# Trace and diffeomorphism anomalies of the classical Liouville theory,
Virasoro algebras, Weyl-gauging and all that

Pavel Haman<EMAIL_ADDRESS>Faculty of Mathematics and Physics,
Charles University, V Holešovičkách 2, 18000 Prague 8, Czech Republic  Alfredo
Iorio<EMAIL_ADDRESS>Faculty of Mathematics and Physics, Charles
University, V Holešovičkách 2, 18000 Prague 8, Czech Republic

###### Abstract

To fully clarify the invariance of the classical Liouville field theory under
the Virasoro algebra, we first elucidate in detail the concept of classical
anomaly, discuss the occurrence of two symmetry algebras associated to this
problem, and provide some new formulae to compute the classical center in a
general fashion. We apply this to the study of the symmetries of the free
boson in two dimensions. Moving to Liouville, we see how this gives rise to an
energy-momentum tensor with non-tensorial conformal transformations, in flat
space, and a non-vanishing trace, in curved space. We provide a variety of
improvements of the (local) theory, that restore Weyl invariance. With
explicit computations, we show that the covariant conservation of the Weyl-
invariance-improved energy-momentum tensor is lost, in general, and relate the
chosen improvement with the corresponding subset of preserved diffeomorphisms.
The non-tensorial transformation rule of the Weyl-invariance-improved energy-
momentum tensor in curved space is explicitly back-traced to the Virasoro
center.

Gravitational anomalies; diffeomorphic invariance; conformal symmetry;
classical Virasoro algebra; Liouville field theory.

##  I Introduction

In this paper we expand and deepen the scope of the investigation reported in
[1], give a detailed explanation for many of the results reported there and
provide new general results and discussions. Although the focus is still on
Liouville field theory, what we say here can pave the way to further
investigations in different models and with different scopes than ours here.

Liouville field theory is an exactly solvable two-dimensional model that
enjoys a prominent role in many fields of the theoretical and mathematical
investigation.

The story started in the 1850s, when Joseph Liouville posed and solved a
problem of the differential geometry of two-dimensional surfaces of constant
Gaussian curvature, $K$, both positive and negative [2]. That is to know the
conformal factor of the metric, say
$g_{\mu\nu}(x,y)=\delta_{\mu\nu}e^{2\rho(x,y)}$. The equation that $\rho$ has
to satisfy is [2]

$\triangle\rho+Ke^{2\rho}=0\,,$ (1)

with $\triangle=\partial_{x}^{2}+\partial_{y}^{2}$.

The story kept going on and on, to the point that Liouville equation is
nowadays ubiquitous in mathematical physics. We find it in lower dimensional
quantum field theories, see, e.g., [3], in condensed matter models, see, e.g.,
[4, 5], and, most notably, in the early models of two-dimensional (quantum)
gravity, see, e.g., [6], governed by

$R-\Lambda=0\,.$ (2)

Indeed, this equation, written in isothermal coordinates, turns out to be
exactly Liouville equation111The Ricci scalar $R$ in two dimensions is related
to $K$ as $R=2K$.

$\Box\rho+\frac{\Lambda}{2}e^{2\rho}=0\,.$ (3)

Perhaps the most known field of application in mathematical physics, though,
is string theory. From the early days of Polyakov’s work [7], till the
nowadays very intense activity of the AdS/CFT correspondence [8]. Indeed, as
well known, turning the point of view the other way around, Liouville theory
is one of the conformal field theories corresponding to the three-dimensional
gravity [9].

It is then important to know its symmetries in all details, already at the
classical level. In particular, Liouville theory is known to be a model
enjoying both scale and full (global) conformal symmetries in flat space,
hence it belongs to the cases studied in [10]. There the reasons why and when
scale invariance of the classical theory (for fields of arbitrary spin in
arbitrary dimensions, including Liouville) implies full conformal invariance
are explained. A crucial ingredient of the recipe is the assumption that Weyl
and diffemorphism invariances of such theories may hold together. Even though
full (global) conformal symmetry is known to be in place, a more recent work
[11] makes the conjecture that Liouville theory might not be made both
diffeomorphic and Weyl invariant, evoking a generic “classical anomaly” as the
reason for that.

In [1] Jackiw’s conjecture is proved and the solutions of those issues are
presented. Here we study the problem from its basic building blocks, starting
by identifying a precise mathematical definition of the suggestive idea of
classical anomaly evoked in [11]. We do so, by first defining a classically
anomalous theory as a theory with genuine central charge in the algebra of
Noether charges, elucidating the key role of the mismatch between the
Hamiltonian and Lagrangian formulations, see, e.g., [12]. This way we clarify
many ambiguities present in the literature and provide some new results, such
as a way to generate the central charge in the case of spatiotemporal
symmetries.

The focus here is on the gravitational anomalies, that we shall show to be
present also at the classical level, and even in flat spacetime. Among those,
Weyl or trace anomaly, found vast applications in quantized theories embedded
in curved spacetimes [13]. There it is quantization that forces the breakdown
of the invariance in the choice of regularization, and the best one can do is
to keep the other invariance, i.e., the one under diffeomorphisms [14]. This
is a spacetime instance of the phenomenon discovered in the early days of the
Standard Model of particle physics, known as the axial or chiral anomaly [15,
16]. Back then, these were seen as mere obstructions in the construction of
consistent theories [17, 18]. The modern approach, though, appreciates that
anomalies are related to fluxes of the energy-momentum, hence could be at the
core of important phenomena, such as Hawking radiation. This is the case of
the two-dimensionial Weyl/trace anomaly [13], and of the breaking of
diffeomorphism invariance in higher dimensions [19] which, together with the
requirement of cancellation of the gauge anomaly and the existence of an event
horizon, necessarily leads to a non-vanishing Hawking flux.

Besides the physical side of the story just recalled, mathematical advances in
geometry and representation theory [20, 21] link anomalies with central
extensions of the algebra of symmetry generators, hence with an exact symmetry
that is softly broken, in a very specific way. Although such breaking is
usually of quantum origin, see, e.g. [22], the algebra extensions are not
exclusive to quantized theories and may emerge already at the classical level
[12].

The paper is organized as follows. To set up the notation, we start off with
Noether theorem in Section II, to then move to symmetries in Hamiltonian
formalism in Section III. This discussion will help clarifying, in Section IV,
that the origin of classical anomalies is the mismatch between Lagrangian and
Hamiltonian formulations. In Section V conformal transformations in two
dimensions are recalled in the light of the previous Sections. In Section VI
we show that an affine center is there already for the free scalar in two
dimensions, it can be set to zero, but it is genuinely there. This paves the
way to Section VII, where Liouville theory in flat space is shown to have a
genuine center, that this time cannot be set to zero. The Section is closed by
remarks of the general validity of some results. This anomalous conformal
symmetry in flat space becomes important, for our analysis, on curved
backgrounds, as shown in Sections VIII and IX. In the crucial Section X, the
energy-momentum tensor (EMT) is studied in detail. The breaking of
diffeomorphism invariance, that is among the most important results of this
paper, is presented in Section XI. Last Section is devoted to our conclusions
and five Appendices provide more details.

##  II Symmetries in Lagrangian formalism and Noether theorem

A classical dynamical system has an anomalous symmetry when it enjoys both a
Hamiltonian and a Lagrangian description, but the algebra obeyed by the
symmetry generators is a centrally extended version of the commutator algebra
obeyed by the symmetry transformations [12].

The first algebra uses Poisson brackets, for which an Hamiltonian description
is necessary, while the symmetry generators are those of Noether theorem,
descending from the Lagrangian formulation. It is then the subtle
discrepancies between the Lagrangian and the Hamiltonian descriptions at the
core of the phenomenon of the classical anomaly. For the sake of clarifying
these issues and to provide the notation that we shall later use, let us now
start with the Noether theorem [23].

Consider the action

$A[\Phi^{i}]=\int\differential[d]{x}\mathcal{L}(\Phi^{i}(x),\partial_{\mu}\Phi^{i}(x))\,,$
(4)

with $\Phi^{i}(x)$ a generic field, or set of fields, and $\mathcal{L}$ the
Lagrangian density, assumed to depend only on fields and their first
derivatives. The latter, although not necessary, simplifies calculations and
it is general enough.

A _symmetry_ is a transformation $\delta\Phi^{i}(x)$ such that

$\delta
A[\Phi^{i},\delta\Phi^{i}]=\int\differential[d]{x}\differential_{\mu}\quantity[K^{\mu}(\Phi^{i}(x),\delta\Phi^{i}(x))]\,,$
(5)

holds for every reasonable input $\Phi^{i}(x)$. Here $K^{\mu}$ is some
functional of $\Phi^{i}(x)$, of $\delta\Phi^{i}$ and of their derivatives, and
$\differential_{\mu}$ denotes a total derivative wrt $x^{\mu}$. Such
transformations are denoted as $\delta_{s}\Phi^{i}(x)$. The symmetries we
consider here belong to _Lie groups_.

Varying the action, first by fixing the transformation and changing the
inputs, and then by fixing the input and changing the transformations, finally
fixing both, the inputs and the transformations, and comparing the two
variations, one gets

$\delta
A[\Phi^{i},\delta_{s}\Phi^{i}]=\int\differential[d]{x}\differential_{\mu}\quantity[K^{\mu}(\Phi^{i}(x),\delta_{s}\Phi^{i}(x))]\approx\int\differential[d]{x}\differential_{\mu}\quantity(\partialderivative{\mathcal{L}}{(\partial_{\mu}\Phi^{i})}\delta_{s}\Phi^{i})\,,$
(6)

where the use of the Euler-Lagrange equations,
$\differential_{\mu}\partialderivative{\mathcal{L}}{(\partial_{\mu}\Phi^{i})}-\partialderivative{\mathcal{L}}{\Phi^{i}}=0$,
is indicated by “$\approx$”.

_Noether theorem_ then states that if an action has an off-shell symmetry,
then there is an on-shell conserved quantity,
$\differential_{\mu}J^{\mu}\approx 0$, where

$J^{\mu}\equiv\partialderivative{\mathcal{L}}{(\partial_{\mu}\Phi^{i})}\delta_{s}\Phi^{i}-K^{\mu}\,,$
(7)

is the _Noether current_. In other words, the Noether charge

$Q[\Phi^{i},\dot{\Phi}{}^{i}](t)=\int_{V}\differential[d-1]{x}J^{0}\,,$ (8)

is conserved, $\derivative*{Q(t)}{t}\approx 0$.

The general form of the Noether current (7) applies to spatiotemporal and
internal symmetries alike. It can be seen that $K^{\mu}$ is nonzero in the
first case, and zero in the second. For the interesting case of supersymmetry,
see [24, 25, 26]

As for spatiotemporal symmetries, we take the active transformations point of
view, hence will compare the original field and the transported field at the
same point222More details on this are in the Appendix A.. The natural
operation to consider is then the _Lie derivative_ $(\mathcal{L}_{f}\Phi)(p)$,
which is the measure of how much a tensor field changes along the flow
generated by $f^{\mu}$ at a given spacetime point $p$. In other words, our
spatiotemporal transformation of a generic functional $F[\Phi^{i}]$ is given
by $\delta_{s}F[\Phi^{i}]\equiv\epsilon\mathcal{L}_{f}F[\Phi^{i}]\equiv
F[\Phi^{i}+\epsilon\mathcal{L}_{f}\Phi^{i}]-F[\Phi^{i}]=F[\Phi^{\prime
i}]-F[\Phi^{i}]$.

Lie derivatives evaluate all objects at the same point, hence commute with
ordinary derivatives, see, e.g., [27]. Spatiotemporal symmetries of special
interest are the isometries $\mathcal{L}_{f}g_{\mu\nu}=0$ that, for Minkowski
spacetime, give for $f^{\mu}$ the Poincaré group. The isometries equation can
be recast into the covariant _Killing equation_

$\nabla_{\mu}f_{\nu}+\nabla_{\nu}f_{\mu}=0\,.$ (9)

Of special interest here are the conformal transformations,
$g^{\prime}_{\mu\nu}(x)=e^{2\lambda(x)}g_{\mu\nu}(x)$, that infinitesimally
may be written as $\mathcal{L}_{f}g_{\mu\nu}(x)=2\lambda(x)g_{\mu\nu}(x)$,
where $\lambda$ is uniquely given by tracing, yielding the covariant
_conformal Killing equation_

$\nabla_{\mu}f_{\nu}+\nabla_{\nu}f_{\mu}=\frac{2}{d}g_{\mu\nu}\nabla_{\rho}f^{\rho}\,.$
(10)

In flat spacetime the solutions form the conformal group [22, 28].

Internal transformations only act on the fields, thus the passive
interpretation based on the transport of the observer does not apply here, see
Appendix A and [29]. In accordance with spatiotemporal transformations, a
local variation of the field can be introduced
$\Delta\Phi^{i}(x)\equiv\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\quantity[{\Phi^{\prime}_{\epsilon}}{}^{i}(x)-\Phi^{i}(x)]$
and this too commutes with ordinary derivatives, hence it fits our approach to
Noether theorem.

Weyl transformation, important here, is a spatiotemporal transfromation that
can be recast as internal as

$g_{\mu\nu}(x)\rightarrow
g^{\prime}_{\mu\nu}(x)=e^{2\omega(x)}g_{\mu\nu}(x)\,,$ (11)

and other fields get a scaling factor

$\Phi^{i}(x)\rightarrow\Phi^{\prime
i}(x)=e^{d_{\Phi^{i}}\omega(x)}\Phi^{i}(x)\,,$ (12)

where $d_{\Phi^{i}}$ is the scaling dimension of the field. Conformal and Weyl
transformations are similar and often connected, see, e.g., [10], but they are
not the same for their different origin and meaning. Nonetheless, the names
“conformal”, “Weyl” and, most of all, “scale” transformations are too often,
if not mistakenly, used interchangeably in the literature, as told, e.g., in
[30].

With this, we can now safely treat both types of transformations,
spatiotemporal and internal, in a unified manner, through the _geometric
transformation_ of [28]. For any infinitesimal transformation we have

$\delta_{f}\Phi^{i}(x)\equiv\Phi^{\prime
i}(x)-\Phi^{i}(x)=\begin{cases}\epsilon\mathcal{L}_{f}\Phi^{i}&\text{spatiotemporal,}\\\
\epsilon\Delta_{f}\Phi^{i}&\text{internal,}\end{cases}$ (13)

where $f$ represents any characterization of the transformation (e.g. a vector
field $f^{\mu}$ for spatiotemporal flows) and $\epsilon$ is an infinitesimal
parameter.

Having a spatiotemporal symmetry transformation characterized by a vector
field $f^{\mu}$, the Noether current can be simplified into a Bessel-Hagen
form, [31], defining the EMT333With $\Theta_{\mu\nu}$ we indicate the EMT in
flat space, while $T_{\mu\nu}$ is for curved space. $\Theta_{\mu\nu}$

$J_{f}^{\mu}=\Theta^{\mu\nu}f_{\nu}\,.$ (14)

For a translations, $K^{\mu}$ in (7) is $K^{\mu}=f^{\mu}\mathcal{L}$, which
defines the _canonical EMT_

${\Theta_{\text{can}}^{\mu\nu}}=\sum_{i}\partialderivative{\mathcal{L}}{(\partial_{\mu}\Phi^{i})}\partial^{\nu}\Phi^{i}-\eta^{\mu\nu}\mathcal{L}\,,$
(15)

that, in general, is not symmetric nor traceless, thus _improvements_ , such
as

$K^{\mu}\rightarrow K^{\mu}+\differential_{\nu}V^{\mu\nu}\,,$ (16)

where $V^{\mu\nu}=-V^{\nu\mu}$, are necessary to make it so. The Belinfante
tensor is a well-known example [28].

For diffeomorphism invariant theories, the EMT can be derived in a variational
way

$T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\functionalderivative{\mathcal{A}}{g_{\mu\nu}}\,,$
(17)

it is inherently symmetric and, in the flat limit, it gives the improved
version of the canonical EMT of (15).

With this procedure the invariances of the theory are immediately inherited by
the EMT, so no improvements are necessary. Indeed,

$\delta_{s}\mathcal{A}\approx-\frac{1}{2}\int\differential[d]{x}\sqrt{-g}T^{\mu\nu}\delta_{s}g_{\mu\nu}=0\,,$
(18)

and for diffeomorphism invariance,
$\delta_{s}g_{\mu\nu}=\mathcal{L}_{f}g_{\mu\nu}$, it gives
$\int\differential[d]{x}\sqrt{-g}f_{\nu}\nabla_{\mu}T^{\mu\nu}=0$, where
$\nabla_{\mu}$ is the covariant derivative. Since the above equation holds for
any $f^{\mu}$, the diffeomorphism invariance implies the conservation
$\nabla_{\mu}T^{\mu\nu}=0$. Similarly, for Weyl transformation (11),
$\delta_{s}g_{\mu\nu}=2\omega g_{\mu\nu}$, it gives

$\int\differential[d]{x}\sqrt{-g}\omega T^{\mu}{}_{\mu}=0\,,$ (19)

and the arbitrariness of $\omega(x)$ implies the vanishing trace of the EMT
$T^{\mu}{}_{\mu}=0$.

For infinitesimal transformations an important tool is the algebra of symmetry
transformations. A commutator of two consecutive geometric transformations
defines another geometric transformation

$[\delta_{f},\delta_{g}]\Phi^{i}=\delta_{h}\Phi^{i}\,,$ (20)

with composition law $h^{\mu}=h^{\mu}(f,g)$.

For spatiotemporal symmetry transformations the algebra is given by
commutators of Lie derivatives
$\commutator{\mathcal{L}_{g}}{\mathcal{L}_{f}}\Phi^{i}=\mathcal{L}_{\commutator{g}{f}}\Phi^{i}$
where the composition law
$\commutator{g}{f}^{\mu}=g^{\nu}\partial_{\nu}f^{\mu}-f^{\nu}\partial_{\nu}g^{\mu}=\mathcal{L}_{g}f^{\mu}$
is the _Lie bracket_. The generators of the transformations, $G_{a}$, can be
introduced [22]

$\delta\Phi^{i}(x)=\Phi^{\prime
i}(x)-\Phi^{i}(x)\equiv\sum_{a}\epsilon^{a}G_{a}\Phi^{i}(x)\,,$ (21)

to simplify the algebra to

$\commutator{G_{a}}{G_{b}}\Phi^{i}={f_{ab}}^{c}G_{c}\Phi^{i}\,,$ (22)

where ${f_{ab}}^{c}$ are the structure constants of the Lie algebra.

##  III Symmetries in Hamiltonian formalism

A theory in Hamiltonian mechanics is characterized by its Hamiltonian $H$ and
the Poisson brackets operator $\poissonbracket{\cdot}{\cdot}$ built upon a
phase space. Then, the transformations of phase-space variables, or associated
functionals, are ruled by the Poisson brackets operator and by the appropriate
generator (e.g., the Hamiltonian $H$ is the generator of the time
translations). For us, the Hamiltonian density, $\mathcal{H}$, depends only on
fields, $\Phi^{i}$, conjugated momenta, $\Pi_{i}$, and spatial derivatives
$\gradient$ of the fields

$H[\Phi^{i},\Pi_{i}](t)=\int\differential[d-1]{\mathbf{x}}\mathcal{H}(\Phi^{i}(\mathbf{x},t),\Pi_{i}(\mathbf{x},t),\gradient\Phi^{i}(\mathbf{x},t)).$
(23)

For non-singular Lagrangians that condition descends from the Legendre
transformation, while for singular Lagrangians we take this condition as a
reasonable assumption, as explained in Appendix B. With the usual definition
of Poisson brackets between two local functionals, for a field and its
conjugated momentum we have

$\poissonbracket{\Phi^{i}(\mathbf{x},t)}{\Pi_{j}(\mathbf{y},t)}=\delta^{i}_{j}\delta(\mathbf{x}-\mathbf{y})\,.$
(24)

The dynamics of a generic quantity $F[\Phi^{i},\Pi_{i}](t)$ is governed by

$\derivative{F[\Phi^{i},\Pi_{i}](t)}{t}\approx\poissonbracket{F(t)}{H(t)}+\partialderivative{F[\Phi^{i},\Pi_{i}](t)}{t}\,,$
(25)

where $\approx$ denote the use of Hamilton equations.

In this Section, we want to prove a somewhat “inverse statement” of the
Noether theorem, namely that a conserved charge generates a symmetry
transformation via Poisson brackets. More precisely, if the transformation
$\delta_{f}\Phi^{i}$ is a symmetry of an action $A$ and
$Q[\Phi^{i},\Pi_{i}|f](t)$ is the associated Noether charge, then

$\delta_{f}\Phi^{i}=\poissonbracket{\Phi^{i}}{Q[f]}\,,$ (26)

which is stated in _Hamiltonian formalism_ while the Noether charge is
obtained in _Lagrangian formalism_. Therefore, the proof requires
reformulating quantities in Hamiltonian version. It is assumed there exists an
invertible relation between conjugated momenta $\Pi_{i}$ and time derivative
of the fields $\dot{\Phi}{}^{i}$. The only quantity containing time
derivatives of the fields is the _Hamiltonian Lagrangian density_
$\mathcal{L}_{H}(\Phi^{i},\dot{\Phi}{}^{i},\gradient\Phi^{i},\Pi_{i})=\dot{\Phi}{}^{i}\Pi_{i}-\mathcal{H}(\Phi^{i},\gradient\Phi^{i},\Pi_{i})$.

Before moving to the proof of (26) we derive a useful off-shell identity for
conserved quantities. Noether charge $Q[f]$ satisfies
$\derivative*{Q[f]}{t}\approx 0$, but on-shell $Q[f]$ also satisfies (25),
$\poissonbracket{Q[f]}{H}+\partialderivative*{Q[f]}{t}\approx 0$. Since there
are no time derivatives of the fields, the equations of motion are not
actually needed, hence if that expression is zero on-shell, it is so off-shell

$\poissonbracket{Q[f]}{H}+\partialderivative{Q[f]}{t}=0\,.$ (27)

With this, we can now prove that the Noether charge is a symmetry generator.
We shall compute the total time derivative of the charge, through Poisson
brackets, and then taking the time derivative of its definition. By comparing
the two results we should be able to see the result.

The total time derivative of a generic functional $F$ is

$\derivative{t}F[\Phi^{i},\Pi_{i}](t)=\int\differential[d-1]{\mathbf{x}}\quantity(\functionalderivative{F}{\Phi^{i}}\dot{\Phi}{}^{i}+\functionalderivative{F}{\Pi_{i}}\dot{\Pi}_{i})+\partialderivative{t}F[\Phi^{i},\Pi_{i}](t)\,.$
(28)

If $F$ is the Noether charge, hence it satisfies (27), and

$\derivative{t}Q[\Phi^{i},\Pi_{i}|f](t)=\int\differential[d-1]{\mathbf{x}}\quantity[\functionalderivative{Q}{\Phi^{i}}\quantity(\dot{\Phi}{}^{i}-\functionalderivative{H}{\Pi_{i}})+\functionalderivative{Q}{\Pi_{i}}\quantity(\dot{\Pi}_{i}+\functionalderivative{H}{\Phi^{i}})]\,.$
(29)

To compute the total time derivative of $Q[f]$ from its definition (8), the
Hamiltonian version is taken as the starting point

$Q[\Phi^{i},\Pi_{i}|f](t)=\int\differential[d-1]{\mathbf{x}}J^{0}=\int\differential[d-1]{\mathbf{x}}\quantity(\partialderivative{\mathcal{L}_{H}}{(\partial_{0}\Phi^{i})}\delta_{f}\Phi^{i}-K_{H}^{0})\,,$
(30)

hence

$\derivative{t}Q[\Phi^{i},\Pi_{i}|f](t)=\int\differential[d-1]{\mathbf{x}}\quantity[-\delta_{f}\Pi_{i}\quantity(\dot{\Phi}{}^{i}-\functionalderivative{H}{\Pi_{i}})+\delta_{f}\Phi^{i}\quantity(\dot{\Pi}_{i}+\functionalderivative{H}{\Phi^{i}})].$
(31)

Comparing (31) to (29), and taking into account that both equations are off-
shell, hence the fields $\Pi_{i}$ and $\Phi^{i}$ are arbitrary and
independent, the quantities in the parentheses are arbitrary, non-zero and
independent. Therefore

$\begin{split}\delta_{f}\Pi_{i}=&-\functionalderivative{Q}{\Phi^{i}}\quad\Rightarrow\quad\delta_{f}\Pi_{i}=\poissonbracket{\Pi_{i}}{Q[f]}\,,\\\
\delta_{f}\Phi^{i}=&~{}\functionalderivative{Q}{\Pi_{i}}\phantom{-}\quad\Rightarrow\quad\delta_{f}\Phi^{i}=\poissonbracket{\Phi^{i}}{Q[f]}\,.\end{split}$
(32)

This proves (26).

Here we assumed smooth boundary conditions, which may not always be ensured.
In case of more general asymptotic behavior and boundary te rms, it is often
necessary to use equations of motion. Nonetheless, in the general case, the
Noether charges always generate symmetry transformations _on-shell_.

##  IV Classical anomalies, a tale of two algebras

Although, the study of centrally extended algebras of symmetries are of great
interest in quantum physics, there has been only a handful of studies on the
central extension at the classical level, [32]. The strong emphasis on
quantized theories might induce to believe that central charges are quantum
effects only. The scope of this work is to show how a center appears in a very
important classical model, Liouville theory, and to study the physical
implications of these on classical anomalies.

Symmetries of a theory manifests themselves as an algebra of symmetry
transformations, on the one hand, and as an algebra of Noether charges, on the
other. If they are manifestations of the same phenomenon, only written within
different formalisms, these two algebras need to be isomorphic. This is what
we want to explore here, and show how this is at the core of the phenomenon of
classical anomalies [12].

In the Lagrangian formalism, the infinitesimal symmetries manifest themselves
in the algebra of transformations. It is only natural to expect an equivalent
structure in Hamiltonian formalism. Indeed, it can be shown that Poisson
brackets of two conserved charges is again a conserved quantity,
$\derivative*{\poissonbracket{Q[f]}{Q[g]}}{t}\approx 0$, hence Noether charges
form a closed algebra under Poisson brackets. Thus, in correspondence to (20),
the algebra of charges can be written as

$\poissonbracket{Q[f]}{Q[g]}=Q[k]\,,$ (33)

where $k^{\mu}=k^{\mu}(f,g)$ is the composition law. Assuming a basis of
Noether charges $Q_{a}$ reduces the previous equation to

$\poissonbracket{Q_{a}}{Q_{b}}={g_{ab}}^{c}Q_{c}\,,$ (34)

where ${g_{ab}}^{c}$ are the structure constants of the Lie algebra of
charges. It can be proved that, if $Q_{a}$ and $Q_{b}$ are well-defined, then
the result of their Poisson brackets is well-defined too [33].

As shown in the previous Section, the transformations on the phase space can
be generated via Noether charges, (26). A general object $F[\Phi^{i},\Pi_{i}]$
transforms as $\delta_{f}F[\Phi^{i},\Pi_{i}]=\poissonbracket{F}{Q[f]}$.
Applying to this $\delta_{g}$, leads to the question of how the $f^{\mu}$
transforms.

If $f^{\mu}$ does not transform, $\delta_{g}f^{\mu}=0$, then $\delta_{f}$
defines a group of internal transformations $\Delta_{f}$ on the phase space,
hence
$\Delta_{g}\Delta_{f}F[\Phi^{i},\Pi_{i}]=\poissonbracket{\poissonbracket{F}{Q[f]}}{Q[g]}$.
Using the Jacobi identity for Poisson brackets, gives the wanted connection
between algebras

$\commutator{\Delta_{g}}{\Delta_{f}}F=\poissonbracket{F}{\poissonbracket{Q[f]}{Q[g]}}\,,$
(35)

and composition rules (20),
$\commutator{\Delta_{g}}{\Delta_{f}}F=\Delta_{h(g,f)}F$, and (33),
$\poissonbracket{Q[f]}{Q[g]}=Q[k(f,g)]$, given by

$h^{\mu}(g,f)=k^{\mu}(f,g)\,.$ (36)

This can be seen from
$\commutator{\Delta_{g}}{\Delta_{f}}F=\Delta_{h}F=\poissonbracket{F}{Q[h]}$,
compared with (35).

We found here that the two composition rules have opposite order or,
equivalently, opposite sign $k^{\mu}(f,g)=-h^{\mu}(f,g)$, due to
$h^{\mu}(f,g)=-h^{\mu}(g,f)$. It is true that one can always redefine the
charges, $Q[f]\rightarrow-Q[f]$, to have the composition law in the same
order, that proves the isomorphism between the algebras. However, this
redefinition does modify the transformation rule (26).

When we deal with spatiotemporal transformations, $\delta_{g}f^{\mu}\neq 0$
mingles the spacetime upon which the phase space is built together with
Poisson brackets. A rule for spatiotemporal transformations can be found by
comparing the Lie derivative $\mathcal{L}_{f}$ with a specifically chosen
internal transformation $\Delta_{f}$ with the following transformation rules

$\Delta_{f}\Phi^{i}=\mathcal{L}_{f}\Phi^{i}\,.$ (37)

Although they transform dynamical fields the same, the two differ
significantly while acting on another non-dynamical spacetime vector field
$g^{\mu}$

$\displaystyle\Delta_{f}g^{\mu}=$ $\displaystyle~{}0\,,$ (38)
$\displaystyle\mathcal{L}_{f}g^{\mu}=$
$\displaystyle~{}f^{\nu}\partial_{\nu}g^{\mu}-g^{\nu}\partial_{\nu}f^{\mu}=\commutator{f}{g}^{\mu}\,.$
(39)

This allows to find the composition rule $h^{\mu}(g,f)$, for a generic field
$\Phi^{i}$

$\displaystyle\commutator{\Delta_{g}}{\Delta_{f}}\Phi^{i}=\Delta_{h(g,f)}\Phi^{i}\,,$
(40)

with
$h^{\mu}(g,f)=f^{\nu}\partial_{\nu}g^{\mu}-g^{\nu}\partial_{\nu}f^{\mu}=-h^{\mu}(f,g)=\mathcal{L}_{f}g^{\mu}$,
the last equality coming from (39).

To find how the algebra of spatiotemporal transformations and the algebra of
Noether charges are related, it is helpful to realize that the previous result
implies

$\mathcal{L}_{f}\mathcal{L}_{g}\Phi^{i}=\mathcal{L}_{f}\Delta_{g}\Phi^{i}=\Delta_{f}\Delta_{g}\Phi^{i}-\Delta_{h(f,g)}\Phi^{i}=\Delta_{g}\Delta_{f}\Phi^{i}\,,$
(41)

where the last equality was obtained by the use of the commutator of two
transformations $\Delta$. Thus, two spatiotemporal transformations follows the
opposite order to the “internalized” version of the transformations.

Since transformations $\Delta_{f}$ are generated in Hamiltonian mechanics via
Poisson brackets, and since the equation (41) connects these and Lie
derivatives $\mathcal{L}_{f}$, a rule for generating $\mathcal{L}_{f}$ without
altering Poisson brackets under the action of the transformation can be
derived. Clearly,
$\commutator{\mathcal{L}_{g}}{\mathcal{L}_{f}}\Phi=\commutator{\Delta_{f}}{\Delta_{g}}\Phi=\Delta_{h(f,g)}\Phi$
and
$\mathcal{L}_{f}\mathcal{L}_{g}\Phi=\poissonbracket{\poissonbracket{\Phi}{Q[f]}}{Q[g]}$.

The general rule can be deduced from these results. For transformations not
changing the phase space the transformations are generated via (26) and
multiple transformations are adding up as

$\Delta_{f}\cdots\Delta_{g}\Delta_{h}\Phi=\poissonbracket{\poissonbracket{\poissonbracket{\poissonbracket{\Phi}{Q[h]}}{Q[g]}}{\cdots}}{Q[f]}\,.$
(42)

On the other hand, the spatiotemporal transformations may lead to additional
structures or changes in the phase space. Thus, the final rule for generating
spatiotemporal transformations is

$\mathcal{L}_{f}\mathcal{L}_{g}\cdots\mathcal{L}_{h}\Phi=\poissonbracket{\poissonbracket{\poissonbracket{\poissonbracket{\Phi}{Q[f]}}{Q[g]}}{\cdots}}{Q[h]}\,.$
(43)

The opposite order in the ladder of Noether charges for internal and
spatiotemporal transformations should be noted.

The above shows how the composition rule of the algebra of Noether charges and
the composition rule of algebra of transformations are connected

$\commutator{\Delta_{f}}{\Delta_{g}}\Phi=\poissonbracket{\Phi}{\poissonbracket{Q[g]}{Q[f]}},\quad\commutator{\mathcal{L}_{f}}{\mathcal{L}_{g}}\Phi=\poissonbracket{\Phi}{\poissonbracket{Q[f]}{Q[g]}}\,.$
(44)

However, the algebra of charges can be centrally extended, still respecting
the algebra of transformations the charges generate. Indeed, the Noether
charges may obey

$\poissonbracket{Q[f]}{Q[g]}=Q[k]+K[f,g]\,,$ (45)

where $k$ is given by the composition rule as in (33) and $K$ is a quantity
independent of fields and conjugated momenta, thus, it has vanishing Poisson
brackets with any field

$\poissonbracket{K[f,g]}{\Phi^{i}}=0\,.$ (46)

Moreover, $\poissonbracket{K[f,g]}{Q[h]}=0$, for any $f,g,h$. This defines $K$
as the _central charge_ and it shows that the transformations generated
through (26) by $\poissonbracket{Q[f]}{Q[g]}$ and by $Q[k(f,g)]$ are
identical, and both charge algebras, without the center (33) or with the
center (45), give rise to the same algebra of transformations (20). Thus, the
presence of a central charge is not reflected in the algebra of
transformations.

Now, Noether charges are not unique, as shown by (16), and this has an effect
on central charges too, since a class of improved charges $Q^{\prime}$ can
always be found

$Q^{\prime}[f]=Q[f]+q[f],$ (47)

with $q$ being a field independent quantity. Although they both generate the
same transformations, $Q^{\prime}$ obeys a centrally extended version of the
algebra obeyed by $Q$, (33),
$\poissonbracket{Q^{\prime}[f]}{Q^{\prime}[g]}=Q^{\prime}[k(f,g)]-q[k(f,g)]$.
Such object $q$ has vanishing Poisson brackets with any charge $Q^{\prime}$,
thus, it is a central charge. We see, then that redefining the charges, a
central charge can be added or removed at will from the algebra, but this does
not provide any new information. Such center is not a general functional of
$f^{\mu}$ and $g^{\mu}$, rather a functional of their specific combination
$k^{\mu}(f,g)$, the composition law of the algebra. As this procedure is
invertible, any part of central charge proportional only to the $k$ is
removable, by a simple redefinition of the charges. If the whole center is
removable, it is called a _trivial_ central charge. If it cannot be removed,
in the way just explained, it is a _genuine_ central charge.

Theories that require genuine extension of algebra of charges are called
_anomalous_ [12]. In what follows we shall always remove trivial central
charges, if not stated otherwise.

The central term appeared in the Hamiltonian reformulation of the theory, in
the algebra of charges. Nonetheless, central term can be witnessed even in
Lagrangian formulations by realizing that on-shell
$\Delta_{g}Q[f]\approx\poissonbracket{Q[f]}{Q[g]}$. Thus, the center may be
studied already in the Lagrangian formalism, [34], by studying the
transformations of charges

$\Delta_{g}Q[f]\approx Q[h(g,f)]+K[f,g].$ (48)

To close this Section, let us show an intriguing property that holds for
spatiotemporal transformations

$\mathcal{L}_{g}Q[f]\approx K[f,g]\,,$ (49)

and let us stress that, as a part of the algebra of charges, the central
charge has to be conserved, i.e.,
$\poissonbracket{K}{H}+\partialderivative{K}{t}=0$ implies
$\partialderivative{K}{t}=0$.

##  V Conformal transformations in two dimensions

In a two-dimensional flat spacetime the conformal Killing equation is

$\partial_{\mu}f_{\nu}+\partial_{\nu}f_{\mu}=\eta_{\mu\nu}\partial_{\rho}f^{\rho}\,,$
(50)

with $\eta_{\mu\nu}$ the flat metric in Euclidean or Minkowski signature, see
Appendix C. Taking a derivative, this becomes
$\partial_{\nu}\partial^{\nu}f_{\mu}=0$, that is the wave equation, in
Minkowski signature, or the Laplace equation, in Euclidean signature.

The focus here is on two-dimensional scalar field theories444Classical
anomalies are not restricted to scalar theories, though. See [12] and [23]..
The geometric conformal transformation of a scalar field $\phi(x)$ is

$\delta_{f}\phi(x)=\phi^{\prime}(x)-\phi(x)=\epsilon
f^{\mu}\partial_{\mu}\phi\,.$ (51)

For the Euclidean plane, the solutions to the conformal Killing equation are
(anti)holomorphic functions $f$ ($\bar{f}$). Hence, they can be expanded as a
Laurent series

$f(z)=\sum_{n\in\mathds{Z}}f_{n}z^{n},\qquad\bar{f}(\bar{z})=\sum_{n\in\mathds{Z}}\bar{f}_{n}\bar{z}^{n}\,.$
(52)

For the Minkowski spacetime the solutions are the right-going, $f^{-}$, and
the left-going, $f^{+}$, waves. Assuming the $P$-periodicity, see Appendix C,
the waves can be expanded into a Fourier basis

$f^{\pm}(x^{\pm})=a_{0}+\sum_{k\in\mathbb{N}}\quantity[a_{k}\cos(\frac{2\pi}{P}kx^{\pm})+b_{k}\sin(\frac{2\pi}{P}kx^{\pm})]=\sum_{n\in\mathbb{Z}}f^{\pm}_{n}e^{i\frac{2\pi}{P}nx^{\pm}}\,.$
(53)

These allow to introduce the generators as (see (21))

$l_{n}=-z^{n+1}\partial,\qquad\bar{l}_{n}=-\bar{z}^{n+1}\bar{\partial}\,,$
(54)

in Euclidean (complex) spacetime, and as

$l_{n}^{\pm}\equiv\frac{iP}{2\pi}e^{i\frac{2\pi}{P}nx^{\pm}}\partial_{\pm}\,,$
(55)

in Minkowski spacetime, where $iP/(2\pi)$ is a normalization factor.

Computing the commutators one gets the Witt algebras [20] in both notations

$\commutator{l_{n}}{l_{m}}=(n-m)\,l_{n+m}\,,\quad\commutator{\bar{l}_{n}}{\bar{l}_{m}}=(n-m)\,\bar{l}_{n+m}\,,\quad\commutator{l_{n}}{\bar{l}_{m}}=0\,,$
(56)

and

$\commutator{l_{n}^{+}}{l_{m}^{+}}=(n-m)\,l_{n+m}^{+}\,,\quad\commutator{l_{n}^{-}}{l_{m}^{-}}=(n-m)\,l_{n+m}^{-}\,,\quad\commutator{l_{n}^{+}}{l_{m}^{-}}=0\,,$
(57)

respectively, whose unique genuine central extension are the _Virasoro
algebras_ [20], e.g.,

$\displaystyle\commutator{l_{n}^{\pm}}{l_{m}^{\pm}}=$
$\displaystyle~{}(n-m)l_{n+m}^{\pm}+\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}\,,$
(58) $\displaystyle\commutator{l_{n}^{\pm}}{c}=$ $\displaystyle~{}0\,,$ (59)

for Minkowski spacetime, where $c$ is the central term, $1/12$ is a
normalization factor.

We need not focus on other metrics than the Euclidean and Minkowski, because
all two-dimensional spacetimes are _locally conformally flat_ [35, 27]. Thus,
for every two-dimensional (pseudo-)Riemannian manifold, _isothermal
coordinates_ exist such that

$\hat{g}_{\mu\nu}(x)=e^{2\rho(x)}\eta_{\mu\nu}\,,$ (60)

where the hat indicates quantities in the isothermal coordinate frame.

When a theory, on top of conformal invariance, also has Weyl invariance, a
Weyl transformation suppresses the conformal factor, hence the computations
can be restricted to flat metric spacetimes just discussed here and in
Appendix C. The connection between scale, conformal, diffeomorphism and Weyl
invariances is explained in [10], for classical theories in arbitrary
dimensions, including two, and for fields of arbitrary spin. The case of the
two dimensional scalar Liouville theory, though, became the focus of more
attention in [11]. It is the scope of the rest of this paper to investigate
these issues in all details.

##  VI The free scalar in two dimensions and its affine center

Before diving into the Liouville theory, it is instructive to apply the
previous results to a simpler model, the free scalar in two dimensions

$A_{\scriptscriptstyle{FS}}[\varphi]=\int\differential[2]{x}\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi\,,$
(61)

and Minkowski metric, $\eta_{\mu\nu}=\operatorname{diag}(1,-1)$. This gives as
equation of motion

$\Box_{\text{M}}\varphi=\partial_{\mu}\partial^{\mu}\varphi=0\,.$ (62)

The theory is invariant under the infinitesimal conformal transformations (51)
where $\epsilon$ is the infinitesimal parameter, and $f^{\mu}$ satisfies the
conformal Killing equation (10), hence

$\Box_{\text{M}}f_{\mu}=0\,.$ (63)

According to Noether theorem, there is a conserved current (7), $J^{\mu}_{f}$,
associated to the symmetry

$J^{\mu}_{f}=\quantity(\partial^{\mu}\varphi\partial^{\nu}\varphi-\frac{1}{2}\eta^{\mu\nu}\partial_{\lambda}\varphi\partial^{\lambda}\varphi)\epsilon
f_{\nu}\,.$ (64)

From the Bessel-Hagen formula (14), $J_{f}^{\mu}=\theta^{\mu\nu}f_{\nu}$, the
canonical EMT of the free field is

$\theta^{\mu\nu}=\partial^{\mu}\varphi\partial^{\nu}\varphi-\frac{1}{2}\eta^{\mu\nu}\partial_{\lambda}\varphi\partial^{\lambda}\varphi\,,$
(65)

and it is symmetric and traceless, as required.

Besides conformal transformations, the theory is symmetric under an affine
transformation [3], that is a field shift,

$\delta_{\chi}\varphi(x)=\epsilon\chi(x)\,,$ (66)

with associated current given by

$J^{\mu}_{\chi}=\epsilon\quantity(\partial^{\mu}\varphi\chi-\varphi\partial^{\mu}\chi)\,.$
(67)

For this to happen, the shift too must satisfy the wave equation

$\Box_{\text{M}}\chi=0\,.$ (68)

All the involved quantities, $\varphi$, $f^{\mu}$ and $\chi$, obey the wave
equation, (62), (63), (68). Thus light-cone coordinates, discussed in Appendix
C, are the most natural:
$\varphi(x)\approx\varphi^{+}(x^{+})+\varphi^{-}(x^{-})$,
$\chi(x)=\chi^{+}(x^{+})+\chi^{-}(x^{-})$, and
$f^{\pm}(x^{\pm})=\frac{1}{\sqrt{2}}\quantity(f^{t}(x)\pm f^{x}(x))$.

The derived quantities, currents $J^{\mu}_{f}$, $J^{\mu}_{\chi}$ and EMT
$\theta^{\mu\nu}$, split to $x^{+}$ and $x^{-}$ dependent parts
correspondingly, on-shell, under the natural assumption,
$\delta_{\chi}\varphi^{+}(x^{+})=\epsilon\chi^{+}(x^{+})$,
$\delta_{\chi}\varphi^{-}(x^{-})=\epsilon\chi^{-}(x^{-})$.

Then it can be seen that
$\theta_{++}=\partial_{+}\varphi\partial_{+}\varphi\approx\theta_{++}(x^{+})$,
$\theta_{+-}=0$,
$\theta_{--}=\partial_{-}\varphi\partial_{-}\varphi\approx\theta_{--}(x^{-})$
and ${J_{f}}_{\pm}=\theta_{\pm\pm}f^{\pm}\approx{J_{f}}_{\pm}(x^{\pm})$,
${J_{\chi}}_{\pm}\approx\partial_{\pm}\varphi^{\pm}\chi^{\pm}-\varphi^{\pm}\partial_{\pm}\chi^{\pm}={J_{\chi}}_{\pm}(x^{\pm})$.
Notice that, e.g., $J_{+}=J_{+}(x^{+})$, but $J^{+}\neq J^{+}(x^{+})$, becasue
$J^{+}=J_{-}=J_{-}(x^{-})$.

Therefore, the problem at hand splits in two independent parts, each with its
own Noether charges:
$\partial_{\mu}J^{\mu}=\partial_{+}J_{-}+\partial_{-}J_{+}\approx 0$, where
each term vanishes independently. The charges can be defined as555Here we
explicitly write the “time” variable. E.g., for the charge $Q^{\pm}$ the
“space” variable, i.e., the integration variable of the definition (8), is
$x^{\pm}$, while the other variable, $x^{\mp}$, plays the role of “time”. This
explicit notation is omitted in the following, but should be understood in the
evaluation of “equal-time” Poisson brackets.

$\displaystyle Q^{+}[f](x^{-})=$
$\displaystyle\int\differential{x^{+}}f^{+}(x^{+})\theta_{++}(x^{+})\,,$ (69)
$\displaystyle Q^{-}[f](x^{+})=$
$\displaystyle\int\differential{x^{-}}f^{-}(x^{-})\theta_{--}(x^{-})\,,$ (70)
$\displaystyle Q[f](t)=$ $\displaystyle~{}Q^{+}[f]+Q^{-}[f]\,,$ (71)

for the transformation $\delta_{f}$ and

$\displaystyle q^{+}[\chi](x^{-})=$
$\displaystyle~{}2\int\differential{x^{+}}\chi^{+}\partial_{+}\varphi^{+}\,,$
(72) $\displaystyle q^{-}[\chi](x^{+})=$
$\displaystyle~{}2\int\differential{x^{-}}\chi^{-}\partial_{-}\varphi^{-}\,,$
(73) $\displaystyle q[\chi](t)=$ $\displaystyle~{}q^{+}[\chi]+q^{-}[\chi]\,,$
(74)

for the transformation $\delta_{\chi}$.

The algebra of these charges can be only obtained by first constructing
Poisson brackets in the light-cone Hamiltonian formalism. Since the Lagrangian
is

$\mathcal{L}=\partial_{+}\varphi\partial_{-}\varphi\,,$ (75)

there are two obstacles. First, there is no natural time variable. Second, and
more direct, the Lagrangian is linear in $\partial_{\pm}\varphi$, therefore
the standard Legendre transformation fails to provide any meaningful
Hamiltonian. Although, not knowing the Hamiltonian is not a complication for
the reconstruction of the charge-algebra, not having a phase space is. In
Appendix B we discuss the issues and derive Poisson brackets for light-cone
Lagrangian, which turn out to be

$\displaystyle\evaluated{\poissonbracket{\varphi(x)}{\varphi(y)}}_{x^{+}=y^{+}}=$
$\displaystyle~{}\poissonbracket{\varphi^{-}(x^{-})}{\varphi^{-}(y^{-})}=-\frac{1}{4}\operatorname{sgn}(x^{-}-y^{-})\,,$
(76)
$\displaystyle\evaluated{\poissonbracket{\varphi(x)}{\varphi(y)}}_{x^{-}=y^{-}}=$
$\displaystyle~{}\poissonbracket{\varphi^{+}(x^{+})}{\varphi^{+}(y^{+})}=-\frac{1}{4}\operatorname{sgn}(x^{+}-y^{+})\,,$
(77) $\displaystyle\poissonbracket{A[\varphi]}{B[\varphi]}=$
$\displaystyle-\frac{1}{4}\int\differential{z}\differential{z^{\prime}}\operatorname{sgn}(z-z^{\prime})\cdot\quantity(\functionalderivative{A[\varphi]}{\varphi^{+}(z)}\functionalderivative{B[\varphi]}{\varphi^{+}(z^{\prime})}+\functionalderivative{A[\varphi]}{\varphi^{-}(z)}\functionalderivative{B[\varphi]}{\varphi^{-}(z^{\prime})})\,,$
(78)

where $\operatorname{sgn}(x)=+1(-1)$ for $x>0~{}(x<0)$, and
$\operatorname{sgn}(x)=0$ for $x=0$, is the sign function. It can be
equivalently rewritten in terms of Haeviside step function
$\operatorname{sgn}(x)=2H(x)-1$, revealing a useful, distributive, property
for calculations, $\operatorname{sgn}^{\prime}(x)=2\delta(x)$. The provided
pair of Poisson brackets (equal $x^{+}$ brackets and equal $x^{-}$ brackets),
displays the split and separate evolutions along the $x^{+}$ and $x^{-}$
lines.

Having Poisson brackets it can be verified that the charges generate
transformations as in (26),
$\delta_{f}\Phi^{i}=\poissonbracket{\Phi^{i}}{Q[f]}$. With the use of the
previous results and derivative of Poisson brackets,
$\poissonbracket{\varphi^{\pm}(x^{\pm})}{\partial_{\pm}\varphi^{\pm}(y^{\pm})}=\frac{1}{2}\delta(x^{\pm}-y^{\pm})$,
the transformations can be derived

$\displaystyle\poissonbracket{\varphi^{+}(x^{+})}{Q^{+}[f]}=$
$\displaystyle~{}f^{+}(x^{+})\partial_{+}\varphi^{+}(x^{+})\,,$ (79)
$\displaystyle\poissonbracket{\varphi^{-}(x^{-})}{Q^{-}[f]}=$
$\displaystyle~{}f^{-}(x^{-})\partial_{-}\varphi^{-}(x^{-})\,,$ (80)
$\displaystyle\poissonbracket{\varphi^{+}(x^{+})}{q^{+}[\chi]}=$
$\displaystyle~{}\chi^{+}(x^{+})\,,$ (81)
$\displaystyle\poissonbracket{\varphi^{-}(x^{-})}{q^{-}[\chi]}=$
$\displaystyle~{}\chi^{-}(x^{-})\,.$ (82)

We can now construct the algebra of charges. In the following, conformal and
affine charges are discussed separately, to emphasize their specific
properties. At the end of the Section both algebras are brought together to
reconstruct the Virasoro algebra.

The conformal charges $Q[f]$ close the conformal algebra

$\displaystyle\poissonbracket{Q^{\pm}[f]}{Q^{\pm}[g]}=$
$\displaystyle~{}Q^{\pm}[k]\,,$ (83)
$\displaystyle\poissonbracket{Q^{\pm}[f]}{Q^{\mp}[g]}=$ $\displaystyle~{}0\,,$
(84)

where $k^{\mu}$ is the Lie bracket (39) of $f^{\mu}$ and $g^{\mu}$,
$k^{\mu}=f^{\nu}\partial_{\nu}g^{\mu}-g^{\nu}\partial_{\nu}f^{\mu}$.

By expanding $f^{\pm}$ into Fourier series (53) the conformal charges can be
decomposed as follows

$Q^{\pm}[f]=\int\differential{x^{\pm}}\theta_{\pm\pm}f^{\pm}=\sum_{n\in\mathbb{Z}}f^{\pm}_{n}\int\differential{x^{\pm}}\theta_{\pm\pm}e^{i\frac{2\pi}{P}nx^{\pm}}=-\frac{2\pi
i}{P}\sum_{n\in\mathbb{Z}}f^{\pm}_{n}Q^{\pm}_{n}\,,$

where

$Q^{\pm}_{n}\equiv\frac{P}{2\pi}\int\differential{x^{\pm}}\theta_{\pm\pm}e^{i\frac{2\pi}{P}nx^{\pm}}\,,$
(85)

form a basis of conformal charges. The factor $P/2\pi$ in the definition of
$Q^{\pm}_{n}$ is chosen so that the confromal charges give the Witt algebra

$\displaystyle i\poissonbracket{Q^{\pm}_{n}}{Q^{\pm}_{m}}=$
$\displaystyle~{}(n-m)Q^{\pm}_{n+m}\,,$ (86) $\displaystyle
i\poissonbracket{Q^{\pm}_{n}}{Q^{\mp}_{m}}=$ $\displaystyle~{}0\,.$ (87)

The presence of complex unit $i$ should not be over-looked. Nonetheless, the
choice of having $i$ here is to obtain a non-imaginary central charge, as
explained later. One can obtain the Witt algebra (57) by a redefinition
$Q^{\pm}_{n}\rightarrow-iQ^{\pm}_{n}$. Aspects of such redefinition are
mentioned in Section IV and alter the connection between conformal charges
(85) and generators (55)

$l_{n}^{\pm}\varphi=\poissonbracket{\varphi}{Q^{\pm}_{n}}\,,$ (88)

as seen from (79) and (80).

The definition (85) shows that the charges $Q^{\pm}_{n}$ are Fourier modes of
the EMT with suitable normalization666In the radial parametrization approach
(see Appendix C), the charges are the coefficients of the Laurent series of
the EMT.. When we sum all of them up, the full information about the EMT is
recovered

$\theta_{\pm\pm}=\sum_{n\in\mathbb{Z}}\frac{2\pi}{P^{2}}Q^{\pm}_{-n}e^{i\frac{2\pi}{P}nx^{\pm}}\,.$
(89)

The affine charges $q[\chi]$, on the other hand, close a centrally extended
algebra

$\displaystyle\poissonbracket{q^{\pm}[\chi]}{q^{\pm}[\zeta]}=$
$\displaystyle~{}K^{\pm}[\chi,\zeta]\,,$ (90)
$\displaystyle\poissonbracket{q^{\pm}[\chi]}{q^{\mp}[\zeta]}=$
$\displaystyle~{}0\,,$ (91)

where

$K^{\pm}[\chi,\zeta]=\int\differential{x^{\pm}}\quantity[\chi^{\pm}(x^{\pm})\partial_{\pm}\zeta^{\pm}(x^{\pm})-\zeta^{\pm}(x^{\pm})\partial_{\pm}\chi^{\pm}(x^{\pm})]\,.$
(92)

The central charge $K$ introduced by the bracket (90) is a field independent
object, therefore, $\poissonbracket{\cdot}{K^{\pm}[\chi,\zeta]}=0$, and, in
particular, $\poissonbracket{\varphi}{K^{\pm}[\chi,\zeta]}=0$. Thus, $K$
itself does not generate any transformation. The waves $\chi$ and $\zeta$ can
be expanded into Fourier modes too

$\chi^{\pm}(x^{\pm})=\sum_{n\in\mathbb{Z}}\chi^{\pm}_{n}e^{i\frac{2\pi}{P}nx^{\pm}}\,,\qquad\zeta^{\pm}(x^{\pm})=\sum_{n\in\mathbb{Z}}\zeta^{\pm}_{n}e^{i\frac{2\pi}{P}nx^{\pm}}\,,$

leading to

$K^{\pm}[\chi,\zeta]=\sum_{n,m\in\mathbb{Z}}\chi^{\pm}_{n}\zeta^{\pm}_{m}\,2\pi
i(m-n)\delta_{n+m,0}\,.$ (93)

Introducing a basis for affine generators $q^{\pm}_{n}\equiv
q^{\pm}\quantity[e^{i\frac{2\pi}{P}nx}]$ the equation (90) can be written as

$i\poissonbracket{q^{\pm}_{n}}{q^{\pm}_{m}}=4\pi n\,\delta_{n+m,0}\,,$ (94)

that is nothing else than a representation of the _Heisenberg algebra_ , [22],

$\commutator{a_{n}}{a_{m}}=n\,\delta_{n+m,0}\,.$ (95)

The Poisson brackets between the charges close

$\displaystyle\poissonbracket{Q^{\pm}[f]}{q^{\pm}[\chi]}=$
$\displaystyle~{}q^{\pm}[f\cdot\partial\chi]\,,$ (96)
$\displaystyle\poissonbracket{Q^{\pm}[f]}{q^{\mp}[\chi]}=$
$\displaystyle~{}0\,.$ (97)

Thus a combined transformation can be chosen. Let us consider the following
choice for $\chi$

$\chi\equiv\partial\cdot f\quad\iff\quad\chi^{\pm}_{n}=f^{\pm}_{n}\frac{2\pi
i}{P}n\,,$ (98)

that is available becasue $f^{\mu}$ itself is a solution of the wave equation.
Now we can combine the two transformations, $\delta_{f}$ and $\delta_{\chi}$,
parametrized only by $f^{\mu}$, into a new improved transformation,
$\tilde{\delta}_{f}$, in the following way

$\tilde{\delta}_{f}\varphi\equiv\delta_{f}\varphi+\frac{1}{\gamma}\delta_{\partial
f}\varphi=f^{\mu}\partial_{\mu}\varphi+\frac{1}{\gamma}\partial_{\mu}f^{\mu},$
(99)

where $\gamma$ is a parameter. The conserved current associated to the new
transformation $\tilde{\delta}_{f}$ is

$\tilde{J}{}^{\mu}=J^{\mu}_{f}+\frac{1}{\gamma}J^{\mu}_{\partial f}\,,$ (100)

and it can be recast in the Bessel-Hagen form (14),
$\tilde{J}^{\mu}=\tilde{\theta}^{\mu\nu}f_{\nu}$, by improving the current
$\tilde{J}^{\mu}$ by a superpotential-like term (16)

$\partial_{\nu}V^{\mu\nu}=2\partial_{\nu}(f^{\mu}\partial^{\nu}\varphi-f^{\nu}\partial^{\mu}\varphi)+\partial_{\nu}(\varphi\partial^{\mu}f^{\nu}-\varphi\partial^{\nu}f^{\mu})\,.$
(101)

By using (10), it can be shown that $J^{\mu}_{\partial
f}+\partial_{\nu}V^{\mu\nu}=2f_{\nu}\quantity(\eta^{\mu\nu}\Box_{\text{M}}-\partial^{\mu}\partial^{\nu})\varphi$.
Thus, the new EMT is

$\tilde{\theta}_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\eta_{\mu\nu}\partial_{\alpha}\varphi\partial^{\alpha}\varphi+\frac{2}{\gamma}(\eta_{\mu\nu}\Box_{\text{M}}-\partial_{\mu}\partial_{\nu})\varphi\,.$
(102)

This new improved EMT $\tilde{\theta}_{\mu\nu}$ is not identically traceless,
but it is so only on-shell. However, the split into $x^{\pm}$ parts is not
violated,

$\displaystyle\tilde{\theta}_{\pm\pm}=$
$\displaystyle~{}\partial_{\pm}\varphi\partial_{\pm}\varphi-\frac{2}{\gamma}\partial_{\pm}^{2}\varphi\approx\tilde{\theta}_{\pm\pm}(x^{\pm})\,,$
(103) $\displaystyle\tilde{\theta}_{+-}=$
$\displaystyle~{}\frac{4}{\gamma}\Box_{\text{M}}\varphi\approx 0\,,$ (104)

and the new improved charges

$\tilde{Q}_{\gamma}^{\pm}[f]=\int\differential{x^{\pm}}\tilde{\theta}_{\pm\pm}f^{\pm}(x^{\pm})=\int\differential{x^{\pm}}\quantity[\partial_{\pm}\varphi^{\pm}\partial_{\pm}\varphi^{\pm}-\frac{2}{\gamma}\partial_{\pm}^{2}\varphi^{\pm}]f^{\pm}=Q^{\pm}[f]+\frac{1}{\gamma}q^{\pm}[\partial\cdot
f]\,,$ (105)

are still well defined.

From the previous results, the algebra of improved charges immediately follows

$\poissonbracket{\tilde{Q}_{\gamma}[f]}{\tilde{Q}_{\gamma}[g]}=\tilde{Q}_{\gamma}[k]+\frac{1}{\gamma^{2}}\Delta[f,g],$
(106)

where

$\Delta[f,g]=K[\partial f,\partial
g]=\underbrace{\int\differential{x^{+}}(g^{+}\partial_{+}^{3}f^{+}-f^{+}\partial_{+}^{3}g^{+})}_{\equiv\Delta^{+}[f,g]}+\underbrace{\int\differential{x^{-}}(g^{-}\partial_{-}^{3}f^{-}-f^{-}\partial_{-}^{3}g^{-})\,}_{\equiv\Delta^{-}[f,g]}.$
(107)

Combining (93) and (98), the central term immediately follows

$\Delta^{\pm}[f,g]=\sum_{n,m\in\mathbb{Z}}g^{\pm}_{m}f^{\pm}_{n}\quantity(\frac{2\pi
i}{P})^{2}4\pi in^{3}\delta_{m+n,0}\,.$ (108)

Furthermore, the improved charges $\tilde{Q}^{\pm}_{n}$ can be introduced
again as the Fourier decomposition of the improved EMT
$\tilde{\theta}_{\mu\nu}$ with a proper normalization

$\tilde{Q}^{\pm}_{n}\equiv\frac{P}{2\pi}\int\differential{x^{\pm}}\tilde{\theta}_{\pm\pm}e^{i\frac{2\pi}{P}nx^{\pm}}\,.$
(109)

The centrally extended algebra (106) can, thus, be recast as

$i\poissonbracket{\tilde{Q}_{n}^{\pm}}{\tilde{Q}_{m}^{\pm}}=(n-m)\tilde{Q}_{n+m}^{\pm}+\frac{4\pi}{\gamma^{2}}n^{3}\delta_{n+m,0}\,.$
(110)

Notice that the previously mentioned redefinition $Q^{\pm}_{n}\rightarrow-
iQ^{\pm}_{n}$ results in $c\rightarrow-ic$.

The composition rule of the Virasoro algebra (58)

$\commutator{l_{n}^{\pm}}{l_{m}^{\pm}}=(n-m)l_{n+m}^{\pm}+\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}\,,$

differs from (110) by a term proportional to
$n\,\delta_{n+m,0}=(n-m)/2\,\delta_{n+m,0}$. Nonetheless, it is easy to see,
that it is a trivial (removable) center. First, note that for
$k^{\mu}=f^{\nu}\partial_{\nu}g^{\mu}-g^{\nu}\partial_{\nu}f^{\mu}$ it holds

$\int\differential{x^{\pm}}k^{\pm}[f,g]=\sum_{n,m\in\mathbb{Z}}g^{\pm}_{m}f^{\pm}_{n}\,2\pi
i(m-n)\delta_{n+m,0}\,.$ (111)

Second, as indicated in (47), we have the freedom to redefine
$\tilde{Q}_{\gamma}[f]$ by adding terms proportional to $f$. A suitable choice
is

$\tilde{Q}_{\gamma}^{\pm}[f^{\pm}]\rightarrow\tilde{Q}_{\gamma}^{\pm}[f^{\pm}]+\frac{1}{\gamma^{2}}\quantity(\frac{2\pi
i}{P})^{2}\int\differential{x^{\pm}}f^{\pm}\,,$ (112)

leading to

$\Delta^{\pm}[f,g]\rightarrow\Delta^{\pm}[f,g]-\quantity(\frac{2\pi
i}{P})^{2}\int\differential{x^{\pm}}k^{\pm}[f,g]=\sum_{n,m\in\mathbb{Z}}g^{\pm}_{m}f^{\pm}_{n}\quantity(\frac{2\pi
i}{P})^{2}4\pi in(n^{2}-1)\delta_{n+m,0}\,.$ (113)

This reconstructs the Virasoro algebra

$i\poissonbracket{\tilde{Q}_{n}^{\pm}}{\tilde{Q}_{m}^{\pm}}=(n-m)\tilde{Q}^{\pm}_{n+m}+\frac{4\pi}{\gamma^{2}}n(n^{2}-1)\delta_{n+m,0}\,,$
(114)

with a center

$c=\frac{48\pi}{\gamma^{2}}\,.$ (115)

Let us conclude this Section with several comments. The conformal symmetry led
to a simple Witt algebra, thus, the conformal symmetry of the classical free
scalar is not anomalous. It was the introduction of the combined, improved,
transformation, $\tilde{\delta}$, that gave rise to the genuine center and
resulted in the Virasoro algebra. Tracking the $1/\gamma$ factors suggests
that the center of the Virasoro algebra is the affine center in new clothes,
dressed upon the demand that the field shift has to be a conformal Killing
field. Nonetheless, the theory itself does not restrict the choice of $\gamma$
in the definition and the choice $\gamma\rightarrow\infty$ restores the
anomaly free conformal transformations.

Although seemingly artificially constructed, the center of improved charges
differs significantly from the center of affine charges. While the value of
the affine center can be changed by the renormalization of the affine charges
$q^{\pm}_{n}$, the value of the Virasoro central charge cannot be normalized
at will by the redefinition of the improved charges $\tilde{Q}_{n}^{\pm}$.
This rigidity has physical consequences illustrated later.

In order to restore the Virasoro algebra with the usual composition law (58),
the improved charges had to be “shifted”. The mathematical justification of
trivial shifts of charges (47) was given in the previous Section and from its
form (112) it can be seen that only the charge $\tilde{Q}_{0}$ changes its
value in this case. Correspondingly, from the definition of charges

$\tilde{Q}_{\gamma}[f]\sim\int\tilde{\theta}\cdot f\,,$ (116)

the above shown shift can be understood as a shift of the EMT by a constant
factor

$\tilde{\theta}_{\mu\nu}\rightarrow\tilde{\theta}_{\mu\nu}-\frac{4\pi^{2}}{\gamma^{2}P^{2}}\eta_{\mu\nu}\,.$
(117)

Such procedure thus corresponds to a shift of the mean value of the EMT.

##  VII Liouville theory in flat space and its genuine center

Classical Liouville theory is genuinely conformally anomalous. In fact,
contrary to the free scalar, Liouville theory

$A_{\scriptscriptstyle{L}}[\Phi]=\int\differential[2]{x}\quantity(\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})\,,$
(118)

with $\eta_{\mu\nu}=\operatorname{diag}(1,-1)$, with equation of motion

$\Box_{\text{M}}\Phi+\frac{m^{2}}{\beta}e^{\beta\Phi}=0\,,$ (119)

is only symmetric under the improved version of the conformal transformations.

Indeed, while the kinetic term is symmetric under standard conformal symmetry,
as the free scalar, the potential term is not, because
$\delta_{f}\quantity(e^{\beta\Phi})=f^{\mu}\partial_{\mu}\quantity(e^{\beta\Phi})$
does not turn (118) into a pure divergence, as required by the Noether theorem
to be a symmetry.

However, under the improvements extensively discussed earlier

$\tilde{\delta}_{f}\Phi=f^{\mu}\partial_{\mu}\Phi+\frac{1}{\beta}\partial_{\mu}f^{\mu}\,,$
(120)

we have
$\tilde{\delta}_{f}\quantity(e^{\beta\Phi})=\partial_{\mu}\quantity(f^{\mu}e^{\beta\Phi})$,
and the theory is symmetric.

Contrary to the free scalar, (99), the parameter $\gamma$ is now fixed by the
theory, $\gamma=\beta$. Moreover, the previously derived EMT only gets an
extra contribution from the potential term

$\tilde{\Theta}_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-\eta_{\mu\nu}\quantity(\frac{1}{2}\partial_{\alpha}\Phi\partial^{\alpha}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})+\frac{2}{\beta}(\eta_{\mu\nu}\Box_{\text{M}}-\partial_{\mu}\partial_{\nu})\Phi\,,$
(121)

and it is symmetric and traceless on-shell

$\tilde{\Theta}^{\mu}_{\mu}=\frac{2}{\beta}\quantity(\Box_{\text{M}}\Phi+\frac{m^{2}}{\beta}e^{\beta\Phi})\approx
0\,.$ (122)

While the field $\Phi$ does not split on-shell into $\Phi^{\pm}$, as is the
case for the free scalar, the EMT still does. By taking a derivative, it can
be shown that $\partial_{\mp}\tilde{\Theta}_{\pm\pm}\approx 0$ and
tracelessness implies $\tilde{\Theta}_{+-}\approx 0$. The conformal Killing
vector $f^{\mu}$ also splits, thus meaningful Noether charges can be defined
in the light-cone formulation

$\tilde{Q}^{\pm}[f](x^{\mp})=\int\differential{x^{\pm}}\tilde{\Theta}_{\pm\pm}f^{\pm}=\int\differential{x^{\pm}}\quantity((\partial_{\pm}\Phi)^{2}-\frac{2}{\beta}\partial^{2}_{\pm}\Phi)f^{\pm}\,.$
(123)

Given that the potential term does not interfere with the definition of
conjugated momentum, Poisson brackets have the same form as for the free
scalar

$\displaystyle\evaluated{\poissonbracket{\Phi(x)}{\Phi(y)}}_{x^{+}=y^{+}}=$
$\displaystyle-\frac{1}{4}\operatorname{sgn}(x^{-}-y^{-})\,,$ (124)
$\displaystyle\evaluated{\poissonbracket{\Phi(x)}{\Phi(y)}}_{x^{-}=y^{-}}=$
$\displaystyle-\frac{1}{4}\operatorname{sgn}(x^{+}-y^{+})\,.$ (125)

Again, it can be verified that the charges correctly generate the
transformations

$\poissonbracket{\Phi(x^{+},x^{-})}{\tilde{Q}^{\pm}[f](x^{\mp})}=f^{\pm}(x^{\pm})\partial_{\pm}\Phi(x^{+},x^{-})+\frac{1}{\beta}\partial_{\pm}f^{\pm}(x^{\pm})\,,$
(126)

and, from the previous discussion of the algebra of charges, it immediately
follows

$\poissonbracket{\tilde{Q}^{\pm}[f]}{\tilde{Q}^{\pm}[g]}=\tilde{Q}^{\pm}[k]+\frac{1}{\beta^{2}}\Delta^{\pm}[f,g]\,,$
(127)

where $k^{\mu}=f^{\nu}\partial_{\nu}g^{\mu}-g^{\nu}\partial_{\nu}f^{\mu}$ and
$\Delta^{\pm}[f,g]=\int\differential{x^{\pm}}(g^{\pm}\partial_{\pm}^{3}f^{\pm}-f^{\pm}\partial_{\pm}^{3}g^{\pm})$.

Introducing the generators

$\tilde{Q}^{\pm}_{n}\equiv\frac{P}{2\pi}\int\differential{x^{\pm}}\tilde{\Theta}_{\pm\pm}e^{i\frac{2\pi}{P}nx^{\pm}}=\frac{P}{2\pi}\tilde{Q}^{\pm}[e^{i\frac{2\pi}{P}nx^{\pm}}]\,,$
(128)

the algebra (127) can be recast into

$i\poissonbracket{\tilde{Q}^{\pm}_{n}}{\tilde{Q}^{\pm}_{m}}=(n-m)\tilde{Q}^{\pm}_{n+m}+\frac{4\pi}{\beta^{2}}n^{3}\delta_{n+m,0}\,.$
(129)

Comparing this to the Virasoro algebra (58) gives the central charge for the
classical Liouville theory

$c=\frac{48\pi}{\beta^{2}}\,.$ (130)

This time the center is genuine since the symmetry of Liouville field, (120),
requires simultaneous shift to the field. That connects the two symmetries of
free scalar field theory in a non-trivial and fixed way.

Let us now see the effect of the center on the transformations of the EMT.
From (123) and (127) it follows that

$\poissonbracket{\tilde{\Theta}_{\pm\pm}(x)}{\tilde{\Theta}_{\pm\pm}(y)}\evaluated{}_{x^{\mp}=y^{\mp}}=\tilde{\Theta}^{\prime}_{\pm\pm}(x)\delta(x^{\pm}-y^{\pm})+2\tilde{\Theta}_{\pm\pm}(x)\delta^{\prime}(x^{\pm}-y^{\pm})-\frac{2}{\beta^{2}}\delta^{\prime\prime\prime}(x^{\pm}-y^{\pm})\,,$
(131)

and the term proportional to $\delta^{\prime\prime\prime}$ stems directly from
the presence of the central charge. This, through (26) with (123), immediately
gives the infinitesimal conformal transformation of the EMT

$\tilde{\delta}_{f}\tilde{\Theta}_{\pm\pm}=f^{\pm}\partial_{\pm}\tilde{\Theta}_{\pm\pm}+2\tilde{\Theta}_{\pm\pm}\partial_{\pm}f^{\pm}-\frac{2}{\beta^{2}}\partial^{3}_{\pm}f^{\pm}\,,$
(132)

which is directly affected by the central charge. Thus, the central charge can
be revealed by a direct computation of $\delta_{f}\tilde{\Theta}_{\mu\nu}$
already at the Lagrangian level, in analogy to (48) and (49).

The infinitesimal transformation (132) can be “exponentiated” to a finite
conformal transformation $x^{\pm}\rightarrow
x^{\prime\pm}=x^{\pm}-f^{\pm}(x^{\pm})$ which gives the following interesting
result

$\tilde{\Theta}^{\prime}_{\pm\pm}(x^{\prime})=\quantity(\partialderivative{x^{\pm}}{x^{\prime\pm}})^{2}\quantity[\tilde{\Theta}_{\pm\pm}(x)-\frac{2}{\beta^{2}}(\mathcal{S}x^{\prime\pm})(x^{\pm})]\,,$
(133)

with $(\mathcal{S}x^{\prime})(x)$ the Schwarzian derivative

$(\mathcal{S}g)(x)\equiv\frac{g^{\prime\prime\prime}(x)}{g^{\prime}(x)}-\frac{3}{2}\quantity(\frac{g^{\prime\prime}(x)}{g^{\prime}(x)})^{2}\,.$
(134)

An interesting consequence for Liouville theory is that there exist a
conformal transformation such that the EMT vanishes, as we now show. Since the
solution to (119) can be written as [2]

$\Phi(x)=\frac{1}{\beta}\ln\frac{F^{\prime}(x^{+})G^{\prime}(x^{-})}{\quantity[1+\frac{1}{4}m^{2}F(x^{+})G(x^{-})]^{2}}\,,$
(135)

where $F$ and $G$ are two arbitrary smooth enough functions, the EMT (121) is

$\displaystyle\tilde{\Theta}_{++}\approx$
$\displaystyle~{}3\quantity(\frac{F^{\prime\prime}}{F^{\prime}})^{2}-2\frac{F^{\prime\prime\prime}}{F^{\prime}}=-2(\mathcal{S}F)(x^{+})\,,$
(136) $\displaystyle\tilde{\Theta}_{--}\approx$
$\displaystyle~{}3\quantity(\frac{G^{\prime\prime}}{G^{\prime}})^{2}-2\frac{G^{\prime\prime\prime}}{G^{\prime}}=-2(\mathcal{S}G)(x^{-})\,.$
(137)

This can clearly be canceled by a suitable conformal transformation (144).

###  VII.1 On the general case

We end this Section by exploring the general validity of the results just
illustrated. Let us consider a two-dimensional theory in Minkowski
spacetime777Analogous results hold in Euclidean spacetimes. enjoying anomalous
conformal symmetry.

Poincaré symmetry requires the EMT, $\Theta_{\mu\nu}$, to be symmetric,
$\Theta_{\mu\nu}\approx\Theta_{\nu\mu}$, and conserved,
$\partial_{\mu}\Theta^{\mu\nu}\approx 0$, on-shell, whereas invariance under
scale (and special conformal) transformations implies zero-trace on-shell,
$\Theta^{\mu}{}_{\mu}\approx 0$. From these it follows that

$\partial_{\mp}\Theta_{\pm\pm}\approx 0\,.$ (138)

Therefore, not only does $f^{\mu}$ split, as a result of (63), but the EMT as
well splits into right-moving and left-moving components. This holds for any
two-dimensional conformally symmetric theory, whether or not the equations of
motion imply the split of fields. Therefore, this result does not depend from
the details of the theory, but only from its symmetries.

This property is immediately passed on to the Noether charges, through their
very definition (8) and (14), so they also split in general

$Q^{\pm}[f]\approx\int\differential{x^{\pm}}\Theta_{\pm\pm}(x^{+})\,f^{\pm}(x^{+})\,,$
(139)

and they are always independent from the “time” variable $x^{\mp}$, on-shell.
Another, alternative argument is that Noether charges should be expected to
split also from the fact that conformal generators split, see (55), and
through (26) they are directly connected to the charges.

As previously shown, if a theory is anomalous, a center must take care of that
in the algebra obeyed by the charges $Q[f]$, which is a centrally extended
version of the algebra of generators, (44). For conformal symmetry the
generators have to follow the Witt algebra (57)

$\commutator{l_{n}^{\pm}}{l_{m}^{\pm}}=(n-m)\,l_{n+m}^{\pm}\,,$

and the only possible central extension of the Witt algebra is the Virasoro
algebra, [20]. Therefore, the only possible conformally anomalous form of the
algebra of charges is888This is true up to trivial central charges, that can
be removed. See Section IV and also (112).

$i\poissonbracket{Q^{\pm}_{n}}{Q^{\pm}_{m}}=(n-m)Q^{\pm}_{n+m}+\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}\,.$
(140)

The interdependence between charges and the EMT implies that the above
propagates to the Poisson brackets of the EMT as follows999In the Euclidean
the factor of $\delta^{\prime\prime\prime}$ is $c/12$. The extra $2\pi$ factor
in the denominator, in the following results, is due to the different choices
of bases, the trigonometric basis for Minkowski, (53), and the “power” basis
for Euclidean, (52). See Appendix C.

$\poissonbracket{\Theta_{\pm\pm}(x)}{\Theta_{\pm\pm}(y)}\evaluated{}_{x^{\mp}=y^{\mp}}=\Theta^{\prime}_{\pm\pm}(x)\delta(x^{\pm}-y^{\pm})+2\Theta_{\pm\pm}(x)\delta^{\prime}(x^{\pm}-y^{\pm})-\frac{c}{24\pi}\delta^{\prime\prime\prime}(x^{\pm}-y^{\pm})\,.$
(141)

Using this, again it follows from (26) that the infinitesimal conformal
transformation of the EMT

$\delta_{f}\Theta_{\pm\pm}=\poissonbracket{\Theta_{\pm\pm}}{Q[f]}\,,$ (142)

for a generic anomalous theory necessarily involve the central charge

$\delta_{f}\Theta_{\pm\pm}=f^{\pm}\partial_{\pm}\Theta_{\pm\pm}+2\Theta_{\pm\pm}\partial_{\pm}f^{\pm}-\frac{c}{24\pi}\partial^{3}_{\pm}f^{\pm}\,.$
(143)

This is a general consequence of the fact that the symmetry algebra is the
Virasoro, rather than the Witt, hence it holds for any two-dimensional
conformally anomalous theory, regardless of the details of the theory.
Noticeably, this is the very same expression one obtains in the quantum
context, see, e.g., [22]. Here, though, we are fully classical.

The “exponentiation” of the infinitesimal transformation to a finite conformal
transformation $x^{\pm}\rightarrow x^{\prime\pm}=x^{\pm}-f^{\pm}(x^{\pm})$
leads to

$\tilde{\Theta}^{\prime}_{\pm\pm}(x^{\prime})=\quantity(\partialderivative{x^{\pm}}{x^{\prime\pm}})^{2}\quantity[\tilde{\Theta}_{\pm\pm}(x)-\frac{c}{24\pi}(\mathcal{S}x^{\prime\pm})(x^{\pm})]\,,$
(144)

with $(\mathcal{S}x^{\prime})(x)$ the Schwarzian derivative, (134). This
result too holds for any conformally anomalous classical theory. The EMT does
not transform as a tensor under a general conformal transformation due to the
center, i.e., it is not a _primary field_ [22]. Nonetheless, it does so for
Möbius transformations

$x^{\prime\pm}=\frac{ax^{\pm}+b}{dx^{\pm}+e}\,,$ (145)

where $ae-bd\neq 0$, because these transformations have vanishing Schwarzian
derivative and are generated through charges
$Q^{\pm}_{-1},Q^{\pm}_{0},Q^{\pm}_{1}$ which close a centerless subalgebra
isomorphic to the global conformal algebra [22].

##  VIII Liouville theory on curved backgrounds

Knowing that, in the quantum case, anomalous EMTs in flat space are
responsible for interesting phenomena when the theory is embedded in curved
backgrounds, it is time we embed classical Liouville theory in a curved
spacetime. Although this has been done extensively in the literature, only
recently we have shown [1] that in this process a gravitational anomaly is
hiding, as suspected by Jackiw [11]. In particular, our goal here is to
explicitly show the details of the behavior of the EMT under diffeomorphism
and Weyl transformations.

Throughout the Section, several versions of Liouville action in curved
spacetime are scrutinized, and we list them all in Appendix D. Let us start
with the diffeomorphism invariant version of flat101010 We should use
different indices for flat spacetime, where form invariance holds under
Lorentz transformations, say Latin letters $a,b,c,\ldots$, and curved
spacetime, where form invariance holds under general coordinate
transformations, say Greek letters $\mu,\nu,\lambda,\ldots$. Here we do not do
that since all four flat-related situations are specifically addressed and no
ambiguity should arise. Details on how to deal with different indices are
given in the Appendix E. Liouville action (118)

$\mathcal{A}_{\scriptscriptstyle{D}}[\Phi]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})\,,$
(146)

with equation of motion

$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Phi+\frac{m^{2}}{\beta}e^{\beta\Phi}=0\,,$
(147)

and EMT, obtained as in (17), given by

$T^{\mu\nu}_{\scriptscriptstyle{D}}=\nabla^{\mu}\Phi\nabla^{\nu}\Phi-g^{\mu\nu}\quantity(\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\Phi\nabla_{\beta}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})\,.$
(148)

This tensor is covariantly conserved,
$\nabla_{\mu}T^{\mu\nu}_{\scriptscriptstyle{D}}\approx 0$, and symmetric, but
its trace is non-vanishing

${T_{\scriptscriptstyle{D}}}^{\mu}{}_{\mu}=\frac{2m^{2}}{\beta^{2}}e^{\beta\Phi}\,,$
(149)

even in the flat limit, which clashes with the just discussed conformal
invariance of previous Section.

The simplest _non-minimal, ad hoc_ improvement of the action is

$\mathcal{A}{}_{\scriptscriptstyle{NM}}[\Phi]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\alpha
R\Phi)\,,$ (150)

where $\alpha$ is a constant real parameter and
$R=g^{\sigma\nu}R^{\rho}{}_{\sigma\rho\nu}$ is the Ricci scalar, descending
from the Riemann tensor
${R^{\rho}}_{\sigma\mu\nu}\equiv\partial_{\mu}{\Gamma^{\rho}}_{\nu\sigma}-\partial_{\nu}{\Gamma^{\rho}}_{\mu\sigma}+{\Gamma^{\rho}}_{\mu\lambda}{\Gamma^{\lambda}}_{\nu\sigma}-{\Gamma^{\rho}}_{\nu\lambda}{\Gamma^{\lambda}}_{\mu\sigma}$.
The equation of motion is

$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Phi+\frac{m^{2}}{\beta}e^{\beta\Phi}=\alpha
R\,,$ (151)

and the EMT is “improved” to

$T^{\mu\nu}_{\scriptscriptstyle{NM}}=~{}\nabla^{\mu}\Phi\nabla^{\nu}\Phi-g^{\mu\nu}\quantity(\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\Phi\nabla_{\beta}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})+2\alpha\quantity(g^{\mu\nu}\nabla_{\rho}\nabla^{\rho}-\nabla^{\mu}\nabla^{\nu})\Phi\,.$
(152)

Comparing its flat limit with the flat EMT (121) gives the additional
condition $\alpha=1/\beta$. The trace is still non-vanishing for a general
metric

${T_{\scriptscriptstyle{NM}}}^{\mu}{}_{\mu}\approx\frac{2}{\beta^{2}}R\,,$
(153)

however it does vanish in the flat limit, so (152), with $\alpha=1/\beta$ is
the correct EMT111111The subscript “$L$” refers to the commonly used Liouville
theory, discussed later and in the Appendix D.
$T_{\scriptscriptstyle{L}}^{\mu\nu}$

$T^{\mu\nu}_{\scriptscriptstyle{L}}=~{}\nabla^{\mu}\Phi\nabla^{\nu}\Phi-g^{\mu\nu}\quantity(\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\Phi\nabla_{\beta}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi})+\frac{2}{\beta}\quantity(g^{\mu\nu}\nabla_{\rho}\nabla^{\rho}-\nabla^{\mu}\nabla^{\nu})\Phi\,.$
(154)

Thus flat Liouville (118) enjoys Poincaré symmetry as well as scale and
conformal symmetries, while the curved Liouville in (150) is diffeomorphism
invariant, but the trace (153) shows that to promote scale and conformal
symmetry to curved spacetime requires additional efforts.

All of the above is classical, but still related to the quantum trace
anomalies in two dimensions [14]

$T^{\mu}{}_{\mu}=\frac{c}{24\pi}R\,,$ (155)

since the known form of the flat EMT (121) requires the additional $R\Phi$
term in the action to obtain the improvement
$\frac{2}{\beta}\quantity(g^{\mu\nu}\nabla_{\rho}\nabla^{\rho}-\nabla^{\mu}\nabla^{\nu})\Phi$,
which survives the flat limit, although the $R\Phi$ term itself does not. It
is this improvement which reflects the presence of central charge

$c=\frac{48\pi}{\beta^{2}},$ (156)

of the Virasoro algebra. Thus, the non-minimal coupling to the curvature
$R\Phi$ introduces the central charge in the action and the coupling constant
$\alpha$ has to be proportional to the center $c$. At the same time, the term
$\alpha R\Phi$ modifies the trace of (17) to the exact form (155). It can be
proved that, for any two-dimensional conformal quantum field theory, the
existence of Virasoro algebra in the flat spacetime implies the trace of the
EMT to be as of (155), [36].

### Weyl gauging

The natural generalization of the scale transformation to curvilinear
coordinates is the rigid Weyl transformation (11). For Liouville this gives

$g_{\mu\nu}\xrightarrow{\text{Weyl}}e^{2\omega}g_{\mu\nu}\,,\qquad\Phi\xrightarrow{\text{Weyl}}\Phi-\frac{2}{\beta}\omega\,,$
(157)

and it can be verified that both actions, (146) and (150), are invariant under
this rigid transformation. Promoting $\omega$ to $\omega(x)$, is the first
step of the Weyl-gauging procedure, an ancestor of modern gauging [29].

The combination of a diffeomorphism leaving the metric rescaled

$g^{\prime}_{\mu\nu}(x^{\prime})=\partialderivative{x^{\alpha}}{x^{\prime\mu}}\partialderivative{x^{\beta}}{x^{\prime\nu}}g_{\alpha\beta}(x)=\Omega\quantity(x(x^{\prime}))g_{\mu\nu}\quantity(x(x^{\prime}))\,,$
(158)

with a Weyl transformation that gets rid of the factor $\Omega$, leaves the
metric untouched and it might be a symmetry. Such transformation coincides
with the definition of conformal transformation (10) in curved spacetime.
Then, the simultaneous invariance under Weyl transformation and diffeomorphism
implies the conformal symmetry. In the case of Liouville theory, the conformal
symmetry transformation (120) is

$\tilde{\delta}_{f}\Phi=\underbrace{\vphantom{\frac{1}{\beta}}f^{\mu}\partial_{\mu}\Phi}_{\text{Diff}}+\underbrace{\frac{1}{\beta}\partial_{\mu}f^{\mu}}_{\text{Weyl}}\,,$
(159)

with the Weyl transformation parameter in (157) restricted by the conformal
diffeomorphism to be

$\omega=-\frac{1}{2}\partial_{\mu}f^{\mu}\,.$ (160)

In the process of gauging, a Weyl covariant derivative $D_{\mu}$ has to be
introduced. The choice

$\nabla_{\mu}\Phi\rightarrow
D_{\mu}\Phi\equiv\nabla_{\mu}\Phi-\frac{2}{\beta}W_{\mu}\,,$ (161)

where $W_{\mu}$ is the (diffeomorphism covariant) Weyl gauge vector field,
transforming under Weyl transformation as

$W_{\mu}\xrightarrow{\text{Weyl}}W_{\mu}-\partial_{\mu}\omega\,,$ (162)

ensures $D_{\mu}\Phi$ to be a Weyl scalar. This promotes curvilinear Liouville
theory to the Weyl-gauged action

$\mathcal{A}_{\scriptscriptstyle{W}}[\Phi,W_{\mu}]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\frac{2}{\beta}\Phi\nabla_{\mu}W^{\mu}+\frac{2}{\beta^{2}}g^{\mu\nu}W_{\mu}W_{\nu})\,.$
(163)

The action $\mathcal{A}_{\scriptscriptstyle{W}}$ is invariant under both
diffeomorphism and Weyl transformations, at the price of the appearance of the
Weyl field. This is undesirable as the goal is to formulate Liouville theory
solely in terms of the dynamical $\Phi$ and geometric tensors. In [10] it is
shown that, for flat scale invariant theories that also enjoy full conformal
symmetry, the expressions in terms of Weyl fields can be traded of for
expressions in terms of curvature tensors, a process called there Ricci-
gauging.

Indeed certain combinations of the Weyl field and its derivatives transform
like curvature tensors. In two dimensions

$2\nabla_{\mu}W^{\mu}\xrightarrow{\text{Weyl}}e^{-2\omega}\quantity(2\nabla_{\mu}W^{\mu}-2\nabla_{\mu}\nabla^{\mu}\omega)\,,$
(164)

should be compared to

$R[g_{\mu\nu}]\xrightarrow{\text{Weyl}}e^{-2\omega}\quantity(R[g_{\mu\nu}]-2\nabla_{\mu}\nabla^{\mu}\omega)\,,$
(165)

while in higher dimensions, $d>2$,

$\Omega_{\mu\nu}[W]\equiv\nabla_{\mu}W_{\nu}-W_{\mu}W_{\nu}+\frac{1}{2}g_{\mu\nu}W_{\rho}W^{\rho}\,,$
(166)

transforms identically to

$\frac{1}{d-2}\quantity[R_{\mu\nu}-\frac{1}{2(d-1)}g_{\mu\nu}R]\,,$ (167)

that is

$\Omega_{\mu\nu}[W]\xrightarrow{\text{Weyl}}\Omega_{\mu\nu}[W]-\nabla_{\mu}\nabla_{\nu}\omega+\partial_{\mu}\omega\partial_{\nu}\omega-\frac{1}{2}g_{\mu\nu}\partial_{\rho}\omega\partial^{\rho}\omega\,.$
(168)

Thus, if $W_{\mu}$ appears in the Weyl-gauged theory _only_ in the previous
combinations (164) or (166), their trading for (165) or (167), respectively,
does not violate the invariance under diffeomorphism and Weyl transformations.
Yet, it does change the equations of motion, hence, the meaning of the theory.
Furthermore, this allows to systematically generate improvements for the EMT,
then computed via (17).

An implicit argument in [10] suggests that the trading of (164) for (165)
always provides a theory invariant under rigid Weyl and local Weyl
transformations, up to field independent terms. If we drop such terms, though,
we lose track of the central charge $c$, that is an important task here. We
cannot omit terms that may actually be the manifestations of non-trivial
centers. They actually are the focus of the rest of this paper.

The trading of $2\nabla_{\mu}W^{\mu}$ for $R$ introduces the non-minimal
coupling, with correct factor $1/\beta$, in the Weyl-gauged action (163)

$\mathcal{A}_{\scriptscriptstyle{R}}[\Phi]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\frac{1}{\beta}\Phi
R+\frac{2}{\beta^{2}}g^{\mu\nu}W_{\mu}W_{\nu})\,,$ (169)

but here, contrary to what is done in [10], we keep the term independent from
$\Phi$ which ensures the Weyl invariance, provided

$2\nabla_{\mu}W^{\mu}=R\,,$ (170)

holds as an equation for $W^{\mu}$. Moreover, the equation of motion for
$\Phi$ now reproduces the standard one in (151) with the coupling
$\alpha=1/\beta$

$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Phi+\frac{m^{2}}{\beta}e^{\beta\Phi}=\frac{1}{\beta}R\,,$
(171)

and we also recover the right flat limit of (119), with no additional demand
of vanishing Weyl field.

Since anomalies are routinely considered as purely quantum phenomena, hence
the concerns of this paper are not shared by many, it is customary to simply
drop $W^{\mu}W_{\mu}$. In fact, it does not affect the equation of motion and
the quantum trace anomaly is to be expected. That is why the most common form
of Liouville action in curved spacetime is [10]

$\mathcal{A}_{\scriptscriptstyle{L}}[\Phi]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\frac{1}{\beta}R\Phi)\,,$
(172)

that is (150) for $\alpha=1/\beta$. It enjoys diffeomorphism invariance, and,
in the flat spacetime, conformal symmetry. It also gives rise to the EMT
$T_{\scriptscriptstyle{L}}^{\mu\nu}$ in (154).

Nonetheless, the question of what to make of this extra term is of great
importance in this paper and the subject matter of the next Section.

##  IX The extra term $W^{\mu}W_{\mu}$

Weyl gauging provided yet another additional term, proportional to
$W^{\mu}W_{\mu}$, to the curvilinear Liouville action. The independence from
the field $\Phi$ and the request $2\nabla_{\mu}W^{\mu}=R$, reduce this extra
term to a purely geometrical term, which cannot affect the equation of motion
(151). This goes hand in hand with the fact that, under Weyl transformation of
the action $\mathcal{A}_{\scriptscriptstyle{R}}$ in (169), this term provides
the cancellation of the extra field-independent contributions.

The restored Weyl invariance means that the trace of the EMT vanishes again.
Since the $W^{\mu}$ is the sole cause of that it is clear that the
$W^{\mu}W_{\mu}$ term carries the center in the form of the “mass”121212 In
the gauged version of the theory the $W^{\mu}W_{\mu}$ term gives the mass to
the field $W^{\mu}$, as the term is accompanied by mass factor $m^{2}/2$.  as
it has to cancel the trace (155) also proportional to the center. In fact,
comparing with (130) the extra term is exactly

$\frac{c}{24\pi}W^{\mu}W_{\mu}\,.$ (173)

This implies that either the trace anomaly is a purely quantum effect or that
this extra term cancels the center completely from the theory. The impact on
the Virasoro algebra in flat spacetime can be understood with no computations.
The term (173) provides only field independent improvement to the EMT. If it
survives in the flat limit (similarly to the improvement generated by $R\Phi$
term) this improvement is invisible to the Poisson brackets which implies that
this additional improvement gives rise only to _trivial_ central charge and
does not change the _genuine_ center $c$ of the Virasoro algebra in (130).

To understand how (173) ensures the tracelessness and to see the subtleties
accompanying this hunt for Weyl invariance one has to look under the hood of
(170) and solve the equation.

A solution of (170) can be provided in the language of Green’s functions [11],
i.e. by finding $K(x,y)$ such that
$\nabla^{2}_{x}K(x,y)=\frac{1}{\sqrt{-g(x)}}\delta^{(2)}(x-y)$. Assuming that
$W_{\mu}$ is a gradient of a scalar field $w$, transforming as
$w\xrightarrow{\text{Weyl}}w-\omega$, the solution to the (170) is

$w(x)=\frac{1}{2}\int\differential[2]{y}\sqrt{-g(y)}K(x,y)R(y)\,.$ (174)

It follows that the extra $W^{\mu}W_{\mu}$ term in the action is

$\int\differential[2]{x}W^{\mu}(x)W_{\mu}(x)=\frac{1}{4}\int\differential[2]{x}\differential[2]{y}\sqrt{-g(x)}R(x)K(x,y)\sqrt{-g(y)}R(y)\,,$
(175)

which is the well-known Polyakov effective action [7], describing a quantized
two-dimensional bosonic string with vanishing cosmological constant.

This action appears in various places in physics, namely in the context of
two-dimensional gravity, where one of the first proposed models was the
Liouville gravity, [6], governed by (2)

$R-\Lambda=0\,,$ (176)

where $\Lambda$ is the cosmological constant. Polyakov effective action yields
this equation provided an additional cosmological term,
$\int\differential[2]{x}\sqrt{-g}\Lambda$ is added.

Correspondingly, the Polyakov effective action provides a simple model
exhibiting the trace anomaly, since it is diffeomorphism but not Weyl
invariant131313This is to be expected as it was derived here from the
$W^{\mu}W_{\mu}$ improvement of exactly such transformation
properties.141414In the quantum case, there is no regularization for which
Weyl and diffeomorphism invariance hold together [37]., [38, 39, 40].
Moreover, the trace of an EMT derived from the Polyakov effective action (175)
has the same form as the trace of Liouville EMT (153).

Such solution, then gives

$\begin{split}\mathcal{A}_{\scriptscriptstyle{LP}}[\Phi]=&~{}\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\frac{1}{\beta}R\Phi)\\\
&+\frac{1}{2\beta^{2}}\int\differential[2]{x}\differential[2]{y}\sqrt{-g(x)}R(x)K(x,y)\sqrt{-g(y)}R(y)\,,\end{split}$
(177)

which is both diffeomorphism and Weyl invariant, at the price of the _loss of
locality_.

A local solution to (170) was found by Deser and Jackiw151515In fact, Deser
and Jackiw arrived at this local solution by looking into the Polyakov action
and its associated EMT. A particular solution for Reissner-Nordström black
hole was derived by Iso et al in [41] using the Green function approach. in
[42] and is

$W^{\mu}_{\scriptscriptstyle{DJ}}=\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}\quantity[\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\Gamma_{\beta\alpha\nu}+(\cosh\sigma-1)\partial_{\nu}\gamma]\,,$
(178)

where $\varepsilon^{tx}=+1$ is the Levi-Civita symbol161616Alternatively, we
choose $\varepsilon^{-+}=+1$ in the light-cone coordinates as explained in
Appendix C. and a “conformal” parametrization of the metric is introduced

$\frac{g_{++}}{\sqrt{-g}}=e^{\gamma}\sinh\sigma\,,\qquad\frac{g_{+-}}{\sqrt{-g}}=\cosh\sigma\,,\qquad\frac{g_{--}}{\sqrt{-g}}=e^{-\gamma}\sinh\sigma\,,$
(179)

and from there

$\gamma=\ln\sqrt{g_{++}/{g_{--}}}\,.$ (180)

Two peculiar properties might be immediately spotted. First, the equation
(170) is blind to additional improvements of the form

$W^{\mu}_{\scriptscriptstyle{DJ}}\rightarrow
W^{\mu}_{\scriptscriptstyle{DJ}}+\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}\partial_{\nu}r,$
(181)

where $r$ is any Weyl scalar. Thus, the solution (178) is not unique. Second,
the first term of $W_{\scriptscriptstyle{DJ}}^{\mu}$ is proportional to the
Levi-Civita connection while the second term requires parametrization of the
metric. This indicates non-vectorial transformation properties. In fact, under
an infinitesimal diffeomorphism

$x^{\mu}\rightarrow x^{\mu}-f^{\mu}(x),$ (182)

it transforms as

$W^{\prime}{}_{\scriptscriptstyle{DJ}}^{\mu}(x^{\prime})=~{}\partialderivative{x^{\prime\mu}}{x^{\nu}}W_{\scriptscriptstyle{DJ}}^{\nu}(x)+\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}\partial_{\nu}\quantity[\quantity(\partial_{-}-e^{-\gamma}\tanh\frac{\sigma}{2}\partial_{+})f^{-}-\quantity(\partial_{+}-e^{\gamma}\tanh\frac{\sigma}{2}\partial_{-})f^{+}]\,,$
(183)

as shown in [43]. This proves that the local solution to
$2\nabla_{\mu}W^{\mu}=R$ is not a contravariant vector, even though its
divergence $\nabla_{\mu}W^{\mu}_{\scriptscriptstyle{DJ}}$ is still a scalar.
On the other hand, under Weyl transformation

$g_{\mu\nu}W_{\scriptscriptstyle{DJ}}^{\nu}\xrightarrow{\text{Weyl}}g_{\mu\nu}W_{\scriptscriptstyle{DJ}}^{\nu}-\partial_{\mu}\omega\,.$
(184)

That is the required transformation (162).

A solution that bypasses the parametrization (179) is possible. It requires
the introduction of _vielbein_ and of _spin connection_ (see Appendix E). If
one does that $W^{\mu}$ takes a compact and appealing form

$W^{\mu}=\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}s_{\nu ab}\varepsilon^{ab}\,,$
(185)

where $s_{\nu ab}$ is the spin connection, the Latin indices $a,b$ represent
the local Lorentz frame and the Greek indices $\mu,\nu$ represent the general
coordinate frame as is explained in detail in Appendix E. Again it can be seen
that $W^{\mu}$ is not a contravariant vector.

It follows from (183) that the improvement term
$g_{\mu\nu}W_{\scriptscriptstyle{DJ}}^{\mu}W_{\scriptscriptstyle{DJ}}^{\nu}$
of the action cannot be a scalar anymore and while this solution keeps Weyl
invariance and locality of the whole action (169), it breaks the
diffeomorphism invariance.

The handling of the $W^{\mu}W_{\mu}$ term closely resembles the situation one
encounters dealing with quantum trace anomaly, even though we never left the
classical theory. Here and there, one is forced to keep only one of the two
invariances, Weyl or diffeomorphism, but cannot have both, [44]. In the rest
of the paper the implications and the “amount” of diffeomorphism breakdown are
studied, focusing on the $W_{\scriptscriptstyle{DJ}}^{\mu}$ solution

$W^{\mu}_{\scriptscriptstyle{DJ}}=\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}\quantity[\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\Gamma_{\beta\alpha\nu}+(\cosh\sigma-1)\partial_{\nu}\gamma+\partial_{\nu}r]\,,$
(186)

where the term (181), that makes explicit the mentioned ambiguity, is
included. The label “$DJ$” is omitted in the following.

##  X Energy-momentum tensor

Let us now look at the impact of (186) on the EMT. First consider the
variational identity

$\delta\quantity[(\cosh\sigma-1)\partial_{\nu}\gamma]-\partial_{\nu}\quantity[(\cosh\sigma-1)\delta\gamma]=\delta\quantity(\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}})\Gamma_{\beta\alpha\nu}+\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\Gamma_{\beta\lambda\nu}g^{\lambda\sigma}\delta
g_{\alpha\sigma}\,,$ (187)

giving

$\delta W^{\mu}=-\frac{1}{2}W^{\mu}g^{\alpha\beta}\delta
g_{\alpha\beta}+\frac{\varepsilon^{\mu\nu}}{2\sqrt{-g}}\quantity(\frac{\varepsilon^{\alpha\lambda}}{\sqrt{-g}}\quantity(\Gamma^{\beta}{}_{\lambda\nu}\delta
g_{\alpha\beta}+\partial_{\alpha}\delta
g_{\nu\lambda})+\partial_{\nu}\quantity[\quantity(\cosh\sigma-1)\delta\gamma+\delta
r])\,.$ (188)

Then the extra contribution

$T^{\mu\nu}_{\text{extra}}\equiv-\frac{2}{\sqrt{-g}}\functionalderivative{g_{\mu\nu}}\int\differential[2]{x}\frac{2}{\beta^{2}}\sqrt{-g}W^{\mu}W_{\mu},$

was derived in [43] as

$\begin{split}\beta^{2}T^{\mu\nu}_{\text{extra}}=&~{}2g^{\mu\nu}W_{\rho}W^{\rho}-4W^{\mu}W^{\nu}-2Rg^{\mu\nu}+2\nabla^{\mu}W^{\nu}+2\nabla^{\nu}W^{\mu}\\\
&+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\partial_{\beta}W_{\alpha}[(\cosh\sigma-1)\Gamma^{\mu\nu}+r^{\mu\nu}]\,,\end{split}$
(189)

where $W_{\rho}\equiv g_{\rho\lambda}W^{\lambda}$ and

$\Gamma^{\mu\nu}=\frac{1}{2\sqrt{g_{--}g_{++}}}\begin{pmatrix}-\sinh\gamma&\cosh\gamma\\\
\cosh\gamma&-\sinh\gamma\end{pmatrix}\,,$ (190)

and $r^{\mu\nu}\equiv\delta r/\delta g_{\mu\nu}$.

As wanted, the trace of this improvement171717Assuming
$r^{\mu\nu}g_{\mu\nu}=0$.

$g_{\mu\nu}T^{\mu\nu}_{\text{extra}}=-\frac{2}{\beta^{2}}R\,,$ (191)

exactly cancels the trace (155) and the complete EMT,
$T_{\scriptscriptstyle{L}}^{\mu\nu}+T^{\mu\nu}_{\text{extra}}$, is traceless,
proving the Weyl invariance of the theory.

The diffeomorphism invariance, that manifested itself as a conservation
equation $\nabla_{\mu}T^{\mu\nu}=0$, is unclear as181818The covariant
derivative here is computed as if the $W^{\mu}$ and consequently
$T^{\mu\nu}_{\text{extra}}$ were proper tensors.

$\begin{split}\beta^{2}\nabla_{\mu}T^{\mu\nu}_{\text{extra}}=&~{}2\nabla_{\rho}\nabla^{\rho}W^{\nu}-RW^{\nu}-\nabla^{\nu}R+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\partial_{\mu}\bigg{(}\partial_{\beta}W_{\alpha}\quantity[(\cosh\sigma-1)\Gamma^{\mu\nu}+r^{\mu\nu}]\bigg{)}\\\
&+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\partial_{\beta}W_{\alpha}\,g^{\mu\nu}\partial_{\sigma}g_{\lambda\mu}\quantity(\quantity[(\cosh\sigma-1)\Gamma^{\sigma\lambda}+r^{\sigma\lambda}]+\frac{1}{2}\frac{\varepsilon^{\sigma\lambda}}{\sqrt{-g}})\,,\end{split}$
(192)

where
$2\nabla_{\rho}\nabla^{\rho}W^{\nu}=2g^{\alpha\beta}(\partial_{\alpha}\partial_{\beta}W^{\nu}+2\Gamma^{\nu}{}_{\beta\lambda}\partial_{\alpha}W^{\lambda}-\Gamma^{\lambda}{}_{\alpha\beta}\partial_{\lambda}W^{\nu}+W^{\lambda}\partial_{\lambda}\Gamma^{\nu}{}_{\alpha\beta})-RW^{\nu}$.

Due to the parametrization (179), we were not able to derive a simpler form of
(192) to put in contact with known expressions of the quantum gravitational
anomalies, such as the so-called consistent anomaly [45] (our notation follows
[19], see also [46])

$\nabla_{\mu}T^{\mu}{}_{\nu}=\frac{1}{48\pi}\frac{\varepsilon^{\sigma\rho}}{2\sqrt{-g}}\partial_{\rho}\partial_{\lambda}\Gamma^{\lambda}{}_{\nu\sigma}\,,$
(193)

or the so-called covariant anomaly [45] (our notation follows [40])

$\nabla_{\mu}T^{\mu}{}_{\nu}=\frac{1}{48\pi}\partial_{\nu}R\,.$ (194)

Let us then rewrite (192) in isothermal coordinates and let us denote all
quantities evaluated in these coordinates as endowed with a hat. We have

$\hat{g}_{\mu\nu}(x)=e^{2\rho(x)}\eta_{\mu\nu}=e^{2\rho(x)}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,,$ (195)

where the indices are labeled $\mu=(t,r)$, and in light-cone coordinates

$\hat{g}_{\pm\pm}(x)=e^{2\rho(x)}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,.$
(196)

The Levi-Civita connection, contracted in the covariant indices, vanishes

$\hat{g}^{\mu\nu}\hat{\Gamma}{}^{\lambda}{}_{\mu\nu}=0\,,$ (197)

as they are

$\hat{\Gamma}{}^{t}{}_{\mu\nu}=\begin{pmatrix}\dot{\rho}&\rho^{\prime}\\\
\rho^{\prime}&\dot{\rho}\end{pmatrix},\qquad\hat{\Gamma}{}^{r}{}_{\mu\nu}=\begin{pmatrix}\rho^{\prime}&\dot{\rho}\\\
\dot{\rho}&\rho^{\prime}\end{pmatrix},$ (198)

or even simpler in the light-cone version

$\hat{\Gamma}{}^{\pm}{}_{\pm\pm}=2\partial_{\pm}\rho\,,$ (199)

where the other combinations vanish. The Ricci scalar is

$\hat{R}=-2\hat{g}^{\mu\nu}\partial_{\mu}\partial_{\nu}\rho\,.$ (200)

Moreover, the relation between Levi-Civita connection and Ricci scalar
simplifies to

$2\hat{g}^{\alpha\beta}\partial_{\alpha}\hat{\Gamma}{}^{\lambda}{}_{\beta\mu}=-\hat{R}\hat{g}^{\lambda}{}_{\mu}\,,$
(201)

since

$\hat{g}{}^{\alpha\beta}\hat{\Gamma}{}^{\lambda}{}_{\alpha\sigma}\hat{\Gamma}{}^{\sigma}{}_{\mu\beta}=0\,.$
(202)

In these coordinates, since the parametrization of metric gives

$\cosh\hat{\sigma}=\hat{g}_{+-}/\sqrt{-\hat{g}}=1\,\qquad\tanh\frac{\hat{\sigma}}{2}=0\,,$
(203)

the solution (186) can be written as simply as

$\hat{W}{}^{\mu}=\frac{1}{2}\partial_{\nu}\hat{g}^{\mu\nu}+\frac{\varepsilon^{\mu\nu}}{2\sqrt{-\hat{g}}}\partial_{\nu}\hat{r}\,,$
(204)

Another simple identity holds

$\varepsilon^{\alpha\beta}\partial_{\beta}\hat{W}{}_{\alpha}=-\frac{\sqrt{-\hat{g}}}{2}\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\hat{r}\,.$
(205)

It should be stressed, that $\gamma$ in (180) is an undefined function in
isothermal coordinates. We assume that $\gamma$ and its derivatives are
bounded.

Substituting the previous identities and setting $r=0$ for a moment, the
improvement of EMT (189) becomes

$\frac{\beta^{2}}{4}\hat{T}{}^{\mu\nu}_{\text{extra}}=(\hat{g}^{\mu\alpha}\partial_{\alpha}\rho)(\hat{g}^{\nu\beta}\partial_{\beta}\rho)-\frac{1}{2}\hat{g}^{\mu\nu}(\hat{g}^{\alpha\beta}\partial_{\alpha}\rho\partial_{\beta}\rho)+\quantity[\hat{g}^{\mu\nu}(\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta})-\hat{g}^{\mu\alpha}\hat{g}^{\nu\beta}\partial_{\alpha}\partial_{\beta}]\rho\,.$
(206)

Notice the close resemblance to the Liouville EMT191919Equivalent situation
appears for the Polyakov effective action, where $w(x)$ in the EMT follows the
same structure, [40]. (152) upon taking $\beta=2$ and $m^{2}=0$. The “field”
$\rho$ even transforms as a Liouville field under Weyl transformation (157)

$\rho\xrightarrow{\text{Weyl}}\rho+\omega\,,$

and the equation (200) replicates the Liouville equation of motion (151).
However, it should not be forgotten that $\rho$ is part of the metric tensor
$\hat{g}_{\mu\nu}$ and that $\rho$ is still a fixed background field. This is
where the analogy with the dynamical Liouville scalar field stops.

The divergence (192) of $T^{\mu\nu}_{\text{extra}}$ simplifies in the
isothermal coordinates to

$\beta^{2}\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}\frac{\varepsilon^{\nu\mu}}{\sqrt{-\hat{g}}}\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\partial_{\mu}\hat{r}-2\hat{g}^{\alpha\beta}\partial_{\mu}\quantity(\hat{r}^{\mu\nu}\partial_{\alpha}\partial_{\beta}\hat{r})+2\partial_{\mu}\hat{g}^{\alpha\beta}\hat{r}^{\mu\nu}\partial_{\alpha}\partial_{\beta}\hat{r}\,.$
(207)

Not only it does not coincide with (193) or (194), but it does vanish for $r$
satisfying

$\hat{\Box}_{M}\hat{r}\equiv\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\hat{r}=0\,,$
(208)

and $T^{\mu\nu}_{\text{extra}}$, thus, may produce a traceless EMT which also
has a vanishing divergence in isothermal coordinates. Notice that

$\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\hat{r}=\frac{1}{\sqrt{-g}}\hat{\Box}_{M}\hat{r}\,,$
(209)

and that for this choice, (208), the right-hand side of (205) also vanishes

$\varepsilon^{\alpha\beta}\partial_{\beta}\hat{W}{}_{\alpha}=0\,.$ (210)

Although, equation (207) looks like a covariant conservation for the choice
(208)

$\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}\evaluated{}_{\hat{\Box}_{M}\hat{r}=0}=0,$
(211)

it has to be emphasized that $T^{\mu\nu}_{\text{extra}}$ is not a tensor.
Under a general coordinate transformation the (211) does not have to
necessarily hold. Therefore, to investigate diffeomorphism invariance one more
step has to be taken and the transformation of
$\hat{\nabla}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}$ has to be found. In the
rest of the paper the choice (208) is assumed, since such $r$ may lead to
simultaneous Weyl and diffeomorphism invariance.

##  XI Breaking diffeomorphism invariance

The loss of diffeomorphism invariance is the loss of general covariance. Thus,
the equations depend on the chosen coordinate frame and the results cannot be
easily transformed between coordinates. Furthermore, the choice of the initial
coordinate frame determines what physics the observer sees, hence it stands as
a privileged frame. With the choice of isothermal coordinates, as initial
coordinate frame, we might have just been lucky. The transformation of (192)
under a general coordinate transformation, or more precisely, of its non-
tensorial part

$\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}(x)\equiv\nabla^{\prime}_{\mu}T^{\prime}{}^{\mu\nu}_{\text{extra}}(x^{\prime})-\partialderivative{x^{\prime\nu}}{x^{\rho}}\hat{\nabla}{}_{\sigma}\hat{T}{}^{\sigma\rho}_{\text{extra}}(x)\,,$
(212)

should be derived, to prove diffeomorphism (non-)invariance. This has to be,
at least partially, expressible in terms of $\Delta\hat{W}{}^{\mu}(x)\equiv
W^{\prime\mu}(x^{\prime})-(\partial x^{\prime\mu}/\partial
x^{\nu})\hat{W}{}^{\nu}(x)$.colorblue For the infinitesimal diffeomorphism
(182) the transformation of $W^{\mu}$ follows (183) and the non-tensorial part
is

$\Delta\hat{W}{}^{\mu}=\frac{\varepsilon^{\mu\nu}}{2\sqrt{-\hat{g}}}\partial_{\nu}\quantity(\partial_{-}f^{-}-\partial_{+}f^{+})+\quantity(\delta^{\mu}_{\alpha}-\partial_{\alpha}f^{\mu})\frac{\varepsilon^{\alpha\beta}}{2\sqrt{-\hat{g}}}\partial_{\beta}\Delta
r\,,$ (213)

where $\Delta r$ was defined as the deviation of $r$ from the scalar
transformation $\Delta r(x)\equiv r^{\prime}(x^{\prime})-r(x)$.

As $r(x)$ is only assumed to be Weyl scalar, (181), its transformation under
diffeomorphism is not specified. To improve the transformation properties of
$W^{\mu}$, $r(x)$ has to tackle the extra terms of (183), which are
proportional to $f^{\mu}$. It is, thus, natural to assume that $\Delta r$
contains at least the first power of $f^{\mu}$ in all of its terms, since
finite terms (terms not containing $f^{\mu}$) in $\Delta r$ would necessarily
inflate the deviation of $W^{\mu}$ from a contravariant vector. Under this
assumption the previous equation is

$\Delta\hat{W}{}^{\mu}=\frac{\varepsilon^{\mu\nu}}{2\sqrt{-\hat{g}}}\partial_{\nu}\,\partial_{\alpha}\quantity(\frac{\varepsilon^{\alpha\beta}}{\sqrt{-\hat{g}}}f_{\beta})+\frac{\varepsilon^{\mu\nu}}{2\sqrt{-\hat{g}}}\partial_{\nu}\Delta
r\equiv\frac{\varepsilon^{\mu\nu}}{2\sqrt{-\hat{g}}}\partial_{\nu}\xi(f,r)\,,$
(214)

and $\Delta\hat{W}{}^{\mu}$ is proportional to the infinitesimal factor
$f^{\mu}$.

Taking this as a starting point, it is clear that several terms in (192) do
not contribute to the infinitesimal transformation. In the isothermal
coordinates and under the assumption (208), $\hat{\Box}_{M}\hat{r}=0$, the
factors $(\cosh\hat{\sigma}-1)$ and
$\varepsilon^{\alpha\beta}\partial_{\beta}W_{\alpha}$ separetely vanish. Terms
in (192) containing both factors effectively disappear since two vanishing
factors are multiplied and only one at a time can be replaced by a factor
containing $f^{\mu}$. The effective part of
$\nabla_{\mu}T^{\mu\nu}_{\text{extra}}$ of concern in (212) is then

$\begin{split}\beta^{2}\nabla_{\mu}T^{\mu\nu}_{\text{extra}}\overset{\text{eff}}{=}&~{}2\nabla_{\rho}\nabla^{\rho}W^{\nu}-RW^{\nu}+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\partial_{\mu}\quantity(\partial_{\beta}W_{\alpha}r^{\mu\nu})\\\
&+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-g}}\partial_{\beta}W_{\alpha}g^{\mu\nu}\partial_{\sigma}g_{\lambda\mu}\quantity(r^{\sigma\lambda}+\frac{1}{2}\frac{\varepsilon^{\sigma\lambda}}{\sqrt{-g}})\,.\end{split}$
(215)

The non-tensorial part of the transformation is

$\begin{split}\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}=&~{}2\hat{\nabla}{}_{\rho}\hat{\nabla}{}^{\rho}\Delta\hat{W}{}^{\nu}-\hat{R}\Delta\hat{W}{}^{\nu}+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-\hat{g}}}\partial_{\mu}\quantity[\partial_{\beta}(\hat{g}_{\alpha\rho}\Delta\hat{W}{}^{\rho})\hat{r}^{\mu\nu}]\\\
&+4\frac{\varepsilon^{\alpha\beta}}{\sqrt{-\hat{g}}}\partial_{\beta}\quantity(\hat{g}_{\alpha\rho}\Delta\hat{W}{}^{\rho})\hat{g}{}^{\mu\nu}\partial_{\sigma}\hat{g}_{\lambda\mu}\quantity(\hat{r}^{\sigma\lambda}+\frac{1}{2}\frac{\varepsilon^{\sigma\lambda}}{\sqrt{-\hat{g}}})\,.\end{split}$
(216)

Using (214), and carefully evaluating each term, it follows

$\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}=\frac{\varepsilon^{\nu\mu}}{\sqrt{-\hat{g}}}\hat{g}^{\alpha\beta}\partial_{\mu}\partial_{\alpha}\partial_{\beta}\xi-2\hat{g}^{\alpha\beta}\partial_{\mu}\quantity(\hat{r}^{\mu\nu}\partial_{\alpha}\partial_{\beta}\xi)+2\hat{r}^{\mu\nu}\partial_{\mu}\hat{g}{}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\xi\,.$
(217)

Using Jackiw’s choice (178), $r=0$, this simplifies significantly to

$\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}=\frac{\varepsilon^{\nu\mu}}{\sqrt{-\hat{g}}}\hat{g}^{\alpha\beta}\partial_{\mu}\partial_{\alpha}\partial_{\beta}\quantity(\partial_{-}f^{-}-\partial_{+}f^{+})\,.$
(218)

This does not vanish for a general $f^{\mu}$, proving the loss of
diffeomorphism invariance in the Weyl invariant formulation of Liouville
theory (169), with a choice of Deser-Jackiw solution (178)

$\mathcal{A}_{\scriptscriptstyle{LDJ}}[\Phi]=\int\differential[2]{x}\sqrt{-g}\quantity(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{m^{2}}{\beta^{2}}e^{\beta\Phi}+\frac{1}{\beta}\Phi
R+\frac{2}{\beta^{2}}g_{\mu\nu}W^{\mu}_{\scriptscriptstyle{DJ}}W^{\nu}_{\scriptscriptstyle{DJ}})\,.$
(219)

Therefore for a general local solution of $2\nabla_{\mu}W^{\mu}=R$ the Weyl
invariance is achieved at the cost of diffeomorphism invariance.

Whether a diffeomorphism can be restored through a suitable choice of $r$ is
unclear to us. Hopefully the equations (207) and (217) may hint appropriate
choices of $r$, as well as the fact, that $r$ needs to be Weyl scalar,
$r\xrightarrow{\text{Weyl}}r$, to preserve Weyl transformation of $W^{\mu}$
(162).

One set of metric-based Weyl scalars is formed by $\sqrt{-g}g^{\alpha\beta}$
and $g_{\alpha\beta}/\sqrt{-g}$ as suggest by the parametrization (179).
Moreover, in isothermal coordinates these are constants and any $r$ built out
of them automatically satisfies (208).

The loss of diffeomorphism, together with the non-trivial form of
$T^{\mu\nu}_{\text{extra}}$, jeopardize the simple transformations between
coordinate frames. Moreover, the complete lack of diffeomorphism hinders from
using our results with isothermal coordinates as a midpoint, since
infinitesimal transformation between the coordinate frames may not exist. The
form of (218) suggests there might be a subset of diffeomorphisms which do not
violate the general covariance. Such subset is of great importance since
suitable choice of coordinate frame can reduce the degrees of freedom of the
problem, as the illustrated choice of isothermal coordinates. A general two-
dimensional metric $g_{\mu\nu}$ given by three independent functions $g_{tt}$,
$g_{tx}=g_{xt}$ and $g_{xx}$ was reduced to a single function $\rho$ and the
two degrees of freedom were removed by the diffeomorphism
$x^{\mu}\rightarrow\hat{x}^{\mu}(x)$.

Compared to the diffeomorphisms, the supplied Weyl invariance can reduce only
one degree of freedom. Nonetheless, some theories allow keeping the Weyl
invariance, while reducing the diffeomorphism invariance to invariance under
area-preserving diffeomorphisms, [39, 40]. These are the diffeomorphisms (182)
such that

$\partial_{\mu}f^{\mu}=0\,.$ (220)

Such subset of diffeomorphisms can remove one degree of freedom, as $f^{\mu}$
may be written as $f^{\mu}=\varepsilon^{\mu\nu}\partial_{\nu}f$, where $f$ is
an arbitrary function.

As a starting point to find the transformations which respect the tensorial
behavior of $\hat{\nabla}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}$ we move to
the flat limit by taking $\rho\rightarrow 0$ in (217)

$\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}\evaluated{}_{\rho\rightarrow
0}=\varepsilon^{\nu\mu}\partial_{\mu}\Box_{M}\xi(r,f)-2\partial_{\mu}\quantity(r^{\mu\nu}\Box_{M}\xi(r,f))\,.$
(221)

The function $\xi(r,f)$ depends only on $f^{\mu}$ and the exact dependence on
$f^{\mu}$ is determined by the choice of $r$, but regardless of the choice,
$f^{\mu}$ is accompanied by at least first order derivative in each term of
$\xi(r,f)$. Thus, the form of (221) suggests that quadratic polynomial
transformations

$f^{\mu}=a^{\mu}{}_{\alpha\beta}x^{\alpha}x^{\beta}+b^{\mu}{}_{\alpha}x^{\alpha}+c^{\mu}\,,$
(222)

which include the Poincaré transformations

$f^{\mu}=\omega^{\mu}{}_{\nu}x^{\nu}+c^{\mu}\,,$ (223)

do not violate the tensorial transformation of
$\hat{\nabla}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}$.

From the linearity of $\xi(r,f)$ in $f^{\mu}$ and commutativity of partial
derivatives it follows that for conformal transformations

$\Box_{M}f^{\mu}=0\,,$

the right-hand side of the (221) also vanish.

_We conclude that for infinitesimal conformal and Poincaré transformations in
flat spacetime the extra term of the EMT,_ $T^{\mu\nu}_{\text{extra}}$_, is
conserved, regardless of the choice of_ $r$. Thus, the extra term does not
violate the symmetries of the EMT in the flat limit.

It should be noted that the subset of conformal diffeomorphisms leaves the
metric rescaled and their effect on metric is equivalent to Weyl
transformations and compared to the area-preserving diffeomorphisms (220) it
does not provide coordinates simplifying a general metric in already Weyl
invariant theory. Therefore, we hope to find another transformations for which
$\hat{\nabla}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}$ transforms tensorially.
It is clear, that such additional transformations depend on the choice of $r$.

For the choice $r=0$ (221) simplifies to

$\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}\evaluated{}_{\rho\rightarrow
0}=\varepsilon^{\nu\mu}\partial_{\mu}\Box_{M}\quantity(\varepsilon^{\alpha\beta}\partial_{\alpha}f_{\beta})\,.$
(224)

This immediately shows that the diffeomorphisms (220) do not respect tensorial
transformations. Whether there exists a choice of $r$ such that the area-
preserving diffeomorphisms are satisfied is unclear to us. However, a
transformation

$f_{\mu}=\partial_{\mu}f\,,$ (225)

where $f$ is an arbitrary function, does respect the tensorial
transformations. It should be noted that this is rather a hint and more work
has to be done to generalize this transformation back to curved background in
such a way that $W{}_{\scriptscriptstyle{DJ}}^{\mu}(x)$ transforms
vectorially.

As demonstrated, the Weyl invariance produces traceless EMT

$\tilde{T}{}^{\mu\nu}\equiv
T^{\mu\nu}_{\scriptscriptstyle{L}}+T^{\mu\nu}_{\text{extra}}\,.$ (226)

That means the center manifested in the “anomalous” trace vanishes.
Nonetheless, the center is still present in the transformation of the curved
space EMT (226), as it was in the transformation of the flat spacetime version
of the EMT, (132). For the infinitesimal diffeomorphism (182),
$x^{\mu}\rightarrow x^{\mu}-f^{\mu}(x)$, the non-tensorial transformation of
the extra term $T^{\mu\nu}_{\text{extra}}$ is

$\begin{split}\beta^{2}\Delta\hat{T}{}^{\mu\nu}_{\text{extra}}(x)\equiv&~{}\beta^{2}\left(\hat{T}^{\prime\mu\nu}_{\text{extra}}(x^{\prime})-\partialderivative{x^{\prime\mu}}{x^{\rho}}\partialderivative{x^{\prime\nu}}{x^{\sigma}}T^{\rho\sigma}_{\text{extra}}(x)\right)\\\
=&~{}\quantity(\hat{g}^{\mu\alpha}\frac{\varepsilon^{\nu\beta}}{\sqrt{-\hat{g}}}+\hat{g}^{\nu\alpha}\frac{\varepsilon^{\mu\beta}}{\sqrt{-\hat{g}}})\partial_{\alpha}\partial_{\beta}\xi\\\
&+(\hat{g}^{\mu\nu}\hat{g}^{\alpha\beta}-\hat{g}^{\mu\beta}\hat{g}^{\nu\alpha}-\hat{g}^{\mu\alpha}\hat{g}^{\nu\beta})\partial_{\alpha}\hat{r}\partial_{\beta}\xi-2\hat{r}^{\mu\nu}\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\xi\,,\end{split}$
(227)

which reduces for the choice $r=0$ to

$\beta^{2}\Delta\hat{T}{}^{\mu\nu}_{\text{extra}}(x)=\quantity(\hat{g}^{\mu\alpha}\frac{\varepsilon^{\nu\beta}}{\sqrt{-\hat{g}}}+\hat{g}^{\nu\alpha}\frac{\varepsilon^{\mu\beta}}{\sqrt{-\hat{g}}})\partial_{\alpha}\partial_{\beta}\xi$
(228)

Assuming conformal diffeomorphisms and taking the flat limit202020Let us
remind here that $\varepsilon^{-+}=+1$. we have

$\Delta\hat{T}{}^{\pm\pm}_{\text{extra}}(x)\evaluated{}_{\rho\rightarrow
0}=-\frac{2}{\beta^{2}}\partial^{3}_{\mp}f^{\mp}\,.$ (229)

Thus, the full improved EMT (226) transforms non-tensorially since

$\Delta\tilde{T}{}^{\mu\nu}=\Delta T^{\mu\nu}_{\scriptscriptstyle{L}}+\Delta
T^{\mu\nu}_{\text{extra}}=\Delta T^{\mu\nu}_{\text{extra}}\,,$ (230)

and it follows that

$\Delta\hat{\tilde{T}}{}^{\pm\pm}(x)\evaluated{}_{\rho\rightarrow
0}=-\frac{2}{\beta^{2}}\partial^{3}_{\mp}f^{\mp}\,.$ (231)

The measure of non-tensoriality $\Delta$ is a universal operation and we can
apply it on the EMT of flat Liouville theory $\tilde{\Theta}_{\mu\nu}$, (121),

$\Delta\tilde{\Theta}_{\pm\pm}\equiv\tilde{\Theta}^{\prime}_{\pm\pm}(x^{\prime})-\partialderivative{x^{\mu}}{x^{\prime\pm}}\partialderivative{x^{\nu}}{x^{\prime\pm}}\tilde{\Theta}_{\mu\nu}(x)=-\frac{2}{\beta^{2}}\partial^{3}_{\pm}f^{\pm}\,,$
(232)

where the last equality follows from (132). To make contact between the two
EMTs, just notice $\Theta^{\pm\pm}=\Theta_{\mp\mp}$. Thus (231) is exactly the
extra anomalous part of conformal transformation of the flat Liouville EMT
$\tilde{\Theta}_{\mu\nu}$, (232),

$\Delta\hat{\tilde{T}}{}^{\pm\pm}\evaluated{}_{\rho\rightarrow
0}=\Delta\tilde{\Theta}{}^{\pm\pm}\,,$ (233)

with center given by $c=48\pi/\beta^{2}$.

This way the center is removed from the trace of the EMT, restoring Weyl
invariance, but it appears in its transformations, breaking general
covariance.

##  XII Conclusions

In the framework of the general results on conformal symmetry of [10], based
on the assumption that diffeomorphism and Weyl invariances for classical field
theories can be made to hold together, and motivated by Jackiw’s conjecture
[11], proved in [1], here we enlarged the scope of [1] and gave a detailed
explanation for many of the results reported in that paper.

We did so by searching for the classical central charges and by studying the
implications of their presence in a classically anomalous theory, that is a
theory with genuine central charge in the algebra of Noether charges. We then
studied conformal transformations in two dimensions, where they satisfy the
Witt algebra. The emergence of a center $c=48\pi/\beta^{2}$ of the Virasoro
algebra of conformal Noether charges for Liouville theory in Minkowski
spacetime is demonstrated in detail. Such center is responsible for an
anomalous transformation of the EMT under the conformal transformations.
Moreover, the form of the anomalous transformation is the same as an anomalous
transformation of an EMT known in conformal _quantum_ field theories.

We then moved to Liouville theory in curved spacetime. The diffeomorphism
invariant version of the theory is Weyl-gauged, leading to diffeomorphism and
Weyl invariant Liouville theory with a dynamical Weyl-gauge field, $W^{\mu}$.
The condition (170), $2\nabla_{\mu}W^{\mu}=R$, on the Weyl-gauge field allows
to eliminate the gauge field while introducing a non-minimal coupling to the
scalar curvature, $R$, and a purely geometrical improvement, independent of
the dynamical Liouville field. This trade-off formally seems to keep the
diffeomorphism and Weyl invariance. In fact, it is not so, as the solution to
condition (170) is either non-local (leading to the Polyakov effective action)
or it leads to a quantity that does not transform as a contravariant vector.
Therefore, if one wants to preserve Weyl invariance, either locality or
diffeomorphism invariance is lost. Notice that the very same geometrical
improvement was found in [11], where Weyl invariant Liouville theory was
constructed from a dimensional limit.

Choosing the local, non-tensorial, Deser-Jackiw solution to the condition,
(170), we proceeded with calculating the extra contributions to the EMT
stemming from the geometrical improvement. Since diffeomorphism and Weyl
invariances of a theory are translated to properties of its EMT, the
tracelessness, implied by the latter invariance, is demonstrated for an
arbitrary metric in Weyl invariant Liouville theory. Following with explicit
calculations, in isothermal coordinates, we showed that for a general local
solution to the condition (170) the EMT is not conserved because the
transformation

$\beta^{2}\Delta\hat{\nabla}{}_{\mu}\hat{T}{}^{\mu\nu}_{\text{extra}}=\frac{\varepsilon^{\nu\mu}}{\sqrt{-\hat{g}}}\hat{g}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\partial_{\mu}\quantity(\partial_{-}f^{-}-\partial_{+}f^{+})\,,$

does not vanish for a general $f^{\mu}$. This implies the loss of
diffeomorphism invariance for Weyl invariant Liouville theory. This proves
Jackiw’s conjecture [11].

Deser-Jackiw solution to the condition (170) contains an undetermined function
$r$. This ambiguity and its relation to the preservation of diffeomorphism
invariance were studied. In (207) and (217) we explicitly derived the
necessary conditions $r$ must satisfy for diffeomorphism invariance to hold.
However, we were not able to find a solution to the conditions and whether
diffeomorphism invariance is achievable, through a particular choice of $r$,
is unclear. The formula (217) also shows how the choice of $r$ is related to
the preserved subset of diffeomorphisms. Regardless of the choice, though,
Poincaré and global conformal symmetries are preserved, in the flat limit.
This substantiates the results of [10] on scale invariance and full conformal
invariance of Liouville theory in flat space.

In parallel, we tracked the presence of the central charge in curved
background. Starting with the commonly used Liouville theory, which is
diffeomorphism invariant and non-minimally coupled to curvature, the trace of
its EMT provides a classical instance of the trace anomaly formula, since it
is proportional only to the center $c$ and curvature $R$. When the theory is
made Weyl invariant the trace vanishes. The center does not disappear from the
theory, though, but rather moves into the transformation of the EMT (227),
which, in the flat limit, gives back the anomalous conformal transformation of
the EMT

$\Delta\hat{T}{}^{\pm\pm}_{\text{extra}}(x)\evaluated{}_{\rho\rightarrow
0}=-\frac{c}{24\pi}\partial^{3}_{\mp}f^{\mp}.$

Although the value of the central charge of Liouville theory changes in the
process of quantization due to the additional contributions from normal
ordering, the forms of Virasoro algebra, anomalous transformation of the EMT
and the trace anomaly remain the same. Liouville theory, thus, may provide a
starting point for further studies to discern the effects of central charge
from the effect of quantization.

In our study we focused mostly on Liouville theory. Whether it is possible, in
general, to derive trace or diffeomorphism anomalies directly from Virasoro
algebra in a classical system is still an open question. The equivalence of
central charge and trace anomaly, in quantum theories is established by using
methods of path integration, which are unavailable for classical theories.
Nonetheless, it is reasonable to suppose that the similarity of the results,
in the quantum case and in the classical case, is not coincidental. Further
work on the central charge of Virasoro algebra in classical theories could
give a definite answer.

A last direction for further research that we would like to mention here, is
the investigation of the connection of the anomalous transformation of the EMT
with classical analog of Unruh and Hawking effects.

## Acknowledgments

We gladly acknowledge support from Charles University Research Center
(UNCE/SCI/013).

## References

  * Haman and Iorio [2023] P. Haman and A. Iorio, Classical gravitational anomalies of Liouville theory (2023), arXiv:2306.09825 [hep-th] .
  * Liouville [1853] J. Liouville, J. Math. Pures Appl. , 71 (1853).
  * D’Hoker and Jackiw [1982] E. D’Hoker and R. Jackiw, Phys. Rev. D 26, 3517 (1982).
  * Iorio [2011] A. Iorio, Ann. of Phys. 326, 1334 (2011).
  * Iorio and Kus [2020] A. Iorio and P. Kus, Int. J. Geom. Methods Mod. Phys. 17, 2050106 (2020).
  * Jackiw [1984] R. Jackiw, in  _Quantum Theory of Gravity_ , edited by S. Christensen (Adam-Hilger Ltd., 1984) pp. 403–420.
  * Polyakov [1981] A. M. Polyakov, Phys. Lett. B 103, 207 (1981).
  * Maldacena [1999] J. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999).
  * Penedones [2017] J. Penedones, in  _New Frontiers in Fields and Strings_, edited by J. Polchinski, P. Vieira, and O. DeWolfe (World Scientific, 2017) Chap. Chapter 2, pp. 75–136.
  * Iorio  _et al._ [1997] A. Iorio, L. O’Raifeartaigh, I. Sachs, and C. Wiesendanger, Nucl. Phys. B 495, 433 (1997).
  * Jackiw [2006] R. Jackiw, Theor. Math. Phys. 148, 941 (2006).
  * Toppan [2001] F. Toppan, J. Nonlin. Math. Phys. 8, 518 (2001).
  * Christensen and Fulling [1977] S. M. Christensen and S. A. Fulling, Phys. Rev. D 15, 2088 (1977).
  * Birrell and Davies [1986] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_ (Cambridge University Press, 1986).
  * Adler [1969] S. L. Adler, Phys. Rev. 177, 2426 (1969).
  * Bell and Jackiw [1969] J. S. Bell and R. Jackiw, Il Nuovo Cimento A 60, 47 (1969).
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An introduction to quantum field theory_ , Frontiers in Physics (Addison-Wesley Pub. Co, 1995).
  * Schwartz [2013] M. D. Schwartz,  _Quantum Field Theory and the Standard Model_ , illustrated ed., Quantum Field Theory and the Standard Model (Cambridge University Press, 2013).
  * Robinson and Wilczek [2005] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005).
  * Schottenloher [2008] M. Schottenloher,  _A Mathematical Introduction to Conformal Field Theory_ (Springer, 2008).
  * Jackiw [1985] R. Jackiw, Comments Nucl. Part. Phys 15, 99 (1985).
  * Di Francesco  _et al._ [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal,  _Conformal Field Theory_ (Springer-Verlag, 1997).
  * Bañados and Reyes [2016] M. Bañados and I. Reyes, Int. J. Mod. Phys. D 25 (2016), arXiv:1601.03616 [hep-th] .
  * Iorio [2000] A. Iorio, Phys. Lett. B 487, 171 (2000).
  * O’Raifeartaigh [2000] L. O’Raifeartaigh, Int. J. Mod. Phys. B 14, 2411 (2000).
  * Iorio  _et al._ [2001] A. Iorio, L. O’Raifeartaigh, and S. Wolf, Ann. Phys. 290, 156 (2001).
  * Nakahara [2003] M. Nakahara,  _Geometry, topology and physics_ (CRC press, 2003).
  * Jackiw [1995a] R. Jackiw,  _Diverse Topics in Theoretical and Mathematical Physics_ (World Scientific Publishing Company, 1995).
  * O’Raifeartaigh [1997] L. O’Raifeartaigh,  _The Dawning of Gauge Theory_ (Princeton University Press, 1997).
  * Nakayama [2015] Y. Nakayama, Phys. Rep. 569, 1 (2015).
  * Bessel-Hagen [1921] E. Bessel-Hagen, Math. Ann. 84, 258 (1921).
  * Brown and Henneaux [1986a] J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986a).
  * Brown and Henneaux [1986b] J. D. Brown and M. Henneaux, J. Math. Phys. 27, 489 (1986b).
  * Blagojević and Vasilić [2005] M. Blagojević and M. Vasilić, Class. Quantum Gravity 22, 3891 (2005).
  * Chern [1955] S.-S. Chern, Proc. Amer. Math. Soc. 6, 771 (1955).
  * Kiritsis [2007] E. Kiritsis,  _String theory in a nutshell_ , rev.ed., pup ed. (Princeton University Press, 2007).
  * Leutwyler [1985] H. Leutwyler, Physics Letters B 153, 65 (1985).
  * Torre [1989] C. G. Torre, Phys. Rev. D 40, 2588 (1989).
  * Karakhanyan  _et al._ [1994] D. R. Karakhanyan, R. P. Manvelyan, and R. L. Mkrtchyan, Phys. Lett. B 329, 185 (1994).
  * Jackiw [1995b] R. Jackiw 10.48550/arxiv.hep-th/9501016 (1995b).
  * Iso  _et al._ [2006] S. Iso, H. Umetsu, and F. Wilczek, Phys. Rev. D 74, 044017 (2006).
  * Deser and Jackiw [1996] S. Deser and R. Jackiw, Int. J. Mod. Phys. B 495, 1499 (1996).
  * Haman [2019] P. Haman,  _Weyl and Conformal Symmetries_ , Bachelor thesis, Charles University in Prague (2019).
  * Guadagnini [1988] E. Guadagnini, Phys. Rev. D 38, 2482 (1988).
  * Bertlmann [2000] R. A. Bertlmann,  _Anomalies in Quantum Field Theory_ (Oxford University Press, 2000).
  * Banerjee and Kulkarni [2008] R. Banerjee and S. Kulkarni, Phys. Rev. D 77, 024018 (2008).
  * Carroll [2003] S. Carroll,  _Spacetime and Geometry: An introduction to General Relativity_ (Pearson, 2003).
  * Dirac [1950] P. A. M. Dirac, Can. J. Math. 2, 129–148 (1950).
  * Jackiw [1993] R. Jackiw, (constrained) quantization without tears (1993), arXiv:hep-th/9306075 [hep-th] .

##  Appendix A Passive and active transformations

For spatiotemporal symmetries we can take the _passive transformations_ point
of view: If $x^{\mu}$ is a coordinate system and $f^{\mu}(x)$ a smooth vector
field over the spacetime, starting at point $p$, with coordinates
$x^{\mu}(p)\equiv p^{\mu}$, one can move along the flow lines of the vector
field $f^{\mu}$ by an infinitesimal distance parametrized by $\epsilon$ to a
new point $q$ with coordinates $x^{\mu}(q)\equiv q^{\mu}$

$p^{\mu}\rightarrow q^{\mu}=p^{\mu}+\epsilon f^{\mu}(p)\,.$ (234)

In differential geometry this is known as the _flow generated by the vector
field $f^{\mu}$_ and it is denoted by $\phi_{\epsilon}$, [27]. This can be
then recast into the coordinate-free identification $\phi_{\epsilon}(p)=q$.

Instead of moving the observer in a given direction, the same result can be
achieved by transporting the fields in the inverse direction, i.e. from the
point $q$ to the point $p$. That is the _active transformations_ point of
view. The point $q$ in a new coordinate frame, $x^{\prime\mu}$, can be chosen
to have the same coordinate values as the point $p$ in the original frame,
$x^{\mu}$, that is, $x^{\prime\mu}(q)=x^{\mu}(p)$ or, equivalently,

$q^{\prime\mu}=p^{\mu}\,.$ (235)

Solving this equation up to the first order gives
$x^{\prime\mu}(p)=x^{\mu}(p)-\epsilon f^{\mu}(p)$.

E.g., taking a vector field $V$, in the coordinate frame $x^{\mu}$, and
applying coordinate transformation rules at point $q$ for vectors one obtains

$V^{\prime\alpha}(x^{\prime}(q))=\partialderivative{x^{\prime\alpha}(q)}{x^{\beta}(q)}V^{\beta}(q).$
(236)

Using the condition (235),
$V^{\prime\alpha}(x^{\prime}(q))=V^{\prime\alpha}(x(p))$, which introduces a
“new” vector field $V^{\prime}$, with components $V^{\prime\alpha}$ in the
_original_ coordinate frame $x^{\mu}$ and equation (236) as its definition.

The new field $V^{\prime}$ can be compared with the original field $V$ at the
point $p$ by expanding the equation (236)

$V^{\prime\alpha}(p)=\partialderivative{x^{\prime\alpha}(q)}{x^{\beta}(q)}V^{\beta}(q)=V^{\alpha}(p)+\epsilon
f^{\beta}\partial_{\beta}V^{\alpha}(p)-\epsilon\partial_{\beta}f^{\alpha}(p)V^{\beta}(p)\,,$
(237)

where in the last equation everything was expanded up to the first order in
$\epsilon$.

The solution to the coordinate transformation (235) is the _Lie transport_
along the flow lines of $f^{\mu}$ from point $q$ back to the point $p$. In
general relativity this backward transport of field is called _pullback_ , as
it is “pulling” the field back opposed to the direction of flow
$\phi_{\epsilon}$, and it is denoted as $\phi^{*}_{\epsilon}$, [47]. Within
this more abstract notation the equation (236) can be rewritten as

$V^{\prime}(p)=\phi^{*}_{\epsilon}[V(\phi_{\epsilon}(p))],$ (238)

which consists of two purely geometrical operations, a flow from the point $p$
to the point $\phi_{\epsilon}(p)$ and a pull of the field $V$ back to the
point $p$. This is a _diffeomorphism_ , and it is an active transformation,
since the fields are transformed, rather than the observer.

For transformations not leaving or changing the spacetime, the equation (238)
holds for a general tensor field $T$ as

$T^{\prime}(p)=\phi^{*}_{\epsilon}[T(\phi_{\epsilon}(p))]\,.$ (239)

Again, starting with a field $T$ a new field $T^{\prime}$ was obtained such
that its domain is the same as of the original tensor field.

##  Appendix B On singular Lagrangians

The case of Lagrangian density linear in the first order time derivatives of
fields (linear in generalized velocities) is often called singular. Singular
in a way that212121There is no summation over the index $i$ in the formula.,

$\partialderivative{\mathcal{L}}{\dot{\Phi}{\vphantom{\Phi}}^{i}}{\dot{\Phi}{\vphantom{\Phi}}^{i}}=0\,.$
(240)

In such situation the standard Legendre transformation is problematic, since
it is not generally possible to invert the definition of conjugated momenta to
obtain $\dot{\Phi}{}^{i}$ and switch to Hamiltonian formalism. This leads to
constrained systems and the way to deal with them was found by Paul
Dirac,[48], by a generalization of Poisson brackets giving Dirac brackets.

Examples of “singular” Lagrangians may be the Schrödinger Lagrangian or Dirac
Lagrangian, where in both theories a Hamiltonian and Poisson brackets (or
equivalently commutators) are well-defined, and the theories are not
considered being singular.

For a subset of singular Lagrangians, there exists a complementary “light-
weight” method to Dirac’s approach introduced by Faddeev and Jackiw, [49].
This method, if possible, derives Poisson brackets. Brief introduction to the
method in the context of the field theory is given here, to find Poisson
brackets for light-cone (and similar) Lagrangians. It is assumed Lagrangian
depends on one scalar field $\phi$ for simplicity, however, the method can be
generalized to other fields and for multiple fields.

A general form of a Lagrangian linear in $\dot{\phi}$ is

$L[\phi,\dot{\phi}](t)=\int\differential[d-1]{\mathbf{x}}\quantity(F[\phi](\mathbf{x},t)\dot{\phi}(\mathbf{x},t)-V[\phi](t))\,,$
(241)

where $F[\phi](\mathbf{x},t)$ is some non-local functional. A naive Legendre
transformation to obtain Hamiltonian gives

$H=\functionalderivative{L}{\dot{\phi}}\dot{\phi}-L=V\,.$ (242)

Therefore, the Lagrangian can be rewritten as

$L[\phi,\dot{\phi}](t)=\int\differential[d-1]{\mathbf{x}}\quantity(F[\phi](\mathbf{x},t)\dot{\phi}(\mathbf{x},t)-H[\phi](t))\,.$
(243)

Since Poisson brackets encompass the equations of motion, they have to be
properly derived to trace carefully the connection between Poisson brackets
and equations of motion for the case of a singular Lagrangian. The non-local
functional $F[\phi](\mathbf{x},t)$ forbids the use of field theoretic
generalization of Euler-Lagrange equation and the equations of motion have to
be derived variationally

$\derivative{t}\functionalderivative{L[\phi,\dot{\phi}](t)}{\dot{\phi}(\mathbf{x},t)}-\functionalderivative{L[\phi,\dot{\phi}](t)}{\phi(\mathbf{x},t)}=0\,.$
(244)

A careful use of the chain rule for functional derivatives gives

$\derivative{t}F[\phi](\mathbf{x},t)-\int\differential[d-1]{\mathbf{y}}\dot{\phi}(\mathbf{y},t)\functionalderivative{F[\phi](\mathbf{y},t)}{\phi(\mathbf{x},t)}+\functionalderivative{H[\phi](t)}{\phi(\mathbf{x},t)}=0\,.$
(245)

Expanding the total time derivative leads to

$\functionalderivative{H[\phi](t)}{\phi(\mathbf{x},t)}=\int\differential[d-1]{\mathbf{y}}\dot{\phi}(\mathbf{y},t)\quantity(\functionalderivative{F[\phi](\mathbf{y},t)}{\phi(\mathbf{x},t)}-\functionalderivative{F[\phi](\mathbf{x},t)}{\phi(\mathbf{y},t)})-\partialderivative{F[\phi](\mathbf{x},t)}{t}\,.$
(246)

In case of Lagrangians independent of explicit time dependence the last term,
$\partialderivative*{F}{t}$, is zero, and since the scope of the paper is in
specific Lagrangians without explicit time dependence, this term is omitted in
the following. Thus,

$\functionalderivative{H[\phi](t)}{\phi(\mathbf{x},t)}=\int\differential[d-1]{\mathbf{y}}\dot{\phi}(\mathbf{y},t)\Xi(\mathbf{x},\mathbf{y})\,,$
(247)

where

$\Xi(\mathbf{x},\mathbf{y})\equiv\functionalderivative{F[\phi](\mathbf{y},t)}{\phi(\mathbf{x},t)}-\functionalderivative{F[\phi](\mathbf{x},t)}{\phi(\mathbf{y},t)}\,.$
(248)

The crucial point is to realize, that by finding a function
$\Xi^{-1}(\mathbf{x},\mathbf{y})$, inverse to $\Xi(\mathbf{x},\mathbf{y})$,

$\int\differential[d-1]{\mathbf{y}}\Xi(\mathbf{x},\mathbf{y})\Xi^{-1}(\mathbf{y},\mathbf{z})=\int\differential[d-1]{\mathbf{y}}\Xi^{-1}(\mathbf{x},\mathbf{y})\Xi(\mathbf{y},\mathbf{z})=\delta(\mathbf{x}-\mathbf{z})\,,$

the previous equation can be turned into

$\dot{\phi}(\mathbf{x},t)=\int\differential[d-1]{\mathbf{y}}\Xi^{-1}(\mathbf{x},\mathbf{y})\functionalderivative{H[\phi](t)}{\phi(\mathbf{y},t)}\,.$
(249)

In Hamiltonian formalism the time evolution of a field is given by Poisson
brackets of that field and Hamiltonian

$\dot{\phi}=\poissonbracket{\phi}{H}\,.$ (250)

Thus, the right-hand side of (249) contains the information about Poisson
brackets. Expanding $H[\phi]$ in a functional Taylor series, it can be shown

$\poissonbracket{\phi(\mathbf{x},t)}{H[\phi](t)}=\int\differential[d-1]{\mathbf{y}}\poissonbracket{\phi(\mathbf{x},t)}{\phi(\mathbf{y},t)}\functionalderivative{H[\phi](t)}{\phi(\mathbf{y},t)}\,.$
(251)

Comparing the equations (249), (250) and (251) it immediately follows

$\poissonbracket{\phi(\mathbf{x},t)}{\phi(\mathbf{y},t)}=\Xi^{-1}(\mathbf{x},\mathbf{y})\,.$
(252)

Poisson brackets are, thus, defined by the $\Xi^{-1}$ and the relation

$\poissonbracket{B[\phi](t)}{C[\phi](t)}=\int\differential[d-1]{\mathbf{x}}\differential[d-1]{\mathbf{y}}\functionalderivative{B[\phi](t)}{\phi(\mathbf{x},t)}\Xi^{-1}(\mathbf{x},\mathbf{y})\functionalderivative{C[\phi](t)}{\phi(\mathbf{y},t)}\,,$
(253)

where $B[\phi](t)$ and $C[\phi](t)$ are functionals of $\phi$.

#### Light-cone Lagrangian