aainstitutetext: Instituto Galego de Física de Altas Enerxías (IGFAE) and
Departamento de Física de Partículas,
Universidade de Santiago de Compostelabbinstitutetext: Center for Quantum
Mathematics and Physics (QMAP),
Department of Physics & Astronomy, University of California, Davis, CA 95616,
USAccinstitutetext: Dipartimento di Fisica e Astronomia,
Universita degli Studi di Padova,
& Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, via Marzolo 8, 35131 Padova, Italy.ddinstitutetext:
Institute for Advanced Study
Einstein Drive, Princeton, New Jersey, 08540 USA.

# On the Classical Integrability of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ Flows

Riccardo Borsato b Christian Ferko c,d,1 Alessandro Sfondrini111IBM Einstein
Fellow<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow
was recently introduced as a universal and classically marginal deformation of
any two-dimensional translation-invariant field theory. The flow commutes with
the (irrelevant) $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flow and it can be integrated explicitly for a large class of actions,
leading to non-analytic Lagrangians reminiscent of the four-dimensional
Modified-Maxwell theory (ModMax). It is not a priori obvious whether the
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow
preserves integrability, like it is the case for the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow. In this paper we
demonstrate that this is the case for a large class of classical models by
explicitly constructing a deformed Lax connection. We discuss the principal
chiral model and the non-linear sigma models on symmetric and semi-symmetric
spaces, without or with Wess-Zumino term. We also construct Lax connections
for the two-parameter families of theories deformed by both Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ for all of these
models.

##  1 Introduction

Given the action $\mathcal{S}_{0}$ of a quantum field theory (QFT) and an
integrated local operator $\mathcal{O}$, it is possible to formally deform the
action infinitesimally by setting

$\mathcal{S}_{\lambda}=\mathcal{S}_{0}+\lambda\,\mathcal{O}+O(\lambda^{2})\,.$
(1)

For instance, in the case where $\mathcal{S}_{0}$ describes a conformal field
theory (CFT) and $\mathcal{O}$ is a relevant operator in the spectrum of the
theory, it is natural to interpret $\mathcal{S}_{0}$ as an ultraviolet (UV)
theory, and $\mathcal{O}$ as the source of a renormalisation group (RG) flow.
Geometrically, $\mathcal{O}$ is a tangent vector to the RG flow at a fixed
point, and the RG flow equations determine the remainder of the
$\lambda$-expansion in (1). Determining the flow completely is a very
difficult problem in general. Remarkably, in two-dimensions it is sometimes
possible to investigate such flows quite explicitly in the special setup where
the perturbing operator preserves “sufficiently many symmetries”. In this
case, the flow is integrable and can be studied e.g. by thermodynamic Bethe
ansatz techniques, see e.g. Zamolodchikov:1989cf ; Zamolodchikov:1991vx ;
Mussardo:1992uc ; Bazhanov:1994ft .

For two-dimensional QFTs it is possible to construct other deformations quite
explicitly. If $\mathcal{S}_{0}$ enjoys at least two conservation laws with
currents $j^{\mu}_{0}$ and $k^{\mu}_{0}$,

$\partial_{\mu}j^{\mu}_{0}=\partial_{\mu}k^{\mu}_{0}=0\,,$ (2)

then we may define the combination

$\mathcal{O}_{0}^{j\wedge
k}=\int\text{d}\tau\int\text{d}\sigma\;j^{\mu}_{0}(\tau,\sigma)\,k^{\nu}_{0}(\tau,\sigma)\,\epsilon_{\mu\nu}\,.$
(3)

It turns out that this composite operator can be defined in the quantum theory
without incurring any short-distance singularities Smirnov:2016lqw . Then, it
is possible to define a deformed action by the equation

$\partial_{\lambda}\mathcal{S}_{\lambda}=\mathcal{O}_{\lambda}^{j\wedge k}\,,$
(4)

where $\mathcal{O}_{\lambda}^{j\wedge k}$ is defined in the deformed theory,
out of the deformed currents $j^{\mu}_{\lambda}$ and $k^{\mu}_{\lambda}$
(which are conserved with respect to $\mathcal{S}_{\lambda}$). This is a
rather general definition that encompasses many well-studied examples,
including the marginal $J\mkern
1.5mu\overline{\mkern-1.5muJ\mkern-1.5mu}\mkern 1.5mu$ deformation (when
$J^{\mu}=j^{\mu}$ and $\epsilon^{\mu\nu}J_{\nu}=k^{\mu}$ are both conserved),
and the irrelevant $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformation Zamolodchikov:2004ce ; Smirnov:2016lqw ; Cavaglia:2016oda ,
which arises when $T^{0\mu}=j^{\mu}$ and $T^{1\mu}=k^{\mu}$. In that case

$\mathcal{O}_{\lambda}^{(2)}=-\int\text{d}\tau\int\text{d}\sigma\;\det\left[T^{\mu\nu}_{\lambda}(\tau,\sigma)\right]\,.$
(5)

The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation is universal: it can be constructed for any translation invariant
QFT. Remarkably, it can be studied very explicitly both at the classical and
at the quantum level. Moreover, if the seed theory $\mathcal{S}_{0}$ happens
to be a CFT, or an integrable QFT (IQFT), infinitely-many analogues of
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ may be
constructed Smirnov:2016lqw : both CFTs and IQFTs possess infinitely-many
higher-spin conserved currents, which may be used to define more and more
irrelevant Poincaré-invariant operators of the form (4). Namely, the currents
related to spin $\pm s$ operators define an operator $\mathcal{O}^{s}$ of
dimension $(2s-2)$. Very remarkably, each of these operators
$\mathcal{O}^{(s)}$, including $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, defines a flow that
preserves all higher-spin charges Smirnov:2016lqw ; Hernandez-Chifflet:2019sua
. In other terms, $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ and its higher-spin counterparts preserve integrability. This allows
for a quite detailed study of these deformations, e.g. yielding flow equations
for various physical quantities along the deformation, most notably the
finite-volume energy spectrum Smirnov:2016lqw ; Cavaglia:2016oda .

Recently, a new type of classically marginal deformation of two-dimensional
field theories has been considered Ferko:2022lol , see also Conti:2022egv ;
Babaei-Aghbolagh:2022kfz . Here, the deforming operator is taken to be222In
what follows, we will always denote the parameter of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow as $\gamma$, in
contrast with the parameter $\lambda$ of a generic deformation. The parameter
of the irrelevant $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformation will also be indicated by $\lambda$.

$\mathcal{R}_{\gamma}=-\int\text{d}\tau\int\text{d}\sigma\;\sqrt{\det\left[\widetilde{T}^{\mu\nu}_{\gamma}(\tau,\sigma)\right]}\,,\qquad\widetilde{T}^{\mu\nu}_{\gamma}=T^{\mu\nu}_{\gamma}-\frac{1}{2}g^{\mu\nu}\,T^{\rho\sigma}_{\gamma}g_{\rho\sigma}\,,$
(6)

where $g_{\mu\nu}$ is the two-dimensional metric. In light-cone
coordinates333We take $g_{\pm\mp}=-2$ and $g^{\pm\mp}=-\tfrac{1}{2}$ so that
for any vector $V^{\mu}$, $V^{\pm}=\tfrac{1}{2}(V^{0}\pm V^{1})$ and
$V_{\pm}=V_{0}\pm V_{1}$.

$\mathcal{R}_{\gamma}=\int\text{d}\sigma_{+}\int\text{d}\sigma_{-}\;\sqrt{\widetilde{T}^{++}_{\gamma}(\sigma_{+},\sigma_{-})\,\widetilde{T}^{--}_{\gamma}(\sigma_{+},\sigma_{-})}\,.$
(7)

It is worth emphasising that $\widetilde{T}^{\pm\pm}$ are chiral only if
$\mathcal{S}_{\gamma}$ defines a CFT — otherwise they genuinely depend on both
$\sigma_{+}$ and $\sigma_{-}$. The classical flow is generated by setting

$\partial_{\gamma}\mathcal{S}_{\gamma}=\mathcal{R}_{\gamma}\,,$ (8)

and it has has several intriguing properties Ferko:2022lol :

  1. 1.

It preserves classical conformal invariance, i.e. if the stress-energy tensor
of the seed theory obeys $T^{\mu\nu}_{0}g_{\mu\nu}=0$, then in the deformed
theory $T^{\mu\nu}_{\gamma}g_{\mu\nu}=0$.

  2. 2.

It commutes with the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flow generated by $\mathcal{O}^{(2)}$, which allows us to define a two-
parameter flow resulting in an action $\mathcal{S}_{(\lambda,\gamma)}$.

  3. 3.

If $\mathcal{S}_{0}$ is given by a member of a large class of Lagrangian
densities $L_{0}$, it is possible to express $L_{\gamma}$ (as well as
$L_{(\lambda,\gamma)}$) in closed form.

  4. 4.

The resulting deformed Lagrangian is not analytic in a derivative expansion,
and is very reminiscent of that of modified Maxwell theories in four
dimensions Bandos:2020jsw ; Bandos:2020hgy .

To illustrate the last two points, let us consider as seed action the one for
$N$ non-interacting massless fields $\phi^{i}$, $i=1,\dots N$,

$\mathcal{S}_{0}=\int\text{d}\tau\int\text{d}\sigma\;\frac{1}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{i}\,.$
(9)

Then we have Ferko:2022lol ; Conti:2022egv

$\mathcal{S}_{\gamma}=\int\text{d}\tau\int\text{d}\sigma\left(\frac{\cosh\gamma}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{i}+\frac{\sinh\gamma}{2}\sqrt{2\partial_{\mu}\phi^{i}\partial^{\nu}\phi^{i}\partial_{\nu}\phi^{j}\partial^{\mu}\phi^{j}-\left(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{i}\right)^{2}}\right).$
(10)

With the exception of the case of the single boson, when

$\mathcal{S}_{\gamma}=e^{\gamma}\,\mathcal{S}_{0}\,,\qquad(N=1),$ (11)

the action is non-analytic for small gradients $\partial_{\mu}\phi^{i}$ and in
fact quite intricate. For instance, although it still admits plane waves such
as $\phi^{\mu}=n^{\mu}e^{ip_{\nu}\sigma^{\nu}}$ as solutions, linear
combinations of plane waves are not solutions. This is very similar to ModMax
theories Bandos:2020jsw ; Bandos:2020hgy ; Bandos:2021rqy ; Lechner:2022qhb
(see Sorokin:2021tge  for a review) and in fact the action (10) can be
obtained as the dimensional reduction of the ModMax action Conti:2022egv .
This non-linear and non-analytic structure makes the quantisation of
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformed
theories, as well as of ModMax theories, an outstanding problem. This is in
stark contrast with $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformed theories where at least some features of the model – like the
spectrum Smirnov:2016lqw ; Cavaglia:2016oda , S matrix Dubovsky:2017cnj ,
partition function Cardy:2018sdv ; Datta:2018thy ; Aharony:2018bad ;
Griguolo:2022xcj ; Griguolo:2022hek ; Griguolo:2022vdx , and correlation
functions Cardy:2019qao ; Kraus:2018xrn  – may be studied quite thoroughly at
the quantum level.

In an effort to better understand Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ theories (and perhaps
ModMax theories too), it is natural to ask if this deformation has any further
property, at least at the classical level. In particular, does it preserve
integrability, like it is the case for $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and its higher-spin
analogues generated by $\mathcal{O}^{(s)}$, and how can this be seen?

In this work we will not answer this question in full generality, but instead
give evidence for a (rather large) class of theories. These are the principal
chiral model (PCM) on a Lie group $G$, and two classes of non-linear sigma
models (NLSMs) on coset spaces $G/H$, where $G$ is a Lie (super)group and $H$
a bosonic subgroup thereof, as well as the extensions of these models by a
Wess-Zumino (WZ) term. These NLSMS are the (bosonic) symmetric-space sigma
model (SSSM) and its supersymmetric extension, the “semi-symmetric space”
sigma model (sSSSM). For all these models we construct the deformed Lax
connection $\mathfrak{L}_{\mu}$, and check that imposing its flatness

$\partial_{+}\mathfrak{L}_{-}-\partial_{-}\mathfrak{L}_{+}+\left[\mathfrak{L}_{+},\,\mathfrak{L}_{-}\right]=0\,,$
(12)

is equivalent to the equations of motion of the deformed model. We also
construct the Lax connection for the $(\gamma,\lambda)$ doubly-deformed
models, where we combine $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations. Our
construction follows a rather explicit case-by-case analysis, similarly to the
construction of $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformed Lax connections of Chen:2021aid .

It should be noted that the semi-symmetric space sigma model reduces to the
symmetric space sigma model by setting its odd (fermionic) part to zero. In
turn, the PCM can also be thought of as a symmetric space coset $(G\times
G)/G$. In a sense, we might as well only describe the most general sSSSM case
(with the addition of a WZ term), and all the others would follow from it.
However, especially given that the construction of the sSSSM is somewhat
technical, we find it clearer to first work out in detail the PCM, which is
substantially simpler, and then “work our way up” to the SSSM and sSSSM.

Let us stress that the identification of a Lax connection is only the first
step in the proof of classical integrability of the deformed models. One
should further check that the conserved charges constructed from the monodromy
matrix have mutually vanishing Poisson brackets, i.e. they are in involution.
In this regard, the models that we are deforming are known to possess some
technical complications due to a non-ultralocal Poisson structure. That means
that when taking the Poisson brackets not only the Dirac delta function
appears but also its derivative. Nevertheless, it is still possible to recast
the Poisson brackets of the Lax connection in a useful form known as Maillet
brackets Maillet:1985ec ; Maillet:1985ek  that allows one to construct the
mutually conserved charges and argue for the classical integrability of the
models. We expect the deformed models analysed here to display a modification
of the original structure of Maillet brackets of the undeformed models. It
would be very important, and at the same time very interesting, to construct
the Maillet brackets explicitly.

This paper is structured as follows. We begin in Section 2 by reviewing the
constructions of Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flows presented in Ferko:2022lol  to the extent that we will need later
in the paper. Next, in Section 3 we discuss the case of the PCM (possibly with
WZ term). We actually discuss the case of the deformation of the PCM in quite
some detail (Section 3.2) as it illustrates nicely the mechanism by which
integrability is preserved in the other models too, but it is less involved
technically. Adding the WZ term also allows us to consider explicitly the
deformation of the classical WZW model, see Section 3.5. In Section 4 we
discuss first the case of symmetric-space sigma models and later that of semi-
symmetric space sigma models. Even though the constructions get more and more
involved, the logic follows quite closely the one we illustrate for the PCM in
the previous section. We conclude and present some open questions in Section
5. The derivation of the equations of motion for the various (deformed) models
of interest, which is straightforward but a little tedious, is presented in
Appendix A.

##  2 Review of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ Flows

Let us briefly review the construction of the deformed Lagrangian presented in
Ferko:2022lol , and fix some notation which we will use later in the paper. To
begin with, we are interested in a Lagrangian featuring $N$ interacting Bosons
$\phi^{i}$, $i=1,\dots N$, of the form

$L_{0}=\frac{1}{2}\left[g^{\mu\nu}G_{ij}(\phi)+\varepsilon^{\mu\nu}B_{ij}(\phi)\right]\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}-V(\phi)\,,$
(13)

where $g^{\mu\nu}$ is the two dimensional inverse-metric, which in light-cone
coordinates reads $g^{\pm\mp}=-\tfrac{1}{2}$ (while $g_{\pm\mp}=-2$) and
$\varepsilon^{\mu\nu}$ is the Levi-Civita tensor with
$\varepsilon^{+-}=-\varepsilon^{-+}=\tfrac{1}{2}$. Notice that the potential
term $V(\phi)$ will not be needed for the models which we study below, but it
turns out that it is easy to include it.

For the purpose of evaluating the Hilbert444The Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation may also be
defined for non-relativistic theories by using the Noether stress tensor.
However, in this work we will restrict attention to theories which enjoy boost
invariance. stress-energy tensor

$T_{\mu\nu}=-2\frac{\partial L}{\partial g^{\mu\nu}}+g_{\mu\nu}L\,,$ (14)

it is important to single out the terms in (13) which couple to the two-
dimensional metric $g_{\mu\nu}$. Hence we define

$(X_{1})_{\mu}{}^{\nu}=G_{ij}(\phi)\,\partial_{\mu}\phi^{i}\partial^{\nu}\phi^{j}\,.$
(15)

When constructing powers of the stress-energy tensor, we will generate new
tensor structures by contracting $(X_{1})_{\mu}{}^{\nu}$ with itself and with
the metric in various ways. Due to the symmetry of the stress-energy tensor
and of the metric, it is immediate to see that these terms will be of the form

$(X_{2})_{\mu}{}^{\nu}=(X_{1})_{\mu}{}^{\rho}(X_{1})_{\rho}{}^{\nu}\,,\qquad(X_{3})_{\mu}{}^{\nu}=(X_{2})_{\mu}{}^{\rho}(X_{1})_{\rho}{}^{\nu}\,,\qquad\dots\,,$
(16)

or generally

$(X_{n+1})_{\mu}{}^{\nu}=(X_{n})_{\mu}{}^{\rho}(X_{1})_{\rho}{}^{\nu}\,,\qquad
n\geq 1\,.$ (17)

The deformed action will be a scalar and therefore will depend on the traces
of the various $(X_{n})_{\mu}{}^{\nu}$, which we denote by

$x_{n}=(X_{n})_{\mu}{}^{\mu}\,,\qquad n\geq 1\,.$ (18)

The deformed Lagrangian will also be a function of the terms in (13) which do
not couple to the metric, which we denote by

$x_{0}=\frac{1}{2}\varepsilon^{\mu\nu}B_{ij}(\phi)\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}-V(\phi)\,.$
(19)

Finally, we note that many of the trace terms $x_{n}$, $n\geq 1$ are related
by trace identities. In fact, $(X_{n})_{\mu}{}^{\nu}$ is a $2\times 2$ matrix
and as such we may express any trace term as a polynomial in $x_{1}$ and
$x_{2}$,

$x_{n}=P_{n}(x_{1},x_{2})\,,\qquad n\geq 3\,.$ (20)

This is sufficient to conclude that the deformed Lagrangian takes the form

$L_{\gamma}(x_{0},x_{1},x_{2})\,,$ (21)

with the seed action being

$L_{0}(x_{0},x_{1},x_{2})\,=\frac{1}{2}x_{1}+x_{0}\,.$ (22)

Once the action is recast in this form, an explicit computation Ferko:2022lol
allows us to integrate the flow equation obtaining simply

$L_{\gamma}(x_{0},x_{1},x_{2})=L(x_{0},x_{1}^{(\gamma)},x_{2}^{(\gamma)})\,.$
(23)

In other words, the deformed Lagrangian takes the same form as the original
Lagrangian in terms of the deformed quantities $x_{1}^{(\gamma)}$ and
$x_{2}^{(\gamma)}$ (while $x_{0}$ remains undeformed). These are given by

$\displaystyle x_{1}^{(\gamma)}$
$\displaystyle=\operatorname{\text{ch}}(\gamma)x_{1}+\operatorname{\text{sh}}(\gamma)\sqrt{2x_{2}-x_{1}^{2}}\,,$
(24) $\displaystyle x_{2}^{(\gamma)}$
$\displaystyle=\operatorname{\text{ch}}(2\gamma)x_{2}+\operatorname{\text{sh}}(2\gamma)x_{1}\sqrt{2x_{2}-x_{1}^{2}}\,.$

In conclusion, for the Bosonic models below, the deformed Lagrangian will be
simply

$L_{\gamma}=\frac{1}{2}x_{1}^{(\gamma)}+x_{0}=\frac{\operatorname{\text{ch}}(\gamma)}{2}x_{1}+\frac{\operatorname{\text{sh}}(\gamma)}{2}\sqrt{2x_{2}-x_{1}^{2}}+x_{0}\,.$
(25)

It is also worth noting that the combination appearing under the square roots
factorises when expressed in light-cone coordinates,

$2x_{2}-x_{1}^{2}=\left(G_{ij}(\phi)\partial_{+}\phi^{i}\partial_{+}\phi^{j}\right)\left(G_{kl}(\phi)\partial_{-}\phi^{k}\partial_{-}\phi^{l}\right),$
(26)

which will be useful later on. Finally we remark that there exists a conserved
quantity along this flow, namely

$(x_{1}^{(\gamma)})^{2}-x_{2}^{(\gamma)}=(x_{1})^{2}-x_{2}\,,$ (27)

which can be used to show that this flow preserves tracelessness of the
stress-energy tensor Ferko:2022lol . This discussion can also be extended to
models with Bosons and Fermions Ferko:2022lol . However, we will not need the
details of the fermionic constructions in what follows. In fact, the only
model where we might expect complications due to the presence of Fermions is
the semi-symmetric space sigma model. In that case, however, the fermionic
currents do not couple to the metric (a Fermionic bilinear appears in
$x_{0}$), while the rest of the dependence on Fermions is packaged inside
bosonic currents, as we shall see below.

###  2.1 Combining $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ and Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ Deformations

Since the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
flow commutes with the (irrelevant) $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow, it is easy to
define a joint deformation which depends on two parameters: $\gamma$ for
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformations, and $\lambda$ for $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations
Ferko:2022lol . In fact, it is possible to construct a doubly-deformed
Lagrangian, $L_{(\lambda,\gamma)}$ so that

$\partial_{\lambda}L_{(\lambda,\gamma)}=-\text{det}\left[T^{\mu\nu}_{(\lambda,\gamma)}\right],\qquad\partial_{\gamma}L_{(\lambda,\gamma)}=\sqrt{\widetilde{T}^{++}_{(\lambda,\gamma)}\widetilde{T}^{--}_{(\lambda,\gamma)}}\,.$
(28)

A simple way to do so is to first perform a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation, and then
deform the resulting action along the lines illustrated above. The actions
resulting from $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
flows can be expressed in closed form Bonelli:2018kik . To do so, it is
convenient to restrict to a slightly more narrow Lagrangian than (13). In
fact, while the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flow is completely insensitive to $x_{0}$ of eq. (19) this is not the
case for $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$. In
particular, the potential $V(\phi)$ enters the flow, which is described by the
Burgers’ equation, as an initial condition; this results in a fairly intricate
dependence on $V(\phi)$ at finite $\lambda$. Conversely, the topological term
is left unchanged by any continuous deformation. For the class of models that
we want to consider here, it suffices to restrict to $V(\phi)=0$, i.e.

$x_{0}=\frac{1}{2}\varepsilon^{\mu\nu}B_{ij}(\phi)\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}\,,\qquad
x_{1}=g^{\mu\nu}G_{ij}(\phi)\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}\,,$
(29)

In this case, starting from a seed Lagrangian

$L_{0}(x_{0},x_{1},x_{2})=\frac{1}{2}x_{1}+x_{0}\,,$ (30)

we have the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformed Lagrangian Bonelli:2018kik

$L_{(\lambda,0)}(x_{0},x_{1},x_{2})=\frac{1}{2\lambda}\left(\sqrt{1+2\lambda
x_{1}+2\lambda^{2}\big{(}(x_{1})^{2}-x_{2}\big{)}}-1\right)+x_{0}\,.$ (31)

Then, according to our general recipe,

$L_{(\lambda,\gamma)}(x_{0},x_{1},x_{2})=L_{(\lambda,0)}(x_{0},x_{1}^{(\gamma)},x_{2}^{(\gamma)}),$
(32)

or more explicitly

$L_{(\lambda,\gamma)}(x_{0},x_{1},x_{2})=\frac{1}{2\lambda}\left(\sqrt{1+2\lambda
x_{1}^{(\gamma)}+2\lambda^{2}\left((x_{1}^{(\gamma)})^{2}-x_{2}^{(\gamma)}\right)}-1\right)+x_{0},$
(33)

with $x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$ as in (24).

##  3 The Principal Chiral Model with Wess-Zumino Term

The Principal Chiral Model (PCM) is a particularly simple sigma model where
the target space is a Lie group $G$. In this section we briefly review the
construction of the Lax connection for the PCM (see also the recent review
Hoare:2021dix ), and then we extend it to the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformed model. We
later show that similar arguments apply also to the case with a Wess-Zumino
(WZ) term and to the double Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation.

###  3.1 The Principal Chiral Model

Consider a Lie group $G$ and its Lie algebra $\mathfrak{g}$. The left- and
right-invariant Maurer-Cartan forms are

$j=g^{-1}\text{d}g\,,\qquad\tilde{\jmath}=-(\text{d}g)g^{-1}\,,$ (34)

with $j\in\mathfrak{g}$ and $\tilde{\jmath}\in\mathfrak{g}$. These are related
by the adjoint action
$\tilde{\jmath}=-\operatorname{\text{Ad}}_{g}{j}=-g^{-1}jg$. Let
$g=g(\tau,\sigma)\in G$ be a $G$-valued field. We will drop the
$(\tau,\sigma)$ dependence to keep our notation lighter. Define $j_{\mu}$ and
$\tilde{\jmath}_{\mu}$ to be the pull-back of the Maurer-Cartan forms

$j_{\mu}=g^{-1}\partial_{\mu}g\,,\qquad\tilde{\jmath}_{\mu}=-\left(\partial_{\mu}g\right)g^{-1}\,,$
(35)

which both take values in $\mathfrak{g}$. Here $j_{\mu}$ is invariant under
left-multiplication by a constant group element $g_{0}\in G$,
$g^{\prime}=g_{0}\,g$, and $\tilde{\jmath}_{\mu}$ is invariant under right-
multiplication by a constant group element, $g^{\prime}=g\,g_{0}$. Both
$j_{\mu}$ and $\tilde{\jmath}_{\mu}$ satisfy the flatness condition, which we
express in light-cone coordinates,

$\partial_{+}j_{-}-\partial_{-}j_{+}+\left[j_{+},\,j_{-}\right]=0=\partial_{+}\tilde{\jmath}_{-}-\partial_{-}\tilde{\jmath}_{+}+\left[\tilde{\jmath}_{+},\,\tilde{\jmath}_{-}\right]\,.$
(36)

The Lagrangian of the principal chiral model takes a simple form in terms of
$j_{\mu}$, namely

$L_{\text{PCM}}=\frac{1}{2}g^{\mu\nu}\operatorname{\text{tr}}\left[j_{\mu}\,j_{\nu}\right]=-\frac{1}{2}\operatorname{\text{tr}}\left[j_{+}\,j_{-}\right]\,,$
(37)

which can be equivalently written as

$L_{\text{PCM}}=\frac{1}{2}g^{\mu\nu}\operatorname{\text{tr}}\left[\tilde{\jmath}_{\mu}\,\tilde{\jmath}_{\nu}\right]=-\frac{1}{2}\operatorname{\text{tr}}\left[\tilde{\jmath}_{+}\,\tilde{\jmath}_{-}\right]\,.$
(38)

Both expressions are manifestly invariant under left-multiplication and right-
multiplication by $g_{0}\in G$, hence the action is invariant under $G\times
G$.

As it is easy to verify explicitly, the Noether current associated to the
invariance under right-multiplication is $j_{\mu}$, and its conservation
equation

$\partial_{\mu}j^{\mu}=0\,,$ (39)

is equivalent to the equations of motion of the model, see also Appendix A.1
for the explicit computation. Therefore, we may define the Lax connection

$\mathfrak{L}_{\pm}=\frac{j_{\pm}}{1\mp z}\,,$ (40)

which depends on the spectral parameter $z\in\mathbb{C}$. Requiring
$\mathfrak{L}_{\mu}$ to be flat for any $z$ amounts to imposing the equations
of motions of the model (39)

$\displaystyle 0$
$\displaystyle=\partial_{+}\mathfrak{L}_{-}-\partial_{-}\mathfrak{L}_{+}+\left[\mathfrak{L}_{+},\mathfrak{L}_{-}\right]$
(41)
$\displaystyle=\frac{1}{1-z^{2}}\Big{(}\partial_{+}j_{-}-\partial_{-}j_{+}+\left[j_{+},j_{-}\right]-z\left(\partial_{+}j_{-}+\partial_{-}j_{+}\right)\Big{)},$

as it may be verified explicitly by using the Maurer-Cartan equation (36).
Similarly, the Noether current related to left-multiplication is
$\tilde{\jmath}_{\mu}$, whose conservation equation

$\partial_{\mu}\tilde{\jmath}{}^{\mu}=0\,,$ (42)

is also equivalent to the equations of motion, which yields an equivalent
construction of the Lax connection.

###  3.2 Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
Deformation

It is immediate to write down a deformed Lagrangian, by plugging into eq. (25)
the appropriate expressions for the PCM, which we may read off (37). We have

$x_{0}=0,\qquad x_{1}=-\operatorname{\text{tr}}[j_{+}j_{-}]\,,\qquad
x_{2}=\frac{1}{2}\left(\operatorname{\text{tr}}[j_{+}j_{+}]\operatorname{\text{tr}}[j_{-}j_{-}]+\left(\operatorname{\text{tr}}[j_{+}j_{-}]\right)^{2}\right)\,,$
(43)

so that

$L_{\text{PCM}}^{(\gamma)}=\frac{1}{2}\left(-\operatorname{\text{ch}}(\gamma)\operatorname{\text{tr}}[j_{+}j_{-}]+\operatorname{\text{sh}}(\gamma)\sqrt{\operatorname{\text{tr}}[j_{+}j_{+}]\operatorname{\text{tr}}[j_{-}j_{-}]}\right).$
(44)

We can equivalently write

$L_{\text{PCM}}^{(\gamma)}=\frac{1}{2}\left(-\operatorname{\text{ch}}(\gamma)\operatorname{\text{tr}}[\tilde{\jmath}_{+}\tilde{\jmath}_{-}]+\operatorname{\text{sh}}(\gamma)\sqrt{\operatorname{\text{tr}}[\tilde{\jmath}_{+}\tilde{\jmath}_{+}]\operatorname{\text{tr}}[\tilde{\jmath}_{-}\tilde{\jmath}_{-}]}\right),$
(45)

either by expressing the seed Lagrangian in terms of $\tilde{\jmath}_{\mu}$ in
the first place, or by using the explicit form of $j_{\mu}$ and
$\tilde{\jmath}_{\mu}$.

The above expressions make it manifest that the deformed model still has
$G\times G$ symmetry. It is possible to derive the resulting conserved current
by Noether’s theorem. In Appendix A.1 we do so for any Lagrangian which
depends on $x_{1}$ and $x_{2}$ as in eq. (43), $L(x_{1},x_{2})$, obtaining for
the current related to the right-multiplication symmetry

$\mathfrak{J}_{\mu}=2\frac{\partial L}{\partial x_{1}}j_{\mu}+4\frac{\partial
L}{\partial
x_{2}}g^{\nu\rho}\operatorname{\text{tr}}[j_{\mu}j_{\nu}]\,j_{\rho}\,,\qquad\partial_{\mu}\mathfrak{J}^{\mu}=0\,.$
(46)

We immediately see that for the seed Lagrangian $L_{0}=\tfrac{1}{2}x_{1}$ this
reduces to the current $j^{\mu}$. For the action (44) this gives

$\mathfrak{J}_{\pm}=\operatorname{\text{ch}}(\gamma)\,j_{\pm}-\operatorname{\text{sh}}(\gamma)\sqrt{\frac{\operatorname{\text{tr}}[j_{\pm}j_{\pm}]}{\operatorname{\text{tr}}[j_{\mp}j_{\mp}]}}\,j_{\mp}\,,\qquad\partial_{+}\mathfrak{J}_{-}+\partial_{-}\mathfrak{J}_{+}=0\,.$
(47)

Note that we may rewrite the Lagrangian (44) as

$L_{\gamma}=-\frac{1}{2}\operatorname{\text{tr}}\left[j_{+}\mathfrak{J}_{-}\right]=-\frac{1}{2}\operatorname{\text{tr}}\left[\mathfrak{J}_{+}j_{-}\right]\,.$
(48)

The current $\mathfrak{J}_{\mu}$ is conserved; it is however not flat. To see
this, let us consider the commutator of the $\mathfrak{J}_{\pm}$, which
actually obeys a nice relation:

$[\mathfrak{J}_{+},\mathfrak{J}_{-}]=\operatorname{\text{ch}}(\gamma)^{2}[j_{+},j_{-}]+\operatorname{\text{sh}}(\gamma)^{2}[j_{-},j_{+}]=[j_{+},j_{-}]\,.$
(49)

This relation will be crucial for our construction below. To begin with, it
means that

$\partial_{+}\mathfrak{J}_{-}-\partial_{-}\mathfrak{J}_{+}+[\mathfrak{J}_{+},\mathfrak{J}_{-}]\neq
0\,,$ (50)

unless $\gamma=0$, because the first two terms are $\gamma$-dependent while
the commutator is not. However, this structure suggests a modification of the
Lax connection which is reminiscent of that discussed in Chen:2021aid ,

$\mathfrak{L}_{\pm}^{(\gamma)}=\frac{j_{\pm}\pm
z\,\mathfrak{J}_{\pm}}{1-z^{2}}\,.$ (51)

By construction, this expression reduces to (40) when $\gamma=0$ and
$\mathfrak{J}_{\mu}=j_{\mu}$. Noting the commutators

$[\mathfrak{J}_{\pm},j_{\mp}]=\operatorname{\text{ch}}(\gamma)[j_{\pm},j_{\mp}]\,,$
(52)

which in particular imply that

$[\mathfrak{J}_{+},j_{-}]-[j_{+},\mathfrak{J}_{-}]=0\,,$ (53)

and exploiting the important relation (49), we can work out the condition for
$\mathfrak{L}_{\mu}^{(\gamma)}$ to be flat,555We use the shorthands
$A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$ and
$A_{(\mu}B_{\nu)}=A_{\mu}B_{\nu}+A_{\nu}B_{\mu}$.

$\displaystyle 0$
$\displaystyle=\partial_{+}\mathfrak{L}^{(\gamma)}_{-}-\partial_{-}\mathfrak{L}^{(\gamma)}_{+}+[\mathfrak{L}^{(\gamma)}_{+},\mathfrak{L}^{(\gamma)}_{-}]$
(54)
$\displaystyle=\frac{1}{1-z^{2}}\left(\partial_{[+}j_{-]}-z\,\partial_{(+}\mathcal{J}_{-)}+\frac{[j_{+},j_{-}]-z([j_{+},\mathfrak{J}_{-}]-[\mathfrak{J}_{+},j_{-}])-z^{2}[\mathfrak{J}_{+},\mathfrak{J}_{-}]}{1-z^{2}}\right)$
$\displaystyle=\frac{1}{1-z^{2}}\left(\partial_{[+}j_{-]}+[j_{+},j_{-}]-z\,\partial_{(+}\mathcal{J}_{-)}\right)\,.$

Using the Maurer-Cartan equation (36) we see that
$\mathfrak{L}_{\mu}^{(\gamma)}$ is flat for any $z$ if and only if the
equations of motion (47) are satisfied.

It is quite clear that we could have performed an analogous construction for
the current related to left-multiplication by $g_{0}$, which would have given

$\widetilde{\mathfrak{J}}_{\pm}=\operatorname{\text{ch}}(\gamma)\,\tilde{\jmath}_{\pm}-\operatorname{\text{sh}}(\gamma)\sqrt{\frac{\operatorname{\text{tr}}[\tilde{\jmath}_{\pm}\tilde{\jmath}_{\pm}]}{\operatorname{\text{tr}}[\tilde{\jmath}_{\mp}\tilde{\jmath}_{\mp}]}}\,\tilde{\jmath}_{\mp}\,,\qquad\partial_{+}\widetilde{\mathfrak{J}}_{-}+\partial_{-}\widetilde{\mathfrak{J}}_{+}=0\,.$
(55)

This current is related to $\mathfrak{J}_{\pm}$ much in the same way as
$\tilde{\jmath}_{\pm}$ is related to $j_{\pm}$, namely

$\widetilde{\mathfrak{J}}_{\mu}=-\text{Ad}_{g}\mathfrak{J}_{\mu}\,.$ (56)

In a similar way, we can define the Lax connection in terms of
$\widetilde{\mathfrak{J}}_{\pm}$, namely

$\displaystyle\widetilde{\mathfrak{L}}_{\pm}(z)=\frac{\tilde{\jmath}_{\pm}\pm
z\widetilde{\mathfrak{J}}_{\pm}}{1-z^{2}}\,.$ (57)

As in the undeformed case, the two choices are related by a gauge
transformation of the Lax connection

$\displaystyle
g(\mathfrak{L}_{\pm}(z)+\partial_{\pm})g^{-1}=\widetilde{\mathfrak{L}}(z^{-1})\,.$
(58)

###  3.3 Two-Parameter Deformation

An important property of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow is that it
commutes with the usual (irrelevant) $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow Ferko:2022lol .
Following the discussion of Section 2.1, for a theory whose “seed” Lagrangian
is

$L_{0}(x_{0},x_{1},x_{2})=\frac{1}{2}x_{1}\,,$ (59)

we have from eq. (33)

$L_{(\lambda,\gamma)}(x_{0},x_{1},x_{2})=\frac{1}{2\lambda}\left(\sqrt{1+2\lambda
x_{1}^{(\gamma)}+2\lambda^{2}\left((x_{1}^{(\gamma)})^{2}-x_{2}^{(\gamma)}\right)}-1\right),$
(60)

with $x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$ as in (24). This can be
immediately adapted to the case of the PCM by substituting the values (43) for
$x_{1}$ and $x_{2}$. Then, we get a deformed action of the type discussed in
Section 3.1. In particular, the equations of motion are still given by the
conservation of $\mathfrak{J}_{\mu}$ as in eq. (46). We can write
$\mathfrak{J}_{\pm}$ a little more explicitly in light-cone coordinates,

$\displaystyle\mathfrak{J}_{+}$ $\displaystyle=2\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{2}}\right)j_{+}-2\frac{\partial
L_{(\lambda,\gamma)}}{\partial
x_{2}}\operatorname{\text{tr}}[j_{+}j_{+}]j_{-}\,,$ (61)
$\displaystyle\mathfrak{J}_{-}$ $\displaystyle=2\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{2}}\right)j_{-}-2\frac{\partial
L_{(\lambda,\gamma)}}{\partial
x_{2}}\operatorname{\text{tr}}[j_{-}j_{-}]j_{+}\,,$

where $j_{\mu}$ is again the pull-back of the left-invariant Maurer-Cartan one
form.666As before, the whole discussion could be repeated in terms of the
right-invariant current $\tilde{\jmath}_{\mu}$.

We will now see that the Lax connection takes once again the form

$\mathfrak{L}_{\pm}^{(\lambda,\gamma)}=\frac{j_{\pm}\pm
z\,\mathfrak{J}_{\pm}}{1-z^{2}}\,,$ (62)

just like in eq. (51), where now $\mathfrak{J}_{\mu}$ is given by (61). To see
why this is the case, we first observe that

$\displaystyle\big{[}\mathfrak{J}_{+},j_{-}\big{]}=2\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{2}}\right)\big{[}j_{+},j_{-}\big{]},$ (63)
$\displaystyle\big{[}j_{+},\mathfrak{J}_{-}\big{]}=2\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{2}}\right)\big{[}j_{+},j_{-}\big{]},$

so that in particular

$\big{[}\mathfrak{J}_{+},j_{-}\big{]}-\big{[}j_{+},\mathfrak{J}_{-}\big{]}=0\,,$
(64)

like we had before (53). Next, we compute again

$\big{[}\mathfrak{J}_{+},\mathfrak{J}_{-}\big{]}=4\left[\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{(\lambda,\gamma)}}{\partial x_{2}}\right)^{2}-\left(\frac{\partial
L_{(\lambda,\gamma)}}{\partial
x_{2}}\right)^{2}\big{(}2x_{2}-x_{1}^{2}\big{)}\right]\,\big{[}j_{+},j_{-}\big{]}.$
(65)

where we used that
$\operatorname{\text{tr}}[j_{+}j_{+}]\operatorname{\text{tr}}[j_{-}j_{-}]=2x_{2}-x_{1}^{2}$
to rewrite the whole coefficient on the right-hand side in terms of $x_{1}$
and $x_{2}$. An explicit computation using the form of $L_{(\lambda,\gamma)}$
in (33) then shows that, for any $\gamma$ and $\lambda$, the whole
coeffiecient on the right-hand side simplifies to one. In other words,

$\big{[}\mathfrak{J}_{+},\mathfrak{J}_{-}\big{]}=\big{[}j_{+},j_{-}\big{]},$
(66)

just like in (49). Using these two observations, the derivation of eq. (54)
goes through immediately, showing that for this model too the flatness of
$\mathfrak{L}_{\pm}^{(\lambda,\gamma)}$ is equivalent to the equations of
motion.777Conversely, we could obtain a whole class of integrability-
preserving Lagrangians $L(x_{1},x_{2})$ by demanding that the coefficient on
the right-hand side of (65) equals one. Unfortunately, it is not immediate to
construct the most general solution of this nonlinear partial differential
equation.  We note that, when $\gamma=0$, we recover the Lax connection for
the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformed
PCM which was obtained in Chen:2021aid .

###  3.4 Adding a Wess-Zumino Term

The PCM on the Lie group $G$ can be supplemented by a Wess-Zumino (WZ) term.
The resulting action, prior to any (Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$) deformation, reads

$\mathcal{S}_{\text{PCMWZ}}=\hay\int\limits_{\partial B}\text{d}^{2}\sigma
L_{\text{PCM}}+\kay\int\limits_{B}\text{d}^{3}\sigma\frac{1}{6}\varepsilon^{ijk}\operatorname{\text{tr}}\big{[}j_{i}\,[j_{j},j_{k}]\big{]}\,.$
(67)

A few remarks are in order. First of all, we have expressed the WZ term as a
three-dimensional integral on a space $B$ whose boundary $\partial B$ is the
usual $(1+1)$-dimensional Minkowski space.888Here $\varepsilon^{ijk}$ is the
three-dimensional Levi-Civita tensor. For this to lead to a well-defined
(quantum) theory it may be necessary to impose a constraint for the
coefficient . In particular, if $G$ is a compact simple Lie group, we should
take

$4\pi\kay\in\mathbb{Z}\,.$ (68)

The coefficient , instead, is unconstrained. We could have of course
introduced in the discussion of the PCM above, but it would only have amounted
to a rescaling of the action.

Let us briefly review the construction of the Lax connection in the undeformed
model, see also e.g. the review Hoare:2021dix . First of all $j_{\mu}$ (and
$\tilde{\jmath}_{\mu}$) still satisfy the flatness condition (36). However,
the equations of motion are now modified and read

$(\hay+\kay)\partial_{+}j_{-}+(\hay-\kay)\partial_{-}j_{+}=0\,,$ (69)

or equivalently

$(\hay-\kay)\partial_{+}\tilde{\jmath}_{-}+(\hay+\kay)\partial_{-}\tilde{\jmath}_{+}=0\,.$
(70)

It is easy to see that the Lax connection

$\mathfrak{L}_{\pm}=\left(1\mp\frac{\kay}{\hay}\right)\,\frac{j_{\pm}}{1\mp
z}$ (71)

reduces to (40) for $\kay=0$ and that its flatness is equivalent to the
equations of motion. We review the computation as it will be useful in a
moment. We have

$\displaystyle 0$
$\displaystyle=\partial_{[+}\mathfrak{L}_{-]}+[\mathfrak{L}_{+},\mathfrak{L}_{-}]$
(72)
$\displaystyle=\frac{1}{1-z^{2}}\left((1-z)(1+\tfrac{\kay}{\hay})\partial_{+}j_{-}-(1+z)(1-\tfrac{\kay}{\hay})\partial_{-}j_{+}+(1-\tfrac{{}^{2}}{{}^{2}})[j_{+},j_{-}]\right)$
$\displaystyle=\frac{1}{1-z^{2}}\left(\tfrac{\kay}{\hay}-z\right)\left[(1+\tfrac{\kay}{\hay})\partial_{+}j_{-}+(1-\tfrac{\kay}{\hay})\partial_{-}j_{+}\right],$

where in the last step we used the Maurer-Cartan equation to eliminate the
commutator. As a result, eq. (72) is equivalent to the equations of motion
(69).

Let us now consider the case of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation (or the
combined Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
and $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation) of the PCM with WZ term. The action of the deformed model is
simply

$\mathcal{S}_{\text{PCMWZ}}^{(\gamma)}=\hay\int\limits_{\partial
B}\text{d}^{2}\sigma
L_{\text{PCM}}^{(\gamma)}+\kay\int\limits_{B}\text{d}^{3}\sigma\frac{1}{6}\varepsilon^{ijk}\operatorname{\text{tr}}\big{[}j_{i}\,[j_{j},j_{k}]\big{]}\,,$
(73)

where $L_{\text{PCM}}^{(\gamma)}$ is the very same Lagrangian as in (44). This
construction also applies verbatim if $L_{\text{PCM}}^{(\gamma)}$ is replaced
by the doubly-deformed Lagrangian $L_{\text{PCM}}^{(\lambda,\gamma)}$ of (33).
In fact, as we discussed in Section 2, the terms of the action which do not
couple to the two-dimensional metric do not affect the flow. This is the case
for the WZ term, which is topological. The derivation of the equations of
motion splits now in two parts. The part scaling with is exactly as in Section
3.2 (see also Appendix A.1 for the detailed derivation). The part scaling with
goes like in the standard WZ term construction. All in all, the equations take
the form

$\partial_{+}\left(\hay\mathfrak{J}_{-}+\kay
j_{-}\right)+\partial_{-}\left(\hay\mathfrak{J}_{+}-\kay j_{+}\right)=0\,,$
(74)

which should be compared to eq. (69) above; a similar equation holds for the
right-invariant currents. These equations of motion suggest a straightforward
guess for the Lax connection of the deformed model, namely

$\mathfrak{L}_{\pm}^{(\gamma)}=\frac{\left(j_{\pm}\mp\frac{\kay}{\hay}\mathfrak{J}_{\pm}\right)\pm
z\,\left(\mathfrak{J}_{\pm}\mp\frac{\kay}{\hay}j_{\pm}\right)}{1-z^{2}}\,.$
(75)

Let us now prepare to study the flatness condition for (75). By making use of
the identities of the previous section, in particular eqs. (49) and (53), we
can compute

$\big{[}\mathfrak{L}^{(\gamma)}_{+},\mathfrak{L}^{(\gamma)}_{-}\big{]}=\frac{1-\frac{{}^{2}}{{}^{2}}}{1-z^{2}}\,[j_{+},j_{-}]\,.$
(76)

Then the computation follows closely that of eq. (72), and reads

$\displaystyle 0$
$\displaystyle=\partial_{[+}\mathfrak{L}_{-]}+[\mathfrak{L}_{+},\mathfrak{L}_{-}]$
(77)
$\displaystyle=\frac{1}{1-z^{2}}\Big{(}(1-\tfrac{\kay}{\hay}z)\partial_{+}j_{-}+(\tfrac{\kay}{\hay}-z)\partial_{+}\mathfrak{J}_{-}$
$\displaystyle\qquad\qquad\qquad-(1-\tfrac{\kay}{\hay}z)\partial_{-}j_{+}+(\tfrac{\kay}{\hay}-z)\partial_{-}\mathfrak{J}_{+}+(1-\tfrac{{}^{2}}{{}^{2}})[j_{+},j_{-}]\Big{)}$
$\displaystyle=\frac{1}{1-z^{2}}\left(\tfrac{\kay}{\hay}-z\right)\left[\partial_{+}\left(\mathfrak{J}_{-}+\tfrac{\kay}{\hay}j_{-}\right)+\partial_{-}\left(\mathfrak{J}_{+}-\tfrac{\kay}{\hay}j_{+}\right)\right],$

where again in the last step we used the Maurer-Cartan equation to eliminate
the term involving $[j_{+},j_{-}]$. The above equation is therefore equivalent
to the equations of motion (74). This construction can be repeated in terms of
the right-invariant current too.

###  3.5 About the Wess-Zumino-Witten Point

The physics of the PCM with a WZ term depends on the ratio of $\kay/\hay$. In
the undeformed model, it is interesting to consider the equations of motion in
the limit when $\hay\to\kay$. One finds that the equations of motion imply
that the left- and right-invariant currents become chiral,

$\partial_{+}j_{-}=0\,,\qquad\partial_{-}\tilde{\jmath}_{+}=0\,.$ (78)

In the opposite limit, when $\hay\to-\kay$, the currents $j_{\mu}$ and
$\tilde{\jmath}_{\mu}$ swap roles. In the quantum theory, these points
correspond to the WZW model. At these points, the chiral conservation laws
(78) lead to the existence of two (chiral and anti-chiral) Kač-Moody algebras
built out of $G$. It is natural to wonder whether this is the case in the
deformed model. In the limit $\hay\to\kay$, the equations of motion (74)
become

$\partial_{+}\left(\mathfrak{J}_{-}+j_{-}\right)+\partial_{-}\left(\mathfrak{J}_{+}-j_{+}\right)=0\,,$
(79)

which is not chiral. This is still a classical statement, but it has
intriguing implications when we consider a Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of the WZW
model. It points to a rather radical deformation of the current algebra
underlying the WZW model, which would be very interesting to study at the
quantum level.

##  4 Non-Linear Sigma Models

Let us consider now a sigma model having as target space the quotient $G/H$,
where $G$ is a Lie group and $H$ is a subgroup, see also Arutyunov:2009ga ;
Zarembo:2017muf ; Seibold:2020ouf  for reviews. Indicating the Lie algebra as
$\mathfrak{g}$ and its subalgebra as $\mathfrak{h}$, we have the vector-space
decomposition

$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{f}\,.$ (80)

Symmetric space sigma models correspond to the special case where the Lie
algebra $\mathfrak{g}$ admits a $\mathbb{Z}_{2}$ automorphism such that999The
reason for choosing subscript indices $0$ and $2$ rather than $0$ and $1$ will
become clear in a few lines, when we will introduce the semi-symmetric space
sigma model.

$\mathfrak{g}_{0}=\mathfrak{h}\,,\qquad\mathfrak{g}_{2}=\mathfrak{f}\,,\qquad[\mathfrak{g}_{n},\mathfrak{g}_{m}]\subset\mathfrak{g}_{(n+m)\text{mod}2}\,.$
(81)

As we will briefly review below, this fact is sufficient to guarantee the
existence of a Lax connection for the model, and we will show that the same is
true even if we perform a Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of the
model.

A further generalisation of such cosets is given by semi-symmetric space sigma
models. These are constructed from Lie supergroups, whose tangent spaces are
Lie superalgebras. Let $G$ be a Lie supergroup, and $\mathfrak{g}$ its
superalgebra. By definition, the superalgebra has a $\mathbb{Z}_{2}$ grading,
given by the “Fermion sign” $(-1)^{F}$. This fact by itself is not sufficient
to guarantee the existence of a Lax connection on a suitably-defined
supercoset. If, however, the algebra can be endowed with a $\mathbb{Z}_{4}$
automorphism $\Omega$, such that $\Omega^{2}=(-1)^{F}$, then it is possible to
construct such an integrable supercoset. We decompose the Lie superalgebra in
its eigenspaces under $\Omega$,

$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\mathfrak{g}_{3},\qquad\Omega\,\mathfrak{g}_{n}=i^{n}\,\mathfrak{g}_{n}\,,\qquad[\mathfrak{g}_{n},\mathfrak{g}_{m}]\subset\mathfrak{g}_{(n+m)\text{mod}4},$
(82)

where the odd eigenspaces corresponds to fermionic generators and the even
ones to bosonic generators. It follows in particular that
$\mathfrak{h}=\mathfrak{g}_{0}$ is a (bosonic) subalgebra of $\mathfrak{g}$.
Calling $H$ the corresponding subgroup, the coset space $G/H$ is the target
manifold of the semi-symmetric space sigma model. This structure is sufficient
to construct a Lax connection for this model, as well as for its Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation.

###  4.1 Symmetric-Space Sigma Model (SSSM)

Here we consider the coset $G/H$, with $H$ a subgroup of $G$ such that the
corresponding Lie algebras enjoy the decomposition (81). Similarly to what we
did for the PCM in Section 3.1, let us consider an element $g\in G$ and denote
the pull-back of the (left-invariant) Maurer-Cartan form as
$j_{\mu}=g^{-1}\partial_{\mu}g$. Since
$j_{\mu}\in\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{2}$, it can be
decomposed as

$j_{\mu}=j^{(0)}_{\mu}+j^{(2)}_{\mu}\,,\qquad
j^{(n)}_{\mu}\in\mathfrak{g}_{n}\,.$ (83)

Hence $j^{(0)}_{\mu}$ takes value in the subalgebra
$\mathfrak{g}_{0}=\mathfrak{h}$, while $j^{(2)}_{\mu}$ takes value in the
orthogonal complement. It is worth noting how these two components transform
under the subalgebra $\mathfrak{h}=\mathfrak{g}_{0}$. Letting
$h\in\mathfrak{h}$, and acting on $g$ by right multiplication,
$g^{\prime}=g\,h$, we see that

$(g^{\prime})^{-1}\partial_{\mu}g^{\prime}=\underbrace{h^{-1}j^{(0)}_{\mu}h+h^{-1}\partial_{\mu}h}_{(j^{\prime})_{\mu}^{(0)}\in\mathfrak{g}_{0}}+\underbrace{h^{-1}j^{(2)}_{\mu}h}_{(j^{\prime})_{\mu}^{(2)}\in\mathfrak{g}_{2}}\,,$
(84)

so that $j^{(0)}_{\mu}$ transforms like a connection while $j^{(2)}_{\mu}$
transforms in the adjoint representation of $\mathfrak{h}$. This also makes it
natural to introduce the “covariant derivative”

$D_{\mu}=\partial_{\mu}+\left[j^{(0)}_{\mu},\,\cdot\,\right]\,,$ (85)

as well as the “field strength”

$F_{\mu\nu}^{(0)}=\partial_{\mu}j^{(0)}_{\nu}-\partial_{\nu}j^{(0)}_{\mu}+[j^{(0)}_{\mu},j^{(0)}_{\nu}]\,.$
(86)

The Lagrangian for the SSSM is then

$L_{\text{SSSM}}=\frac{1}{2}g^{\mu\nu}\operatorname{\text{tr}}\left[j^{(2)}_{\mu}\,j^{(2)}_{\nu}\right]=-\frac{1}{2}\operatorname{\text{tr}}\left[j^{(2)}_{+}\,j^{(2)}_{-}\right].$
(87)

This is immediately invariant under (local) transformations of $H$ from the
right, as well as under global $G$ transformations from the left. To see the
left $G$-invariance it is sufficient to write

$\operatorname{\text{tr}}\left[j^{(2)}_{\mu}\,j^{(2)}_{\nu}\right]=\operatorname{\text{tr}}\left[g^{-1}\partial_{\mu}g\,P^{(2)}\,g^{-1}\partial_{\nu}g\right]\,,$
(88)

where $P^{(2)}$ projects onto $\mathfrak{g}_{2}$. This expression is
manifestly invariant under $g\to g^{\prime}=g_{0}\,g$. The equations of motion
are equivalent to the covariant conservation of $j^{(2)}_{\mu}$

$g^{\mu\nu}D_{\mu}\,j^{(2)}_{\nu}=0\,,$ (89)

see Appendix A.2. The appearance of the covariant derivative is natural to
ensure symmetry with respect to $H$. Note that now $j^{(2)}_{\mu}$ is
(covariantly) conserved but, unlike the case of the PCM, it is not flat —
unlike $j_{\mu}$, which still satisfies the Maurer-Cartan equation

$0=\partial_{[+}j_{-]}+[j_{+},j_{-}]=\underbrace{F_{+-}^{(0)}+\left[j^{(2)}_{+},j^{(2)}_{-}\right]}_{\mathfrak{g}_{0}}+\underbrace{D_{+}j^{(2)}_{-}-D_{-}j^{(2)}_{+}}_{\mathfrak{g}_{2}}\,.$
(90)

Note that we decomposed the right-hand side under $\mathbb{Z}_{2}$, and that
the two underbraced terms must separately vanish.

For this model it is possible to define a Lax connection

$\mathfrak{L}_{\pm}=j^{(0)}_{\pm}+\frac{z\mp 1}{z\pm 1}j^{(2)}_{\pm}\,.$ (91)

Let us sketch the computation of the flatness condition, in preparation for
the case of the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformed model. We have

$\displaystyle 0$
$\displaystyle=\partial_{[+}\mathfrak{L}_{-]}+\left[\mathfrak{L}_{+},\mathfrak{L}_{-}\right]$
(92)
$\displaystyle=F^{(0)}_{+-}+\left[j^{(2)}_{+},j^{(2)}_{-}\right]+\frac{(z+1)^{2}D_{+}j^{(2)}_{-}-(z-1)^{2}D_{-}j^{(2)}_{+}}{z^{2}-1}\,.$

We see that the $z$-independent part vanishes due to the
$\mathfrak{g}_{0}$-projection of the Maurer-Cartan equation (90), while the
remaining terms arrange themselves into the $\mathfrak{g}_{1}$-projection of
the Maurer-Cartan equation and the equations of motion (89), which is
precisely what we need.

###  4.2 Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation of the SSSM

The SSSM Lagrangian (87) is again of the form $L(x_{0},x_{1},x_{2})$ (see
Section 2) with $x_{0}=0$ and

$x_{1}=-\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{-}]\,,\qquad
x_{2}=\frac{1}{2}\left(\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{+}]\operatorname{\text{tr}}[j^{(2)}_{-}j^{(2)}_{-}]+\left(\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{-}]\right)^{2}\right)\,,$
(93)

which is the same form as (43). As a result, the deformed Lagrangian
$L_{\text{SSSM}}^{(\gamma)}$ takes the same functional form as in (44), that
is

$L_{\text{SSSM}}^{(\gamma)}=\frac{1}{2}\left(-\operatorname{\text{ch}}(\gamma)\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{-}]+\operatorname{\text{sh}}(\gamma)\sqrt{\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{+}]\operatorname{\text{tr}}[j^{(2)}_{-}j^{(2)}_{-}]}\right).$
(94)

This Lagrangian is manifestly invariant under local $H$ transformations as
well as under global $G$ transformations from the left $g^{\prime}=g_{0}g$,
with $g_{0}$ a constant element of $G$. The equations of motion can be derived
as in Appendix A.2, and they read

$D_{\mu}\mathfrak{J}_{(2)}^{\mu}=0\,,\qquad\mathfrak{J}^{(2)}_{\mu}=2\frac{\partial
L}{\partial x_{1}}j^{(2)}_{\mu}+4g^{\nu\rho}\frac{\partial L}{\partial
x_{2}}\operatorname{\text{tr}}[j^{(2)}_{\mu}j^{(2)}_{\nu}]\,j^{(2)}_{\rho}\,.$
(95)

By analogy with the discussion of Section 3.2, it is natural to define the Lax
connection

$\mathfrak{L}_{\pm}^{(\gamma)}=j^{(0)}_{\pm}+\frac{(z^{2}+1)\,j^{(2)}_{\pm}\pm
2z\,\mathfrak{J}_{\pm}^{(2)}}{z^{2}-1}\,,$ (96)

which reduces to $\mathfrak{L}_{\pm}$ in the undeformed model where
$\mathfrak{J}_{\pm}^{(2)}=j_{\pm}^{(2)}$.

To check the flatness condition, let us start by evaluating more explicitly
$\mathfrak{J}^{(2)}_{\mu}$. This takes the same form as in (61),

$\displaystyle\mathfrak{J}^{(2)}_{+}$ $\displaystyle=2\left(\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial x_{2}}\right)j^{(2)}_{+}-2\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{2}}\operatorname{\text{tr}}[j^{(2)}_{+}j^{(2)}_{+}]j^{(2)}_{-}\,,$ (97)
$\displaystyle\mathfrak{J}^{(2)}_{-}$ $\displaystyle=2\left(\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial x_{2}}\right)j^{(2)}_{-}-2\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{2}}\operatorname{\text{tr}}[j^{(2)}_{-}j^{(2)}_{-}]j^{(2)}_{+}\,.$

Like in the case of PCM, we note the analogue of the identities (49) and (53),
namley

$\big{[}\mathfrak{J}^{(2)}_{\pm},j^{(2)}_{\mp}\big{]}=2\left(\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial x_{1}}+x_{1}\frac{\partial
L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{2}}\right)\big{[}j^{(2)}_{\pm},j^{(2)}_{\mp}\big{]},$ (98)

so that in particular

$\big{[}\mathfrak{J}^{(2)}_{+},j^{(2)}_{-}\big{]}-\big{[}j^{(2)}_{+},\mathfrak{J}^{(2)}_{-}\big{]}=0\,.$
(99)

Additionally

$\big{[}\mathfrak{J}^{(2)}_{+},\mathfrak{J}^{(2)}_{-}\big{]}=\big{[}j^{(2)}_{+},j^{(2)}_{-}\big{]}\,,$
(100)

which can be checked by using that $L_{\text{SSSM}}^{(\gamma)}$ obeys the
differential equation

$4\left[\left(\frac{\partial L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{1}}+x_{1}\frac{\partial L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{2}}\right)^{2}-\left(\frac{\partial L_{\text{SSSM}}^{(\gamma)}}{\partial
x_{2}}\right)^{2}\big{(}2x_{2}-(x_{1})^{2}\big{)}\right]=1\,,$ (101)

exactly as in (65). The same differential equation is satisfied by the doubly-
deformed Lagrangian $L_{SSSM}^{(\lambda,\gamma)}$ of (33), so this entire
argument can be repeated for the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformed SSSM, defining
the currents (97) appropriately (see Section 4.5 below). Using these
observations we find that

$\displaystyle 0$
$\displaystyle=\partial_{[+}\mathfrak{L}_{-]}+\left[\mathfrak{L}_{+},\mathfrak{L}_{-}\right]$
(102)
$\displaystyle=F^{(0)}_{+-}+\left[j^{(2)}_{+},j^{(2)}_{-}\right]+\frac{z^{2}+1}{z^{2}-1}\left(D_{+}j^{(2)}_{-}-D_{-}j^{(2)}_{+}\right)+\frac{2z}{z^{2}-1}\left(D_{+}j^{(2)}_{-}+D_{-}j^{(2)}_{+}\right)\,,$

where the various pieces reproduce the Maurer-Cartan equation (more precisely,
its $\mathfrak{g}_{0}$ and $\mathfrak{g}_{2}$ components), and the equations
of motion.

###  4.3 Semi-Symmetric-Space Sigma Model (sSSSM)

In the sSSSM, $G$ is a supergroup with superalgebra $\mathfrak{g}$, and it
enjoys a $\mathbb{Z}_{4}$ decomposition of the form (82). We consider the
coset $G/H$ with $\mathfrak{h}=\mathfrak{g}_{0}$. Accordingly, we decompose
the pull-back of the Maurer-Cartan form as

$j_{\mu}=j_{\mu}^{(0)}+j_{\mu}^{(1)}+j_{\mu}^{(2)}+j_{\mu}^{(3)}\,,\qquad
j_{\mu}^{(n)}\in\mathfrak{g}_{n}.$ (103)

The Maurer-Cartan equation itself can be decomposed as

$\mathfrak{g}_{n}:\qquad
0=\partial_{+}j^{(n)}_{-}-\partial_{-}j^{(n)}_{+}+\sum_{j=0}^{3}\left[j_{+}^{(j)},j_{-}^{(n-j)}\right]\,.$
(104)

for $n=0,\dots,3$, where the indices of the current are understood to take
values modulo 4. It is convenient to highlight the special role played by
$\mathfrak{h}=\mathfrak{g}_{0}$ by expressing the Maurer-Cartan equation in
terms of the covariant derivative (85) and field strength (86),101010Recall
that in our conventions $\varepsilon^{+-}=-\varepsilon^{-+}=\tfrac{1}{2}$,
while $g^{+-}=g^{-+}=-\tfrac{1}{2}$ and $g_{+-}=g_{-+}=-2$.

$\displaystyle\mathfrak{g}_{0}:$ $\displaystyle\qquad
0=F_{+-}^{(0)}+\left[j_{+}^{(2)},j_{-}^{(2)}\right]+2\varepsilon^{\mu\nu}\left[j_{\mu}^{(1)},j_{\nu}^{(3)}\right],$
(105) $\displaystyle\mathfrak{g}_{1}:$ $\displaystyle\qquad
0=D_{+}j^{(1)}_{-}-D_{-}j^{(1)}_{+}+2\varepsilon^{\mu\nu}\left[j_{\mu}^{(2)},j_{\nu}^{(3)}\right],$
$\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
0=D_{+}j^{(2)}_{-}-D_{-}j^{(2)}_{+}+\varepsilon^{\mu\nu}\left(\left[j_{\mu}^{(1)},j_{\nu}^{(1)}\right]+\left[j_{\mu}^{(3)},j_{\nu}^{(3)}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$ $\displaystyle\qquad
0=D_{+}j^{(3)}_{-}-D_{-}j^{(3)}_{+}+2\varepsilon^{\mu\nu}\left[j_{\mu}^{(2)},j_{\nu}^{(1)}\right].$

The Lagrangian of the sSSSM can be written as Metsaev:1998it 111111The two
terms in the sum are separately $\mathbb{Z}_{4}$-invariant. The model is only
integrable if their relative coefficient is $\pm 1$; this is also the only
choice that guarantees $\kappa$-symmetry when one wants to interpret this
models as a string theory Arutyunov:2009ga .

$L_{\text{sSSSM}}=\frac{1}{2}\text{str}\left[g^{\mu\nu}j^{(2)}_{\mu}j^{(2)}_{\nu}+\varepsilon^{\mu\nu}j^{(1)}_{\mu}j^{(3)}_{\nu}\right],$
(106)

which is written in terms of the supertrace. The equations of motion then take
the form (see Appendix A.3)

$\displaystyle\mathfrak{g}_{1}:$
$\displaystyle\qquad\varepsilon^{\mu\nu}D_{\mu}j^{(1)}_{\nu}=\left(\varepsilon^{\mu\nu}-2g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(3)}_{\nu}\right],$
(107) $\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
g^{\mu\nu}D_{\mu}j^{(2)}_{\nu}=\frac{1}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$
$\displaystyle\qquad\varepsilon^{\mu\nu}D_{\mu}j^{(3)}_{\nu}=\left(\varepsilon^{\mu\nu}+2g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(1)}_{\nu}\right].$

These expressions can further be simplified by using (105), giving

$\displaystyle\mathfrak{g}_{1}:$ $\displaystyle\qquad 0=$
$\displaystyle\left(\varepsilon^{\mu\nu}-g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(3)}_{\nu}\right],$
(108) $\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
g^{\mu\nu}D_{\mu}j^{(2)}_{\nu}=$
$\displaystyle\,\frac{1}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$ $\displaystyle\qquad 0=$
$\displaystyle\left(\varepsilon^{\mu\nu}+g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(1)}_{\nu}\right].$

The Lax connection is Bena:2003wd

$\mathfrak{L}_{\pm}=j^{(0)}_{\pm}+\sqrt{\frac{z+1}{z-1}}\,j^{(1)}_{\pm}+\frac{z\mp
1}{z\pm 1}\,j^{(2)}_{\pm}+\sqrt{\frac{z-1}{z+1}}\,j^{(3)}_{\pm}\,.$ (109)

The flatness condition for $\mathfrak{L}_{\pm}$ can then be expressed over the
components $\mathfrak{g}_{n}$, yielding:

$\displaystyle\mathfrak{g}_{0}:$ $\displaystyle\qquad
0=F_{+-}^{(0)}+\left[j_{+}^{(2)},j_{-}^{(2)}\right]+2\varepsilon^{\mu\nu}\left[j_{\mu}^{(1)},j_{\nu}^{(3)}\right],$
(110) $\displaystyle\mathfrak{g}_{1}:$ $\displaystyle\qquad
0=D_{+}j^{(1)}_{-}-D_{-}j^{(1)}_{+}+\frac{(z-1)^{2}}{(z+1)^{2}}\left[j_{+}^{(2)},j_{-}^{(3)}\right]+\left[j_{+}^{(3)},j_{-}^{(2)}\right],$
$\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
0=\frac{z+1}{z-1}\left(D_{+}j^{(2)}_{-}+\left[j_{+}^{(1)},j_{-}^{(1)}\right]\right)-\frac{z-1}{z+1}\left(D_{-}j^{(2)}_{+}-\left[j_{+}^{(3)},j_{-}^{(3)}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$ $\displaystyle\qquad
0=D_{+}j^{(3)}_{-}-D_{-}j^{(3)}_{+}+\left[j_{+}^{(2)},j_{-}^{(1)}\right]+\frac{(z+1)^{2}}{(z-1)^{2}}\left[j_{+}^{(1)},j_{-}^{(2)}\right].$

We see that the $\mathfrak{g}_{0}$ component matches with the 0-th component
of the Maurer-Cartan equation. The $\mathfrak{g}_{1}$ component, evaluted at
$z=0$, gives the corresponding component of the Maurer-Cartan equation, while
its residue at the double-pole gives the equations of motion (108); similarly
for the $\mathfrak{g}_{3}$ component. Finally, evaluating the
$\mathfrak{g}_{2}$ component at $z=0$ and $z=i$ gives the Maurer-Cartan and
equations of motion, respectively.

###  4.4 Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation of the sSSSM

The action of the sSSSM is still of the form $L(x_{0},x_{1},x_{2})$. Moreover,
$x_{1}$ and $x_{2}$ take the same form as in the (bosonic) SSSM if we express
them in terms of $j^{(2)}_{\mu}$ up to exchaging the trace for the supertrace,

$x_{1}=-\text{str}\left[j^{(2)}_{+}j^{(2)}_{-}\right]\,,\qquad
x_{2}=\frac{1}{2}\left(\text{str}\left[j^{(2)}_{+}j^{(2)}_{+}\right]\text{str}\left[j^{(2)}_{-}j^{(2)}_{-}\right]+\left(\text{str}\left[j^{(2)}_{+}j^{(2)}_{-}\right]\right)^{2}\right)\,,$
(111)

see also eq. (93). The main new ingredient is that here $x_{0}\neq 0$, and
instead

$x_{0}=\frac{1}{2}\varepsilon^{\mu\nu}\text{str}\left[j^{(1)}_{\mu}\,j^{(3)}_{\nu}\right].$
(112)

Still, the deformed Lagrangian $L_{\text{sSSSM}}^{(\gamma)}$ can be easily
found from (25), and the doubly-deformed version
$L_{\text{sSSSM}}^{(\lambda,\gamma)}$ from (33). The equations of motion of
the deformed model can be expressed in terms of

$\mathfrak{J}^{(2)}_{\mu}=2\frac{\partial
L_{\text{sSSSM}}^{(\gamma)}}{\partial x_{1}}j^{(2)}_{\mu}+4\frac{\partial
L_{\text{sSSSM}}^{(\gamma)}}{\partial
x_{2}}\,g^{\nu\rho}\text{str}\left[j^{(2)}_{\mu}j^{(2)}_{\nu}\right]j^{(2)}_{\rho}\,,$
(113)

which generalises the expression in eq. (95). Note however that the form of
the equations of motion is rather more complicated now, due to the
contributions from the odd currents in $x_{0}$. We refer the reader to
appendix A.3 for their derivation. They are

$\displaystyle\mathfrak{g}_{1}:$ $\displaystyle\qquad 0=$
$\displaystyle\left[\left(\varepsilon^{\mu\nu}j^{(2)}_{\mu}-g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(3)}_{\nu}\right],$
(114) $\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
g^{\mu\nu}D_{\mu}\mathfrak{J}^{(2)}_{\nu}=$
$\displaystyle\,\frac{1}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$ $\displaystyle\qquad 0=$
$\displaystyle\left[\left(\varepsilon^{\mu\nu}j^{(2)}_{\mu}+g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(1)}_{\nu}\right],$

and it is immediate to see that they reduce to (108) in the undeformed limit
when $\mathfrak{J}^{(2)}_{\mu}=j^{(2)}_{\mu}$. We now define the Lax
connection

$\mathfrak{L}_{\pm}^{(\gamma)}=j^{(0)}_{\pm}+\sqrt{\frac{z+1}{z-1}}\,j^{(1)}_{\pm}+\frac{\left(z^{2}+1\right)\,j^{(2)}_{\pm}\mp
2z\,\mathfrak{J}^{(2)}_{\pm}}{z^{2}-1}+\sqrt{\frac{z-1}{z+1}}\,j^{(3)}_{\pm}\,.$
(115)

Clearly the manipulations involving terms which do not involve $j^{(1)}_{\mu}$
or $j^{(3)}_{\mu}$ go through exactly like for the SSSM. In particular, we can
still use equations (99)–(100). The flatness condition for
$\mathfrak{L}_{\pm}^{(\gamma)}$ then becomes121212Recall that in our
conventions $A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\nu}$.

$\displaystyle\mathfrak{g}_{0}:$ $\displaystyle\
0=F_{+-}^{(0)}+\left[j_{+}^{(2)},j_{-}^{(2)}\right]+2\varepsilon^{\mu\nu}\left[j_{\mu}^{(1)},j_{\nu}^{(3)}\right],$
(116) $\displaystyle\mathfrak{g}_{1}:$ $\displaystyle\
0=D^{\phantom{|}}_{[+}j^{(1)}_{-]}+\left[\frac{(z^{2}+1)j_{+}^{(2)}-2z\mathfrak{J}_{+}^{(2)}}{(z+1)^{2}},j_{-}^{(3)}\right]+\left[j_{+}^{(3)},\frac{(z^{2}+1)j_{+}^{(2)}+2z\mathfrak{J}_{-}^{(2)}}{(z+1)^{2}}\right],$
$\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\
0=\frac{z^{2}+1}{z^{2}-1}D^{\phantom{|}}_{[+}j^{(2)}_{-]}+\frac{2z}{z^{2}-1}D^{\phantom{|}}_{(+}\mathfrak{J}^{(2)}_{-)}+\frac{z+1}{z-1}\left[j_{+}^{(1)},j_{-}^{(1)}\right]+\frac{z-1}{z+1}\left[j_{+}^{(3)},j_{-}^{(3)}\right],$
$\displaystyle\mathfrak{g}_{3}:$ $\displaystyle\
0=D^{\phantom{|}}_{[+}j^{(3)}_{-]}+\left[\frac{(z^{2}+1)j_{+}^{(2)}-2z\mathfrak{J}_{+}^{(2)}}{(z-1)^{2}},j_{-}^{(3)}\right]+\left[j_{+}^{(3)},\frac{(z^{2}+1)j_{-}^{(2)}+2z\mathfrak{J}_{-}^{(2)}}{(z-1)^{2}}\right].$

We now require these equations, in particular the last three, to hold for any
$z\in\mathbb{C}$. We can see that the $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{3}$ components give the components of the Maurer-Cartan
equations at $z=0$, and give the equations of motion when taking the residues
of the double poles. The component along $\mathfrak{g}_{2}$ gives the Maurer-
Cartan equation for $z=0$ and the equations of motion for $z=\pm i$. This
shows the equivalence of the flatness condition for
$\mathfrak{L}_{\pm}^{(\gamma)}$ with the equations of motion.

###  4.5 sSSSM with Wess-Zumino Term

For certain semi-symmetric superspaces, it is possible to add a non-trivial
Wess-Zumino coupling while preserving integrability Cagnazzo:2012se . As
before, let $G$ be a supergroup with Lie superalgebra $\mathfrak{g}$. If $G$
is a simple supergroup, the candidate Wess-Zumino term vanishes identically,
so we will not consider models of this form in the current section. However, a
Wess-Zumino coupling exists and is non-trivial if $G$ is a direct product
$G=G^{\prime}\times G^{\prime}$ where $G^{\prime}$ is another supergroup. For
the present discussion, we will assume that $G$ admits such a direct product
structure. In particular, this means that
$\mathfrak{g}=\mathfrak{g}^{\prime}\oplus\mathfrak{g}^{\prime}$, and such a
Lie algebra always admits a $\mathbb{Z}_{4}$ action given by the semi-graded
permutation of the two factors. This choice of grading goes under the name of
permutation cosets, see Cagnazzo:2012se  for further details. We use this
structure to decompose
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\mathfrak{g}_{3}$
as in equation (82). We again decompose the current $j_{\mu}$ using the
grading of $\mathfrak{g}$ as

$j_{\mu}=j_{\mu}^{(0)}+j_{\mu}^{(1)}+j_{\mu}^{(2)}+j_{\mu}^{(3)}\,,\qquad
j_{\mu}^{(n)}\in\mathfrak{g}_{n}\,.$ (117)

In terms of the components in this decomposition, the action for the semi-
symmetric space sigma model with Wess-Zumino term takes the form
Cagnazzo:2012se

$\displaystyle\mathcal{S}_{\text{SMWZ}}$
$\displaystyle=\hay\int\limits_{\partial
B}\text{d}^{2}\sigma\frac{1}{2}\operatorname{\text{str}}\left(g^{\mu\nu}j_{\mu}^{(2)}j_{\nu}^{(2)}+\ell\epsilon^{\mu\nu}j_{\mu}^{(1)}j_{\nu}^{(3)}\right)$
(118)
$\displaystyle\qquad+\kay\int\limits_{B}\text{d}^{3}\sigma\frac{1}{3}\varepsilon^{ijk}\operatorname{\text{str}}\left(j^{(2)}_{i}[j^{(2)}_{j},j^{(2)}_{k}]+3j^{(1)}_{i}[j^{(3)}_{j},j^{(2)}_{k}]\right)\,.$

A few comments about this action are in order. The first is that, although the
current $j_{\mu}$ itself admits a decomposition according to the grading of
$\mathfrak{g}$, the supertrace is not compatible with this new
$\mathbb{Z}_{4}$ grading. In particular, one may have a non-vanishing
supertrace $\operatorname{\text{str}}(\mathfrak{g}_{m}\mathfrak{g}_{n})$ not
only if131313In Section 4.3, for simplicity, when deriving the equations of
motion we were implicitly assuming compatibility of the supertrace with the
$\mathbb{Z}_{4}$ grading, i.e.
$\operatorname{\text{str}}(\mathfrak{g}_{m}\mathfrak{g}_{n})$ only if
$(m+n)\text{ mod }4=0$. See also appendix A.3. However, it is worth stressing
that the final expression for the equations of motion is the same even when
relaxing this assumption, thanks to the fact that the Lagrangian is given by
the supertrace of an element of grading 0. $(m+n)\text{ mod }4=0$ but also if
$(m+n)\text{ mod }4=2$. This is in fact reflected in the Wess-Zumino term on
the second line of (118), which _does not_ respect the $\mathbb{Z}_{4}$
grading. In particular, this means that the equations of motion for this
theory will mix terms from different subspaces $\mathfrak{g}_{n}$, so that we
can no longer decompose equations according to the grading and demand that the
components in each subspace vanish separately. However, it is important to
remark that the the second line of (118) is chosen so that it gives rise to a
well-defined WZ term, whose variation reduces to a 2-dimensional integral.

The second comment is that we have introduced an additional parameter $\ell$
which controls the relative scaling between the sigma model term and the odd-
current term. Taking $\kay\to 0$ and $\ell\to 1$ recovers the sSSSM with the
usual conventions, up to the overall scaling of the action controlled by .
Although we fixed $\ell=1$ in the preceding discussion, we will see that it is
not consistent to do so when we include the Wess-Zumino term, and in fact the
parameters are required to satisfy a constraint
$\ell^{2}=1-\frac{{}^{2}}{{}^{2}}$ in order to maintain integrability.141414In
the case where the sigma model should be interpreted as a string theory, this
requirement can be understood as enforcing $\kappa$-symmetry Cagnazzo:2012se .

The equations of motion arising from (118) can be written as

$\displaystyle 0$
$\displaystyle=\left(\ell\varepsilon^{\mu\nu}-g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(3)}_{\nu}\right]-\frac{\kay}{\hay}\varepsilon^{\mu\nu}[j^{(2)}_{\mu},j^{(1)}_{\nu}],$
(119) $\displaystyle g^{\mu\nu}D_{\mu}j^{(2)}_{\nu}$
$\displaystyle=\frac{\ell}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right)+\frac{\kay}{\hay}\varepsilon^{\mu\nu}\left([j^{(1)}_{\mu},j^{(3)}_{\nu}]+[j^{(2)}_{\mu},j^{(2)}_{\nu}]\right),$
$\displaystyle 0$
$\displaystyle=\left(\ell\varepsilon^{\mu\nu}+g^{\mu\nu}\right)\left[j^{(2)}_{\mu},j^{(1)}_{\nu}\right]+\frac{\kay}{\hay}\varepsilon^{\mu\nu}[j_{\mu}^{(2)},j_{\nu}^{(3)}],$

and we can see that the terms proportional to $\frac{\kay}{\hay}$ explicitly
break the grading. However, the Maurer-Cartan identity for $j_{\mu}$ is
unchanged, so we may still use the same equations (105) as in discussion of
the sSSSM without WZ term.

The Lax connection for this model takes the form Cagnazzo:2012se

$\displaystyle\mathfrak{L}_{\pm}$
$\displaystyle=j_{\pm}^{(0)}+\ell\frac{z^{2}+1}{z^{2}-1}j_{\pm}^{(2)}\pm\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)j_{\pm}^{(2)}+\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\frac{\ell\left(1-\frac{\kay}{\hay}\right)}{z^{2}-1}}j_{\pm}^{(1)}$
(120)
$\displaystyle\quad+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\frac{\ell\left(1+\frac{\kay}{\hay}\right)}{z^{2}-1}}j_{\pm}^{(3)}\,.$

Although the equations of motion (and thus the flatness condition for the Lax
connection) do not respect the $\mathbb{Z}_{4}$ grading of $\mathfrak{g}$,
they of course still respect the $\mathbb{Z}_{2}$ grading of the algebra into
bosonic and fermionic subalgebras $\mathfrak{g}_{\text{B}}$ and
$\mathfrak{g}_{\text{F}}$. Projecting the equation
$\partial_{+}\mathfrak{L}_{-}-\partial_{-}\mathfrak{L}_{+}+[\mathfrak{L}_{+},\mathfrak{L}_{-}]=0$
onto these subspaces gives the equations

$\displaystyle\mathfrak{g}_{\text{B}}:\;0$
$\displaystyle=F_{+-}^{(0)}+\left(\ell\frac{z-1}{z+1}+\frac{\kay}{\hay}\right)\left(\ell\frac{z+1}{z-1}-\frac{\kay}{\hay}\right)[j^{(2)}_{+},j^{(2)}_{-}]$
(121) $\displaystyle\
+\left(\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\frac{\ell\sqrt{1-\frac{{}^{2}}{{}^{2}}}}{z^{2}-1}\right)\left([j_{+}^{(1)},j_{-}^{(3)}]+[j_{+}^{(3)},j_{-}^{(1)}]\right)$
$\displaystyle\
+\left(\ell\frac{z+1}{z-1}-\frac{\kay}{\hay}\right)D_{+}j_{-}^{(2)}-\left(\ell\frac{z-1}{z+1}+\frac{\kay}{\hay}\right)D_{-}j_{+}^{(2)}$
$\displaystyle\
+\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)^{2}\frac{\ell\left(1-\frac{\kay}{\hay}\right)}{z^{2}-1}[j_{+}^{(1)},j_{-}^{(1)}]+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)^{2}\frac{\ell\left(1+\frac{\kay}{\hay}\right)}{z^{2}-1}[j_{+}^{(3)},j_{-}^{(3)}]\,.$

and

$\displaystyle\mathfrak{g}_{\text{F}}:\;0$
$\displaystyle=\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}\left(D_{+}j_{-}^{(1)}-D_{-}j_{+}^{(1)}\right)$
(122) $\displaystyle\
+\left(\ell\frac{z-1}{z+1}+\frac{\kay}{\hay}\right)\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}[j_{+}^{(2)},j_{-}^{(3)}]$
$\displaystyle\
+\left(\ell\frac{z+1}{z-1}-\frac{\kay}{\hay}\right)\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}[j_{+}^{(3)},j_{-}^{(2)}]$
$\displaystyle\quad+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}\left(D_{+}j_{-}^{(3)}-D_{-}j_{+}^{(3)}\right)$
$\displaystyle\quad+\left(\ell\frac{z-1}{z+1}+\frac{\kay}{\hay}\right)\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}[j_{+}^{(2)},j_{-}^{(1)}]$
$\displaystyle\quad+\left(\ell\frac{z+1}{z-1}-\frac{\kay}{\hay}\right)\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}[j_{+}^{(1)},j_{-}^{(2)}]\,.$

By substituting the equations of motion (119), as well as the Maurer-Cartan
identities (105) into the right sides of equations (121) and (122), one finds
that the resulting expressions vanish identically if and only if

$\displaystyle\ell^{2}=1-\frac{{}^{2}}{{}^{2}}\,,$ (123)

as we mentioned above. When the parameters are related in this way, the
equations of motion and flatness of $j_{\mu}$ imply the flatness of the Lax
connection for any value of $z$. The converse is also true, as one can show by
evaluating the flatness conditions at several values of $z$ to obtain several
independent equations and then solving the resulting system. For instance, one
can obtain three independent equations from evaluating the bosonic projection
(121) at $z=0$ and extracting the two residues at $z=1$ and at $z=-1$. These
three equations can be solved simultaneously to yield expressions for
$F_{+-}$, $D_{+}j^{(2)}_{-}$, and $D_{-}j_{+}^{(2)}$ which are equivalent to
the $\mathfrak{g}_{0}$ and $\mathfrak{g}_{2}$ parts of the Maurer-Cartan
identity, along with the equation of motion for $j_{\mu}^{(2)}$. Similarly,
one can extract four independent equations from the fermionic projection (122)
by computing the two residues at $z=\pm 1$ and then evaluating the equation at
two other values of $z$, for instance $z=0$ and
$z=\ell/(1+\frac{\kay}{\hay})$. The resulting system of four equations is
equivalent to the $\mathfrak{g}_{1}$ and $\mathfrak{g}_{3}$ components of the
Maurer-Cartan identity, along with the two equations of motion for
$j_{\mu}^{(1)}$ and $j_{\mu}^{(3)}$ on the first and third lines of (119).
This shows that the flatness of the Lax connection for any $z$ implies the
equations of motion and the Maurer-Cartan identities, which completes the
second part of the equivalence.

###  4.6 Two-Parameter Deformation of the sSSSM with WZ Term

We now consider a two-parameter family of models which arise from deforming
the semi-symmetric space sigma model action (119) by both $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$. Our Lagrangian
$L_{(\lambda,\gamma)}$ satisfies the two simultaneous flow equations in
equation (28), which are written in terms of the Hilbert stress tensor of the
theory. It is easy to see from the definition of the Hilbert stress tensor,

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{S}}{\delta
g^{\mu\nu}}\,,$ (124)

that the only term in (118) which will contribute to $T_{\mu\nu}$ is the first
term which involves $g^{\mu\nu}j_{\mu}^{(2)}j_{\nu}^{(2)}$. The two remaining
terms are written in terms of $\varepsilon^{\mu\nu}$ and $\varepsilon^{ijk}$,
respectively, and therefore do not couple to the metric either via an explicit
metric contraction nor through implicit dependence on the measure. They are
topological and they are not deformed along our combined $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow.

The deformed action can be written as

$\displaystyle\mathcal{S}_{\text{SMWZ}}^{(\lambda,\gamma)}$
$\displaystyle=\hay\int\limits_{\partial
B}\text{d}^{2}\sigma\frac{1}{2}\left(L_{\text{SMWZ}}^{(\lambda,\gamma)}(x_{1},x_{2})+\ell\,\operatorname{\text{str}}\left(\varepsilon^{\mu\nu}j_{\mu}^{(1)}j_{\nu}^{(3)}\right)\right)$
(125)
$\displaystyle\qquad+\kay\int\limits_{B}\text{d}^{3}\sigma\frac{1}{3}\varepsilon^{ijk}\operatorname{\text{str}}\left(j^{(2)}_{i}[j^{(2)}_{j},j^{(2)}_{k}]+3j^{(1)}_{i}[j^{(3)}_{j},j^{(2)}_{k}]\right)\,.$

where

$\displaystyle
x_{1}=g^{\mu\nu}\operatorname{\text{str}}\left(j_{\mu}^{(2)}j_{\nu}^{(2)}\right),\qquad
x_{2}=g^{\mu\rho}g^{\nu\sigma}\operatorname{\text{str}}\left(j_{\mu}^{(2)}j_{\nu}^{(2)}\right)\operatorname{\text{str}}\left(j_{\rho}^{(2)}j_{\sigma}^{(2)}\right),$
(126)

as before, and $L_{\text{SMWZ}}^{(\lambda,\gamma)}(x_{1},x_{2})$ is the same
Lagrangian (33) that we considered above. The only feature of this Lagrangian
which we will need for the present analysis is that the corresponding current

$\mathfrak{J}^{(2)}_{\mu}=2\frac{\partial
L_{\text{SMWZ}}^{(\lambda,\gamma)}}{\partial
x_{1}}j^{(2)}_{\mu}+4g^{\nu\rho}\frac{\partial
L_{\text{SMWZ}}^{(\lambda,\gamma)}}{\partial
x_{2}}\operatorname{\text{tr}}[j^{(2)}_{\mu}j^{(2)}_{\nu}]\,j^{(2)}_{\rho}\,,$
(127)

built from $L_{\text{SMWZ}}^{(\lambda,\gamma)}(x_{1},x_{2})$ (where $x_{0}$
does not play a role) satisfies

$\big{[}\mathfrak{J}^{(2)}_{+},j^{(2)}_{-}\big{]}-\big{[}j^{(2)}_{+},\mathfrak{J}^{(2)}_{-}\big{]}=0\,,\qquad\big{[}\mathfrak{J}^{(2)}_{+},\mathfrak{J}^{(2)}_{-}\big{]}=\big{[}j^{(2)}_{+},j^{(2)}_{-}\big{]}\,,$
(128)

as we saw before, cf. (99)–(100). In terms of this current, the equations of
motion for the deformed model are

$\displaystyle 0$
$\displaystyle=\left[\left(\ell\varepsilon^{\mu\nu}j^{(2)}_{\mu}-g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(3)}_{\nu}\right]-\frac{\kay}{\hay}\varepsilon^{\mu\nu}[j^{(2)}_{\mu},j^{(1)}_{\nu}],$
(129) $\displaystyle g^{\mu\nu}D_{\mu}\mathfrak{J}^{(2)}_{\nu}$
$\displaystyle=\frac{\ell}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right)+\frac{\kay}{\hay}\varepsilon^{\mu\nu}\left([j^{(1)}_{\mu},j^{(3)}_{\nu}]+[j^{(2)}_{\mu},j^{(2)}_{\nu}]\right),$
$\displaystyle 0$
$\displaystyle=\left[\left(\ell\varepsilon^{\mu\nu}j^{(2)}_{\mu}+g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(1)}_{\nu}\right]+\frac{\kay}{\hay}\varepsilon^{\mu\nu}[j_{\mu}^{(2)},j_{\nu}^{(3)}]\,.$

We now claim that these equations of motion, along with the Maurer-Cartan
identities, are equivalent to the flatness of the Lax connection

$\displaystyle\mathfrak{L}_{\pm}^{(\gamma)}$
$\displaystyle=j_{\pm}^{(0)}+\ell\frac{z^{2}+1}{z^{2}-1}j_{\pm}^{(2)}\pm\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}_{\pm}^{(2)}+\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\frac{\ell\left(1-\frac{\kay}{\hay}\right)}{z^{2}-1}}j_{\pm}^{(1)}$
$\displaystyle\quad+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\frac{\ell\left(1+\frac{\kay}{\hay}\right)}{z^{2}-1}}j_{\pm}^{(3)}\,,$

at any value of the spectral parameter $z$. As in the undeformed case, we will
decompose this flatness condition into its components along the bosonic and
fermionic subalgebras. In order to compute the commutator
$[\mathfrak{L}_{+},\mathfrak{L}_{-}]$, it is convenient to note that the
contribution arising from $j_{\pm}^{(2)}$ and $\mathfrak{J}_{\pm}^{(2)}$,
namely

$\displaystyle\left[\ell\frac{z^{2}+1}{z^{2}-1}j_{+}^{(2)}+\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{+}\,,\,\ell\frac{z^{2}+1}{z^{2}-1}j_{-}^{(2)}-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{-}\right]$
$\displaystyle\qquad=\left(\ell^{2}\left(\frac{z^{2}+1}{z^{2}-1}\right)^{2}-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)^{2}\right)[j_{+}^{(2)},j_{-}^{(2)}]\,,$ (130)

is identical to that of the undeformed theory, see eq. (121). Here we have
used the commutator property (127). The bosonic part of the flatness condition
for our candidate Lax connection is then

$\displaystyle\mathfrak{g}_{\text{B}}:\;0$
$\displaystyle=F_{+-}+\left(\ell^{2}\left(\frac{z^{2}+1}{z^{2}-1}\right)^{2}-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)^{2}\right)[j_{+}^{(2)},j_{-}^{(2)}]$ (131) $\displaystyle\
+\left(\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\frac{\ell\sqrt{1-\frac{{}^{2}}{{}^{2}}}}{z^{2}-1}\right)\left([j_{+}^{(1)},j_{-}^{(3)}]+[j_{+}^{(3)},j_{-}^{(1)}]\right)$
$\displaystyle\
+\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)\left(D_{+}j_{-}^{(2)}-D_{-}j_{+}^{(2)}\right)-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\left(D_{+}\mathfrak{J}^{(2)}_{-}+D_{-}\mathfrak{J}^{(2)}_{+}\right)$
$\displaystyle\
+\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)^{2}\frac{\ell\left(1-\frac{\kay}{\hay}\right)}{z^{2}-1}[j_{+}^{(1)},j_{-}^{(1)}]+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)^{2}\frac{\ell\left(1+\frac{\kay}{\hay}\right)}{z^{2}-1}[j_{+}^{(3)},j_{-}^{(3)}]\,.$

Likewise, the fermionic component of the flatness condition is

$\displaystyle\mathfrak{g}_{\text{F}}:\;0$
$\displaystyle=\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}\left(D_{+}j_{-}^{(1)}-D_{-}j_{+}^{(1)}\right)$
(132) $\displaystyle\
+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}\left[\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)j_{+}^{(2)}+\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{+},j_{-}^{(3)}\right]$ $\displaystyle\
+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}\left[j_{+}^{(3)},\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)j_{-}^{(2)}-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{-}\right]$ $\displaystyle\
+\left(z-\frac{\ell}{1+\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1+\frac{\kay}{\hay}\right)}\left(D_{+}j_{-}^{(3)}-D_{-}j_{+}^{(3)}\right)$
$\displaystyle\
+\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}\left[\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)j_{+}^{(2)}+\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{+},j_{-}^{(1)}\right]$ $\displaystyle\
+\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)\left(z+\frac{\ell}{1-\frac{\kay}{\hay}}\right)\sqrt{\ell\left(1-\frac{\kay}{\hay}\right)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left[j_{+}^{(1)},\left(\ell\frac{z^{2}+1}{z^{2}-1}\right)j_{-}^{(2)}-\left(\frac{\kay}{\hay}-\frac{2\ell
z}{z^{2}-1}\right)\mathfrak{J}^{(2)}_{-}\right].$

By an explicit calculation, substituting in the deformed equations of motion
(129) and the Maurer-Cartan identity (105), one finds that (131) and (132)
both vanish identically, assuming that we again impose the condition for
$\ell$ (123) precisely as in the undeformed theory. Conversely, one can obtain
four independent equations from extracting the residues of (131) and (132) at
$z=\pm 1$, two additional constraints from evaluating both equations at $z=0$,
and one more relation from evaluating the $\mathfrak{g}_{\text{F}}$ equation
at another value of $z$, for instance $z=\ell/(1+\frac{\kay}{\hay})$. This
gives a system of seven equations relating the various $j_{\mu}^{(n)}$,
$\mathfrak{J}_{\mu}^{(2)}$, and their derivatives. The solution to this system
satisfies the three equations of motion (129) and the four Maurer-Cartan
identities. Therefore, we see that the implication goes in both directions, so
the flatness of the Lax connection at any $z$ is equivalent to the equations
of motion and Maurer-Cartan identities, as claimed.

##  5 Conclusions and Outlook

In this paper we have shown that a large class of classical integrable models
still admit a Lax connection after an arbitrary combination of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and (irrelevant)
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations.
While this is not an indication of integrability for the deformation of all
classically integrable models, it is still a striking and non-trivial
statement about the interplay of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations with
symmetries. Moreover, having explicit Lax connections at our disposal for a
large class of models provides us with a powerful tool to study properties of
classical solutions in the deformed theory, such as the value of the conserved
charges under deformations.

There are many natural and pressing questions. Firstly, properly establishing
integrability at the classical level requires not only the existence of a Lax
connection, but also its compatibility with the Poisson structure of the
model. This is what allows us to construct charges that are not just conserved
under time-evolution, but also mutually Poisson-commuting Sklyanin:1980ij ;
Maillet:1985ec ; Maillet:1985ek . This is a natural step to truly establish
classical integrability of these models (this is sometimes called “strong
integrability”). Naturally, it would also be interesting to study the
classical integrability of even more general classes of theories such as Toda
field theories, see e.g. Corrigan:1994nd , or more general sigma models such
as the ones that arise as Yang-Baxter deformations of (semi-)symmetric sigma
models Klimcik:2008eq ; Delduc:2013qra ; Kawaguchi:2014qwa ;
vanTongeren:2015soa  (see also Seibold:2020ouf ). This would be a strong hint
of integrability in general, and provide an even larger playground for the
study of these deformations.

For the irrelevant $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformation, it is possible to generate solutions by recasting the flow
as a dynamical change of coordinates Conti:2018tca . It would be interesting
to see if a similar construction is possible here, as this might provide a way
to prove that Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ preserves integrability in general. It would also be interesting to
extend this construction to non-relativistic models and one-dimensional
(quantum-mechanical) systems, like it was done for $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, see Cardy:2018jho ;
Ceschin:2020jto ; Esper:2021hfq  and Marchetto:2019yyt ; Pozsgay:2019ekd ;
Gross:2019ach ; Gross:2019uxi ; Ebert:2022ehb , respectively. In fact, it was
recently shown that a Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of a simple
quantum-mechanical system preserves integrability too Garcia:2022wad . Again,
it would be important to understand whether this is a general feature of such
deformations.

An important outstanding question is how to quantise Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ theories (as well as
ModMax theories Bandos:2020jsw ; Bandos:2020hgy ; Bandos:2021rqy ;
Lechner:2022qhb ; Sorokin:2021tge ). It might be worth undertaking this study
on some particular examples which are integrable. For instance, we have seen
in Section 3.5 that the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of a
$G$-valued Wess-Zumino-Witten model is still classically integrable and
conformal, but it does not enjoy the conservation of two (anti-)chiral
$\mathfrak{g}$-currents. This suggests that, at the very least, the Kač-Moody
symmetry of the quantum model is modified — if the model remains conformal at
the quantum level, which is not obvious. It would be interesting to perform an
analysis similar to what was done for current-current deformations
Chaudhuri:1988qb ; Forste:2003km ; Borsato:2018spz  to determine the effect of
the deformation.151515Another direction worth exploring with the aim of
quantisation, following recent work on ModMax Lechner:2022qhb , would be to
understand the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ model by adding auxiliary fields, which should make the deformed action
analytic.

An immediate by-product of our construction is the observation that
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformations “play well” with symmetries. In particular, they manifestly
preserve the $G$-symmetries of the sigma models that we have considered. In
the case where $G$ is a supergroup, this means that they preserve
integrability and target-space supersymmetry. One of the crucial applications
of the semi-symmetric sigma models has been the study of string theory on
curved backgrounds — the most celebrated example being $AdS_{5}\times S^{5}$
Metsaev:1998it ; Bena:2003wd ; Arutyunov:2009ga , as well as other
$AdS_{n}\times M_{10-n}$ backgrounds Klose:2010ki ; Sfondrini:2014via
,161616In fact, the introduction of the WZ term in the semi-symmetric sigma
model by Cagnazzo:2012se  has been a key ingredient in the development of
integrability for $AdS_{3}/CFT_{2}$ Hoare:2013pma ; Lloyd:2014bsa . and
deformations thereof. Even if the deformed actions are not of the sigma-model
type, it would be very interesting to consider them as potential deformations
of a string action. In particular, they may still enjoy a local fermionic
symmetry inherited from the $\kappa$-symmetry of the original sigma-model,
which is usually related to the condition of integrability (see e.g.
Arutyunov:2009ga ; Cagnazzo:2012se ).171717It is worth remarking that, even
for sigma-model actions, $\kappa$-symmetry does not imply Weyl invariance of
the action Wulff:2016tju , see e.g. Arutyunov:2015qva ; Arutyunov:2015mqj ;
Borsato:2018idb ; Hoare:2018ngg  for some examples. Hence, it would actually
be interesting, though possibly harder, to check the stronger property of Weyl
invariance.  On top of the obvious physical interest of this study, this would
probably allow for the perturbative quantisation of the deformed models once a
suitable gauge is fixed. In fact, we would expect the gauge-fixed theory to
become analytic in the small-field expansion of the transverse modes of the
string (like it is the case for the Nambu-Goto action). In the case of
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation, a
similar analysis was performed in the uniform light-cone gauge
Arutyunov:2005hd ; Arutyunov:2009ga , see e.g. Baggio:2018gct ; Frolov:2019nrr
; Frolov:2019xzi ; Sfondrini:2019smd . For Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ this might offer a way,
however roundabout, to interpret the deformation at the level of the S matrix
(like it was done for the irrelevant $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation
Smirnov:2016lqw ; Cavaglia:2016oda ; Dubovsky:2017cnj ) and obtain some
information on the deformed spectrum via the thermodynamic Bethe ansatz
Zamolodchikov:1989cf ; Dorey:1996re  for a suitable mirror model
Arutyunov:2007tc . In a first approximation one could ask this question for
bosonic strings in light-cone gauge, but it would be quite intriguing to
understand it for $AdS$-type backgrounds, as it might have important
implications for their holographic duals.

Another natural question is whether these deformations “play well” also with
$(1+1)$-dimensional worldsheet supersymmetry. Encouragingly, it is possible to
supersymmetrise the ModMax model Bandos:2021rqy . Since the dimensional
reduction of ModMax yields a Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow Babaei-
Aghbolagh:2022uij ; Ferko:2022iru ; Conti:2022egv  with a particularly simple
seed action, one may suspect that supersymmetry persists for any
supersymmetric seed. This is certainly the case for the irrelevant $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation
Baggio:2018rpv ; Chang:2018dge ; Jiang:2019hux ; Chang:2019kiu ; Ferko:2019oyv
; Ferko:2021loo ; Ebert:2022xfh . In that case, it was useful to understand
the whole deformation in terms of superfields; it may be important to do the
same here.

All the above questions are very interesting in and of themselves, but they
also open the road to applying these deformations to stringy and holographic
setups (both on the worldsheet and in the dual CFT) and to a deeper
understanding of the ModMax theories Bandos:2020jsw ; Bandos:2020hgy ;
Bandos:2021rqy ; Lechner:2022qhb ; Sorokin:2021tge , to which Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations are
intimately connected.

## Acknowledgements

R.B. and A.S. would like to dedicate this work to Kurt Lechner, from whom they
first learned field theory.

We are grateful to Ben Hoare, Fiona Seibold, Dima Sorokin, Gabriele
Tartaglino-Mazzucchelli, and Roberto Tateo for discussions related to this
work.

We are grateful to the Kavli Institute for Theoretical Physics in Santa
Barbara for hosting two of us (RB & AS) during the Integrable22 workshop. The
stimulating atmosphere of the program led to the discussions that eventually
resulted in this work.

The work of RB was supported by the fellowship of “la Caixa Foundation” (ID
100010434) with code LCF/BQ/PI19/11690019, by AEI-Spain (under project
PID2020-114157GB-I00 and Unidad de Excelencia María de Maetzu MDM-2016-0692),
Xunta de Galicia (Centro singular de investigación de Galicia accreditation
2019-2022, and project ED431C-2021/14), and by the European Union FEDER. C. F.
is supported by U.S. Department of Energy grant DE-SC0009999 and by funds from
the University of California. This research was supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958. A. S.
acknowledges support from the European Union – NextGenerationEU, and from the
program STARS@UNIPD, under project “Exact-Holography”, A new exact approach to
holography: harnessing the power of string theory, conformal field theory, and
integrable models.

##  Appendix A Derivation of the Equations of Motion

We collect here the derivation of the equations for the (deformed) models of
our interest.

###  A.1 Deformed Principal Chiral Model

If we consider an infinitesimal variation of the group element $g\in G$ as
$\delta g=g\epsilon$ with $\epsilon\in\mathfrak{g}$, then the variation of the
Maurer-Cartan current reads $\delta
j_{\mu}=\partial_{\mu}\epsilon+[j_{\mu},\epsilon]$. Taking
$x_{1}=\operatorname{\text{tr}}(j_{\mu}j^{\mu})$ and
$x_{2}=\operatorname{\text{tr}}(j_{\mu}j_{\nu})\operatorname{\text{tr}}(j^{\mu}j^{\nu})$
(as done in (43)) one finds

$\displaystyle\delta x_{1}$
$\displaystyle=2\operatorname{\text{tr}}(j^{\mu}\delta
j_{\mu})=2\operatorname{\text{tr}}(j^{\mu}(\partial_{\mu}\epsilon+[j_{\mu},\epsilon]))=2\operatorname{\text{tr}}(j^{\mu}\partial_{\mu}\epsilon),$
(133)

where we used that the term with $[j_{\mu},\epsilon]$ vanishes thanks to the
cyclicity of the trace. Similarly, we have

$\displaystyle\delta x_{2}$
$\displaystyle=4\operatorname{\text{tr}}(j^{\mu}j^{\nu})\operatorname{\text{tr}}(j_{\mu}\partial_{\nu}\epsilon).$
(134)

It follows that the variation of the generic Lagrangian $L(x_{1},x_{2})$ is

$\displaystyle\delta L$ $\displaystyle=2\frac{\partial L}{\partial
x_{1}}\operatorname{\text{tr}}\left(\partial_{\mu}\epsilon\,j^{\mu}\right)+4\frac{\partial
L}{\partial
x_{2}}\operatorname{\text{tr}}\left(j_{\mu}j_{\nu}\right)\operatorname{\text{tr}}\left(\partial^{\mu}\epsilon\,j^{\nu}\right)\,.$
(135)

When considering the variation of the action
$\mathcal{S}=\int\text{d}^{2}\sigma\,L$, we can integrate by parts and
conclude that

$\delta\mathcal{S}=-\int\text{d}^{2}\sigma\,\operatorname{\text{tr}}\left[\epsilon\,\partial_{\mu}\mathfrak{J}^{\mu}\right]\,,$
(136)

where $\mathfrak{J}_{\mu}$ is

$\mathfrak{J}_{\mu}=2\frac{\partial L}{\partial x_{1}}j_{\mu}+4\frac{\partial
L}{\partial x_{2}}\operatorname{\text{tr}}[j_{\mu}j_{\nu}]\,j^{\nu}\,,$ (137)

as we presented in eq. (46) of the main text.

#### Adding a Wess-Zumino Term.

When adding a WZ term to the action, of the form

$\displaystyle\mathcal{S}_{\text{WZ}}=\frac{\kay}{6}\int\text{d}^{3}\sigma\,\epsilon^{ijk}\operatorname{\text{tr}}\left(j_{i}[j_{j},j_{k}]\right),$
(138)

the equations of motion are modified by the additional contribution

$\displaystyle\delta\mathcal{S}_{\text{WZ}}=\frac{\kay}{2}\int\text{d}^{2}\sigma\,\operatorname{\text{tr}}\left(\epsilon(\partial_{+}j_{-}-\partial_{-}j_{+})\right).$
(139)

Therefore we have that they take the form of the conservation of

$\partial_{+}\left(\hay\mathfrak{J}_{-}+\kay
j_{-}\right)+\partial_{-}\left(\hay\mathfrak{J}_{+}-\kay j_{+}\right)=0\,,$
(140)

with $\mathfrak{J}_{\mu}$ as in (137), that is eq. (74) of the main text.

###  A.2 Deformed Symmetric Space Sigma Model

In the case of the SSSM it is conventient to rewrite

$\delta j_{\mu}=D_{\mu}\epsilon+\left[j^{(2)}_{\mu},\,\epsilon\right]\,,$
(141)

where $D_{\mu}$ is the covariant derivative defined in (85). Then, writing
(93) as

$x_{1}=g^{\mu\nu}\operatorname{\text{tr}}[j^{(2)}_{\mu}j^{(2)}_{\nu}]\,,\qquad
x_{2}=g^{\mu\rho}g^{\nu\sigma}\operatorname{\text{tr}}[j^{(2)}_{\mu}j^{(2)}_{\nu}]\operatorname{\text{tr}}[j^{(2)}_{\rho}j^{(2)}_{\sigma}]\,,$
(142)

one obtains

$\displaystyle\delta x_{1}$
$\displaystyle=2g^{\mu\nu}\operatorname{\text{tr}}[j^{(2)}_{\mu}\delta
j^{(2)}_{\nu}]=2g^{\mu\nu}\operatorname{\text{tr}}[j^{(2)}_{\mu}\delta
j_{\nu}]=2g^{\mu\nu}\operatorname{\text{tr}}[j^{(2)}_{\mu}D_{\nu}\epsilon],$
(143)

where we used that $\operatorname{\text{tr}}(x^{(0)}y^{(2)})=0$ for any
$x^{(0)}\in\mathfrak{g}_{0}$, $y^{(2)}\in\mathfrak{g}_{2}$, and the cyclicity
of the trace, and all indices are contracted with the metric. Similarly, we
have

$\displaystyle\delta x_{2}$
$\displaystyle=4g^{\mu\rho}g^{\nu\sigma}\operatorname{\text{tr}}[j^{(2)}_{\mu}j^{(2)}_{\nu}]\operatorname{\text{tr}}[j^{(2)}_{\rho}D_{\sigma}\epsilon].$
(144)

This is enough to conclude that the variation of a Lagrangian $L(x_{1},x_{2})$
is

$\displaystyle\delta L$ $\displaystyle=2g^{\mu\nu}\frac{\partial L}{\partial
x_{1}}\operatorname{\text{tr}}\left[D_{\mu}\epsilon\,j^{(2)}_{\nu}\right]+4g^{\mu\rho}g^{\nu\sigma}\frac{\partial
L}{\partial
x_{2}}\operatorname{\text{tr}}\left[j^{(2)}_{\mu}j^{(2)}_{\nu}\right]\operatorname{\text{tr}}\left[D_{\rho}\epsilon\,j^{(2)}_{\sigma}\right]\,,$
(145)

and that the variation of the action $\mathcal{S}=\int\text{d}^{2}\sigma\,L$
is

$\delta\mathcal{S}=-\int\text{d}^{2}\sigma\,g^{\mu\nu}\operatorname{\text{tr}}\left[\epsilon\,\partial_{\mu}\mathfrak{J}^{(2)}_{\nu}\right],$
(146)

where

$\mathfrak{J}^{(2)}_{\mu}=2\frac{\partial L}{\partial
x_{1}}j^{(2)}_{\mu}+4g^{\nu\rho}\frac{\partial L}{\partial
x_{2}}\operatorname{\text{tr}}[j^{(2)}_{\mu},j^{(2)}_{\nu}]\,j^{(2)}_{\rho}\,,$
(147)

as in eq. (95) of the main text.

###  A.3 Deformed Semi-Symmetric Space Sigma Model

The calculations in the case of the sSSSM follow closely what was done above.
In this case we want to rewrite

$\delta
j_{\mu}=D_{\mu}\epsilon+\left[j^{(1)}_{\mu}+j^{(2)}_{\mu}+j^{(3)}_{\mu},\,\epsilon\right]\,.$
(148)

Now we take $x_{1}$ and $x_{2}$ as in (142) above, and compute the variations

$\displaystyle\delta x_{1}$
$\displaystyle=2g^{\mu\nu}\operatorname{\text{str}}\left[j^{(2)}_{\mu}D_{\nu}\epsilon+\epsilon[j^{(2)}_{\mu},j^{(1)}_{\nu}+j^{(3)}_{\nu}]\right],$
(149) $\displaystyle\delta x_{2}$
$\displaystyle=4g^{\mu\rho}g^{\nu\sigma}\operatorname{\text{str}}\left[j^{(2)}_{\mu}j^{(2)}_{\nu}\right]\operatorname{\text{str}}\left[j^{(2)}_{\rho}D_{\sigma}\epsilon+\epsilon[j^{(2)}_{\rho},j^{(1)}_{\sigma}+j^{(3)}_{\sigma}]\right].$

The Lagrangian of the sSSSM can be written as the sum
$L_{\text{sSSSM}}=L_{\text{sSSSM}}^{g}+L_{\text{sSSSM}}^{\varepsilon}$ where

$\displaystyle
L_{\text{sSSSM}}^{g}=\frac{1}{2}\,\operatorname{\text{str}}\left[g^{\mu\nu}j^{(2)}_{\mu}j^{(2)}_{\nu}\right],\qquad\qquad
L_{\text{sSSSM}}^{\varepsilon}=\frac{1}{2}\,\operatorname{\text{str}}\left[\varepsilon^{\mu\nu}j^{(1)}_{\mu}j^{(3)}_{\nu}\right],$
(150)

and after promoting $L_{\text{sSSSM}}^{g}$ to
$L^{g,\gamma}_{\text{sSSSM}}(x_{1},x_{2})$ one finds that

$\displaystyle\delta
S^{g,\gamma}_{\text{sSSSM}}=-\int\text{d}^{2}\sigma\,g^{\mu\nu}\operatorname{\text{str}}\left[\epsilon\,(\partial_{\mu}\mathfrak{J}^{(2)}_{\nu}+[j^{(1)}_{\mu}+j^{(3)}_{\mu},\mathfrak{J}^{(2)}_{\nu}])\right],$
(151)

where $\mathfrak{J}^{(2)}_{\mu}$ is still given by the same expression.
Finally, one also has

$\delta
S^{\varepsilon}_{\text{sSSSM}}=\frac{1}{2}\int\text{d}^{2}\sigma\,\varepsilon^{\mu\nu}\operatorname{\text{str}}\left[\epsilon\left(D_{\mu}(j^{(1)}_{\nu}-j^{(3)}_{\nu})+\left[j^{(1)}_{\mu}+j^{(2)}_{\mu}+j^{(3)}_{\mu},j^{(1)}_{\nu}-j^{(3)}_{\nu}\right]\right)\right].$
(152)

We conclude by noting that the equations of motion for the deformed sSSSM can
be written in the form

$\displaystyle
0=D_{\mu}\Lambda^{\mu}+\left[j_{\mu},\Lambda^{\mu}\right],\qquad\text{where}\quad\Lambda^{\mu}=g^{\mu\nu}\mathfrak{J}^{(2)}_{\nu}-\frac{1}{2}\varepsilon^{\mu\nu}\left(j^{(1)}_{\nu}-j^{(3)}_{\nu}\right).$
(153)

Projecting this expression over the four subspaces of $\mathfrak{g}$ gives

$\displaystyle\mathfrak{g}_{1}:$
$\displaystyle\qquad\varepsilon^{\mu\nu}D_{\mu}j^{(1)}_{\nu}=\left[\left(\varepsilon^{\mu\nu}j^{(2)}_{\mu}-2g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(3)}_{\nu}\right],$
(154) $\displaystyle\mathfrak{g}_{2}:$ $\displaystyle\qquad
g^{\mu\nu}D_{\mu}\mathfrak{J}^{(2)}_{\nu}=\frac{1}{2}\varepsilon^{\mu\nu}\left(\left[j^{(1)}_{\mu},j^{(1)}_{\nu}\right]-\left[j^{(3)}_{\mu},j^{(3)}_{\nu}\right]\right),$
$\displaystyle\mathfrak{g}_{3}:$
$\displaystyle\qquad\varepsilon^{\mu\nu}D_{\mu}j^{(3)}_{\nu}=\left[\left(\varepsilon^{\mu\nu}j^{(2)}_{\mu}+2g^{\mu\nu}\mathfrak{J}^{(2)}_{\mu}\right),j^{(1)}_{\nu}\right],$

while the equation is automatically satisfied along $\mathfrak{g}_{0}$. Using
the Maurer-Cartan equation (105), this reduces to (114) and, in the undeformed
case, to (108).

#### Adding a Wess-Zumino term.

If we consider the action (125), we have an additonal contribution to the
equations of motion, which is proportional to . In particular, let us define

$\mathcal{S}_{\text{WZ}}=\kay\int\limits_{B}\text{d}^{3}\sigma\frac{1}{3}\varepsilon^{ijk}\operatorname{\text{str}}\left(j^{(2)}_{i}[j^{(2)}_{j},j^{(2)}_{k}]+3j^{(1)}_{i}[j^{(3)}_{j},j^{(2)}_{k}]\right).$
(155)

Now, to compute the variation of $\mathcal{S}_{\text{WZ}}$, we cannot use the
compatibility of the supertrace with the $\mathbb{Z}_{4}$ grading. One finds
that the variation of the integrand can be separated into two contributions,
namely a total derivative and an expression to be integrated in 3 dimensions

$\delta\mathcal{S}_{\text{WZ}}=\kay\int\limits_{B}\text{d}^{3}\sigma\
\varepsilon^{ijk}\operatorname{\text{str}}\left(\partial_{i}B_{jk}+Z_{ijk}\right),$
(156)

where

$B_{ij}=\epsilon^{(2)}([j^{(2)}_{i},j^{(2)}_{j}]+[j^{(1)}_{i},j^{(3)}_{j}])+\epsilon^{(1)}[j^{(3)}_{i},j^{(2)}_{j}]+\epsilon^{(3)}[j^{(1)}_{i},j^{(2)}_{j}].$
(157)

Importantly, $Z_{ijk}$ identically vanishes thanks to the Maurer-Cartan
identity and the Jacobi identity on the superalgebra. This is crucial in order
to have a well-defined WZ term, yielding equations of motion that only depend
on the boundary (i.e. 2-dimensional) degrees of freedom. To conclude, the
equations of motion receive the additional contribution of
$\kay\varepsilon^{\mu\nu}B_{\mu\nu}$, and they read as in (129).

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