aainstitutetext: University of Wrocław, Faculty of Physics and Astronomy, pl.
M. Borna 9, 50-204 Wrocław, Polandbbinstitutetext: National Centre for Nuclear
Research, Pasteura 7, 02-093 Warsaw, Poland

# A tale of two theories of gravity in asymptotically Anti-de Sitter spacetime

Remigiusz Durka<EMAIL_ADDRESS>a,b Jerzy Kowalski-Glikman
<EMAIL_ADDRESS>
###### Abstract

We consider two BF formulations of the theory of gravity with a negative
cosmological constant, of Plebanski and of MacDowell–Mansouri. Both give the
standard Einstein equations in the bulk but differ in expressions of edge
charges. We compute the asymptotic charges explicitly in both theories for
AdS–Schwarzschild, AdS–Kerr, and AdS–Taub–NUT solutions. We find that while in
the case of the Plebanski theory the charges are divergent, they are finite
for MacDowell–Mansouri theory. Furthermore, we show that in the case of the
arbitrary asymptotically AdS spacetimes, MacDowell–Mansouri asymptotic
charges, action, and symplectic form are all finite. Therefore
MacDowell–Mansouri theory of gravity in asymptotically AdS spaces does not
need any counterterms.

###### Keywords:

gravity, corner charges, BF theory, Plebanski theory, MacDowell–Mansouri
theory, asymptotically AdS spaces

##  1 Introduction

In recent years we have witnessed an important shift in the front line of
gravity research, both classical and quantum. Rather than investigate the bulk
structure of gravity, described by solutions of Einstein (Wheeler–De Witt)
equations, motivated by the AdS/CFT correspondence and discoveries of
remarkable relations between asymptotic symmetries and soft theorems more and
more interest is attracted by boundaries, finite asymptotic of spacetime
regions.

These boundaries are commonly called corners and it was claimed that the
Noether charges Kijowski:1976ze ; Lee:1990nz ; Iyer:1994ys ; Brown:1992br ;
Crnkovic:1987tz ; Wald:1999wa ; Freidel:2015gpa ; Frodden:2017qwh ;
Frodden:2019ylc ; DePaoli:2018erh ; Oliveri:2019gvm  associated with them, and
their representations might carry relevant information, essential for the
program of quantizing gravity Donnelly:2016auv , Donnelly:2020xgu .

In the recent series of papers Freidel:2020xyx ; Freidel:2020svx ;
Freidel:2020ayo , using the covariant phase space technique Crnkovic:1986ex ;
Crnkovic:1987tz ; Julia:2002df ; Harlow:2019yfa , the authors analyze the
corner charges structure in the case of two formulations of gravity, the
traditional metric one and the Plebanski tetrad formulation expressed as a BF
theory Plebanski:1977zz . Following these investigations, in our paper
Durka:2021ftc  we discussed the corner charges of gravity in the case of yet
another formulation of gravity known as a constrained BF theory, proposed in
Smolin:2003qu , Freidel:2005ak  and based on the MacDowell and Mansouri
approach MacDowell:1977jt . These two theories give rise to the same bulk
Einstein equations, but their Lagrangians differ by topological terms, which
makes the corner charges very different. It is the aim of this paper to
investigate these differences in detail, in the case of asymptotically Anti-de
Sitter (AAdS) spacetime.

In the following section, we start our investigations of these two theories by
writing their actions in the case of a non-vanishing negative cosmological
constant. To this end the addition of the cosmological term is needed in the
case of Plebanski theory, whereas the MacDowell–Mansouri model has a
cosmological constant built-in. We normalize both theories so that their
Lagrangians, up to the topological terms, agree with the Holst Lagrangian.

In Section 3 we recall the derivation of conserved corner charges in both
theories. In the following Section 4, we use these obtained expressions to
compute the asymptotic charges of AdS–Schwarzschild, AdS–Kerr, and
AdS–Taub–NUT solutions. We find that these charges diverge for Plebanski
theory, but they are finite in the case of MacDowell–Mansouri model.

In Section 5 we further investigate the MacDowell–Mansouri theory and we show
that for the general asymptotic behavior of the metric all the charges of this
theory are finite. We then show that the MacDowell–Mansouri action is finite.
It follows that the symplectic structure is finite as well. This shows that,
contrary to the Plebanski one, this theory does not require any counterterms
at infinity Compere:2008us ; Anastasiou:2020zwc .

We finish the paper with short conclusions.

##  2 The two BF theories

In this section, we define two BF theories of gravity (for the extended
discussion see Freidel:2012np ) in the case of spacetime with a negative
cosmological constant and point out their similarities and differences. To
distinguish them, we shall call the one considered in Freidel:2020xyx ;
Freidel:2020svx ; Freidel:2020ayo  ‘Plebanski BF theory’, while the one
discussed in Durka:2021ftc  will be dubbed ‘MacDowell–Mansouri BF theory’.

###  2.1 Plebanski BF theory

The starting point here is the BF Lagrangian with the so-called ’cosmological
term’

$32\pi G\,\mathcal{L}_{Pl}=a\,B_{ij}\wedge
R^{ij}(\omega)-\frac{b}{2}\,B_{ij}\wedge B^{ij}\,.$ (1)

In this Lagrangian $B_{ij}$ is a two-form valued in the Lorentz algebra
$\mathsf{so}(3,1)$, $R^{ij}(\omega)$ is the curvature of Lorentz connection
$\omega^{ij}$

$R^{ij}(\omega)=d\omega^{ij}+\omega^{i}{}_{k}\wedge\omega^{kj}\,,$ (2)

and $a$, $b$ are parameters to be adjusted in a moment. The indices
$i,j,\ldots=0,\ldots,3$ are lowered and raised with the help of the Minkowski
metric $\eta_{ij}$. In the next step of the construction, we impose the
simplicity constraint, which expresses the two form $B$ in terms of the
tetrads, to wit

$B^{ij}=c\,e^{i}\wedge e^{j}+\epsilon^{ij}{}_{kl}e^{k}\wedge e^{l}\,.$ (3)

Substituting (3) to (1) we obtain

$32\pi G\,\mathcal{L}_{Pl}=a\,\epsilon_{ijkl}\,e^{i}\wedge e^{j}\wedge
R^{kl}+ac\,e^{i}\wedge e^{j}\wedge R_{ij}+{bc}\,\epsilon_{ijkl}e^{i}\wedge
e^{j}\wedge e^{k}\wedge e^{l}\,.$ (4)

Comparing this with Holst Lagrangian Holst:1995pc ,

$32\pi G\,\mathcal{L}_{Holst}=\epsilon_{ijkl}\,e^{i}\wedge e^{j}\wedge
R^{kl}+\frac{2}{\gamma}\,e^{i}\wedge e^{j}\wedge
R_{ij}-\frac{\Lambda}{6}\,\epsilon_{ijkl}e^{i}\wedge e^{j}\wedge e^{k}\wedge
e^{l}\,,$ (5)

we see that $a=1$, $b=\Lambda\gamma/12$, $c=2/\gamma$. The second term in the
action, known as the Holst term, is not topological but, as a result of
vanishing torsion, it does not contribute to the bulk Einstein vacuum field
equations. It influences, however, the symplectic form and, in turn, the form
and algebra of (asymptotic) conserved charges.

The variation of Plebanski action is

$32\pi G\delta S_{Pl}=\delta\int B_{ij}\wedge
R^{ij}(\omega)-\frac{\Lambda\gamma}{24}\,B_{ij}\wedge B^{ij}\,.$ (6)

Inserting the simplicity constraint (3) we see that the $\delta e$ field
equations are the standard Einstein equations, while the equations resulting
from the variation of Lorentz connection $\delta\omega$ become the torsion
vanishing condition. In the course of deriving this last equation, we have to
integrate by parts, so that there is a boundary contribution to the variation
of the action

$32\pi G\delta
S_{Pl}\approx\int_{\Sigma}B_{ij}\wedge\delta\omega^{ij}=\int_{\Sigma}\left(\frac{2}{\gamma}\,e^{i}\wedge
e^{j}+\epsilon^{ij}{}_{kl}e^{k}\wedge e^{l}\right)\wedge\delta\omega_{ij}\,,$
(7)

where the weak equality $\approx$ denotes equality on-shell. As explained in
details in Freidel:2020svx  the resulting symplectic form further decomposes
into the bulk $\Sigma$ and the corner $S=\partial\Sigma$ contributions.

###  2.2 MacDowell–Mansouri BF theory

In the case of MacDowell–Mansouri BF theory, the starting point is the BF
Lagrangian with the cosmological term for the Anti-de Sitter gauge group
appended by an additional deformation term that breaks the gauge symmetry down
to the local Lorentz one111The more general case when the last term in the
action depends on the background field ${\mathcal{v}}^{I}$ was considered in
Durka:2021ftc . It turns out that such generalization does not lead to any new
effects, and therefore, without any loss of generality we restrict ourselves
here to the case ${\mathcal{v}}^{I}=(0,0,0,0,-1)$.

$32\pi G\,\mathcal{L}_{MM}=B_{IJ}\wedge
F^{IJ}(A)-\frac{\beta}{2}\,B_{IJ}\wedge
B^{IJ}-\frac{\alpha}{4}\,\epsilon_{ijkl}\,B^{ij}\wedge B^{kl}\,,$ (8)

where $F^{IJ}(A)$ is the curvature of AdS connection $A^{IJ}$ and the indices
$I,J,\ldots$ run from 0 to 4, $I=(i,4)$. The Anti-de Sitter connection
$A^{IJ}$ decomposes into the Lorentz connection and the tetrad

$A^{ij}=\omega^{ij}\,,\quad A^{i4}=\frac{1}{\ell}\,e^{i}$ (9)

with the parameter $\ell$ introduced for dimensional reasons to be given an
interpretation in a moment. The $B$-field equations can be decomposed to give

$\displaystyle B^{ij}$
$\displaystyle=\frac{\alpha}{(\alpha^{2}+\beta^{2})}\left(\frac{1}{2}\frac{\beta}{\alpha}\,\delta^{ij}_{kl}-\frac{1}{2}\epsilon^{ij}{}_{kl}\right)F^{kl}\,,$
(10) $\displaystyle B^{i4}$ $\displaystyle=\frac{1}{\beta}\,F^{i4}\,,$ (11)

where
$\delta^{ij}_{kl}=\delta^{i}_{k}\delta^{j}_{l}-\delta^{i}_{l}\delta^{j}_{k}$
and the curvature $F^{IJ}=dA^{IJ}+A^{I}{}_{K}\wedge A^{KJ}$ components are

$\displaystyle F^{ij}$
$\displaystyle=R^{ij}(\omega)+\frac{1}{\ell^{2}}\,e^{i}\wedge e^{j}\,,$ (12)
$\displaystyle F^{i4}$ $\displaystyle=\frac{1}{\ell}T^{i}\,.$ (13)

In what follows, we will call $F^{ij}$ the AdS-curvature, to distinguish it
from the Lorentz curvature $R^{ij}(\omega)$. Substituting these equations to
the original Lagrangian and using the explicit expressions for the
parameters222In the formulas below, we make the change
$\gamma\mapsto-1/\gamma$ as compared to the standard formulation of MM BF
theory Freidel:2005ak ; Durka:2012wd ; Durka:2021ftc  to have the inverse
Barbero-Immirzi parameter as a coefficient of Holst term, in agreement with
most of the literature.

$\displaystyle\alpha=\frac{\Lambda}{3}\frac{\gamma^{2}}{1+{\gamma^{2}}}\,,\qquad{\beta}=-\frac{\alpha}{\gamma}\,,\qquad\Lambda=-\frac{3}{\ell^{2}}\,,$
(14)

we obtain

$\displaystyle 32\pi G\,\mathcal{L}_{MM}$
$\displaystyle=\epsilon_{ijkl}R^{ij}\wedge
e^{k}\wedge\,e^{l}-\frac{\Lambda}{6}\epsilon_{ijkl}e^{i}\,\wedge e^{j}\wedge
e^{k}\,\wedge e^{l}-{2}{\gamma}R_{ij}\wedge e^{i}\wedge e^{j}$
$\displaystyle-2\frac{1+\gamma^{2}}{\gamma}\,\left(T^{i}\wedge
T_{i}-R_{ij}\wedge e^{i}\wedge
e^{j}\right)-\frac{3}{\Lambda\gamma}R^{ij}\wedge
R_{ij}-\frac{3}{2\Lambda}\epsilon_{ijkl}R^{ij}\wedge R^{kl}\,.$ (15)

The first line in $\mathcal{L}_{MM}$ reproduces the Holst Lagrangian (5),
albeit with the inverse power of the Barbero-Immirzi parameter, while the
second line is a combination of Nieh-Yan, Pontryagin, and Euler forms. One
then combines the second term of the Nieh-Yan form with the Holst term in the
first line to finally get the right $2/\gamma$ coefficient.

The Lagrangian (15) can be conveniently rewritten as a sum of squares of
torsion and AdS curvature

$\displaystyle 32\pi G\,\mathcal{L}_{MM}$
$\displaystyle=-\frac{3}{2\Lambda}\left(\frac{1}{\gamma}\delta_{ijkl}+\epsilon_{ijkl}\right)F^{ij}\wedge
F^{kl}-2\frac{1+\gamma^{2}}{\gamma}\,T^{i}\wedge T_{i}\,.$ (16)

Let us pause here to comment on why the theories considered here can be, as it
is often done, called “constrained BF theories”. The Lagrangians of Plebanski
BF theory with a cosmological constant (6) and the MacDowell–Mansouri BF
theory (8) can be both written as a sum of the topological BF Lagrangian
accompanied by the Lagrange multiplier enforcing an appropriate constraint

$\displaystyle 32\pi G\,\mathcal{L}_{Pl}$ $\displaystyle\approx B_{ij}\wedge
R^{ij}-\frac{\Lambda\gamma}{24}\,B_{ij}\wedge
B^{ij}+\lambda_{ij}\left(B^{ij}-\left(\frac{1}{\gamma}\delta^{ij}_{kl}+\epsilon^{ij}{}_{kl}\right)e^{k}\wedge
e^{l}\right)$ (17) $\displaystyle 32\pi G\,\mathcal{L}_{MM}$
$\displaystyle\approx B_{IJ}\wedge
F^{IJ}+\lambda_{ij}\left(B^{ij}+\frac{3}{2\Lambda}\left(\frac{1}{\gamma}\delta^{ij}_{kl}+\epsilon^{ij}{}_{kl}\right)F^{kl}\right)+\lambda_{i}\left(B^{i4}-F^{i4}\right)\,.$
(18)

In spite of the fact that the theories are very different, it follows from
(17) and (18) that to derive the symplectic form and conserved charges
resulting from the first BF term in the Lagrangian, one can handle both
theories in an essentially the same way inserting (the same) bulk field
equations and (different) fields $B$ at the very end of the calculations.

Now that we have described the formulation of the two BF theories, let us turn
to investigate their further properties.

##  3 The conserved charges

In this section, we will describe the conserved surface charges associated
with local gauge symmetries. Both BF theories described in the preceding
section are invariant under local Lorentz transformations and diffeomorphisms,
and there are conserved charges associated with these symmetries. These
charges are defined by the formula

$\displaystyle-\delta{\cal H}_{s}=\delta_{s}\intprod\Omega\,,$ (19)

where ${\cal H}_{s}$ is the charge associated with symmetry $\delta_{s}$ and
$\Omega$ is the symplectic form, which has the form

$\displaystyle\Omega_{Pl}$ $\displaystyle=\frac{1}{32\pi G}\int_{\Sigma}\delta
B^{ij}\wedge\delta\omega_{ij}$ (20) $\displaystyle\Omega_{MM}$
$\displaystyle=\frac{1}{32\pi G}\int_{\Sigma}\delta B^{IJ}\wedge\delta
A_{IJ}\,.$ (21)

These symplectic forms can be further decomposed into the bulk and boundary
parts, but since we will investigate the resulting charges in the case of the
given on-shell asymptotic expansion of the gravitational field, we do not need
to make use of this decomposition here.

###  3.1 Plebanski theory

Let us start with Lorentz symmetry with parameter $\lambda^{ij}$. Since

$\displaystyle\delta\omega^{ij}=d\lambda^{ij}+[\omega,\lambda]^{ij}\,,\quad\delta
B^{ij}=[B,\lambda]^{ij}\,,$ (22)

we have

$\displaystyle-\delta{\cal H}_{Pl}[\lambda]=\frac{1}{32\pi
G}\int_{\Sigma}[B,\lambda]^{ij}\wedge\delta\omega_{ij}-\delta
B_{ij}\wedge\left(d\lambda^{ij}+[\omega,\lambda]^{ij}\right)=-\frac{1}{32\pi
G}\int_{S}\delta\left(B_{ij}\lambda^{ij}\right)\,,$

so that the Lorentz charge has only the corner component

${\cal H}_{Pl}[\lambda]=\frac{1}{32\pi
G}\int_{S}B_{ij}\lambda^{ij}=\frac{1}{32\pi
G}\int_{S}\left(\frac{2}{\gamma}\,e^{i}\wedge
e^{j}+\epsilon^{ij}{}_{kl}e^{k}\wedge e^{l}\right)\lambda_{ij}\,,$ (23)

where to get the second equality we used the simplicity constraint.

Next, consider the charge associated with the diffeomorphisms. Following
Freidel:2020xyx  we have

$\delta{\cal H}_{Pl}[\xi]=-\mathcal{L}_{\xi}\intprod\Omega=\frac{1}{32\pi
G}\int_{\Sigma}\delta
B_{ij}\wedge\mathcal{L}_{\xi}\omega^{ij}-\mathcal{L}_{\xi}B_{ij}\wedge\delta\omega^{ij}\,,$
(24)

where $\mathcal{L}_{\xi}$ is a Lie derivative and we used an identity
$\mathcal{L}_{\xi}(\cdot)=\xi\lrcorner(d\,\cdot)+d(\xi\lrcorner\,\cdot)$. This
can be further rewritten as

$\delta{\cal H}_{Pl}[\xi]=-\mathcal{L}_{\xi}\intprod\,\Omega=\frac{1}{32\pi
G}\delta\left(\int_{\Sigma}B_{ij}\wedge\mathcal{L}_{\xi}\omega^{ij}\right)-\frac{1}{32\pi
G}\int_{S}\xi\intprod\left(B_{ij}\wedge\delta\omega^{ij}\right)\,.$ (25)

If we assume that the only non-vanishing components of the vector field $\xi$
on the corner $S$ are those tangent to $S$, the second term is a
diffeomorphism acting on an integral of a scalar density and vanishes
identically. Therefore, in this case, the diffeomorphism charge has the form

$\mathcal{H}_{Pl}[\xi]=\frac{1}{32\pi
G}\int_{\Sigma}B_{ij}\wedge\mathcal{L}_{\xi}\omega^{ij}\,.$ (26)

However, for the diffeomorphism generated by an arbitrary vector field $\xi$,
the last term in (25) does not vanish, and the charge ${\cal H}_{\xi}$ is not
well defined. This case was discussed in detail in Freidel:2021cjp . Here we
resolve this problem by using the definition of the diffeomorphism charge
proposed by Wald Wald:1993nt , in which the definition (24) is modified by the
addition of the corner term that cancels the trouble-making one in (25)

$\delta{\cal H}_{Pl}^{W}[\xi]=\delta{\cal H}_{\xi}+\frac{1}{32\pi
G}\int_{S}\xi\intprod\left(B_{ij}\wedge\delta\omega^{ij}\right)\,.$ (27)

In this way, (26) becomes the expression for the diffeomorphism charge for
arbitrary vector field $\xi$ that we are going to adopt here. In what follows,
we will use the Wald definition for diffeomorphism charge, and we will not use
the label $W$ anymore.

Using the identity $\mathcal{L}_{\xi}\omega^{ij}=\xi\lrcorner
R(\omega)^{ij}+D^{\omega}(\xi\lrcorner\omega^{ij})$, which is a consequence of
the field equations $D^{\omega}B_{ij}=0$, we write (26) as

$\mathcal{H}_{Pl}[\xi]=\frac{1}{32\pi G}\int_{\Sigma}B_{ij}\wedge\xi\lrcorner
R(\omega)^{ij}+d(B_{ij}\wedge\xi\lrcorner\omega^{ij})\,,$ (28)

which decomposes into the bulk part

${\cal H}_{Pl}^{\Sigma}[\xi]=\frac{1}{32\pi
G}\int_{\Sigma}B_{ij}\wedge\xi\lrcorner R(\omega)^{ij}$ (29)

and the corner component

$\mathcal{H}_{Pl}^{S}[\xi]=\frac{1}{32\pi
G}\int_{S}B_{ij}\wedge\xi\lrcorner\omega^{ij}\,.$ (30)

Using $B^{ij}=\left(\frac{2}{\gamma}\,e^{i}\wedge
e^{j}+\epsilon^{ij}{}_{kl}e^{k}\wedge e^{l}\right)$ we can write latter as

$\mathcal{H}_{Pl}^{S}[\xi]=\frac{1}{32\pi
G}\int_{S}\left(\frac{2}{\gamma}\,e^{i}\wedge
e^{j}+\epsilon^{ij}{}_{kl}e^{k}\wedge
e^{l}\right)\wedge\xi\lrcorner\omega^{ij}\,,$ (31)

which encapsulates the dual (’magnetic’) and standard (’electric’) parts of
the charge Godazgar:2020kqd ; Godazgar:2020gqd .

One immediately sees the potential problem that could arise in the case of
asymptotic charges of Plebanski theory (36). The integrand of this charge goes
to zero rather slowly as $r\rightarrow\infty$ and one may wonder if the
asymptotic charges are finite. As we will see in the next section this is
indeed the case, even for the simplest asymptotically AdS solutions of
Einstein equations.

###  3.2 MacDowell–Mansoury theory

In the case of MacDowell–Mansouri BF theory, the derivation of the expressions
for charges goes in exactly the same way as above, so we can readily write for
the Lorentz charges

${\cal H}_{MM}[\lambda]=\frac{1}{32\pi G}\int_{S}B_{ij}\lambda^{ij}$ (32)

and for the diffeomorphisms ones

$\mathcal{H}_{MM}[\xi]\thickapprox\frac{1}{32\pi
G}\int_{\Sigma}B_{IJ}\wedge\xi\lrcorner F(A)^{IJ}+d(B_{IJ}\wedge\xi\lrcorner
A^{IJ})\,,$ (33)

which again can be decomposed into the bulk and corner parts. Using the fact
that $B_{I4}=0$ from field equations (vanishing of torsion), we have

${\cal H}_{MM}^{\Sigma}[\xi]=\frac{1}{32\pi
G}\int_{\Sigma}B_{ij}\wedge\xi\lrcorner F(A)^{ij}$ (34)

and the corner component

$\mathcal{H}_{MM}^{S}[\xi]=\frac{1}{32\pi
G}\int_{S}B_{ij}\wedge\xi\lrcorner\omega^{ij}\,.$ (35)

In these equations $B_{ij}$ should be replaced by the on-shell value of the
expression on the right-hand side of (10), ie.,

$\displaystyle
B^{ij}=-\frac{1}{2}\,\frac{3}{\Lambda}\left(\frac{1}{\gamma}\delta_{kl}^{ij}+\epsilon^{ij}{}_{kl}\right)F^{kl}\,,$

which for the corner charge gives

$\displaystyle\mathcal{H}_{MM}^{S}[\xi]$ $\displaystyle=-\frac{3}{64\pi\Lambda
G}\int_{S}\left(\frac{1}{\gamma}\delta_{kl}^{ij}+\epsilon^{ij}{}_{kl}\right)\,F^{kl}\xi\lrcorner\omega_{ij}\,.$
(36)

Contrary to the Plebanski BF theory, here the integrand goes to zero fast
asymptotically, which suggests that the asymptotic charges are finite. Below
we will show that this is indeed the case.

##  4 The charges of AdS–Schwarzschild, AdS–Kerr, and AdS–Taub–NUT

Before discussing the most general case of the arbitrary asymptotically AdS
spacetime, as a warm-up, let us compute the asymptotic charges of two simple
solutions, AdS–Schwarzschild and AdS–Kerr, whose physical interpretation is
clear. Then we consider the more complicated AdS–Taub–NUT. As we will see, the
charges of MacDowell–Mansouri BF theory have the expected finite form, while
the ones of Plebanski BF theory are divergent. This suggests that Plebanski BF
theory requires addition of the counterterms that render the charges finite,
see e.g. Compere:2008us , Anastasiou:2020zwc , and references therein.

###  4.1 AdS–Schwarzschild

The AdS–Schwarzschild metric reads

$\displaystyle
ds^{2}=-\left(1-\frac{2GM}{r}-\frac{\Lambda}{3}r^{2}\right)dt^{2}+\left(1-\frac{2GM}{r}-\frac{\Lambda}{3}r^{2}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\,.$
(37)

The corresponding tetrads are defined as

$\displaystyle e^{0}=f(r)dt,\qquad e^{1}=f(r)^{-1}dr,\qquad
e^{2}=rd\theta,\qquad e^{3}=r\sin\theta d\varphi$ (38)

with

$f(r)^{2}=1-\frac{2GM}{r}-\frac{\Lambda}{3}r^{2}\,,$ (39)

and one derives expressions for the connection $\omega^{ij}$

$\displaystyle\omega^{01}$ $\displaystyle=f^{\prime}(r)e^{0}$
$\displaystyle\omega^{23}$
$\displaystyle=-\frac{1}{r}\frac{\cos\theta}{\sin\theta}e^{3}$
$\displaystyle\omega^{12}$ $\displaystyle=-\frac{f(r)}{r}e^{2}$
$\displaystyle\omega^{13}$ $\displaystyle=-\frac{f(r)}{r}e^{3}\,.$ (40)

The AdS curvature $F^{ij}=R^{ij}+\frac{1}{\ell^{2}}e^{i}\wedge e^{j}$ of the
AdS–Schwarzschild spacetime can be written in a simple form as

$\displaystyle F^{01}$ $\displaystyle=-2he^{0}\wedge e^{1}$ $\displaystyle
F^{02}$ $\displaystyle=he^{0}\wedge e^{2}$ $\displaystyle F^{03}$
$\displaystyle=he^{0}\wedge e^{3}$ $\displaystyle F^{12}$
$\displaystyle=he^{1}\wedge e^{2}$ $\displaystyle F^{13}$
$\displaystyle=he^{1}\wedge e^{3}$ $\displaystyle F^{23}$
$\displaystyle=-2he^{2}\wedge e^{3}$ (41)

with the $h=-{GM}/{r^{3}}$. Notice that the AdS-curvature goes to zero
asymptotically.

For the calculation concerning mass, we take $\partial_{t}$ Killing vector,
which naturally leads to the asymptotic corner diffeomorphism charge, computed
on the sphere at infinity $S_{\infty}$

$\displaystyle\mathcal{H}_{MM}^{S_{\infty}}[\partial_{t}]$
$\displaystyle=M\,,$ (42)

providing the expected value of the charge at infinity. The Plebanski theory
gives instead the divergent result

$\displaystyle\mathcal{H}_{Pl}^{S_{\infty}}[\partial_{t}]$
$\displaystyle=\frac{M}{2}-\lim_{r\to\infty}\frac{\Lambda}{3}\frac{r^{3}}{2G}$
(43)

with the finite part now being half of the mass parameter. This shows that
Plebanski theory requires holographic renormalization, which not only renders
the asymptotic charges finite but also rescales the finite part of the charge.
It can be checked that adding the Euler term to Plebanski action, with the
coefficient $-3/2\Lambda$ being exactly that of MacDowell–Mansouri theory cf.
(15), does the trick here Aros:1999id ; Aros:1999kt .

It can be further checked, that all other diffeomorphism charges vanish for
both theories.

###  4.2 AdS–Kerr

Tetrads of the AdS–Kerr are given by

$\displaystyle e^{0}$
$\displaystyle=\frac{\sqrt{\Delta_{r}}}{\rho}\left(dt-\frac{a}{\Xi}\sin^{2}\theta
d\varphi\right),\quad e^{1}=\rho\frac{dr}{\sqrt{\Delta_{r}}}$ (44)
$\displaystyle e^{2}$
$\displaystyle=\rho\frac{d\theta}{\sqrt{\Delta_{\theta}}},\quad
e^{3}=\frac{\sqrt{\Delta_{\theta}}}{\rho}\sin\theta\left(\frac{\left(r^{2}+a^{2}\right)}{\Xi}d\varphi-
adt\right)\,,$ (45)

where we use the standard abbreviations

$\displaystyle\rho^{2}$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,,$ (46)
$\displaystyle\Delta_{r}$
$\displaystyle=\left(r^{2}+a^{2}\right)\left(1-\frac{\Lambda
r^{2}}{3}\right)-2GMr\,,$ (47) $\displaystyle\Delta_{\theta}$
$\displaystyle=1+\frac{\Lambda a^{2}}{3}\cos^{2}\theta\,,$ (48)
$\displaystyle\Xi$ $\displaystyle=1+\frac{\Lambda a^{2}}{3}\,.$ (49)

The volume element reads

$\displaystyle e^{0}\wedge e^{1}\wedge e^{2}\wedge
e^{3}=\frac{r^{2}+a^{2}\cos^{2}\theta}{\Xi}dt\wedge dr\wedge d\theta\wedge
d\varphi\,.$

The non-vanishing connection components read

$\displaystyle\omega^{01}$
$\displaystyle=\left(\frac{(r^{2}+a^{2}\cos^{2}\theta)(M-r+(2r^{2}+a^{2})\frac{\Lambda}{3}r)+r\Delta_{r})}{\sqrt{\Delta_{r}}\rho^{3/2}}\right)e^{0}-\frac{ar\sqrt{\Delta_{\theta}}\sin\theta}{\rho^{3/2}}e^{3}$
$\displaystyle\omega^{02}$
$\displaystyle=-\frac{a^{2}\sqrt{\Delta_{\theta}}\cos\theta\sin\theta}{\rho^{3/2}}e^{0}-\frac{a\sqrt{\Delta_{r}}\cos\theta}{\rho^{3/2}}e^{3}$
$\displaystyle\omega^{03}$
$\displaystyle=-\frac{ar\sqrt{\Delta_{\theta}}\sin\theta}{\rho^{3/2}}e^{1}+\frac{a\sqrt{\Delta_{r}}\cos\theta}{\rho^{3/2}}e^{2}$
$\displaystyle\omega^{12}$
$\displaystyle=-\frac{a^{2}\sqrt{\Delta_{\theta}}\cos\theta\sin\theta}{\rho^{3/2}}e^{1}-\frac{r\sqrt{\Delta_{r}}}{\rho^{3/2}}e^{2}$
$\displaystyle\omega^{13}$
$\displaystyle=-\frac{ar\sqrt{\Delta_{\theta}}\sin\theta}{\rho^{3/2}}e^{0}-\frac{r\sqrt{\Delta_{r}}}{\rho^{3/2}}e^{3}$
$\displaystyle\omega^{23}$
$\displaystyle=-\frac{a\sqrt{\Delta_{r}}\cos\theta}{\rho^{3/2}}e^{0}-\left(\frac{\left(a^{2}+r^{2}\right)+a^{2}\frac{\Lambda}{3}\left(a^{2}+r^{2}-2\left(a^{2}+r^{2}\right)\sin^{2}\theta+a^{2}\sin^{4}\theta\right)}{\rho^{3/2}\sqrt{\Delta_{\theta}}}\right)\frac{\cos\theta}{\sin\theta}e^{3}$

Quite unexpectedly, the AdS curvatures
$F^{ij}=R^{ij}-\frac{\Lambda}{3}e^{i}\wedge e^{j}$ components have a simple
form Blagojevic:2020edq ; Blagojevic:2020ymf

$\displaystyle F^{01}$ $\displaystyle=-2he^{0}\wedge e^{1}-2be^{2}\wedge
e^{3}$ $\displaystyle F^{02}$ $\displaystyle=he^{0}\wedge e^{2}-be^{1}\wedge
e^{3}$ $\displaystyle F^{03}$ $\displaystyle=he^{0}\wedge e^{3}+be^{1}\wedge
e^{2}$ $\displaystyle F^{12}$ $\displaystyle=he^{1}\wedge e^{2}-be^{0}\wedge
e^{3}$ $\displaystyle F^{13}$ $\displaystyle=he^{1}\wedge e^{3}+be^{0}\wedge
e^{2}$ $\displaystyle F^{23}$ $\displaystyle=-2he^{2}\wedge
e^{3}+2be^{0}\wedge e^{1}$ (50)

with the functions $h$, $b$ computed to be

$\displaystyle h$
$\displaystyle=\frac{\left(3a^{2}\cos^{2}\theta-r^{2}\right)r}{\left(r^{2}+a^{2}\cos^{2}\theta\right)^{3}}M$
(51) $\displaystyle b$
$\displaystyle=\frac{\left(3r^{2}-a^{2}\cos^{2}\theta\right)a\cos\theta}{\left(r^{2}+a^{2}\cos^{2}\theta\right)^{3}}M\,.$
(52)

We focus on the asymptotic Killing vectors of time translation $\partial_{t}$
and rotation around the $z$-axis of axial symmetry $\partial_{\varphi}$. In
the case of the MacDowell–Mansouri theory, we obtain the following expressions
for the mass and angular momentum

$\displaystyle\mathcal{H}^{S}_{MM}[\partial_{t}]$
$\displaystyle=\frac{M}{\Xi}$ (53)
$\displaystyle\mathcal{H}_{MM}^{S}[\partial_{\varphi}]$
$\displaystyle=\frac{-aM}{\Xi^{2}}\,.$ (54)

It can be checked that the corner charges associated with other asymptotic
Killing vectors vanish.

There is an ambiguity discussed, for example in Gibbons:2004ai ; Durka:2012wd
concerning the proper definition of the mass. Taking into account the presence
of an non-zero angular velocity at infinity, instead of the Killing vector
$\partial_{t}$ it is more proper to use
$\tilde{\partial}_{t}=\partial_{t}+\frac{a\Lambda}{3}\partial_{\varphi}$,
which leads to the modified expression for the mass

$\displaystyle\mathcal{H}_{MM}^{S}[\tilde{\partial}_{t}]=\frac{M}{\Xi^{2}}\,.$
(55)

In the case of the Palatini BF theory, for the asymptotic charge associated
with time translation we find

$\displaystyle\mathcal{H}_{Pl}^{S}[\partial_{t}]$
$\displaystyle=\lim_{r\to\infty}\left(\frac{M/2}{\Xi}-\frac{\Lambda}{3}\frac{1}{2G}\frac{(r^{2}+a^{2})r}{\Xi}\right)=\frac{M}{2\Xi}-\lim_{r\to\infty}\frac{\Lambda}{6G}\frac{r^{3}}{\Xi}\,.$
(56)

We see again that the mass is divergent and the finite part is half of the
expected value, as it was in the Schwarzschild case. Exactly as it was in the
Schwarzschild case, adding the Euler counterterm with the coefficient of MM
theory to the Plebanski Lagrangian renders the finite part correct and cancels
the divergent term.

As for the angular momentum, the result agrees with the one for
MacDowell–Mansouri theory

$\displaystyle\mathcal{H}_{Pl}^{S}[\partial_{\varphi}]=\frac{-aM}{\Xi^{2}}\,.$
(57)

Interestingly, in the Plebanski BF theory, in the leading and subleading $r$
order, we have a non-vanishing, divergent charges of the boost
$\mathcal{N}_{3}$ and translation $\mathcal{P}_{3}$ along the $z$ axis

$\displaystyle\mathcal{H}_{Pl}^{S}[\mathcal{N}_{3}]$
$\displaystyle=-\frac{1}{\gamma}\sin\left(\sqrt{-\Lambda/3}\,t\right)\lim_{r\to\infty}\frac{\frac{\Lambda}{3}a(r^{2}+a^{2})}{6G\Xi}$
(58) $\displaystyle\mathcal{H}_{Pl}^{S}[\mathcal{P}_{3}]$
$\displaystyle=-\frac{1}{\gamma}\cos\left(\sqrt{-\Lambda/3}\,t\right)\lim_{r\to\infty}\frac{\left(-\frac{\Lambda}{3}\right)^{3/2}a(r^{2}+a^{2})}{6G\Xi}\,.$
(59)

These expressions can be again canceled by adding the topological term. All
the other asymptotic charges vanish.

###  4.3 AdS–Taub–NUT solution

In both cases discussed above, the asymptotic charges did not depend on the
Barbero-Immirzi parameter. This changes in the case of AdS–Taub–NUT solution,
as we are going to show now. This solution attracted some attention recently
in the context of holography; see for example Kalamakis:2020aaj  and
references therein. In what follows we do not impose any conditions on the
AdS-Taub-NUT parameters, contrary, for example, to the Euclidean approach of
Liko:2011cq ; Araneda:2016iiy ; Ciambelli:2020qny ; Corral:2021xsu .

The AdS–Taub–NUT spacetime is defined by the tetrad one forms

$\displaystyle e^{0}=f(r)(dt-2n(\cos\theta-k)d\varphi),$ $\displaystyle
e^{1}=f(r)^{-1}dr,$ $\displaystyle e^{2}=\sqrt{r^{2}+n^{2}}d\theta,$
$\displaystyle e^{3}=\sqrt{r^{2}+n^{2}}\sin\theta d\varphi$ (60)

with

$f(r)^{2}=\frac{r^{2}-2GMr-n^{2}-\Lambda\left(r^{4}+6n^{2}r^{2}-3n^{4}\right)/3}{n^{2}+r^{2}}\,.$
(61)

Parameter $k=\\{-1,0,1\\}$ encodes location of the so-called Misner string(s)
Misner:1963fr  with $k=0$ representing case with strings on both poles,
whereas $k=-1$ one string up, and $k=1$ one string down.

The spin connection components are given by

$\displaystyle\omega^{01}$ $\displaystyle=f^{\prime}(r)e^{0}$
$\displaystyle\omega^{02}$ $\displaystyle=\frac{nf(r)}{n^{2}+r^{2}}e^{3}$
$\displaystyle\omega^{03}$ $\displaystyle=-\frac{nf(r)}{n^{2}+r^{2}}e^{2}$
$\displaystyle\omega^{12}$ $\displaystyle=-\frac{rf(r)}{n^{2}+r^{2}}e^{2}$
$\displaystyle\omega^{13}$ $\displaystyle=-\frac{rf(r)}{n^{2}+r^{2}}e^{3}$
$\displaystyle\omega^{23}$
$\displaystyle=\frac{nf(r)}{n^{2}+r^{2}}e^{0}-\frac{1}{\sqrt{n^{2}+r^{2}}}\frac{\cos\theta}{\sin\theta}e^{3}\,,$
(62)

whereas AdS-curvatures are

$\displaystyle F^{01}$ $\displaystyle=-2he^{0}\wedge e^{1}-2be^{2}\wedge
e^{3}$ $\displaystyle F^{02}$ $\displaystyle=he^{0}\wedge e^{2}-be^{1}\wedge
e^{3}$ $\displaystyle F^{03}$ $\displaystyle=he^{0}\wedge e^{3}+be^{1}\wedge
e^{2}$ $\displaystyle F^{12}$ $\displaystyle=he^{1}\wedge e^{2}-be^{0}\wedge
e^{3}$ $\displaystyle F^{13}$ $\displaystyle=he^{1}\wedge e^{3}+be^{0}\wedge
e^{2}$ $\displaystyle F^{23}$ $\displaystyle=-2he^{2}\wedge
e^{3}+2be^{0}\wedge e^{1}\,,$ (63)

where $h$ and $b$ are two auxiliary functions

$\displaystyle h$
$\displaystyle=\frac{\left(3n^{2}-r^{2}\right)r}{\left(n^{2}+r^{2}\right)^{3}}\,GM+\frac{\left(n^{2}-3r^{2}\right)n}{\left(n^{2}+r^{2}\right)^{3}}\left(1-\frac{4\Lambda
n^{2}}{3}\right)n\,,$ (64) $\displaystyle b$
$\displaystyle=-\frac{\left(3n^{2}-r^{2}\right)r}{\left(n^{2}+r^{2}\right)^{3}}\left(1-\frac{4\Lambda
n^{2}}{3}\right)n+\frac{\left(n^{2}-3r^{2}\right)n}{\left(n^{2}+r^{2}\right)^{3}}\,GM\,.$
(65)

We notice that the AdS–Taub–NUT solution is defined by two parameters, the
mass $M$ and the dual mass $N=n(1-4n^{2}{\Lambda}/{3})/G$ Godazgar:2022jxm .
When $\Lambda\to 0$ we recover Taub–NUT solution, while in the limit $n\to 0$
we get AdS–Schwarzschild spacetime with $b=0$ and $h=-{GM}/{r^{3}}$.
Asymptotically, for $r\to\infty$ the leading order expansion is

$h\to-\frac{GM}{r^{3}}\,,\qquad b\to-\frac{N}{r^{3}}\,,\quad
f\to\sqrt{\frac{|\Lambda|}{3}}r\,.$ (66)

As it was pointed out in Misner:1963fr ; Zanelli:2009kra , the Killing vectors
are now modified

$\displaystyle\begin{aligned} \xi^{t}&=\frac{\partial}{\partial t}\\\
\xi^{x}&=-\sin\varphi\frac{\partial}{\partial\theta}-\cos\varphi\cot\theta\frac{\partial}{\partial\varphi}-2nk\left(\cos\varphi\cot\theta-\frac{\cos\varphi}{\sin\theta}\right)\frac{\partial}{\partial
t}\\\
\xi^{y}&=\cos\varphi\frac{\partial}{\partial\theta}-\sin\varphi\cot\theta\frac{\partial}{\partial\varphi}-2nk\left(\sin\varphi\cot\theta-\frac{\sin\varphi}{\sin\theta}\right)\frac{\partial}{\partial
t}\\\ \xi^{z}&=\frac{\partial}{\partial\varphi}+2nk\frac{\partial}{\partial
t}\,.\end{aligned}$ (67)

The corner charge computed for the time-like vector $\xi^{t}$ has
contributions from the mass (resulting from Einstein-Cartan + Euler
Aros:1999kt ; Aros:1999id ) and from the dual mass (appearing as a consequence
of the Holst and Pontryagin terms Durka:2011yv )

$\displaystyle\mathcal{H}_{MM}^{S}[\xi^{t}]$
$\displaystyle=M+\frac{1}{\gamma}\left(1-\frac{4\Lambda
n^{2}}{3}\right)\frac{n}{G}\,.$ (68)

The charge associated with the rotational Killing vector, somewhat similar to
the AdS–Kerr case above, requires not taking $\partial_{\phi}$ but $\xi^{z}$.
This gives the quantity associated with angular momentum

$\displaystyle\mathcal{H}_{MM}^{S}[\xi^{z}]$
$\displaystyle=-kn\left(M+\frac{1}{\gamma}\,\left(1-\frac{4\Lambda
n^{2}}{3}\right)\frac{n}{G}\right)\,.$ (69)

It was stressed in Bonnor:1969ala ; Manko:2005nm ; Durka:2019ajz ;
Frodden:2021ces , the Misner strings could be considered as the source of
angular momentum. In the case of a single string (either up, $k=-1$ or down,
$k=+1$) the angular momentum is proportional to $n\cdot(\mbox{Mass})$. This is
analogous to the case of Kerr metric, where for the Kerr parameter $a$, we
have angular momentum proportional to $a\cdot(\mbox{Mass})$. Choosing $k=0$
leads to cancellation of the total angular momentum, as then we deal with two
counter-rotating Misner stings with angular velocity being
$\Omega=\pm\frac{1}{2n}$.

The corresponding charges calculated for the Plebanski theory are

$\displaystyle\mathcal{H}_{Pl}^{S}[\xi^{t}]$
$\displaystyle=\frac{1}{\gamma}\left(\frac{n}{2G}-\frac{\Lambda
n\left(5n^{2}+r^{2}\right)}{6G}\right)-\frac{\Lambda
r\left(n^{2}+r^{2}\right)}{6G}+\frac{M}{2}$ (70)

and

$\displaystyle\mathcal{H}_{Pl}^{S}[\xi^{z}]$
$\displaystyle=\frac{kn^{2}}{\gamma}\left(\frac{5\Lambda n^{2}+\Lambda
r^{2}-3}{3G}\right)-krn\left(\frac{5\Lambda n^{2}+\Lambda
r^{2}-3}{3G}\right)-2kMn\,.$ (71)

As before, these charges are not only divergent but also their finite parts
come with the wrong factors. As expected, adding the Euler and Pontryagin
terms to Plebanski action with the coefficients as in (15) removes the
divergencies and regularizes the finite part.

Note that for the Taub–NUT spacetime there is explicit dual charge
contribution proportional to the Barbero-Immirzi parameter, which was absent
in the case of AdS–Kerr spacetime. This contribution can be traced back to the
presence of the Holst and Pontryagin terms in the Lagrangian, effectively
repeating the MacDowell–Mansouri scheme of factors.

The contribution to the charges proportional to inverse $\gamma$ is
subleading, and such terms are going to be present in the expressions for
charges associated with corners at a finite distance. Once again we stress the
regularizing effect of the topological terms; without them, the values of
masses and angular momenta would not be correct (see for more details
Aros:1999id ; Aros:1999kt ; Durka:2011yv ; Durka:2012wd ).

As for the spacetimes considered before, in the AdS–Schwarzschild case the
Pontryagin/Holst contribution to charges vanishes identically. In the case of
AdS–Kerr the Holst/Pontryagin contributions to the charges disappear as a
result of integration over $S^{2}$. The situation is very different for the
Taub–NUT scenario, which produces in addition the dual mass and non-zero
angular momenta.

###  4.4 Summary

In this section, we have considered the asymptotic charges of some exact
asymptotically AdS solutions of Einstein equations associated with the
asymptotic Killing vectors. We found that for all of them the
MacDowell–Mansouri BF theory gives the finite results, while in the case of
Plebanski BF theory the charges, as a rule, contain divergences and their
finite part does not lead to the correct result. This means that this theory
must be modified by adding terms that render the charges finite and have the
right value. As it turns out the necessary counterterms are exactly the
appropriate topological invariants present in the MacDowell–Mansouri BF theory
action. One can speculate therefore that the proper holographic
renormalization of Plebanski BF theory is the MacDowell–Mansouri BF model of
gravity. To confirm this claim in the next section we will show that in the
case of MM theory the asymptotic charges, the action and the symplectic
structure of an arbitrary asymptotically AdS spacetimes are finite.

Finally, let us mention that the three discussed solutions of the Lagrangian
(16) have a very simple form

$\displaystyle\mathcal{L}_{MM}$ $\displaystyle=-\frac{9}{4\pi
G\Lambda}\left(-\frac{2}{\gamma}hb+(h^{2}-b^{2})\right)e^{0}\wedge e^{1}\wedge
e^{2}\wedge e^{3}\,,$ (72)

where the functions $h$ and $b$ are defined for the AdS–Kerr by (51) and (52),
for AdS–Taub–NUT by (64) and (65), whereas for AdS–Schwarzschild by
$h=-GM/r^{3}$ and $b=0$. In all these cases we see that value of
$\mathcal{L}_{MM}$ is finite.

At the same time, for these solutions the Plebanski Lagrangian (4) is
proportional to the volume element

$\displaystyle 32\pi G\,\mathcal{L}_{Pl}$
$\displaystyle=\left(\frac{2}{\gamma}4(4b-4b+\Lambda-\Lambda)+(4(2\Lambda+4h-4h)-4\Lambda)\right)e^{0}\wedge
e^{1}\wedge e^{2}\wedge e^{3}$ $\displaystyle=4\Lambda\,e^{0}\wedge
e^{1}\wedge e^{2}\wedge e^{3}\,,$ (73)

which again shows the necessity of adding counterterms to this theory.

##  5 General asymptotically Anti-de Sitter spacetimes

In this section, we first briefly review the asymptotic structure of
asymptotically AdS spacetime at spatial infinity following closely the
classical discussion presented in Henneaux:1985tv . Then we compute the
asymptotic charges for MacDowell–Mansouri BF theory of gravity.

The AdS asymptotic metric in the coordinates $(t,r,\theta,\varphi)$ has the
form

$\displaystyle
ds^{2}=\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}=-\left(1-\frac{r^{2}\Lambda}{3}\right)dt^{2}+\frac{1}{\left(1-\frac{r^{2}\Lambda}{3}\right)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\,.$ (74)

The perturbations are assumed to have the following large $r$ behavior

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+g^{(1)}_{\mu\nu}+\ldots$ (75)

with

$\displaystyle g^{(1)}_{tt}$ $\displaystyle=\frac{h_{tt}}{r}\,,\quad
g^{(1)}_{tr}=\frac{h_{tr}}{r^{4}}\,,\quad
g^{(1)}_{t\theta}=\frac{h_{t\theta}}{r}\,,\quad
g^{(1)}_{t\varphi}=\frac{h_{t\varphi}}{r}\,,$ $\displaystyle g^{(1)}_{rr}$
$\displaystyle=\frac{h_{rr}}{r^{5}}\,,\quad
g^{(1)}_{r\theta}=\frac{h_{r\theta}}{r^{4}}\,,\quad
g^{(1)}_{r\varphi}=\frac{h_{r\varphi}}{r^{4}}\,,$ $\displaystyle
g^{(1)}_{\theta\theta}$ $\displaystyle=\frac{h_{\theta\theta}}{r}\,,\quad
g^{(1)}_{\theta\varphi}=\frac{h_{\theta\varphi}}{r}\,,\quad
g^{(1)}_{\varphi\varphi}=\frac{h_{\varphi\varphi}}{r}\,,$ (76)

where all the coefficients $h_{\mu\nu}$ are functions of $(t,\theta,\varphi)$.

Let us consider the asymptotic corner charges, which we compute at the sphere
at infinity $S=S^{2}_{\infty}$. Rewriting the general expression (36) in term
of the metric the AdS curvature tensor
$F^{\mu\nu}{}_{\alpha\beta}=\bar{F}^{\mu\nu}{}_{\alpha\beta}+F^{(1)\mu\nu}{}_{\alpha\beta}$
leads to

$\displaystyle\mathcal{H}_{MM}^{S}[\xi]$ $\displaystyle=-\frac{3}{64\pi\Lambda
G}\int_{S}F^{\mu\nu}{}_{\theta\varphi}\,\xi^{\rho}\,e^{k}_{\mu}\,e^{l}_{\nu}\left(\frac{1}{\gamma}\delta_{kl}^{ij}+\epsilon^{ij}{}_{kl}\right)\,\omega_{\rho
ij}d\theta d\varphi\,.$ (77)

Let us now notice that since the background AdS part $\bar{F}$ of the AdS
curvature tensor vanishes asymptotically, we can write the leading order of
perturbations of (77) as

$\displaystyle\mathcal{H}_{MM}^{S}[\xi]$
$\displaystyle\approx-\frac{3}{64\pi\Lambda
G}\int_{S}F^{(1)\,\mu\nu}{}_{\theta\varphi}\,\xi^{\rho}\,\bar{e}^{k}_{\mu}\,\bar{e}^{l}_{\nu}\left(\frac{1}{\gamma}\delta_{kl}^{ij}+\epsilon^{ij}{}_{kl}\right)\,\bar{\omega}_{\rho
ij}d\theta d\varphi\,.$ (78)

The AdS tetrads are defined as

$\displaystyle\bar{e}^{0}=\sqrt{1-\frac{\Lambda}{3}r^{2}}\,dt,\qquad\bar{e}^{1}=1/\sqrt{1-\frac{\Lambda}{3}r^{2}}\,dr,\qquad\bar{e}^{2}=r\,d\theta,\qquad\bar{e}^{3}=r\,\sin\theta
d\varphi$ (79)

and one derives expressions for connection $\omega^{ij}$:

$\displaystyle\bar{\omega}^{01}$ $\displaystyle=-\frac{\Lambda}{3}r\,dt$
$\displaystyle\bar{\omega}^{23}$ $\displaystyle=-\cos\theta d\varphi$
$\displaystyle\bar{\omega}^{12}$
$\displaystyle=-\sqrt{1-\frac{\Lambda}{3}r^{2}}\,d\theta$
$\displaystyle\bar{\omega}^{13}$
$\displaystyle=-\sqrt{1-\frac{\Lambda}{3}r^{2}}\,\sin\theta d\varphi\,.$ (80)

At the same time, we have the following relevant components for the leading
order asymptotic AdS curvature (all other curvature components can be found in
the accompanying supplement file)

$\displaystyle F^{(1)\,t,r}{}_{\theta,\varphi}$
$\displaystyle=\frac{3}{2r^{2}}\left(\partial_{\varphi}h_{t\theta}-\partial_{\theta}h_{t\varphi}\right)$
(81) $\displaystyle F^{(1)\,t,\theta}{}_{\theta,\varphi}$
$\displaystyle=-\frac{3}{2r^{3}}\,h_{t\varphi}$ (82) $\displaystyle
F^{(1)\,t,\varphi}{}_{\theta,\varphi}$
$\displaystyle=\frac{3}{2r^{3}}\,h_{t\theta}$ (83) $\displaystyle
F^{(1)\,r,\theta}{}_{\theta,\varphi}$
$\displaystyle=\frac{\Lambda}{2r^{2}}\left(\frac{\Lambda}{9}\partial_{\varphi}h_{{rr}}-\partial_{\varphi}h_{\theta\theta}+\partial_{\theta}h_{\theta\varphi}+\cot\theta
h_{\theta\varphi}\right)$ (84) $\displaystyle
F^{(1)\,r,\varphi}{}_{\theta,\varphi}$
$\displaystyle=\frac{\Lambda}{2r^{2}}\left(-\frac{\Lambda}{9}\partial_{\theta}h_{\text{rr}}+\frac{1}{\sin^{2}\theta}(\partial_{\theta}h_{\varphi\varphi}-\partial_{\varphi}h_{\theta\varphi})-\cot\theta(h_{\theta\theta}+\frac{1}{\sin^{2}\theta}h_{\varphi\varphi})\right)$
(85) $\displaystyle F^{(1)\,\theta,\varphi}{}_{\theta,\varphi}$
$\displaystyle=\frac{\Lambda}{2r^{3}}\left(-\frac{2\Lambda}{9}h_{\text{rr}}+h_{\theta\theta}+\frac{1}{\sin^{2}\theta}h_{\varphi\varphi}\right)\,.$
(86)

We can further evaluate the formula (78) to be of the form of

$\displaystyle\mathcal{H}_{MM}^{S}[\xi]$
$\displaystyle\approx\int_{S}\xi^{\rho}\,\Xi_{\rho}\,d\theta\,d\varphi\,,$
(87)

where to the leading order

$\displaystyle\Xi_{t}$ $\displaystyle=\frac{\Lambda}{32\pi
G}\sin\theta\left(\frac{2}{9}\Lambda
h_{\text{rr}}-h_{\theta\theta}-h_{\varphi\varphi}\csc^{2}\theta\right)+O\left(\frac{1}{r}\right)$
(88) $\displaystyle\Xi_{r}$ $\displaystyle=0$ (89) $\displaystyle\Xi_{\theta}$
$\displaystyle=\frac{3}{32\pi G}\sin\theta
h_{t\theta}+O\left(\frac{1}{r}\right)$ (90) $\displaystyle\Xi_{\varphi}$
$\displaystyle=\frac{3}{32\pi G}\sin\theta
h_{t\varphi}+O\left(\frac{1}{r}\right)$ (91)

and the metric components $h_{\mu\nu}$ are subject to the asymptotic Einstein
equations (see Appendix). As we see the Barbero-Immirzi parameter completely
disappears from these leading order asymptotic expressions. One may find this
fact surprising since in the previous section we found an explicit $1/\gamma$
contribution to the charges for the AdS–Taub–NUT solution. However, this
solution does not satisfy the asymptotic boundary conditions (76) and
therefore is not included in the general frame of this section.

Let us note in passing that for the AdS–Kerr, as pointed out in
Henneaux:1985tv , the coordinates we used in Section 4.2 also do not have
proper asymptotic behavior, which is a consequence of non-vanishing rotation
at infinity. To obtain the correct value of the mass (55) a coordinate
transformation is required. Alternatively, this can be achieved by the
redefinition of the Killing vector at infinity Gibbons:2004ai ; Durka:2012wd .

The asymptotic $r\rightarrow\infty$ Killing vectors are given by the following
expressions that we borrowed from Henneaux:1985tv . We took the liberty to
adopt the notation used in the case of Poincaré Killing vectors, that could be
obtained from them in the limit $\Lambda\rightarrow\infty$. In the expressions
below we neglect all the terms that fall at infinity as $1/r$ or faster and we
use an abbreviation $\tau\equiv\sqrt{-\Lambda/3}\,t$

  * •

Time translation

$\displaystyle\mathcal{E}=\frac{\partial}{\partial t}$ (92)

  * •

Space translations

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=-\sin\tau\sin\theta\cos\varphi\,\frac{\partial}{\partial
t}+r\sqrt{-\frac{\Lambda}{3}}\cos\tau\sin\theta\cos\varphi\frac{\partial}{\partial
r}$
$\displaystyle+\sqrt{-\frac{\Lambda}{3}}\cos\tau\left(\cos\theta\cos\varphi\frac{\partial}{\partial\theta}-\frac{\sin\varphi}{\sin\theta}\frac{\partial}{\partial\varphi}\right)$
(93) $\displaystyle\mathcal{P}_{2}$
$\displaystyle=-\sin\tau\sin\theta\sin\varphi\,\frac{\partial}{\partial
t}+r\sqrt{-\frac{\Lambda}{3}}\cos\tau\sin\theta\sin\varphi\frac{\partial}{\partial
r}$
$\displaystyle+\sqrt{-\frac{\Lambda}{3}}\cos\tau\left(\cos\theta\sin\varphi\frac{\partial}{\partial\theta}-\frac{\cos\varphi}{\sin\theta}\frac{\partial}{\partial\varphi}\right)$
(94) $\displaystyle\mathcal{P}_{3}$
$\displaystyle=-\sin\tau\cos\theta\,\frac{\partial}{\partial
t}+r\sqrt{-\frac{\Lambda}{3}}\cos\tau\cos\theta\frac{\partial}{\partial r}$
$\displaystyle-\sqrt{-\frac{\Lambda}{3}}\cos\tau\sin\theta\frac{\partial}{\partial\theta}$
(95)

  * •

Rotations

$\displaystyle\mathcal{M}_{1}$
$\displaystyle=\frac{\partial}{\partial\varphi}$ (96)
$\displaystyle\mathcal{M}_{2}$
$\displaystyle=-\sin\varphi\frac{\partial}{\partial\theta}-\operatorname{cot}\theta\cos\varphi\frac{\partial}{\partial\varphi}$
(97) $\displaystyle\mathcal{M}_{3}$
$\displaystyle=\cos\varphi\frac{\partial}{\partial\theta}-\operatorname{cot}\theta\sin\varphi\frac{\partial}{\partial\varphi}$
(98)

  * •

Boosts

$\displaystyle\mathcal{N}_{1}$
$\displaystyle=\sqrt{-\frac{3}{\Lambda}}\cos\tau\sin\theta\cos\varphi\,\frac{\partial}{\partial
t}+r\sin\tau\sin\theta\cos\varphi\frac{\partial}{\partial r}$
$\displaystyle+\sin\tau\left(\cos\theta\cos\varphi\frac{\partial}{\partial\theta}-\frac{\sin\varphi}{\sin\theta}\frac{\partial}{\partial\varphi}\right)$
(99) $\displaystyle\mathcal{N}_{2}$
$\displaystyle=\sqrt{-\frac{3}{\Lambda}}\cos\tau\sin\theta\sin\varphi\,\frac{\partial}{\partial
t}+r\sin\tau\sin\theta\sin\varphi\frac{\partial}{\partial r}$
$\displaystyle+\sin\tau\left(\cos\theta\sin\varphi\frac{\partial}{\partial\theta}-\frac{\cos\varphi}{\sin\theta}\frac{\partial}{\partial\varphi}\right)$
(100) $\displaystyle\mathcal{N}_{3}$
$\displaystyle=\sqrt{-\frac{3}{\Lambda}}\cos\tau\cos\theta\,\frac{\partial}{\partial
t}+r\sin\tau\cos\theta\frac{\partial}{\partial r}$
$\displaystyle-\sin\tau\sin\theta\frac{\partial}{\partial\theta}$ (101)

Using the formula (87) we can now compute all ten asymptotic charges. For
example, we find the energy to be

$\displaystyle E=\frac{\Lambda}{32\pi G}\int\left(\frac{2}{9}\Lambda
h_{\text{rr}}-h_{\theta\theta}-h_{\varphi\varphi}\csc^{2}\theta\right)\sin\theta
d\theta d\varphi\,,$ (102)

whereas the angular momentum is

$\displaystyle J=\frac{3}{32\pi G}\int h_{t\varphi}\sin\theta d\theta
d\varphi\,.$ (103)

With the help of the formulas (87) and (88)–(91) one can check that the all
the remaining charges are finite, which indicates that the MacDowell–Mansouri
BF theory does not require holographic renormalization. To check the
finiteness of the theory we must find out further if both the Lagrangian and
symplectic structure are finite as well. Let us consider the Lagrangian first.
To this end, we return to the formula (16), which in the case of vanishing
torsion can be rewritten as

$\displaystyle 32\pi G\,\mathcal{L}_{MM}$
$\displaystyle=-\frac{3}{2\Lambda}\left(\frac{1}{\gamma}g_{\mu\rho}g_{\nu\sigma}-\frac{1}{\gamma}g_{\nu\rho}g_{\mu\sigma}+\epsilon_{\mu\nu\rho\sigma}\right)\frac{1}{4}F^{(1)\mu\nu}{}_{\alpha\beta}F^{(1)\rho\sigma}{}_{\gamma\delta}\,\epsilon^{\alpha\beta\gamma\rho}\sqrt{-\bar{g}}d^{4}x\,.$
(104)

This leads to the following asymptotic expansion of the Lagrangian being
second perturbation order

$\displaystyle\mathcal{L}_{MM}$ $\displaystyle=\frac{1}{r^{4}}\frac{2}{32\pi
G}\sin\theta\left(\frac{\Lambda^{3}}{9}h_{{rr}}{}^{2}-2\Lambda
h_{{rr}}\left(h_{{tt}}+\frac{\Lambda}{3}h_{\theta\theta}\right)-9h_{t\theta}{}^{2}+9h_{\theta\theta}h_{{tt}}\right)$
$\displaystyle+\frac{1}{r^{4}}\frac{2}{32\pi
G}\frac{1}{\sin\theta}\left(h_{\phi\phi}\left(-\frac{2\Lambda^{2}}{3}h_{{rr}}+3\Lambda
h_{\theta\theta}+9h_{{tt}}\right)-3\Lambda
h_{\theta\phi}{}^{2}-9h_{t\phi}{}^{2}\right)$
$\displaystyle+\frac{1}{\gamma}\frac{2}{32\pi G}O(1/r^{5})\,.$ (105)

It follows from (105) that the action is finite (assuming appropriate boundary
conditions at the horizons in the bulk, if any are present). As a consequence,
the symplectic structure that can be derived from the action is also finite.
This concludes our proof that the MacDowell–Mansouri BF action for gravity is
well suited for describing physics in asymptotically AdS spacetime.

##  6 Conclusions

In this paper, we considered two constrained BF theories of gravity, one
formulated by Plebanski and another based on MacDowell–Mansouri theory. Both
give the same bulk Einstein equations with the negative cosmological constant,
but since their actions differ by the presence of topological invariants in
the second case, the expressions for charges associated with corners, at
infinity, and in the final distance are different. We show that as a rule the
asymptotic charges are divergent for Plebanski theory with the finite part
also giving incorrect results. On the other hand for the MacDowell–Mansouri
theory, all the charges are finite and have the correct value. Moreover, the
action of this theory and the symplectic structure are both finite. This
indicates that to properly describe gravity in asymptotically AdS spacetime
one should use the MacDowell–Mansouri theory and not the Plebanski one.

##  Appendix A Asymptotic Einstein equations

The Einstein equations can be easily produced from the evaluation of the
contacted AdS curvature, namely

$\displaystyle F^{\lambda}{}_{\mu\lambda\nu}$
$\displaystyle=R^{\lambda}{}_{\mu\lambda\nu}-\frac{\Lambda}{3}(g^{\lambda}{}_{\lambda}g_{\mu\nu}-g^{\lambda}{}_{\nu}g_{\mu\lambda})=R_{\mu\nu}-\frac{\Lambda}{3}(4g_{\mu\nu}-g_{\mu\nu})$
$\displaystyle=R_{\mu\nu}-\frac{3\Lambda}{3}g_{\mu\nu}=G_{\mu v}\,,$ (106)

as the Einstein tensor is then $G_{\mu v}=R_{\mu v}-\frac{1}{2}g_{\mu
v}R+\Lambda g_{\mu v}=R_{\mu v}-\Lambda g_{\mu v}$.

The Einstein equations decompose accordingly to
$0=G_{\mu\nu}=\bar{G}_{\mu\nu}+G^{(1)}_{\mu\nu}$, where AdS part is
identically vanishing. Their leading order in the powers of $r$ are:

  * •

$G^{(1)}_{tt}$ of order $1/r$

$\displaystyle
0=\frac{\Lambda^{2}}{3}h_{rr}-3h_{tt}-\Lambda\left(h_{\theta\theta}+\csc^{2}\theta
h_{\varphi\varphi}\right)$

  * •

$G^{(1)}_{tr}$ of order $1/r^{4}$

$\displaystyle
0=\frac{2\Lambda}{9}\partial_{t}h_{rr}-\partial_{t}h_{\theta\theta}+\partial_{\theta}h_{t\theta}+\cot\theta
h_{t\theta}-\csc^{2}\theta\partial_{t}h_{\varphi\varphi}+\csc^{2}\theta\partial_{\varphi}h_{t\varphi}$

  * •

$G^{(1)}_{t\theta}$ of order $1/r^{3}$

$\displaystyle 0$
$\displaystyle=\frac{2\Lambda}{3}\partial_{t}h_{r\theta}+\frac{\Lambda}{3}\partial_{t}\partial_{\theta}h_{{rr}}-2h_{t\theta}+\frac{2}{3}\Lambda\partial_{\theta}h_{{tr}}+\cot\theta\partial_{t}h_{\theta\theta}$
$\displaystyle+\csc^{2}\theta\left(\partial_{t}\partial_{\varphi}h_{\theta\varphi}-\partial_{t}\partial_{\theta}h_{\varphi\varphi}+\cot\theta\partial_{t}h_{\varphi\varphi}-\partial^{2}_{\varphi}h_{t\theta}+\partial_{\theta}\partial_{\varphi}h_{t\varphi}\right)$

  * •

$G^{(1)}_{t\varphi}$ of order $1/r^{3}$

$\displaystyle 0$
$\displaystyle=\Lambda\partial_{t}\partial_{\varphi}h_{rr}+2\Lambda\partial_{t}h_{r\varphi}-3\partial_{t}\partial_{\varphi}h_{\theta\theta}+3\partial_{t}\partial_{\theta}h_{\theta\varphi}+3\cot\theta\partial_{t}h_{\theta\varphi}$
$\displaystyle+3\partial_{\theta}\partial_{\varphi}h_{t\theta}-3\cot\theta\partial_{\varphi}h_{t\theta}+2\Lambda\partial_{\varphi}h_{tr}-6h_{t\varphi}-3\partial^{2}_{\theta}h_{t\varphi}+3\cot(\theta)\partial_{\theta}h_{t\varphi}$

  * •

$G^{(1)}_{rr}$ of order $1/r^{5}$

$\displaystyle
0=\frac{\Lambda^{2}}{3}h_{rr}-3h_{tt}-\Lambda\left(h_{\theta\theta}+\csc^{2}\theta
h_{\varphi\varphi}\right)$

  * •

$G^{(1)}_{r\theta}$ of order $1/r^{4}$

$\displaystyle 0$
$\displaystyle=\frac{2\Lambda^{2}}{9}\partial_{\theta}h_{rr}+3\partial_{t}h_{t\theta}-3\partial_{\theta}h_{tt}+\Lambda\cot\theta
h_{\theta\theta}+\Lambda\cot(\theta)\csc^{2}\theta h_{\varphi\varphi}$
$\displaystyle+\Lambda\csc^{2}\theta\partial_{\varphi}h_{\theta\varphi}-\Lambda\csc^{2}\theta\partial_{\theta}h_{\varphi\varphi}$

  * •

$G^{(1)}_{r\varphi}$ of order $1/r^{4}$

$\displaystyle 0$
$\displaystyle=2\Lambda^{2}\partial_{\varphi}h_{rr}-9\Lambda\partial_{\varphi}h_{\theta\theta}+9\Lambda\partial_{\theta}h_{\theta\varphi}+9\Lambda\cot\theta
h_{\theta\varphi}-27\partial_{\varphi}h_{tt}+27\partial_{t}h_{t\varphi}$

  * •

$G^{(1)}_{\theta\theta}$ of order $1/r$

$\displaystyle 0$
$\displaystyle=\frac{\Lambda^{2}}{3}h_{rr}-3h_{tt}-\Lambda\left(h_{\theta\theta}+\csc^{2}\theta
h_{\varphi\varphi}\right)$

  * •

$G^{(1)}_{\theta\varphi}$ of order $1/r^{3}$

$\displaystyle 0$
$\displaystyle=2\Lambda^{2}\partial_{\varphi}h_{r\theta}+\Lambda^{2}\partial_{\theta}\partial_{\varphi}h_{rr}+2\Lambda^{2}\partial_{\theta}h_{r\varphi}-18\Lambda
h_{\theta\varphi}$
$\displaystyle-9\partial^{2}_{t}h_{\theta\varphi}+9\partial_{t}\partial_{\varphi}h_{t\theta}-9\partial_{\theta}\partial_{\varphi}h_{tt}+9\partial_{t}\partial_{\theta}h_{t\varphi}$
$\displaystyle-\Lambda^{2}\cot\theta\partial_{\varphi}h_{rr}-4\Lambda^{2}\cot\theta
h_{r\varphi}+9\cot\theta\partial_{\varphi}h_{tt}-18\cot\theta\partial_{t}h_{t\varphi}$

  * •

$G^{(1)}_{\varphi\varphi}$ of order $1/r$

$\displaystyle 0$
$\displaystyle=\frac{\Lambda^{2}}{3}h_{rr}-3h_{tt}-\Lambda\left(h_{\theta\theta}+\csc^{2}\theta
h_{\varphi\varphi}\right)$

Note the repeated equation in the diagonal part, which validity can be tested
for instance by using the AdS-Schwarzschild perturbation scheme with

$\displaystyle h_{tt}=2GM\,,\qquad h_{rr}=2GM\frac{9}{\Lambda^{2}}\,.$ (107)

## Acknowledgment

For JKG, this work was supported by funds provided by the National Science
Center, project number 2019/33/B/ST2/00050.

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