?

#  Revised version  Improvement of a conserved current density
versus
adding a total derivative to a Lagrangian density

François<EMAIL_ADDRESS>
###### Abstract

For classical relativistic field theory in Minkowski space-time, the addition
of a superpotential term to a conserved current density is trivial in the
sense that it does not modify the local conservation law nor change the
conserved charge, though it may allow us to obtain a current density with some
improved properties. The addition of a total derivative term to a Lagrangian
density is also trivial in the sense that it does not modify the equations of
motion of the theory. These facts suggest that both operations are related and
possibly equivalent to each other for any global symmetry of an action
functional. We address this question following the study of two quite
different (and well known) instances: the Callan-Coleman-Jackiw improvement of
the canonical energy-momentum tensor for scalar and vector fields (providing
an on-shell traceless energy-momentum tensor) and the construction of a
current density satisfying a zero curvature condition for two-dimensional
sigma models on deformed spaces (notably the squashed three-sphere and warped
AdS spaces). These instances correspond to fairly different implementations of
the general results. An appendix addresses the precise relationship between
the approaches to local conservation laws based on active and passive symmetry
transformations, respectively.

Institut de Physique des $2$ Infinis de Lyon,

Université de Lyon, Université Claude Bernard Lyon 1 and CNRS/IN2P3,

Bat. P. Dirac, 4 rue Enrico Fermi, F-69622-Villeurbanne (France)

Dedicated to the memory of Krzysztof Gawȩdzki (1947-2022)

_who made a large variety of original contributions to diverse fields of
theoretical and mathematical physics, and in particular to classical and
quantum field theory. The discussions with him have always been quite pleasant
and enlightening. By his humbleness, attentiveness, kindness and generosity he
has been a great example and steady encouragement. His precious advice, fine
mind, humor and reassuring presence are deeply missed._

###### Contents

  1. 1 Introduction
  2. 2 Some reminders
    1. 2.1 Lagrangian density given by a total derivative
    2. 2.2 Noether’s first theorem and improvement of currents
    3. 2.3 Different implementations of Noether’s first theorem
  3. 3 Scale invariance for relativistic fields
    1. 3.1 Reminder 1: Scale invariance and canonical dilatation current
    2. 3.2 Reminder 2: New improved EMT for a scalar field
    3. 3.3 Derivation of the new improved EMT for a free scalar field
    4. 3.4 Generalization to a multiplet of self-interacting real or complex scalar fields
    5. 3.5 Summary and assessment for a scalar field
    6. 3.6 Scale invariance for a vector field
  4. 4 Conformal transformations of scalar fields
  5. 5 Application in supersymmetric field theory
  6. 6 Main point: Current improvement induced by a total derivative Lagrangian
  7. 7 Two-dimensional integrable models based on a flat improved current
    1. 7.1 Reminder: Two-dimensional $SU(2)$ principal chiral model
    2. 7.2 Two-dimensional sigma model on the squashed $3$-sphere
    3. 7.3 Further examples of the same nature
      1. 7.3.1 Sigma models on warped $AdS_{3}$ and on Schrödinger space-time
      2. 7.3.2 Deformed WZNW models on the squashed $3$-sphere (and on warped $AdS_{3}$)
      3. 7.3.3 Two-dimensional sigma models on para-complex $\mathds{Z}_{T}$-cosets
  8. 8 Concluding remarks
  9. A Variation of Lagrangian
  10. B Noether’s first theorem
  11. C Passive symmetry transformations
  12. D Procedure of Gell-Mann and Lévy for determining Noether currents
  13. E Proof of relation (7.45)

##  1 Introduction

Noether’s first theorem [1, 2] establishing a general relationship between
global symmetries of an action functional and local conservation laws has
become a pillar of modern physics, e.g. see reference [3] for some reviews and
reference [4] for an historical account up to recent developments. Though the
main result is by now part of the standard physics curriculum, it has taken
some time for achieving a deeper conceptual and mathematical understanding in
terms of equivalence classes of currents and of symmetry transformations [5,
6, 7, 8, 9].

The present article addresses a particular instance of Noether’s first theorem
namely the relationship between Lagrangians which are given by a total
derivative and locally conserved current densities which are derivatives of a
superpotential (so-called superpotential terms). The upshot is that the
superpotential terms which are generally introduced by hand (in order to
obtain conserved current densities having improved properties with respect to
the model under consideration) also follow from Noether’s first theorem as
applied to a Lagrangian density which is given by a total derivative. Some
related results or examples have previously appeared in the vast literature on
field theory (and will be explicitly indicated in our discussion), but we are
not aware of a complete and general treatment including illustrations of
different nature. Such an investigation represents the main subject of the
present text. We have also included a discussion of the precise relationship
between the approaches based on active and passive symmetry transformations,
respectively. In fact, different authors generally choose either of these two
approaches, but the detailed relationship between both of them requires a bit
of care.

We emphasize that we will only be concerned with theories defined on unbounded
Minkowski space-time $\mathds{R}^{n}$ and not with subsets thereof, henceforth
not with boundaries of the latter subsets. (A simple example [10] of a subset
of $\mathds{R}^{4}$ is given by the “spatial upper half-space”
$\Omega=\\{(t,x,y,z)\,|\,z\geq 0\\}$ which has a boundary described by $z=0$.)
The presence of such boundaries generally breaks symmetries like translation
invariance and the discussion of local or global conservation laws then has to
take into account boundary conditions of fields as well as boundary terms. For
a discussion of this subject in the framework of supersymmetric field
theories, we refer to [10, 11] and references therein.

Our article is organized as follows. To set the stage, we first recall in
section 2 some well known facts concerning Lagrangian densities, Noether’s
first theorem and equivalence classes of conserved current densities. In
section 3, we outline the results which follow from the scale invariance of
the action functional for a real scalar field in $n$ space-time dimensions:
for this model we discuss the fact that the addition of a particular total
derivative to the Lagrangian density describing the dynamics yields the so-
called new improved or Callan-Coleman-Jackiw energy-momentum tensor (EMT) [12]
as well as the fact that this tensor differs from the canonical EMT by a
superpotential term. In the subsequent section, the results which hold for
scale invariance of scalar fields are generalized to the full group of
conformal transformations in $n$-dimensional space-time. These results allow
us to apprehend more fully those which hold for scale symmetry that has been
the main focus in the literature in relationship with the EMT. In section 5,
we show that the total derivative Lagrangian which naturally occurs in the
four-dimensional supersymmetric Wess-Zumino model yields the familiar
improvements of the EMT and of the supersymmetry current (which are part of
the supermultiplet of currents) of this model. In section 6, we consider
active symmetry transformations to derive a simple _general formula for the
current density which is associated to a global symmetry of a Lagrangian
density that is given by a total derivative._ This allows us to recover the
new improved EMT for a scalar field, but this also leads (by application of
the method of Gell-Mann and Lévy [13, 14] for deriving Noether current
densities) to general expressions for the currents appearing in other classes
of models. The latter include the two-dimensional sigma models with different
target spaces that have previously been investigated in the literature and
that we address in section 7. More precisely, we will provide a short
introduction to these models while emphasizing that the addition of a
particular total derivative to the Lagrangian density induces a superpotential
term in the conserved current density: this addition ensures that the total
Lie algebra-valued current density satisfies the zero curvature condition and
thereby permits to establish straightforwardly the integrability of these
field theories. The appendices gather some derivations as well as the
discussion of the general relationship between active and passive symmetry
transformations in the implementation of Noether’s first theorem (appendix C).
For the sake of completeness, we have also included a short presentation of
the procedure of Gell-Mann and Lévy which is not always described in great
detail or generality in the literature (appendix D).

##### Notation:

We consider the natural system of units ($c\equiv 1$) and we use standard
notation for the coordinates of $n$-dimensional space-time (with $n\geq 2$):
$x=(t,\vec{x}\,)\equiv(x^{\mu})_{\mu=0,1,\dots,n-1}$ and
$\vec{x}\equiv(x^{i})_{i=1,\dots,n-1}$ for the spatial coordinates, the
Minkowski metric $(\eta_{\mu\nu})$ being assumed to be mostly ‘mostly minus’.

##  2 Some reminders

###  2.1 Lagrangian density given by a total derivative

Suppose the Lagrangian density ${\cal L}$ for some classical relativistic
fields $\varphi$ is given by a total derivative, i.e. ${\cal
L}=\partial_{\mu}k^{\mu}$ where $k^{\mu}$ depends on $\varphi$ and/or its
derivatives up to some finite order. A variation
$\delta\varphi(x)\equiv\varphi^{\prime}(x)-\varphi(x)$ then induces a
variation of the action functional $S\equiv\int_{\Omega}d^{n}x\,{\cal L}$
defined on a space-time domain $\Omega\subset\mathds{R}^{n}$:

$\displaystyle\delta S=\int_{\Omega}d^{n}x\,\delta{\cal
L}=\int_{\Omega}d^{n}x\,\partial_{\mu}(\delta
k^{\mu})=\oint_{\partial\Omega}d^{n-1}x_{\mu}\,\delta k^{\mu}\,.$ (2.1)

Here, Stokes’ theorem was applied for the last equality, see reference [15]
for the notation of the hypersurface integration measure. Thus, if the
variation $\delta\varphi$ and its derivatives vanish at the boundary
$\partial\Omega$ of $\Omega$, the variation $\delta S$ vanishes identically
for all of these field configurations and so does its functional derivative
with respect to $\varphi$, i.e. we have the _identity_ $\delta
S/\delta\varphi=0$.

###  2.2 Noether’s first theorem and improvement of currents

##### Generalities:

For a Lagrangian which is at most of second order, i.e. ${\cal L}={\cal
L}(\varphi,\partial_{\mu}\varphi,\partial_{\mu}\partial_{\nu}\varphi)$,
Noether’s first theorem states: if $\delta{\cal L}=\partial_{\mu}\Omega^{\mu}$
under the infinitesimal variation
$\delta\varphi(x)\equiv\varphi^{\prime}(x)-\varphi(x)$, then

$\displaystyle 0=\frac{\delta
S}{\delta\varphi}\,\delta\varphi+\partial_{\mu}j^{\mu}\,,\quad\mbox{with}\
\;\left\\{\begin{array}[]{l}\frac{\delta S}{\delta\varphi}=\frac{\partial{\cal
L}}{\partial\varphi}-\partial_{\mu}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\right)+\partial_{\mu}\partial_{\nu}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\partial_{\nu}\varphi)}\right)\\\
j^{\mu}=\left[\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}-\partial_{\rho}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\partial_{\rho}\varphi)}\right)\right]\delta\varphi+\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\partial_{\rho}\varphi)}\,\partial_{\rho}\delta\varphi-\Omega^{\mu}\,.\end{array}\right.$
(2.4)

We note that these expressions reduce to the familiar results for a first
order Lagrangian. Since the standard textbook presentations focus on first
order Lagrangians, we outline the derivation of (2.4) in the appendices A and
B. Of course, these results and derivations straightforwardly generalize to a
Lagrangian density which depends on higher than second order derivatives [1,
2], but we focused on second order derivatives here in view of the physical
applications to be addressed.

As a matter of fact, the general formulation [5, 6, 7, 8, 9] of Noether’s
first theorem states that there is a one-to-one correspondence between
equivalence classes of (global) variational symmetries and equivalence classes
of (on-shell) conserved currents. (For a review, see for instance references
[9, 15].) More precisely, two infinitesimal global symmetry transformations
are considered to be equivalent if they differ by a gauge symmetry
transformation and/or an “equation of motion symmetry transformation”, i.e. a
symmetry transformation which is a linear combination of Euler-Lagrange
derivatives and their space-time derivatives up to a finite order (with
possibly field-dependent coefficients).

The _equivalence of current densities_ is defined by

$\displaystyle\boxed{\ j^{\mu}\sim
j^{\mu}+\underbrace{\partial_{\rho}B^{\rho\mu}}_{\textrm{superpot.
term}}+\underbrace{t^{\mu}}_{\approx\,0}\ }\,,\qquad\mbox{where}\
\;B^{\rho\mu}=-B^{\mu\rho}\,.$ (2.5)

Here, the so-called _superpotential_ $B^{\rho\mu}$ defines a current density
$\partial_{\rho}B^{\rho\mu}$ which is identically conserved due to the
antisymmetry of $B^{\rho\mu}$. Moreover, here and in the following, we use
Dirac’s notation $F\approx 0$ to denote an on-shell equality, i.e. a relation
which holds by virtue of the equations of motion.

For two equivalent currents, say $(j_{1}^{\mu})$ and $(j_{2}^{\mu})$, we have
$\partial_{\mu}j_{1}^{\mu}\approx\partial_{\mu}j_{2}^{\mu}$ which implies that
$(j_{1}^{\mu})$ is on-shell conserved if and only if $(j_{2}^{\mu})$ is on-
shell conserved. The addition of a trivial term
$\partial_{\rho}B^{\rho\mu}+t^{\mu}$ to a given (on-shell) conserved current
$(j^{\mu})$ is generally referred to as an _improvement of the current_ since
this addition eventually allows us to obtain a conserved current which has
“better properties” than $(j^{\mu})$, e.g. in a gauge field theory it may be
gauge invariant if $(j^{\mu})$ does not have this property. In this respect it
is worth recalling the following example. (Quite generally, in this context we
also mention the important fact that the (on-shell value of the) Noether
charge $Q\equiv\int_{\mathds{R}^{n-1}}d^{n-1}x\,j^{0}$ is not modified by the
addition of an improvement term to the current density $j^{\mu}$ provided the
field $B^{i0}$ decays sufficiently fast at spatial infinity
$\partial\mathds{R}^{n-1}$.)

##### Example of EMT of the electromagnetic field:

The _translation invariance of the action for free Maxwell theory in
$n$-dimensional Minkowski space-time,_ i.e. of the functional
$S_{Max}[A]\equiv-\frac{1}{4}\int_{\mathds{R}^{n}}d^{n}x\,F^{\mu\nu}F_{\mu\nu}$
(with $F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
equation of motion $\partial_{\mu}F^{\mu\nu}=0$) leads, by virtue of Noether’s
first theorem (2.4) to the local conservation law
$\partial_{\mu}T_{\textrm{can}}^{\mu\nu}\approx 0$ for the _canonical energy-
momentum tensor (EMT)_ of the electromagnetic field:

$\displaystyle
T_{\textrm{can}}^{\mu\nu}=-F^{\mu\rho}\partial^{\nu}A_{\rho}+\frac{1}{4}\,\eta^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\,.$
(2.6)

Since the first term of this expression is not gauge invariant,
$(T_{\textrm{can}}^{\mu\nu})$ cannot be viewed as a physically acceptable
representative for the EMT of the electromagnetic field (the components of
this tensor being measurable quantities). This raises the question whether the
equivalence class of the on-shell conserved currents
$(T_{\textrm{can}}^{\mu\nu})_{\nu=0,1,\dots,n-1}$ contain a representative
which is gauge invariant. To find such a representative, we simply express the
derivatives $\partial^{\nu}A_{\rho}$ in terms of ${F^{\nu}}_{\rho}$:

$\displaystyle-F^{\mu\rho}\partial^{\nu}A_{\rho}=F^{\mu\rho}{F_{\rho}}^{\nu}-F^{\mu\rho}\partial_{\rho}A^{\nu}\,.$
(2.7)

After applying the Leibniz rule to the last term,

$\displaystyle-F^{\mu\rho}\partial_{\rho}A^{\nu}=\partial_{\rho}(-F^{\mu\rho}A^{\nu})+(\partial_{\rho}F^{\mu\rho})A^{\nu}\,,$
(2.8)

we find that

$\displaystyle
T_{\textrm{can}}^{\mu\nu}=T_{\textrm{phys}}^{\mu\nu}+\underbrace{\partial_{\rho}\chi^{\rho\mu\nu}}_{\textrm{superpot.
term}}+\underbrace{t^{\mu\nu}}_{\approx\,0}\qquad\mbox{with}\quad\left\\{\begin{array}[]{c}\chi^{\rho\mu\nu}\equiv
F^{\rho\mu}A^{\nu}=-\chi^{\mu\rho\nu}\\\
t^{\mu\nu}\equiv-(\partial_{\rho}F^{\rho\mu})A^{\nu}\approx
0\,,\end{array}\right.$ (2.11)

and

$\displaystyle\boxed{\ T_{\textrm{phys}}^{\mu\nu}\equiv
F^{\mu\rho}{F_{\rho}}^{\nu}+\frac{1}{4}\,\eta^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\
}\,.$ (2.12)

Thus, for each value of $\nu$, the currents $(T_{\textrm{can}}^{\mu\nu})$ and
$(T_{\textrm{phys}}^{\mu\nu})$ are equivalent from the point of view of
Noether’s first theorem since they differ only by trivial terms. While the
representative $(T_{\textrm{phys}}^{\mu\nu})$ of the equivalence class is
_gauge invariant_ and _symmetric_ as well as _traceless for $n=4$_, the
representative $(T_{\textrm{can}}^{\mu\nu})$ does not have any of these
properties. As a matter of fact, the symmetry of the EMT is also a desired
property if the theory in Minkowski space-time is viewed as the flat space
limit of the theory in curved space-time described by general relativity: the
EMT in Minkowski space-time should then coincide with the Einstein-Hilbert
EMT, i.e. the flat space limit of the metric EMT

$\displaystyle T^{\mu\nu}\equiv\frac{-2}{\sqrt{|g|}}\,\frac{\delta
S_{Max}[A,\textbf{{g}}]}{\delta g_{\mu\nu}}\,,$ (2.13)

where $S_{Max}[A,\textbf{{g}}]$ represents the coupling of the gauge field
$(A^{\mu})$ to an external gravitational field described by a fixed, symmetric
metric tensor field $\textbf{{g}}(x)\equiv(g_{\mu\nu}(x))$ and
$g\equiv\textrm{det}\,\textbf{{g}}$ (see [16, 17] and references therein for
further details and subtleties). As a matter of fact, the improvement (2.11)
has already been discussed by F. J. Belinfante and L. Rosenfeld in the 1930s
and is usually referred to by their names. Here, we simply emphasized the
mathematical and physical vision brought about the general formulation of
Noether’s first theorem which describes a correspondence between equivalence
classes of global symmetries and on-shell conserved current densities.

###  2.3 Different implementations of Noether’s first theorem

The fact that relation (2.4), i.e. $0=\frac{\delta
S}{\delta\varphi}\,\delta\varphi+\partial_{\mu}j^{\mu}$, does not yield a
gauge invariant current density $j^{\mu}=T^{\mu\nu}a_{\nu}$ for the case of
translations of a gauge field, i.e. of the infinitesimal symmetry
transformations $\delta\varphi=\delta A_{\mu}=a^{\nu}\partial_{\nu}A_{\mu}$,
does not come as a surprise since the latter variation is not gauge invariant.
For this reason various authors have looked for alternative implementations of
Noether’s first theorem which automatically yield a gauge invariant EMT. A
natural procedure (which was rediscovered numerous times over the last
decades, e.g. in reference [18]) was put forward by E. Bessel-Hagen in his
pioneering work [2] from 1921 in which he introduced divergence symmetries
(following the advice of E. Noether) and applied Noether’s theorems to the
invariance of four-dimensional Maxwell’s equations under the conformal group.
This procedure (qualified as _“Kunstgriff”_ , i.e. trick, by E. Bessel-Hagen)
consists in “covariantizing” the variation $\delta
A_{\mu}=a^{\nu}\partial_{\nu}A_{\mu}$ with the help of the gauge invariant
tensor $F_{\nu\mu}=\partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu}$, i.e.
replacing the gauge variant expression $\delta A_{\mu}$ by the gauge invariant
one

$\displaystyle\delta_{cov}A_{\mu}\equiv a^{\nu}F_{\nu\mu}=\delta
A_{\mu}-\partial_{\mu}(a^{\nu}A_{\nu})\,.$ (2.14)

Here, the last term represents a local gauge transformation (with field
dependent parameter $a^{\nu}A_{\nu}$) and thereby it is a trivial contribution
to the global symmetry transformation $\delta A_{\mu}$ (in the sense of the
equivalences of global symmetry transformations defined above). This procedure
directly leads to a gauge invariant EMT, namely to the result (2.12). When
applied to the conformal Killing vector fields
$\xi\equiv\xi^{\mu}(x)\partial_{\mu}$ (of the Minkowski metric) which
parametrize the Lie algebra of the conformal group (rather than the
translations $(a^{\mu})$ alone), it yields the _Bessel-Hagen form_
$T^{\mu\nu}_{\textrm{phys}}\xi_{\nu}$ for all of the conserved current
densities associated to conformal invariance (see pages 271-272 of the
original work [2] and reference [19] for a recent assessment).

##  3 Scale invariance for relativistic fields

###  3.1 Reminder 1: Scale invariance and canonical dilatation current

A _scale transformation_ (or _dilatation_ or _dilation_) of the space-time
coordinates is defined by $x\mapsto x^{\prime}=\textrm{e}^{\rho}x$ where
$\rho$ is a constant real number. The induced change of the Minkowski metric
is also a rescaling with a positive factor:

$\displaystyle ds^{2}\equiv\eta_{\mu\nu}dx^{\mu}dx^{\nu}\ \leadsto\
ds^{\prime\,2}=\textrm{e}^{2\rho}ds^{2}\,.$ (3.1)

A classical relativistic field $\varphi$ (like a scalar field $\phi$, a vector
field $(A^{\mu})$ or a spinor field $\psi$) transforms under such a rescaling
according to 222More precisely, fields transforming in this manner are
referred to as _scaling fields_ [20] or as _“quasi-primary” fields_ in $n$
space-time dimensions [21].

$\displaystyle\boxed{\
\varphi^{\prime}(x^{\prime})=\textrm{e}^{-\rho\,d_{\varphi}}\varphi(x)\
}\qquad\textrm{for}\ \;\boxed{\ x^{\prime}=\textrm{e}^{\rho}x\ }\,.$ (3.2)

Here, the natural number $d_{\varphi}$ denotes the so-called _scale dimension_
of the field $\varphi$. If one chooses this dimension to coincide with the
canonical (engineering) dimension of the field $\varphi$ in $n$ space-time
dimensions (i.e. $d_{\phi}=\frac{n-2}{2}$ for a scalar field $\phi$ or for a
vector field $(A^{\mu})$, and $d_{\psi}=\frac{n-1}{2}$ for a spinor field
$\psi$), then the action for a _free massless field_ $\varphi$ in $n$
dimensions,

$\displaystyle S[\varphi]\equiv\int d^{n}x\,{\cal
L}(\varphi,\partial_{\mu}\varphi)\,,\qquad{\cal
L}^{\prime}(x^{\prime})=\textrm{e}^{-n\rho}\,{\cal L}(x)\qquad\textrm{for}\
\;x^{\prime}=\textrm{e}^{\rho}x\,,$ (3.3)

is scale invariant. However mass terms and in general also interaction terms
involving dimensionful coupling constants violate scale invariance so that one
is not simply dealing with dimensional analysis.

From the invariance of the action under infinitesimal scale transformations,

$\displaystyle\delta_{\rho}x^{\mu}=\rho\,x^{\mu}\,,\qquad\delta_{\rho}\varphi=-\rho\,(x\cdot\partial+d_{\varphi})\varphi\quad\textrm{with}\
\;x\cdot\partial\equiv x^{\mu}\partial_{\mu}\,,$ (3.4)

and

$\displaystyle\delta_{\rho}{\cal L}=-\rho\,(x\cdot\partial+n){\cal
L}=\partial_{\mu}\Omega^{\mu}\qquad\mbox{with}\quad\Omega^{\mu}\equiv-\rho\,x^{\mu}{\cal
L}\,,$ (3.5)

it follows by virtue of Noether’s first theorem that we have an on-shell
conserved _canonical dilatation current_ density of the form

$\displaystyle\boxed{\
j^{\mu}_{\textrm{dil,can}}=T^{\mu\nu}_{\textrm{can}}\,x_{\nu}+d_{\varphi}\,\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\varphi\ }\qquad\textrm{with}\
\;\partial_{\mu}j^{\mu}_{\textrm{dil,can}}\approx 0\,.$ (3.6)

Here, $T^{\mu\nu}_{\textrm{can}}\equiv\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\partial^{\nu}\varphi-\eta^{\mu\nu}{\cal
L}$ denotes the _canonical EMT_ whose conservation law
$\partial_{\mu}T^{\mu\nu}_{\textrm{can}}\approx 0$ follows from the invariance
of the action under space-time translations.

The result (3.6) is reminiscent of the expression for the _canonical angular
momentum tensor:_ For non-scalar fields the latter not only involves the
moments of the canonical EMT, but also an additional term, namely the spin
density tensor. This motivated C. Callan, S. Coleman and R. Jackiw [12] to
search for an improvement such that its addition to
$j^{\mu}_{\textrm{dil,can}}$ eliminates the second term in expression (3.6).
To achieve this goal, they added an appropriate superpotential term to the
canonical EMT $T^{\mu\nu}_{\textrm{can}}$ so as to obtain a “new improved” EMT
$T^{\mu\nu}_{\textrm{conf}}$ which is (on-shell) traceless so that the
improved dilatation current $j^{\mu}_{\textrm{dil,conf}}$ is simply given by
the “moments of the EMT”:

$\displaystyle\boxed{\
j^{\mu}_{\textrm{dil,conf}}=T^{\mu\nu}_{\textrm{conf}}\,x_{\nu}\
}\,,\qquad\textrm{hence}\ \quad\boxed{\
\partial_{\mu}j^{\mu}_{\textrm{dil,conf}}=T^{\mu}_{\textrm{conf}\,\mu}\approx
0\ }\,.$ (3.7)

Thus, _the on-shell tracelessness of the new improved EMT
$T^{\mu\nu}_{\textrm{conf}}$ represents a mathematical reflection of the scale
invariance of the theory under consideration._

###  3.2 Reminder 2: New improved EMT for a scalar field

Let us consider the case of a _real free massless scalar field $\phi$ in $n$
space-time dimensions,_ i.e. the action functional

$\displaystyle S[\phi]\equiv\int_{M}d^{n}x\,{\cal
L}(\partial_{\mu}\phi)\equiv\frac{1}{2}\int_{M}d^{n}x\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)\,.$
(3.8)

Then, expression (3.6) writes

$\displaystyle
j^{\mu}_{\textrm{dil,can}}=x_{\nu}T^{\mu\nu}_{\textrm{can}}+d_{\phi}\,\phi\partial^{\mu}\phi\,,\quad\textrm{with}\
\;T_{\textrm{can}}^{\mu\nu}=(\partial^{\mu}\phi)(\partial^{\nu}\phi)-\frac{1}{2}\,\eta^{\mu\nu}(\partial^{\rho}\phi)(\partial_{\rho}\phi)\quad\mbox{and}\
\;d_{\phi}=\frac{n-2}{2}\,.$ (3.9)

Following Callan, Coleman and Jackiw [12] (who studied the four dimensional
case), one adds a particular derivative term to the canonical EMT
$T_{\textrm{can}}^{\mu\nu}$ of $\phi$ so as to obtain the so-called _new
improved EMT_ or _CCJ tensor_

$\displaystyle\boxed{\ T^{\mu\nu}_{\textrm{conf}}\equiv
T^{\mu\nu}_{\textrm{can}}-\xi_{n}\,(\partial^{\mu}\partial^{\nu}-\eta^{\mu\nu}\Box)\phi^{2}\
}\qquad\textrm{with}\quad\boxed{\ \xi_{n}\equiv\frac{1}{4}\,\frac{n-2}{n-1}\
}\,,$ (3.10)

and $\Box\equiv\partial^{\mu}\partial_{\mu}$. The tensor (3.10) is still
_symmetric, on-shell conserved_ and yields the _same conserved charge as_
$T^{\mu\nu}_{\textrm{can}}$ (upon the assumption that the fields fall off
sufficiently fast at spatial infinity). We have labeled it by ‘conformal’
since it is _on-shell traceless,_ i.e. $T^{\mu}_{\textrm{conf}\,\mu}\approx 0$
by virtue of the equation of motion $\Box\phi=0$, and it is directly related
to the conformally invariant coupling of scalar fields to gravity [12, 24].
With (3.10) and the redefinition

$\displaystyle j^{\mu}_{\textrm{dil,conf}}\equiv
j^{\mu}_{\textrm{dil,can}}+\xi_{n}\,\partial_{\rho}\left[(x^{\mu}\partial^{\rho}-x^{\rho}\partial^{\mu})\phi^{2}\right]\,,$
(3.11)

we get the expression (3.7) for the dilatation current that we looked for,
i.e. $j^{\mu}_{\textrm{dil,conf}}$ is _on-shell conserved,_ yields the _same
conserved charge as_ $j^{\mu}_{\textrm{dil,can}}$ and is simply _given by the
“moments of the EMT”._

###  3.3 Derivation of the new improved EMT for a free scalar field

The improved expressions, i.e. $T^{\mu\nu}_{\textrm{conf}}$ for the EMT and
$j^{\mu}_{\textrm{dil,conf}}$ for the dilatation current, respectively, can be
straightforwardly obtained as follows. (Our derivation has been motivated by
the appendix of reference [25], but we note that the results are implicit in
reference [26] though its author argues in a different manner. As we realized
quite recently, the result for the EMT in four space-time dimensions is
explicit in a work devoted to a supersymmetric model [27] which we will
discuss in section 5 below.)

First, we recall that, for a given Lagrangian ${\cal
L}(\phi,\partial_{\mu}\phi,\partial_{\mu}\partial_{\nu}\phi)$, the addition of
a total derivative ${\cal L}_{1}\equiv\partial_{\mu}k^{\mu}$ (with $k^{\mu}$
depending on $x,\phi,\partial_{\mu}\phi$) does not modify the equation of
motion determined by ${\cal L}$. Such a trivial addition eventually leads to
the addition of a superpotential term to a conserved current associated to an
invariance of the action $S[\phi]\equiv\int d^{n}x\,{\cal L}$ (e.g. to the EMT
associated to translation invariance).

For the scale invariant action (3.8) describing a real free massless scalar
field in $n$ space-time dimensions, we can obtain a scale invariant integral
over a total derivative by partial integration:

$\displaystyle\frac{1}{2}\int_{M}d^{n}x\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)=-\frac{1}{2}\int_{M}d^{n}x\,\phi\,\Box\phi+\frac{1}{2}\int_{M}d^{n}x\,\partial^{\mu}(\phi\,\partial_{\mu}\phi)\,.$
(3.12)

Henceforth, we will consider the _scale invariant action_
$S_{1}[\phi]\equiv\int d^{n}x\,{\cal L}_{1}$ with

$\displaystyle\boxed{\ {\cal L}_{1}\equiv\partial_{\mu}k^{\mu}\
}\,,\qquad\mbox{where}\quad\boxed{\ k^{\mu}\equiv-\xi\,\partial^{\mu}\phi^{2}\
}\quad(\xi\in\mathds{R})\,,$ (3.13)

i.e.

$\displaystyle
k^{\mu}=-2\xi\,\phi\,\partial^{\mu}\phi\qquad\mbox{and}\quad{\cal
L}_{1}=-2\xi\,[(\partial^{\mu}\phi)(\partial_{\mu}\phi)+\phi\,\Box\phi]\,.$
(3.14)

Here, we have introduced an arbitrary real factor $\xi$ to keep track of the
surface term in the subsequent calculations and in accordance with the
arbitrariness of this term for the equation of motion. For the symmetries of
the second order Lagrangian (3.13), we will now apply Noether’s first theorem
(2.4).

##### Space-time translations:

Since ${\cal L}_{1}$ does not explicitly depend on $x$, it is invariant under
_space-time translations_ given at the infinitesimal level by
$\delta\phi=a_{\nu}\partial^{\nu}\phi$ and

$\displaystyle\delta{\cal L}_{1}=a_{\nu}\partial^{\nu}{\cal
L}_{1}=\partial_{\mu}\Omega_{1}^{\mu}\qquad\mbox{with}\
\;\Omega_{1}^{\mu}=a_{\nu}\eta^{\mu\nu}{\cal L}_{1}\,.$ (3.15)

By virtue of Noether’s theorem (2.4), we thus have a conserved current
$j_{1}^{\mu}\equiv T_{1}^{\mu\nu}a_{\nu}$ with

$\displaystyle
T_{1}^{\mu\nu}=\partial^{\nu}k^{\mu}-\eta^{\mu\nu}\partial_{\rho}k^{\rho}\,,$
(3.16)

i.e.

$\displaystyle\boxed{\ T_{1}^{\mu\nu}=-\partial_{\rho}\chi^{\rho\mu\nu}\
}\qquad\mbox{with}\ \;\boxed{\ \chi^{\rho\mu\nu}\equiv
k^{\rho}\eta^{\mu\nu}-k^{\mu}\eta^{\rho\nu}\ }\,.$ (3.17)

Since $\chi^{\rho\mu\nu}=-\chi^{\mu\rho\nu}$, this EMT is simply a
superpotential term, i.e. it is identically conserved:
$\partial_{\mu}T_{1}^{\mu\nu}=0$. We note that the tensor (3.16) is symmetric
since the definition of $k^{\mu}$ implies that
$\partial^{\nu}k^{\mu}=\partial^{\mu}k^{\nu}$. This symmetry may be rendered
manifest in eqn. (3.17) by symmetrizing the second term of
$\chi^{\rho\mu\nu}$, i.e. by considering
$\chi^{\rho\mu\nu}=k^{\rho}\eta^{\mu\nu}-\frac{1}{2}(k^{\mu}\eta^{\rho\nu}+k^{\nu}\eta^{\rho\mu})$
[26].

By adding the EMT (3.17) to $T^{\mu\nu}_{\textrm{can}}$, we obtain a total EMT
$T^{\mu\nu}_{\textrm{tot}}\equiv T^{\mu\nu}_{\textrm{can}}+T^{\mu\nu}_{1}$
whose trace is given by

$\displaystyle
T^{\mu}_{\textrm{tot}\,\mu}=-\frac{1}{2}\,\left[(n-2)-4\xi\,(n-1)\right](\partial^{\mu}\phi)(\partial_{\mu}\phi)+2\xi\,(n-1)\,\phi\underbrace{\Box\phi}_{\approx\,0}\,.$
(3.18)

Thus, the tensor $T^{\mu\nu}_{\textrm{tot}}$ is on-shell traceless for
$\xi=\xi_{n}\equiv\frac{1}{4}\,\frac{n-2}{n-1}$: expression $T^{\mu\nu}_{1}$
with $\xi=\xi_{n}$ obviously coincides with the additional term in (3.10) and
$T^{\mu\nu}_{\textrm{tot}}=T^{\mu\nu}_{\textrm{conf}}$.

##### Scale transformations:

For the _scale transformation_ (3.4), we have (3.5), i.e. $\delta{\cal
L}_{1}=\partial_{\mu}\Omega_{1}^{\mu}$ with
$\Omega_{1}^{\mu}=-\rho\,x^{\mu}{\cal L}_{1}$: by virtue of (2.4) this leads
to the conserved Noether current

$\displaystyle
j^{\mu}_{1}=T^{\mu\nu}_{1}x_{\nu}+(n-1)\,k^{\mu}=\left(\partial^{\nu}k^{\mu}-\eta^{\mu\nu}\partial_{\rho}k^{\rho}\right)x_{\nu}+(n-1)\,k^{\mu}\,,$
(3.19)

i.e.

$\displaystyle\boxed{\ j^{\mu}_{1}=\partial_{\rho}B^{\rho\mu}\
}\qquad\mbox{with}\ \;\boxed{\
B^{\rho\mu}\equiv-\chi^{\rho\mu\nu}\zeta_{\nu}=x^{\rho}k^{\mu}-x^{\mu}k^{\rho}\
}\,.$ (3.20)

Here, $\zeta^{\nu}\equiv x^{\nu}$ represents the vector field
$\zeta\cdot\partial$ generating scale transformations. Since
$B^{\rho\mu}=-B^{\mu\rho}$, the current $j_{1}^{\mu}$ also defines a
superpotential term, i.e. $\partial_{\mu}j^{\mu}_{1}=0$.

For $j^{\mu}_{\textrm{tot}}\equiv j^{\mu}_{\textrm{dil,can}}+j^{\mu}_{1}$
(with $j^{\mu}_{\textrm{dil,can}}$ given by (3.9) and $j^{\mu}_{1}$ given by
(3.19)), we get

$\displaystyle
j^{\mu}_{\textrm{tot}}=\big{[}\underbrace{T^{\mu\nu}_{\textrm{can}}+T^{\mu\nu}_{1}}_{=\,T^{\mu\nu}_{\textrm{tot}}}\big{]}x_{\nu}+\frac{1}{2}\,\left[(n-2)-4\xi\,(n-1)\right]\,\phi\,\partial^{\mu}\phi\,,$
(3.21)

hence $j^{\mu}_{\textrm{tot}}=T^{\mu\nu}_{\textrm{tot}}x_{\nu}$ for
$\xi=\xi_{n}$. In summary, for $\xi=\xi_{n}$ we have
$j^{\mu}_{\textrm{tot}}=j^{\mu}_{\textrm{dil,conf}}$ and expression
$j^{\mu}_{1}$ coincides with the additional term in (3.11).

##### Summary:

For the “conformal value” $\xi=\xi_{n}$, the total Lagrangian has the form

$\displaystyle{\cal L}_{\textrm{tot}}\equiv{\cal L}+{\cal
L}_{1}=\frac{1}{n-1}\,\Big{[}\frac{1}{2}\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)-d_{\phi}\,\phi\,\Box\phi\Big{]}\qquad\mbox{for}\
\ \xi=\xi_{n}\equiv\frac{1}{4}\,\frac{n-2}{n-1}\,.$ (3.22)

The total EMT is on-shell traceless and the dilatation current is given by the
moments of the the total EMT.

We note that the result (3.18) implies that an _off-shell traceless EMT_
$T^{\mu\nu}_{\textrm{off}}$ can be obtained from
$T^{\mu\nu}_{\textrm{tot}}=T^{\mu\nu}_{\textrm{conf}}$ by the addition of the
trivial current
$-2\xi\,\frac{n-1}{n}\,\eta^{\mu\nu}\phi\,\Box\phi=-\frac{n-2}{2n}\,\eta^{\mu\nu}\phi\,\Box\phi$
which vanishes on-shell (see eqn. (2.5) with $j^{\mu}\equiv T^{\mu\nu}a_{\nu}$
in the present case). In this respect, we mention that the off-shell traceless
EMT $T^{\mu\nu}_{\textrm{off}}$ can also be obtained by a modification of the
Gell-Mann and Lévy procedure [22]. In fact, the usual procedure consists in
promoting the translation parameters $a^{\mu}$ in the infinitesimal
transformation law $\delta\phi=a^{\mu}\partial_{\mu}\phi$ to space-time
dependent parameters $a^{\mu}(x)$, but one can also add, more generally, a
derivative term $(\partial_{\mu}a_{\nu})\psi^{\mu\nu}(x)$ to
$a^{\mu}\partial_{\mu}\phi$, this derivative term vanishing if $a^{\mu}(x)$
reduces to a constant $a^{\mu}$ (see [22] as well as [23] for earlier work
along the same lines). A careful exploitation of the derivative term then
yields $T^{\mu\nu}_{\textrm{off}}$ in a constructive way.

###  3.4 Generalization to a multiplet of self-interacting real or complex
scalar fields

For Minkowski space-time of dimension $n>2$ (as well as for a space of
dimension $n=1$) the previous considerations can be generalized to the case
where the Lagrangian density for the real scalar field $\phi$ involves a scale
invariant self-interaction term [12, 28]:

$\displaystyle{\cal
L}\equiv\frac{1}{2}\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)-V(\phi)\,,\qquad\mbox{with}\quad
V(\phi)\equiv\lambda\,\phi^{\frac{2n}{n-2}}\,,$ (3.23)

$\lambda$ being a dimensionless coupling constant. The term $-V(\phi)$ yields
a contribution $\eta^{\mu\nu}V$ to the canonical EMT
$T^{\mu\nu}_{\textrm{can}}$ and thus a contribution $nV$ to its trace
$T^{\,\mu}_{\textrm{can}\,\mu}$. Thereby, the new improved EMT has a trace
given by $T^{\mu\nu}_{\textrm{conf}}=\frac{n-2}{2}\,\phi\,\Box\phi+nV$. The
equation of motion following from the Lagrangian density (3.23), i.e.
$\Box\phi+V^{\prime}=0$, now implies that $\frac{n-2}{2}\,\phi\,\Box\phi=-nV$.
By way of consequence, the EMT $T^{\mu\nu}_{\textrm{conf}}$ is still _on-shell
traceless_ and thereby reflects the scale invariance of this model.

Instead of a single real self-interacting scalar field $\phi$, we can consider
a _multiplet_ $\Phi$ of such fields as well as a _complex-valued field_ $\phi$
or a _multiplet $\Phi$ of such fields_ with a dynamics described by the scale
invariant Lagrangian density

$\displaystyle{\cal
L}\equiv(\partial^{\mu}\Phi^{\dagger})(\partial_{\mu}\Phi)-\lambda\,(\Phi^{\dagger}\Phi)^{\frac{n}{n-2}}\,.$
(3.24)

As a matter of fact, we will consider the case of a complex-valued field in
section 5 below in the context of a four-dimensional supersymmetric model.

###  3.5 Summary and assessment for a scalar field

The results presented above for the case of a real free massless scalar field
$\phi$ (i.e. the Lagrangian density ${\cal
L}=\frac{1}{2}\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)$) can be summarized
as follows without reference to the method of derivation (addition of the
Lagrangian density ${\cal L}_{1}$ to ${\cal L}$).

For such a field, the on-shell conserved canonical dilatation current (3.6)
has the form

$\displaystyle
j^{\mu}_{\textrm{dil,can}}=T^{\mu\nu}_{\textrm{can}}\,x_{\nu}-\tilde{J}^{\mu}\,,\qquad\mbox{with}\
\;\tilde{J}^{\mu}\equiv\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}\,\tilde{\delta}_{\textrm{dil}}\phi\,.$
(3.25)

Here, $\tilde{\delta}_{\textrm{dil}}\phi=-d_{\phi}\,\phi$ describes the
_passive_ scale transformation
$\tilde{\delta}\phi\equiv\rho\,\tilde{\delta}_{\textrm{dil}}\phi=-\rho\,d_{\phi}\phi$
(see equation (C.6) for the passive point of view of scale transformations).
The quantity $(\tilde{J}^{\mu})$ may be viewed as the “passive symplectic
potential” current density (cf. (A.5) for the general expression of the
symplectic potential)333For a scalar field, the spin matrix
$(\Sigma^{\alpha\beta})$ vanishes and thereby the current $(\tilde{J}^{\mu})$
coincides for these fields with the so-called _virial field_ $(V^{\mu})$ which
plays an important role for the invariance of physical models under both scale
and special conformal transformations, e.g. see reference [28]. For a vector
field $(A^{\mu})$, the spin matrix $(\Sigma^{\alpha\beta})$ does not vanish
and the contribution $(\tilde{J}^{\mu})$ to the canonical dilatation current
(which we will consider in eqn. (3.29) below) only coincides with the Maxwell
virial field up to a (space-time dependent) numerical factor.. More
explicitly, for the scalar field $\phi$, we have
$\tilde{J}^{\mu}=(n-1)\,k^{\mu}$ where
$k^{\mu}\equiv-\xi_{n}\,\partial^{\mu}\phi^{2}$ is the vector field (3.13) on
which our derivation of improvements (using the Lagrangian density ${\cal
L}_{1}=\partial_{\mu}k^{\mu}$) was based.

The results for the improvements are summarized by

$\displaystyle\boxed{\
j^{\mu}_{\textrm{dil,conf}}=T^{\mu\nu}_{\textrm{conf}}\,x_{\nu}\
}\,,\quad\boxed{\
j^{\mu}_{\textrm{dil,conf}}=j^{\mu}_{\textrm{dil,can}}+\partial_{\rho}B^{\rho\mu}\
}\,,\quad\boxed{\
T^{\mu\nu}_{\textrm{conf}}=T^{\mu\nu}_{\textrm{can}}-\partial_{\rho}\chi^{\rho\mu\nu}\
}\,,$ (3.26)

with $B^{\rho\mu}$ and $\chi^{\rho\mu\nu}$ given by (3.20) and (3.17),
respectively. These three relations imply that we have

$\displaystyle
j^{\mu}_{\textrm{dil,can}}=T^{\mu\nu}_{\textrm{can}}\,x_{\nu}-\left[(\partial_{\rho}\chi^{\rho\mu\nu})\,x_{\nu}+\partial_{\rho}B^{\rho\mu}\right]\,.$
(3.27)

Comparison with (3.25) now yields the following decomposition for
$\tilde{J}^{\mu}=(n-1)\,k^{\mu}$:

$\displaystyle(n-1)\,k^{\mu}=\partial_{\rho}(k^{\rho}\eta^{\mu\nu}-k^{\mu}\eta^{\rho\nu})\,x_{\nu}+\partial_{\rho}(x^{\rho}k^{\mu}-x^{\mu}k^{\rho})\,.$
(3.28)

Thus, this relation can be used (and in fact was used in reference [26]) as a
starting point for improving the canonical current densities
$T^{\mu\nu}_{\textrm{can}}$ and $j^{\mu}_{\textrm{dil,can}}$ of a scalar
field.

Relation (3.26) with $k^{\mu}=\frac{1}{n-1}\tilde{J}^{\mu}$ and
$\tilde{J}^{\mu}\equiv\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\varphi)}\,\tilde{\delta}_{\textrm{dil}}\varphi$
can also be applied to the case of a vector field $\varphi=(A^{\mu})$, see
next subsection.

###  3.6 Scale invariance for a vector field

##### General results:

The Lagrangian density for the free Maxwell field, i.e. ${\cal
L}=-\frac{1}{4}\,F^{\mu\nu}F_{\mu\nu}$, is scale invariant in $n$ space-time
dimensions, the vector field $(A^{\mu})$ having a scale dimension
$d_{A}=\frac{n-2}{2}$. By virtue of relation (3.25) and $\frac{\partial{\cal
L}}{\partial(\partial_{\mu}A_{\nu})}=-F^{\mu\nu}$, the on-shell conserved
canonical dilatation current density for the Maxwell field $(A^{\mu})$ reads

$\displaystyle
j^{\mu}_{\textrm{dil,can}}=T^{\mu\nu}_{\textrm{can}}\,x_{\nu}-\tilde{J}^{\mu}\,,\qquad\mbox{where}\
\;\tilde{J}^{\mu}=d_{A}\,F^{\mu\nu}A_{\nu}\,,$ (3.29)

and where the canonical EMT $T^{\mu\nu}_{\textrm{can}}$ is given by (2.6). The
latter EMT is not on-shell traceless, but an EMT with this property may be
obtained by virtue of the improvement (3.26):

$\displaystyle
T^{\mu\nu}_{\textrm{conf}}=T^{\mu\nu}_{\textrm{can}}-\frac{1}{n-1}\,\partial_{\rho}\left(\tilde{J}^{\rho}\eta^{\mu\nu}-\tilde{J}^{\mu}\eta^{\rho\nu}\right)\,.$
(3.30)

An explicit expression readily follows from
$\tilde{J}^{\mu}=d_{A}\,F^{\mu\nu}A_{\nu}=\frac{n-2}{2}\,F^{\mu\nu}A_{\nu}$:

$\displaystyle\boxed{\
(n-1)\,T^{\mu\nu}_{\textrm{conf}}=-\frac{n}{2}\,F^{\mu\rho}\partial^{\nu}A_{\rho}+\frac{n-2}{2}\,(\partial^{\nu}F^{\mu\rho})A_{\rho}+\frac{1}{4}\,\eta^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\
}\,.$ (3.31)

By construction, this EMT is _on-shell conserved and on-shell traceless_ [26],
but it is neither symmetric nor gauge invariant. For the construction of an
_off-shell_ traceless EMT (by application of the generalized Gell-Mann and
Lévy procedure mentioned above) and the ensuing lack of symmetry of this
tensor, we refer to the general discussion in [22].

We note that the combination of expressions (2.11) and (3.30) yields the
relation

$T^{\mu\nu}_{\textrm{conf}}\approx
T^{\mu\nu}_{\textrm{phys}}-\partial_{\rho}\xi^{\rho\mu\nu}\quad\mbox{with}\quad\xi^{\rho\mu\nu}\equiv\frac{1}{2}\frac{n-2}{n-1}\,\left(F^{\rho\sigma}A_{\sigma}\eta^{\mu\nu}-F^{\mu\sigma}A_{\sigma}\eta^{\rho\nu}\right)-F^{\rho\mu}A^{\nu}=-\xi^{\mu\rho\nu}\,,$

where $T^{\mu\nu}_{\textrm{phys}}$ is the physical EMT of the Maxwell field,
the latter being on-shell conserved, gauge invariant and symmetric as well as
traceless for $n=4$. In four space-time dimensions, the EMT
$T^{\mu\nu}_{\textrm{phys}}$ generalizes to the case of a pure non-Abelian
Yang-Mills (YM) field, the Lagrangian for the latter theory being only scale
invariant for $n=4$ due to the fact that the YM coupling constant is
dimensionful for $n\neq 4$ (e.g. see appendix of reference [29] for further
discussion).

##### Derivation from a total derivative:

By combining our previous discussions of scalar and vector fields, we conclude
that the on-shell traceless EMT (3.31) for the free Maxwell field in $n$
space-time dimensions can also be obtained from the total Lagrangian ${\cal
L}_{\textrm{tot}}={\cal L}+{\cal L}_{1}$ with ${\cal
L}=-\frac{1}{4}\,F^{\mu\nu}F_{\mu\nu}$ and

$\displaystyle\boxed{\ {\cal L}_{1}\equiv\partial_{\mu}k^{\mu}\
}\,,\qquad\mbox{with}\quad\boxed{\ k^{\mu}\equiv 2\xi_{n}\,F^{\mu\nu}A_{\nu}\
}\ \quad(\,\xi_{n}\equiv\frac{1}{4}\,\frac{n-2}{n-1}\,)\,.$ (3.32)

This Lagrangian is quasi invariant (i.e. $\delta{\cal
L}_{\textrm{tot}}=\partial_{\mu}\Omega^{\mu}$ for some vector field
$(\Omega^{\mu})$) under translations and under scale transformations and it
yields the total EMT $T^{\mu\nu}_{\textrm{tot}}=T^{\mu\nu}_{\textrm{conf}}$
given in equations (3.30),(3.31).

To conclude, we emphasize that the addition of a total derivative to a
Lagrangian density (which is quasi invariant under some global symmetry
transformations) only induces the addition of a superpotential term to the
Noether currents $(j^{\mu})$: since the equation of motion following from a
total derivative Lagrangian vanishes identically, we cannot generate a
contribution $t^{\mu}\approx 0$ to currents $(j^{\mu})$, i.e. an equation of
motion term (see eqn. (2.5) for the equivalence relation defining currents).
Accordingly, the physical EMT of the Maxwell field cannot be obtained from the
Maxwell Lagrangian by adding a total derivative – see eqn. (2.11) which
involves an equation of motion term $t^{\mu\nu}\approx 0$.

##  4 Conformal transformations of scalar fields

In the following we address the invariance of the action for a real, free or
self-interacting, massless scalar field in $n$ space-time dimensions under
general conformal transformations following the spirit of the previous
section444We wish to thank the anonymous referee for raising this interesting
question.. Remarkably enough, one obtains analogous results and the broader
point of view of conformal symmetry sheds further light on the general
relationships.

##### Generalities:

We recall that the scale transformations discussed in the previous section
represent a particular instance of conformal transformations in Minkowski
space-time $(\mathds{R}^{n},\eta)$. By definition, the _conformal group
associated to $(\mathds{R}^{n},\eta)$_ consists of all transformations
$x\leadsto x^{\prime}(x)$ which preserve the angles, i.e. the Minkowski metric
is preserved under these transformations up to a scale factor:
$ds^{2}\leadsto\textrm{e}^{\lambda}ds^{2}$ where $\lambda$ represents a real
parameter [3]. The associated infinitesimal transformations
$x^{\prime\mu}(x)\simeq x^{\mu}-\Xi^{\mu}(x)$ are generated by _conformal
Killing vector fields_
$\Xi\equiv\Xi^{\mu}\partial_{\mu}\equiv\Xi\cdot\partial$, i.e. solutions of
the

$\displaystyle\mbox{conformal Killing equation
:}\qquad\partial_{\mu}\Xi_{\nu}+\partial_{\nu}\Xi_{\mu}-\frac{2}{n}\,(\partial_{\rho}\Xi^{\rho})\,\eta_{\mu\nu}=0\,.$
(4.1)

The general solution of this equation reads

$\displaystyle\Xi_{\mu}=a_{\mu}+\varepsilon_{\mu\nu}x^{\nu}+\rho\,x_{\mu}+2\,(c\cdot
x)\,x_{\mu}-c_{\mu}x^{2}\,,$ (4.2)

where $a_{\mu}\,,\rho\,,c_{\mu}$ and
$\varepsilon_{\mu\nu}=-\varepsilon_{\nu\mu}$ are constant real parameters.
More precisely, the variables $a_{\mu},\,\varepsilon_{\mu\nu}$ parametrize
infinitesimal Poincaré transformations and $\rho$ labels _scale
transformations (dilatations)_ while $(c_{\mu})$ labels _special conformal
transformations (SCTs)_ which are also referred to as _conformal boosts._ We
note that expression (4.2) implies that
$\partial_{\mu}\Xi^{\mu}=n\rho+2n\,(c\cdot x)$.

Under a conformal transformation generated by the vector field
$\Xi=\Xi^{\mu}\partial_{\mu}$ (with $\Xi^{\mu}$ given by (4.2)), a
relativistic scalar field $\phi$ transforms according to

$\displaystyle\boxed{\ \delta_{\stackrel{{\scriptstyle\
}}{{\Xi}}}\phi=\Xi\cdot\partial\,\phi+\frac{1}{n}\,d\,(\partial_{\mu}\Xi^{\mu})\,\phi\
}\,.$ (4.3)

As before (see eqn. (3.2)), $d\equiv d_{\phi}\equiv\frac{n-2}{2}$ denotes the
scale dimension of the scalar field $\phi$ and the contribution
$\Xi\cdot\partial\,\phi=L_{\Xi}\phi$ to $\delta_{\stackrel{{\scriptstyle\
}}{{\Xi}}}\phi$ may be viewed as the action of the Lie derivative $L_{\Xi}$
with respect to the vector field $\Xi\cdot\partial$ on the field $\phi$. For
instance, for a dilatation, we have $\Xi^{\mu}=\rho x^{\mu}$ (whence
$\partial_{\mu}\Xi^{\mu}=n\rho$) and the transformation law (4.3) thus reduces
(up to a global sign factor for $\Xi^{\mu}$) to the one encountered before,
see eqn. (3.4).

While we dealt with translations and dilatations in the previous section, we
will now be interested as well in Lorentz transformations and (in particular)
in SCTs. We denote the vector field generating SCTs by $\zeta\cdot\partial$:
by virtue of (4.2), the relation (4.3) then yields the following
_transformation law of $\phi$ under an infinitesimal SCT parametrized by
$c^{\mu}$:_

$\displaystyle\delta_{\stackrel{{\scriptstyle\
}}{{\zeta}}}\phi=\zeta\cdot\partial\,\phi+2d\,(c\cdot
x)\,\phi\,,\qquad\mbox{with}\ \;\zeta^{\mu}=2\,(c\cdot
x)\,x^{\mu}-c^{\mu}x^{2}\,.$ (4.4)

Here, the last term is related [28] to the so-called _virial vector field_
$V^{\nu}\equiv\frac{\partial{\cal
L}}{\partial(\partial^{\mu}\phi)}\,(\eta^{\mu\nu}d-\Sigma^{\mu\nu})$ that we
already mentioned in subsection 3.5. In fact, for the dynamics of a real, free
(or self-interacting), massless scalar field $\phi$, i.e. for the Lagrangian
density

$\displaystyle{\cal
L}=\frac{1}{2}\,(\partial^{\mu}\phi)(\partial_{\mu}\phi)-\lambda\,\phi^{\frac{n}{d}}\qquad(\lambda\in\mathds{R})\,,$
(4.5)

the virial field represents the divergence of a _“virial potential”_ :

$\displaystyle V^{\nu}=\partial_{\mu}\sigma^{\mu\nu}\,,\qquad\mbox{with}\
\;\sigma^{\mu\nu}=\frac{1}{4}\,(n-2)\,\eta^{\mu\nu}\phi^{2}\,.$ (4.6)

This potential frequently shows up in the context of SCTs and it is worthwhile
recalling that the condition $V^{\nu}=\partial_{\mu}\sigma^{\mu\nu}$ (for some
tensor field $(\sigma^{\mu\nu})$) ensures the invariance under SCTs for a
field theory which is Poincaré and scale invariant [28].

##### Conformal invariance:

The Lagrangian density (4.5) is quasi invariant under the conformal
transformations (4.3):

$\displaystyle\delta_{\stackrel{{\scriptstyle\ }}{{\Xi}}}{\cal
L}=\partial_{\mu}\Omega^{\mu}\,,\qquad\mbox{with}\
\;\Omega^{\mu}\equiv\Xi^{\mu}{\cal L}+d\,c^{\mu}\,\phi^{2}\,.$ (4.7)

By virtue of Noether’s first theorem, the _canonical current density_
(involving the arbitrary symmetry parameters
$a^{\mu},\varepsilon^{\mu\nu},\dots$) _which is associated to the conformal
invariance of the action_ corresponding to (4.5) is given by
$j^{\mu}_{\textrm{can}}=\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}\,\delta_{\stackrel{{\scriptstyle\
}}{{\Xi}}}\phi-\Omega^{\mu}$ and reads

$\displaystyle\boxed{\
j^{\mu}_{\textrm{can}}=T^{\mu\nu}_{\textrm{can}}\,\Xi_{\nu}+d\,\Big{[}\frac{1}{2n}\,(\partial_{\nu}\Xi^{\nu})\,\partial^{\mu}-c^{\mu}\Big{]}\phi^{2}\
}\,.$ (4.8)

For dilatations, i.e. $\Xi^{\mu}=\rho x^{\mu}$, this expression coincides with
the canonical dilatation current density (3.9) (with
$T^{\mu\nu}_{\textrm{can}}$ including the self-interaction term).

If we express the canonical EMT $T^{\mu\nu}_{\textrm{can}}$ on the right hand
side of relation (4.8) in terms of the new improved EMT
$T^{\mu\nu}_{\textrm{conf}}$ of the scalar field $\phi$ (as given by eqn.
(3.10)), then one expects to obtain (up to a superpotential term) the
Besselhagen form $j^{\mu}_{\textrm{tot}}\equiv j^{\mu}_{\textrm{conf}}\equiv
T^{\mu\nu}_{\textrm{conf}}\,\Xi_{\nu}$ for the canonical current
$j^{\mu}_{\textrm{can}}$: indeed, since $T^{\mu\nu}_{\textrm{conf}}$ is
symmetric as well as on-shell conserved and traceless, we then have (by virtue
of the conformal Killing equation (4.1))

$\displaystyle\partial_{\mu}j^{\mu}_{\textrm{tot}}=(\underbrace{\partial_{\mu}T^{\mu\nu}_{\textrm{conf}}}_{\approx\,0})\,\Xi_{\nu}+T^{\mu\nu}_{\textrm{conf}}\,(\partial_{\mu}\Xi_{\nu})=\frac{1}{2}\,T^{\mu\nu}_{\textrm{conf}}\,(\underbrace{\partial_{\mu}\Xi_{\nu}+\partial_{\nu}\Xi_{\mu}}_{=\,\frac{2}{n}\,(\partial_{\rho}\Xi^{\rho})\,\eta_{\mu\nu}})=\frac{1}{n}\,(\partial_{\rho}\Xi^{\rho})\,\underbrace{T^{\mu}_{\textrm{conf}\,\mu}}_{\approx\,0}\,.$
(4.9)

We saw that this is indeed the case for the dilatation current, see eqn.
(3.11). In the following we address this issue in complete generality for all
conformal transformations.

##### Derivation of Besselhagen form by adding a total derivative to the
Lagrangian:

The Lagrangian density (3.13) with $\xi$ taking the conformal value
$\xi_{n}\equiv\frac{1}{4}\,\frac{n-2}{n-1}$, i.e. ${\cal
L}_{1}\equiv\partial_{\mu}k^{\mu}$ with
$k^{\mu}\equiv-\xi_{n}\,\partial^{\mu}\phi^{2}$, is quasi invariant under the
conformal transformations (4.3):

$\displaystyle\delta_{\stackrel{{\scriptstyle\ }}{{\Xi}}}{\cal
L}_{1}=\partial_{\mu}\Omega_{1}^{\mu}\,,\qquad\mbox{with}\
\;\Omega_{1}^{\mu}\equiv\Xi^{\mu}{\cal
L}_{1}-2\xi_{n}\,(n-2)\,c^{\mu}\,\phi^{2}\,.$ (4.10)

Application of relation (2.4) for the Noether current density associated to
${\cal L}_{1}$ yields a conformal group current density which is a
superpotential term (the latter being determined by the EMT superpotential
$\chi^{\rho\mu\nu}$ introduced in eqn. (3.17)):

$\displaystyle\boxed{\
j^{\mu}_{1}=-\partial_{\rho}(\chi^{\rho\mu\nu}\,\Xi_{\nu})\ }\,.$ (4.11)

The quasi invariance of the total Lagrangian density ${\cal
L}_{\textrm{tot}}\equiv{\cal L}+{\cal L}_{1}$ under conformal transformations
now yields the _total conformal group current density_

$\displaystyle\boxed{\ j^{\mu}_{\textrm{tot}}\equiv
j^{\mu}_{\textrm{can}}+j^{\mu}_{1}=T^{\mu\nu}_{\textrm{conf}}\,\Xi_{\nu}+\partial_{\nu}Y^{\nu\mu}\
}\,,$ (4.12)

with

$\displaystyle\boxed{\
Y^{\nu\mu}\equiv-\xi_{n}\,\varepsilon^{\nu\mu}\,\phi^{2}-2\xi_{n}\,\big{(}x^{\nu}\,c^{\mu}-x^{\mu}c^{\nu}\big{)}\,\phi^{2}=-Y^{\mu\nu}\
}\,.$ (4.13)

Thus, $j^{\mu}_{\textrm{tot}}$ _has the Besselhagen form up to a
superpotential term._ The latter term shows up for Lorentz transformations and
for SCTs. This results from the second term in expression (4.8) as well as
from the $x$-dependence of $\Xi^{\mu}$ which contributes to the total
derivative (4.11): for dilatations these two different contributions to
$j^{\mu}_{\textrm{tot}}$ compensate each other so that it does not contain a
superpotential term (as we already found in the previous section).

_In summary,_ the addition of the total derivative ${\cal
L}_{1}\equiv\partial_{\mu}k^{\mu}$ (with
$k^{\mu}\equiv-\xi_{n}\,\partial^{\mu}\phi^{2}$) to the Lagrangian density
(4.5) of a scalar field allows us to derive the improvements for the conserved
currents associated to conformal invariance in a constructive way.

##### Return to special conformal transformations:

For SCTs, the current $j^{\mu}_{\textrm{can}}$ becomes the so-called
_canonical conformal current_ which we denote by $K^{\mu}_{\textrm{can}}$:

$\displaystyle
K^{\mu}_{\textrm{can}}=T^{\mu\nu}_{\textrm{can}}\,\zeta_{\nu}+d\,\big{[}(c\cdot
x)\,\partial^{\mu}-c^{\mu}\big{]}\phi^{2}\,.$ (4.14)

In this case, relation (4.12) implies the following expression for the
canonical current density $K^{\mu}_{\textrm{can}}$ in terms of the new
improved EMT $T^{\mu\nu}_{\textrm{conf}}$:

$\displaystyle\boxed{\
K^{\mu}_{\textrm{can}}=T^{\mu\nu}_{\textrm{conf}}\,\zeta_{\nu}+\partial_{\nu}(X^{\nu\mu}-X^{\mu\nu})\
}\,,\qquad\mbox{with}\ \;\boxed{\
X^{\nu\mu}\equiv\xi_{n}\big{(}\zeta^{\nu}\,\partial^{\mu}+2c^{\nu}x^{\mu}\big{)}\phi^{2}\
}\,.$ (4.15)

The results (4.14) and (4.15) coincide in four space-time dimensions with the
corresponding results given in reference [30] (where they are derived by
expressing $T^{\mu\nu}_{\textrm{can}}$ in terms of
$T^{\mu\nu}_{\textrm{conf}}$).

##  5 Application in supersymmetric field theory

Invariance under scale transformations and more generally under the
superconformal group (which also involves the conformal group as well as
supersymmetry transformations) plays an important role in supersymmetric field
theories. This fact has already been pointed out in the pioneering work of J.
Wess and B. Zumino [31, 32] and has been further explored later on [33], e.g.
see [34] and references therein. In these investigations, the currents (or the
current superfield englobing these currents) are improved by hand rather than
dynamically, i.e. by adding a total derivative to the Lagrangian density. (An
exception is the article by S. V. Kuzmin and D. G. C. McKeon [27] (which has
essentially gone unnoticed in the literature) which has motivated the
procedure that we will follow in the sequel and on which we will comment at
the end of this section.) More precisely, we will show that the results and
expressions presented in section 3 provide a simple derivation of the
supermultiplet of currents for the free, massless Wess-Zumino (WZ) model in
four space-time dimensions, i.e. the supersymmetric extension of the
Lagrangian density ${\cal
L}\equiv(\partial^{\mu}\bar{\phi})(\partial_{\mu}\phi)$ for a complex scalar
field $\phi$. We limit ourselves to a concise presentation while postponing a
more comprehensive discussion of currents in supersymmetry to a separate work
[35]. In this section, we rely on the well-known basics of global
supersymmetry as presented for instance in the textbooks [36, 37, 38, 39, 40]
whose notation and conventions are also used here555Thus, in this section, we
consider the mostly plus signature for the Minkowski metric. However, by
contrast to the mentioned textbooks, we denote the indices of space-time
coordinates by a greek letter $\mu,\nu,\dots$ (as in the rest of our paper)
rather than a latin letter $m,n,\dots$.

##### Supermultiplet of currents for the WZ model:

For $n=4$, a _scalar superfield_ $\Phi$ satisfies the chirality constraint
$\bar{D}_{{\dot{\alpha}}}\Phi=0$ and thereby admits the component field
expansion

$\displaystyle\Phi(y,\theta)=A(y)+\sqrt{2}\,\theta\,\psi(y)+\theta^{2}F(y)\qquad\mbox{with}\
\;y^{\mu}\equiv x^{\mu}+\textrm{i}\,\theta\sigma^{\mu}\bar{\theta}\,.$ (5.1)

Here, $A$ and $F$ denote complex scalar fields and
$\psi\equiv(\psi^{\alpha})_{\alpha=1,2}$ a Weyl $2$-spinor. By complex
conjugation, we obtain an anti-chiral superfield $\Phi^{\dagger}$ which
satisfies $D_{\alpha}\Phi^{\dagger}=0$ and which gathers the space-time fields
$\bar{A},\bar{F}$ and
$\bar{\psi}\equiv(\bar{\psi}_{\dot{\alpha}})_{{\dot{\alpha}}=1,2}$. The
product of $\Phi^{\dagger}$ and $\Phi$ yields the _vector superfield_
$V\equiv\Phi^{\dagger}\Phi$ which is real-valued, i.e. $V^{\dagger}=V$. The
component field expansion of a generic vector superfield reads

$\displaystyle
V(x,\theta,\bar{\theta})=C(x)+\theta\chi(x)+\bar{\theta}\bar{\chi}(x)+\cdots+\theta^{2}\,\bar{\theta}\bar{\lambda}(x)+\bar{\theta}^{2}\,\theta{\lambda}(x)+\theta^{2}\,\bar{\theta}^{2}\,D(x)\,.$
(5.2)

(Here, we do not consider the reparametrization of the higher component fields
$\lambda,\bar{\lambda},D$ of $V$ in terms of the lower ones
$C,\chi,\bar{\chi}$ which is usually chosen in relationship with
supersymmetric gauge field theories [36] so as to ensure the supergauge
invariance of the photino field $\lambda$ and of the auxiliary field $D$
[39].) For concreteness and for later reference, we spell out [36] the
explicit expressions of some components of the superfield
$V\equiv\Phi^{\dagger}\Phi$ following from the expansion (5.1):

$\displaystyle C$
$\displaystyle=\bar{A}A\,,\qquad\chi=\sqrt{2}\,\bar{A}\psi\,,$ (5.3a)
$\displaystyle\lambda$
$\displaystyle=-\frac{\textrm{i}}{\sqrt{2}}\,\sigma^{\mu}\left[\bar{\psi}(\partial_{\mu}A)-(\partial_{\mu}\bar{\psi})A\right]+\sqrt{2}\,\psi\bar{F}\,,\dots\,,$
(5.3b) $\displaystyle D$ $\displaystyle=\frac{1}{4}\,\bar{A}\,\Box
A+\frac{1}{4}\,(\Box\bar{A})A-\frac{1}{2}\,(\partial^{\mu}\bar{A})(\partial_{\mu}A)-\frac{\textrm{i}}{2}\,\psi\sigma^{\mu}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\\!\bar{\psi}+\bar{F}F\,.$
(5.3c)

Under an _infinitesimal global supersymmetry variation_ parametrized by
constant Weyl $2$-spinors $\zeta\equiv(\zeta^{\alpha})_{\alpha=1,2}$ and
$\bar{\zeta}\equiv(\bar{\zeta}_{\dot{\alpha}})_{{\dot{\alpha}}=1,2}$, the
auxiliary field $D$ transforms into a total derivative:

$\displaystyle\delta_{\zeta}D=\frac{\textrm{i}}{2}\,\partial_{\mu}\left(\zeta\sigma^{\mu}\bar{\lambda}-\lambda\sigma^{\mu}\bar{\zeta}\right)\,.$
(5.4)

Thus, a Lagrangian density ${\cal L}$ (for the supermultiplet $\Phi$) which is
quasi invariant under global supersymmetry transformations is obtained by
considering the highest (i.e. $D$) component of the real superfield
$V\equiv\Phi^{\dagger}\Phi$: by virtue of (5.3c) we have

$\displaystyle{\cal
L}\equiv\left.(\Phi^{\dagger}\Phi)\right|_{\theta^{2}\bar{\theta}^{2}}=D={\cal
L}_{\textrm{WZ}}+\tilde{{\cal L}}_{1}\,,$ (5.5)

with

$\displaystyle{\cal L}_{\textrm{WZ}}$
$\displaystyle\equiv-\partial^{\mu}\bar{A}\,\partial_{\mu}A-\frac{\textrm{i}}{2}\,\psi\sigma^{\mu}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\\!\bar{\psi}+\bar{F}F$
(5.6a) $\displaystyle\tilde{{\cal L}}_{1}$
$\displaystyle\equiv\partial_{\mu}\tilde{k}^{\mu}\,,\qquad\qquad\qquad\mbox{where}\quad\tilde{k}^{\mu}\equiv\frac{1}{4}\,\partial^{\mu}(\bar{A}A)\,.$
(5.6b)

Here, ${\cal L}_{\textrm{WZ}}$ is the standard Lagrangian density for the
_free, massless WZ-model_ and $\tilde{{\cal L}}_{1}$ represents a total
derivative which is usually discarded in the literature due to the fact that
it does not contribute to the equations of motion. If we do not discard the
contribution $\tilde{{\cal L}}_{1}$ to the Lagrangian density (5.5), then we
conclude from the considerations of section 3.3 that the EMT following from
the translation invariance of ${\cal L}$ is the _CCJ-improved EMT for the WZ-
model_ ,

$\displaystyle\boxed{\
\begin{array}[]{rl}-T^{\mu\nu}_{\textrm{CCJ}}=(\partial^{\mu}\bar{A})(\partial^{\nu}A)\\!\\!\\!\\!&+\,(\partial^{\nu}\bar{A})(\partial^{\mu}A)-\frac{1}{3}\,(\partial^{\mu}\partial^{\nu}-\eta^{\mu\nu}\Box)(A\bar{A})\\\
&+\,\frac{\textrm{i}}{4}\,[\psi\sigma^{\mu}\partial^{\nu}\bar{\psi}+\bar{\psi}\bar{\sigma}^{\mu}\partial^{\nu}{\psi}+(\mu\leftrightarrow\nu)]+\eta^{\mu\nu}{\cal
L}_{\textrm{WZ}}\end{array}\ }$ (5.9)

provided we rescale $\tilde{{\cal L}}_{1}$ (i.e. $\tilde{k}^{\mu}$) by the
numerical factor $\frac{4}{3}$, i.e. replace $\tilde{{\cal L}}_{1}$ by

$\displaystyle{\cal L}_{1}\equiv\frac{4}{3}\,\tilde{{\cal
L}}_{1}=\partial_{\mu}{k}^{\mu}\,,\qquad\mbox{with}\quad{k}^{\mu}\equiv\frac{1}{3}\,\partial^{\mu}(\bar{A}A)\,.$
(5.10)

In other words, we consider the total Lagrangian density

$\displaystyle\boxed{\ {\cal L}_{\textrm{tot}}\equiv{\cal
L}_{\textrm{WZ}}+{\cal L}_{1}\ }\qquad\mbox{with ${\cal L}_{1}$ given
by~{}\eqref{eq:ConfTD}}\,.$ (5.11)

Since $\frac{1}{3}=\frac{1}{4}+\frac{1}{12}$, this Lagrangian density may as
well be thought of as resulting from ${\cal L}={\cal
L}_{\textrm{WZ}}+\tilde{{\cal L}}_{1}$ by adding a term
$\frac{1}{12}\,\Box(\bar{A}A)$ which is of the same form as $\tilde{{\cal
L}}_{1}$: this trivial remark will be exploited in a non-trivial manner at the
end of this section.

Next, we determine the supersymmetry current density
$(j_{\textrm{tot}}^{\mu})$, i.e. the on-shell conserved current associated to
the invariance of ${\cal L}_{\textrm{tot}}$ under supersymmetry
transformations of the multiplet $\Phi$, the latter transformations being
given by

$\displaystyle\delta_{\zeta}A=\sqrt{2}\,\zeta\psi\,,\qquad\delta_{\zeta}\psi_{\alpha}=\textrm{i}\sqrt{2}\,(\sigma^{\mu}\bar{\zeta})_{\alpha}\partial_{\mu}A+\sqrt{2}\,\zeta_{\alpha}F\,,\qquad\delta_{\zeta}F=\textrm{i}\sqrt{2}\,\bar{\zeta}\bar{\sigma}^{\mu}\partial_{\mu}\psi\,.$
(5.12)

For the part ${\cal L}_{\textrm{WZ}}$ of the Lagrangian density ${\cal
L}_{\textrm{tot}}$, the results are given in the literature [31, 37]: by
writing the on-shell conserved _supersymmetry currents_ as
$j^{\mu}\equiv\zeta^{\alpha}\,q^{\mu}_{\
\alpha}+\bar{\zeta}_{{\dot{\alpha}}}\,\bar{q}^{\mu\,{\dot{\alpha}}}$, we have
the _supersymmetry current associated to ${\cal L}_{\textrm{WZ}}$_:

$\displaystyle\frac{1}{\sqrt{2}}\,q_{\textrm{WZ}}^{\mu}=(\sigma^{\nu}\bar{\sigma}^{\mu}{\psi})\,\partial_{\nu}\bar{A}\,,\qquad\frac{1}{\sqrt{2}}\,\bar{q}_{\textrm{WZ}}^{\,\mu}=(\bar{\sigma}^{\nu}{\sigma}^{\mu}\bar{\psi})\,\partial_{\nu}{A}\,.$
(5.13)

In this respect, we only note that the derivation of these expressions
involves the supersymmetry variation of ${\cal L}_{\textrm{WZ}}$ [37]:

$\displaystyle-\delta_{\zeta}{\cal
L}_{\textrm{WZ}}=-\partial_{\mu}{\Omega}^{\mu}\equiv\partial_{\mu}\left[\sqrt{2}\,(\zeta\psi)\,\partial^{\mu}\bar{A}-\frac{\textrm{i}}{\sqrt{2}}\,(\zeta\sigma^{\mu}\bar{\psi})\,F+\frac{1}{\sqrt{2}}\,(\zeta\sigma^{\nu}\bar{\sigma}^{\mu}{\psi})\,\partial_{\nu}\bar{A}\right]+\mbox{c.c.}\,.$
(5.14)

The supersymmetry current $\tilde{j}^{\mu}_{1}$ associated to the Lagrangian
density $\tilde{{\cal L}}_{1}\equiv\partial_{\mu}\tilde{k}^{\mu}$ (satisfying
$\delta_{\zeta}\tilde{{\cal L}}_{1}=\partial_{\mu}\tilde{\Omega}_{1}^{\mu}$)
has the general form given by eqn. (6.4) below, i.e.
$\tilde{j}^{\mu}_{1}=\delta_{\zeta}\tilde{k}^{\mu}-\tilde{\Omega}_{1}^{\mu}$.
Here, the contribution $\delta_{\zeta}\tilde{k}^{\mu}$ readily follows from
(5.12):

$\displaystyle\delta_{\zeta}\tilde{k}^{\mu}=\frac{1}{2\sqrt{2}}\,\zeta\,\partial^{\mu}(\psi\bar{A})+\mbox{c.c.}$
(5.15)

The quantity $\tilde{\Omega}_{1}^{\mu}$ is best determined by returning to
relation (5.5) which implies that

$\displaystyle\delta_{\zeta}\tilde{{\cal
L}}_{1}=\delta_{\zeta}D-\delta_{\zeta}{\cal L}_{\textrm{WZ}}\qquad\mbox{with}\
\;D\equiv\left.(\Phi^{\dagger}\Phi)\right|_{\theta^{2}\bar{\theta}^{2}}\,.$
(5.16)

The supersymmetry transformation of $D$ is given by (5.4) with $\lambda$ and
$\bar{\lambda}$ expressed in terms of the components of the superfields
$\Phi^{\dagger}$ and $\Phi$ (see eqn. (5.3b)) which leads to

$\displaystyle\delta_{\zeta}D=$
$\displaystyle\;\partial_{\mu}\left[\frac{1}{2\sqrt{2}}\,\zeta\,(\partial^{\mu}\psi)\bar{A}-\frac{1}{2\sqrt{2}}\,(\zeta\psi)\,\partial^{\mu}\bar{A}\right.$
(5.17) $\displaystyle\qquad\qquad\qquad\qquad\left.\
-\frac{1}{\sqrt{2}}\,(\zeta\sigma^{\mu\nu}\partial_{\nu}\psi)\bar{A}+\frac{1}{\sqrt{2}}\,(\zeta\sigma^{\mu\nu}\psi)\,\partial_{\nu}\bar{A}+\frac{\textrm{i}}{\sqrt{2}}\,({\zeta}{\sigma}^{\mu}\bar{\psi})\,F\right]+\mbox{c.c.}\,.$

Substitution of this expression and of (5.14) into (5.16) yields

$\displaystyle\delta_{\zeta}\tilde{{\cal
L}}_{1}=\partial_{\mu}\tilde{\Omega}_{1}^{\mu}\,,\qquad\mbox{with}\quad\tilde{\Omega}_{1}^{\mu}=\frac{1}{2\sqrt{2}}\,\zeta\,\partial^{\mu}(\psi\bar{A})-\frac{1}{\sqrt{2}}\,(\zeta\sigma^{\mu\nu}\partial_{\nu}({\psi}\bar{A})+\mbox{c.c.}\,.$
(5.18)

The supersymmetry current associated to the Lagrangian density ${\cal L}_{1}$
given by (5.10) now follows from this result and (5.15):

$\displaystyle
j^{\mu}_{1}\equiv\frac{4}{3}\,\tilde{j}^{\mu}_{1}=\frac{4}{3}\,(\delta_{\zeta}\tilde{k}^{\mu}-\tilde{\Omega}_{1}^{\mu})=\frac{4}{3}\,\frac{1}{\sqrt{2}}\,\zeta\sigma^{\mu\nu}\partial_{\nu}({\psi}\bar{A})+\mbox{c.c.}\,.$
(5.19)

As expected on general grounds (see next section) it represents a
superpotential term. In conclusion, the supersymmetry current associated to
the total Lagrangian density $\hat{{\cal L}}_{\textrm{tot}}={\cal
L}_{\textrm{tot}}+{\cal L}_{1}={\cal L}_{\textrm{WZ}}+2{\cal L}_{1}$ has the
form
$\hat{j}^{\mu}_{\textrm{tot}}=\zeta\hat{q}^{\mu}_{\textrm{tot}}+\bar{\zeta}\,\hat{\bar{q}}_{\textrm{tot}}^{\mu}$
with

$\displaystyle\hat{q}^{\mu}_{\textrm{tot}}=q_{\textrm{WZ}}^{\mu}+\frac{4}{3}\,{\sqrt{2}}\,\sigma^{\mu\nu}\partial_{\nu}({\psi}\bar{A})\qquad\mbox{and
c.c.}\,,$ (5.20)

where $q_{\textrm{WZ}}^{\mu}$ is given by (5.13). This expression is gamma-
traceless, i.e. $\sigma_{\mu}\hat{q}^{\mu}_{\textrm{tot}}=0$, and it coincides
with the one for the _improved supersymmetry current_ which was defined by S.
Ferrara and B. Zumino [33] and which is part of the supermultiplet of
conserved currents for the (free massless) Wess-Zumino model.

##### Assessment:

In summary, the only details that we have put in by hand is the choice of the
numerical factor in front of the total derivative $\tilde{{\cal L}}_{1}$ in
eqn. (5.5): for the discussion of the dynamics this factor is usually chosen
to vanish (i.e. the Lagrangian density $\tilde{{\cal L}}_{1}$ is simply
discarded/ignored), but here we choose this factor in such a way that it
ensures the tracelessness of the total EMT (which includes the improvement
term determined by $\tilde{{\cal L}}_{1}$). This choice then yields the proper
expression for the total EMT (namely an on-shell traceless EMT) and, upon
multiplication by an extra factor two (i.e. addition of an identical term),
the proper expression of the supersymmetry current density (namely a gamma-
traceless expression)666We do not have a plausible explanation for the extra
factor of two apart from the fact that it appears to preclude a reformulation
in terms of superfields.. Henceforth, this derivation of the supermultiplet of
conserved currents is both simple and constructive as well as directly based
on a Lagrangian density without the need for ad hoc improvements of the
currents. We will discuss it further in a separate work [35].

##### Comparison with the literature:

As we indicated at the beginning of this section, a similar, albeit more
intricate line of reasoning has been followed by the authors of reference
[27]. For the higher component fields $\lambda,\bar{\lambda},D$ of the real
superfield $V$, the latter authors considered redefinitions in terms of the
lower order components $C,\chi,\bar{\chi}$ by virtue of two real parameters
$a,b$: more precisely, instead of the expansion (5.2) they introduced the
following expansion (where we put primes on the redefined component fields),

$\displaystyle V(x,\theta,\bar{\theta})=$
$\displaystyle\,C(x)+\theta\sigma^{\mu}\bar{\theta}\,v_{\mu}(x)+\theta^{2}\,M(x)+\bar{\theta}^{2}\,\bar{M}(x)$
(5.21)
$\displaystyle\qquad+\left\\{\theta\chi(x)+\bar{\theta}^{2}\,\theta[{\lambda}^{\prime}(x)-\textrm{i}b\,\sigma^{\mu}\partial_{\mu}\bar{\chi}(x)]\,+\,\mbox{c.c.}\right\\}+\theta^{2}\,\bar{\theta}^{2}\,[D^{\prime}(x)+a\,\Box
C(x)]\,.$

The relation $V=\Phi^{\dagger}\Phi$ with $\Phi$ given by (5.1) yields a result
of the form (5.5)-(5.6b), but involving an additional contribution coming from
the term $a\,\Box C=a\,\Box(\bar{A}A)$ in the highest order component of $V$
(see equation (5.21)): we presently have

$\displaystyle D^{\prime}={\cal
L}_{\textrm{WZ}}+(\frac{1}{4}-a)\,\Box(\bar{A}A)\,.$ (5.22)

By choosing $a=-\frac{1}{12}$, this total Lagrangian density takes the form
(5.11), i.e. we have

$\displaystyle\boxed{\ {\cal L}_{\textrm{tot}}\equiv{\cal
L}_{\textrm{WZ}}+\frac{1}{3}\,\Box(\bar{A}A)\ }\,.$ (5.23)

Henceforth, the translation invariance again leads to the CCJ-improved EMT
(5.9) which is on-shell traceless. In comparison to the pedestrian procedure
followed above, the present approach simply requires to fix the parameter $a$
to the particular value $a=-\frac{1}{12}$.

The parameter $b$ does not appear in the Lagrangian density, but it shows up
in the supersymmetry variation of $D^{\prime}$: instead of the transformation
law (5.4), we presently have

$\displaystyle\delta_{\zeta}D^{\prime}=\frac{\textrm{i}}{2}\,\partial_{\mu}(\zeta\sigma^{\mu}\bar{\lambda}^{\prime})-\frac{1}{2}\,(b+2a)\,\Box(\zeta\chi)+\mbox{c.c.}\equiv\partial_{\mu}K^{\prime\mu}\,,$
(5.24)

where $a$ has already been fixed. The total supersymmetry current
$(j_{\textrm{tot}}^{\mu})$ can be determined by working out the general
expression following from Noether’s first theorem as applied to a second order
Lagrangian density, see eqn. (2.4). Since $\delta_{\zeta}{\cal
L}_{\textrm{tot}}=\delta_{\zeta}D^{\prime}=\partial_{\mu}K^{\prime\mu}$, the
expression for $(j_{\textrm{tot}}^{\mu})$ involves the contribution
$K^{\prime\mu}$ and thereby the parameter $b$:

$\zeta{q}^{\mu}_{\textrm{tot}}+\bar{\zeta}\,{\bar{q}}_{\textrm{tot}}^{\mu}\equiv
j_{\textrm{tot}}^{\mu}\equiv\sum_{\varphi}\left\\{\delta_{\zeta}\varphi\left[\frac{\partial{\cal
L}_{\textrm{tot}}}{\partial(\partial_{\mu}\varphi)}-\partial_{\rho}\left(\frac{\partial{\cal
L}_{\textrm{tot}}}{\partial(\partial_{\mu}\partial_{\rho}\varphi)}\right)\right]+\partial_{\rho}(\delta_{\zeta}\varphi)\,\frac{\partial{\cal
L}_{\textrm{tot}}}{\partial(\partial_{\mu}\partial_{\rho}\varphi)}\right\\}-K^{\prime\mu}\,,$

where the sum runs over the fields
$\varphi\in\\{A,\bar{A},\psi,\bar{\psi},F,\bar{F}\\}$. The authors of
reference [27] then argued that a judicious choice of the parameter $b$ yields
a total supersymmetry current which is on-shell gamma-traceless. However,
substitution of $\bar{\lambda}^{\prime}$ (as a function of $b$) and of
$\chi=\sqrt{2}\,\psi\bar{A}$ into (5.24) shows that $\delta_{\zeta}D^{\prime}$
does _not_ depend on $b$. This conclusion is also consistent with the fact
that $D^{\prime}-D=-a\,\Box(\bar{A}A)=\frac{1}{12}\,\Box(\bar{A}A)$ (compare
for instance (5.2) and (5.21) or (5.5) and (5.22)) and that the supersymmetry
variations of $A,\bar{A}$ and of $D$ do not depend on the parameter $b$.

##  6 Main point: Current improvement induced by a total derivative Lagrangian

##### General result:

Consider a Lagrangian density ${\cal L}$ which is quasi invariant under an
infinitesimal symmetry transformation
${\delta}\varphi(x)\equiv\varphi^{\prime}(x)-\varphi(x)$, i.e.

$\displaystyle\delta{\cal L}=\partial_{\mu}\Omega^{\mu}\quad\mbox{(off-
shell)}\,.$ (6.1)

Variation of ${\cal L}$ yields (see equations (A.2)-(A.3))

$\displaystyle\delta{\cal L}=\frac{\delta
S}{\delta\varphi}\,\delta\varphi+\partial_{\mu}J^{\mu}\,.$ (6.2)

Combination of these two equations results in the on-shell conservation
equation $\partial_{\mu}j^{\mu}\approx 0$ for the current density
$j^{\mu}\equiv J^{\mu}-\Omega^{\mu}$.

Now suppose the Lagrangian density is given by a total derivative, i.e. ${\cal
L}=\partial_{\mu}k^{\mu}$. In this case, the derivative $\delta
S/\delta\varphi$ vanishes identically and we thus have the _off-shell
relation_ $\partial_{\mu}j^{\mu}=0$ for $j^{\mu}\equiv J^{\mu}-\Omega^{\mu}$.
Moreover, the variation (6.2) of ${\cal L}$ now writes

$\displaystyle\delta{\cal
L}=\delta(\partial_{\mu}k^{\mu})=\partial_{\mu}(\delta k^{\mu})\,.$ (6.3)

In summary, the _current density $(j^{\mu})$ that is associated to a
Lagrangian ${\cal L}=\partial_{\mu}k^{\mu}$ which is quasi invariant (i.e.
$\delta{\cal L}=\partial_{\mu}\Omega^{\mu}$)_ reads

$\displaystyle\boxed{\ j^{\mu}=\delta k^{\mu}-\Omega^{\mu}\ }\,,$ (6.4)

and it is _conserved off-shell_. More precisely, by virtue of its derivation
and of the property $\partial_{\mu}j^{\mu}=0$, _the current density
$(j^{\mu})$ is defined up to a superpotential term and it represents itself a
superpotential term_.

For instance, if $k^{\mu}$ does not explicitly depend on $x$, the Lagrangian
density ${\cal L}=\partial_{\mu}k^{\mu}$ is quasi invariant under translations
and for these symmetry transformations we have $\Omega^{\mu}=a^{\mu}{\cal L}$
as well as $\delta k^{\mu}=a_{\nu}\partial^{\nu}k^{\mu}$: expression (6.4) now
yields the result

$\displaystyle j^{\mu}=T^{\mu\nu}a_{\nu}\qquad\mbox{with}\quad
T^{\mu\nu}=\partial^{\nu}k^{\mu}-\eta^{\mu\nu}\,\partial_{\rho}k^{\rho}\,,$
(6.5)

i.e.

$\displaystyle\boxed{\ T^{\mu\nu}=-\partial_{\rho}\psi^{\rho\mu\nu}\
}\,,\qquad\mbox{with}\quad\boxed{\ \psi^{\rho\mu\nu}\equiv
k^{\rho}\eta^{\mu\nu}-k^{\mu}\eta^{\rho\nu}=-\psi^{\mu\rho\nu}\ }\,.$ (6.6)

Thus, _the EMT associated to a Lagrangian given by a total derivative
represents a superpotential term._

The conserved current density (6.4) can also be rewritten in terms of _passive
symmetry transformations_ of coordinates and fields (see appendix C for a
discussion of the latter),

$\displaystyle\tilde{\delta}x^{\mu}\equiv
x^{\prime\mu}-x^{\mu}\,,\qquad\tilde{\delta}\varphi(x)\equiv\varphi^{\prime}(x^{\prime})-\varphi(x)\,.$
(6.7)

It then follows from $\delta
k^{\mu}=\tilde{\delta}k^{\mu}-\tilde{\delta}x_{\nu}\,\partial^{\nu}k^{\mu}$
and ${\Omega}^{\mu}=\tilde{\Omega}^{\mu}-\tilde{\delta}x^{\mu}\,{\cal L}$ that

$\displaystyle
j^{\mu}=-T^{\mu\nu}\,\tilde{\delta}x_{\nu}+\tilde{\delta}k^{\mu}-\tilde{\Omega}^{\mu}\,,\qquad\mbox{with
$T^{\mu\nu}$ given by~{}\eqref{eq:EMTtotDer1} }\,.$ (6.8)

This result coincides with the expression for $j^{\mu}$ which is given without
a detailed derivation in reference [26]. A concise derivation from scratch is
presented at the end of appendix C.

We note that, by virtue of its derivation, a superpotential term (6.4) (which
is always conserved without use of the field equations) can only result from a
trivial Lagrangian density, i.e. from a total derivative ${\cal
L}=\partial_{\mu}k^{\mu}$ for which the field equations are trivially
satisfied.

##### Example of scale invariance:

For the example of a real free massless scalar field $\phi$ in $n$ space-time
dimensions and the Lagrangian density ${\cal
L}_{1}\equiv\partial_{\mu}k^{\mu}$ with
$k^{\mu}\equiv-\xi\,\partial^{\mu}\phi^{2}$ (see eqn. (3.13)), relation (6.4)
(or equivalently (6.8)) gives the result (3.17) with
$\chi^{\rho\mu\nu}=\psi^{\rho\mu\nu}$ for the EMT. Similarly, with
$\Omega^{\mu}=-\rho\,x^{\mu}{\cal L}$, $\delta
k^{\mu}=-\rho\,[x\cdot\partial+(n-1)]k^{\mu}$ and $-\rho\,j^{\mu}\equiv\delta
k^{\mu}-\Omega^{\mu}$, we readily obtain the expression (3.20) for the
dilatation current. In this respect we note that the scale transformation of
the scalar field $\phi$ identifies it as a scaling or “quasi-primary” field in
$n$ space-time dimensions [21] and similarly for $\phi^{2}$, but its
derivative $k^{\mu}\propto\partial^{\mu}\phi^{2}$ does not represent such a
field: this reflects itself in the fact that the scale factor $(n-1)$ in the
transformation law of $k^{\mu}$ does not coincide with the canonical scale
dimension $\frac{n-2}{2}$ of a vector field $(A^{\mu})$.

##### Particular instance:

Though the result (6.4) is valid quite generally, it may not be conclusive in
some instances in the sense that it may entail a vanishing expression for the
representative $(j^{\mu})$ of the current density. For instance, in two space-
time dimensions, the Lagrangian density may have a trivial topological form
involving the Levi-Civita symbol $\varepsilon^{\mu\nu}=-\varepsilon^{\nu\mu}$:

$\displaystyle\boxed{\ {\cal L}=\partial_{\nu}k^{\nu}\qquad\mbox{with}\quad
k^{\nu}=\varepsilon^{\nu\mu}{\cal L}_{\mu}\,,\quad\delta{\cal L}_{\mu}=0\ }$
(6.9)

Then, we have $\delta k^{\mu}=0$ and $\Omega^{\mu}=0$, whence
$j^{\mu}\equiv\delta k^{\mu}-\Omega^{\mu}=0$. In this case, the off-shell
conserved current density associated to the invariance of ${\cal L}_{\mu}$
(and thus of ${\cal L}$) is expected to have the topological form

$\displaystyle\boxed{\
j^{\mu}=\partial_{\nu}(\varepsilon^{\nu\mu}R)\qquad\mbox{for some $R$}\ }\,.$
(6.10)

This result may eventually be derived by applying the method of Gell-Mann and
Lévy [13, 14] to determine conserved currents, i.e. one considers symmetry
parameters which are space-time dependent. As a matter of fact, we will
encounter this instance in subsection 7.2 below (see expressions in eqn.
(7.47)) for the two-dimensional sigma model with values in the Lie group
$SU(2)\simeq S^{3}$, the $3$-sphere $S^{3}$ being endowed with the so-called
Berger metric (as well as for various generalizations of this model).

In our concluding remarks we will come back to the general results discussed
in this section.

##  7 Two-dimensional integrable models based on a flat improved current

In this section, we will discuss several classes of two-dimensional sigma
models where the addition of a total derivative to the Lagrangian density
yields an improvement of the conserved Noether current $(j^{\mu})$ which is
associated to the natural symmetry of the action functional: this improvement
ensures that the current $(j^{\mu})$ satisfies a zero curvature (i.e.
flatness) condition whereas the non-improved current does not do so. The
existence of such a flat conserved current allows one to construct an infinite
number of non-local conserved charges and thereby to establish the classical
integrability of the model (by applying for instance the BIZZ-algorithm [41]).
For the sake of clarity, we first address in some detail a $1$-parameter
deformation of the two-dimensional sigma model with target space $SU(2)\simeq
S^{3}$. A $2$-parameter deformation of this model has already been introduced
in 1981 by I. V. Cherednik [42] who argued its classical integrability by
viewing the classical equation of motion as the quasi-classical limit of a
quantum model which is integrable by the quantum inverse scattering method.
The Lagrangian for the $1$-parameter deformation of the $SU(2)$ sigma model
has been spelled out and discussed in references [43, 44, 45, 46]. The
improvement of the current which ensures the flatness of this current has
first been introduced in reference [46]. Our presentation rather relies on the
more recent work of I. Kawaguchi and K. Yoshida [47, 48, 49] who considered
the very same improvement for this current so as to derive the classical
integrability of the model and who added a total derivative to the Lagrangian
density so as to generate this improved current. As a matter of fact, the
latter argument has been generalized to a variety of similar models and we
will briefly outline these results in section 7.3 so as to emphasize the
analogies between these fairly different theories. Our presentation is self-
contained and includes a general proof of the zero curvature condition for the
improved current (appendix E). For further background and details on sigma
models, we refer to the works [50]. It is worth noting that the afore-
mentioned deformations of the two-dimensional $SU(2)$ principal model have
triggered the introduction and study of other families of deformations besides
those considered in section 7.3 (in particular following the works [51, 52]),
e.g. see the reviews [53].

###  7.1 Reminder: Two-dimensional $SU(2)$ principal chiral model

##### Geometric framework:

The two-dimensional $SU(2)$ principal chiral model [54, 55, 56] (also referred
to as $SU(2)\otimes SU(2)$-invariant or $O(4)$ non-linear sigma model) is a
non-linear sigma model on the _source space_ $(\mathds{R}^{2},\eta)$ (i.e.
two-dimensional Minkowski space-time endowed with the metric tensor
$\eta\equiv\textrm{diag}\,(+1,-1)$) and with the _target space_ $G=SU(2)$,
i.e. the compact matrix Lie group $SU(2)$ which can identified geometrically
with the unit sphere $S^{3}$ (homeomorphism $SU(2)\simeq S^{3}$). For the Lie
algebra $\textbf{{g}}=su(2)$ associated to the Lie group $SU(2)$ we consider a
basis $(T^{a})_{a=1,2,3}$ consisting of anti-Hermitean matrices $T^{a}$
satisfying

$\displaystyle[T^{a},T^{b}]={\varepsilon^{ab}}_{c}T^{c}\,,\qquad\kappa^{ab}\equiv\textrm{Tr}\,(T^{a}T^{b})=-\frac{1}{2}\,\delta^{ab}\,.$
(7.1)

Here, $\varepsilon^{abc}$ denotes the components of the Levi-Civita symbol
normalized by $\varepsilon^{123}=1$ and $(\kappa^{ab})$ represents the Cartan-
Killing scalar product in $su(2)$ for which we have chosen the same
normalisation as in references [48, 57]. The latter implies that

$\displaystyle A\in su(2)\qquad\Longleftrightarrow\qquad
A=A^{a}T^{a}\quad\mbox{with}\ \;A^{a}=-2\,\textrm{Tr}\,(AT^{a})\,.$ (7.2)

An explicit realization of the matrices $T^{a}$ (fundamental representation of
$su(2)$) is given by $T^{a}=-\frac{\textrm{i}}{2}\,\sigma^{a}$ where
$(\sigma^{a})_{a=1,2,3}$ denote the Pauli matrices.

Thus, the _fields_ of the model are given by maps

$\displaystyle\Phi\,:\,\mathds{R}^{2}\ \longrightarrow$ $\displaystyle\ \;G$
$\displaystyle x\ \longmapsto$ $\displaystyle\ \Phi(x)\equiv g\,,$ (7.3)

the space-time coordinates being labeled by
$x\equiv(x^{\mu})_{\mu=0,1}\equiv(t,\sigma)$. The action of the model is
defined by considering the _Maurer-Cartan form on $G$,_ i.e. the left-
invariant g-valued $1$-form on $G$ given by $\omega\equiv
g^{-1}dg\in\Omega^{1}(G,\textbf{{g}})$. More precisely, a field
$\Phi:\mathds{R}^{2}\to G$ allows us to pull back this $1$-form on $G$ to a
$1$-form on $\mathds{R}^{2}$:

$\displaystyle\Phi^{*}\,:\,\Omega^{1}(G,\textbf{{g}})\ \longrightarrow$
$\displaystyle\ \;\Omega^{1}(\mathds{R}^{2},\textbf{{g}})$
$\displaystyle\omega\ \longmapsto$ $\displaystyle\
\Phi^{*}\omega=(g^{-1}\partial_{\mu}g)\,dx^{\mu}\,.$ (7.4)

In the last expression and in the sequel, the fields are denoted by $x\mapsto
g(x)\in G$ (cf. (7.3)). For these fields one imposes the usual _boundary
condition_ $\lim_{\sigma\to\pm\infty}g(t,\sigma)=\mathds{1}_{2}$ or [48],
somewhat more generally, $\lim_{\sigma\to\pm\infty}g(t,\sigma)=g_{\pm}$ (=
given constant group element). Here, the convergence is assumed to be given by
a _rapid_ decrease. This implies that the

$\displaystyle\mbox{ $\textbf{{g}}$-valued \emph{current density}
components}\qquad\boxed{\ J^{\mu}\equiv J^{\mu}_{a}T^{a}\equiv
g^{-1}\partial^{\mu}g\ }\,,$ (7.5)

vanish rapidly at spatial infinity, i.e. for $\sigma\to\pm\infty$. Since the
covariant vector field $(J_{\mu})$ corresponds to the components of the
Maurer-Cartan $1$-form, it satisfies the so-called _Maurer-Cartan equation_ or
_flatness condition_ or

$\displaystyle\mbox{zero curvature condition for $(J_{\mu})$ :}\qquad\boxed{\
\partial_{\mu}J_{\nu}-\partial_{\nu}J_{\mu}+[J_{\mu},J_{\nu}]=0\ }\,,$ (7.6)

(as can readily be checked). Accordingly, the g-valued $1$-form
$J_{\mu}dx^{\mu}$ on $\mathds{R}^{2}$ can be viewed as a flat connection
(associated with the symmetry group $SU(2)$).

##### Dynamics:

The _action functional_ of the model is quadratic in $\Phi^{*}\omega$ and
reads

$\displaystyle
S_{0}[g]\equiv\int_{\mathds{R}^{2}}\textrm{Tr}\,\big{[}(\Phi^{*}\omega)\wedge\star(\Phi^{*}\omega)\big{]}\,,$
(7.7)

where $\star$ denotes the Hodge dual of differential forms (associated to the
Minkowski metric on $\mathds{R}^{2}$). More explicitly, we have
$S_{0}[g]=\int_{\mathds{R}^{2}}dt\,d\sigma\,{\cal L}_{0}$ with the _Lagrangian
density_

$\displaystyle\boxed{\ {\cal
L}_{0}=\textrm{Tr}\,\big{[}(g^{-1}\partial^{\mu}g)(g^{-1}\partial_{\mu}g)\big{]}=\kappa_{ab}(g^{-1}\partial^{\mu}g)^{a}(g^{-1}\partial_{\mu}g)^{b}=\kappa_{ab}J^{a\mu}J_{\mu}^{b}\
}\,.$ (7.8)

Under an arbitrary variation $\delta g(x)$ of the field $x\mapsto g(x)$
subject to the considered boundary condition, the action changes by an amount

$\displaystyle\delta
S_{0}[g]=-2\int_{\mathds{R}^{2}}d^{2}x\,\textrm{Tr}\,\big{[}(g^{-1}\delta
g)\,\partial^{\mu}(g^{-1}\partial_{\mu}g)\big{]}\,.$ (7.9)

Thus, the field $x\mapsto g(x)$ admits the

$\displaystyle\mbox{equation of
motion}\quad\partial^{\mu}(g^{-1}\partial_{\mu}g)=0\,,\qquad\mbox{i.e.}\quad\boxed{\
\partial^{\mu}J_{\mu}\approx 0\ }\,.$ (7.10)

Here, $J_{\mu}$ denotes the g-valued current introduced in (7.5) and we use
Dirac’s notation $\approx$ for an equality which holds by virtue of the
equations of motion (weak equality).

##### Equivalent formulation:

Instead of the Maurer-Cartan $1$-form which is left-invariant, one can rewrite
[57] the Lagrangian density of the model in terms of the components
$K_{\mu}\equiv(\partial_{\mu}g)g^{-1}$ of the _right-invariant $1$-form_
$(dg)g^{-1}$, i.e. of $\Phi^{*}(dgg^{-1})=K_{\mu}dx^{\mu}$: by virtue of the
cyclicity of the trace we have

$\displaystyle\boxed{\ {\cal L}_{0}=\textrm{Tr}\,(K^{\mu}K_{\mu})\
}\,,\qquad\mbox{whence}\quad\boxed{\ \partial^{\mu}K_{\mu}\approx 0\
}\,,\qquad\mbox{with}\quad\boxed{\ K_{\mu}\equiv(\partial_{\mu}g)g^{-1}\ }\,.$
(7.11)

The covariant vector field $(K_{\mu})$ satisfies the

$\displaystyle\mbox{zero curvature condition for $(K_{\mu})$ :}\qquad\boxed{\
\partial_{\mu}K_{\nu}-\partial_{\nu}K_{\mu}-[K_{\mu},K_{\nu}]=0\ }\,,$ (7.12)

i.e. the zero curvature condition (7.6) with the opposite sign in front of the
commutator: relation (7.6) is obtained [57] for $K_{\mu}$ by choosing the
opposite sign for $K_{\mu}$, i.e. by considering
$\tilde{K}_{\mu}\equiv-(\partial_{\mu}g)g^{-1}=g(\partial_{\mu}g^{-1})$. The
variable $\tilde{K}_{\mu}$ turns into $J_{\mu}$ upon the replacement
$g\leadsto g^{-1}$ which represents a discrete symmetry of ${\cal
L}_{0}=-\textrm{Tr}\,\big{[}(\partial^{\mu}g)(\partial_{\mu}g^{-1})\big{]}$.

##### Symmetries and conservation laws:

The Lagrangian density ${\cal L}_{0}$ given by (7.8) (and thereby the
corresponding action) admits two natural global symmetries. The fact that the
current $J_{\mu}=g^{-1}\partial_{\mu}g$ is invariant under the left action of
the group $G$ (i.e. under $g\leadsto hg$ with $h\in G$) implies that ${\cal
L}_{0}$ is manifestly _left-invariant_. Analogously, the fact that the current
$K_{\mu}\equiv(\partial_{\mu}g)g^{-1}$ is invariant under the right action of
the group $G$ (i.e. under $g\leadsto gh$ with $h\in G$) implies that ${\cal
L}_{0}$ as rewritten in terms of $K_{\mu}$ (see eqn. (7.11)) is manifestly
_right-invariant_.

At the infinitesimal level, the left/right symmetry transformations of the
fields $x\mapsto g(x)$ are parametrized by $\varepsilon_{L,R}\in\mathds{R}$
and read

$\displaystyle SU(2)_{L}\;:\qquad$
$\displaystyle\delta^{a}_{L}g=\varepsilon_{L}\,T^{a}\,g$ (7.13a)
$\displaystyle SU(2)_{R}\;:\qquad$
$\displaystyle\delta^{a}_{R}g=\varepsilon_{R}\,g\,T^{a}\,.$ (7.13b)

By virtue of Noether’s first theorem, the invariance of ${\cal L}_{0}$ under
these global transformations implies the existence of on-shell conserved
current densities. In the present setting, the latter are conveniently derived
by applying the method of Gell-Mann and Lévy [13, 14], i.e. by rendering the
infinitesimal symmetry parameters $\varepsilon_{L,R}$ space-time dependent:
one readily finds that

$\displaystyle\delta^{a}_{L}{\cal L}_{0}$
$\displaystyle=2\,(\partial_{\mu}\varepsilon_{L})\,\textrm{Tr}\,\big{[}(\partial^{\mu}g)g^{-1}\,T^{a}\big{]}=2\,(\partial_{\mu}\varepsilon_{L})\,\textrm{Tr}\,\big{[}K^{\mu}T^{a}\big{]}$
(7.14a) $\displaystyle\delta^{a}_{R}{\cal L}_{0}$
$\displaystyle=2\,(\partial_{\mu}\varepsilon_{R})\,\textrm{Tr}\,\big{[}g^{-1}(\partial^{\mu}g)\,T^{a}\big{]}=2\,(\partial_{\mu}\varepsilon_{R})\,\textrm{Tr}\,\big{[}J^{\mu}T^{a}\big{]}\,,$
(7.14b)

for $a\in\\{1,2,3\\}$.

##### Summary:

The Lagrangian density

$\displaystyle{\cal
L}_{0}(\partial_{\mu}g,\partial_{\mu}g^{-1})\equiv\textrm{Tr}\,\big{[}J^{\mu}J_{\mu}\big{]}=\textrm{Tr}\,\big{[}K^{\mu}K_{\mu}\big{]}=-\textrm{Tr}\,\big{[}(\partial^{\mu}g)(\partial_{\mu}g^{-1})\big{]}\,,$
(7.15)

is invariant under the symmetry group $SU(2)_{L}\times SU(2)_{R}$ (and under
the discrete symmetry $g\leadsto g^{-1}$): the associated (on-shell conserved)
Noether current densities read

$\displaystyle SU(2)_{L}\;:\qquad$ $\displaystyle
K_{\mu}=(\partial_{\mu}g)g^{-1}=gJ_{\mu}g^{-1}\,,\qquad\
\partial^{\mu}K_{\mu}\approx 0\,,$ (7.16a) $\displaystyle SU(2)_{R}\;:\qquad$
$\displaystyle
J_{\mu}=g^{-1}\partial_{\mu}g\,,\qquad\qquad\qquad\qquad\partial^{\mu}J_{\mu}\approx
0\,.$ (7.16b)

The local conservation laws for $(K_{\mu})$ and $(J_{\mu})$ obviously coincide
with the equations of motion of the model, see equations (7.10) and (7.11),
which reflects the transitivity of the action of symmetry transformations.
Moreover, the components $K_{\mu}$ of the right-invariant $1$-form
$K_{\mu}dx^{\mu}$ satisfy the zero curvature condition (7.12) while the
components $J_{\mu}$ of the left-invariant $1$-form $J_{\mu}dx^{\mu}$ satisfy
the zero curvature condition (7.6). This symmetry structure of the $SU(2)$
principal chiral model is at the origin of the integrability of the model at
the classical and quantum levels. In fact, the described symmetries allow us
to construct a Lax pair and thereby an infinite number of (non-local) charges
satisfying the so-called _Yangian algebra_ [57]. This construction can
equivalently be based on the conserved, flat $SU(2)_{L}$-current $(K_{\mu})$
or on the conserved, flat $SU(2)_{R}$-current $(J_{\mu})$, both currents being
related by the

$\displaystyle\mbox{left-right duality transformation}\qquad
J_{\mu}=g^{-1}K_{\mu}g\,.$ (7.17)

##### The group manifold as a Riemannian space:

For later reference, we recall [58] that the Maurer-Cartan $1$-form $\omega$
can also be used to define the Cartan-Killing metric on $G$, i.e. the natural
Riemannian metric on the group manifold $G$. In this respect, one introduces
local coordinates $\vec{\varphi}\equiv(\varphi^{i})$ to parametrize the
elements of $G$: $\omega\in\Omega^{1}(G,\textbf{{g}})$ then writes

$\displaystyle\omega=\omega^{a}T^{a}\,,\qquad\mbox{with}\
\;\omega^{a}={\omega^{a}}_{i}(\vec{\varphi}\,)\,d\varphi^{i}\in\Omega^{1}(G)\,,$
(7.18)

where the variables $\omega^{a}$ are referred to as frame fields for $G$. The
Cartan-Killing metric on the compact Lie group $G$ is now given by the line
element

$\displaystyle
ds^{2}=-2\,\textrm{Tr}\,(\omega\omega)=-2\,\textrm{Tr}\,(g^{-1}dg\,g^{-1}dg)\,,$
(7.19)

i.e. (with $\textrm{Tr}\,(T^{a}T^{b})=-\frac{1}{2}\,\delta^{ab}$)

$\displaystyle
ds^{2}=\omega^{a}\omega^{a}=g_{ij}d\varphi^{i}d\varphi^{j}\,,\qquad\mbox{with}\
\;g_{ij}\equiv{\omega^{a}}_{i}\,{\omega^{a}}_{j}\,.$ (7.20)

The pull-back of this metric from $G$ to $\mathds{R}^{2}$ yields (cf.
equations (7.4) and (7.19))

$\displaystyle\Phi^{*}(ds^{2})=-2\,\textrm{Tr}\,(J_{\mu}\,J_{\nu})\,dx^{\mu}\,dx^{\nu}\,.$
(7.21)

Contraction of the components $\textrm{Tr}\,(J_{\mu}\,J_{\nu})$ of this
expression with $\eta^{\mu\nu}$ then yields the sigma model Lagrangian ${\cal
L}_{0}$ given by eqn. (7.8). The invariance of this Lagrangian under the
symmetry group $SU(2)_{L}\times SU(2)_{R}$ is tantamount to the left/right
(i.e. bi-invariance) of the Cartan-Killing metric, i.e. the symmetry
transformations (7.13a)-(7.13b) represent _isometries of the metric._

We note that $su(2)_{L}\oplus su(2)_{R}\simeq so(4)$ and that $S^{3}\simeq
SU(2)$ can also be viewed as the _symmetric coset space_ $SO(4)/SO(3)$.

A convenient local parametrization of $g\in SU(2)\simeq S^{3}$ is given [59,
57, 26] by Euler angles $(\varphi^{i})_{i=1,2,3}\equiv(\phi,\theta,\psi)$:

$\displaystyle g=\textrm{e}^{\phi T^{3}}\,\textrm{e}^{\theta
T^{2}}\,\textrm{e}^{\psi T^{3}}\,,\qquad\mbox{with}\quad 0\leq\phi<2\pi\,,\
0\leq\theta\leq\pi\,,\ 0\leq\psi\leq 4\pi\,.$ (7.22)

In terms of $(\phi,\theta,\psi)$, the pullback of the canonical $1$-forms,
$(\Phi^{*}\omega^{a})(x)\equiv J^{a}_{\mu}(x)dx^{\mu}\equiv J^{a}(x)$,
reads777For $\Theta\equiv\frac{\pi}{2}-\theta$, we have
$\cos\Theta=\sin\theta$ and $\sin\Theta=\cos\theta$ corresponding to the
expressions considered in reference [47]. [59, 57]

$\displaystyle J^{1}=-\sin\theta\,\cos\psi\;d\phi+\sin\psi\;d\theta$ (7.23a)
$\displaystyle J^{2}=\sin\theta\,\sin\psi\;d\phi+\cos\psi\;d\theta$ (7.23b)
$\displaystyle J^{3}=\cos\theta\;d\phi+d\psi\,.$ (7.23c)

This implies that the target space metric (7.21) takes the form

$\displaystyle
ds^{2}=\underbrace{d\theta^{2}+\sin^{2}\theta\,d\phi^{2}}_{S^{2}}+\underbrace{(d\psi+\cos\theta\,d\phi)^{2}}_{S^{1}\
\mbox{fibration}}\,.$ (7.24)

This form reflects the $U(1)$-fibration (labeled by $\psi$) of $S^{3}$ over
$S^{2}$ (labeled by $\phi,\theta$), i.e. the so-called _Hopf fibration_
$\pi:S^{3}\to S^{2}$ (principal fiber bundle $S^{3}$ over the base space
$S^{2}$ with fiber $U(1)\simeq S^{1}$ as structure group), see Figure 1.

$\mathds{R}^{2}$compactifyfield $g$$SU(2)\simeq S^{3}$$\pi\,=$Hopf
fibration$S^{2}$acts on$S^{1}\simeq U(1)$ Figure 1: Hopf fibration
$\pi:S^{3}\to S^{2}$.

###  7.2 Two-dimensional sigma model on the squashed $3$-sphere

One may wonder whether the symmetry structure (and thereby the integrability)
of the $SU(2)$-principal model survives a deformation thereof. This motivated
the authors of references [42, 43, 44, 45, 46] and later on I. Kawaguchi and
K. Yoshida [47, 48, 49] to consider a deformation of the group manifold
$SU(2)$ (i.e. of the sphere $S^{3}$) which is no longer a symmetric coset
space and to study the symmetry structure of the corresponding principal
chiral model. As a matter of fact, the deformed $3$-sphere under consideration
was first introduced in 1961 by the French mathematician Marcel Berger (in his
work on the classification of simply connected, normal, homogeneous Riemannian
manifolds with strictly positive sectional curvature [60]). The so-called
_Riemannian Berger sphere_ of dimension $3$ is the $3$-sphere endowed with the
_Berger metric,_ i.e. a $1$-parameter deformation of the standard (“round
sphere”) metric along the Hopf fibers: the metric (7.24) (expressed in terms
of Euler coordinates) thus becomes

$\displaystyle
ds^{2}=\underbrace{d\theta^{2}+\sin^{2}\theta\,d\phi^{2}}_{S^{2}}+\;\alpha\,\underbrace{(d\psi+\cos\theta\,d\phi)^{2}}_{S^{1}\
\mbox{fibration}}\,,$ (7.25)

where $\alpha\in\mathds{R}^{*}$ denotes the deformation parameter. (The case
$\alpha<0$ is referred to as the _Lorentzian Berger sphere_ [61].) Thereby the
volume form of $S^{3}$ is simply rescaled by $\sqrt{|\alpha|}$ and similarly
the constant scalar curvature of $S^{3}$ is rescaled by a constant factor
[62]. The Berger $3$-sphere may be viewed as the most symmetric sphere after
the round sphere [63]: this Riemannian manifold (and more generally the odd-
dimensional spheres endowed with a metric of this type) have been investigated
in mathematics, in particular for constructing counterexamples to several
geometric conjectures, e.g. see the textbooks [64] as well as the work [65]
and references therein. In the physics literature, the Berger spheres are
known as _squashed spheres_ and they have been considered in various contexts,
notably in relationship with integrable models, string theory and black holes
(see [66, 67, 62, 68, 69] and references therein), condensed matter physics
[70] or supersymmetric models [71].

##### Geometric set-up and action functional:

Let us consider a deformation of $G=SU(2)$ described by a $1$-parameter
deformation of the Cartan-Killing metric of $su(2)$: instead of the diagonal
metric $\kappa^{ab}\equiv-\frac{1}{2}\,\delta^{ab}$, we consider a metric of
the form
$(\tilde{\kappa}^{ab}(\alpha))\equiv-\frac{1}{2}\,\textrm{diag}\,(1,1,\alpha)$
where $\alpha\in\mathds{R}^{*}$ denotes a real parameter. The resulting
deformation of the Lagrangian (7.8) can then be written in an invariant form
as

$\displaystyle\boxed{\ {\cal
L}=\textrm{Tr}\,(J_{\mu}J^{\mu})-2C\,\textrm{Tr}\,(T^{3}J_{\mu})\,\textrm{Tr}\,(T^{3}J^{\mu})\
}\,,$ (7.26)

with $C\equiv\alpha-1\neq-1$. With $J_{\mu}=J_{\mu}^{a}T^{a}$ and relations
(7.1)-(7.2), one can equivalently write

$\displaystyle{\cal
L}=-\frac{1}{2}\,\Big{[}\sum_{a=1}^{2}J_{\mu}^{a}J^{a\mu}+(1+C)\,J_{\mu}^{3}J^{3\mu}\Big{]}\,.$
(7.27)

In the sequel, we will consider $\alpha>0$ (i.e. $C>-1$) which ensures that
$(\tilde{\kappa}^{ab})$ is negative definite and that all kinetic terms in the
Lagrangian density (7.26) have the same global sign. The assumption that
$\alpha>0$ also implies that the Berger metric (7.25) is Riemannian, i.e.
positive definite.

We note that I. V. Cherednik [42] originally considered the generalized
$SU(2)$ principal chiral sigma model described by the Lagrangian

$\displaystyle{\cal
L}_{Ch}\equiv\textrm{Tr}\,(J_{\mu}PJ^{\mu})\,,\qquad\mbox{with}\
\;P\equiv\textrm{diag}\,(p_{1},p_{2},p_{3})\,,$ (7.28)

where $p_{1},p_{2},p_{3}$ are real constants. By factoring out one of the
elements of the diagonal matrix $P$, e.g. $p_{1}$, one gets

$\displaystyle P=p_{1}\;\textrm{diag}\,(1,\beta,\alpha)\,,\qquad\mbox{with}\ \
\beta\equiv\frac{p_{2}}{p_{1}}\,,\ \alpha\equiv\frac{p_{3}}{p_{1}}\,.$ (7.29)

The constant $p_{1}$ then represents an overall factor in the Lagrangian
(7.28) and defines a coupling constant (which we ignore in our discussion of
the classical theory except when adding a Wess-Zumino term in subsection
7.3.2). By substituting (7.29) and $J_{\mu}=J_{\mu}^{a}T^{a}$ into (7.28), we
conclude that

$\displaystyle{\cal L}_{Ch}\propto
K_{ab}(\alpha,\beta)\,J_{\mu}^{a}J^{b\mu}\,,\qquad\mbox{with}\
\;K_{ab}(\alpha,\beta)\equiv\textrm{Tr}\,(PT^{a}T^{b})\,.$ (7.30)

Thus, the Lagrangian (7.28) represents a $2$-parameter deformation of the
$SU(2)$ sigma model (as noted for instance in reference [52]). For $\beta=1$
(i.e. $p_{1}=p_{2}$), we recover the $1$-parameter deformation of the $SU(2)$
sigma model given by (7.27) with $1+C=\alpha$. In the literature [42, 44, 46,
51, 72], the latter model is referred to as _asymmetric_ or _(diagonal)
anisotropic_ $SU(2)$ principal chiral model.

The Lagrangian (7.26) was explicitly introduced in the works [43, 45, 46, 48]
and I. Kawaguchi and K. Yoshida [48] referred to it for short as the _squashed
sigma model_. In fact, the underlying one-parameter deformation (labeled by
$C\in\mathds{R}$) of the group manifold $SU(2)\simeq S^{3}$ amounts to
considering the _squashed $3$-sphere,_ i.e. the manifold $S^{3}$ endowed with
a metric such that888With a different overall factor, this corresponds to the
expression given in reference [48], namely
$ds^{2}\equiv-\frac{L^{2}}{2}\,\big{\\{}\textrm{Tr}\,J^{2}-2C\,\left[\textrm{Tr}\,(T^{3}J)\right]^{2}\big{\\}}$.
(cf. equations (7.19)-(7.21) for the undeformed case)

$\displaystyle\Phi^{*}(ds^{2})=-2\left\\{\textrm{Tr}\,(J_{\mu}J_{\nu})-2C\,\textrm{Tr}\,(T^{3}J_{\mu})\,\textrm{Tr}\,(T^{3}J_{\nu})\right\\}dx^{\mu}\,dx^{\nu}\,.$
(7.31)

This Riemannian manifold can still be viewed as a principal fiber bundle over
the base space $S^{2}$, but with $U(1)$-fibration labeled by
$\sqrt{1+C}T^{3}$, i.e. the constant $1+C$ parametrizes the size of the
fibers. In the limit $C\to 0$, expressions (7.26),(7.31) as well those
discussed in the sequel (which is based on reference [48]) reduce to the ones
of the $SU(2)$ principal model considered above.

It is instructive [70] to consider an explicit parametrization of the elements
$g(x)\in SU(2)$ in terms of the components of a unit vector
$n(x)\equiv\left[\begin{array}[]{c}n_{0}(x)\\\
n_{1}(x)\end{array}\right]\in\mathds{C}^{2}$:

$\displaystyle g=\left[\begin{array}[]{rr}\bar{n}_{1}&n_{0}\\\
-\bar{n}_{0}&n_{1}\end{array}\right]\,,\qquad\mbox{with}\
\;1=\mbox{det}\,g=|n_{0}|^{2}+|n_{1}|^{2}\equiv n^{\dagger}n\,.$ (7.34)

The Lagrangian (7.26) then writes

$\displaystyle\boxed{\ {\cal
L}=-2\left[(\partial^{\mu}n^{\dagger})(\partial_{\mu}n)+C\,n^{\dagger}(\partial^{\mu}n)\,(\partial_{\mu}n^{\dagger})n\right]\
}\,,\qquad\mbox{with}\ \;n^{\dagger}n=1\,.$ (7.35)

For $C=0$, one recognizes the $O(4)$-model, i.e. the two-dimensional
$\sigma$-model on $S^{3}$. For $C=-1$, we have the Lagrangian describing the
two-dimensional $\sigma$-model on $CP^{1}$, i.e. the one-dimensional complex
projective space which is homeomorphic to the $2$-sphere $S^{2}$. (The latter
model is also known as _$O(3)$ model_ or _classical Heisenberg model_ since it
represents the continuum limit of the corresponding lattice model.) We remark
that these limiting cases are already encoded in the geometry, i.e. in the
Berger metric (7.25) where $\alpha=1+C$: for $\alpha=1$, this metric is the
standard one of the $3$-sphere $S^{3}$ and for $\alpha\to 0$ it reduces to the
standard metric of the $2$-sphere $S^{2}$. Thus [43, 44, 45, 46, 70], _for
$C\in\,]-1,0[$, the Lagrangian for the two-dimensional $\sigma$-model on the
squashed $3$-sphere interpolates between the $\sigma$-model on $CP^{1}$ and
the $\sigma$-model on $S^{3}$:_ the $1$-parameter deformation is performed in
the direction of the $U(1)$-fibers of the Hopf (principal fiber) bundle
$\pi:S^{3}\to S^{2}$. Such an interpolation is notably of interest in view of
the fact that the $\sigma$-models on $CP^{1}$ and on $S^{3}$ have different
properties (e.g. as far as the possibility of adding topological terms is
concerned [70]). The interpolation can be viewed [44] as a natural realisation
of the Hamiltonian reduction from the $O(4)$ to the $O(3)$ non-linear
$\sigma$-model [73]. We note that expression (7.35) also makes sense for
$x\in\mathds{R}^{d}$ and $n\in\mathds{C}^{N}$, but in the sequel we will not
consider this case nor the parametrization (7.34)-(7.35) and we will rather
rely on expression (7.26).

##### Equation of motion:

Variation of the field $g(x)$ subject to the same boundary condition as for
the undeformed model yields the

$\displaystyle\mbox{equation of motion:}\qquad\boxed{\
0\approx\partial^{\mu}J_{\mu}-2C\,\textrm{Tr}\,(T^{3}J^{\mu})\,[J_{\mu},T^{3}]-2C\,\textrm{Tr}\,(T^{3}\partial_{\mu}J^{\mu})\,T^{3}\
}\,.$ (7.36)

By multiplying this relation with $T^{3}$ and taking the trace, we obtain the
$T^{3}$-component of the previous equation,

$\displaystyle 0\approx(1+C)\,\partial_{\mu}\textrm{Tr}\,(T^{3}J^{\mu})\,,$
(7.37)

which will be related to the symmetries of the model below. Since $C\neq-1$,
eqn. (7.37) implies the relation
$\partial_{\mu}\textrm{Tr}\,(T^{3}J^{\mu})\approx 0$ by virtue of which the
equation (7.36) takes a simpler form:

$\displaystyle
0\approx\partial^{\mu}J_{\mu}-2C\,\textrm{Tr}\,(T^{3}J^{\mu})\,[J_{\mu},T^{3}]\,.$
(7.38)

##### Symmetries:

For $C\neq 0$, the isometries of the metric and thereby the symmetries of the
model described by the Lagrangian ${\cal L}$ still include $SU(2)_{L}$ due to
the left-invariance of $J_{\mu}$, but (due to the $C$-dependent term in ${\cal
L}$) the right-invariance is now broken down to an Abelian $U(1)$-symmetry
corresponding to $T^{3}$: we presently have the invariances

$\displaystyle SU(2)_{L}\;:\qquad$
$\displaystyle\delta^{a}_{L}g=\varepsilon_{L}\,T^{a}g$ (7.39a) $\displaystyle
U(1)_{R}\;:\qquad$ $\displaystyle\delta_{R}g=-\varepsilon_{R}\,g\,T^{3}\,,$
(7.39b)

with $\varepsilon_{L,R}\in\mathds{R}$ and with a conventional minus sign in
the last equation. From a geometric point of view (i.e. considering the
squashed $3$-sphere as a principal fiber bundle over the base space $S^{2}$
with rescaled $U(1)$-fibers), the $SU(2)_{L}$-invariance reflects the
symmetries of the base space $S^{2}$ and the $U(1)_{R}$-invariance the ones of
the $U(1)$-fibers [74].

The (on-shell conserved) Noether current densities associated to the global
symmetries (7.39a)-(7.39b) can again be determined by the procedure of Gell-
Mann and Lévy and they read

$\displaystyle SU(2)_{L}\;:\qquad j^{\mu}\equiv\;$ $\displaystyle
j^{\mu}_{L}=gJ^{\mu}g^{-1}-2C\,\textrm{Tr}\,(T^{3}J^{\mu})\,gT^{3}g^{-1}\,,\qquad\
\partial_{\mu}j^{\mu}\approx 0\,,$ (7.40a) $\displaystyle
U(1)_{R}\;:\qquad\qquad\ \ $ $\displaystyle
j_{R}^{\mu}=(1+C)\,\textrm{Tr}\,(T^{3}J^{\mu})\,,\qquad\qquad\qquad\qquad\partial_{\mu}j_{R}^{\mu}\approx
0\,.$ (7.40b)

The conservation law $\partial_{\mu}j_{R}^{\mu}\approx 0$ coincides with eqn.
(7.37) which followed from the equation of motion (7.36) by projecting out its
$T^{3}$-component. Similarly, from the identities

$\displaystyle\partial^{\mu}(gJ_{\mu}g^{-1})=$
$\displaystyle\,g\,(\partial^{\mu}J_{\mu})\,g^{-1}\,,$ (7.41a)
$\displaystyle\partial^{\mu}(gT^{3}g^{-1})=$
$\displaystyle\,g\,[J^{\mu},T^{3}]\,g^{-1}=[j^{\mu},g\,T^{3}g^{-1}]\,,$
(7.41b)

one readily infers that $\partial_{\mu}j^{\mu}=g\,\textrm{EM}\,g^{-1}$ where
EM denotes the equation of motion function appearing on the right hand side of
eqn. (7.36). Henceforth _the equation of motion of the model is again
equivalent to the local conservation law associated to the left-invariance of
the Lagrangian._ In the limit $C\to 0$, the Noether current $(j^{\mu}_{L})$
reduces to the one of the undeformed model and similarly for $(j^{\mu}_{R})$
and the $U(1)_{R}$-subgroup of $SU(2)_{R}$, see equations (7.14a)-(7.14b) and
(7.16a)-(7.16b).

##### Improved current and modified Lagrangian:

While the left current $(j^{\mu}_{L})$ is on-shell conserved as in the
undeformed model, it does no longer coincide with the components
$K^{\mu}\equiv gJ^{\mu}g^{-1}$ of the right-invariant $1$-form and thereby it
does not satisfy the zero curvature condition (7.12): with
$\varepsilon^{\mu\nu}=-\varepsilon^{\nu\mu}$ and $\varepsilon_{12}\equiv 1$,
the _violation of the zero curvature condition ( 7.12)_ can readily be derived
by using the identity (7.41b) and by taking into account that
$K_{\mu}=gJ_{\mu}g^{-1}$ satisfies (7.12) while $J_{\mu}$ satisfies (7.6): one
finds the relation [48]

$\displaystyle\varepsilon^{\mu\nu}(\partial_{\mu}j_{\nu}-j_{\mu}j_{\nu})=C\varepsilon^{\mu\nu}\,\textrm{Tr}\,(T^{3}[J_{\mu},J_{\nu}])\,g\,T^{3}g^{-1}\,.$
(7.42)

For the model under consideration, this violation of the zero curvature
condition can be eliminated [46, 48, 49] for $C>0$ by adding a _superpotential
term_ to $(j^{\mu})$: for $\epsilon=\pm 1$, the

$\displaystyle\mbox{improved left current density}\qquad\boxed{\
\tilde{j}^{\mu}\equiv
j^{\mu}-\epsilon\,\sqrt{C}\,\varepsilon^{\mu\nu}\,\partial_{\nu}(g\,T^{3}g^{-1})\
}\,,$ (7.43)

is _on-shell conserved_ (since the superpotential term is identically
conserved) and it satisfies the

$\displaystyle\mbox{on-shell zero curvature condition}\qquad\boxed{\
\varepsilon^{\mu\nu}(\partial_{\mu}\tilde{j}_{\nu}-\tilde{j}_{\mu}\tilde{j}_{\nu})\approx
0\ }$ (7.44)

by virtue of the relation

$\displaystyle\varepsilon^{\mu\nu}(\partial_{\mu}\tilde{j}_{\nu}-\tilde{j}_{\mu}\tilde{j}_{\nu})=-\epsilon\,\sqrt{C}\,[\partial_{\mu}j^{\mu},g\,T^{3}g^{-1}]\,,$
(7.45)

and the conservation law $\partial_{\mu}j^{\mu}\approx 0$. Relation (7.45) is
established in appendix E. In the literature, the superpotential term is also
referred to as _topological current_ since it is identically conserved. The
case where $-1<C<0$ will be discussed in subsection 7.3.2.

The improvement (7.43) of the left current density (which ensures the validity
of the on-shell zero curvature condition (7.44)) can be implemented
dynamically by adding a _boundary term_ to the action (i.e. a total derivative
to the Lagrangian density ${\cal L}$): the

$\displaystyle\mbox{modified Lagrangian density }\qquad\boxed{\ \tilde{{\cal
L}}\equiv{\cal
L}-2\epsilon\,\sqrt{C}\,\varepsilon^{\mu\nu}\partial_{\nu}\,\textrm{Tr}\,(J_{\mu}T^{3})\
}\,,$ (7.46)

is left-invariant and it yields the Noether current (7.43). Thus, the
modification of the Lagrangian density does not modify the equation of motion
of the model, but it yields a Noether current which satisfies the zero
curvature condition on-shell (by contrast to the original Lagrangian for which
this condition is not fulfilled). The fact that $\tilde{j}_{\mu}dx^{\mu}$
represents an (on-shell) flat connection $1$-form can be used to construct a
Lax pair and thus to derive an infinite number of conserved charges satisfying
the Yangian algebra [47], thereby establishing the classical integrability of
the model. For this purpose, one can for instance use the so-called BIZZ-
construction [41], see reference [57] for the construction and [47, 48, 49]
for application to the model under consideration.

To conclude, we note that the authors of references [47, 48] only considered
the sign $\epsilon=+1$ in expressions (7.43) and (7.46), but realized later on
[76, 75] that different signs can be chosen. Actually, the latter are of
interest for exploring the integrable structure of the models, in particular
in the case of a two-dimensional sigma model whose target space is three-
dimensional Schrödinger space-time [69], see section 7.3.

##### General structure of modifications:

In relationship with the general formulae put forward in section 6, we note
that the structure of the addition ${\cal L}_{\text{add}}$ to the Lagrangian
density ${\cal L}$ in eqn. (7.46) as well as the associated addition
$j^{\mu}_{\text{add}}\equiv j^{\mu}_{\text{add},a}T^{a}$ to the left current
density $j^{\mu}$ in eqn. (7.43) are of the general form (6.9) and (6.10),
respectively:

$\displaystyle{\cal L}_{\text{add}}$
$\displaystyle=\partial_{\nu}k^{\nu}\qquad\mbox{with}\quad
k^{\nu}=\varepsilon^{\nu\mu}{\cal L}_{\mu},\,\quad{\cal L}_{\mu}\equiv
2\epsilon\,\sqrt{C}\,\textrm{Tr}\,(J_{\mu}T^{3})$ $\displaystyle
j_{\text{add},a}^{\mu}$
$\displaystyle=\partial_{\nu}(\varepsilon^{\nu\mu}R^{a})\qquad\mbox{with}\quad
R^{a}\equiv\epsilon\,\sqrt{C}\,(g\,T^{3}g^{-1})^{a}\,.$ (7.47)

The quantity ${\cal L}_{\mu}$ (and thereby the Lagrangian density ${\cal
L}_{\text{add}}$) is invariant under the global left $SU(2)$-transformations
(i.e. $\delta_{L}^{a}{\cal L}_{\mu}=0$), but it is not invariant if the
symmetry parameter $\varepsilon_{L}$ is considered to be space-time dependent
(as one assumes in the approach of Gell-Mann and Lévy for determining Noether
currents):

$\displaystyle\delta^{a}_{\textrm{loc}}{\cal
L}_{\mu}=(\partial_{\mu}\varepsilon_{L})\,R^{a}\,,$ (7.48)

henceforth

$\displaystyle\delta^{a}_{\textrm{loc}}{\cal
L}_{\text{add}}=\varepsilon^{\nu\mu}\partial_{\nu}(\delta^{a}_{\textrm{loc}}{\cal
L}_{\mu})=\varepsilon^{\nu\mu}\partial_{\nu}\left[(\partial_{\mu}\varepsilon_{L})\,R^{a}\right]=(\partial_{\mu}\varepsilon_{L})\,j^{\mu}_{\text{add},a}\,,$
(7.49)

with $j^{\mu}_{\text{add},a}$ given by (7.47). Thus, we presently have
expressions which admit the general form advocated in section 6.

##### Remark on the dual descriptions of the integrability:

A short calculation shows that the $U(1)_{R}$-current $(j_{R}^{\mu})$ is
related to the $SU(2)_{L}$-current $(j^{\mu})$ or its improvement
$(\tilde{j}^{\mu})$ by

$\displaystyle
j_{R}^{\mu}=\textrm{Tr}\,[(g^{-1}j^{\mu}g)T^{3}]=\textrm{Tr}\,[(g^{-1}\tilde{j}^{\mu}g)T^{3}]\,,$
(7.50)

where the last equality follows from the identity (7.41b). If we write
$j_{R}^{\mu}\equiv-\frac{1}{2}\,j_{R}^{\mu,3}$, then the last equation reads

$\displaystyle\boxed{\
j_{R}^{\mu,3}=(g^{-1}{j}^{\mu}g)^{3}=(g^{-1}\tilde{j}^{\mu}g)^{3}\ }\,.$
(7.51)

This relation generalizes the $T^{3}$-component of the duality transformation
(7.17) which holds for the $SU(2)$ principal model. The discussion of the
integrability of the quashed sigma model [48, 49] can either be based on the
$SU(2)_{L}$-current $(\tilde{j}^{\mu})$ (leading to classical $r/s$-matrices
of _rational type_ that satisfy the extended classical Yang-Baxter equation)
or on the $U(1)_{R}$-current $({j}_{R}^{\mu})$. Indeed, in the latter case, a
specific non-local field constructed from $j_{R}^{0,3}$ can be introduced in
order to obtain non-local conserved currents associated to the broken
components $T^{1},T^{2}$ of $SU(2)_{R}$: thereby one gets currents which
satisfy a current algebra that is associated to a $q$-deformation of the Lie
algebra $su(2)_{R}$ with parameter
$q\equiv\textrm{exp}(\frac{\sqrt{C}}{1+C})$. In this setting, the classical
$r/s$-matrices are of _trigonometric type_ and the duality transformation
(7.51) for the $T^{3}$-component $(j_{R}^{\mu,3})$ of the conserved current
admits a $q$-deformed non-local extension to the $T^{1,2}$-components. _In
summary,_ the broken $SU(2)_{R}$-symmetry of the squashed sigma model can be
promoted to a $q$-deformed $SU(2)_{R}$-symmetry (referred to as “enhanced
$U(1)_{R}$-symmetry” in reference [77]) which provides a description of the
integrable structure of the model which is equivalent to the one based on the
$SU(2)_{L}$-symmetry. (This feature has been referred to as “hybrid classical
integrability” [48, 49].) The squashing of the three-sphere (i.e. of the
$SU(2)$ principal chiral model) represents an integrable deformation despite
the fact that the squashed three-sphere does not define a symmetric coset
space.

##### Remark on the geometric interpretation:

In terms of the Euler angles $(\phi,\theta,\psi)$ defined by eqn. (7.22), the
_improvement term_
$t^{\mu}\equiv\sqrt{C}\,\varepsilon^{\mu\nu}\,\partial_{\nu}(g\,T^{3}g^{-1})\equiv
t^{\mu}_{a}T^{a}$ reads

$\displaystyle\boxed{\
t^{\mu}_{a}=\sqrt{C}\,\varepsilon^{\mu\nu}\,\partial_{\nu}e^{a}\ }\,,$ (7.52)

where the functions $e^{a}$ are the components of a unit vector $\vec{e}\in
S^{2}\subset\mathds{R}^{3}$ parametrized by

$\displaystyle\vec{e}\equiv(e^{a})\equiv(\sin\theta\,\cos\psi,\sin\theta\,\sin\psi,\cos\theta)\,.$
(7.53)

Similarly, in terms of the Euler angles, the _total derivative term_ which was
added to the Lagrangian density ${\cal L}$ in eqn. (7.46) yields the following
contribution to the action functional [47]:

$\displaystyle
H\equiv-2\sqrt{C}\int_{\mathds{R}^{2}}d^{2}x\,\varepsilon^{\mu\nu}\partial_{\nu}\,\textrm{Tr}\,(J_{\mu}T^{3})=\sqrt{C}\int_{\mathds{R}^{2}}d^{2}x\,\varepsilon^{\mu\nu}\sin\theta\,(\partial_{\mu}\theta)(\partial_{\nu}\phi)\,.$
(7.54)

Thus, we have (cf. [59, 57])

$\displaystyle\boxed{\ H=\sqrt{C}\int_{\mathds{R}^{2}}\Omega\
}\,,\qquad\mbox{with}\ \;\boxed{\ \Omega\equiv\sin\theta\,d\theta\wedge
d\phi=\frac{1}{2}\,\varepsilon_{abc}\,n^{a}(dn^{b}\wedge dn^{c})\ }\,.$ (7.55)

Here, $\Omega$ represents a local expression for the _area form on_ $S^{2}$,
written in terms of spherical coordinates $(\phi,\theta)$, and $\vec{n}\in
S^{2}$ is a unit vector written in terms of these coordinates,

$\displaystyle\vec{n}\equiv(n^{a})\equiv(\sin\theta\,\cos\phi,\sin\theta\,\sin\phi,\cos\theta)\,.$
(7.56)

The area $2$-form $\Omega$ on $S^{2}$ is locally exact and can be viewed as
the exterior derivative of the (opposite of the) canonical $1$-form $J^{3}$
given in eqn. (7.23c):

$\displaystyle-dJ^{3}=\sin\theta\,d\theta\wedge d\phi=\Omega\,.$ (7.57)

Concerning the result (7.54), it is worthwhile to recall the Hopf fiber bundle
map [78]:

$\displaystyle\pi\,:\,SU(2)\simeq S^{3}\ \longrightarrow$ $\displaystyle\
\;S^{2}$ $\displaystyle g\ \ \longmapsto$ $\displaystyle\ \
\vec{S}\qquad\mbox{with}\quad\vec{S}\cdot\vec{\sigma}\equiv
g^{-1}\sigma^{3}g\,,$ (7.58)

i.e. $S^{a}=\frac{1}{2}\,\textrm{Tr}\,(\sigma^{a}g^{-1}\sigma^{3}g)$. Then,
the $1$-form ${\cal A}$ on $G=SU(2)$ given by $-\textrm{i}{\cal
A}=\textrm{Tr}\,(\omega T^{3})$ with $\omega\equiv
g^{-1}dg\in\Omega^{1}(G,\textbf{{g}})$ may be viewed as a $U(1)$-connection on
the total space $SU(2)$ of the Hopf bundle. Its pullback to the base manifold
$S^{2}$ yields the connection $1$-form $-\textrm{i}A_{\mu}dx^{\mu}$ with
$A_{\mu}=\textrm{Tr}\,(J_{\mu}T^{3})$: the associated curvature $2$-form
$dA\equiv F\equiv\frac{1}{2}\,F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ (which is
often referred to as a _flux_ $2$-form) has a single component
$F_{01}=\varepsilon^{\mu\nu}\partial_{\mu}A_{\nu}$ and is known [78] to
coincide with the area form of $S^{2}$: this is the fact that we noticed by an
explicit calculation in equations (7.54)-(7.55).

###  7.3 Further examples of the same nature

####  7.3.1 Sigma models on warped $AdS_{3}$ and on Schrödinger space-time

In section 7.2 we considered the Riemannian Berger $3$-sphere, i.e. the
$3$-sphere $S^{3}\simeq SU(2)$ endowed with a Riemannian metric involving a
squashing along the Hopf fibers. The $3$-sphere endowed with the standard
metric represents a $3$-dimensional maximally symmetric Riemannian manifold
with a constant positive scalar curvature. In this respect we recall (e.g. see
reference [79]) that a $3$-dimensional maximally symmetric Lorentzian manifold
with a constant negative scalar curvature is given by the _three-dimensional
anti-de Sitter space-time_ $AdS_{3}\simeq SL(2,\mathds{R})$. In this case, one
can consider a squashing or stretching along fibers that are either space-like
or time-like. The geometry and interest of these spaces are discussed for
instance in references [80, 62, 81, 67, 76, 82]. In relationship with our
foregoing considerations, we note the following. The set-up of section 7.1 and
of section 7.2 concerning two-dimensional sigma models on the source space
$\mathds{R}^{2}$ with target space $G=SU(2)\simeq S^{3}$ as well as the
deformation of the latter (the squashed $3$-sphere) can be generalized to the
case of the _target space_ $G=SL(2,\mathds{R})$ and a $1$-parameter
deformation thereof [47, 77, 57]. Here, $SL(2,\mathds{R})$ represents a three-
dimensional non-compact matrix Lie group which is homeomorphic to anti-de
Sitter space-time $AdS_{3}\simeq SO(2,2)/SO(1,2)$: the latter admits the
isometry group $SO(2,2)$ with $so(2,2)\simeq sl(2,\mathds{R})_{L}\oplus
sl(2,\mathds{R})_{R}$. The metric (7.24) on $S^{3}\simeq SU(2)$ expressed in
terms of Euler angles $(\phi,\theta,\psi)$ (with
$\theta\equiv\frac{\pi}{2}-\Theta$) goes over to the metric of $AdS_{3}$ by
the double Wick rotation $(\phi,\Theta,\psi)\equiv(\tau,\textrm{i}\sigma,iu)$:

$\displaystyle-
ds^{2}=\underbrace{d\sigma^{2}-\cosh^{2}\sigma\,d\tau^{2}}_{AdS_{2}}+(1+C)\,\underbrace{(du+\sinh\sigma\,d\tau)^{2}}_{\mbox{fibration}}\,.$
(7.59)

In the latter equation [66, 81, 67], we already introduced the deformation
parameter $C\in\mathds{R}$. The first two terms in (7.59) represent the metric
of $AdS_{2}$, hence the metric (7.59) of squashed $AdS_{3}$ describes this
space as a real line bundle over Lorentzian $AdS_{2}$ (analogous to the Hopf
bundle $\pi:S^{3}\to S^{2}$ with $U(1)$-fibers), the deformation being
performed along the fiber direction. As a matter of fact, $AdS_{3}$ admits
three types of anisotropic deformations (corresponding to the hyperbolic,
elliptic and parabolic elements of the group $SL(2,\mathds{R})\simeq AdS_{3}$
[62]), namely deformations along space-like, time-like and light-like
directions: expression (7.59) represents the metric of _space-like warped
$AdS_{3}$_ [67]. The metric of _time-like warped $AdS_{3}$_ can be obtained by
a similar Wick rotation, namely
$(\phi,\Theta,\psi)\equiv(\textrm{i}u,\textrm{i}\sigma,\tau)$. The light-like
deformation of $AdS_{3}$ follows from the metric (7.59) (or from its time-like
version) by taking a certain scaling limit and it describes the metric of
_three-dimensional Schrödinger space-time_ [77]. For these deformed sigma
models, the global $SL(2,\mathds{R})_{L}\times SL(2,\mathds{R})_{R}$-symmetry
of the undeformed $SL(2,\mathds{R})$-sigma model is broken to
$SL(2,\mathds{R})_{L}\times U(1)_{R}$. The whole discussion presented in
section 7.2 now carries over to the present setting [77]. In particular, one
can again _improve the on-shell conserved $SL(2,\mathds{R})_{L}$-current so as
to satisfy the zero curvature condition on-shell_ and implement this
improvement dynamically by _adding a total derivative to the Lagrangian
density._ Alternatively, the $U(1)_{R}$-symmetry can be enhanced by
introducing a non-local field in order to get a dual description of the
classical integrability for the deformed sigma model under consideration [77,
69].

####  7.3.2 Deformed WZNW models on the squashed $3$-sphere (and on warped
$AdS_{3}$)

Let us first come back to the sigma model on the squashed $3$-sphere discussed
in section 7.2. For the Lagrangian density (7.26), the deformation parameter
$C$ was assumed to satisfy $C>-1$, but it was further restricted to the range
$C\geq 0$ for the improvement (7.43) of the $SU(2)_{L}$-current and for the
corresponding modification of the Lagrangian as given by eqn. (7.46). The
authors of reference [76] have addressed the generalization of these arguments
to the range of $C$ given by $-1<C<0$ which is the preferred one for certain
physical applications999E.g. it was noted in the first of references [45] that
the quantized theory with $-1\leq C\leq 0$ is asymptotically free. . In this
respect, they added a Wess-Zumino (WZ) term to the action. The resulting model
is then referred to as the _squashed Wess-Zumino-Novikov-Witten (WZNW) model_
and its action reads

$\displaystyle S_{sWZNW}=S_{s\sigma m}+S_{WZ}\,,$ (7.60)

with

$\displaystyle S_{s\sigma m}$
$\displaystyle=\frac{1}{\lambda^{2}}\int_{\mathds{R}^{2}}d^{2}x\,{\cal
L}\qquad\qquad\qquad\qquad\qquad\qquad\ \,\mbox{with ${\cal L}$ given
by~{}\eqref{eq:LagrSSM} }\,,$ (7.61a) $\displaystyle S_{WZ}$
$\displaystyle=\frac{n}{12\pi}\int_{0}^{1}ds\int_{\mathds{R}^{2}}d^{2}x\;\varepsilon_{\hat{\mu}\hat{\nu}\hat{\rho}}\;\textrm{Tr}\\!\left(J_{s}^{\hat{\mu}}J_{s}^{\hat{\nu}}J_{s}^{\hat{\rho}}\right)\qquad\mbox{with}\
n\in\mathds{Z}\,.$ (7.61b)

Here, $\lambda^{2}$ represents the bare coupling constant and the coefficient
of the WZ-term is fixed by dimensionality and quantum consistency (which
implies that it has the same expression as for the usual $SU(2)$ WZNW-model,
e.g. see reference [70] for further details on this point). The integral
defining the WZ-term is done over a (fictitious) three-dimensional base space
parametrized by $(x^{\hat{\mu}})\equiv(x^{\mu},s)\equiv(x,s)$ with
$s\in[0,1]$. The current density $J_{s}^{\hat{\mu}}\equiv
g_{s}^{-1}\partial^{\hat{\mu}}g_{s}$ is defined in terms of a group element
$g_{s}(x)\equiv g(x,s)$ which interpolates continuously between the unit
element and $g(x)\in SU(2)$, i.e. $g_{0}(x)=\mathds{1}$ and $g_{1}(x)=g(x)$.
The Levi-Civita symbol $\varepsilon_{\hat{\mu}\hat{\nu}\hat{\rho}}$ is totally
antisymmetric and normalized by $\varepsilon_{t\sigma s}=1$, henceforth the
integrant of the WZ-term is a three-form. Accordingly, the latter is
proportional to the volume form, e.g. in term of the Euler angles

$\displaystyle\varepsilon_{\hat{\mu}\hat{\nu}\hat{\rho}}\;\textrm{Tr}\\!\left(J_{s}^{\hat{\mu}}J_{s}^{\hat{\nu}}J_{s}^{\hat{\rho}}\right)\,dt\,d\sigma\,ds\propto\varepsilon_{\hat{\mu}\hat{\nu}\hat{\rho}}\,\cos\theta_{s}\,(\partial^{\hat{\mu}}\phi_{s})(\partial^{\hat{\nu}}\theta_{s})(\partial^{\hat{\rho}}\psi_{s})\,dt\,d\sigma\,ds\,.$
(7.62)

Let us briefly summarize the results pertaining to the construction of a flat
current density and thereby of the integrability of the model [76]. The
inclusion of the WZ-term yields an extra contribution to the equation of
motion function (i.e. the right hand side of eqn. (7.36)), namely a
contribution $-\frac{K}{2}\,\varepsilon^{\mu\nu}\,[J_{\mu},J_{\nu}]$ with
$K\equiv\frac{n\lambda^{2}}{8\pi}$. The model with $C\neq 0$ still enjoys the
$SU(2)_{L}\times U(1)_{R}$-invariance of the squashed sigma model, but the
(on-shell conserved) $SU(2)_{L}$-current $j^{\mu}$ given by (7.40a) presently
involves an extra term
$-K\varepsilon^{\mu\nu}(\partial_{\nu}g)g^{-1}=-K\varepsilon^{\mu\nu}gJ_{\nu}g^{-1}$.
The addition of a total derivative of the form (7.46) to the total Lagrangian,
i.e.

$\displaystyle{\cal L}_{sWZNW}\;\leadsto\;{\cal
L}_{sWZNW}+2\,\frac{A}{\lambda^{2}}\,\varepsilon^{\mu\nu}\partial_{\nu}\,\textrm{Tr}\,(J_{\mu}T^{3})\qquad\mbox{with}\
\;A\in\mathds{R}\,,$ (7.63)

implies that the $SU(2)_{L}$-current is improved as in eqn. (7.43), i.e.
$j^{\mu}\leadsto j^{\mu}+A\varepsilon^{\mu\nu}\,\partial_{\nu}(gT^{3}g^{-1})$.
Now the violation (7.42) of the zero curvature condition is no longer
proportional to $C$, but rather to the following combination of factors:

$\displaystyle\beta\equiv C\left(1-\frac{K^{2}}{1+C}\right)-A^{2}\,.$ (7.64)

The vanishing of $\beta$ (i.e the flatness of the conserved current) can be
achieved for $A=0$ as well as for $A\neq 0$, i.e. in the absence as well as in
the presence of the improvement. In the first case (i.e. for $A=0$), we have a
flat current if the deformation parameter $C$ of the squashed WZNW model takes
the particular value $C=K^{2}-1$: this value can be considered if $C\geq-1$,
i.e. even _for negative values of $C$._ Thus, _the sigma model on squashed
$S^{3}$ admits a non improved current which is flat if one includes a WZ term
in the action_ (and chooses the overall coefficient $\lambda^{2}\propto K$
such that $C=K^{2}-1$).

In the second case (i.e. for $A\neq 0$), one obtains a flat current if the
parameters $A,C$ and $K$ are related by the condition

$\displaystyle A^{2}=C\left(1-\frac{K^{2}}{1+C}\right)\,.$ (7.65)

For $K=0$ (no WZ-term), the deformation parameter $C=A^{2}$ is strictly
positive and the choice $A=\epsilon\,\sqrt{C}$ with $\epsilon=\pm 1$ yields
the expressions of section 7.2, see equations (7.43) and (7.46). For $K\neq 0$
and $C>-1$, one can find two solutions $C^{\pm}_{A}(K)$ of the quadratic
algebraic relation (7.65) for $C$: thus, _for negative values of the
deformation parameter ( $-1<C<0$), there is a flat improved current for the
squashed WZNW model._ By way of consequence, this model is classically
integrable, see references [76, 70] for further details. By considering a
double Wick rotation (cf. preceding subsection), the previous results can be
generalized to a _WZNW on warped $AdS_{3}$_ [76].

####  7.3.3 Two-dimensional sigma models on para-complex
$\mathds{Z}_{T}$-cosets

The authors of reference [83] considered a two-dimensional sigma model on
Minkowski space-time $\mathds{R}^{2}$ with a target space given by a certain
para-complex $\mathds{Z}_{T}$-coset $G/H$. The action which was initially
considered for these cosets by C. A. S. Young [84] (i.e. equation (3.1) of
reference [83]) can conveniently be decomposed as a term which reflects the
para-complex structure on $G/H$ and a total derivative term (given by the
relation between equations (3.4) and (3.5) of [83]). Thus, _the total
derivative term in the Lagrangian density yields an improvement term_ (given
in the last equation of section 3.2 of [83]) _for the current density
associated to the global left $G$-symmetry of the action._ This improvement
term (together with an appropriate overall factor) ensures that the resulting
current is not only on-shell conserved and gauge invariant, but also on-shell
_flat,_ i.e. on-shell it satisfies the zero curvature condition. The latter
current can then be used to define a Lax connection of Zakharov-Mikhailov type