# Most general theory of 3d gravity:
Covariant phase space, dual diffeomorphisms, and more

Marc Geiller1, Christophe Goeller1, Nelson Merino2

( 1Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,
CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
2Instituto de Ciencias Exactas y Naturales (ICEN), Facultad de Ciencias,
Universidad Arturo Prat, Iquique, Chile
)

###### Abstract

We show that the phase space of three-dimensional gravity contains two layers
of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms”
on the one hand, and between first order curvature and torsion on the other
hand. This is most elegantly revealed and understood when studying the most
general Lorentz-invariant first order theory in connection and triad
variables, described by the so-called Mielke–Baekler Lagrangian. By analyzing
the quasi-local symmetries of this theory in the covariant phase space
formalism, we show that in each sector of the torsion/curvature duality there
exists a well-defined notion of dual diffeomorphism, which furthermore follows
uniquely from the Sugawara construction. Together with the usual
diffeomorphisms, these duals form at finite distance, without any boundary
conditions, and for any sign of the cosmological constant, a centreless double
Virasoro algebra which in the flat case reduces to the BMS3 algebra. These
algebras can then be centrally-extended via the twisted Sugawara construction.
This shows that the celebrated results about asymptotic symmetry algebras are
actually generic features of three-dimensional gravity at any finite distance.
They are however only revealed when working in first order connection and
triad variables, and a priori inaccessible from Chern–Simons theory. As a
bonus, we study the second order equations of motion of the Mielke–Baekler
model, as well as the on-shell Lagrangian. This reveals the duality between
Riemannian metric and teleparallel gravity, and a new candidate theory for
three-dimensional massive gravity which we call teleparallel topologically
massive gravity.

###### Contents

  1. 1 Introduction
  2. 2 Mielke–Baekler Lagrangian
    1. 2.1 Equations of motion
    2. 2.2 Symmetries
  3. 3 Covariant phase space and charge algebra
    1. 3.1 Generators and charges
    2. 3.2 Dual diffeomorphisms
    3. 3.3 Charge algebra
    4. 3.4 Classical Sugawara construction
    5. 3.5 Teleparallel dual diffeomorphisms
  4. 4 Chern–Simons formulation
    1. 4.1 Chern–Simons formulation with $(J,P)$
    2. 4.2 Chern–Simons formulation with $(J,T)$
    3. 4.3 Charges in Chern–Simons theory
    4. 4.4 Dual diffeomorphisms again
  5. 5 Asymptotic charges and symmetries
    1. 5.1 BTZ black hole
    2. 5.2 AdS spacetimes and flat limit
  6. 6 Metric, teleparallel, and massive sectors
    1. 6.1 Second order equations of motion
    2. 6.2 Second and higher order Lagrangians
  7. 7 Perspectives
  8. A Notations and conventions
  9. B Covariant phase space
    1. B.1 Notations and conventions
    2. B.2 Noether and Hamiltonian charges
  10. C Brackets and algebra
    1. C.1 Poisson brackets involving $\mathcal{C}(\xi)$
    2. C.2 Fourier representation of the algebra
    3. C.3 Details on the Sugawara construction
    4. C.4 Twisted Sugawara construction and central extensions
  11. D Symplectic symmetries, Kosmann derivative, and Fefferman–Graham gauge

##  Introduction

In spite of the apparent simplicity owing to its topological nature, three-
dimensional gravity is a very rich theory [1]. It contains black hole
solutions with thermodynamical behavior [2, 3, 4], admits non-perturbative
quantizations following different yet complementary techniques [5, 6, 7, 8, 9,
10], allows to describe topology change [11, 12, 13, 14], possesses remarkable
holographic properties [15, 16, 17, 18, 19, 20, 21, 22], gives rise to quantum
groups [23, 24, 25, 26, 27, 28, 29], and can be generalized to include e.g.
massive propagating gravitons [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43].

Gravity in three spacetime dimensions has also proven very useful to
understand the symmetry content of general relativity in the presence of
boundaries. The seminal work of Brown and Henneaux on AdS3 [15] is now
understood as part of a much bigger picture relating asymptotic symmetries and
boundary dynamics, and many recent studies have for example focused on
developing an understanding of the BMS3 group associated with flat spacetimes
[44, 45, 46, 47, 48, 49, 20, 50, 51, 52, 53]. Studies such as [54, 55] have
also started to classify possible boundary conditions and related asymptotic
symmetries.

This wealth of results is in part due to the existence of many (possibly
inequivalent) formulations of three-dimensional gravity. The most notorious
one is the Chern–Simons formulation (CS hereafter), which itself is, loosely
speaking, based on a change of variables from the first order connection-triad
formulation of Cartan. At the difference with the metric formulation, the
Chern–Simons one puts forward the role of algebraic structures and of internal
gauge transformations as opposed to diffeomorphisms.

In order to understand some general and systematic results about three-
dimensional gravity, we revisit here the study of the most general
diffeomorphism and Lorentz-invariant theory in connection and triad variables.
This is described by the Mielke–Baekler (MB) Lagrangian (2.1), which is given
by the sum of a volume (or cosmological constant) term, the usual
Hilbert–Palatini term, a Chern–Simons term for the Lorentz connection, and a
torsion term [56, 57]. This four-parameter topological theory has first order
equations of motion which contain a source of curvature measured by a
parameter $p$, and a source of torsion measured by a parameter $q$. When
solving for the torsion, the second order equations of motion reduce to the
Riemannian Einstein equations $R_{\mu\nu}=2\Lambda g_{\mu\nu}$, where
$\Lambda(p,q)$ is the effective cosmological constant. This simple observation
has previously led authors to conclude that the MB formulation of three-
dimensional gravity is simply a CS formulation whose gauge
algebra111Explicitly, as we work in Lorentzian signature we have
$\mathfrak{g}_{\Lambda}=\mathfrak{iso}(2,1)$ when $\Lambda=0$,
$\mathfrak{so}(2,2)$ when $\Lambda<0$, and $\mathfrak{so}(3,1)$ when
$\Lambda>0$. $\mathfrak{g}_{\Lambda}$ depends on the sign of $\Lambda$ [58,
59]. This CS formulation has in turn been used in many studies of holography,
asymptotic symmetries, and black hole entropy in the MB model [60, 61, 62, 63,
64, 65, 66, 67, 68].

Here we take a closer look at the theory by focusing on its covariant phase
space (using the eponymous formalism) and its charges. More precisely, we are
interested in performing the Sugawara construction and in the study of a
certain set of “dual” diffeomorphism charges. Together with the “usual”
diffeomorphisms, these enable to define, at finite distance, for tangent
diffeomorphisms, and without any boundary conditions, an algebra of charges
consisting in two copies of the Witt algebra (below we will often call this
the centreless Virasoro algebra). This algebra is defined even with a positive
cosmological constant, and in the flat limit reduces to BMS3. We furthermore
explain why this result is actually not attainable from CS theory. The
advantage of performing this construction on the MB model is that it allows
for an exploration of the four-dimensional space of parameters, in which one
finds for example the first order Hilbert–Palatini Lagrangian for three-
dimensional gravity with cosmological constant, but also Witten’s “exotic”
Lagrangian [5]. We will show in fact that different dual charges can be
defined in different sectors of the MB model, in particular the ones with
$(p\neq 0,q=0)$ and $(p=0,q\neq 0)$, thereby even defining a curvature/torsion
duality between dual charges. The dual charges agree when $(p=0,q=0)$, in
which case their algebra with the usual diffeomorphisms is that of BMS3.

Being diffeomorphism-invariant and topological, the MB theory has the property
that its diffeomorphisms can be written on-shell as field-dependent gauge
transformations. There are two types of such internal gauge transformations,
namely the Lorentz transformations, and the “translations”222The name comes
from the fact that in the flat case the corresponding generators commute.. We
start this work by actually identifying these translations. Their algebra with
Lorentz transformations has three central extensions which depend on the
parameters of the theory. Although it is isomorphic to
$\mathfrak{g}_{\Lambda}$, this algebra does actually not have the familiar
“canonical form” unless one performs a redefinition of the generators. While
this is of course always allowed, since it amounts to redefining what we mean
by the translations to define “translations prime”, we explain why it is not
natural. Working with the initial translations, we are therefore led to
introduce a different CS formulation of MB gravity than the
$\mathfrak{g}_{\Lambda}$ one introduced previously in [58, 59]. Besides this
elementary remark and exercise in change of generators, the interest in the
translations lies in their ability to provide a simple expression for the
Sugawara construction.

Setting aside aspects of conformal field theory, in gravitational terms the
Sugawara construction amounts to defining diffeomorphisms as quadratic
generators starting from the internal gauge transformations. This reflects the
above-mentioned relationship which exists for topological theories and which
states that, on-shell, diffeomorphisms are nothing but field-dependent gauge
transformations. In algebraic terms, this amounts to starting from the
$\mathfrak{g}_{\Lambda}$ current algebra, and constructing a Virasoro algebra
from quadratics in the currents. These quadratics are of course related to the
Casimirs, while considering higher order operators leads to $W$-algebras [69].
In this work we write down the Sugawara construction leading to the
diffeomorphisms of the MB model. Since $\mathfrak{g}_{\Lambda}$ has two
quadratic Casimirs, it is then natural to ask whether there exists another,
dual Sugawara construction, and which gauge generator it corresponds to. Here
we show that one is naturally led to this discussion when thinking about the
relationship between diffeomorphisms and internal gauge transformations. Since
this involves field-dependent gauge transformations, one should be careful
about issues of integrability in the covariant phase space. As we will
explain, analyzing the condition of integrability for quadratic generators
built from field-dependent Lorentz transformations and translations
automatically leads to the identification of the above-mentioned dual
diffeomorphism charges.

In the familiar constructions of the BMS3 or double Virasoro asymptotic
symmetries, the diffeomorphism charges depend on two directions of the vector
field (e.g. $u$ and $\varphi$ in Bondi coordinates). This is, in a sense, the
reason for the presence of two factors in the associated asymptotic symmetry
algebras. Moreover, it is the non-tangentiality of such diffeomorphisms (i.e.
the fact that they probe directions other than $\varphi$) which is responsible
for the appearance of a classical central extension in the symmetry algebras.
For the sake of completeness we recall here these known facts in the case of
the MB model. In light of this, one can then understand better our new result,
which is the construction at finite distance and from tangent vector fields of
a centreless double Virasoro algebra, which can further be reduced to BMS3 in
the flat case. More precisely, the reason for which we obtain centreless
algebras is because we focus on tangential vector fields. However, in spite of
these vector fields being only along the boundary circle, we still obtain
algebras with two factors because of the presence of the dual diffeomorphism
charges. Let us stress that these dual charges are well-defined already in the
usual first order formulation of three-dimensional gravity. Here we simply
study the MB model in order to have, once and for all, an understanding of the
most general Lagrangian and of the curvature/torsion duality which it encodes.

The interest in studying these dual charges and their algebra with the usual
diffeomorphisms, the Lorentz transformations, and the translations, comes from
the following question: What are the boundary symmetries of a given
formulation of gravity? If, as argued in [70, 71, 72], quantum gravity arises
from a quantization of the quasi-local symmetries of gravity, it is desirable
to unravel and understand the nature of the largest possible symmetry algebra.
In a first order theory like the MB model (or even BF theory), this seemingly
simple question has no clear answer. Indeed, even though they are equivalent
as symmetries of the Lagrangian, it is not clear at the level of the charge
algebra whether one should consider the diffeomorphisms as independent from
the Lorentz transformations and the translations. One the one hand, we have
explained above and will show in detail that the Lorentz transformations and
the translations parametrize the independent gauge symmetries. In the presence
of boundaries they give rise to a centrally-extended $\mathfrak{g}_{\Lambda}$
current algebra of charges333The $\mathfrak{g}_{\Lambda}$ current algebra is
also the asymptotic algebra which appears with the most general AdS3 and flat
space boundary conditions of [54, 55].. It is these symmetries which play a
physical role e.g. in discussions of surface states and entanglement entropy
in condensed matter444There is no doubt that all these results can also be
reinterpreted in terms of diffeomorphisms, although to our knowledge this has
not been presented anywhere. [73, 74, 75, 76]. On the other hand, we are also
encouraged by intuition from metric gravity to work with the diffeomorphism
transformations and their charges, and at infinity they indeed carry important
physical information such as the energy and the angular momentum. It is clear
however that diffeomorphisms are in a sense not fundamental at finite
distance. First, they are not always integrable, while the Lorentz
transformations and translations always are. Second, since they are only
(particular) quadratics built from the Lorentz and translational charges,
there is no reason not to consider other possible quadratics, and even higher
order operators. In summary, in this work we take a first step towards
extending the quasi-local charge algebra of three-dimensional gravity. We do
so by studying carefully the relationship between diffeomorphisms and internal
gauge transformations, and show that this naturally leads to the introduction
of dual diffeomorphism charges.

After having studied the finite distance covariant phase space of the MB model
and introduced the dual charges, we conclude this work with a study of the
metric and teleparallel formulations of the theory, both at the level of the
equations of motion and of the Lagrangian itself. To start with, one can
combine the first order equations of motion of the MB theory into a single
second order equation. This is done by decomposing the connection into a
reference connection and a contorsion tensor, and then solving for the
contorsion. When doing so, one can choose the reference connection to be
either torsionless or flat. Choosing the reference connection to be
torsionless, the combined first order equations of motion reduce to the usual
second order Riemannian Einstein equations $R_{\mu\nu}=2\Lambda g_{\mu\nu}$
mentioned above. Choosing instead the reference connection to be flat, one
obtains the equations of motion of teleparallel gravity [77]. Interestingly,
the metric and teleparallel formulations of gravity can also be understood as
a form of duality [78], which corresponds to choosing to encode the dynamics
either in the curvature or in the torsion of the geometry.

Going further, one can then inject the solution to the torsion equation of
motion back into the Lagrangian. In the case of usual first order gravity with
only the Hilbert–Palatini term, this leads to the second order metric
Lagrangian or to the teleparallel Lagrangian, depending on whether the
reference connection is chosen to be torsionless or flat [79]. In the case of
the MB model however, the presence of a CS term in the Lagrangian has dramatic
consequences for the theory when solving and injecting the torsion equation of
motion. Indeed, when the reference connection is chosen to be the torsionless
Levi–Civita connection, the CS term for this latter becomes third order in
derivatives of the metric. This is responsible for the appearance of a massive
propagating mode, i.e. a massive graviton, and the corresponding Lagrangian is
known as that of topologically massive gravity (TMG) [30, 31, 57]. We complete
this picture by deriving the analogue result for the teleparallel sector of
the theory, and show that when choosing a flat connection the on-shell
Lagrangian has a CS term for the Weitzenböck torsion. It is then natural to
conjecture that the so-obtained Lagrangian is that of a teleparallel version
of topologically massive gravity, which we call TTMG. We postpone its detailed
study to future work.

This paper is organized as follows. In section 2 we present the Lagrangian of
the MB model and study its symmetries. This enables us to identify the
translations as the transformations which, when combined with Lorentz
transformations, reproduce on-shell the diffeomorphisms. In section 3 we then
study the covariant phase space and the algebra of charges. We also present
the Sugawara construction and relate it to diffeomorphisms and their dual. We
explain how different dual charges can be identified on subspaces of the MB
model, and show how they lead at finite distance to a double Virasoro algebra
with BMS3 as the flat limit. Section 4 is then devoted to the study of the CS
formulation of the MB model, using the algebra of Lorentz transformations and
translations. We then present a general discussion on the integrability of
quadratic charges, and a set of dual charges which can be considered for
different values of the couplings of the MB model. This explains how the dual
charges can be defined for non-tangent diffeomorphisms as well. In section 5
we recall the study of asymptotic symmetries in the MB model using Bondi
gauge. Section 6 contains the study of the second order metric and
teleparallel sectors of the theory, as well as their massive generalizations
obtained when injecting the solution of the torsion equation back into the
Lagrangian. This section is independent from the previous ones, and can be
read separately for the interested reader. We finally present our conclusions
and discuss many directions for future work in section 7.

In addition to the new results, this article covers very general facts about
three-dimensional gravity, and fills-in some gaps about elementary aspects
which are often not discussed in the literature. It can therefore be
considered also as a pedagogical introduction to the covariant phase space in
three-dimensional gravity. In order to make this self-contained, we have
recalled details about our notations and conventions in appendix A, and
details about covariant phase space methods in appendix B. Appendix C contains
detailed proofs of mosts calculations.

##  Mielke–Baekler Lagrangian

The Mielke–Baekler (MB) Lagrangian is the most general Lorentz-invariant
three-form constructed with a triad one-form $e$ and a connection one-form
$\omega$ [57, 56]. It is defined as the sum of the volume, Hilbert–Palatini,
Chern–Simons and torsion Lagrangians, each with their own coupling
$\sigma_{i}$, and takes the simple form

$\displaystyle L_{\text{MB}}[e,\omega]$
$\displaystyle=\sigma_{0}L_{\text{V}}[e]+\sigma_{1}L_{\text{HP}}[e,\omega]+\sigma_{2}L_{\text{CS}}[\omega]+\sigma_{3}L_{\text{T}}[e,\omega]$
(2.1) $\displaystyle=\frac{\sigma_{0}}{3}e\wedge[e\wedge e]+2\sigma_{1}e\wedge
F+\sigma_{2}\omega\wedge\left(\mathrm{d}\omega+\frac{1}{3}[\omega\wedge\omega]\right)+\sigma_{3}e\wedge\mathrm{d}_{\omega}e.$
(2.2)

Here and throughout this article we work in Lorentzian signature, and use an
index-free differential form notation recalled in appendix A.

###  Equations of motion

The variation of the MB Lagrangian is

$\displaystyle\delta L_{\text{MB}}[e,\omega]$
$\displaystyle=\mathrm{d}\Big{(}2\sigma_{1}\delta\omega\wedge
e+\sigma_{2}\delta\omega\wedge\omega+\sigma_{3}\delta e\wedge e\Big{)}$ (2.3)
$\displaystyle\phantom{\ =}+\delta
e\wedge\Big{(}2\sigma_{1}F+2\sigma_{3}\mathrm{d}_{\omega}e+\sigma_{0}[e\wedge
e]\Big{)}$ (2.4) $\displaystyle\phantom{\
=}+\delta\omega\wedge\Big{(}2\sigma_{2}F+2\sigma_{1}\mathrm{d}_{\omega}e+\sigma_{3}[e\wedge
e]\Big{)}.$ (2.5)

This reveals the symplectic potential, to which we will come back shortly, as
well as the equations of motion. One can see that both equations of motion
contain all possible two-forms in a “symmetrized” manner, and that likewise
the symplectic potential contains all possible polarizations. This evidently
results from the presence of all four terms/couplings in the MB Lagrangian.

To understand the physical meaning of the first order equations of motion, it
is useful to rewrite them in the form

$\displaystyle 2F+p[e\wedge e]\approx 0,\qquad\qquad
2\mathrm{d}_{\omega}e+q[e\wedge e]\approx 0,$ (2.6)

with

$\displaystyle
p\coloneqq\frac{\sigma_{0}\sigma_{1}-\sigma_{3}^{2}}{\sigma_{1}^{2}-\sigma_{2}\sigma_{3}},\qquad\qquad
q\coloneqq\frac{\sigma_{1}\sigma_{3}-\sigma_{0}\sigma_{2}}{\sigma_{1}^{2}-\sigma_{2}\sigma_{3}}.$
(2.7)

This shows that the theory has a source of Lorentzian curvature and torsion
measured respectively by the parameters $p$ and $q$. We use here the adjective
“Lorentzian” because these are properties of the first order connection
variable $\omega$. Later on we will talk about Riemannian curvature and
torsion when going to second order variable by decomposing $\omega$ into a
reference connection and a contorsion tensor555In particular, as we will see
below, the theory can have vanishing Lorentzian curvature but non-vanishing
Riemannian curvature. It is this latter which matters when talking about the
metric geometry of spacetime.. Note that we require of course that
$\sigma_{1}^{2}-\sigma_{2}\sigma_{3}\neq 0$ otherwise the theory is trivial
(or more precisely admits only degenerate triads as solutions). Two useful
relations which we will use in many places below are
$\sigma_{0}=p\sigma_{1}+q\sigma_{3}$ and $\sigma_{3}=p\sigma_{2}+q\sigma_{1}$.

It is now enlightening to combine the first order equations of motion into a
single second order equation. For this, one can decompose the Lorentz
connection as $\omega=\Gamma+k$, where $\Gamma$ is the torsionless Levi–Civita
connection and $k$ is the contorsion. With this decomposition, the curvature
and the torsion become

$\displaystyle F=R+\mathrm{d}_{\Gamma}k+\frac{1}{2}[k\wedge
k],\qquad\qquad\mathrm{d}_{\omega}e=[k\wedge e],$ (2.8)

where $R$ is the Riemannian curvature of $\Gamma$. Using the torsion equation
of motion (2.6), this implies that $k\approx-qe/2$, and the curvature equation
of motion then leads to

$\displaystyle 2R\approx\Lambda[e\wedge
e],\qquad\qquad\Lambda\coloneqq-\left(p+\frac{q^{2}}{4}\right).$ (2.9)

In metric terms this means that $R_{\mu\nu}\approx 2\Lambda g_{\mu\nu}$, i.e.
that the MB model describes constant curvature spacetimes with cosmological
constant $\Lambda$. This is the result which has led authors to conclude that
the MB model can be written as a CS theory with gauge algebra
$\mathfrak{g}_{\Lambda}$ [58, 59]. We will make this statement more precise in
section 4, and see that it actually contains subtleties.

Note that here we have obtained a metric second order equation of motion
because we have used the torsionless connection $\Gamma$ as the reference
connection when decomposing $\omega$. As mentioned in the introduction, one
can also consider a flat reference connection instead, and this choice leads
to teleparallel equations of motion. We come back to this and analyse in more
details the metric and teleparallel formulations of the theory in section 6.

Finally, let us observe that there are two particular cases for specific
values of the couplings. For $\sigma_{1}\sigma_{3}=\sigma_{0}\sigma_{2}$ we
get $q=0$, and the MB Lagrangian reduces to Witten’s “exotic” Lagrangian [5].
For $\sigma_{0}\sigma_{1}>0$ and $\sigma_{3}=\pm\sqrt{\sigma_{0}\sigma_{1}}$,
which implies $p=0$, we get a first order Lagrangian with Lorentzian torsion
but vanishing Lorentzian curvature. Notice that in this latter case we
necessarily have $\Lambda<0$, and therefore an AdS Riemannian geometry.

###  Symmetries

We now turn to the study of the gauge symmetries of the theory. First, we have
the internal Lorentz transformations, which act infinitesimally as

$\displaystyle\delta^{\text{j}}_{\alpha}e=[e,\alpha],\qquad\qquad\delta^{\text{j}}_{\alpha}\omega=\mathrm{d}_{\omega}\alpha,\qquad\qquad\delta^{\text{j}}_{\alpha}F=[F,\alpha],\qquad\qquad\delta^{\text{j}}_{\alpha}\mathrm{d}_{\omega}e=[\mathrm{d}_{\omega}e,\alpha].$
(2.10)

Then, we have the translations acting on the triad and on the connection as

$\displaystyle\delta^{\text{t}}_{\phi}e=\mathrm{d}_{\omega}\phi+q[e,\phi],\qquad\qquad\delta^{\text{t}}_{\phi}\omega=p[e,\phi],$
(2.11)

and on the curvature and torsion as

$\displaystyle\delta^{\text{t}}_{\phi}F=p\mathrm{d}_{\omega}[e,\phi],\qquad\qquad\delta^{\text{t}}_{\phi}\mathrm{d}_{\omega}e=\frac{1}{2}\big{[}2F+p[e\wedge
e],\phi\big{]}+q\mathrm{d}_{\omega}[e,\phi].$ (2.12)

Using (2.3) one can indeed check that these are symmetries as they leave the
Lagrangian invariant up to the boundary terms
$\delta^{\text{j}}_{\alpha}L_{\text{MB}}=\sigma_{2}\mathrm{d}(\alpha\mathrm{d}\omega)$
and $\delta^{\text{t}}_{\phi}L_{\text{MB}}=\mathrm{d}\big{(}2\sigma_{1}\phi
F+\sigma_{3}\phi\mathrm{d}_{\omega}e+p[e,\phi]\wedge(\sigma_{1}e+\sigma_{2}\omega)\big{)}$.

While the Lorentz transformations have their usual form, one can observe at
this point that there is a priori nothing canonical about the transformations
which we have called translations. Indeed, since both parameters $\alpha$ and
$\phi$ are Lie algebra-valued scalars, any combination of (2.10) and (2.11) is
also a legitimate independent gauge symmetry which we could have identified as
the translation. Part of the reason for which (2.11) is actually the natural
form for the translations comes from the expression for the diffeomorphisms as
field-dependent gauge transformations, which will then be the starting point
for the Sugawara construction. Indeed, for diffeomorphisms, which act as the
Lie derivative $\delta^{\text{d}}_{\xi}=\pounds_{\xi}$, a simple rewriting
reveals that

$\displaystyle\delta^{\text{d}}_{\xi}e$
$\displaystyle=\mathrm{d}(\xi\lrcorner\,e)+\xi\lrcorner\,(\mathrm{d}e)$
(2.13a)
$\displaystyle=\mathrm{d}_{\omega}(\xi\lrcorner\,e)+\xi\lrcorner\,(\mathrm{d}_{\omega}e)+[e,\xi\lrcorner\,\omega]$
(2.13b)
$\displaystyle=\delta^{\text{j}}_{\xi\lrcorner\,\omega}e+\delta^{\text{t}}_{\xi\lrcorner\,e}e+\frac{1}{2}\xi\lrcorner\,\big{(}2\mathrm{d}_{\omega}e+q[e\wedge
e]\big{)},$ (2.13c) $\displaystyle\delta^{\text{d}}_{\xi}\omega$
$\displaystyle=\mathrm{d}(\xi\lrcorner\,\omega)+\xi\lrcorner\,(\mathrm{d}\omega)$
(2.13d)
$\displaystyle=\mathrm{d}_{\omega}(\xi\lrcorner\,\omega)+\xi\lrcorner\,F$
(2.13e)
$\displaystyle=\delta^{\text{j}}_{\xi\lrcorner\,\omega}\omega+\delta^{\text{t}}_{\xi\lrcorner\,e}\omega+\frac{1}{2}\xi\lrcorner\,\big{(}2F+p[e\wedge
e]\big{)},$ (2.13f)

where one can recognize the equations of motion (2.6). As we will see in the
next section, yet another reason for considering (2.10) and (2.11) is that
their Hamiltonian generators are, as expected, the spatial pullbacks of the
equations of motion enforced respectively by $\omega$ and $e$ in (2.3).
Therefore, using the Lorentz transformations and the translations to
parametrize the set of gauge transformations is the only choice which exhibits
the two symmetrical forms (2.3) and (2.6) of the equations of motion.

By definition of the Lie derivative, we have that
$\delta^{\text{d}}_{\xi}L_{\text{MB}}=\mathrm{d}(\xi\lrcorner\,L_{\text{MB}})$.
Consistently, one can check that the same result is also obtained when acting
with the right-hand side of (2.13). This uses the fact that the equations of
motion appearing there act as so-called trivial gauge transformations [80]. We
note that the diffeomorphisms as field-dependent gauge transformations in the
MB model were also discussed in [81, 82, 80], although in a much more
convoluted way and without explicitly identifying the translations.

Now, by acting twice on the dynamical fields with the translations and Lorentz
transformations, one can derive the commutators

$\displaystyle\big{[}\delta^{\text{j}}_{\alpha},\delta^{\text{t}}_{\phi}\big{]}=\delta^{\text{t}}_{[\alpha,\phi]},\qquad\qquad\big{[}\delta^{\text{j}}_{\alpha},\delta^{\text{j}}_{\beta}\big{]}=\delta^{\text{j}}_{[\alpha,\beta]},\qquad\qquad\big{[}\delta^{\text{t}}_{\phi},\delta^{\text{t}}_{\chi}\big{]}=p\delta^{\text{j}}_{[\phi,\chi]}+q\delta^{\text{t}}_{[\phi,\chi]}.$
(2.14)

This motivates the introduction of generators $(J_{i},T_{i})_{i=1,2,3}$
forming the algebra

$\displaystyle[J_{i},T_{j}]={\varepsilon_{ij}}^{k}T_{k},\qquad\qquad[J_{i},J_{j}]={\varepsilon_{ij}}^{k}J_{k},\qquad\qquad[T_{i},T_{j}]={\varepsilon_{ij}}^{k}(pJ_{k}+qT_{k}),$
(2.15)

whose quadratic Casimir operators are

$\displaystyle C_{1}=pJ^{i}J_{i}+T^{i}T_{i},\qquad\qquad
C_{2}=J^{i}T_{i}+T^{i}J_{i}-qJ^{i}J_{i}.$ (2.16)

We will come back to these Casimirs when studying the Sugawara construction in
section 3.4. Defining the new generators

$\displaystyle P\coloneqq T-\frac{q}{2}J$ (2.17)

then maps this algebra to

$\displaystyle[J_{i},P_{j}]={\varepsilon_{ij}}^{k}P_{k},\qquad\qquad[J_{i},J_{j}]={\varepsilon_{ij}}^{k}J_{k},\qquad\qquad[P_{i},P_{j}]=-\Lambda{\varepsilon_{ij}}^{k}J_{k},$
(2.18)

which are the usual $\mathfrak{g}_{\Lambda}$ commutation relations used in the
CS formulation of three-dimensional gravity. The construction of the MB model
based on the algebras spanned by $(J,T)$ and $(J,P)$ will be discussed in
section 4, where we will see once again why the $(J,T)$ parametrization of the
algebra is more natural.

The commutation relations (2.15) only capture the global part of the algebra
of gauge symmetries, whose fine structure is revealed upon inspection of the
charges in e.g. the covariant phase space formalism.

##  Covariant phase space and charge algebra

The study of the charges and their algebra is best carried out in the
covariant phase space formalism, whose ingredients are recalled in appendix B.
The starting point is the symplectic potential, which from (2.3) is given by

$\displaystyle\theta=2\sigma_{1}\delta\omega\wedge
e+\sigma_{2}\delta\omega\wedge\omega+\sigma_{3}\delta e\wedge e.$ (3.1)

This in turn determines the symplectic structure

$\displaystyle\Omega=-\int_{\Sigma}2\sigma_{1}\delta\omega\wedge\delta
e+\sigma_{2}\delta\omega\wedge\delta\omega+\sigma_{3}\delta e\wedge\delta e.$
(3.2)

We are now ready to contract this symplectic structure with the various gauge
transformations introduced above. This will identify the Hamiltonian
generators as well as the boundary charges. We will then compute the algebra
of these charges, and present the Sugawara construction which relates
diffeomorphism charges to internal gauge transformations. Along the way we
will introduce the dual diffeomorphism charges.

###  Generators and charges

Let us consider initially that the parameters $(\alpha,\phi,\xi)$ are field-
independent. Contracting the gauge transformations with the symplectic
structure gives

$\displaystyle-\delta^{\text{j}}_{\alpha}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega=\delta\mathcal{J}(\alpha),\qquad\qquad-\delta^{\text{t}}_{\phi}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega=\delta\mathcal{T}(\phi),\qquad\qquad-\delta^{\text{d}}_{\xi}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega=\not{\delta}\mathcal{D}(\xi),$
(3.3)

where the notation $\not{\delta}$ means that, for diffeomorphisms, the field-
space 1-form obtained with the contraction is not necessarily integrable, i.e.
not necessarily a total variation as we will see below. By construction of the
covariant phase space, the generators on the right-hand side of these
expressions are the sum of a bulk piece vanishing on-shell and a boundary
piece called the charge. Let us now identify these two pieces for the various
gauge transformations.

For Lorentz transformations, after computing the contraction with the
symplectic structure we find that the Hamiltonian generator is given by

$\displaystyle\mathcal{J}(\alpha)$
$\displaystyle=\int_{\Sigma}2\mathrm{d}_{\omega}\alpha\wedge(\sigma_{1}e+\sigma_{2}\omega)+\sigma_{2}\alpha[\omega\wedge\omega]-\sigma_{3}\alpha[e\wedge
e]$ (3.4)
$\displaystyle=-\int_{\Sigma}\alpha\big{(}2\sigma_{2}F+2\sigma_{1}\mathrm{d}_{\omega}e+\sigma_{3}[e\wedge
e]\big{)}+2\oint_{S}\alpha(\sigma_{1}e+\sigma_{2}\omega).$ (3.5)

For the translations we find that the Hamiltonian generator is

$\displaystyle\mathcal{T}(\phi)$
$\displaystyle=\int_{\Sigma}2\mathrm{d}_{\omega}\phi\wedge(\sigma_{1}\omega+\sigma_{3}e)+\sigma_{1}\phi[\omega\wedge\omega]-\sigma_{0}\phi[e\wedge
e]$ (3.6)
$\displaystyle=-\int_{\Sigma}\phi\big{(}2\sigma_{1}F+2\sigma_{3}\mathrm{d}_{\omega}e+\sigma_{0}[e\wedge
e]\big{)}+2\oint_{S}\phi(\sigma_{1}\omega+\sigma_{3}e),$ (3.7)

where we have used the fact that $\sigma_{0}=p\sigma_{1}+q\sigma_{3}$ and
$\sigma_{3}=p\sigma_{2}+q\sigma_{1}$. As usual, we have written here the
generators in their manifestly differentiable form, which is the first line of
each expression. We have then integrated by parts to reveal the constraints
(which are the spatial pullbacks of the equations of motion), and therefore
the boundary charges which are left when going on-shell.

For the diffeomorphisms, we write the Lie derivative as the second line of
(2.13) for each variable, and find

$\displaystyle\not{\delta}\mathcal{D}(\xi)$
$\displaystyle=2\int_{\Sigma}\Big{(}\sigma_{1}\big{(}\mathrm{d}_{\omega}(\xi\lrcorner\,\omega)+\xi\lrcorner\,F\big{)}+\sigma_{3}\big{(}\mathrm{d}_{\omega}(\xi\lrcorner\,e)+\xi\lrcorner\,(\mathrm{d}_{\omega}e)+[e,\xi\lrcorner\,\omega]\big{)}\Big{)}\wedge\delta
e$ (3.8)
$\displaystyle\phantom{=\int}+\Big{(}\sigma_{2}\big{(}\mathrm{d}_{\omega}(\xi\lrcorner\,\omega)+\xi\lrcorner\,F\big{)}+\sigma_{1}\big{(}\mathrm{d}_{\omega}(\xi\lrcorner\,e)+\xi\lrcorner\,(\mathrm{d}_{\omega}e)+[e,\xi\lrcorner\,\omega]\big{)}\Big{)}\wedge\delta\omega$
(3.9) $\displaystyle=\int_{\Sigma}\xi\lrcorner\,\Big{(}2\delta
e\wedge(\sigma_{1}F+\sigma_{3}\mathrm{d}_{\omega}e)+\delta\omega\wedge\big{(}2\sigma_{2}F+2\sigma_{1}\mathrm{d}_{\omega}e+\sigma_{3}[e\wedge
e]\big{)}\Big{)}$ (3.10)
$\displaystyle\phantom{=\int}-\delta\Big{(}2(\xi\lrcorner\,e)(\sigma_{1}F+\sigma_{3}\mathrm{d}_{\omega}e)+(\xi\lrcorner\,\omega)\big{(}2\sigma_{2}F+2\sigma_{1}\mathrm{d}_{\omega}e+\sigma_{3}[e\wedge
e]\big{)}\Big{)}$ (3.11)
$\displaystyle\phantom{=\int}+\oint_{S}\delta\big{(}2\sigma_{1}(\xi\lrcorner\,\omega)e+\sigma_{2}(\xi\lrcorner\,\omega)\omega+\sigma_{3}(\xi\lrcorner\,e)e\big{)}-\xi\lrcorner\,\theta.$
(3.12)

We see that half of the equations of motion appear in the bulk (second and
fourth terms), while for the other half the term in $\sigma_{0}$ is missing.
However this is consistent because of the identity
$\xi\lrcorner\,\big{(}\delta e\wedge[e\wedge
e]\big{)}=\delta\big{(}(\xi\lrcorner\,e)[e\wedge e]\big{)}$, which means that
the corresponding term can indeed be re-introduced. As usual, we see that
generic diffeomorphisms are not integrable. The non-integrable piece in the
bulk vanishes on-shell, or off-shell for $\xi$ tangent to $\Sigma$, while for
the boundary charge we need $\xi$ to be tangent to $S$. If $\xi$ is tangent to
both $\Sigma$ and $S$ we have that

$\displaystyle\mathcal{D}(\xi)=\int_{\Sigma}2\sigma_{1}\pounds_{\xi}\omega\wedge
e+\sigma_{2}\pounds_{\xi}\omega\wedge\omega+\sigma_{3}\pounds_{\xi}e\wedge e.$
(3.13)

Let us now go back to the case of a generic vector field $\xi$ and study more
closely the non-integrability of the diffeomorphisms. From now on we will
focus on the boundary charges. For convenience, we will use a slight abuse of
notation and write all the generators on-shell, so that they coincide with
their boundary charge and we can discard the bulk once and for all.

We have seen earlier that there is a relationship, given by (2.13), between
the diffeomorphisms and the internal gauge transformations. Since this
involves field-dependent gauge transformations, which are therefore
generically non-integrable, we should go back to the notation $\not{\delta}$
which keeps track of this fact. With this the Lorentz and translation charges
are

$\displaystyle\not{\delta}\mathcal{J}(\alpha)\approx
2\oint_{S}\alpha\delta(\sigma_{1}e+\sigma_{2}\omega),\qquad\qquad\not{\delta}\mathcal{T}(\phi)\approx
2\oint_{S}\phi\delta(\sigma_{1}\omega+\sigma_{3}e),$ (3.14)

and the diffeomorphism charge is

$\displaystyle\not{\delta}\mathcal{D}(\xi)\approx\oint_{S}\delta\big{(}2\sigma_{1}(\xi\lrcorner\,\omega)e+\sigma_{2}(\xi\lrcorner\,\omega)\omega+\sigma_{3}(\xi\lrcorner\,e)e\big{)}-\xi\lrcorner\,\theta.$
(3.15)

Using the fact that
$2(\xi\lrcorner\,\omega)\delta\omega=\delta\big{(}(\xi\lrcorner\,\omega)\omega\big{)}-\xi\lrcorner\,(\delta\omega\wedge\omega)$,
and similarly for $e$, it is then straightforward to see that we have

$\displaystyle\not{\delta}\mathcal{D}(\xi)=\not{\delta}\mathcal{J}(\xi\lrcorner\,\omega)+\not{\delta}\mathcal{T}(\xi\lrcorner\,e),$
(3.16)

which reflects (2.13). This shows, as expected, how the non-integrability of
the diffeomorphism charge comes from the non-integrable contributions of the
field field-dependent translations and Lorentz transformations.

A natural question to ask is whether the integrability of
$\not{\delta}\mathcal{D}(\xi)$ implies separately the integrability of
$\not{\delta}\mathcal{J}(\xi\lrcorner\,\omega)$ and
$\not{\delta}\mathcal{T}(\xi\lrcorner\,e)$. In the general case this statement
is evidently not true, as for example taking $\xi$ tangent to $S$ renders
$\not{\delta}\mathcal{D}(\xi)$ integrable but neither
$\not{\delta}\mathcal{J}(\xi\lrcorner\,\omega)$ nor
$\not{\delta}\mathcal{T}(\xi\lrcorner\,e)$. We will come back to this question
in section 4 in the case of more generic boundary conditions for non-tangent
vector fields at finite distance.

###  Dual diffeomorphisms

Let us now jump a little bit ahead of ourselves, and introduce a new charge
also constructed from field-dependent Lorentz transformations and
translations, but with a different field-dependency as that of
diffeomorphisms. We will come back to the general idea behind the
identification of this charge (and its teleparallel dual) in section 4.4. For
the time being, we are free to define it, to analyse its integrability, and to
consider it along with the other charges introduced so far on the covariant
phase space.

More precisely, the object which we want to introduce is the dual
diffeomorphism charge. For this, in full generality and without any boundary
conditions, let us consider the quantity

$\displaystyle\not{\delta}\mathcal{C}(\xi)$ $\displaystyle\coloneqq
p\,\not{\delta}\mathcal{J}(\xi\lrcorner\,e)+\not{\delta}\mathcal{T}(\xi\lrcorner\,\omega)$
(3.17)
$\displaystyle\phantom{:}=\oint_{S}\delta\big{(}2p\sigma_{2}(\xi\lrcorner\,\omega)e+\sigma_{1}(\xi\lrcorner\,\omega)\omega+p\sigma_{1}(\xi\lrcorner\,e)e\big{)}-\xi\lrcorner\,\big{(}2p\sigma_{2}\delta\omega\wedge
e+\sigma_{1}\delta\omega\wedge\omega+p\sigma_{1}\delta e\wedge e\big{)}$
(3.18)
$\displaystyle\phantom{\coloneqq\oint_{S}}+2(\sigma_{3}-p\sigma_{2})(\xi\lrcorner\,\omega)\delta
e.$ (3.19)

Several important things should be noticed about this definition.

  1. 1.

We have introduced this charge by analogy with the diffeomorphism charge
(3.16), but swapped the fields used in the field-dependent gauge
transformations. This is a first justification for the name “dual
diffeomorphism”, which will be further clarified below.

  2. 2.

It is important to notice that this charge is for the moment non-integrable.
However, we see that integrability is achieved without restricting the
dynamical fields when $\xi$ is tangent to $S$ and when
$\sigma_{3}=p\sigma_{2}$ (which means at this stage that either $q=0$ or
$\sigma_{1}=0$). Notice also that for the familiar Hilbert–Palatini
Lagrangian, obtained for $\sigma_{2}=\sigma_{3}=0$, this new charge does
indeed exist.

  3. 3.

To define the dual charge we have combined the field-dependent gauge
transformations with a relative factor of $p$, whose role will be clear when
computing the algebra below. However, it is important to point out that,
instead of the definition (3.17), we could have defined the new charge as
$\not{\delta}\mathcal{C}(\xi)=\sigma_{3}\not{\delta}\mathcal{J}(\xi\lrcorner\,e)+\sigma_{2}\not{\delta}\mathcal{T}(\xi\lrcorner\,\omega)$.
It is easy to see that this quantity is always integrable, without the need to
constrain the couplings $\sigma_{i}$ of the MB model. This definition however
has the disadvantage that it makes not obvious the $\sigma_{2}=\sigma_{3}=0$
limit giving the usual Hilbert–Palatini Lagrangian, since it gives the false
impression that in this case the dual charge does not exist. Moreover, we will
see in a moment and explain in appendix C.1 that, no mater which coefficients
are taken to define $\mathcal{C}$, consistency of its algebra with the gauge
charges requires to take $q=0$, which in turn implies $\sigma_{3}=p\sigma_{2}$
anyways. This is why it is coherent to simply start with the definition
(3.17). We should keep this remark in mind because it will also apply to the
other dual charge introduced below in section 3.5.

Now, recall that for diffeomorphisms we had the relations (2.13) which were
reflected in (3.16). It is then natural, given the definition of
$\mathcal{C}$, to look at a transformation of the fields of the form

$\displaystyle\delta^{\text{c}}_{\xi}e$
$\displaystyle=p\delta^{\text{j}}_{\xi\lrcorner\,e}e+\delta^{\text{t}}_{\xi\lrcorner\,\omega}e$
(3.20a)
$\displaystyle=p[e,\xi\lrcorner\,e]+\mathrm{d}_{\omega}(\xi\lrcorner\,\omega)+q[e,\xi\lrcorner\,\omega]$
(3.20b)
$\displaystyle=\pounds_{\xi}\omega-\xi\lrcorner\,F+p[e,\xi\lrcorner\,e]+q[e,\xi\lrcorner\,\omega]$
(3.20c) $\displaystyle\approx\pounds_{\xi}\omega+q[e,\xi\lrcorner\,\omega],$
(3.20d) $\displaystyle\delta^{\text{c}}_{\xi}\omega$
$\displaystyle=p\delta^{\text{j}}_{\xi\lrcorner\,e}\omega+\delta^{\text{t}}_{\xi\lrcorner\,\omega}\omega$
(3.20e)
$\displaystyle=p\mathrm{d}_{\omega}(\xi\lrcorner\,e)+p[e,\xi\lrcorner\,\omega]$
(3.20f) $\displaystyle=p\pounds_{\xi}e-p\xi\lrcorner\,(\mathrm{d}_{\omega}e)$
(3.20g) $\displaystyle\approx p\pounds_{\xi}e-pq[e,\xi\lrcorner\,e],$ (3.20h)

where in both cases for the last equality we have gone on-shell. We see that,
up to terms proportional to $q$, the object $\mathcal{C}$ which we have
introduced is the charge of a transformation sending $e$ to the Lie derivative
of $\omega$, and vice-versa. This is yet another reason behind the name “dual
diffeomorphism”. To understand what is special about the value $q=0$, for
which (3.20) really reduces to a Lie derivative, we have to study the algebra
of the charges.

###  Charge algebra

Let us now go back to the general case of arbitrary couplings $\sigma_{i}$.
Given the Hamiltonian generator
$\delta\mathcal{H}(\alpha)=-\delta_{\alpha}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega$
of an integrable transformation $\delta_{\alpha}$, the Poisson bracket between
two generators is given by
$\big{\\{}\mathcal{H}(\alpha),\mathcal{H}(\beta)\big{\\}}=\delta_{\alpha}\mathcal{H}(\beta)=\delta_{\alpha}\lrcorner\,\\!\\!\\!\lrcorner\,\big{(}\delta\mathcal{H}(\beta)\big{)}=-\delta_{\alpha}\lrcorner\,\\!\\!\\!\lrcorner\,\delta_{\beta}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega$.
Focusing first on the Lorentz transformations and the translations we
find666Notice that the bracket of translations can be rewritten as
$\big{\\{}\mathcal{T}(\phi),\mathcal{T}(\chi)\big{\\}}=p\big{\\{}\mathcal{J}(\phi),\mathcal{J}(\chi)\big{\\}}+q\big{\\{}\mathcal{J}(\phi),\mathcal{T}(\chi)\big{\\}}$.

$\displaystyle\big{\\{}\mathcal{J}(\alpha),\mathcal{T}(\phi)\big{\\}}$
$\displaystyle=\mathcal{T}([\alpha,\phi])-2\sigma_{1}\oint_{S}\alpha\mathrm{d}\phi,$
(3.21a)
$\displaystyle\big{\\{}\mathcal{J}(\alpha),\mathcal{J}(\beta)\big{\\}}$
$\displaystyle=\mathcal{J}([\alpha,\beta])-2\sigma_{2}\oint_{S}\alpha\mathrm{d}\beta,$
(3.21b) $\displaystyle\big{\\{}\mathcal{T}(\phi),\mathcal{T}(\chi)\big{\\}}$
$\displaystyle=p\mathcal{J}([\phi,\chi])+q\mathcal{T}([\phi,\chi])-2\sigma_{3}\oint_{S}\phi\mathrm{d}\chi.$
(3.21c)

These Poisson brackets form as expected an ultra-local extension of the
algebra (2.15), and contain three classical central extensions. The central
charges are simply given by three of the couplings of the MB Lagrangian, which
is also a consequence of having chosen (2.11) as the translations. Defining
$\mathcal{K}_{i}\coloneqq\mathcal{J}+k_{i}\mathcal{T}$ for $i=1,2$ and
computing the new brackets, one can easily see that it is always possible to
kill any two central extensions with an appropriate choice of $k_{i}$’s.

Following (2.17), we can also define the new generator

$\displaystyle\mathcal{P}(\alpha)\coloneqq\mathcal{T}(\alpha)-\frac{q}{2}\mathcal{J}(\alpha),$
(3.22)

which maps the algebra to

$\displaystyle\big{\\{}\mathcal{J}(\alpha),\mathcal{P}(\phi)\big{\\}}$
$\displaystyle=\mathcal{P}([\alpha,\phi])+(q\sigma_{2}-2\sigma_{1})\oint_{S}\alpha\mathrm{d}\phi,$
(3.23a)
$\displaystyle\big{\\{}\mathcal{J}(\alpha),\mathcal{J}(\beta)\big{\\}}$
$\displaystyle=\mathcal{J}([\alpha,\beta])-2\sigma_{2}\oint_{S}\alpha\mathrm{d}\beta,$
(3.23b) $\displaystyle\big{\\{}\mathcal{P}(\phi),\mathcal{P}(\chi)\big{\\}}$
$\displaystyle=-\Lambda\left(\mathcal{J}([\phi,\chi])-2\sigma_{2}\oint_{S}\phi\mathrm{d}\chi\right).$
(3.23c)

As expected we recognize here the $\mathfrak{g}_{\Lambda}$ current algebra,
whose global part is (2.18), and find once again three central extensions. In
this parametrization of the algebra however, the central charges are more
complicated functions of the initial couplings $\sigma_{i}$ of the Lagrangian.
We will see below that these central charges are actually (up to signs) the
ones appearing in the asymptotic symmetry algebra (5.21) of AdS spacetimes in
Bondi gauge. It is intriguing that these asymptotic central charges, which
depend on boundary conditions, can already be read off from the algebra (3.23)
which is independent of boundary conditions (and is not even an algebra of
diffeomorphisms).

We can now complete the algebra of Lorentz transformations and translations by
including the diffeomorphisms. However, since generic diffeomorphisms are not
integrable without specifying boundary conditions, we focus as explained above
on vector fields $\xi$ tangent to $S$. We then find the usual brackets

$\displaystyle\big{\\{}\mathcal{D}(\xi),\mathcal{J}(\alpha)\big{\\}}$
$\displaystyle=-\mathcal{J}(\pounds_{\xi}\alpha),$ (3.24a)
$\displaystyle\big{\\{}\mathcal{D}(\xi),\mathcal{T}(\phi)\big{\\}}$
$\displaystyle=-\mathcal{T}(\pounds_{\xi}\phi),$ (3.24b)
$\displaystyle\big{\\{}\mathcal{D}(\xi),\mathcal{D}(\zeta)\big{\\}}$
$\displaystyle=-\mathcal{D}([\xi,\zeta]).$ (3.24c)

The brackets (3.21) and (3.24) describe the algebra of charges of the MB
model, with the only assumption being that of tangentiality of the
diffeomorphisms.

Finally, in the particular case $\sigma_{3}=p\sigma_{2}$, it is natural to
continue the logic of extending the symmetry algebra by including as well the
dual diffeomorphisms. As we are about to see, their algebra with the charges
introduced so far in (3.21) and (3.24) contains an important information.
Considering again tangential vector fields, the new brackets are

$\displaystyle\big{\\{}\mathcal{C}(\xi),\mathcal{J}(\alpha)\big{\\}}$
$\displaystyle=-\mathcal{T}(\pounds_{\xi}\alpha),$ (3.25a)
$\displaystyle\big{\\{}\mathcal{C}(\xi),\mathcal{T}(\phi)\big{\\}}$
$\displaystyle=-p\mathcal{J}(\pounds_{\xi}\phi)+q\mathcal{T}([\xi\lrcorner\,\omega,\phi]),$
(3.25b) $\displaystyle\big{\\{}\mathcal{C}(\xi),\mathcal{D}(\zeta)\big{\\}}$
$\displaystyle=-\mathcal{C}([\xi,\zeta]),$ (3.25c)
$\displaystyle\big{\\{}\mathcal{C}(\xi),\mathcal{C}(\zeta)\big{\\}}$
$\displaystyle=-p\mathcal{D}([\xi,\zeta]),$ (3.25d)

for which we give detailed proofs in appendix C.1. In these brackets we see
the appearance of a new object which requires scrutiny. Indeed, at the
difference with all the other charges appearing in (3.25) and in the brackets
above, the object $\mathcal{T}([\xi\lrcorner\,\omega,\phi])$ in (3.25b) has a
field-dependent smearing. This means that it cannot actually be obtained as
the charge of a field-dependent translation, since this latter would not be
integrable. However, we can reverse the logic and ask under which condition
$\not{\delta}\mathcal{T}([\xi\lrcorner\,\omega,\phi])$ does become integrable.
Writing this charge explicitly we find

$\displaystyle\not{\delta}\mathcal{T}([\xi\lrcorner\,\omega,\phi])=2\oint_{S}[\xi\lrcorner\,\omega,\phi]\delta(\sigma_{1}\omega+\sigma_{3}e)=\oint_{S}\sigma_{1}\delta([\omega,\xi\lrcorner\,\omega]\phi)+2\sigma_{3}[\xi\lrcorner\,\omega,\phi]\delta
e,$ (3.26)

where we have used the tangentiality of $\xi$ to show that the first term is
always integrable. We now recall that the charge $\mathcal{C}$, which leads to
the above algebra, only exists when $\sigma_{3}=p\sigma_{2}$, and that this
condition implies either that $q=0$ or $\sigma_{1}=0$. If $q=0$, we see that
the translation term with field-dependent smearing in (3.25b) simply drops. On
the other hand, if instead we have $\sigma_{1}=0$, this term survives, and we
have to discuss its integrability. Without imposing boundary conditions on the
fields, this can only be achieved with $\sigma_{3}=0$. This in turn
trivializes (3.26), but most importantly implies once again that $q=0$. In
summary, we see that consistency of the brackets (3.25) always implies that
$q=0$. Moreover, we explain at the end of appendix (C.1) that it is always the
condition $q=0$ which is singled out, regardless of which coefficients are
used to define $\mathcal{C}$ in (3.17). Here we have chosen the definition
which leads to the simplest form of the algebra.

It is also enlightening to rewrite the above Poisson brackets in Fourier
space. This will be useful in the next section to study the Sugawara
construction, and it also reveals the particular role played by the condition
$q=0$. With the notations and detailed proofs given in appendix C.2, the
Fourier version of the algebra (3.21), (3.24), and (3.25) is

$\displaystyle\big{\\{}\mathcal{J}^{i}_{n},\mathcal{T}^{j}_{m}\big{\\}}$
$\displaystyle={\varepsilon^{ij}}_{k}\mathcal{T}^{k}_{n+m}-2\mathrm{i}\sigma_{1}m\eta^{ij}\delta_{n+m,0},$
(3.27a)
$\displaystyle\big{\\{}\mathcal{J}^{i}_{n},\mathcal{J}^{j}_{m}\big{\\}}$
$\displaystyle={\varepsilon^{ij}}_{k}\mathcal{J}^{k}_{n+m}-2\mathrm{i}\sigma_{2}m\eta^{ij}\delta_{n+m,0},$
(3.27b)
$\displaystyle\big{\\{}\mathcal{T}^{i}_{n},\mathcal{T}^{j}_{m}\big{\\}}$
$\displaystyle={\varepsilon^{ij}}_{k}\big{(}p\mathcal{J}^{k}_{n+m}+q\mathcal{T}^{k}_{n+m}\big{)}-2\mathrm{i}\sigma_{3}m\eta^{ij}\delta_{n+m,0},$
(3.27c) $\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{J}^{i}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}m\mathcal{J}^{i}_{n+m},$ (3.27d)
$\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{T}^{i}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}m\mathcal{T}^{i}_{n+m},$ (3.27e)
$\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}_{n+m},$ (3.27f)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{C}_{n+m},$ (3.27g)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{C}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}p(m-n)\mathcal{D}_{n+m},$ (3.27h)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{J}^{i}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}m\mathcal{T}^{i}_{n+m},$ (3.27i)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{T}^{i}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}pm\mathcal{J}^{i}_{n+m}-\frac{pq\sigma_{2}}{2(p\sigma_{2}^{2}-\sigma_{1}^{2})}{\varepsilon^{i}}_{jk}\sum_{k\in\mathbb{Z}}\mathcal{J}^{j}_{n+m+k}\mathcal{T}^{k}_{-k}.$
(3.27j)

Note that $\sigma_{2}\sigma_{3}-\sigma_{1}^{2}\neq 0$ is a condition which we
have from the beginning, and which here becomes
$p\sigma_{2}^{2}-\sigma_{1}^{2}\neq 0$ since we are considering $\mathcal{C}$
in the algebra. Once again, we see that the bracket (3.27j) produces an
unusual term, namely a quadratic in the currents which has an open index
produced with the epsilon symbol. This is the Fourier space version of the
field-dependent translation in (3.25b). From this Fourier expression, it is
easier to see that the presence of this term prevents the algebra from
closing, as the iterated Poisson bracket of (3.27j) with $\mathcal{C}_{n}$
produces higher and higher order contractions of the currents. We therefore
see once again that the well-definiteness of this algebra singles out the
condition $q=0$ (recall that $q=0$ in turn implies $\sigma_{3}=p\sigma_{2}$).

The remarkable result now comes from the algebra formed by the dual
diffeomorphism $\mathcal{C}_{n}$ together with the usual diffeomorphism
$\mathcal{D}_{n}$. In the case $q=0$, the cosmological constant becomes
$\Lambda=-p$. We can then discuss two particular cases of interest. First, if
$p=0$, one can see from the above Poisson brackets that the couple
$(\mathcal{D}_{n},\mathcal{C}_{n})$ forms the algebra

$\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}_{n+m},$ (3.28a)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{C}_{n+m},$ (3.28b)
$\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{C}_{m}\big{\\}}$
$\displaystyle=0,$ (3.28c)

which we recognize as the BMS3 algebra with vanishing central charge. When $p$
is non-vanishing, we can define

$\displaystyle\mathcal{D}_{n}^{\pm}\coloneqq\frac{1}{2}\left(\frac{1}{\sqrt{p}}\mathcal{C}_{\pm
n}\pm\mathcal{D}_{\pm
n}\right)_{\\!\\!p>0},\qquad\qquad\mathcal{D}_{n}^{\pm}\coloneqq\frac{1}{2}\left(\frac{1}{\mathrm{i}\sqrt{-p}}\mathcal{C}_{\pm
n}\pm\mathcal{D}_{\pm n}\right)_{\\!\\!p<0},$ (3.29)

and in both cases find the algebra

$\displaystyle\big{\\{}\mathcal{D}_{n}^{\pm},\mathcal{D}_{m}^{\pm}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}_{n+m}^{\pm},$ (3.30a)
$\displaystyle\big{\\{}\mathcal{D}_{n}^{+},\mathcal{D}_{m}^{-}\big{\\}}$
$\displaystyle=0.$ (3.30b)

In the AdS case $p>0$ these are two centreless Virasoro algebras. In the dS
case, i.e. when $p<0$, this is the algebra of the conformal group of the
complex plane, which is the infinite-dimensional enhancement of the
$\mathfrak{g}_{\Lambda}=\mathfrak{so}(3,1)$ isometries.

As announced, the brackets (3.28) and (3.30) are one of the main results of
this paper, namely the derivation, at finite distance and without boundary
conditions, of a centreless BMS or Witt algebra depending on the sign of the
cosmological constant. This result is surprising because it is obtained for a
generic boundary. It shows that the BMS and Virasoro structures are not solely
properties of asymptotic boundaries, but general features of the boundary
algebra of quadratic charges. This also reveals an important difference
between the two setups. In the asymptotic case, the BMS or Virasoro structure
(and their central charge) comes entirely from the diffeomorphism
$\mathcal{D}_{n}$ alone, which has two independent components due to the non-
tangentiality of the asymptotic vector field. Here however, the tangentiality
of the vector field requires us to use the dual charge $\mathcal{C}_{n}$ in
addition to $\mathcal{D}_{n}$ to construct the algebras, and is also
responsible for the lack of central charge. It is also interesting that this
finite distance construction works for any value of the cosmological constant
$\Lambda=-p$, which in particular includes the less-studied dS case. Finally,
we note that the algebra of $(\mathcal{C}_{n},\mathcal{D}_{n})$ is
automatically obtained in such a way that the flat limit is well-defined. This
is to be contrasted with studies of the asymptotic symmetry algebra, where the
Fefferman–Graham gauge does not lead to a well-defined flat limit while the
Bondi gauge does [83, 84].

It is important to note that finite distance Virasoro and BMS algebras were
introduced before in [85, 86] using the notion of symplectic symmetries. There
it is however the algebra (5.21) formed by the diffeomorphisms $\mathcal{D}$
alone which is brought to any finite distance $r$. We comment below (5.18) and
in appendix D on the definition of the symplectic symmetries in the first
order formulation.

In light of these results, a natural question is therefore that of the fate of
$\mathcal{C}$ in the asymptotic case and in the case of finite boundaries with
non-tangential vector fields. This also evidently raises the question of the
geometrical and metric interpretation of the dual diffeomorphism
$\mathcal{C}$. We will come back to this in future work, but give brief
comments in section 5. Here we have established a first contact with this dual
charge, and presented the simplest and natural setup in which it arises and
can be understood. This understanding is deepened by the Sugawara
construction, which we now turn to.

###  Classical Sugawara construction

We now want to construct objects which are quadratics in the basic currents
$\mathcal{J}^{i}_{n}$ and $\mathcal{T}^{i}_{n}$, and in particular understand
their relationship with the generators $\mathcal{D}_{n}$ and $\mathcal{C}_{n}$
of diffeomorphisms and their dual. This is the essence of the Sugawara
construction, which provides a concrete algebraic implementation of the
relations (3.16) and (3.17). In the case of diffeomorphisms this is well-know
for usual three-dimensional gravity. Here we extend this result to the case of
the MB model. In addition, we give an interpretation of the dual
diffeomorphism as a Sugawara quadratic generator. This is best achieved in the
Fourier representation (3.27). The various brackets for this section are given
in appendix C.3.

To start with, let us introduce the three quadratics

$\displaystyle\mathcal{Q}_{n}^{1}\coloneqq
2\sum_{k\in\mathbb{Z}}\mathcal{J}^{i}_{n+k}\mathcal{T}^{i}_{-k},\qquad\qquad\mathcal{Q}_{n}^{2}\coloneqq\sum_{k\in\mathbb{Z}}\mathcal{T}^{i}_{n+k}\mathcal{T}^{i}_{-k},\qquad\qquad\mathcal{Q}_{n}^{3}\coloneqq\sum_{k\in\mathbb{Z}}\mathcal{J}^{i}_{n+k}\mathcal{J}^{i}_{-k}.$
(3.31)

Their brackets with $\mathcal{J}^{i}_{m}$ and $\mathcal{T}^{i}_{m}$ are given
in (C.60), while the brackets of $\mathcal{Q}_{n}$’s with themselves are given
in (C.61) and (C.62). Matching (C.71) with the action of $\mathcal{D}_{n}$ on
$\mathcal{J}^{i}_{m}$ and $\mathcal{T}^{i}_{m}$, we find a unique solution
which is that

$\displaystyle\mathcal{D}_{n}=\frac{-1}{4(\sigma_{1}^{2}-\sigma_{2}\sigma_{3})}\big{(}\sigma_{1}\mathcal{Q}^{1}_{n}-\sigma_{2}\mathcal{Q}^{2}_{n}-\sigma_{3}\mathcal{Q}^{3}_{n}\big{)}.$
(3.32)

Then (C.72) gives (3.27f) as expected. This is the expression of the Sugawara
construction for the diffeomorphisms in the MB model.

Now, matching (C.71) with the action of $\mathcal{C}_{n}$ on
$\mathcal{J}^{i}_{m}$ and $\mathcal{T}^{i}_{m}$ requires to take
$\sigma_{3}=p\sigma_{2}$, and we then find a unique solution given by

$\displaystyle\mathcal{C}_{n}=\frac{-1}{4(\sigma_{1}^{2}-p\sigma_{2}^{2})}\big{(}\sigma_{1}\mathcal{Q}^{2}_{n}+p\sigma_{1}\mathcal{Q}^{3}_{n}-p\sigma_{2}\mathcal{Q}^{1}_{n}\big{)}.$
(3.33)

This equation is the Sugawara construction for the dual diffeomorphisms in the
MB model with the extra condition $\sigma_{3}=p\sigma_{2}$. Once again, note
that this includes the case of usual first order three-dimensional gravity,
which is recovered for $\sigma_{2}=\sigma_{3}=0$. The existence of this dual
Sugawara construction is yet another argument for why these dual charges are
natural to consider.

Finally, we can also connect these Sugawara constructions with the expression
(2.16) for the Casimirs of the $(J,T)$ algebra, which is the global part of
the $(\mathcal{J},\mathcal{T})$ one. Indeed, the existence of two Sugawara
constructions, and therefore the existence of the dual charges, can in a sense
also be understood as a consequence of the existence of two Casimirs.
Identifying the terms in (2.16) with the quadratics in (3.31) shows that

$\displaystyle\mathcal{D}_{n}\sim\frac{-1}{4(\sigma_{1}^{2}-\sigma_{2}\sigma_{3})}\big{(}\sigma_{1}C_{2}-\sigma_{2}C_{1}\big{)},\qquad\qquad\mathcal{C}_{n}\sim\frac{-1}{4(\sigma_{1}^{2}-p\sigma_{2}^{2})}\big{(}\sigma_{1}C_{1}-p\sigma_{2}C_{2}-pq\sigma_{2}\mathcal{Q}^{3}_{n}\big{)}.$
(3.34)

Interestingly, we see here resurfacing the condition $q=0$ which has been
identified in the previous subsection in order for the action of $\mathcal{C}$
on $\mathcal{T}$ to be stable. While this condition does not appear in (3.33),
it appears here when trying to connect the quadratic generators to the
Casimirs. When $q=0$, we get as expected a (formal) linear relationship
between the Casimirs of the initial algebra and the generators of
diffeomorphisms and their dual.

In summary, we have shown in this section that all the known results about
diffeomorphisms, internal Lorentz transformations, and translations, can be
extended to the MB model. We have done so by computing the charge algebra, and
related the currents to the quadratics by the Sugawara construction. In
addition, we have also shown that the $(p\neq 0,q=0)$ subspace of the MB model
contains a dual diffeomorphism charge. This latter admits a Sugawara
interpretation, and forms with the usual diffeomorphisms a finite distance
centreless double Virasoro or BMS algebra. In section 4 we will see how these
algebras can be centrally-extended when considering non-tangent vector fields.
This can also be interpreted in terms of the so-called twisted Sugawara
construction.

The twisted Sugawara construction relies on a first order linear twist of the
quadratics $\mathcal{Q}_{n}$ introduced above in (3.31). We define these
twists in appendix C.4 in terms of fixed internal vectors $\lambda^{i}$ and
$\mu^{i}$, and also compute all the relevant brackets between the twisted
quadratics. To illustrate this construction and the result, we can focus for
simplicity on the case $\sigma_{2}=\sigma_{3}=0$ (and therefore $q=0$) of
usual three-dimensional gravity. In this case, the diffeomorphisms and their
dual are given by

$\displaystyle\mathcal{D}_{n}=\frac{-1}{4\sigma_{1}}\check{\mathcal{Q}}^{1}_{n},\qquad\qquad\mathcal{C}_{n}=\frac{-1}{4\sigma_{1}}\big{(}\check{\mathcal{Q}}^{2}_{n}+p\check{Q}^{3}_{n}\big{)},$
(3.35)

where the $\check{\mathcal{Q}}_{n}$’s are the twisted quadratics defined in
(C.75). With this twist, the algebra of the diffeomorphisms and their dual
picks up central extensions, and takes the form

$\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}_{m+n}+\mathrm{i}\frac{\lambda\cdot\mu}{\sigma_{1}}m^{3}\delta_{n+m,0},$
(3.36a) $\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{C}_{m+n}+\mathrm{i}\frac{1}{2\sigma_{1}}(\lambda\cdot\lambda+p\mu\cdot\mu)m^{3}\delta_{n+m,0},$
(3.36b) $\displaystyle\big{\\{}\mathcal{C}_{n},\mathcal{C}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}p(m-n)\mathcal{D}_{m+n}+\mathrm{i}p\frac{\lambda\cdot\mu}{\sigma_{1}}m^{3}\delta_{n+m,0}.$
(3.36c)

With the redefinition of generators (3.29) this becomes a centrally-extended
double Virasoro algebra. In the flat limit $p=0$ this becomes a centrally-
extended BMS3 algebra. This shows that the finite distance Virasoro and BMS
algebras formed by the diffeomorphisms and their dual can be centrally-
extended in the presence of a twist. In section 4 we will interpret this twist
as a choice of boundary conditions on the connection and the triad.

###  Teleparallel dual diffeomorphisms

We have now had a first contact with the dual charges, and studied in minute
details the example of the dual diffeomorphisms $\mathcal{C}$ defined in the
case $(p\neq 0,q=0)$. We have in particular discovered the algebra which they
form with the usual diffeomorphisms $\mathcal{D}$, and related them to the
Sugawara construction.

Given how the MB model elegantly unifies the Lorentzian curvature and torsion
sectors of three-dimensional gravity, characterized respectively by $p$ and
$q$, it is natural to now ask whether there exists also a dual charge in the
case $(p=0,q\neq 0)$. With a slight abuse of language we choose to call this
the teleparallel sector777Indeed, recall that $p$ and $q$ are measures of the
Lorentzian curvature and torsion, while the Riemannian curvature and torsion
sectors of the MB model will be studied in section 6.. In this sector of the
MB model, the dual charge is given by

$\displaystyle\not{\delta}\mathcal{B}(\xi)\coloneqq
q\,\not{\delta}\mathcal{T}(\xi\lrcorner\,e)+\not{\delta}\mathcal{T}(\xi\lrcorner\,\omega),$
(3.37)

whose integrability can easily be seen to be achieved with a tangent vector
and the condition $\sigma_{3}=q\sigma_{1}$. Much like what has been done above
for the dual charge $\mathcal{C}$, one can explicitly check that consistency
of the bracket of $\mathcal{B}(\xi)$ with $\mathcal{T}(\phi)$ requires that
$p=0$. In this sense the charge $\mathcal{B}$ is the torsion/curvature dual to
the charge $\mathcal{C}$.

The action of $\mathcal{B}$ on the connection and on the triad can be computed
by analogy with (3.20). A direct calculation and a little bit of rewriting
shows that

$\displaystyle\delta^{\text{b}}_{\xi}e$
$\displaystyle=q\delta^{\text{t}}_{\xi\lrcorner\,e}e+\delta^{\text{t}}_{\xi\lrcorner\,\omega}e$
(3.38a)
$\displaystyle=q\mathrm{d}_{\omega}(\xi\lrcorner\,e)+q^{2}[e,\xi\lrcorner\,e]+\mathrm{d}_{\omega}(\xi\lrcorner\,\omega)+q[e,\xi\lrcorner\,\omega]$
(3.38b)
$\displaystyle=\pounds_{\xi}\omega-\xi\lrcorner\,F+q\mathrm{d}_{\omega}(\xi\lrcorner\,e)+q^{2}[e,\xi\lrcorner\,e]+q[e,\xi\lrcorner\,\omega]$
(3.38c)
$\displaystyle=\pounds_{\xi}\omega+q\pounds_{\xi}e-\xi\lrcorner\,F-\frac{q}{2}\xi\lrcorner\,\big{(}2\mathrm{d}_{\omega}e+q[e\wedge
e]\big{)}$ (3.38d)
$\displaystyle\approx\pounds_{\xi}\omega+q\pounds_{\xi}e-p[e,\xi\lrcorner\,e],$
(3.38e) $\displaystyle\delta^{\text{b}}_{\xi}\omega$
$\displaystyle=q\delta^{\text{t}}_{\xi\lrcorner\,e}\omega+\delta^{\text{t}}_{\xi\lrcorner\,\omega}\omega$
(3.38f) $\displaystyle=pq[e,\xi\lrcorner\,e]+p[e,\xi\lrcorner\,\omega].$
(3.38g)

This result shows that when $p=0$ the transformation $\mathcal{B}$ does indeed
have a geometrical action on the dynamical fields. At the difference with
$\mathcal{C}$ which was completely dualizing the diffeomorphisms by
transforming $e$ into the Lie derivative of $\omega$ and vice-versa, we see
that $\mathcal{B}$ transforms $e$ but not $\omega$. This therefore also shows
that these duality structures on the MB space of theories are very subtle,
since e.g. here we could not have intuited the form of (3.38) from the
analogue formula for $\mathcal{D}$ and $\mathcal{C}$. This opens the question
of the geometrical understanding of the dual transformations $\mathcal{B}$ and
$\mathcal{C}$, which we will explore in future work.

Using the results of appendix C, we can now also perform a Sugawara
construction for this new dual charge. Its expression in terms of the
quadratics (3.31) is in fact very simple, and found to be

$\displaystyle\mathcal{B}_{n}=\frac{-1}{4\sigma_{1}}\mathcal{Q}^{2}_{n}.$
(3.39)

Again, we can then relate this Sugawara construction to the Casimirs of the
$(J,T)$ algebra, and we find that

$\displaystyle\mathcal{B}_{n}\sim\frac{-1}{4\sigma_{1}}\big{(}C_{1}-p\mathcal{Q}^{3}_{n}\big{)},$
(3.40)

which bring up again the condition $p=0$ in order to have a correspondance
between the Casimirs and the quadratic generators. The other quadratic in the
case $(p=0,q\neq 0)$ is again given of course by $\mathcal{D}_{n}$, and the
new algebra of diffeomorphisms and their teleparallel dual is found to be

$\displaystyle\big{\\{}\mathcal{D}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}_{m+n},$ (3.41a)
$\displaystyle\big{\\{}\mathcal{B}_{n},\mathcal{D}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{B}_{m+n},$ (3.41b)
$\displaystyle\big{\\{}\mathcal{B}_{n},\mathcal{B}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}q(m-n)\mathcal{B}_{m+n}.$ (3.41c)

In the case $q=0$ of vanishing (Lorentzian) torsion this is the centreless
BMS3 algebra, as we have found in the flat case with the generators
$(\mathcal{D}_{n},\mathcal{C}_{n})$. For $q\neq 0$ however, upon rescaling
$\mathcal{B}_{n}$ we recognize two centreless Virasoro algebras in semi-direct
sum. One should recall that, although the teleparallel dual diffeomorphism
$\mathcal{B}$ is defined when the Lorentzian curvature $p$ is vanishing, the
Riemannian curvature is still non-vanishing and given by $\Lambda=-q^{2}/4$.
From this point of view it is perhaps not surprising to find Virasoro
algebras, although the novelty here is that they appear with a semi-direct
sum. These algebras can also easily be centrally-extended with the twisted
Sugawara construction using the results of appendix C.4.

Consistently, we then find that in the case $(p=0,q=0)$ we get
$\mathcal{B}=\mathcal{C}$, so the dual diffeomorphisms agree. This can be seen
both in the definitions (3.17) and (3.37) and in the Sugawara expressions
(3.33) and (3.39).

Finally, let us close with a remark about the (im)possibility of extending
further the duality. Given how $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$
are defined in terms of field-dependent gauge transformations in equations
(3.37), (3.17), and (3.16), and in particular the fact that $\mathcal{B}$ uses
the translations, one can wonder if it is possible to use instead the sum of
two field-dependent Lorentz transformations. One can indeed consider
$\not{\delta}\mathcal{A}\coloneqq\alpha\not{\delta}\mathcal{J}(\xi\lrcorner\,e)+\beta\not{\delta}\mathcal{J}(\xi\lrcorner\,\omega)$,
which is integrable for tangent vectors if $\beta\sigma_{1}=\alpha\sigma_{2}$.
However, then a quick calculation along the lines of (C.30) reveals that this
charges cannot have a stable action on $\mathcal{T}(\phi)$ unless
$\sigma_{1}^{2}-\sigma_{2}\sigma_{3}=0$, which is not authorized since it
corresponds to the degenerate sector of the MB model. This therefore closes
the study of the quadratic charges obtained as combinations of field-dependent
gauge transformations. We can now step back a little to study the CS
formulation of the MB model, and then relax the condition of tangentiality of
the vector fields.

##  Chern–Simons formulation

We now turn to the Chern–Simons formulation of the MB model. First, we will
explain the difference between the $\mathfrak{g}_{\Lambda}$ basis (2.18) and
(2.15). Then we will focus on the charges and go back to the connection and
triad variables in order to identify the dual charges in the case of non-
tangent vector fields.

Recall that for any connection decomposed as $A=\omega^{i}J_{i}+e^{i}K_{i}$
(we will keep the indices implicit below), for some generators $(J,K)$ whose
algebra is irrelevant for the moment, the CS Lagrangian gives

$\displaystyle L_{\text{CS}}[A]$
$\displaystyle=A\wedge\left(\mathrm{d}A+\frac{1}{3}[A\wedge A]\right)$ (4.1)
$\displaystyle=\omega^{i}\wedge\mathrm{d}\omega^{j}\langle
J_{i},J_{j}\rangle+\omega^{i}\wedge\mathrm{d}e^{j}\langle
J_{i},K_{j}\rangle+e^{i}\wedge\mathrm{d}\omega^{j}\langle
K_{i},J_{j}\rangle+e^{i}\wedge\mathrm{d}e^{j}\langle K_{i},K_{j}\rangle$ (4.2)
$\displaystyle\phantom{\ =}+\omega^{i}\wedge\omega^{j}\wedge e^{k}\langle
J_{i},[J_{j},K_{k}]\rangle+\omega^{i}\wedge e^{j}\wedge e^{k}\langle
J_{i},[K_{j},K_{k}]\rangle$ (4.3) $\displaystyle\phantom{\
=}+\frac{1}{3}\omega^{i}\wedge\omega^{j}\wedge\omega^{k}\langle
J_{i},[J_{j},J_{k}]+\frac{1}{3}e^{i}\wedge e^{j}\wedge e^{k}\langle
K_{i},[K_{j},K_{k}]\rangle,$ (4.4)

where $\langle\cdot\,,\cdot\rangle$ denotes the pairing in the algebra. This
formula is always valid, as for the moment we have not used any commutators or
pairings in the algebra of $(J,K)$. Let us now look at the cases $K=P$ and
$K=T$.

###  Chern–Simons formulation with $\boldsymbol{(J,P)}$

Let us consider the algebra

$\displaystyle[J_{i},P_{j}]={\varepsilon_{ij}}^{k}P_{k},\qquad\qquad[J_{i},J_{j}]={\varepsilon_{ij}}^{k}J_{k},\qquad\qquad[P_{i},P_{j}]=-\mu_{0}{\varepsilon_{ij}}^{k}J_{k},$
(4.5)

and the compatible pairings

$\displaystyle\langle J_{i},P_{j}\rangle=\mu_{1}\eta_{ij},\qquad\qquad\langle
J_{i},J_{j}\rangle=\mu_{2}\eta_{ij},\qquad\qquad\langle
P_{i},P_{j}\rangle=-\mu_{0}\mu_{2}\eta_{ij}.$ (4.6)

With this and $A=\omega J+eP$ we find

$\displaystyle L_{\text{CS}}[A]$
$\displaystyle=L_{\text{W}}[e,\omega]+\mu_{1}\mathrm{d}(e\wedge\omega)$ (4.7)
$\displaystyle=2\mu_{1}e\wedge\left(F-\frac{\mu_{0}}{6}[e\wedge
e]\right)+\mu_{2}\left(\omega\wedge\left(\mathrm{d}\omega+\frac{1}{3}[\omega\wedge\omega]\right)-\mu_{0}e\wedge\mathrm{d}_{\omega}e\right)+\mu_{1}\mathrm{d}(e\wedge\omega).\qquad$
(4.8)

This is the sum of the usual Lagrangian with coupling $\mu_{1}$, and the so-
called “exotic” Lagrangian with coupling $\mu_{2}$. We have denoted the sum
after Witten since the two terms appeared in [5]. The Lagrangian (4.7) differs
from the MB Lagrangian (2.1) in two ways:

  1. 1.

The couplings for the Lagrangians $L_{\text{CS}}[\omega]$ and
$L_{\text{T}}[e,\omega]$ in (4.7) are not independent, since they both appear
in the exotic Lagrangian with the common factor of $\mu_{2}$. One could think
of relaxing this by introducing a new independent coupling and working with
the pairing $\langle P_{i},P_{j}\rangle=\mu_{3}\eta_{ij}$, but then we would
have $\langle J_{i},[P_{j},P_{k}]\rangle\neq\langle
P_{k},[J_{i},P_{j}]\rangle$, so this pairing is not compatible with the
algebra.

  2. 2.

The couplings for the Lagrangians $L_{\text{V}}[e]$ and
$L_{\text{HP}}[e,\omega]$ in (4.7) are not independent. One could think of
relaxing this by replacing $\mu_{0}\mapsto\mu_{0}/\mu_{1}$, but then the limit
$\mu_{1}\rightarrow 0$ would not be well-defined, neither in the algebra nor
in the Lagrangian. One way to make the limit well-defined in the Lagrangian
however would be to also replace $\mu_{2}\mapsto\mu_{1}\mu_{2}$, but then the
Lagrangians $L_{\text{HP}}[e,\omega]$ and $L_{\text{CS}}[\omega]$ would not be
independent.

All this comes from the fact that we actually need a fourth independent
parameter in the game. This is indeed the property which distinguishes the MB
model from (4.7).

As pointed out in [58, 59], a fourth coupling can be introduced consistently
by considering the connection

$\displaystyle A=(\omega+\mu_{3}e)J+eP.$ (4.9)

As an indication, one can notice that the curvature of this connection is
given by

$\displaystyle\mathrm{d}A+\frac{1}{2}[A\wedge
A]=\left(F+\mu_{3}\mathrm{d}_{\omega}e+\frac{1}{2}(\mu_{3}^{2}-\mu_{0})[e\wedge
e]\right)J+\Big{(}\mathrm{d}_{\omega}e+\mu_{3}[e\wedge e]\Big{)}P.$ (4.10)

Setting it to zero gives the equations

$\displaystyle 2F-(\mu_{3}^{2}+\mu_{0})[e\wedge
e]=0,\qquad\qquad\mathrm{d}_{\omega}e+\mu_{3}[e\wedge e]=0,$ (4.11)

which therefore describe a source of Lorentzian curvature and torsion as in
the MB model. The Lagrangian obtained from this connection is

$\displaystyle L_{\text{CS}}[A]$
$\displaystyle=L_{\text{W}}[e,\omega+\mu_{3}e]+\mu_{1}\mathrm{d}(e\wedge\omega)$
(4.12)
$\displaystyle=\sigma_{0}L_{\text{V}}[e]+\sigma_{1}L_{\text{HP}}[e,\omega]+\sigma_{2}L_{\text{CS}}[\omega]+\sigma_{3}L_{\text{T}}[e,\omega]+\sigma_{1}\mathrm{d}(e\wedge\omega),$
(4.13)

with the identifications

$\displaystyle\sigma_{0}=\mu_{1}(3\mu_{3}^{2}-\mu_{0})+\mu_{2}\mu_{3}(\mu_{3}^{2}-3\mu_{0}),\qquad\sigma_{1}=\mu_{1}+\mu_{2}\mu_{3},\qquad\sigma_{2}=\mu_{2},\qquad\sigma_{3}=2\mu_{1}\mu_{3}+\mu_{2}(\mu_{3}^{2}-\mu_{0}).$
(4.14)

Notice how when going from the first to the second line in (4.12) the boundary
term also picks up a contribution from an integration by parts. One can
therefore see that the CS Lagrangian for the shifted connection (4.9) produces
indeed the MB Lagrangian. However, the couplings of the MB Lagrangian then
have the complicated relationship (4.14) with the parameters defining the
algebra and its pairings. Moreover, one can also notice that the connection
$\omega$ of the MB model is not the connection used to build the composite CS
connection (4.9), since there it is the shift $\omega+\mu_{3}e$ which appears.
This means that, in this particular $(J,P)$ CS formulation, the MB model is
not actually determined solely by the algebraic data (4.5) and (4.6), but
requites also the shift of the connection.

It turns out that these two points can be improved at once if we use the
$(J,T)$ algebra (2.15) instead of $(J,P)$. First, notice that one can invert
the relations (4.14) to find $\mu_{i}=f(\sigma_{i})$. This yields in
particular $\mu_{3}=q/2$, meaning that in (4.9) instead of shifting the
connection one can simply change the generators as $(J,P)\mapsto(J,T)$. With
this change of generators the pairings (4.6) lead to the pairings (4.16) given
below.

###  Chern–Simons formulation with $\boldsymbol{(J,T)}$

A more transparent CS formulation of the MB model can be obtained by using the
algebra $(J,T)$. This puts forward more clearly the role of the couplings and
of the curvature and torsion parameters $(p,q)$. Let us consider the algebra

$\displaystyle[J_{i},T_{j}]={\varepsilon_{ij}}^{k}T_{k},\qquad\qquad[J_{i},J_{j}]={\varepsilon_{ij}}^{k}J_{k},\qquad\qquad[T_{i},T_{j}]={\varepsilon_{ij}}^{k}(pJ_{k}+qT_{k}),$
(4.15)

and the pairings

$\displaystyle\langle
J_{i},T_{j}\rangle=\sigma_{1}\eta_{ij},\qquad\qquad\langle
J_{i},J_{j}\rangle=\sigma_{2}\eta_{ij},\qquad\qquad\langle
T_{i},T_{j}\rangle=\sigma_{3}\eta_{ij}.$ (4.16)

For the moment $p$ and $q$ can of course be arbitrary. With this, defining
$A=\omega J+eT$ leads to

$\displaystyle
L_{\text{CS}}[A]=(p\sigma_{1}+q\sigma_{3})L_{\text{V}}[e]+\sigma_{1}L_{\text{HP}}[e,\omega]+\sigma_{2}L_{\text{CS}}[\omega]+\sigma_{3}L_{\text{T}}[e,\omega]+\sigma_{1}\mathrm{d}(e\wedge\omega),$
(4.17)

where in this derivation the only condition which we have to impose on the
pairings in order for the torsion Lagrangian to appear, and therefore to get
this Lorentz-invariant MB form of the total Lagrangian, is

$\displaystyle\langle T_{i},T_{j}\rangle=p\langle J_{i},J_{j}\rangle+q\langle
J_{i},T_{j}\rangle\qquad\Rightarrow\qquad\sigma_{3}=p\sigma_{2}+q\sigma_{1}.$
(4.18)

This relation is in fact precisely the compatibility condition

$\displaystyle\langle J_{i},[T_{j},T_{k}]\rangle=\langle
T_{k},[J_{i},T_{j}]\rangle.$ (4.19)

The other condition, i.e. the one involving two $J$’s and one $T$, is already
satisfied. We therefore have a priori 5 couplings
$(\sigma_{1},\sigma_{2},\sigma_{3},p,q)$ minus this constraint, which gives 4
independent couplings as expected. We can then trade one of these couplings
for $\sigma_{0}$ by further imposing

$\displaystyle\sigma_{0}=p\sigma_{1}+q\sigma_{3}.$ (4.20)

One can then check for consistency that the parameters $p$ and $q$ are indeed
solutions to (4.18) and (4.20), so we finally have the very simple result

$\displaystyle
L_{\text{CS}}[A]=L_{\text{MB}}[e,\omega]+\sigma_{1}\mathrm{d}(e\wedge\omega).$
(4.21)

This shows, as announced, that the MB Lagrangian is a CS theory for the
connection $A=\omega J+eT$ and the $(J,T)$ algebra (2.15).

###  Charges in Chern–Simons theory

We now recall basic facts about gauge and diffeomorphism charges in CS theory,
before going back to connection and triad variables to analyse the
integrability of the diffeomorphisms. The variation of the CS Lagrangian, the
potential and the symplectic structure are given respectively by

$\displaystyle\delta L_{\text{CS}}[A]=2\delta A\wedge F+\mathrm{d}(\delta
A\wedge A),\qquad\qquad\theta_{\text{CS}}=\delta A\wedge
A,\qquad\qquad\Omega_{\text{CS}}=-\int_{\Sigma}\delta A\wedge\delta A.$ (4.22)

Consistently with (4.21), we have that
$\theta_{\text{CS}}=\theta_{\text{MB}}+\sigma_{1}\delta(e\wedge\omega)$. The
generator of the CS gauge transformation
$\delta_{\lambda}A=\mathrm{d}_{A}\lambda$ is computed from
$-\delta_{\lambda}\lrcorner\,\\!\\!\\!\lrcorner\,\Omega=\delta\mathcal{F}(\lambda)$
and given by

$\displaystyle\mathcal{F}(\lambda)=-2\int_{\Sigma}\lambda F+2\oint_{S}\lambda
A,\qquad\qquad\big{\\{}\mathcal{F}(\lambda),\mathcal{F}(\mu)\big{\\}}=\mathcal{F}([\lambda,\mu])-2\oint_{S}\lambda\mathrm{d}\mu.$
(4.23)

We can now decompose $A=\omega J+eT$ and $\lambda=\alpha J+\phi T$. This gives

$\displaystyle\mathcal{F}_{JT}(\alpha,\phi)\approx
2\oint_{S}\alpha(\sigma_{1}e+\sigma_{2}\omega)+\phi(\sigma_{1}\omega+\sigma_{3}e),$
(4.24)

which indeed agrees with the charges (3.4) and (3.6). For diffeomorphisms we
have as expected that

$\displaystyle\not{\delta}\mathcal{D}(\xi)$
$\displaystyle=2\int_{\Sigma}\xi\lrcorner\,(\delta A\wedge
F)-\delta\big{(}(\xi\lrcorner\,A)F\big{)}+2\oint_{S}(\xi\lrcorner\,A)\delta A$
(4.25) $\displaystyle=2\int_{\Sigma}\xi\lrcorner\,(\delta A\wedge
F)-\delta\big{(}(\xi\lrcorner\,A)F\big{)}+\oint_{S}\delta\big{(}(\xi\lrcorner\,A)A\big{)}-\xi\lrcorner\,\theta,$
(4.26)

which is in turn equal to (3.8). This shows the consistency of our CS
formulation.

In CS theory, the relation (3.16) giving the diffeomorphisms as field-
dependent gauge transformations becomes simply
$\not{\delta}\mathcal{D}(\xi)=\not{\delta}\mathcal{F}(\xi\lrcorner\,A)$. Just
as in the connection/triad formulation, generic diffeomorphisms are not
integrable, and integrability can be achieved with either tangent vector
fields or with specific boundary conditions on the dynamical fields. We also
note very interesting recent work on the possibility of obtaining integrable
diffeomorphism charges by considering field-dependent vector fields [87],
which we think could be extended to the MB model as well. So far we have
focused here on tangent diffeomorphisms in order to define the charges
$\mathcal{D}$ and their dual $\mathcal{C}$. Here we would like to relax this
condition and instead use boundary conditions on the dynamical fields
themselves. In the following subsection we therefore recall how this is
traditionally done in CS theory, and then what happens when going back to
connection/triad variables.

###  Dual diffeomorphisms again

Let us now look at generic diffeomorphism charges, and consider an arbitrary
field-independent vector field of the form
$\xi=(\xi^{t},\xi^{r},\xi^{\varphi})$. With this the (a priori non-integrable)
CS diffeomorphism charge is

$\displaystyle\not{\delta}\mathcal{D}(\xi)=2\oint_{S}(\xi^{t}A_{t}+\xi^{r}A_{r}+\xi^{\varphi}A_{\varphi})\delta
A_{\varphi}.$ (4.27)

The angular $\varphi$ piece is always integrable. The $(t,r)$ pieces are
integrable if we choose either $(A_{t},A_{r})\big{|}_{S}\propto A_{\varphi}$,
or $(A_{t},A_{r})\big{|}_{S}=\text{constant}$, or some mixture of these
conditions. Usually this choice is constrained by the form of the boundary
conditions chosen for the variational principle. Typical boundary conditions
are the chiral ones $A_{t}=\tau A_{\varphi}$, which indeed set
$\theta_{\text{CS}}\big{|}_{\Delta}=0$ on the time-like boundary. With a gauge
choice it is also customary to fix $A_{r}$ to be a constant $a$. With these
conditions the diffeomorphism charge becomes888Recall that here $\tau$ is a
number while $a$ is a Lie algebra element.

$\displaystyle\mathcal{D}(\xi)=\oint_{S}\xi^{\tau}(A_{\varphi})^{2}+2\xi^{r}aA_{\varphi},$
(4.28)

where we have introduced $\xi^{\tau}\coloneqq\tau\xi^{t}+\xi^{\varphi}$. To
compute the algebra of these charges, we need two further assumptions. The
first one is that $\xi$ depends only on the angular variable $\varphi$. This
ensures that the derivative of vector fields which appear when computing the
bracket can be rewritten as the Lie bracket. The second one is that
$\xi^{\tau}=\kappa\xi^{\varphi}$. This is required in order for the bracket to
close. With these assumptions, we find that the bracket is

$\displaystyle\big{\\{}\mathcal{D}(\xi),\mathcal{D}(\zeta)\big{\\}}=-\kappa\mathcal{D}([\xi,\zeta])-2a^{2}\oint_{S}\xi^{r}\partial_{\varphi}\zeta^{r}.$
(4.29)

This is the centrally-extended $\text{diff}(S)\oplus\mathfrak{u}(1)$ algebra,
where $r$ is the Abelian direction (which is the one carrying the central
extension). When the radial direction is then related to the angular one with
$\xi=(\xi^{t},\xi^{r},\xi^{\varphi})=(0,\gamma\partial_{\varphi}\bar{\xi},\bar{\xi})$,
this becomes the centrally-extended Virasoro algebra

$\displaystyle\big{\\{}\mathcal{D}(\xi),\mathcal{D}(\zeta)\big{\\}}=-\kappa\mathcal{D}([\xi,\zeta])+2a^{2}\gamma^{2}\oint_{S}\bar{\xi}\partial^{3}_{\varphi}\bar{\zeta}.$
(4.30)

This simple and well-known result shows how the central extension of the
diffeomorphism algebra arises from the non-tangentiality of the vector
fields999Note that in appendix B, when presenting the covariant phase space,
we have reproduced the proof of [88] that the non-integrable piece
$\xi\lrcorner\,\theta$ of the diffeomorphism charge (which is the piece
vanishing when $\xi$ is tangent) is the one responsible for the appearance of
a central extension. In fact in this appendix we have extended this proof to
the presence of a corner term along the lines of [89, 90, 70]. This is
equation (B.31). Note however that this computation does not allow to recover
the Virasoro or BMS central extension, because it assumes that the boundary
term $b$ is covariant. A more general formula for the central extension was
given in [91], and it would be interesting to explicitly show how this formula
reproduces the Virasoro and BMS central extension.. In this case we have to
pick boundary conditions in order for the charge to be integrable, and we have
chosen here the conditions leading to (4.28) as an example. In section 3.4 and
in appendix C.4 we have also explained how the non-tangentiality of the
diffeomorphisms and the appearance of a central charge can be understood in
terms of the so-called twisted Sugawara construction. Note that in (4.30) and
in the twisted Sugawara construction, although the vector fields are not
tangent, they only depend on a single parameter since the radial and angular
components are related.

It is now enlightening to go back to the MB model in connection/triad
variables. Denoting $a=A_{r}=(e_{r},\omega_{r})=(b,c)$, the diffeomorphism
charge (4.28) for the MB model is

$\displaystyle\mathcal{D}(\xi)=\oint_{S}\xi^{\tau}\big{(}2\sigma_{1}\omega_{\varphi}e_{\varphi}+\sigma_{2}(\omega_{\varphi})^{2}+\sigma_{3}(e_{\varphi})^{2}\big{)}+2\xi^{r}\big{(}\sigma_{1}ce_{\varphi}+\sigma_{1}b\omega_{\varphi}+\sigma_{2}c\omega_{\varphi}+\sigma_{3}be_{\varphi}\big{)}.$
(4.31)

For tangent diffeomorphisms, we have seen previously that the field-dependent
Lorentz transformations and translations in (3.16) are not separately
integrable. With the vector fields and boundary conditions chosen here, we
find that these field-dependent charges are also non-integrable and given by

$\displaystyle\not{\delta}\mathcal{J}(\xi\lrcorner\,\omega)$
$\displaystyle=\oint_{S}\delta\Big{(}\xi^{\tau}\sigma_{2}(\omega_{\varphi})^{2}+2\xi^{r}\big{(}\sigma_{1}ce_{\varphi}+\sigma_{2}c\omega_{\varphi}\big{)}\Big{)}+2\xi^{\tau}\sigma_{1}\omega_{\varphi}\delta
e_{\varphi},$ (4.32a) $\displaystyle\not{\delta}\mathcal{T}(\xi\lrcorner\,e)$
$\displaystyle=\oint_{S}\delta\Big{(}\xi^{\tau}\sigma_{3}(e_{\varphi})^{2}+2\xi^{r}\big{(}\sigma_{1}b\omega_{\varphi}+\sigma_{3}be_{\varphi}\big{)}\Big{)}+2\xi^{\tau}\sigma_{1}e_{\varphi}\delta\omega_{\varphi}.$
(4.32b)

The first piece in each expression is integrable, but because of the cross
term in $\sigma_{1}$, it is only the combination of these charges into
diffeomorphisms which is integrable. The charges are therefore generically not
integrable on their own.

Interestingly, the obstruction to separate integrability of the charges in
(4.32) comes from $\sigma_{1}\neq 0$, and we can therefore make these charges
integrable separately by working in the sector $\sigma_{1}=0$ of the theory.
This possibility exists because we are considering the MB model, which is non-
trivial even when $\sigma_{1}=0$, while the charges of usual three-dimensional
Hilbert–Palatini gravity would trivialize in this case. In order to find
charges which are integrable in the case $\sigma_{1}\neq 0$, this then
suggests to look at a natural alternative, which is to consider the “dual”
type of field-dependency for the gauge charges. Instead of (4.32) we are then
led to consider the dual charges

$\displaystyle\not{\delta}\mathcal{J}(\xi\lrcorner\,e)$
$\displaystyle=\oint_{S}\delta\Big{(}\xi^{\tau}\sigma_{1}(e_{\varphi})^{2}+2\xi^{r}\big{(}\sigma_{2}b\omega_{\varphi}+\sigma_{1}be_{\varphi}\big{)}\Big{)}+2\xi^{\tau}\sigma_{2}e_{\varphi}\delta\omega_{\varphi},$
(4.33a) $\displaystyle\not{\delta}\mathcal{T}(\xi\lrcorner\,\omega)$
$\displaystyle=\oint_{S}\delta\Big{(}\xi^{\tau}\sigma_{1}(\omega_{\varphi})^{2}+2\xi^{r}\big{(}\sigma_{3}ce_{\varphi}+\sigma_{1}c\omega_{\varphi}\big{)}\Big{)}+2\xi^{\tau}\sigma_{3}\omega_{\varphi}\delta
e_{\varphi}.$ (4.33b)

Looking at (4.32) and (4.33), it is now evident to recognize the integrable
combinations which we have introduced in the previous section. Indeed, if we
put a condition on the couplings we can get new integrable charges defined as
follows:

$\displaystyle\mathcal{B}(\xi)$ $\displaystyle\coloneqq
q\mathcal{T}(\xi\lrcorner\,e)+\mathcal{T}(\xi\lrcorner\,\omega)\qquad\text{when}\qquad
p=0,$ (4.34a) $\displaystyle\mathcal{C}(\xi)$ $\displaystyle\coloneqq
p\mathcal{J}(\xi\lrcorner\,e)+\mathcal{T}(\xi\lrcorner\,\omega)\qquad\text{when}\qquad
q=0,$ (4.34b) $\displaystyle\mathcal{D}(\xi)$
$\displaystyle\coloneqq\phantom{p}\mathcal{J}(\xi\lrcorner\,\omega)+\mathcal{T}(\xi\lrcorner\,e)\qquad\text{whenever}.$
(4.34c)

The generator $\mathcal{D}$ of diffeomorphisms is of course always integrable,
while the dual charges $\mathcal{B}$ and $\mathcal{C}$ each require an extra
condition. As explained in the previous section we need $p=0$ and $q=0$
respectively for $\mathcal{B}$ and $\mathcal{C}$ to have a consistent algebra
with the rest of the charges, and furthermore we then get
$p=0\Rightarrow\sigma_{3}=q\sigma_{1}$ and
$q=0\Rightarrow\sigma_{3}=q\sigma_{2}$.

In the case of the boundary conditions used in this section, where the vector
fields are not tangent, we have the explicit expressions

$\displaystyle\mathcal{B}(\xi)$
$\displaystyle=\oint_{S}\xi^{\tau}\big{(}2\sigma_{3}\omega_{\varphi}e_{\varphi}+\sigma_{1}(\omega_{\varphi})^{2}+q\sigma_{3}(e_{\varphi})^{2}\big{)}+2\xi^{r}\big{(}\sigma_{3}ce_{\varphi}+\sigma_{3}b\omega_{\varphi}+\sigma_{1}c\omega_{\varphi}+q\sigma_{3}be_{\varphi}\big{)},$
(4.35a) $\displaystyle\mathcal{C}(\xi)$
$\displaystyle=\oint_{S}\xi^{\tau}\big{(}2\sigma_{3}\omega_{\varphi}e_{\varphi}+\sigma_{1}(\omega_{\varphi})^{2}+p\sigma_{1}(e_{\varphi})^{2}\big{)}+2\xi^{r}\big{(}\sigma_{3}ce_{\varphi}+\sigma_{3}b\omega_{\varphi}+\sigma_{1}c\omega_{\varphi}+p\sigma_{1}be_{\varphi}\big{)}.$
(4.35b)

One can compare this to (4.31) to see that these new generators can
schematically be obtained from $\mathcal{D}$ upon changing the coupling
constants according to the pattern

(4.39)

where in addition we should recall that $\sigma_{3}=q\sigma_{1}$ for
$\mathcal{B}$ and $\sigma_{3}=p\sigma_{2}$ for $\mathcal{C}$. Just like the
algebra of the diffeomorphisms gets centrally-extended as in (4.29) because of
the non-tangentiality of the vector field, the brackets involving
$\mathcal{B}$ and $\mathcal{C}$ will also receive central extensions. This
shows how in the case of non-tangent vector fields it is still possible to
define the dual charges, and moreover to obtain central extensions for the
finite distance algebras $(\mathcal{D},\mathcal{C})$ or
$(\mathcal{D},\mathcal{B})$. Moreover this can be understood in terms of a
twisted Sugawara construction, as we explain in appendix C.4. We keep the
detailed investigation of the central charges for future work.

Finally, an important observation following from the construction of the dual
charges is that it is a priori not possible to obtain them directly from CS
theory. Indeed, as we have recalled in the previous subsection, the
diffeomorphisms of the MB model are obtained from its CS formulation by
considering the field dependent gauge charges $\mathcal{F}(\xi\lrcorner\,A)$.
It is however clear that it is not possible to consider any other field
dependency, such as the ones defining $\mathcal{B}$ and $\mathcal{C}$. CS
theory does simply not allow to write down such objects. It could perhaps be
possible to define dualities directly at the level of CS theory, using e.g. a
connection $\tilde{A}=\omega T+eJ$ or defining an operation swapping the
couplings as in (4.39), but this is for the moment speculative and should be
investigated further.

##  Asymptotic charges and symmetries

Now that we have unraveled the finite distance covariant phase space structure
of the MB model, introduced the dual charges, and studied the charge algebra,
we would like to briefly recall results about the asymptotic structure of the
MB model. This paves the road for future work on the study of the dual charges
in the asymptotic context.

###  BTZ black hole

We start by recalling some basic facts about the BTZ black hole in the MB
model. In Schwarzschild coordinates, the Lorentzian line element is given by

$\displaystyle\mathrm{d}s^{2}=-N(r)^{2}\mathrm{d}t^{2}+\frac{1}{N(r)^{2}}\mathrm{d}r^{2}+r^{2}\big{(}\mathrm{d}\varphi+N_{\varphi}(r)\mathrm{d}t\big{)}^{2},$
(5.1)

where

$\displaystyle
N(r)^{2}=-8Gm+\frac{r^{2}}{\ell^{2}}+\frac{16G^{2}j^{2}}{r^{2}},\qquad\qquad
N_{\varphi}(r)=-\frac{4Gj}{r^{2}},$ (5.2)

with $m$ and $j$ the mass and angular momentum. The horizons are located at

$\displaystyle
r_{\pm}^{2}=4Gm\ell^{2}\left(1\pm\sqrt{1-\left(\frac{j}{m\ell}\right)^{2}}\right).$
(5.3)

The cosmological constant here is related to that in the solution (2.9) given
above by

$\displaystyle\frac{1}{\ell^{2}}=-\Lambda=p+\frac{q^{2}}{4}.$ (5.4)

The coordinates are the time $t\in\mathbb{R}$, the radius $0<r<\infty$, and
the angle $0\leq\varphi\leq 2\pi$. With the internal metric
$\eta_{ij}=\text{diag}(-1,1,1)$, an adapted set of triad components such that
$g_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\eta_{ij}$, and the corresponding connection
components

$\displaystyle\omega^{i}_{\mu}=\Gamma^{i}_{\mu}-\frac{q}{2}e^{i}_{\mu},\qquad\qquad\Gamma^{i}_{\mu}=\frac{1}{2}{\varepsilon^{i}}_{jk}g^{\alpha\beta}e^{j}_{\alpha}\nabla_{\mu}e^{k}_{\beta},$
(5.5)

are given by101010The columns and rows correspond respectively to the $i$ and
$\mu$ indices.

$\displaystyle e^{i}_{\mu}=\begin{pmatrix}N&0&rN_{\varphi}\\\ 0&1/N&0\\\
0&0&r\end{pmatrix},\qquad\qquad\omega^{i}_{\mu}=\begin{pmatrix}0&0&r/\ell^{2}\\\
0&-N_{\varphi}/N&0\\\ N&0&rN_{\varphi}\end{pmatrix}-\frac{q}{2}e^{i}_{\mu}.$
(5.6)

Notice how the connection has a contribution from $q$, which comes from the
fact that we have decomposed it as $\omega=\Gamma+k$, where the contorsion is
fixed to $k\approx-qe/2$ by the torsion equation of motion in (2.8).

As usual, the energy and angular momentum are given by the temporal and
angular components of the diffeomorphism charge for the vector field
$\xi^{\mu}=(\xi^{t},0,\xi^{\varphi})$. In order to write down the charge for
such a non-tangent vector field, recall that the boundary condition used on
the time-like boundary is $(\theta+\delta b)\big{|}_{\Delta}=0$, where
$b=\sigma_{1}(e\wedge\omega)$ is the first order Gibbons–Hawking–York (GHY)
term111111Note that, as always, this boundary term is automatically provided
by the CS formulation, since from (4.21) we get
$\theta_{\text{CS}}=\theta_{\text{MB}}+\sigma_{1}\delta(e\wedge\omega)$.. With
this boundary condition, the non-integrable piece $-\xi\lrcorner\,\theta$ for
a generic diffeomorphism charge (3.8) becomes integrable. Indeed, as explained
in the appendix leading to formula (B.23), where $c=0$ because the first order
GHY has no derivative and therefore no corner term, the diffeomorphism charge
is then

$\displaystyle\mathcal{D}(\xi)=\oint_{S}Q+\sigma_{1}\xi\lrcorner\,(e\wedge\omega),$
(5.7)

where the Noether charge and the extra contribution are

$\displaystyle
Q=2\sigma_{1}(\xi\lrcorner\,\omega)e+\sigma_{2}(\xi\lrcorner\,\omega)\omega+\sigma_{3}(\xi\lrcorner\,e)e,\qquad\qquad\xi\lrcorner\,(e\wedge\omega)\big{|}_{S}=-\xi^{t}\left(N^{2}-r^{2}N_{\varphi}^{2}+\frac{r^{2}}{\ell^{2}}\right).$
(5.8)

We then find that the total diffeomorphism charge is finite for any value of
$r$ and given by

$\displaystyle\mathcal{D}(\xi)=\xi^{t}E+\xi^{\varphi}L,$ (5.9)

with

$\displaystyle\frac{E}{16\pi
G}\coloneqq(2\sigma_{1}-q\sigma_{2})\frac{m}{2}-\frac{\sigma_{2}}{\ell^{2}}j,\qquad\qquad\frac{L}{16\pi
G}\coloneqq-(2\sigma_{1}-q\sigma_{2})\frac{j}{2}+\sigma_{2}m.$ (5.10)

These conserved charges for the MB model were derived previously in [61] using
the canonical analysis (i.e. as the boundary terms in the Hamiltonian
formalism) instead of the covariant phase space. We can see here the
efficiency of the latter over the former. It is interesting to note that the
energy and angular momentum are in fact a linear combination of the mass and
spin of the black hole, even in the case of vanishing Lorentzian torsion $q$.

We can complete this analysis with a brief discussion of the thermodynamical
properties of the BTZ black hole in the MB model. In terms of the radii, the
temperature, mass, and spin are given by

$\displaystyle T=\frac{r_{+}^{2}-r_{-}^{2}}{2\pi\ell^{2}r_{+}},\qquad\qquad
m=\frac{r_{+}^{2}+r_{-}^{2}}{8G\ell^{2}},\qquad\qquad
j=\frac{r_{+}r_{-}}{4G\ell}.$ (5.11)

With the angular velocity of the outer horizon given by
$\Omega=N_{\varphi}(r_{+})$, the first law
$\mathrm{d}E=T\mathrm{d}S+\Omega\mathrm{d}L$ gives an entropy [67]

$\displaystyle
S=8\sigma_{1}\pi^{2}r_{+}-4\sigma_{2}\pi^{2}\left(qr_{+}+\frac{2r_{-}}{\ell}\right).$
(5.12)

Carefully keeping track of the conventions to match the first order action
with the Einstein–Hilbert one, as in (A.21), shows that we need
$\sigma_{1}=(16\pi G)^{-1}$. The first term then indeed agrees with the usual
result $S=\text{perimeter}/4G$, but is modified by $\sigma_{2}$ even in the
absence of Lorentzian torsion $q$.

###  AdS spacetimes and flat limit

Having studied the fixed BTZ black hole solution, we can now study asymptotic
symmetries for a family of spacetimes. We would like to do so in a way which
enables to study both the AdS and flat cases by taking a flat limit. For this,
we follow [83] and choose Bondi null coordinates $(u,r,\varphi)$ to write the
line element

$\displaystyle\mathrm{d}s^{2}=\left(\mathscr{M}(u,\varphi)-\frac{r^{2}}{\ell^{2}}\right)\mathrm{d}u^{2}-2\mathrm{d}u\,\mathrm{d}r+\mathscr{N}(u,\varphi)\mathrm{d}u\,\mathrm{d}\varphi+r^{2}\mathrm{d}\varphi^{2},$
(5.13)

where $\ell^{2}\partial_{u}\mathscr{M}=\partial_{\varphi}\mathscr{N}$ and
$\partial_{u}\mathscr{N}=\partial_{\varphi}\mathscr{M}$. This is the most
general solution to the Einstein field equations
$R_{\mu\nu}=-\ell^{-2}g_{\mu\nu}$. With the internal metric

$\displaystyle\eta_{ij}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix},$
(5.14)

a compatible set of triad and connection components can be chosen in the form

$\displaystyle
e^{i}_{\mu}=\begin{pmatrix}\big{(}\mathscr{M}-r^{2}/\ell^{2}\big{)}/2&1&0\\\
-1&0&0\\\
\mathscr{N}/2&0&r\end{pmatrix},\qquad\qquad\omega^{i}_{\mu}=\begin{pmatrix}\mathscr{N}/(2\ell^{2})&0&r/\ell^{2}\\\
0&0&0\\\
\big{(}\mathscr{M}-r^{2}/\ell^{2}\big{)}/2&1&0\end{pmatrix}-\frac{q}{2}e^{i}_{\mu}.$
(5.15)

The on-shell asymptotic Killing vectors are
$\xi^{\mu}=(\xi^{u},\xi^{r},\xi^{\varphi})$ with

$\displaystyle\xi^{u}=f,\qquad\qquad\xi^{r}=\partial^{2}_{\varphi}f-r\partial_{\varphi}g-\mathscr{N}\frac{\partial_{\varphi}f}{2r},\qquad\qquad\xi^{\varphi}=g-\frac{\partial_{\varphi}f}{r},$
(5.16)

and the conditions $\partial_{r}f=0=\partial_{r}g$,
$\ell^{2}\partial_{u}g=\partial_{\varphi}f$, and
$\partial_{u}f=\partial_{\varphi}g$. Note that this expansion of the vector
field is exact, and stops at next-to-next-to-leading order.

With the fields given above, the only non-vanishing component of the
symplectic potential (3.1) is $\theta_{\varphi
u}=\sigma_{1}\delta\mathscr{M}$. This is the component along the time-like
boundary $\Delta$, so we need a boundary condition. This is again given by
$(\theta+\delta b)\big{|}_{\Delta}=0$ with $b=\sigma_{1}(e\wedge\omega)$. With
this we then have

$\displaystyle\xi\lrcorner\,(e\wedge\omega)\big{|}_{S}=\xi^{u}\left(\mathscr{M}-\frac{2r^{2}}{\ell^{2}}\right)-\xi^{r},$
(5.17)

and the full diffeomorphism charge (5.7) is found to be

$\displaystyle\mathcal{D}(\xi)=\oint_{S}f\left((2\sigma_{1}-q\sigma_{2})\frac{\mathscr{M}}{2}+\frac{\sigma_{2}}{\ell^{2}}\mathscr{N}\right)+g\left((2\sigma_{1}-q\sigma_{2})\frac{\mathscr{N}}{2}+\sigma_{2}\mathscr{M}\right)+\mathcal{O}(r^{-1}).$
(5.18)

The subleading term in this charge is exact and of order $r^{-1}$. This is to
be contrasted with the corresponding charges computed when
$\sigma_{2}=\sigma_{3}=0$ in the metric formulation, in which case the $r$
dependency drops completely and the charges can then be defined at any finite
distance $r$ (see e.g. equation (2.13) of [85]). This latter case is an
example of so-called symplectic symmetries [85, 86, 92]. This slight
discrepancy between (5.18) and the corresponding metric result (in the case
$\sigma_{2}=\sigma_{3}=0$), which prevents (5.16) from being interpreted as
symplectic symmetries (although they remain of course asymptotic ones), can be
corrected if we replace the Lie derivative on the first order variables by the
Kosmann derivative. The use of this derivative in first order tetrad gravity
was advocated in [93, 94] and studied in more details in [95, 96, 97]. It
amounts to correcting the ordinary Lie derivative by a field-dependent Lorentz
transformation. We explain how this is realized in appendix D, where we also
illustrate the same result in Fefferman–Graham gauge instead of the Bondi
gauge used here.

We now come back to asymptotic symmetries, and focus on the charges (5.18) in
the limit $r\to\infty$. We are interested in their Poisson bracket. For this,
we need to know how the functions $\mathscr{M}$ and $\mathscr{N}$ transform.
This information can be found by looking at
$\delta^{\text{d}}_{\xi}g_{\mu\nu}=\pounds_{\xi}g_{\mu\nu}$. Denoting the
angular derivative $\partial_{\varphi}$ by a prime, we find

$\displaystyle\delta^{\text{d}}_{\xi}\mathscr{M}$
$\displaystyle=g\mathscr{M}^{\prime}+2\mathscr{M}g^{\prime}-2g^{\prime\prime\prime}+\frac{1}{\ell^{2}}\big{(}2\mathscr{N}f^{\prime}+f\mathscr{N}^{\prime}\big{)},$
(5.19a) $\displaystyle\delta^{\text{d}}_{\xi}\mathscr{N}$
$\displaystyle=f\mathscr{M}^{\prime}+2\mathscr{M}f^{\prime}-2f^{\prime\prime\prime}+2\mathscr{N}g^{\prime}+g\mathscr{N}^{\prime}.$
(5.19b)

To compute the algebra, let us denote the null and angular components of the
diffeomorphism charge by

$\displaystyle\mathcal{E}(f)\coloneqq\oint_{S}f\left((2\sigma_{1}-q\sigma_{2})\frac{\mathscr{M}}{2}+\frac{\sigma_{2}}{\ell^{2}}\mathscr{N}\right),\qquad\qquad\mathcal{L}(g)\coloneqq\oint_{S}g\left((2\sigma_{1}-q\sigma_{2})\frac{\mathscr{N}}{2}+\sigma_{2}\mathscr{M}\right).$
(5.20)

Note that with the identifications $\mathscr{M}=8Gm$ and $\mathscr{N}=-8Gj$
this agrees with the BTZ charges $E$ and $L$ which we have computed in (5.10),
with the factor of $2\pi$ coming from the angular integral. With these charges
we then find the algebra

$\displaystyle\big{\\{}\mathcal{L}(g),\mathcal{E}(f)\big{\\}}$
$\displaystyle=-\mathcal{E}([g,f])+(2\sigma_{1}-q\sigma_{2})\oint_{S}gf^{\prime\prime\prime},$
(5.21a) $\displaystyle\big{\\{}\mathcal{L}(g_{1}),\mathcal{L}(g_{2})\big{\\}}$
$\displaystyle=-\mathcal{L}([g_{1},g_{2}])+2\sigma_{2}\oint_{S}g_{1}g_{2}^{\prime\prime\prime},$
(5.21b) $\displaystyle\big{\\{}\mathcal{E}(f_{1}),\mathcal{E}(f_{2})\big{\\}}$
$\displaystyle=-\frac{1}{\ell^{2}}\left(\mathcal{L}([f_{1},f_{2}])-2\sigma_{2}\oint_{S}f_{1}f_{2}^{\prime\prime\prime}\right),$
(5.21c)

where $[f,g]=fg^{\prime}-f^{\prime}g$. The flat limit $\ell\to\infty$ of this
algebra is well-defined. It reproduces as expected a centrally-extended BMS3
algebra, where the central charges are functions of the couplings of the MB
model. Note that, up to signs, they are the same as in the algebra (3.23). For
a finite cosmological constant, the algebra is most easily understood in
Fourier modes, in terms of which it becomes

$\displaystyle\big{\\{}\mathcal{L}_{n},\mathcal{E}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{E}_{n+m}+(2\sigma_{1}-q\sigma_{2})\mathrm{i}m^{3}\delta_{n+m,0},$
(5.22a) $\displaystyle\big{\\{}\mathcal{L}_{n},\mathcal{L}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{L}_{n+m}+2\sigma_{2}\mathrm{i}m^{3}\delta_{n+m,0},$
(5.22b) $\displaystyle\big{\\{}\mathcal{E}_{n},\mathcal{E}_{m}\big{\\}}$
$\displaystyle=\frac{1}{\ell^{2}}\big{(}\mathrm{i}(m-n)\mathcal{L}_{n+m}+2\sigma_{2}\mathrm{i}m^{3}\delta_{n+m,0}\big{)}.$
(5.22c)

Introducing

$\displaystyle\mathcal{D}^{\pm}_{n}\coloneqq\frac{1}{2}\big{(}\ell\mathcal{E}_{\pm
n}\pm\mathcal{L}_{\pm n}\big{)},$ (5.23)

we obtain

$\displaystyle\big{\\{}\mathcal{D}^{\pm}_{n},\mathcal{D}^{\pm}_{m}\big{\\}}$
$\displaystyle=\mathrm{i}(m-n)\mathcal{D}^{\pm}_{n+m}+\frac{c^{\pm}}{12}\mathrm{i}m^{3}\delta_{m+n,0},$
(5.24a)
$\displaystyle\big{\\{}\mathcal{D}^{\pm}_{n},\mathcal{D}^{\mp}_{m}\big{\\}}$
$\displaystyle=0,$ (5.24b)

which is a double Virasoro algebra with central charges
$c^{\pm}=6\big{(}\ell(2\sigma_{1}-q\sigma_{2})\pm 2\sigma_{2}\big{)}$
depending once again non-trivial on the couplings of the MB model.

These central charges can be used to compute the entropy using the Cardy
formula [62, 65]. For this, let us first notice that in appendix C.2 we have
defined the Fourier representation with angular integrals on
$\varphi\in[0,1]$. We therefore first restore the factor of $2\pi$ in the
central charges and introduce

$\displaystyle c^{\pm}=(2\pi)6\big{(}\ell(2\sigma_{1}-q\sigma_{2})\pm
2\sigma_{2}\big{)},\qquad\qquad D^{\pm}=\frac{1}{2}(\ell E\pm L),$ (5.25)

where $D^{\pm}$ is the zero-mode of (5.23) defined in terms of (5.10). With
this the Cardy formula gives

$\displaystyle
S=2\pi\left(\sqrt{\frac{c^{+}D^{+}}{6}}+\sqrt{\frac{c^{-}D^{-}}{6}}\right)=8\sigma_{1}\pi^{2}r_{+}-4\sigma_{2}\pi^{2}\left(qr_{+}+\frac{2r_{-}}{\ell}\right),$
(5.26)

which matches the result derived from the first law.

In this section we have studied the asymptotic diffeomorphism charges for AdS
spacetimes in Bondi gauge, and shown how the couplings of the MB model affect
the central charges. The asymptotic algebras have two factors (i.e. the two
Virasoros or the rotation/translation parts of BMS3) because the asymptotic
Killing vector fields are along $u$ and $\varphi$, and are centrally-extended
because these vectors fields are not tangent. At finite distance and for
tangent vector fields along $\varphi$ only, we have seen in (3.28) and (3.30)
that the BMS and double Virasoro algebras exist because of the availability of
the dual charges. We have also seen in section 4.4 and with the twisted
Sugawara construction that the algebra of dual charges also becomes centrally-
extended in the case of non-tangent vector fields (although we have not
studied the central extensions in details). Together, these results and
observations point towards an important step for future work, namely the study
of the asymptotic dual charges. Were these charges to be non-trivial, they
could extend the asymptotic symmetry algebra (5.21) by potentially doubling it
and bringing new central extensions. This requires to first analyse whether
there exist spacetimes and vector fields for which $\mathcal{B}(\xi)$ and/or
$\mathcal{C}(\xi)$ can be defined to act non-trivially as asymptotic
symmetries. We keep the detailed investigation of this question for future
work.

##  Metric, teleparallel, and massive sectors

We finally come to the last part of this work121212This part is completely
independent from the previous ones., which concerns the second order metric
and teleparallel formulations of the MB model. The MB Lagrangian (2.1) is
first order in the connection and triad variables. As such, and as we have
seen, it leads to two independent first order equations of motion. They can be
read on (2.3), or rewritten in the elegant form (2.6) displaying the sources
of Lorentzian curvature and torsion. In such a theory, two natural questions
then arise:

  1. 1.

How to combine the two first order equations of motion into a single second
order equation? What is the physical interpretation of this latter?

  2. 2.

Can we inject part or all of the equations of motion back into the initial
Lagrangian? What is the physical interpretation of the newly obtained
Lagrangian?

The goal of this section is to study these questions. Similar work was done in
[79] in the case of the three-dimensional Hilbert–Palatini Lagrangian, and
here we extend the results to the MB model with arbitrary couplings. As we are
about to see, this is the source of new surprises, and in particular of a new
description of three-dimensional massive gravity.

###  Second order equations of motion

We start by decomposing the connection as $\omega=\varpi+k$, where $\varpi$ is
an arbitrary reference connection whose curvature will be denoted by $R$, and
$k$ an arbitrary tensor131313As a remark, we note that picking a non-dynamical
internal vector $n$ (i.e. such that $\delta n=0$) and choosing $k=[e,n]$ leads
to the Iwasawa decomposition which was used in [29] to explain the classical
origin of quantum group symmetry in three-dimensional gravity. This study can
potentially be extended to the MB model, and it would be interesting to see
how its symmetries are deformed. In terms of this decomposed connection, the
MB Lagrangian becomes

$\displaystyle L_{\text{MB}}[e,\varpi+k]$
$\displaystyle=L_{\text{MB}}[e,\varpi]+2\sigma_{2}k\wedge
R+(2\sigma_{1}e+\sigma_{2}k)\wedge\mathrm{d}_{\varpi}k$ (6.1)
$\displaystyle\phantom{\ =}+\sigma_{1}e\wedge[k\wedge
k]+\sigma_{3}k\wedge[e\wedge e]+\frac{\sigma_{2}}{3}k\wedge[k\wedge
k]+\sigma_{2}\mathrm{d}(k\wedge\varpi).$ (6.2)

Notice that choosing $k=\kappa e$ gives again $L_{\text{MB}}[e,\varpi]$, but
with new coupling constants. This is the “stability” property of the MB
Lagrangian. The equations of motion obtained by varying the above decomposed
Lagrangian with respect to $e$ and $k$ respectively are

$\displaystyle
2\sigma_{1}R+2\sigma_{3}\mathrm{d}_{\varpi}e+2\sigma_{1}\mathrm{d}_{\varpi}k+\sigma_{0}[e\wedge
e]+\sigma_{1}[k\wedge k]+2\sigma_{3}[k\wedge e]\approx 0,$ (6.3a)
$\displaystyle
2\sigma_{2}R+2\sigma_{1}\mathrm{d}_{\varpi}e+2\sigma_{2}\mathrm{d}_{\varpi}k+\sigma_{3}[e\wedge
e]+\sigma_{2}[k\wedge k]+2\sigma_{1}[k\wedge e]\approx 0.$ (6.3b)

Notice that varying with respect to $\varpi$ leads of course to the same
equation of motion as varying along $k$. The equations of motion can be
rewritten in the form

$\displaystyle 2R+2\mathrm{d}_{\varpi}k+[k\wedge k]+p[e\wedge e]\approx
0,\qquad\qquad 2\mathrm{d}_{\varpi}e+2[k\wedge e]+q[e\wedge e]\approx 0,$
(6.4)

which is evidently the same as plugging the decomposition $\omega=\varpi+k$
into (2.6).

Although the initial equations of motion were quite “symmetrical”, we see that
in (6.4) they are not exactly on the same footing anymore, as it is the second
one, i.e. the torsion equation, which can easily be solved for $k$. In order
to do so, let us consider the relation $[k\wedge e]=M$, where $M$ is an
arbitrary Lie algebra-valued 2-form. In components, this is

$\displaystyle{\varepsilon^{i}}_{jk}\big{(}k^{j}_{\mu}e^{k}_{\nu}-k^{j}_{\nu}e^{k}_{\mu}\big{)}=M^{i}_{\mu\nu}.$
(6.5)

When the triad is invertible, this relation can be inverted to find

$\displaystyle
k^{i}_{\mu}=\left(\delta^{\alpha}_{\mu}\delta^{i}_{j}-\frac{1}{4}e^{i}_{\mu}\hat{e}^{\alpha}_{j}\right){\varepsilon^{j}}_{kl}M^{k}_{\alpha\beta}\hat{e}^{\beta
l}=\frac{1}{2}{\varepsilon^{i}}_{jl}\left(\delta^{\alpha}_{\mu}\delta^{j}_{k}+\frac{1}{2}e_{\mu
k}\hat{e}^{\alpha j}\right)M^{k}_{\alpha\beta}\hat{e}^{\beta l},$ (6.6)

where $\hat{e}$ denotes the inverse triad. It will be convenient to write this
in index-free notation with a linear operator such that $k=\mathbb{E}(M)$.
With this compact notation at hand, solving the second equation of motion in
(6.4) tells us that

$\displaystyle
k\approx-\frac{q}{2}e-\mathbb{T},\qquad\qquad\mathbb{T}\coloneqq\mathbb{E}(\mathrm{d}_{\varpi}e),\qquad\qquad[\mathbb{T}\wedge
e]=\mathrm{d}_{\varpi}e,$ (6.7)

where we have defined the tensor $\mathbb{T}$ to be the image under the above
map of the torsion of the reference connection. We can then inject this in the
first equation of motion in (6.4) to find

$\displaystyle
2R-2\mathrm{d}_{\varpi}\mathbb{T}+[\mathbb{T}\wedge\mathbb{T}]\approx\Lambda[e\wedge
e].$ (6.8)

This is the second order form of the equations of motion of the MB model. It
encodes a duality, which is that between the metric and teleparallel
formulations of gravity. When the reference connection is the torsionless
Levi–Civita connection $\varpi=\Gamma$, the tensor $k$ is the contorsion, we
have $\mathrm{d}_{\varpi}e=\mathrm{d}_{\Gamma}e=0=\mathbb{T}$, and we recover
the metric equations of motion $R_{\mu\nu}\approx 2\Lambda g_{\mu\nu}$. When
the reference connection is a flat connection,
$\varpi=\Upsilon=\vartheta\mathrm{d}\vartheta^{-1}$, we have $R=0$ and the
quantity $\mathrm{d}_{\varpi}e=\mathrm{d}_{\Upsilon}e\neq 0$ is the
Weitzenböck torsion. We recover in this case the teleparallel equations of
motion with a cosmological constant [77]. We can see that in both cases the
only parameter is $\Lambda$.

###  Second and higher order Lagrangians

Now that we have described the second order equations of motion of the MB
model, we can ask which second order Lagrangian (if any) they follow from.
This can be investigated by injecting the solution (6.7) into the original
Lagrangian. In the usual case of Hilbert–Palatini gravity (i.e. for
$\sigma_{2}=\sigma_{3}=0$), choosing the torsionless Levi–Civita connection as
the reference connection leads to the second order Einstein–Hilbert action in
triad variables. Choosing instead a flat reference connection leads to the
teleparallel Lagrangian as in [79]. As we are about to see, such a
manipulation in the MB model leads to a Lagrangian with drastically different
properties from the initial first order one.

Let us consider again an arbitrary decomposition of the connection as
$\omega=\varpi+k$, and inject the solution (6.7) of the torsion equation of
motion into the decomposed Lagrangian (6.1). A lengthy rewriting making
repeated use of $[\mathbb{T}\wedge e]=\mathrm{d}_{\varpi}e$ shows that this
leads to the “half on-shell” Lagrangian

$\displaystyle L_{\text{h}}[e,\varpi]$
$\displaystyle\approx\frac{1}{2}(2\sigma_{1}-q\sigma_{2})\Big{(}L_{\text{HP}}[e,\varpi]-\Lambda
L_{\text{V}}[e]-e\wedge\mathrm{d}_{\varpi}\mathbb{T}\Big{)}+\sigma_{2}L_{\text{CS}}[\varpi]$
(6.9) $\displaystyle\phantom{\ =}-2\sigma_{2}\mathbb{T}\wedge
R+\sigma_{2}\mathbb{T}\wedge\left(\mathrm{d}_{\varpi}\mathbb{T}-\frac{1}{3}[\mathbb{T}\wedge\mathbb{T}]\right)+\mathrm{d}(\sigma_{1}e\wedge\mathbb{T}+\sigma_{2}k\wedge\varpi),$
(6.10)

where for simplicity we have kept $k$ in the boundary term. This result is the
general form of the MB Lagrangian on-shell of the torsion equation, for an
arbitrary reference connection $\varpi$. Several important things can be
observed.

  1. 1.

Let us first consider the Riemannian metric geometry where $\varpi=\Gamma$ and
$\mathbb{T}=0$. In this case, up to a boundary term we obtain the Lagrangian

$\displaystyle
L_{\text{TMG}}[e]=\frac{1}{2}(2\sigma_{1}-q\sigma_{2})\Big{(}L_{\text{HP}}[e]-\Lambda
L_{\text{V}}[e]\Big{)}+\sigma_{2}L_{\text{CS}}[\Gamma],$ (6.11)

which depends only on $e$ since the reference connection is the torsionless
connection. Because of the presence of the CS term for $\Gamma(e)$, this
Lagrangian is actually third order in derivatives of $e$. This is responsible
for the appearance of a massive propagating mode. The equations of motion
contain the Cotton tensor, and describe the propagation of a massive graviton.
As already pointed out in [57, 56], this result shows that the Riemannian
metric sector of the MB model is a theory of massive gravity, known as
topologically massive gravity (TMG) [30, 31]. The massless sector is obtained
only for $\sigma_{2}=0$. This shows that the Riemannian metric equation of
motion $2R=\Lambda[e\wedge e]$ in (6.8) with an arbitrary $\Lambda$ cannot be
obtained from the Lagrangian (6.11). It can only be obtained when
$\sigma_{2}=0$.

  2. 2.

In the teleparallel sector of the theory, i.e. when the reference connection
is flat (let us choose the representative $\varpi=0$), up to a boundary term
we obtain the Lagrangian

$\displaystyle
L_{\text{TTMG}}[e]=\frac{1}{2}(2\sigma_{1}-q\sigma_{2})\Big{(}e\wedge\mathrm{d}\tilde{\mathbb{T}}-\Lambda
L_{\text{V}}[e]\Big{)}+\sigma_{2}L_{\text{CS}}[\tilde{\mathbb{T}}],$ (6.12)

where we have denoted $\tilde{\mathbb{T}}=-\mathbb{T}$. When $\sigma_{2}=0$,
this is the Lagrangian for teleparallel gravity with a cosmological constant.
When $\sigma_{2}\neq 0$ however, we see the appearance of a Chern–Simons term
for $\tilde{\mathbb{T}}$, and the Lagrangian (6.12) is exactly the
teleparallel dual of (6.11). In particular, because of the CS term it is also
of third order in derivatives of $e$. It is therefore natural to conjecture
that this describes a teleparallel version of topologically massive gravity,
which is why we have chosen to call this Lagrangian TTMG. The detailed study
of this Lagrangian is deferred to future work.

  3. 3.

When $\sigma_{2}=0$, up to a boundary term and an integration by parts the
teleparallel Lagrangian in (6.9) can be written as
$\mathrm{d}_{\varpi}e\wedge\mathbb{T}$. Since
$\mathbb{T}=\mathbb{E}(\mathrm{d}_{\varpi}e)$, this Lagrangian can in turn be
written in the suggestive form
$\mathrm{d}_{\varpi}e\wedge\star\mathrm{d}_{\varpi}e$, where the non-standard
$\star$ operation is defined by the action of the linear operator
$\mathbb{E}$. This is the Yang–Mills form of the teleparallel Lagrangian [77].

  4. 4.

Finally, it is interesting to note that the couplings of (6.9) actually
correspond to the central charges of (3.23) and (5.21).

We therefore see that inserting half of the equations of motion of the MB
model back into the Lagrangian has a drastic effect: it leads to a third order
Lagrangian describing a theory of three-dimensional massive gravity. This
result was known for the Riemannian metric sector of the theory, described by
(6.11), and we have extended it to the dual teleparallel sector (6.12), which
describes the TTMG dual of TMG.

There is a vast literature on three-dimensional massive gravity [30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Among all the available models,
that of [42] resembles a bit the MB model. Indeed, the simple theory of
massive gravity introduced in [42] is a first order Lagrangian built with the
connection and the triad, and contains all possible terms which can be written
in terms of these two fields, much like in the MB model. The only difference
with the MB model is that the Lagrangian of [42] breaks internal Lorentz
invariance, and this is the mechanism responsible for the appearance of a
massive propagating mode. At the level of the MB Lagrangian (2.1), a two-field
first order Lagrangian for massive gravity in the family of [42] can be
obtained by detuning any covariant derivative and replacing e.g.
$\mathrm{d}_{\omega}e\mapsto\mathrm{d}e+\mu[\omega\wedge e]$.

As a last manipulation, we can now insert the second order equation of motion
(6.8) into the half on-shell Lagrangian (6.9). To do so, notice that taking a
wedge product of (6.8) with $e$ implies that

$\displaystyle L_{\text{HP}}[e,\varpi]$ $\displaystyle\approx 3\Lambda
L_{\text{V}}[e]+e\wedge[\mathbb{T}\wedge\mathbb{T}]-2\mathrm{d}(e\wedge\mathbb{T})$
(6.13) $\displaystyle=3\Lambda
L_{\text{V}}[e]+e\wedge\mathrm{d}_{\varpi}\mathbb{T}-\mathrm{d}(e\wedge\mathbb{T}),$
(6.14)

while a wedge product with $\mathbb{T}$ gives

$\displaystyle 2\mathbb{T}\wedge R\approx
2\mathbb{T}\wedge\mathrm{d}_{\varpi}\mathbb{T}-\mathbb{T}\wedge[\mathbb{T}\wedge\mathbb{T}]+\Lambda\mathbb{T}\wedge[e\wedge
e].$ (6.15)

Using these two relations we can go once again on-shell in (6.9) to obtain the
“full on-shell” Lagrangian

$\displaystyle L_{\text{f}}[e,\varpi]$
$\displaystyle\approx(2\sigma_{1}-q\sigma_{2})\Lambda
L_{\text{V}}[e]+\sigma_{2}L_{\text{CS}}[\varpi]-\sigma_{2}\mathbb{T}\wedge\left(\mathrm{d}_{\varpi}\mathbb{T}-\frac{2}{3}[\mathbb{T}\wedge\mathbb{T}]+\Lambda[e\wedge
e]\right)+\sigma_{2}\mathrm{d}(k\wedge\varpi).$ (6.16)

Once again, for $\sigma_{2}=0$ we recover the expected result, namely that the
full on-shell Lagrangian is non-dynamical and just proportional to the volume
term. For $\sigma_{2}\neq 0$ however we obtain a dynamical Lagrangian with an
independent Chern–Simons term for both the metric and teleparallel sectors.

##  Perspectives

We have shown that the phase space of three-dimensional gravity contains two
layers of rich dualities, which are elegantly revealed when considering the
most general first order Lagrangian depending on the connection and triad
variables. This Lagrangian is that of the Mielke–Baekler model (2.1), and it
describes a theory with Lorentzian curvature and torsion measured by the
parameters $p$ and $q$ respectively. This model therefore has two natural
subspaces, corresponding to $(p\neq 0,q=0)$ and $(p=0,q\neq 0)$, which
represents a first layer of duality between Lorentzian curvature and torsion.
In each of these sectors, we have exhibited a second layer of duality encoded
in the existence of new integrable charges, called the dual diffeomorphisms,
which arise as particular field-dependent gauge transformations, or
equivalently as quadratics in the Sugawara construction. We have first studied
the algebra of these dual diffeomorphisms together with the usual
diffeomorphisms in the generic case of finite boundaries with no boundary
conditions, in which case integrability requires tangent vector fields. In the
case $(p\neq 0,q=0)$, the resulting centreless algebra consists in two
commuting copies of Virasoro, and is defined even for a positive cosmological
constant. In the flat limit it reduces to BMS3. In the case $(p=0,q\neq 0)$
the two Virasoro algebras are found to be in semi-direct sum, and also reduce
to BMS3 in the flat limit. To the best of our knowledge, it is the first time
that such algebras, which are typically thought to be properties of asymptotic
boundaries, are seen to appear at finite distance for arbitrary subregions and
without boundary conditions (nor the use of symplectic symmetries [85, 86]).

This result is particularly interesting in light of recent work on asymptotic
dual charges in four-dimensional gravity [98, 99, 100, 101, 102, 97] (see also
[103, 104] for a soft theorem interpretation of these dual charges). It begs
the question of whether the dual diffeomorphisms $\mathcal{B}$ and
$\mathcal{C}$ which we have identified act non-trivially at infinity, and
whether they can be expressed in terms of second order metric variables.
Important future work will now be to understand these dual charges in the case
of asymptotic boundaries, where they could further enlarge the asymptotic
symmetry algebra. In particular, it would be interesting to study the dual
charges using the most general flat and AdS3 boundary conditions presented in
[54, 55].

Coming back to finite distance and arbitrary subregions, this enlargement of
the boundary symmetry algebra could have important consequences for the
quantization of gravity. The program initiated in [70, 72, 71] aims at
constructing quantum gravity from the quantization of the quasi-local algebra
of charges, and from this perspective it is important to understand the nature
and the physical significance of the largest available symmetry algebra. In
terms of representation theory, a related question is that of the relationship
between representations of the $\mathfrak{g}_{\Lambda}$ current algebras
(3.21) or (3.23) and the algebra of diffeomorphisms. In principle, the current
algebra seen as a universal enveloping algebra contains all possible
polynomials in the currents, and therefore includes the diffeomorphisms and
their dual. Understanding this at the level of representation theory is
however still an open issue, especially given the fact that dual
diffeomorphisms do not seem to have been discussed before.

In summary, by a detailed analysis of the covariant phase space and the
boundary symmetries of the MB model, we have shown that if we consider the
stable quadratics in the enveloping algebra (3.21), at finite distance we are
automatically lead to the double Virasoro algebras (3.30) and (3.41) (and
their flat BMS3 limit) depending on which sector of the duality between
$(p\neq 0,q=0)$ and $(p=0,q\neq 0)$ we consider. It is noteworthy that (3.30)
is a direct sum, while (3.41) is a semi-direct one, and that it is therefore
(3.30) which is the finite distance counterpart of the asymptotic result
(5.21) (minus the central extensions).

After having presented a detailed study of the dual charges at finite
distance, we have discussed in section 6 the second order equations of motion
arising from the MB model as well as the on-shell Lagrangians. This represents
the second order version of the duality between Lorentzian curvature and
torsion, which then becomes a duality between the Riemannian metric and
teleparallel formulations of gravity. In the metric formulation the dynamics
of general relativity is encoded in the curvature and the geometry is
torsionless, while in the teleparallel formulation the geometry is flat and
the dynamics is encoded in the torsion. The MB Lagrangian has the
particularity that, when rewritten on-shell, it actually becomes third order
and describes a propagating massive graviton. While this was known in the
metric case [57, 56], we have extended the result to the teleparallel dual.
This has revealed a new candidate theory of massive gravity, described by the
Lagrangian (6.12) and called TTMG for teleparallel topologically massive
gravity.

We close this article with a list of possible directions for future work. It
would be particularly interesting to pursue these directions in the case of
the MB model, because as we have seen here this latter reveals dual structures
which are hard (if not impossible) to guess when working with the usual
Hilbert–Palatini or Chern–Simons Lagrangians for three-dimensional gravity.

  1. 1.

Asymptotic dual charges. As already explained above, it is now important to
study whether the dual charges presented in this work have a realization at
infinity and potential analogues in second order metric variables.

  2. 2.

Holography. Boundary symmetries in three-dimensional gravity take a
particularly important physical meaning when studying holography and the
boundary CFT [105, 106]. Three-dimensional gravity being exactly soluble, one
can expect it to always be dual to a CFT or an integrable model, and to admit
a quasi-local holographic description as constructed in [18, 19, 20, 21, 22].
Some aspects of holography in the MB model were studied previously in [64]. It