generalized metric ${\mathcal{H}}^{\textrm{\tiny
L}}$ can always be defined on such an almost para-Hermitian manifold by
considering a left-invariant Riemannian metric ${\mathcal{G}}$ on
${\mathsf{G}}$ and setting

$g_{-}(X_{-},Y_{-})={\mathcal{G}}\big{(}X^{\textrm{\tiny L}},Y^{\textrm{\tiny
L}}\big{)}\ ,$

where $X_{-},Y_{-}\in\mathsf{\Gamma}(L_{\mathfrak{g}}({\mathsf{D}}))$ are the
sections of $L_{\mathfrak{g}}({\mathsf{D}})$ corresponding to the left-
invariant vector fields $X^{\textrm{\tiny L}},Y^{\textrm{\tiny
L}}\in\mathsf{\Gamma}(T{\mathsf{G}})$ on ${\mathsf{G}}.$ The fiberwise metric
$g_{-}$ on $L_{\mathfrak{g}}({\mathsf{D}})$ induces a fiberwise metric $g_{+}$
on $L_{\mathfrak{m}}({\mathsf{D}})$ in the usual way by

$g_{+}(X_{+},Y_{+})=g_{-}^{-1}\big{(}\eta^{\textrm{\tiny
L}}{}^{\flat}(X_{+}),\eta^{\textrm{\tiny L}}{}^{\flat}(Y_{+})\big{)}$ (6.15)

for all $X_{+},Y_{+}\in\mathsf{\Gamma}(L_{\mathfrak{m}}({\mathsf{D}}))$. Then
the left-invariant compatible generalized metric is

${\mathcal{H}}^{\textrm{\tiny L}}=g_{+}+g_{-}\ ,$

which is indeed Riemannian. It is straightforward to show that this metric is
the unique left-invariant Riemannian metric that can be defined on
${\mathsf{D}}$ for which the basis of left-invariant vector fields is
orthonormal.

######  Remark 6.16.

Whenever ${\mathsf{G}}$ is a closed connected Lie subgroup of the doubled
group ${\mathsf{D}}$, the coset space
${\mathcal{Q}}={\mathsf{D}}/{\mathsf{G}}$ is a smooth manifold and the
quotient map $\pi:{\mathsf{D}}\rightarrow{\mathcal{Q}}$ is a principal
${\mathsf{G}}$-bundle. In this case, $L_{\mathfrak{g}}({\mathsf{D}})$ is the
induced vertical distribution and $L_{\mathfrak{m}}({\mathsf{D}})$ is the
horizontal distribution. Then an alternative way of defining a compatible
generalized metric is by lifting a Riemannian metric defined on
${\mathcal{Q}}$ to ${\mathsf{D}},$ as discussed in Example 2.39.

Para-Hermitian structures on Drinfel’d doubles now arise naturally from the
above discussion.

######  Definition 6.17.

Let $(\mathfrak{d},\mathfrak{g};j)$ be a split Manin pair. If
$\tilde{\mathfrak{g}}={\rm Im}(j)$ closes a Lie subalgebra of $\mathfrak{d},$
then $(\mathfrak{d},\mathfrak{g},\tilde{\mathfrak{g}})$ is a _Manin triple_. A
corresponding triple of integrating Lie groups
$({\mathsf{D}},{\mathsf{G}},\tilde{{\mathsf{G}}})$ is a _Drinfel’d double_ ,
and is denoted

${\mathsf{D}}={\mathsf{G}}\Join\tilde{\mathsf{G}}\ .$

For further details on Drinfel’d doubles, see [116]. For a Manin triple, in
addition to (6.12) there is also the short exact sequence of vector spaces

$0\longrightarrow\mathfrak{g}^{*}\xlongrightarrow{}\mathfrak{d}\xlongrightarrow{}\mathfrak{g}\longrightarrow
0$ (6.18)

since a Manin triple corresponds to the Lie bialgebras
$(\mathfrak{g},\tilde{\mathfrak{g}})$ and
$(\tilde{\mathfrak{g}},\mathfrak{g}).$ Then there is a canonical para-
Hermitian structure induced by the vector space splitting

$\mathfrak{d}=\tilde{\mathfrak{g}}\oplus{\mathfrak{g}}$

and the non-degenerate symmetric pairing $\braket{\,\cdot\,,\,\cdot\,}.$ The
subgroup of non-abelian T-duality transformations
$\vartheta\in{\mathsf{O}}(d,d)({\mathsf{D}})$ which map a Manin triple
$(\mathfrak{d},\mathfrak{g},\tilde{\mathfrak{g}})$ into another Manin triple
$(\mathfrak{d},\mathfrak{g}_{\vartheta},\tilde{\mathfrak{g}}_{\vartheta})$
captures some features of Poisson-Lie T-duality [106].

######  Example 6.19.

Let ${\mathsf{G}}$ be a $d$-dimensional Lie group. Its cotangent bundle
$T^{*}{\mathsf{G}}\simeq{\mathsf{G}}\ltimes\mathbb{R}^{d}$ is a Drinfel’d
double Lie group ${\mathsf{D}}$ with $\tilde{\mathsf{G}}=\mathbb{R}^{d}$.
Denoting the bundle projection by
$\pi:T^{*}{\mathsf{G}}\rightarrow{\mathsf{G}},$ the canonical short exact
sequence of vector bundles

$0\longrightarrow L_{\tt
v}(T^{*}{\mathsf{G}})\xlongrightarrow{}T(T^{*}{\mathsf{G}})\xlongrightarrow{}\pi^{*}(T{\mathsf{G}})\longrightarrow
0$

corresponds fiberwise to the short exact sequence of vector spaces (6.18). A
left-invariant isotropic splitting with respect to the split signature metric
$\eta^{\textrm{\tiny L}},$ induced by the Drinfel’d double structure, of this
short exact sequence defines a left-invariant para-Hermitian structure on
$T^{*}{\mathsf{G}}$; note that the associated fundamental 2-form
$\omega^{\textrm{\tiny L}}$ is not necessarily the canonical symplectic 2-form
$\omega_{0}$ on $T^{*}{\mathsf{G}}$, which is not left-invariant in general. A
left-invariant compatible generalized metric ${\mathcal{H}}^{\textrm{\tiny
L}}$ on $T^{*}{\mathsf{G}}$ can be obtained by the horizontal lift of a left-
invariant Riemannian metric ${\mathcal{G}}$ on ${\mathsf{G}}$ which, in turn,
induces a left-invariant Riemannian metric on $T^{*}{\mathsf{G}}$.

###  6.3. Doubled Group Born Sigma-Models

Let ${\mathsf{D}}$ be a doubled group whose Lie algebra $\mathfrak{d}$ has the
structure of a split Manin pair, and let $(K^{\textrm{\tiny
L}},\eta^{\textrm{\tiny L}})$ be the associated almost para-Hermitian
structure on ${\mathsf{D}}.$ As we have seen, there is a natural compatible
generalized metric ${\mathcal{H}}^{\textrm{\tiny L}}$ induced by a left-
invariant Riemannian metric on the Lie subgroup
${\mathsf{G}}\subset{\mathsf{D}}$, as well as the fundamental 2-form
$\omega^{\textrm{\tiny L}}$ induced by the almost para-Hermitian structure.
Thus the doubled group ${\mathsf{D}}$ naturally serves as the target space for
a Born sigma-model ${\mathcal{S}}({\mathcal{H}}^{\textrm{\tiny
L}},\omega^{\textrm{\tiny L}}).$ Since ${\mathsf{D}}$ is foliated by
${\mathsf{G}}$, we may look for conditions under which the Born sigma-model
${\mathcal{S}}({\mathcal{H}}^{\textrm{\tiny L}},\omega^{\textrm{\tiny L}})$
admits a gauging and thereby yields a conventional sigma-model description of
the quotient ${\mathsf{D}}/{\mathsf{G}}.$ We will study the problem of the
existence of a gauged Born sigma-model with target space ${\mathsf{D}}$ in a
split Manin pair polarization, using the general description of the Lie
algebroid gauging of Section 4.

For this, we consider the generators of left-invariant vector fields
$\\{Z_{I}\\}=\\{Z_{i},\tilde{Z}^{i}\\}$ on ${\mathsf{D}}$ such that
$\\{Z_{i}\\}$ spans the sections of $L_{\mathfrak{m}}({\mathsf{D}})$ and
$\\{\tilde{Z}^{i}\\}$ spans the left-invariant vector fields on
${\mathsf{G}}$. This frame closes a Lie algebra of the form

$\displaystyle[Z_{m},Z_{n}]$
$\displaystyle={f_{mn}}^{k}\,Z_{k}+H_{mnk}\,\tilde{Z}^{k}\ ,$
$\displaystyle[Z_{m},\tilde{Z}^{n}]$
$\displaystyle={f_{km}}^{n}\,\tilde{Z}^{k}+Q_{m}{}^{nk}\,Z_{k}\ ,$ (6.20)
$\displaystyle[\tilde{Z}^{m},\tilde{Z}^{n}]$
$\displaystyle=Q_{k}{}^{mn}\,\tilde{Z}^{k}\ ,$

and admits a dual left-invariant coframe
$\\{\Theta^{I}\\}=\\{\Theta^{i},\tilde{\Theta}_{i}\\}$ such that
$\\{\Theta^{i}\\}$ spans the sections of $L_{\mathfrak{m}}^{*}({\mathsf{D}})$
and $\\{\tilde{\Theta}_{i}\\}$ spans the left-invariant 1-forms on
${\mathsf{G}}$.

A left-invariant Born metric ${\mathcal{H}}^{\textrm{\tiny L}}$ on
${\mathsf{D}}$ is specified by a fiberwise left-invariant metric $g_{+}$ on
$L_{\mathfrak{m}}({\mathsf{D}})$. To see when the Lie algebroid of left-
invariant vector fields on ${\mathsf{G}}$ generates the generalized isometries
of ${\mathcal{H}}^{\textrm{\tiny L}},$ we check when the transverse invariance
condition for ${\mathcal{H}}^{\textrm{\tiny L}}$ is satisfied. Since the left-
invariant vector fields $\tilde{Z}^{k}$ generate the right action of
${\mathsf{G}}$ on ${\mathsf{D}}$, the vanishing requirement

$\pounds_{\tilde{Z}^{k}}\,g_{+}=0$

from Section 4.3 implies that the metric $g_{+}$ is bi-invariant for the
${\mathsf{G}}$-action.

We also need to check the transverse invariance of the fundamental 2-form
$\omega^{\textrm{\tiny L}}$:

$(\pounds_{\tilde{Z}^{k}}\,\omega^{\textrm{\tiny
L}})(X_{+},Y_{+})=\omega^{\textrm{\tiny
L}}\big{(}[\tilde{Z}^{k},X_{+}],Y_{+}\big{)}+\omega^{\textrm{\tiny
L}}\big{(}X_{+},[\tilde{Z}^{k},Y_{+}]\big{)}=0$

for all $X_{+},Y_{+}\in\mathsf{\Gamma}(L_{\mathfrak{m}}({\mathsf{D}}))$. This
holds if and only if the Lie bracketing of the subspace
$\mathfrak{m}\subset\mathfrak{d}$ is given by

$[{\mathfrak{g}},{\mathfrak{m}}]_{\mathfrak{d}}\subseteq\mathfrak{m}\ .$
(6.21)

When ${\mathsf{G}}$ is connected this implies that the splitting (6.13) is
also invariant for the adjoint action of ${\mathsf{G}}$.

Generally, the quotient ${\mathcal{Q}}={\mathsf{D}}/{\mathsf{G}}$ is a
homogeneous space and the quotient map ${\mathsf{D}}\rightarrow{\mathcal{Q}}$
is a principal ${\mathsf{G}}$-bundle. The condition (6.21) then implies that
${\mathcal{Q}}$ is a reductive homogeneous space: it means that there is a
natural ${\mathsf{G}}$-action on ${\mathsf{D}}$ given by right multiplication,
with $\mathfrak{g}$ the Lie algebra of the isotropy subgroup and
$\mathfrak{m}$ the generators of infinitesimal translations of ${\mathcal{Q}}$
[117]. In this case the gaugeable Born sigma-models from Section 4.3 are in
correspondence with ${\mathsf{G}}$-invariant connections191919Strictly
speaking, for this correspondence we should consider a right-invariant para-
Hermitian structure on ${\mathsf{D}}$, but this would not affect any of the
results above. on the principal ${\mathsf{G}}$-bundle
${\mathsf{D}}\rightarrow{\mathcal{Q}}$ which are maximally isotropic with
respect to $\eta^{\textrm{\tiny L}},$ the split signature metric induced by
the split Manin pair structure of $\mathfrak{d}={\sf Lie}({\mathsf{D}})$. In
particular, in a split Manin pair polarization one always obtains a geometric
background for the reduced worldsheet sigma-model $S(g,b)$ for
${\mathcal{Q}}$, where the Riemannian metric $g$ descends from the
${\mathsf{G}}$-bi-invariant metric $g_{+}$ and the Kalb-Ramond field $b$ is
given by the transverse component of the fundamental 2-form
$\omega^{\textrm{\tiny L}}$. In this setting, a non-abelian T-duality
transformation between Born sigma-models, as a change of split Manin pair
polarization, is in the same spirit as the Poisson-Lie T-duality of [8]. We
shall describe these sigma-models explicitly below in the special case of
matrix Lie groups.

######  Example 6.22.

The simplest example of a fiberwise metric $g_{-}$ on
$L_{\mathfrak{g}}({\mathsf{D}})$ induced by a left-invariant metric
${\mathcal{G}}$ on ${\mathsf{G}}$ can be written as

$g_{-}=\delta^{ij}\,\tilde{\Theta}_{i}\otimes\tilde{\Theta}_{j}\ .$

Then the fiberwise metric $g_{+}$ on $L_{\mathfrak{m}}({\mathsf{D}})$ given by
(6.15) reads

$g_{+}=\delta_{ij}\,\Theta^{i}\otimes\Theta^{j}\ ,$

with ${\mathcal{H}}^{\textrm{\tiny L}}=g_{+}+g_{-}.$ In this case from (6.20)
and the Maurer-Cartan equations we find

$\pounds_{\tilde{Z}^{k}}\,g_{+}=\delta_{ij}\,Q_{l}{}^{kj}\,\Theta^{i}\odot\Theta^{l}\
,$

which vanishes when the structure constants $Q_{l}{}^{kj}$ are completely
skew. This implies that the Lie group ${\mathsf{G}}$ is semi-simple and
$g_{+}$ is the lift of the metric on ${\mathsf{G}}$ given by the Cartan-
Killing form

$c^{mn}=\tfrac{1}{2}\,Q_{p}{}^{mq}\,Q_{q}{}^{np}$

on $\mathfrak{g}={\sf Lie}({\mathsf{G}}).$

For the fundamental 2-form

$\omega^{\textrm{\tiny L}}=\Theta^{i}\wedge\tilde{\Theta}_{i}\ ,$

in this case we find

$\pounds_{\tilde{Z}^{k}}\,\omega^{\textrm{\tiny
L}}={f_{ij}}^{k}\,\Theta^{i}\wedge\Theta^{j}\ .$

Hence $\pounds_{\tilde{Z}^{k}}\,\omega^{\textrm{\tiny L}}$ has only one
component which belongs to
$\mathsf{\Gamma}(\text{\Large$\wedge$}^{2}L_{\mathfrak{m}}^{*}({\mathsf{D}})),$
so it has to vanish identically. This implies that

${f_{ij}}^{k}=0\ ,$

or equivalently that the Lie bracketing of the subspace
$\mathfrak{m}\subset\mathfrak{d}$ satisfies

$[{\mathfrak{m}},{\mathfrak{m}}]_{\mathfrak{d}}\subseteq{\mathfrak{g}}\ ,$
(6.23)

in addition to (6.21). This means that the almost para-complex structure
$\kappa$ defined by (6.14) endows the splitting (6.13) with a
$\mathbb{Z}_{2}$-grading by assigning degree $0$ to elements of $\mathfrak{g}$
and degree $1$ to elements of $\mathfrak{m}$. The remaining fluxes in (6.20)
are constrained by the Bianchi identities

$H_{m[np}\,Q_{l]}{}^{km}=0\ ,$

where the brackets denote skew-symmetrization of the enclosed indices.

The extra condition (6.23) implies that the reductive homogeneous space
${\mathcal{Q}}={\mathsf{D}}/{\mathsf{G}}$ is a symmetric space: the quotient
${\mathcal{Q}}$ is invariant under inversion about any chosen origin [117].
Symmetric string backgrounds in this case were also found in [66, 67] as
particular explicit solutions to the strong constraint in the target space
double field theory.

####  6.3.1. Matrix Lie Groups

The Born sigma-model ${\mathcal{S}}({\mathcal{H}}^{\textrm{\tiny
L}},\omega^{\textrm{\tiny L}})$ for a general doubled group ${\mathsf{D}}$
which is a matrix group can be written using the exponential parameterization
from Section 6.1.1 as

${\mathcal{S}}[\phi]=\frac{1}{4}\,\int_{\Sigma}\,\bar{\hat{\mathcal{H}}}_{MN}(x)\,\bar{\Xi}^{M}\wedge\star\,\bar{\Xi}^{N}+\frac{1}{4}\,\int_{\Sigma}\,\bar{\hat{\omega}}_{MN}(x)\,\bar{\Xi}^{M}\wedge\bar{\Xi}^{N}\
,$

where the map $\phi$ embeds a closed string worldsheet $\Sigma$ into the
doubled group ${\mathsf{D}}$, while $\hat{\mathcal{H}}_{MN}$ and
$\hat{\omega}_{MN}$ are the components of the generalized compatible metric
given in (6.9) and of the fundamental 2-form given in (6.8). The sigma-model
${\mathcal{S}}[\phi]$ has a rigid symmetry given by the action of
${\mathsf{D}}$ on itself by left multiplication. Since the $R$-flux $R^{mnp}$
vanishes in a split Manin pair polarization, $p^{m}=\varrho^{m}$ and
$\tilde{\ell}_{m}$ are the left-invariant Maurer-Cartan 1-forms on
${\mathsf{G}}$, as discussed in Section 6.1.1. Since by Example 6.22 the
metric torsion coefficients $f_{mn}{}^{p}$ vanish in this case by the
generalized isometry constraints, it follows that

$e=\mathds{1}\qquad\mbox{and}\qquad\varrho^{m}=\mathrm{d}x^{m}\ .$

The Lie algebroid gauging of the Born sigma-model in this case is achieved by
the minimal coupling of the Maurer-Cartan 1-forms $\tilde{\ell}_{m}$ to a
${\mathsf{G}}$-invariant connection 1-form ${\mathscr{C}}_{m}$, giving the
gauged Born sigma-model actional functional

$\displaystyle{\mathcal{S}}[\phi,{\mathscr{C}}]$
$\displaystyle=\frac{1}{4}\,\int_{\Sigma}\,\big{(}\delta_{mn}+\bar{b}_{mk}\,\delta^{kp}\,\bar{b}_{np}\big{)}\,\mathrm{d}\bar{x}^{m}\wedge\star\,\mathrm{d}\bar{x}^{n}+\frac{1}{2}\,\int_{\Sigma}\,\bar{b}_{mn}\,\mathrm{d}\bar{x}^{m}\wedge\mathrm{d}\bar{x}^{n}$
$\displaystyle\quad\,+\frac{1}{4}\,\int_{\Sigma}\,\big{(}\delta^{mn}+\bar{\beta}^{mk}\,\delta_{kp}\,\bar{\beta}^{np}\big{)}\,\big{(}\bar{\tilde{q}}_{m}+\bar{\mathscr{C}}_{m}\big{)}\wedge\star\,(\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}$
$\displaystyle\hskip
170.71652pt-\frac{1}{2}\,\int_{\Sigma}\,\bar{\beta}^{mn}\,\big{(}\bar{\tilde{q}}_{m}+\bar{\mathscr{C}}_{m}\big{)}\wedge\big{(}\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}$
(6.24)
$\displaystyle\quad\,-\frac{1}{2}\,\int_{\Sigma}\,\big{(}\bar{b}_{mk}\,\delta^{kn}-\delta_{mk}\,\bar{\beta}^{kn}\big{)}\,\mathrm{d}\bar{x}^{m}\wedge\star\,\big{(}\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}$
$\displaystyle\hskip
170.71652pt+\frac{1}{2}\,\int_{\Sigma}\,\big{(}\delta_{m}{}^{n}+\bar{b}_{mk}\,\bar{\beta}^{kn}\big{)}\,\mathrm{d}\bar{x}^{m}\wedge\big{(}\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}\
.$

Varying (6.24) with respect to the gauge fields ${\mathscr{C}}_{m}$ leads to
the self-duality constraints

$\displaystyle\big{(}\delta^{mn}+\bar{\beta}^{mk}\,\delta_{kp}\,\bar{\beta}^{np}\big{)}\,\star\,\big{(}\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}-2\,\bar{\beta}^{mn}\,\big{(}\bar{\tilde{q}}_{n}+\bar{\mathscr{C}}_{n}\big{)}$
$\displaystyle\hskip
113.81102pt=\big{(}\bar{b}_{nk}\,\delta^{km}-\delta_{nk}\,\bar{\beta}^{km}\big{)}\,\star\,\mathrm{d}\bar{x}^{n}-\big{(}\delta_{n}{}^{m}+\bar{b}_{nk}\,\bar{\beta}^{km}\big{)}\,\mathrm{d}\bar{x}^{n}\
,$

which are formally solved by the non-local expression

$\displaystyle\bar{\tilde{q}}_{m}+\bar{\mathscr{C}}_{m}=\bigg{(}\frac{1}{(\mathds{1}-\bar{\beta}\,\star)^{2}}\bigg{)}_{mn}\,\Big{(}(\bar{b}-\bar{\beta})^{n}{}_{k}\,\mathrm{d}\bar{x}^{k}-(\mathds{1}+\bar{\beta}\,\bar{b})^{n}{}_{k}\,\star\,\mathrm{d}\bar{x}^{k}\Big{)}\
.$

Substituting this into the gauged action functional (6.24) eliminates the
dependence on $\bar{\tilde{q}}_{m}$, giving a reduced sigma-model that depends
only on the leaf space coordinates $x^{m}$. The complicated non-local
dependence on the bivector $\beta$ owes to the appearence of $Q$-flux in the
doubled group background; nonetheless, the resulting physical background is
geometric.

We conclude this section by briefly considering some explicit examples which
illustrate how this reproduces well-known backgrounds in the doubled
formalism.

######  Example 6.25 (The Doubled Torus).

The simplest case corresponds to setting the structure constants to zero:

$Q_{k}{}^{ij}=0\ ,$

in addition to $f_{ij}{}^{k}=0$ in (6.20), so that ${\mathsf{D}}$ is a doubled
group integrating a Manin pair corresponding to the abelian Lie group

${\mathsf{G}}=\mathbb{R}^{d}\ .$

In this case one finds [24]

$\varrho_{m}=\tfrac{1}{2}\,H_{mnp}\,x^{p}\,\mathrm{d}x^{n}\
,\quad\tilde{\ell}_{m}=\mathrm{d}\tilde{x}_{m}\
,\quad\beta^{mn}=0\qquad\mbox{and}\qquad b_{mn}=H_{mnp}\,x^{p}\ .$

The reduction of the Born sigma-model then yields the standard non-linear
sigma-model $S(g,b)$ with flat metric

$g=\delta_{mn}\,\mathrm{d}x^{m}\otimes\mathrm{d}x^{n}$

and Kalb-Ramond field $b$, so that the spacetime is locally
${\mathcal{Q}}=\mathbb{R}^{d}$. The $H$-flux of the $B$-field agrees with the
field strength (6.4) of the fundamental 2-form in this case:

${\mathcal{K}}^{\textrm{\tiny
L}}=\tfrac{1}{2}\,H_{mnp}\,\mathrm{d}x^{m}\wedge\mathrm{d}x^{n}\wedge\mathrm{d}x^{p}\
.$

After taking the quotient by a cocompact discrete subgroup
${\mathsf{D}}(\mathbb{Z})$, the spacetime becomes a $d$-dimensional torus
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ with $H$-flux in this split
Manin pair polarization. Thus in this case the compact space
$M={\mathsf{D}}/{\mathsf{D}}(\mathbb{Z})$ reproduces the standard doubled
torus in the geometric $H$-flux polarization [20].

######  Example 6.26 (Doubled WZW Models).

Setting

$H_{mnk}=c_{mi}\,c_{nj}\,Q_{k}{}^{ij}$

in addition to $f_{ij}{}^{k}=0$ in (6.20) recovers the doubled sigma-model
description of the Wess-Zumino-Witten (WZW) model discussed in [103, 104, 25,
67]. The doubled group is

${\mathsf{D}}={\mathsf{G}}\times{\mathsf{G}}\ ,$

and if ${\mathsf{G}}$ is compact then ${\mathsf{D}}$ is embedded into the
maximal compact subgroup ${\mathsf{O}}(d)\times{\mathsf{O}}(d)$ of the
generalized T-duality group ${\mathsf{O}}(d,d)$, where the two copies of the
semi-simple Lie group ${\mathsf{G}}$ are associated to the left-moving and
right-moving worldsheet sectors. In this case we can set $\beta=0$ using an
${\mathsf{O}}(d)\times{\mathsf{O}}(d)$-transformation, and embedding
${\mathsf{G}}$ as the diagonal subgroup of ${\mathsf{D}}$, the gauged Born
sigma-model is a gauged WZW model based on the group ${\mathsf{D}}$ with gauge
group ${\mathsf{G}}$. The field strength (6.4) yields the standard $H$-flux

$H=-c_{mi}\,c_{nj}\,Q_{k}{}^{ij}\,\theta^{m}\wedge\theta^{n}\wedge\theta^{k}$

for the reduced WZW model at level $1$ based on
${\mathsf{D}}/{\mathsf{G}}\simeq{\mathsf{G}}$, where $\theta$ is the left-
invariant Maurer-Cartan 1-form on ${\mathsf{G}}$. The Riemannian submersion
from the Born manifold ${\mathsf{D}}$ to the quotient
${\mathsf{D}}/{\mathsf{G}}$ is given by

${\mathit{\Pi}}:{\mathsf{D}}\longrightarrow{\mathsf{G}}\
,\quad(g,g^{\prime})\longmapsto g^{-1}\,g^{\prime}\ ,$

where $g^{-1}$ and $g^{\prime}$ become the left-moving and right-moving closed
string fields after imposing the self-duality constraint resulting from the
gauged Born sigma-model. Other natural choices of left-invariant almost para-
complex structures on the doubled group
${\mathsf{D}}={\mathsf{G}}\times{\mathsf{G}}$ correspond to subgroups of
${\mathsf{D}}$ in the same conjugacy class as the diagonal subgroup
${\mathsf{G}}\subset{\mathsf{D}}$ [25]. Discrete quotients of this doubled
group in the case ${\mathsf{G}}={\sf SU}(2)$ are studied in [25, 66] in the
context of the T-duals of the 3-sphere $\mathbb{S}^{3}$, viewed as a circle
bundle over the 2-sphere $\mathbb{S}^{2}$.

######  Example 6.27 (Drinfel’d Doubles).

The case of a Drinfel’d double ${\mathsf{D}}$ which is a matrix Lie group
corresponds to setting

$H_{mnk}=0$

in (6.20). The gaugeable Born sigma-models in a Manin triple polarization, for
which $f_{ij}{}^{k}=0$, single out the cotangent bundles

${\mathsf{D}}=T^{*}{\mathsf{G}}={\mathsf{G}}\ltimes\mathbb{R}^{d}$

of semi-simple Lie groups ${\mathsf{G}}$, which reduce to a sigma-model
description of flat space
$T^{*}{\mathsf{G}}/{\mathsf{G}}\simeq\mathbb{R}^{d}$. In this case one has

$\varrho_{m}=0\ ,\quad
b_{mn}=0\qquad\mbox{and}\qquad\beta^{mn}=Q_{p}{}^{mn}\,x^{p}\ ,$

and the reduced sigma-model action $S^{\prime}(g^{\prime},b^{\prime})$ can be
expressed in terms of a metric $g^{\prime}$ and Kalb-Ramond field $b^{\prime}$
defined through $g^{\prime}+b^{\prime}=(\mathds{1}+\beta)^{-1}$, or
equivalently

$g^{\prime}=(\mathds{1}-\beta^{2})^{-1}\qquad\mbox{and}\qquad
b^{\prime}=-(\mathds{1}+\beta)^{-1}\,\beta\,(\mathds{1}-\beta)^{-1}\ .$

Alternatively, considering the change of polarization which interchanges the
roles of the Lie bialgebras $(\mathfrak{g},\mathbb{R}^{d})$ and
$(\mathbb{R}^{d},\mathfrak{g})$ in the Manin triple, which is the simplest