on $\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}$ along with Remark 4.5 about the
simple zero case, and the definition of the operator $\widetilde{\mathcal{E}}$
in (4.7), we see that (4.27) coincides exactly with the expressions in (4.26).
In particular, the relativistic invariance of the affine Gaudin model was
shown in [16] to be ensured by $\epsilon_{y}^{2}=1$ for all $y\in\bm{\zeta}$.
We see that this coincides with the condition $\mathcal{E}^{2}=\textup{id}$
for the relativistic invariance of the $\mathcal{E}$-model, see Remark 2.8.

In Remark 4.5, we also gave a simple condition for the operator $\mathcal{E}$
to be positive in the case when $\omega$ has simple real poles and
$\mathfrak{g}$ is compact, namely that $\varphi^{\prime}(y)$ and
$\epsilon_{y}$ should have opposite signs for every $y\in{\bm{\zeta}}$. This
corresponds to the condition given in [16, Section 2.2.3] for the Hamiltonian
$\int_{\mathbb{R}}{\rm d}\sigma\;T^{\tau}_{\;\;\tau}$ to be positive. See also
Section 2.5.

###  4.7 Symmetries of the model

####  4.7.1 Global $\bm{G^{\text{diag}}}$-symmetry

In this section we show that the $\mathcal{E}$-model action (4.19) for the
edge mode $l\in C^{\infty}(\Sigma,D)$ has a  _global_ diagonal $G$-symmetry.

Let $\Delta\colon G\to G^{\times|{\bm{z}}^{\prime}|}\subset D$ denote the
diagonal embedding of $G$ into $D$. For any $g_{0}\in G$ and
$\mathsf{U}=(\mathsf{u}_{x,p}\otimes\varepsilon_{x}^{p})_{[x,p]\in\bm{[}{\bm{z}}^{\prime}\bm{]}}\in\mathfrak{d}$,
the adjoint action of $\Delta(g_{0})\in D$ on $\mathsf{U}$ reads

$\displaystyle\operatorname{Ad}_{\Delta(g_{0})}\mathsf{U}=\big{(}(\operatorname{Ad}_{g_{0}}\mathsf{u}_{x,p})\otimes\varepsilon_{x}^{p}\big{)}_{[x,p]\in\bm{[}{\bm{z}}^{\prime}\bm{]}}.$

Since $g_{0}\in G$ we have
$\tau\operatorname{Ad}_{g_{0}}=\operatorname{Ad}_{g_{0}}\tau$ and so it
follows from the explicit form (4.14) of the linear operator
$\mathcal{E}\colon\mathfrak{d}\to\mathfrak{d}$ defined in (4.13) that

$\displaystyle\mathcal{E}\operatorname{Ad}_{\Delta(g_{0})}=\operatorname{Ad}_{\Delta(g_{0})}\mathcal{E}.$
(4.28)

######  Proposition 4.8.

The action (4.19) is invariant under $l\mapsto l\Delta(g_{0})$ for any
$g_{0}\in G$.

###### Proof.

By construction, the kernel and image of the projector
$\mathcal{P}_{l\,\Delta(g_{0})}$ are given by

$\displaystyle\ker\mathcal{P}_{l\,\Delta(g_{0})}=\operatorname{Ad}^{-1}_{l\,\Delta(g_{0})}\mathfrak{k}=\operatorname{Ad}_{\Delta(g_{0})}^{-1}\operatorname{Ad}_{l}^{-1}\mathfrak{k}=\operatorname{Ad}_{\Delta(g_{0})}^{-1}\ker\mathcal{P}_{l},$
$\displaystyle\operatorname{im}\mathcal{P}_{l\,\Delta(g_{0})}=\mathcal{E}\operatorname{Ad}^{-1}_{l\,\Delta(g_{0})}\mathfrak{k}=\mathcal{E}\operatorname{Ad}_{\Delta(g_{0})}^{-1}\operatorname{Ad}_{l}^{-1}\mathfrak{k}=\operatorname{Ad}_{\Delta(g_{0})}^{-1}\mathcal{E}\operatorname{Ad}_{l}^{-1}\mathfrak{k}=\operatorname{Ad}_{\Delta(g_{0})}^{-1}\operatorname{im}\mathcal{P}_{l},$

where in the second line we have used (4.28) in the third step. It is a
standard result on projectors that $\mathcal{P}_{l\,\Delta(g_{0})}$ is thus
given by

$\displaystyle\mathcal{P}_{l\,\Delta(g_{0})}=\operatorname{Ad}_{\Delta(g_{0})}^{-1}\mathcal{P}_{l}\operatorname{Ad}_{\Delta(g_{0})}.$

Similar equalities also hold for $\overline{\mathcal{P}}_{l\,\Delta(g_{0})}$
and the transpose of $\mathcal{P}_{l\,\Delta(g_{0})}$ and
$\overline{\mathcal{P}}_{l\,\Delta(g_{0})}$.

Moreover, under $l\mapsto l\Delta(g_{0})$, the Maurer–Cartan current
$l^{-1}\partial_{\mu}l$ transforms as
$l^{-1}\partial_{\mu}l\mapsto\operatorname{Ad}_{\Delta(g_{0})}^{-1}l^{-1}\partial_{\mu}l$.
Putting all of the above together it now follows that the first term in the
action (4.19) is invariant under $l\mapsto l\Delta(g_{0})$.

Finally, since the WZ-term in (4.19) is independent of the choice of extension
$\widehat{l}^{-1}{\rm d}\widehat{l}$ of the $1$-form $l^{-1}{\rm
d}l\in\Omega^{1}(\Sigma,\mathfrak{d})$ to the bulk $\Sigma\times I$, we can
choose this extension for the transformed $1$-form
$\operatorname{Ad}_{\Delta(g_{0})}^{-1}l^{-1}{\rm d}l$ to be
$\operatorname{Ad}_{\Delta(g_{0})}^{-1}\widehat{l}^{-1}{\rm d}\widehat{l}$,
from which it follows that the WZ-term is also invariant under the
transformation $l\mapsto l\Delta(g_{0})$. ∎

######  Remark 4.9.

The only property of the element $\Delta(g_{0})\in D$ which we used in the
proof of Proposition 4.8 is (4.28). Therefore, the statement of the
proposition would also hold for any other Lie subgroup of $D$ with the
property that all its elements $d\in D$ are such that
$\operatorname{Ad}_{d}\colon\mathfrak{d}\to\mathfrak{d}$ commutes with
$\mathcal{E}$.

####  4.7.2 Global symmetries for $\bm{\mathfrak{k}}$ an ideal

We now consider the $\mathcal{E}$-model action (4.19) in a more specific
setup.

######  Proposition 4.10.

If the Lagrangian subalgebra $\mathfrak{k}\subset\mathfrak{d}$ is an ideal,
then the $\mathcal{E}$-model action (4.19) is invariant under $l\mapsto al$
for any $a\in D$.

######  Remark 4.11.

In fact, since the $\mathcal{E}$-model action has a gauge invariance $l\mapsto
kl$ for $k\in C^{\infty}(\Sigma,K)$, see Section 2.3, the global symmetry in
Proposition 4.10 is really only an additional symmetry by the Lie group
$K\backslash D$.

###### Proof.

Proposition 4.10 The currents $l^{-1}\partial_{\mu}l$ and the WZ-term in
(4.19) are both invariant under the transformation $l\mapsto al$. It remains
to check that $\mathcal{P}_{al}=\mathcal{P}_{l}$ and
$\overline{\mathcal{P}}_{al}=\overline{\mathcal{P}}_{l}$. Since $\mathfrak{k}$
is an ideal of $\mathfrak{d}$ we have
$\operatorname{Ad}_{l}^{-1}\mathfrak{k}=\mathfrak{k}$ and thus

$\displaystyle\operatorname{im}\mathcal{P}_{l}=\mathcal{E}^{-1}\mathfrak{k},\qquad\ker\mathcal{P}_{l}=\mathfrak{k}.$

We see that the projector $\mathcal{P}_{l}$, and also
$\overline{\mathcal{P}}_{l}$, is in fact independent of $l$. It is therefore
invariant under $l\mapsto al$, as required. ∎

We can construct simple examples of Lagrangian ideals
$\mathfrak{k}\subset\mathfrak{d}$ in the case when all the multiplicities
$n_{x}$ of the poles $x\in{\bm{z}}$ of $\omega$ are even, i.e., $n_{x}=2r_{x}$
for some $r_{x}\in\mathbb{Z}_{\geq 1}$. Recall the definition (3.13) of the
Lie algebra $\mathfrak{d}$, namely

$\displaystyle\mathfrak{d}=\bigoplus_{x\in\bm{z}^{\prime}_{\rm
r}}\mathfrak{g}\otimes_{\mathbb{R}}\mathcal{T}^{2r_{x}}_{x}\oplus\bigoplus_{x\in\bm{z}_{\rm
c}}\big{(}\mathfrak{g}^{\mathbb{C}}\otimes_{\mathbb{C}}\mathcal{T}^{2r_{x}}_{x}\big{)}_{\mathbb{R}}.$

Consider the ideals $\mathcal{T}^{\geq
r_{x}}_{x}\coloneqq\varepsilon_{x}^{r_{x}}\mathbb{R}[\varepsilon_{x}]/\big{(}\varepsilon_{x}^{2r_{x}}\big{)}\subset\mathcal{T}^{2r_{x}}_{x}$
for real finite poles $x\in{\bm{z}}^{\prime}_{\rm r}$ and $\mathcal{T}^{\geq
r_{x}}_{x}\coloneqq\varepsilon_{x}^{r_{x}}\mathbb{C}[\varepsilon_{x}]/\big{(}\varepsilon_{x}^{2r_{x}}\big{)}\subset\mathcal{T}^{2r_{x}}_{x}$
for complex poles $x\in{\bm{z}}_{\rm c}$. It is easy to check that

$\displaystyle\mathfrak{k}\coloneqq\bigoplus_{x\in\bm{z}^{\prime}_{\rm
r}}\mathfrak{g}\otimes_{\mathbb{R}}\mathcal{T}^{\geq
r_{x}}_{x}\oplus\bigoplus_{x\in\bm{z}_{\rm
c}}\big{(}\mathfrak{g}^{\mathbb{C}}\otimes_{\mathbb{C}}\mathcal{T}^{\geq
r_{x}}_{x}\big{)}_{\mathbb{R}}$

is a Lagrangian ideal of $\mathfrak{d}$.

##  5 Examples

In this section we give a few examples of the above general construction,
including details of the one mentioned in the introduction. In each case, we
make a choice of meromorphic $1$-form $\omega$, Lagrangian subalgebra
$\mathfrak{k}\subset\mathfrak{d}$ and set of parameters $\epsilon_{y}$
associated with each zero $y\in{\bm{\zeta}}$ of $\omega$. This is then fed
into the general construction to produce the action and Lax connection of the
corresponding integrable $\sigma$-models.

###  5.1 Principal chiral model and non-abelian $\bm{T}$-dual

We rederive the well-known actions of the principal chiral model and its non-
abelian $T$-dual as an application of our general construction. The principal
chiral model was already derived in [13, Section 10.2] and then again in [17,
Section 5.1] using the unifying $2$d action valid when $\omega$ has at most
double poles. We derive it again in Section 5.1.3 below since it provides the
simplest illustration of our construction.

However, the derivation of the non-abelian $T$-dual in Section 5.1.4 below
from $4$d Chern–Simons theory is new. This was conjectured but could not be
derived in [17] since the formula for the unifying $2$d action there was only
applicable under a technical condition on the Lagrangian subalgebra
$\mathfrak{k}\subset\mathfrak{d}$, see [17, equation (4.16)]. The latter is
not satisfied by the choice of Lagrangian subalgebra used in Section 5.1.4 to
derive the non-abelian $T$-dual.

We begin by setting up the formalism to discuss both the principal chiral
model and its non-abelian $T$-dual. We let $a>0$ and consider the $1$-form

$\displaystyle\omega=\frac{a^{2}{\rm d}z}{z^{2}}-{\rm d}z.$ (5.1)

In the notation of Section 3.1 we have the set of poles
${\bm{z}}=\\{0,\infty\\}$ and the set of zeroes is ${\bm{\zeta}}=\\{a,-a\\}$.
Both poles are double poles so that $n_{0}=n_{\infty}=2$ and both zeroes are
simple so that $m_{a}=m_{-a}=1$. Moreover, all the zeroes and poles are real
so here ${\bm{z}}={\bm{z}}_{\rm r}$ and ${\bm{\zeta}}={\bm{\zeta}}_{\rm r}$.
The levels are read off from $\omega$ to be $\ell^{0}_{0}=0$,
$\ell^{0}_{1}=a^{2}$ and $\ell^{\infty}_{1}=1$.

We also choose the parameters associated to the set of zeroes in (4.6) to be

$\displaystyle\epsilon_{\pm a}=\pm 1.$ (5.2)

Below we shall construct all the data associated with the choice of $1$-form
(5.1) and parameters (5.2), and then use this data to build the actions for
the principal chiral model in Section 5.1.3 and its non-abelian $T$-dual in
Section 5.1.4.

####  5.1.1 Lie groups $\bm{D}$, $\bm{K}$ and $\bm{\tilde{K}}$

The defect Lie algebra (3.13) is given here by

$\displaystyle\mathfrak{d}=\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{R}[\varepsilon_{0}]/\big{(}\varepsilon_{0}^{2}\big{)}=\mathfrak{g}\ltimes\mathfrak{g}_{\rm
ab},$

where $\mathfrak{g}_{\rm
ab}\coloneqq\mathfrak{g}\otimes_{\mathbb{R}}\varepsilon_{0}\mathbb{R}[\varepsilon_{0}]/\big{(}\varepsilon_{0}^{2}\big{)}$
is isomorphic to the vector space $\mathfrak{g}$ equipped with the trivial Lie
bracket and the adjoint action of $\mathfrak{g}$. By using the abbreviated
notation
$\mathsf{u}^{p}\coloneqq\mathsf{u}^{[0,p]}=\mathsf{u}\otimes\varepsilon^{p}_{0}$
for any $\mathsf{u}\in\mathfrak{g}$ and $p\in\\{0,1\\}$, the Lie algebra
relations in $\mathfrak{d}$ read

$\displaystyle\big{[}\mathsf{u}^{0},\mathsf{v}^{0}\big{]}=[\mathsf{u},\mathsf{v}]^{0},\qquad\big{[}\mathsf{u}^{0},\mathsf{v}^{1}\big{]}=\big{[}\mathsf{u}^{1},\mathsf{v}^{0}\big{]}=[\mathsf{u},\mathsf{v}]^{1},\qquad\big{[}\mathsf{u}^{1},\mathsf{v}^{1}\big{]}=0,$
(5.3)

for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$.

The associated Lie group $D$ is the tangent bundle $TG$ of $G$, which in the
right trivialisation is isomorphic to the Lie group $G\ltimes\mathfrak{g}$
with product and inverse

$\displaystyle(g,\mathsf{u})(h,\mathsf{v})=\big{(}gh,\mathsf{u}+\operatorname{Ad}_{g}\mathsf{v}\big{)},\qquad(g,\mathsf{u})^{-1}=\big{(}g^{-1},-\operatorname{Ad}_{g}^{-1}\mathsf{u}\big{)}$

for every $g,h\in G$ and $\mathsf{u},\mathsf{v}\in\mathfrak{g}$.

We have the two obvious Lie subalgebras

$\displaystyle\tilde{\mathfrak{k}}\coloneqq\mathfrak{g}\oplus\\{0\\}=\big{\\{}\mathsf{u}^{0}\mid\mathsf{u}\in\mathfrak{g}\big{\\}},\qquad\mathfrak{k}\coloneqq\\{0\\}\oplus\mathfrak{g}_{\rm
ab}=\big{\\{}\mathsf{u}^{1}\mid\mathsf{u}\in\mathfrak{g}\big{\\}}$ (5.4)

of $\mathfrak{d}$. These are complementary since we have the direct sum
decomposition

$\displaystyle\mathfrak{d}=\mathfrak{k}\dotplus\tilde{\mathfrak{k}}.$ (5.5)

Let $\tilde{K},K\subset D$ denote the corresponding connected Lie subgroups of
$D$, which are isomorphic to $G\times\\{0\\}$ and
$\\{\textup{id}\\}\times\mathfrak{g}$, respectively. In particular, it is
clear that $K$ is normal in $D$. We have the global decomposition
$D=K\tilde{K}=\tilde{K}K$.

The bilinear form (3.3) on $\mathfrak{d}$ is given explicitly by

$\displaystyle\langle\\!\langle\mathsf{u}^{0},\mathsf{v}^{0}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathsf{u}^{1},\mathsf{v}^{1}\rangle\\!\rangle_{\mathfrak{d}}=0,\qquad\langle\\!\langle\mathsf{u}^{0},\mathsf{v}^{1}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathsf{u}^{1},\mathsf{v}^{0}\rangle\\!\rangle_{\mathfrak{d}}=a^{2}\langle\mathsf{u},\mathsf{v}\rangle,$
(5.6)

for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$ so that $\mathfrak{k}$ and
$\tilde{\mathfrak{k}}$ are both Lagrangian subalgebras of $\mathfrak{d}$.

####  5.1.2 Linear operator $\bm{\mathcal{E}}$

The real vector space (4.3) associated with the two simple zeroes of $\omega$
is given here by
$\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}\oplus\mathfrak{g}$. One
checks by computing the Cauchy matrix (4.9) that the isomorphism
$\mathcal{C}\colon\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}\to\mathfrak{d}$ from
Lemma 4.1 (see also (4.10) and (4.11)) and its inverse are given here by

$\displaystyle\mathcal{C}(\mathsf{u},\mathsf{v})=-\frac{\mathsf{u}^{0}}{a}+\frac{\mathsf{v}^{0}}{a}-\frac{\mathsf{u}^{1}}{a^{2}}-\frac{\mathsf{v}^{1}}{a^{2}},\qquad\mathcal{C}^{-1}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}\big{)}=\bigg{(}{-}\frac{1}{2}a\mathsf{u}-\frac{1}{2}a^{2}\mathsf{v},\frac{1}{2}a\mathsf{u}-\frac{1}{2}a^{2}\mathsf{v}\bigg{)}$

for every $\mathsf{u},\mathsf{v}\in\mathfrak{g}$. Given the choice of
parameters $\epsilon_{y}$ for $y\in{\bm{\zeta}}$ we made in (5.2), the linear
isomorphism (4.7) then reads
$\widetilde{\mathcal{E}}(\mathsf{u},\mathsf{v})=(\mathsf{u},-\mathsf{v})$ and
the linear operator (4.13) is found to act in the present case as

$\displaystyle\mathcal{E}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}\big{)}=a\mathsf{v}^{0}+\frac{\mathsf{u}^{1}}{a}$
(5.7)

for every $\mathsf{u},\mathsf{v}\in\mathfrak{g}$. We have
$\mathcal{E}^{2}=\textup{id}$, corresponding to the fact that the principal
chiral model and its non-abelian $T$-dual are relativistic.

The Lie algebra relations (5.3), bilinear form (5.6) and linear operator (5.7)
agree with those for the principal chiral model given in [32, equations
(19)–(21)].

####  5.1.3 Principal chiral model

Here we apply the general construction of Section 4 with $\mathfrak{k}$
defined in (5.4) as the Lagrangian subalgebra. Since we have the global
factorisation $D=K\tilde{K}$ we can factorise the field $l\in
C^{\infty}(\Sigma,D)$ uniquely as $l=kg$ for some $k\in C^{\infty}(\Sigma,K)$
and $g\in C^{\infty}\big{(}\Sigma,\tilde{K}\big{)}$. Using the gauge
invariance of the action (2.14) under $l\mapsto k^{-1}l$ from Section 2.3, we
can then fix $l=g$. In particular, the action (2.14) now reads

$\displaystyle S_{\rm 2d}(g)=\frac{1}{2}\int_{\Sigma}\big{(}\langle\\!\langle
g^{-1}\partial_{\tau}g,\mathcal{E}\mathcal{P}_{g}\big{(}g^{-1}\partial_{\tau}g\big{)}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle
g^{-1}\partial_{\sigma}g,\mathcal{E}\mathcal{P}_{g}\big{(}g^{-1}\partial_{\sigma}g\big{)}\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle\hphantom{S_{\rm
2d}(g)=\frac{1}{2}\int_{\Sigma}\big{(}}{}+\langle\\!\langle
g^{-1}\partial_{\tau}g,\big{(}\mathcal{P}_{g}-{}^{t}\mathcal{P}_{g}\big{)}\big{(}g^{-1}\partial_{\sigma}g\big{)}\rangle\\!\rangle_{\mathfrak{d}}\big{)}{\rm
d}\sigma\wedge{\rm d}\tau-\frac{1}{2}I^{\rm WZ}_{\tilde{\mathfrak{k}}}[g]$
(5.8)

for the group valued field $g\in C^{\infty}(\Sigma,\tilde{K})$, where we have
used the fact that $\mathcal{E}^{2}=\textup{id}$ together with part $(vi)$ of
Proposition 2.3. Here $\mathcal{P}_{g}$ denotes the projector defined by the
relations (2.4), namely

$\displaystyle\ker\mathcal{P}_{g}=\operatorname{Ad}_{g}^{-1}\mathfrak{k},\qquad\operatorname{im}\mathcal{P}_{g}=\mathcal{E}\operatorname{Ad}_{g}^{-1}\mathfrak{k}.$

Since $[\tilde{\mathfrak{k}},\mathfrak{k}]\subset\mathfrak{k}$ we have
$\operatorname{Ad}_{g}^{-1}\mathfrak{k}=\mathfrak{k}$ and hence

$\displaystyle\ker\mathcal{P}_{g}=\mathfrak{k},\qquad\operatorname{im}\mathcal{P}_{g}=\mathcal{E}\mathfrak{k}=\tilde{\mathfrak{k}},$

where the last equality uses the explicit forms (5.4) and (5.7) of the two
subalgebras $\mathfrak{k},\tilde{\mathfrak{k}}\subset\mathfrak{d}$ and of
$\mathcal{E}$. Thus $\mathcal{P}_{g}$ is simply the projection onto
$\tilde{\mathfrak{k}}$ along $\mathfrak{k}$, relative to the direct sum
decomposition (5.5). In particular, it acts as the identity on
$g^{-1}\partial_{\mu}g\in C^{\infty}(\Sigma,\tilde{\mathfrak{k}})$. Moreover,
since $\tilde{\mathfrak{k}}$ is isotropic with respect to the bilinear form
(5.6), the WZ-term in the action (5.8) vanishes.

Putting all of the above together and noting the identity
$\langle\\!\langle\mathsf{u}^{0},\mathcal{E}\mathsf{v}^{0}\rangle\\!\rangle_{\mathfrak{d}}=a\langle\mathsf{u},\mathsf{v}\rangle$
for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$, the action (5.8) reduces to
the usual principal chiral model action

$\displaystyle S_{\rm 2d}(g)=\frac{1}{2}a\int_{\Sigma}\langle
g^{-1}\partial_{+}g,g^{-1}\partial_{-}g\rangle{\rm d}\sigma\wedge{\rm d}\tau,$

where $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$, for the Lie group
valued field $g\in C^{\infty}\big{(}\Sigma,\tilde{K}\big{)}\cong
C^{\infty}(\Sigma,G)$. The global $G^{\rm diag}$-symmetry from Proposition 4.8
corresponds here to the right $G$-symmetry of the principal chiral model.
Since the Lagrangian subalgebra $\mathfrak{k}\subset\mathfrak{d}$ is an ideal
we also have the global left symmetry of Proposition 4.10 by the Lie group
$K\backslash D\simeq\tilde{K}\simeq G$, corresponding to the left $G$-symmetry
of the principal chiral model.

Let $j\coloneqq g^{-1}{\rm d}g$. Using $\ast{\rm d}\sigma=-{\rm d}\tau$ and
$\ast{\rm d}\tau=-{\rm d}\sigma$, the Lax connection (4.23) is given by
$\mathcal{L}={\bm{p}}\big{(}j^{0}-\frac{1}{a}\ast j^{1}\big{)}$, where the map
${\bm{p}}$ in (4.22) is the inverse of the map ${\bm{j}}_{{\bm{z}}^{\prime}}$
defined in (4.1) and given explicitly in the present case by

$\displaystyle{\bm{j}}_{\bm{z}^{\prime}}\colon\
\frac{\mathsf{u}_{a}}{z-a}+\frac{\mathsf{u}_{-a}}{z+a}\longmapsto\frac{1}{a}(\mathsf{u}_{-a}-\mathsf{u}_{a})^{0}+\frac{1}{a^{2}}(-\mathsf{u}_{-a}-\mathsf{u}_{a})^{1}.$

Its inverse is then given explicitly by

$\displaystyle{\bm{p}}\colon\
\mathsf{v}^{0}+\mathsf{w}^{1}\longmapsto\frac{a\mathsf{v}+a^{2}\mathsf{w}}{2(a-z)}+\frac{a\mathsf{v}-a^{2}\mathsf{w}}{2(a+z)}.$
(5.9)

We therefore obtain the Lax connection of the principal chiral model

$\displaystyle\mathcal{L}={\bm{p}}\big{(}j^{0}-\ast
j^{1}\big{)}=a\frac{aj-z\ast j}{a^{2}-z^{2}}=\frac{aj_{+}}{a-z}{\rm
d}\sigma^{+}+\frac{aj_{-}}{a+z}{\rm d}\sigma^{-},$ (5.10)

where $j_{\pm}\coloneqq g^{-1}\partial_{\pm}g$. This coincides with the usual
Lax connection of the principal chiral model after rescaling the spectral
parameter as $z\mapsto az$.

####  5.1.4 Non-abelian $\bm{T}$-dual

We will now use the reverse factorisation $D=\tilde{K}K$, treating
$\tilde{K}\subset D$ as the Lie subgroup which we quotient by in Section 2.2.
As in Section 5.1.3, we can factorise our field $l\in C^{\infty}(\Sigma,D)$
uniquely as $l=\tilde{k}p$ for some $\tilde{k}\in
C^{\infty}\big{(}\Sigma,\tilde{K}\big{)}$ and $p\in C^{\infty}(\Sigma,K)$. We
may therefore use the gauge invariance by the subgroup $\tilde{K}$ from
Section 2.3 to fix $l=p$, obtaining the action (2.14) for $p\in
C^{\infty}(\Sigma,K)$, where $\tilde{\mathfrak{k}}$ now plays the role of
$\mathfrak{k}$. As in Section 5.1.3, the WZ-term in this action vanishes since
the Lie subalgebra $\mathfrak{k}$ is isotropic. Furthermore, by definition we
can write $p=\big{(}\textup{id},\frac{1}{a}\mathsf{m}\big{)}$ for some
$\mathsf{m}\in C^{\infty}(\Sigma,\mathfrak{g})$, where the factor of
$\frac{1}{a}$ is introduced for later convenience. In particular, we then have
$p^{-1}\partial_{\mu}p=\frac{1}{a}\partial_{\mu}\mathsf{m}^{1}\in
C^{\infty}(\Sigma,\mathfrak{k})$. In the present case, the action (2.14)
therefore simplifies to

$\displaystyle S_{\rm
2d}(\mathsf{m})=\frac{1}{2}a^{-2}\int_{\Sigma}\big{(}\langle\\!\langle\partial_{\tau}\mathsf{m}^{1},\mathcal{E}\tilde{\mathcal{P}}_{p}\big{(}\partial_{\tau}\mathsf{m}^{1}\big{)}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle\partial_{\sigma}\mathsf{m}^{1},\mathcal{E}\tilde{\mathcal{P}}_{p}\big{(}\partial_{\sigma}\mathsf{m}^{1}\big{)}\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle\hphantom{S_{\rm
2d}(\mathsf{m})=\frac{1}{2}a^{-2}\int_{\Sigma}\big{(}}{}+\langle\\!\langle\partial_{\tau}\mathsf{m}^{1},\big{(}\tilde{\mathcal{P}}_{p}-{}^{t}\tilde{\mathcal{P}}_{p}\big{)}\big{(}\partial_{\sigma}\mathsf{m}^{1}\big{)}\rangle\\!\rangle_{\mathfrak{d}}\big{)}{\rm
d}\sigma\wedge{\rm d}\tau,$ (5.11)

where now $\tilde{\mathcal{P}}_{p}$ denotes the projector defined in the same
way as in (2.4) but relative to the Lagrangian subalgebra
$\tilde{\mathfrak{k}}$, namely

$\displaystyle\ker\tilde{\mathcal{P}}_{p}=\operatorname{Ad}_{p}^{-1}\tilde{\mathfrak{k}}=\bigg{(}\textup{id}-\frac{1}{a}\operatorname{ad}_{\mathsf{m}^{1}}\bigg{)}\tilde{\mathfrak{k}}=\bigg{\\{}\mathsf{u}^{0}-\frac{1}{a}[\mathsf{m},\mathsf{u}]^{1}\,\bigg{|}\,\mathsf{u}\in\mathfrak{g}\bigg{\\}},$
$\displaystyle\operatorname{im}\tilde{\mathcal{P}}_{p}=\mathcal{E}\operatorname{Ad}_{p}^{-1}\tilde{\mathfrak{k}}=\mathcal{E}\bigg{(}\bigg{(}\textup{id}-\frac{1}{a}\operatorname{ad}_{\mathsf{m}^{1}}\bigg{)}\tilde{\mathfrak{k}}\bigg{)}=\bigg{\\{}\frac{1}{a}\mathsf{u}^{1}-[\mathsf{m},\mathsf{u}]^{0}\,\bigg{|}\,\mathsf{u}\in\mathfrak{g}\bigg{\\}}.$

It is straightforward to check that the projector
$\tilde{\mathcal{P}}_{p}\colon\mathfrak{d}\to\mathfrak{d}$ with the above
kernel and image is given by

$\displaystyle\tilde{\mathcal{P}}_{p}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}\big{)}=-\bigg{(}\frac{\operatorname{ad}^{2}_{\mathsf{m}}}{\textup{id}-\operatorname{ad}_{\mathsf{m}}^{2}}\mathsf{u}\bigg{)}^{0}-\bigg{(}\frac{a\operatorname{ad}_{\mathsf{m}}}{\textup{id}-\operatorname{ad}_{\mathsf{m}}^{2}}\mathsf{v}\bigg{)}^{0}+\bigg{(}\frac{a^{-1}\operatorname{ad}_{\mathsf{m}}}{\textup{id}-\operatorname{ad}_{\mathsf{m}}^{2}}\mathsf{u}\bigg{)}^{1}+\bigg{(}\frac{1}{\textup{id}-\operatorname{ad}_{\mathsf{m}}^{2}}\mathsf{v}\bigg{)}^{1}$

for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$. In particular, applying this
to $\partial_{\mu}\mathsf{m}^{1}$ we find

$\displaystyle\tilde{\mathcal{P}}_{p}\big{(}\partial_{\mu}\mathsf{m}^{1}\big{)}=\bigg{(}{-}\frac{a\operatorname{ad}_{\mathsf{m}}}{\textup{id}-\operatorname{ad}^{2}_{\mathsf{m}}}\partial_{\mu}\mathsf{m}\bigg{)}^{0}+\bigg{(}\frac{1}{\textup{id}-\operatorname{ad}^{2}_{\mathsf{m}}}\partial_{\mu}\mathsf{m}\bigg{)}^{1}.$
(5.12)

Substituting this into the action (5.11) and using the expressions (5.6) and
(5.7) of the bilinear form
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$ and the linear
operator $\mathcal{E}$ we arrive at the standard action of the non-abelian
$T$-dual of the principal chiral model [24, 25], namely

$\displaystyle S_{\rm
2d}(\mathsf{m})=\frac{1}{2}a\int_{\Sigma}\bigg{\langle}\partial_{+}\mathsf{m},\frac{1}{\textup{id}-\operatorname{ad}_{\mathsf{m}}}\partial_{-}\mathsf{m}\bigg{\rangle}{\rm
d}\sigma\wedge{\rm d}\tau.$

The Lax connection (4.23) is now given by

$\displaystyle\mathcal{L}={\bm{p}}\bigg{(}\frac{1}{a}\tilde{\mathcal{P}}_{p}\big{(}{\rm
d}\mathsf{m}^{1}\big{)}-\frac{1}{a}\mathcal{E}\tilde{\mathcal{P}}_{p}\big{(}{\ast}{\rm
d}\mathsf{m}^{1}\big{)}\bigg{)}.$

Using the explicit form of the inverse ${\bm{p}}$ in (5.9) and using (5.12) we
find

$\displaystyle\mathcal{L}=\frac{a}{a-z}\frac{1}{\textup{id}+\operatorname{ad}_{\mathsf{m}}}\partial_{+}\mathsf{m}\;{\rm
d}\sigma^{+}-\frac{a}{a+z}\frac{1}{\textup{id}-\operatorname{ad}_{\mathsf{m}}}\partial_{-}\mathsf{m}\;{\rm
d}\sigma^{-}.$

This becomes the usual Lax connection of the non-abelian $T$-dual of the
principal chiral model after the rescaling $z\mapsto az$.

###  5.2 Fourth order pole

In this section we give an example of our construction in the case when
$\omega$ has a pole of order $4$. We will let $a>b>0$ and take

$\displaystyle\omega=\frac{\big{(}z^{2}-a^{2}\big{)}\big{(}b^{2}-z^{2}\big{)}}{z^{4}}{\rm
d}z.$ (5.13)

The set of poles is ${\bm{z}}={\bm{z}}_{\rm r}=\\{0,\infty\\}$ with orders
$n_{0}=4$ and $n_{\infty}=2$. The associated levels are
$\ell^{0}_{3}=-a^{2}b^{2}$, $\ell^{0}_{2}=0$, $\ell^{0}_{1}=a^{2}+b^{2}$,
$\ell^{0}_{0}=0$ and $\ell^{\infty}_{1}=1$. The set of zeroes is
${\bm{\zeta}}={\bm{\zeta}}_{\rm r}=\\{a,-a,b,-b\\}$ with all zeroes being
simple.

We let the parameters in (4.6) associated with the set ${\bm{\zeta}}$ of
zeroes of $\omega$ be

$\displaystyle\epsilon_{\pm a}=\pm 1,\qquad\epsilon_{\pm b}=\mp 1.$ (5.14)

This choice ensures that $\mathcal{E}$ will be positive (for compact
$\mathfrak{g}$) and such that $\mathcal{E}^{2}=\textup{id}$. Indeed, the
latter condition follows since $\epsilon_{y}^{2}=1$ for all $y\in{\bm{\zeta}}$
and positivity follows from Remark 4.5 after noting that
$-\epsilon_{y}\varphi^{\prime}(y)>0$ for each $y\in{\bm{\zeta}}$. In what
follows we shall construct all the necessary data associated with the choice
of $1$-form (5.13) and parameters (5.14). We then extract from this data the
action and Lax connection of a new $2$d integrable field theory.

####  5.2.1 Lie groups $\bm{D}$ and $\bm{K}$

Since the pole $0$ is real, the defect Lie algebra (3.13) is

$\displaystyle\mathfrak{d}=\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{R}[\varepsilon_{0}]/\big{(}\varepsilon_{0}^{4}\big{)}.$

As in Section 5.1 we use the abbreviated notation
$\mathsf{u}^{p}\coloneqq\mathsf{u}^{[0,p]}=\mathsf{u}\otimes\varepsilon^{p}_{0}$
for any $\mathsf{u}\in\mathfrak{g}$ and $p\in\\{0,1,2,3\\}$. The Lie algebra
relations in $\mathfrak{d}$ read

$\displaystyle\big{[}\mathsf{u}^{p},\mathsf{v}^{q}\big{]}=[\mathsf{u},\mathsf{v}]^{p+q},$

for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$ and $p,q\in\\{0,1,2,3\\}$. In
particular, recall this vanishes for $p+q\geq 4$.

The Lie group $D$ is given by the $3^{\rm rd}$ order jet bundle $J^{3}G$ of
the Lie group $G$. In the right trivialisation it is isomorphic to
$G\times\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}$ equipped with the
Lie group product and inverse [66]

$\displaystyle(g,\mathsf{u},\mathsf{v},\mathsf{w})(h,\mathsf{x},\mathsf{y},\mathsf{z})=\big{(}gh,\mathsf{u}+\operatorname{Ad}_{g}\mathsf{x},\mathsf{v}+\operatorname{Ad}_{g}\mathsf{y}+[\mathsf{u},\operatorname{Ad}_{g}\mathsf{x}],$
$\displaystyle\hphantom{(g,\mathsf{u},\mathsf{v},\mathsf{w})(h,\mathsf{x},\mathsf{y},\mathsf{z})=\big{(}}{}\mathsf{w}+\operatorname{Ad}_{g}\mathsf{z}+2[\mathsf{u},\operatorname{Ad}_{g}\mathsf{y}]+[\mathsf{v},\operatorname{Ad}_{g}\mathsf{x}]+[\mathsf{u},[\mathsf{u},\operatorname{Ad}_{g}\mathsf{x}]\big{)},$
$\displaystyle(g,\mathsf{u},\mathsf{v},\mathsf{w})^{-1}=\big{(}g^{-1},-\operatorname{Ad}_{g}^{-1}\mathsf{u},-\operatorname{Ad}_{g}^{-1}\mathsf{v},-\operatorname{Ad}_{g}^{-1}\mathsf{w}+\operatorname{Ad}_{g}^{-1}[\mathsf{u},\mathsf{v}]\big{)}$

for every $g,h\in G$ and
$\mathsf{u},\mathsf{v},\mathsf{w},\mathsf{x},\mathsf{y},\mathsf{z}\in\mathfrak{g}$.

We consider the ideal $\mathfrak{k}\subset\mathfrak{d}$ defined in Section
4.7.2, which is given here by

$\displaystyle\mathfrak{k}=\mathfrak{g}\otimes_{\mathbb{R}}\varepsilon_{0}^{2}\mathbb{R}[\varepsilon_{0}]/\big{(}\varepsilon_{0}^{4}\big{)}=\big{\\{}\mathsf{u}^{2}+\mathsf{v}^{3}\mid\mathsf{u},\mathsf{v}\in\mathfrak{g}\big{\\}}.$

Let $K\subset D$ denote the corresponding connected Lie subgroup of $D$. It is
isomorphic to
$\\{\textup{id}\\}\times\\{0\\}\times\mathfrak{g}\times\mathfrak{g}$ which is
normal in $G\times\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}$ since

$\displaystyle(g,\mathsf{u},\mathsf{v},\mathsf{w})(\textup{id},0,\mathsf{y},\mathsf{z})(g,\mathsf{u},\mathsf{v},\mathsf{w})^{-1}=\big{(}\textup{id},0,\operatorname{Ad}_{g}\mathsf{y},\operatorname{Ad}_{g}\mathsf{z}+3[\mathsf{u},\operatorname{Ad}_{g}\mathsf{y}]\big{)},$

for any $g\in G$ and
$\mathsf{u},\mathsf{v},\mathsf{w},\mathsf{y},\mathsf{z}\in\mathfrak{g}$. In
particular, the left coset $K\backslash D$ is naturally a Lie group which, as
a manifold, is diffeomorphic to
$G\times\mathfrak{g}\times\\{0\\}\times\\{0\\}$.

The bilinear form (3.3) on $\mathfrak{d}$ is given explicitly by

$\displaystyle\langle\\!\langle\mathsf{u}^{p},\mathsf{v}^{q}\rangle\\!\rangle_{\mathfrak{d}}=\left\\{\begin{array}[]{ll}-a^{2}b^{2}\langle\mathsf{u},\mathsf{v}\rangle,&\text{if}\quad
p+q=3,\\\
\big{(}a^{2}+b^{2}\big{)}\langle\mathsf{u},\mathsf{v}\rangle,&\text{if}\quad
p+q=1,\\\ 0,&\textup{otherwise}\end{array}\right.$

for any $\mathsf{u},\mathsf{v}\in\mathfrak{g}$. In particular, we see that
$\mathfrak{k}$ is indeed a Lagrangian subalgebra of $\mathfrak{d}$.

####  5.2.2 Linear operator $\bm{\mathcal{E}}$

The real vector space (4.3) associated with the zeroes of $\omega$ is given
here by $\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}^{\oplus 4}$.
With the choice of parameters $\epsilon_{y}$ for $y\in{\bm{\zeta}}$ in (5.14)
we find by explicitly computing the Cauchy matrix (4.9) that the linear
operator $\mathcal{E}\colon\mathfrak{d}\to\mathfrak{d}$ in (4.13) is given by

$\displaystyle\mathcal{E}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}+\mathsf{w}^{2}+\mathsf{x}^{3}\big{)}=\frac{a^{2}-ab+b^{2}}{a-b}\mathsf{v}^{0}-\frac{a^{2}b^{2}}{a-b}\mathsf{x}^{0}+\frac{1}{a-b}\mathsf{u}^{1}-\frac{ab}{a-b}\mathsf{w}^{1}$
$\displaystyle\hphantom{\mathcal{E}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}+\mathsf{w}^{2}+\mathsf{x}^{3}\big{)}=}{}+\frac{1}{a-b}\mathsf{v}^{2}-\frac{ab}{a-b}\mathsf{x}^{2}+\frac{1}{ab(a-b)}\mathsf{u}^{3}-\frac{a^{2}-ab+b^{2}}{ab(a-b)}\mathsf{w}^{3},$
(5.15)

for every $\mathsf{u},\mathsf{v},\mathsf{w},\mathsf{x}\in\mathfrak{g}$. As
previously noted after (5.14) we have $\mathcal{E}^{2}=\textup{id}$. Since
$\mathfrak{k}\subset\mathfrak{d}$ is an ideal we have
$\operatorname{Ad}_{l}^{-1}\mathfrak{k}=\mathfrak{k}$ for any $l\in D$ and
therefore the projector $\mathcal{P}_{l}$ defined as in (2.4) has kernel and
image

$\displaystyle\ker\mathcal{P}_{l}=\mathfrak{k}=\big{\\{}\mathsf{u}^{2}+\mathsf{v}^{3}\mid\mathsf{u},\mathsf{v}\in\mathfrak{g}\big{\\}},$
$\displaystyle\operatorname{im}\mathcal{P}_{l}=\mathcal{E}\mathfrak{k}=\big{\\{}a^{3}b^{3}\mathsf{u}^{0}+a^{2}b^{2}\mathsf{v}^{1}+a^{2}b^{2}\mathsf{u}^{2}+\big{(}a^{2}-ab+b^{2}\big{)}\mathsf{v}^{3}\mid\mathsf{u},\mathsf{v}\in\mathfrak{g}\big{\\}},$

where we used the explicit form of $\mathcal{E}$ in (5.15). Note that
$\mathcal{P}_{l}$ is therefore independent of $l$. Explicitly, we have

$\displaystyle\mathcal{P}_{l}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}+\mathsf{w}^{2}+\mathsf{x}^{3}\big{)}=\mathsf{u}^{0}+\mathsf{v}^{1}+\frac{1}{ab}\mathsf{u}^{2}+\frac{a^{2}-ab+b^{2}}{a^{2}b^{2}}\mathsf{v}^{3},$
$\displaystyle\mathcal{E}\mathcal{P}_{l}\big{(}\mathsf{u}^{0}+\mathsf{v}^{1}+\mathsf{w}^{2}+\mathsf{x}^{3}\big{)}=\frac{b-a}{ab}\mathsf{v}^{2}+\frac{b-a}{a^{2}b^{2}}\mathsf{u}^{3},$
(5.16)

for any $\mathsf{u},\mathsf{v},\mathsf{w},\mathsf{x}\in\mathfrak{g}$.

####  5.2.3 Action

Let $p=(g,\mathsf{u},0,0)\in C^{\infty}(\Sigma,D)$ be a representative of a
class in $K\backslash D$ in $D$. We would like to explicitly determine the
corresponding action (2.14), namely

$\displaystyle S_{\rm 2d}(p)=\frac{1}{2}\int_{\Sigma}\big{(}\langle\\!\langle
p^{-1}{\rm d}p,\mathcal{E}\mathcal{P}_{p}\big{(}{\ast}p^{-1}{\rm
d}p\big{)}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle p^{-1}{\rm
d}p,\mathcal{P}_{p}\big{(}p^{-1}{\rm
d}p\big{)}\rangle\\!\rangle_{\mathfrak{d}}\big{)}-\frac{1}{2}I^{\rm
WZ}_{\mathfrak{d}}[p].$ (5.17)

First, we note that [66]

$\displaystyle p^{-1}{\rm d}p=\big{(}g^{-1}{\rm
d}g\big{)}^{0}+\big{(}\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}\big{)}^{1}-\frac{1}{2}\big{(}\operatorname{Ad}_{g}^{-1}[\mathsf{u},{\rm
d}\mathsf{u}]\big{)}^{2}+\frac{1}{6}\big{(}\\!\operatorname{Ad}_{g}^{-1}[\mathsf{u},[\mathsf{u},{\rm
d}\mathsf{u}]]\big{)}^{3}.$

Applying the operators (5.16) we then find

$\displaystyle\mathcal{P}_{p}\big{(}p^{-1}{\rm d}p\big{)}=\big{(}g^{-1}{\rm
d}g\big{)}^{0}+\big{(}\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}\big{)}^{1}+\frac{1}{ab}\big{(}g^{-1}{\rm
d}g\big{)}^{2}+\frac{a^{2}-ab+b^{2}}{a^{2}b^{2}}(\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u})^{3},$
$\displaystyle\mathcal{E}\mathcal{P}_{p}\big{(}{\ast}p^{-1}{\rm
d}p\big{)}=\frac{b-a}{ab}\big{(}\operatorname{Ad}_{g}^{-1}\ast{\rm
d}\mathsf{u}\big{)}^{2}+\frac{b-a}{a^{2}b^{2}}\big{(}{\ast}g^{-1}{\rm
d}g\big{)}^{3}.$

The first two terms in the action (5.17) then take the form

$\displaystyle\langle\\!\langle p^{-1}{\rm
d}p,\mathcal{E}\mathcal{P}_{p}\big{(}{\ast}p^{-1}{\rm
d}p\big{)}\rangle\\!\rangle_{\mathfrak{d}}=(a-b)\langle{\rm d}gg^{-1},\ast{\rm
d}gg^{-1}\rangle+ab(a-b)\langle{\rm d}\mathsf{u},\ast{\rm
d}\mathsf{u}\rangle,$ $\displaystyle\langle\\!\langle p^{-1}{\rm
d}p,\mathcal{P}_{p}\big{(}p^{-1}{\rm
d}p\big{)}\rangle\\!\rangle_{\mathfrak{d}}=-(a-b)^{2}\langle{\rm
d}gg^{-1},{\rm
d}\mathsf{u}\rangle+\frac{a^{2}b^{2}}{2}\bigg{\langle}[\mathsf{u},{\rm
d}\mathsf{u}],{\rm d}\mathsf{u}+\frac{1}{3}\big{[}\mathsf{u},{\rm
d}gg^{-1}\big{]}\bigg{\rangle}.$

On the other hand, we find that the Wess–Zumino $3$-form is exact since

$\displaystyle\langle\\!\langle\widehat{p}^{-1}{\rm
d}\widehat{p},\big{[}\widehat{p}^{-1}{\rm d}\widehat{p},\widehat{p}^{-1}{\rm
d}\widehat{p}\big{]}\rangle\\!\rangle_{\mathfrak{d}}={\rm
d}\big{(}\big{\langle}{\rm
d}\widehat{g}\widehat{g}^{-1},6\big{(}a^{2}+b^{2}\big{)}{\rm
d}\widehat{\mathsf{u}}-a^{2}b^{2}[\widehat{\mathsf{u}},[\widehat{\mathsf{u}},{\rm
d}\widehat{\mathsf{u}}]]\big{\rangle}$
$\displaystyle\hphantom{\langle\\!\langle\widehat{p}^{-1}{\rm
d}\widehat{p},[\widehat{p}^{-1}{\rm d}\widehat{p},\widehat{p}^{-1}{\rm
d}\widehat{p}]\rangle\\!\rangle_{\mathfrak{d}}={\rm
d}\big{(}}{}-a^{2}b^{2}\big{\langle}[\widehat{\mathsf{u}},[{\rm
d}\widehat{\mathsf{u}},{\rm d}\widehat{\mathsf{u}}]]\big{\rangle}\big{)}.$

Putting all of the above together we then arrive at the action

$\displaystyle S_{\rm
2d}(g,\mathsf{u})=\int_{\Sigma}\bigg{(}\frac{1}{2}(a-b)\langle{\rm
d}gg^{-1},\ast{\rm d}gg^{-1}\rangle+\frac{1}{2}ab(a-b)\langle{\rm
d}\mathsf{u},\ast{\rm d}\mathsf{u}\rangle-ab\langle{\rm d}gg^{-1},{\rm
d}\mathsf{u}\rangle$ $\displaystyle\hphantom{S_{\rm
2d}(g,\mathsf{u})=\int_{\Sigma}\bigg{(}}{}-\frac{1}{6}a^{2}b^{2}\langle\mathsf{u},[{\rm
d}\mathsf{u},{\rm d}\mathsf{u}]\rangle\bigg{)}.$ (5.18)

It is interesting to note that in the limit $b\to 0$ we recover the principal
chiral model action. In particular, the model with action (5.18) can be seen
as a deformation of the principal chiral model to which a new
$\mathfrak{g}$-valued field $\mathsf{u}$ is added. In fact, removing all the
terms involving the field $g$ from the above action we are left with the
action of the pseudo-dual of the principal chiral model for the field
$\mathsf{u}$ [14, 46, 68]. One may therefore view the action (5.18) as
coupling together a principal chiral model field $g$ and a pseudo-dual
principal chiral model field $\mathsf{u}$ in an integrable way.

Note that the pseudo-dual of the principal chiral model was derived very
recently in [4] starting from 6d holomorphic Chern–Simons theory. It was
argued there that such an action could also be derived directly from 4d
Chern–Simons theory where $\omega$ is taken to have a fourth order pole but is
regular at infinity. By contrast, in the present work we explicitly required
$\omega$ to have a double pole at infinity in (3.1) and then used the right
diagonal gauge invariance by $G$ in (3.12) to fix the corresponding edge modes
at infinity in Section 3.6. We expect that by starting instead from a
meromorphic $1$-form $\omega$ with a fourth order pole at the origin and which
is regular at infinity we would obtain a gauged version of the action (5.18).
Moreover, after fixing the gauge invariance by setting $g=\textup{id}$ this
action should reduce to that of the pseudo-dual of the principal chiral model
field, as in [4].

####  5.2.4 Lax connection

The Lax connection (4.23) takes the form

$\displaystyle\mathcal{L}$
$\displaystyle={\bm{p}}\big{(}\mathcal{P}_{p}\big{(}p^{-1}{\rm
d}p\big{)}-\mathcal{E}\mathcal{P}_{p}\big{(}{\ast}p^{-1}{\rm
d}p\big{)}\big{)}$
$\displaystyle={\bm{p}}\bigg{(}j^{0}+\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}^{1}+\frac{1}{ab}j^{2}+\frac{a-b}{ab}\operatorname{Ad}_{g}^{-1}\ast{\rm
d}\mathsf{u}^{2}+\frac{a^{2}-ab+b^{2}}{a^{2}b^{2}}\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}^{3}+\frac{a-b}{a^{2}b^{2}}\ast j^{3}\bigg{)},$

where we have introduced the shorthand $j\coloneqq g^{-1}{\rm d}g$ and
${\bm{p}}$ is the inverse (4.22) of ${\bm{j}}_{{\bm{z}}^{\prime}}$ defined in
(4.1) given explicitly here by

$\displaystyle{\bm{p}}\colon\
\mathsf{u}^{0}+\mathsf{v}^{1}+\mathsf{w}^{2}+\mathsf{x}^{3}$
$\displaystyle\longmapsto\frac{a^{3}(\mathsf{u}+a\mathsf{v}-b^{2}(\mathsf{w}+a\mathsf{x}))}{2(b^{2}-a^{2})(z-a)}+\frac{a^{3}(-\mathsf{u}+a\mathsf{v}+b^{2}(\mathsf{w}-a\mathsf{x}))}{2(b^{2}-a^{2})(z+a)}$
$\displaystyle\qquad-\frac{b^{3}(\mathsf{u}+b\mathsf{v}-a^{2}(\mathsf{w}+b\mathsf{x}))}{2(b^{2}-a^{2})(z-b)}-\frac{b^{3}(-\mathsf{u}+b\mathsf{v}+a^{2}(\mathsf{w}-b\mathsf{x}))}{2(b^{2}-a^{2})(z+b)}.$

The above Lax connection therefore explicitly reads

$\displaystyle\mathcal{L}=\bigg{(}a^{2}\frac{j_{+}+b\operatorname{Ad}_{g}^{-1}\partial_{+}\mathsf{u}}{(a+b)(a-z)}+b^{2}\frac{j_{+}-a\operatorname{Ad}_{g}^{-1}\partial_{+}\mathsf{u}}{(a+b)(b+z)}\bigg{)}{\rm
d}\sigma^{+}$
$\displaystyle\hphantom{\mathcal{L}=\bigg{(}}{}+\bigg{(}a^{2}\frac{j_{-}-b\operatorname{Ad}_{g}^{-1}\partial_{-}\mathsf{u}}{(a+b)(a+z)}+b^{2}\frac{j_{-}+a\operatorname{Ad}_{g}^{-1}\partial_{-}\mathsf{u}}{(a+b)(b-z)}\bigg{)}{\rm
d}\sigma^{-}.$

Note that in the limit $b\to 0$ we recover the Lax connection of the principal
chiral model, in the form given in (5.10). This is in agreement with the
observation made above about the action (5.18).

The flatness of $\mathcal{L}$ is equivalent to

$\displaystyle{\rm d}j+\frac{1}{2}[j,j]=0,$ (5.19a) $\displaystyle{\rm
d}\big{(}\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}\big{)}+\operatorname{Ad}_{g}^{-1}\big{[}{\rm d}\mathsf{u},{\rm
d}gg^{-1}\big{]}=0,$ (5.19b) $\displaystyle{\rm d}\ast{\rm
d}\mathsf{u}+\frac{ab[{\rm d}\mathsf{u},{\rm
d}\mathsf{u}]}{2(a-b)}+\frac{\big{[}{\rm d}gg^{-1},{\rm
d}gg^{-1}\big{]}}{2(a-b)}=0,$ (5.19c) $\displaystyle{\rm d}\big{(}{\ast}{\rm
d}gg^{-1}\big{)}+\frac{ab}{a-b}\big{[}{\rm d}\mathsf{u},{\rm
d}gg^{-1}\big{]}=0.$ (5.19d)

The equations (5.19a) and (5.19b) are both identically true off-shell. The
first is the Maurer–Cartan equation for $j$ and the second holds because

$\displaystyle{\rm d}\big{(}\operatorname{Ad}_{g}^{-1}{\rm
d}\mathsf{u}\big{)}$ $\displaystyle={\rm d}\big{(}g^{-1}{\rm
d}\mathsf{u}g\big{)}=-g^{-1}{\rm d}gg^{-1}\wedge{\rm d}\mathsf{u}g-g^{-1}{\rm
d}\mathsf{u}\wedge{\rm d}g$
$\displaystyle=-\frac{1}{2}\operatorname{Ad}_{g}^{-1}\big{(}\big{[}{\rm
d}\mathsf{u},{\rm d}gg^{-1}\big{]}+\big{[}dgg^{-1},{\rm
d}\mathsf{u}\big{]}\big{)}=-\operatorname{Ad}_{g}^{-1}\big{[}{\rm
d}\mathsf{u},{\rm d}gg^{-1}\big{]}.$

The equations (5.19c) and (5.19d) coincide with the equations of motion
obtained from the action (5.18), as expected.

###  5.3 Real simple zeroes and poles

In this final section we discuss the example mentioned in the introduction.
Let $\omega$ be given as in (1.1), namely

$\displaystyle\omega=-\ell^{\infty}_{1}\frac{\prod_{i=1}^{N}(z-\zeta_{i})}{\prod_{i=1}^{N}(z-z_{i})}{\rm
d}z,$

where the poles and zeroes are all real and distinct. In the notation of
Section 3.1 we then have ${\bm{z}}={\bm{z}}_{\rm r}=\\{z_{i}\\}_{i=1}^{N}$ and
${\bm{\zeta}}={\bm{\zeta}}_{\rm r}=\\{\zeta_{i}\\}_{i=1}^{N}$. And since all
the poles and zeroes are simple we have $n_{x}=1$ for all $x\in{\bm{z}}$ and
$m_{y}=1$ for all $y\in{\bm{\zeta}}$.

As in the introduction, we shall use the shorthand notation
$\epsilon_{i}\coloneqq\epsilon_{\zeta_{i}}$ for every $i=1,\dots,N$ but leave
these real non-zero parameters arbitrary.

Since all the poles of $\omega$ are real and simple, the defect Lie algebra
(3.13) is simply given by the direct sum of Lie algebras
$\mathfrak{d}=\mathfrak{g}^{\oplus N}$. The corresponding Lie group is
$D=G^{\times N}$. As in the introduction, we shall leave the choice of
Lagrangian subalgebra $\mathfrak{k}\subset\mathfrak{d}$ unspecified, and only
assume that it satisfies the technical condition (2.3). We denote the
corresponding connected Lie subgroup by $K\subset D$.

Since all the zeroes of $\omega$ are real and simple, the real vector space
(4.3) associated with these zeroes is given here by
$\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}^{\oplus N}$.

The linear isomorphism
$\mathcal{C}\colon\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}^{\oplus
N}\to\mathfrak{d}=\mathfrak{g}^{\oplus N}$ from Lemma 4.1 is given in
components by (4.9), namely

$\displaystyle C^{i}_{\;j}\coloneqq
C^{[z_{i},0]}_{\quad\;\;(\zeta_{j},0)}=\frac{1}{z_{i}-\zeta_{j}}$ (5.20)

for $i,j=1,\dots,N$. These are simply the components of the usual Cauchy
matrix. It is well know that the inverse of the Cauchy matrix (5.20) has
components

$\displaystyle\big{(}C^{-1}\big{)}^{i}_{\;j}=\frac{\prod_{r\neq
i}(\zeta_{r}-z_{j})\prod_{r}(z_{r}-\zeta_{i})}{\prod_{r\neq
j}(z_{r}-z_{j})\prod_{r\neq i}(\zeta_{r}-\zeta_{i})}$ (5.21)

for $i,j=1,\dots,N$. In particular, since
$\widetilde{\mathcal{E}}\colon\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}^{\oplus
N}\to\mathfrak{g}^{\bm{(}{\bm{\zeta}}\bm{)}}=\mathfrak{g}^{\oplus N}$ defined
in (4.7) is given in components by the diagonal matrix
$\text{diag}(\epsilon_{1},\dots,\epsilon_{N})$, the components of the linear
operator $\mathcal{E}\colon\mathfrak{d}=\mathfrak{g}^{\oplus
N}\to\mathfrak{d}=\mathfrak{g}^{\oplus N}$ defined in (4.13) are given by

$\displaystyle\sum_{j=1}^{N}C^{i}_{\;j}\epsilon_{j}\big{(}C^{-1}\big{)}^{j}_{\;k}=\sum_{j=1}^{N}\epsilon_{j}\frac{\prod_{r\neq
j}(\zeta_{r}-z_{k})\prod_{r\neq i}(z_{r}-\zeta_{j})}{\prod_{r\neq
k}(z_{r}-z_{k})\prod_{r\neq j}(\zeta_{r}-\zeta_{j})}.$

These coincide with the components given in (1.3).

Finally, to compute the Lax connection using the general formula (4.23) we
need to compute the inverse ${\bm{p}}$ in (4.22) of the map
${\bm{j}}_{{\bm{z}}^{\prime}}$ defined in (4.1). The latter reads

$\displaystyle{\bm{j}}_{\bm{z}^{\prime}}\colon\
\sum_{j=1}^{N}\frac{\mathsf{u}_{j}}{z-\zeta_{j}}\longmapsto\bigg{(}\sum_{j=1}^{N}\frac{\mathsf{u}_{j}}{z_{i}-\zeta_{j}}\bigg{)}_{i=1}^{N}=\bigg{(}\sum_{j=1}^{N}C^{i}_{\;j}\mathsf{u}_{j}\bigg{)}_{i=1}^{N},$

where the equality is by definition (5.20) of the Cauchy matrix. Its inverse
is then clearly given by

$\displaystyle{\bm{p}}\colon\
(\mathsf{v}_{i})_{i=1}^{N}\longmapsto\sum_{i,j=1}^{N}\frac{\big{(}C^{-1}\big{)}^{i}_{\;\;j}\mathsf{v}_{j}}{z-\zeta_{i}}.$
(5.22)

According to (4.23), the Lax connection is now given by
$\mathcal{L}={\bm{p}}(\mathcal{J}_{\sigma}{\rm d}\sigma+\mathcal{J}_{\tau}{\rm
d}\tau)$, where the $\mathfrak{d}$-valued fields
$\mathcal{J}_{\sigma},\mathcal{J}_{\tau}\in C^{\infty}(\Sigma,\mathfrak{d})$
are given in components by

$\displaystyle\mathcal{J}_{\sigma}=(\mathcal{J}^{i}_{\sigma})_{i=1}^{N}=l^{-1}\partial_{\sigma}l-\mathcal{E}^{-1}\mathcal{P}_{l}\big{(}\mathcal{E}\big{(}l^{-1}\partial_{\sigma}l\big{)}-l^{-1}\partial_{\tau}l\big{)},$
$\displaystyle\mathcal{J}_{\tau}=(\mathcal{J}^{i}_{\tau})_{i=1}^{N}=\mathcal{E}\big{(}l^{-1}\partial_{\sigma}l\big{)}-\mathcal{P}_{l}\big{(}\mathcal{E}\big{(}l^{-1}\partial_{\sigma}l\big{)}-l^{-1}\partial_{\tau}l\big{)}.$

In other words, using the explicit form (5.22) of the linear map ${\bm{p}}$ we
deduce that the Lax connection reads

$\displaystyle\mathcal{L}=\sum_{i,j=1}^{N}\frac{\big{(}C^{-1}\big{)}^{i}_{\;\;j}\big{(}\mathcal{J}^{j}_{\sigma}{\rm
d}\sigma+\mathcal{J}^{j}_{\tau}{\rm d}\tau\big{)}}{z-\zeta_{i}},$

which corresponds to the expression (1.4) from the introduction using the
explicit inverse of the Cauchy matrix in (5.21).

##  6 Outlook

In this work we constructed a very broad family of integrable
$\mathcal{E}$-models using the formalism of Costello–Yamazaki [13] by starting
from the general $2$d action obtained in [3]. There are a number of
interesting open problems.

###  6.1 Hamiltonian formalism

In this work we focused entirely on constructing the actions of the new family
of $2$d integrable field theories. Indeed, the formalism of Costello–Yamazaki
[13] is most convenient for describing integrable field theories in the
Lagrangian formalism.

By contrast, $2$d integrable field theories can be best described in the
Hamiltonian formalism using the framework of classical dihedral affine Gaudin
models proposed in [64], and further developed in [16, 41]. The formalisms of
[13] and [64] were shown to be intimately related in [65] by performing a
Hamiltonian analysis of $4$d Chern–Simons theory. It would therefore be
interesting to perform the Hamiltonian analysis of the family of integrable
$\mathcal{E}$-model actions described in the present work. In particular, one
should show that the Poisson bracket of the Lax matrix is of the Maillet
$r/s$-form [44, 45] with twist function, which is equivalent to describing
these models as classical dihedral affine Gaudin models. We will come back to
this in a forthcoming paper [43].

###  6.2 Degenerate $\bm{\mathcal{E}}$-model

An important restriction we imposed on the general setting of [3] is that the
$1$-form $\omega$ had a double pole at infinity. This allowed us in Section
3.6 to partially fix the gauge invariance of the $2$d action of [3], by
bringing the component of the edge mode at infinity to the identity.

It would be natural to try to extend our construction to the general setting
of [3] by allowing arbitrary orders at all the poles of $\omega$. The
resulting $2$d integrable field theory would have an additional gauge
invariance and so it is natural to expect that this generalisation would lead
to an integrable family of the class of degenerate $\mathcal{E}$-models
introduced in [38], see also [34, 35, 56, 59], which, in particular, would
include by [35] the bi-Yang–Baxter model with WZ-term [15].

###  6.3 Integrable $\bm{\mathcal{E}}$-model hierarchy

A crucial step in our analysis was imposing the condition (4.6) on the
coefficients in the partial fraction decomposition of the components of the
Lax connection $\mathcal{L}$. Indeed, this condition gave a particular way of
satisfying the admissibility condition $(b)$ from Section 3.5 and we showed
that within this class of admissible $1$-forms $\mathcal{L}$ there was a
unique solution $\mathcal{L}=\mathcal{L}(l)$ to the boundary equation (3.16)
relating $\mathcal{L}$ to the edge mode $l\in C^{\infty}(\Sigma,D)$. Moreover,
the condition (4.6) is at the origin of the introduction of the operator
$\mathcal{E}$ in our construction.

However, (4.6) is by no means the only way to solve the admissibility
condition $(b)$, and it would be very interesting to explore other classes of
admissible $1$-forms $\mathcal{L}$. In the case when the zeroes of $\omega$
are all simple, an obvious alternative way to solve the admissibility
condition $(b)$ is to work in a representation of the complex Lie algebra
$\mathfrak{g}^{\mathbb{C}}$, pick $n\in\mathbb{Z}_{\geq 1}$ and set

$\displaystyle\mathcal{L}_{t_{n}}^{(y,0)}=\epsilon_{n,y}\big{(}\mathcal{L}_{\sigma}^{(y,0)}\big{)}^{n},$
(6.1)

for every $y\in{\bm{\zeta}}$ and some choice of $\epsilon_{n,y}$. On the left
hand side we used the notation $t_{n}$ instead of $\tau$ for the time
coordinate since we expect the corresponding model to be related to a
different flow in the same hierarchy. Indeed, the above solution (6.1) of the
admissibility condition $(b)$ is motivated by the expressions for the Lax
matrices inducing higher flows in the integrable hierarchies of affine Gaudin
models [42]. Explicitly, when $\mathfrak{g}^{\mathbb{C}}$ is of type $B$, for
instance, to each simple zero $y\in{\bm{\zeta}}$ and every odd positive
integer $n$ is associated a higher flow $\partial_{t_{n}}$ with corresponding
Lax matrix

$\displaystyle\mathcal{L}_{t_{n}}=\frac{\big{(}\mathcal{L}_{\sigma}^{(y,0)}\big{)}^{n}}{z-y}.$

We therefore expect from [42] that the Lax connection defined by imposing
(6.1) corresponds to a higher flow of the same integrable $\mathcal{E}$-model
hierarchy. It would be very interesting to investigate this further. In
particular, we expect from [42], see also [21, 22, 23], that when
$\mathfrak{g}^{\mathbb{C}}$ is of type $B$, $C$ or $D$ the condition (6.1)
should give non-trivial commuting flows for all odd $n\in\mathbb{Z}_{\geq 1}$,
corresponding to the set of exponents of the (untwisted) affine Kac–Moody
algebra associated with $\mathfrak{g}^{\mathbb{C}}$. In type $A$ we also
expect that one should have to modify the ansatz (6.1) accordingly to produce
commuting flows [21, 22, 23, 42].

###  6.4 3d Chern–Simons theory

The $\mathcal{E}$-model on the infinite cyclinder $S^{1}\times\mathbb{R}$ was
shown in [52] to arise from $3$d Chern–Simons theory for the Lie group $D$ on
the solid cylinder $\raisebox{-0.56905pt}{ \leavevmode\hbox to7.8pt{\vbox
to7.8pt{\pgfpicture\makeatletter\hbox{\hskip 3.89899pt\lower-5.74847pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\times\mathbb{R}$,
with  a disc, by imposing a suitable boundary condition on the gauge field at
the boundary $\partial\,\raisebox{-0.56905pt}{ \leavevmode\hbox to7.8pt{\vbox
to7.8pt{\pgfpicture\makeatletter\hbox{\hskip 3.89899pt\lower-5.74847pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\simeq S^{1}$.
Moreover, the $\sigma$-model on $K\backslash D$ was also obtained from $3$d
Chern–Simons theory on a hollowed out cylinder $\raisebox{-0.56905pt}{
\leavevmode\hbox to7.8pt{\vbox to7.8pt{\pgfpicture\makeatletter\hbox{\hskip
3.89899pt\lower-5.74847pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{1.29463pt}{-1.84949pt}\pgfsys@curveto{1.29463pt}{-1.13448pt}{0.71501pt}{-0.55486pt}{0.0pt}{-0.55486pt}\pgfsys@curveto{-0.71501pt}{-0.55486pt}{-1.29463pt}{-1.13448pt}{-1.29463pt}{-1.84949pt}\pgfsys@curveto{-1.29463pt}{-2.5645pt}{-0.71501pt}{-3.14412pt}{0.0pt}{-3.14412pt}\pgfsys@curveto{0.71501pt}{-3.14412pt}{1.29463pt}{-2.5645pt}{1.29463pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{1.29463pt}{-1.84949pt}\pgfsys@curveto{1.29463pt}{-1.13448pt}{0.71501pt}{-0.55486pt}{0.0pt}{-0.55486pt}\pgfsys@curveto{-0.71501pt}{-0.55486pt}{-1.29463pt}{-1.13448pt}{-1.29463pt}{-1.84949pt}\pgfsys@curveto{-1.29463pt}{-2.5645pt}{-0.71501pt}{-3.14412pt}{0.0pt}{-3.14412pt}\pgfsys@curveto{0.71501pt}{-3.14412pt}{1.29463pt}{-2.5645pt}{1.29463pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\times\mathbb{R}$,
with  an annulus, by imposing the same boundary condition as before on the
gauge field at the outer boundary and another topological boundary condition
depending on the choice of Lagrangian subalgebra
$\mathfrak{k}\subset\mathfrak{d}$ at the inner boundary.

It would be interesting to understand if, in the integrable case, there is a
relation between the above description of the $\sigma$-model on $K\backslash
D$ from $3$d Chern–Simons derived in [52] and the description from $4$d
Chern–Simons theory obtained here.

Another possible connection to $3$d Chern–Simons theory is suggested by the
results of [49, 50] where the action of the $\lambda$-deformation [57] of the
principal chiral model, in the form of the universal $2$d action (3.19), was
obtained from a certain “doubled” version of $3$d Chern–Simons theory on
$\raisebox{-0.56905pt}{ \leavevmode\hbox to7.8pt{\vbox
to7.8pt{\pgfpicture\makeatletter\hbox{\hskip 3.89899pt\lower-5.74847pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@moveto{3.69899pt}{-1.84949pt}\pgfsys@curveto{3.69899pt}{0.19342pt}{2.04291pt}{1.8495pt}{0.0pt}{1.8495pt}\pgfsys@curveto{-2.04291pt}{1.8495pt}{-3.69899pt}{0.19342pt}{-3.69899pt}{-1.84949pt}\pgfsys@curveto{-3.69899pt}{-3.8924pt}{-2.04291pt}{-5.54848pt}{0.0pt}{-5.54848pt}\pgfsys@curveto{2.04291pt}{-5.54848pt}{3.69899pt}{-3.8924pt}{3.69899pt}{-1.84949pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-1.84949pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\times\mathbb{R}$.

##  Appendix A Proof of Proposition 2.3

As noted in Section 2.1, the direct sum (2.3) is orthogonal with respect to
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d},\mathcal{E}}$.
The corresponding projector $\mathcal{P}_{l}$ is then symmetric with respect
to $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d},\mathcal{E}}$.
Hence, for any $\mathsf{U},\mathsf{V}\in\mathfrak{d}$, we have

$\displaystyle\langle\\!\langle\mathsf{U},{}^{t}\mathcal{P}_{l}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathcal{E}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d},\mathcal{E}}=\langle\\!\langle\mathsf{U},\mathcal{P}_{l}\mathcal{E}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d},\mathcal{E}}=\langle\\!\langle\mathsf{U},\mathcal{E}^{-1}\mathcal{P}_{l}\mathcal{E}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}},$

where we used the symmetry of $\mathcal{P}_{l}$ with respect to
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d},\mathcal{E}}$ in
the third step. Part $(i)$ now follows from the non-degeneracy of
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$.

Using this result and the definition (2.5) of $\overline{\mathcal{P}}_{l}$, we
then get

$\displaystyle\mathcal{P}_{l}\mathcal{E}+\mathcal{E}\overline{\mathcal{P}}_{l}=\mathcal{P}_{l}\mathcal{E}-\mathcal{E}\,{}^{t}\mathcal{P}_{l}+\mathcal{E}=\mathcal{E}.$

This is the first equation in $(ii)$. The second one is simply obtained by
multiplying it on both sides by $\mathcal{E}^{-1}$.

Part $(iii)$ is easily proved by observing that
${}^{t}\overline{\mathcal{P}}_{l}\,\mathcal{P}_{l}=(\textup{id}-\mathcal{P}_{l})\mathcal{P}_{l}=0$
since $\mathcal{P}_{l}$ is a projector, and similarly for
${}^{t}\mathcal{P}_{l}\overline{\mathcal{P}}_{l}=0$.

Let us now prove $(iv)$. Let $\mathsf{U},\mathsf{V}\in\mathfrak{d}$. From the
definition (2.5) of $\overline{\mathcal{P}}_{l}$, we have

$\displaystyle\langle\\!\langle\big{(}\mathcal{P}_{l}-\overline{\mathcal{P}}_{l}\big{)}\mathsf{U},\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle\mathsf{U},(\textup{id}-\mathcal{P}_{l})\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathcal{P}_{l}\mathsf{V}+(\textup{id}-\mathcal{P}_{l})\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle\mathcal{P}_{l}\mathsf{U}+(\textup{id}-\mathcal{P}_{l})\mathsf{U},(\textup{id}-\mathcal{P}_{l})\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathcal{P}_{l}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}-\langle\\!\langle(\textup{id}-\mathcal{P}_{l})\mathsf{U},(\textup{id}-\mathcal{P}_{l})\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}.$

Clearly $(\textup{id}-\mathcal{P}_{l})\mathsf{U}$ and
$(\textup{id}-\mathcal{P}_{l})\mathsf{V}$ belong to
$\ker\mathcal{P}_{l}=\operatorname{Ad}_{l}^{-1}\mathfrak{k}$. By the ad-
invariance of $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$
and the isotropy of $\mathfrak{k}$, we thus get
$\langle\\!\langle(\textup{id}-\mathcal{P}_{l})\mathsf{U},(\textup{id}-\mathcal{P}_{l})\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=0$,
leaving only the first term in the above equation. Moreover, a similar
computation can be performed with $\mathcal{P}_{l}$ and
$\overline{\mathcal{P}}_{l}$ exchanged. In the end, we get

$\displaystyle\langle\\!\langle(\mathcal{P}_{l}-\overline{\mathcal{P}}_{l})\mathsf{U},\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathcal{P}_{l}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=-\langle\\!\langle\overline{\mathcal{P}}_{l}\mathsf{U},\overline{\mathcal{P}}_{l}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}.$
(A.1)

Bringing all operators on the left-hand side of the bilinear form
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$ and using its
non-degeneracy, we get
$\mathcal{P}_{l}-\overline{\mathcal{P}}_{l}={}^{t}\mathcal{P}_{l}\mathcal{P}_{l}=-{}^{t}\overline{\mathcal{P}}_{l}\overline{\mathcal{P}}_{l}$,
proving part $(iv)$.

Part $(v)$ then follows from

$\displaystyle\mathcal{P}_{l}{}^{t}\mathcal{P}_{l}=\big{(}1-{}^{t}\overline{\mathcal{P}}_{l}\big{)}\big{(}1-\overline{\mathcal{P}}_{l}\big{)}=1-{}^{t}\overline{\mathcal{P}}_{l}-\overline{\mathcal{P}}_{l}+{}^{t}\overline{\mathcal{P}}_{l}\overline{\mathcal{P}}_{l}=\mathcal{P}_{l}-\overline{\mathcal{P}}_{l}+\overline{\mathcal{P}}_{l}-\mathcal{P}_{l}=0,$
(A.2)

and from a similar computation for
$\overline{\mathcal{P}}_{l}{}^{t}\overline{\mathcal{P}}_{l}$.

Finally, we note that $\mathcal{P}_{l}\mathsf{U}$ and
$\mathcal{P}_{l}\mathsf{V}$ in (A.1) belong to
$\operatorname{im}\mathcal{P}_{l}=\mathcal{E}\operatorname{Ad}_{l}^{-1}\mathfrak{k}$.
If $\mathcal{E}^{2}=\textup{id}$, the latter is an isotropic subspace and
hence
$\langle\\!\langle\big{(}\mathcal{P}_{l}-\overline{\mathcal{P}}_{l}\big{)}\mathsf{U},\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=\langle\\!\langle\mathcal{P}_{l}\mathsf{U},\mathcal{P}_{l}\mathsf{V}\rangle\\!\rangle_{\mathfrak{d}}=0$.
So $(vi)$ follows from the non-degeneracy of
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$.

##  Appendix B Proof of Proposition 2.7

It is convenient to first consider a general model describing a field $l\in
C^{\infty}(\Sigma,D)$, with an action of the form

$\displaystyle S(l)=\int_{\Sigma}\bigg{(}\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\tau}l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}-\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{O}^{\sigma\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle\hphantom{S(l)=\int_{\Sigma}\bigg{(}}{}+\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}\bigg{)}{\rm
d}\sigma\wedge{\rm d}\tau-\frac{1}{2}I^{\rm WZ}_{\mathfrak{d}}[l],$ (B.1)

where $\mathcal{O}^{\tau\tau}$, $\mathcal{O}^{\sigma\sigma}$ and
$\mathcal{O}^{\tau\sigma}$ are linear operators on $\mathfrak{d}$, which can
depend on the field $l$ but not on its derivatives. Without loss of
generality, we can suppose $\mathcal{O}^{\tau\tau}$ and
$\mathcal{O}^{\sigma\sigma}$ symmetric with respect to
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$. Note that the
model (B.1) is relativistic if and only if
$\mathcal{O}^{\tau\tau}=\mathcal{O}^{\sigma\sigma}$ and
$\mathcal{O}^{\tau\sigma}$ is skew-symmetric with respect to
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathfrak{d}}$. The goal of
this appendix is to prove the following result.

######  Lemma B.1.

The components of the energy-momentum tensor of the model (B.1) are given by

$\displaystyle
T^{\tau}_{\;\;\,\tau}=-T^{\sigma}_{\;\;\,\sigma}=\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\tau}l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}+\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{O}^{\sigma\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}},$
$\displaystyle T^{\tau}_{\;\;\,\sigma}=\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\tau}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}+\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{O}^{\tau\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}},$
$\displaystyle T^{\sigma}_{\;\;\,\tau}=-\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\sigma\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}+\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\sigma}l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}.$

###### Proof.

Let us fix a basis $\\{I_{A}\\}_{A=1,\dots,\dim\mathfrak{d}}$ of the Lie
algebra $\mathfrak{d}$ and a choice of local coordinates
$\big{\\{}y^{M}\big{\\}}_{M=1,\dots,\dim\mathfrak{d}}$ on the group manifold
$D$, parametrising the element $l$ in $D$. We define the vielbeins
$e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}$
through the decomposition

$\displaystyle l^{-1}\frac{\partial l}{\partial
y^{M}}=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>
We may then express the components $l^{-1}\partial_{\mu}l$ for
$\mu=\tau,\sigma$ of the Maurer–Cartan current as

$\displaystyle
l^{-1}\partial_{\mu}l=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}\,\partial_{\mu}y^{M}\,I_{A}.$
(B.2)

For any linear operator $\mathcal{O}$ on $\mathfrak{d}$, let us also define

$\displaystyle\mathcal{O}_{AB}\coloneqq\langle\\!\langle
I_{A},\mathcal{O}I_{B}\rangle\\!\rangle_{\mathfrak{d}}.$

The symmetry of $\mathcal{O}^{\tau\tau}$ and $\mathcal{O}^{\sigma\sigma}$ then
translates to the fact that
$\mathcal{O}^{\tau\tau}_{AB}=\mathcal{O}^{\tau\tau}_{BA}$ and
$\mathcal{O}^{\sigma\sigma}_{AB}=\mathcal{O}^{\sigma\sigma}_{BA}$ for every
$A,B=1,\dots,\dim\mathfrak{d}$. Using the above definitions, we then rewrite
the action (B.1) as the integral over $\Sigma$ of the Lagrangian density

$\displaystyle
L=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\bigg{(}\frac{1}{2}\mathcal{O}^{\tau\tau}_{AB}\,\partial_{\tau}y^{M}\partial_{\tau}y^{N}-\frac{1}{2}\mathcal{O}^{\sigma\sigma}_{AB}\,\partial_{\sigma}y^{M}\partial_{\sigma}y^{N}+(\mathcal{O}^{\tau\sigma}_{AB}+W_{AB})\,\partial_{\tau}y^{M}\partial_{\sigma}y^{N}\bigg{)},$

where $W_{AB}=-W_{BA}$ describes the contribution of the Wess–Zumino term to
the action (such a rewriting of the Wess–Zumino term as a two-dimensional
integral is always possible, at least locally — as we shall see, an explicit
expression for $W_{AB}$ will not be needed in what follows).

The energy-momentum tensor of the model can be computed explicitly in terms of
the Lagrangian density $L$ as

$\displaystyle T^{\mu}_{\;\;\,\nu}=\frac{\partial
L}{\partial(\partial_{\mu}y^{M})}\partial_{\nu}y^{M}-\delta^{\mu}_{\;\,\nu}\,L.$
(B.3)

From the above expression of $L$, the symmetry of
$\mathcal{O}^{\tau\tau}_{AB}$ and $\mathcal{O}^{\sigma\sigma}_{AB}$ and the
skew-symmetry of $W_{AB}$, we get

$\displaystyle\frac{\partial
L}{\partial(\partial_{\tau}y^{M})}=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\big{(}\mathcal{O}^{\tau\tau}_{AB}\partial_{\tau}y^{N}+(\mathcal{O}^{\tau\sigma}_{AB}+W_{AB})\partial_{\sigma}y^{N}\big{)},$
(B.4a) $\displaystyle\frac{\partial
L}{\partial(\partial_{\sigma}y^{M})}=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\big{(}{-}\mathcal{O}^{\sigma\sigma}_{AB}\partial_{\sigma}y^{N}+(\mathcal{O}^{\tau\sigma}_{BA}-W_{AB})\partial_{\tau}y^{N}\big{)}.$
(B.4b)

We then deduce that

$\displaystyle
T^{\tau}_{\;\;\,\tau}=\frac{1}{2}e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\big{(}\mathcal{O}^{\tau\tau}_{AB}\,\partial_{\tau}y^{M}\partial_{\tau}y^{N}+\mathcal{O}^{\sigma\sigma}_{AB}\,\partial_{\sigma}y^{M}\partial_{\sigma}y^{N}\big{)}.$

Note in particular that the parts proportional to
$\mathcal{O}^{\tau\sigma}_{AB}+W_{AB}$ coming from the two terms in the right-
hand side of (B.3) cancel for $\mu=\nu=\tau$. A similar computation yields
$T^{\sigma}_{\;\;\,\sigma}=-T^{\tau}_{\;\;\,\tau}$. In terms of the
Maurer–Cartan currents (B.2), we find

$\displaystyle
T^{\tau}_{\;\;\,\tau}=-T^{\sigma}_{\;\;\,\sigma}=\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\tau}l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}+\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{O}^{\sigma\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}.$

Let us now compute the crossed-term $T^{\tau}_{\;\;\,\sigma}$. Substituting
the expression (B.4a) in (B.3), we obtain

$\displaystyle
T^{\tau}_{\;\;\,\sigma}=e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\big{(}\mathcal{O}^{\tau\tau}_{AB}\,\partial_{\sigma}y^{M}\partial_{\tau}y^{N}+(\mathcal{O}^{\tau\sigma}_{AB}+W_{AB})\,\partial_{\sigma}y^{M}\partial_{\sigma}y^{N}\big{)}.$

The term containing $W_{AB}$ vanishes, as it is given by the contraction of
the symmetric tensor
$e^{A}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A}M}e^{B}_{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}B}N}\partial_{\sigma}y^{M}\partial_{\sigma}y^{N}$
with the skew-symmetric tensor $W_{AB}$. One then rewrites the resulting
expression in terms of the Maurer–Cartan currents (B.2) as

$\displaystyle T^{\tau}_{\;\;\,\sigma}=\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{O}^{\tau\tau}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}+\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{O}^{\tau\sigma}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}.$

A similar computation leads to the announced expression of
$T^{\tau}_{\;\;\,\sigma}$. ∎

Let us now turn to the proof of Proposition 2.7. In the above notations, the
action (2.14) corresponds to the choice of operators

$\displaystyle\mathcal{O}^{\tau\tau}=\mathcal{E}^{-1}\mathcal{P}_{l},\qquad\mathcal{O}^{\sigma\sigma}=\mathcal{E}\overline{\mathcal{P}}_{l}\qquad\text{and}\qquad\mathcal{O}^{\tau\sigma}=\overline{\mathcal{P}}_{l}-{}^{t}\mathcal{P}_{l}.$
(B.5)

Applying Lemma B.1, we then get an explicit expression of the components
$T^{\mu}_{\;\;\,\nu}$ of the energy-momentum tensor. For instance, we have

$\displaystyle
T^{\tau}_{\;\;\,\tau}=-T^{\sigma}_{\;\;\,\sigma}=\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}+\frac{1}{2}\langle\\!\langle
l^{-1}\partial_{\sigma}l,\mathcal{E}\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}.$

On the other hand, from the definition (2.16) of $\mathcal{J}_{\sigma}$ and
the symmetry of $\mathcal{E}$, we get

$\displaystyle\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{E}\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=\frac{1}{2}\langle\\!\langle\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l,\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}+\frac{1}{2}\langle\\!\langle\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l,\mathcal{E}\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle\hphantom{\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{E}\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=}{}+\langle\\!\langle\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l,\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}.$

The last line vanishes due to part $(iii)$ of Proposition 2.3. Moreover, from
part $(i)$, we have

$\displaystyle{}^{t}\mathcal{P}_{l}\mathcal{E}^{-1}\mathcal{P}_{l}=\mathcal{E}^{-1}\mathcal{P}_{l}^{2}=\mathcal{E}^{-1}\mathcal{P}_{l},$

where we have used the fact that $\mathcal{P}_{l}$ is a projector and thus
that $\mathcal{P}_{l}^{2}=\mathcal{P}_{l}$. A similar computation yields
${}^{t}\overline{\mathcal{P}}_{l}\mathcal{E}\overline{\mathcal{P}}_{l}=\mathcal{E}\overline{\mathcal{P}}_{l}$.
Thus, we get

$\displaystyle\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{E}\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=T^{\tau}_{\;\;\,\tau}=-T^{\sigma}_{\;\;\,\sigma}.$

Let us now turn our attention to $T^{\tau}_{\;\;\,\sigma}$. Applying Lemma B.1
with the operators (B.5), we get

$\displaystyle T^{\tau}_{\;\;\,\sigma}=\langle\\!\langle
l^{-1}\partial_{\tau}l,\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}+\langle\\!\langle
l^{-1}\partial_{\sigma}l,\big{(}\overline{\mathcal{P}}_{l}-\mathcal{P}_{l}\big{)}l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}},$

where in particular we transposed the operator ${}^{t}\mathcal{P}_{l}$ in the
last term. On the other hand, it follows from (2.16) that

$\displaystyle\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=\frac{1}{2}\langle\\!\langle\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l,\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l\rangle\\!\rangle_{\mathfrak{d}}+\frac{1}{2}\langle\\!\langle\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l,\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}$
$\displaystyle\hphantom{\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=}{}+\langle\\!\langle\mathcal{E}^{-1}\mathcal{P}_{l}\,l^{-1}\partial_{\tau}l,\overline{\mathcal{P}}_{l}\,l^{-1}\partial_{\sigma}l\rangle\\!\rangle_{\mathfrak{d}}.$

Note from part $(i)$ of Proposition 2.3 that $\mathcal{E}^{-1}\mathcal{P}_{l}$
is symmetric. Using also part $(v)$ we get

$\displaystyle{}^{t}\bigl{(}\mathcal{E}^{-1}\mathcal{P}_{l}\bigr{)}\mathcal{E}^{-1}\mathcal{P}_{l}=\mathcal{E}^{-1}\mathcal{P}_{l}\,{}^{t}\bigl{(}\mathcal{E}^{-1}\mathcal{P}_{l}\bigr{)}=\mathcal{E}^{-1}\mathcal{P}_{l}{}^{t}\mathcal{P}_{l}\mathcal{E}^{-1}=0,$

so that the first term in
$\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}$
vanishes. Using the same properties, we also get

$\displaystyle{}^{t}\bigl{(}\mathcal{E}^{-1}\mathcal{P}_{l}\bigr{)}\overline{\mathcal{P}}_{l}=\mathcal{E}^{-1}\mathcal{P}_{l}\overline{\mathcal{P}}_{l}=\mathcal{E}^{-1}\big{(}\mathcal{P}_{l}-\mathcal{P}_{l}{}^{t}\mathcal{P}_{l}\big{)}=\mathcal{E}^{-1}\mathcal{P}_{l}.$

Finally, using
${}^{t}\overline{\mathcal{P}}_{l}\overline{\mathcal{P}}_{l}=\overline{\mathcal{P}}_{l}-\mathcal{P}_{l}$
(see part $(iv)$ of Proposition 2.3), we find that

$\displaystyle\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=T^{\tau}_{\;\;\,\sigma}.$

A similar computation yields
$-\frac{1}{2}\langle\\!\langle\mathcal{J}_{\sigma},\mathcal{E}^{2}\mathcal{J}_{\sigma}\rangle\\!\rangle_{\mathfrak{d}}=T^{\sigma}_{\;\;\,\tau}$.

#### Acknowledgements

S.L. would like to thank B. Hoare for useful discussions. The work of S.L. is
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe”
– 390833306.

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