# Finite Charges from the Bulk Action

Robert McNees1 and Céline Zwikel2

( 1Department of Physics, Loyola University Chicago Chicago, IL, USA
<EMAIL_ADDRESS>
2Perimeter Institute for Theoretical Physics,
31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
<EMAIL_ADDRESS>
###### Abstract

Constructing charges in the covariant phase space formalism often leads to
formally divergent expressions, even when the fields satisfy physically
acceptable fall-off conditions. These expressions can be rendered finite by
corner ambiguities in the definition of the presymplectic potential, which in
some cases may be motivated by arguments involving boundary Lagrangians. We
show that the necessary corner terms are already present in the variation of
the bulk action and can be extracted in a straightforward way. Once these
corner terms are included in the presymplectic potential, charges derived from
an associated codimension-2 form are automatically finite. We illustrate the
procedure with examples in two and three dimensions, working in Bondi gauge
and obtaining integrable charges. As a by-product, actions are derived for
these theories that admit a well-defined variational principle when the fields
satisfy boundary conditions on a timelike surface with corners. An interesting
feature of our analysis is that the fields are not required to be fully on-
shell.

###### Contents

  1. 1 Introduction
  2. 2 Covariant phase space and charges
    1. 2.1 Sketch of main result
    2. 2.2 Proof of finite charges at the boundary
    3. 2.3 A comment on boundary terms and variational principles
  3. 3 2d Dilaton gravity
    1. 3.1 Solution space and symmetries in linear Bondi gauge
    2. 3.2 Covariant phase space for two dimensional dilaton gravity
    3. 3.3 Charges
    4. 3.4 First variation of the action and holographic renormalization
  4. 4 3d Einstein gravity
    1. 4.1 Solution space and symmetries in Bondi-Weyl gauge
    2. 4.2 Covariant phase space for three dimensional gravity
    3. 4.3 Charges
    4. 4.4 First variation of the action and holographic renormalization
  5. 5 Conclusion
  6. A “Einstein-Hilbert” contribution to $k_{\xi}^{\mu\nu}$ in 2d dilaton gravity
  7. B Counterterms for dilaton gravity
  8. C Foliations of $\mathcal{B}$
  9. D Renormalization of the phase space in Fefferman–Graham gauge

##  1 Introduction

The covariant phase space formalism, based on a Lagrangian description of
gauge theories, can be used to compute charges associated with gauge
symmetries [1, 2, 3, 4, 5, 6, 7, 8], see also the review [9]. These charges
carry physical information about the system and label the states in the phase
space.

In the present work we are interested in theories where the behavior of the
fields is specified around a boundary; for instance the spatial boundary. For
an asymptotic symmetry $\xi$ that preserves these fall-offs, an associated
charge can be defined as the integral of a form $k_{\xi}$ over a closed,
codimension-2 surface $\mathcal{C}$ located on this boundary. The Iyer-Wald
construction [3, 4, 5] can be used to obtain a suitable form $k_{\xi}$, and
the candidate charge is then given by

$\displaystyle\not{\delta}Q_{\xi}=\lim_{r\to\infty}\int_{\mathcal{C}}k_{\xi}~{}.$
(1.1)

The limit in this expression, which refers to a coordinate $r$, is shorthand
for a procedure involving integrals over a sequence of closed surfaces
$\mathcal{C}_{r}$ that approach $\mathcal{C}$.

An immediate problem is that the limit described above may not exist for a
given $k_{\xi}$. One is then faced with the problem of somehow adjusting
$k_{\xi}$ so that the $r\to\infty$ limit of its integral over $\mathcal{C}$
exists. A common approach to this problem is to exploit so-called “corner
ambiguities” in the Iyer-Wald covariant phase space formalism [3, 4, 5, 10].
There, the codimension-2 form satisfies $\mathrm{d}k_{\xi}=\omega_{\xi}$ for a
choice of presymplectic current $\omega$ contracted with the symmetry. The
presymplectic current is itself obtained from a potential that is defined only
up to the addition of a closed form, so there is a natural ambiguity in the
choice of $\omega$ and hence the defining property of $k_{\xi}$. The goal,
then, is to find an appropriate shift in the codimension-2 form, consistent
with this ambiguity, so that the limit (1.1) exists.

Once this hurdle is cleared, there is still the issue of the integrability
property of $\not{\delta}Q_{\xi}$ 111See [11, 12, 13] for another method to
discuss the (non-)integrability of charges.. For theories without propagating
degrees of freedom it is expected that the charge is integrable. But this can
be tedious to show, especially in the presence of “leaks” at the boundary [14,
15, 9]. These leaks might be associated with a flux that could pass across the
boundary, or simply the fact that there are fields on the boundary with
unspecified dynamics, as would be the case for an open system. When there is
flux through the boundary, there is a discussion on how to split the charges
into integrable and non-integrable parts [10, 16, 17, 18].

Identifying a corner ambiguity that renders the charges finite can involve
trial-and-error, or _ad hoc_ justifications that seem to apply only to
specific examples. 222However, see [19] for a renormalization procedure for
electromagnetism for spacetime dimensions higher or equal to six. But in some
cases there are proposals that unambiguously motivate these terms. For
example, in Einstein-Hilbert gravity with asymptotically anti-de Sitter
boundary conditions, one can construct the charges working in Fefferman–Graham
gauge [20]. The corner terms used to render the charges finite for various
boundary conditions are related to symplectic potentials associated with
boundary Lagrangians [21, 22, 23, 24] used for holographic renormalization of
the action [25, 26]. Another proposal for extracting the necessary corner
contributions from a boundary Lagrangian was given in [27, 28], though it may
be difficult to determine the precise form of the boundary Lagrangian, or if
such a Lagrangian exists. Note also that the prescription advocated in [29,
30, 31] is inapplicable in presence of leaks.

However, even in a theory with asymptotically AdS boundary conditions, it may
be desirable to work in a gauge other than Fefferman-Graham. For example, one
might consider AdS spacetimes in Bondi gauge, since this description admits a
well-defined flat limit [32, 33, 34, 35]. But when the boundary metric
dynamics is fluctuating, it does not seem that the asymptotically AdS
construction referenced above can be carried out in Bondi gauge while
simultaneously preserving the flat limit [36, 37]. Specifically, it was shown
that the renormalization of the charges was not connected to the boundary
terms renormalizing the action.

In this paper, we show that there is a straightforward way of ensuring that
the right-hand side of (1.1) exists, using only the variation of the bulk
Lagrangian and the fall-offs of the fields at the boundary to identify the
necessary corner terms. We apply this prescription to two dimensional dilaton
gravity theories and three dimensional Einstein gravity (with or without a
cosmological constant) with leaky boundary conditions. These lower dimensional
theories are widely used as toy models of quantum gravity, tractable examples
of black hole evaporation, and concrete examples of holographic dualities [38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 32, 48]. The resulting charges (1.1) are
not only finite, but also symplectic in the sense that the codimension-2 form
is independent of the coordinate $r$ used to define the limiting procedure
(consistent with previous results obtained in various gauges [49, 50, 36, 37,
51, 52]). With the correct field-dependence assigned to the symmetry
parameters, the charges are integrable and agree with or generalize previous
results [36, 37].

Our approach ensures that the limiting procedure used to define the charges
exists, but a residual ambiguity remains in the finite (as $r\to\infty$) part
of the codimension-2 form. However, at least for the lower-dimensional
examples considered here, we argue that one prescription for identifying the
finite part seems to be more natural in the sense that it leads to integrable
charges. In particular, for three-dimensional Einstein gravity in Bondi-Weyl
gauge (which is an extension of Bondi gauge that satisfies a weaker condition
on the determinant of the transverse metric), this explains the origin of a
finite corner term added by hand in [37] to restore integrability of the
charges.

The construction of the charges does not reference boundary terms in the
action, and there is no requirement that the first variation of the action
vanishes (though we do impose a subset of the bulk equations of motion).
However, the analysis of the asymptotic structure of the presymplectic
potential naturally identifies boundary terms which must be added to the
action to obtain a well-defined variational principle for a particular choice
of boundary conditions. For the two- and three-dimensional theories we
consider, we first obtain the charges, and then use our results for the
presymplectic potential to work out actions for fields satisfying Dirichlet
conditions at the asymptotic boundary. This generalizes results obtained in
[53, 36, 37]. In particular our construction of the action extends to cases
where the boundary includes non-smooth corners.

An additional feature of our analysis is that, unlike the usual covariant
phase space approach, the fields are not assumed to be fully on-shell.
Instead, we enforce a subset of the equations of motion sufficient to
determine the $r$-dependence of the fields. 333In the low-dimensional gravity
theories we consider, the $r$-dependence of the fields and their variations is
completely fixed. In theories with local degrees of freedom, it will only be
determined to some order in a large-$r$ expansion. The other equations of
motion, which are conjugate to fields fixed by our choice of gauge and would
normally be imposed as constraints, are not enforced. Nevertheless, we show
there is a (not unique) codimension-2 form $k_{\xi}$ sourced by a combination
of the presymplectic current and the weakly-vanishing Noether current
associated with diffeomorphism invariance. Although the focus here is lower-
dimensional theories, this result is completely general and holds for any
diffeomorphism invariant theory in any number of spacetime dimensions. It
reduces to the usual definition of the codimension-2 form when the fields are
fully on-shell, but supports additional charges for our slightly weaker
condition. This is relevant, for example, in Jackiw-Teitelboim dilaton gravity
[54, 55], which is dual to a quantum mechanical model that exhibits on-shell
breaking of conformal symmetry [42, 43, 56, 57, 58, 59]. There, relaxing one
of the equations of motion results in charges that realize a Virasoro algebra,
but the conformal symmetry is broken when the fields are fully on-shell.

#### Organization of the paper

In section 2 we state our main result and present the proof. Sections 3 and 4
illustrate our procedure applied to two dimensional dilaton gravity in linear
dilaton Bondi gauge, and on three dimensional gravity in Bondi-Weyl gauge,
respectively. In both cases, we follow up an analysis the charges with the
construction of an action principle using our adapted choice of presymplectic
potential.

Some details of the calculations in sections 3 and 4 have been place in
appendices. A final appendix D translates some of our results for three
dimensional gravity back to Fefferman-Graham gauge.

#### Notation

In general we use Greek letters $\mu,\nu,\ldots$ for indices on the full two-
or three-dimensional spacetime, and letters $a,b,\ldots$ from the beginning of
the Roman alphabet for indices on codimension-1 surfaces at finite $r$ or on
the asymptotic boundary. A generic index $i$ is used to label the field
content. Other notation is explained as it is introduced.

##  2 Covariant phase space and charges

In the first subsection we give a rough overview of the main result. This is
derived in more detail, with a slightly weaker set of assumptions, in the
second subsection. Finally, we comment on the relation between our results and
the boundary terms needed for an action with a well-defined variational
principle.

###  2.1 Sketch of main result

It is easy to see why the $r\to\infty$ limit in the definition (1.1) of the
charge $\not{\delta}Q_{\xi}$ might not exist. Consider a theory on a spacetime
$M$ and some choice of presymplectic current
$\omega^{\mu}(\Psi;\delta_{1}\Psi,\delta_{2}\Psi)$. For now we assume that the
fields and field variations satisfy the equations of motion, so $\omega$ is
closed. In the covariant phase space formalism, the codimension-2 form for an
asymptotic symmetry $\xi$ satisfies

$\displaystyle\partial_{\nu}k_{\xi}^{\mu\nu}=\omega_{\xi}^{\mu}~{}$ (2.1)

where the equality holds on-shell. The right-hand side of this equation is
shorthand for the presymplectic current with one of the field variations given
by the action of $\xi$ on the fields. Working in coordinates where the
boundary is the $r\to\infty$ limit of a constant $r$ surface $\mathcal{B}$,
let $\mathcal{N}$ be a surface that intersects $\mathcal{B}$ at a closed,
codimension-2 surface $\mathcal{C}$. On a neighborhood of $\mathcal{B}$ the
surface $\mathcal{N}$ is an isosurface of some other coordinate $u$, and the
charge associated with $\xi$ is then obtained as

$\displaystyle\not{\delta}Q_{\xi}=\lim_{r\to\infty}\int_{\mathcal{C}}k_{\xi}^{ur}~{}.$
(2.2)

Denoting coordinates on $\mathcal{C}$ by $x^{A}$, this component of the
codimension-2 form satisfies

$\displaystyle\partial_{r}k_{\xi}^{ur}+\partial_{A}k_{\xi}^{uA}=\omega_{\xi}^{u}~{}.$
(2.3)

If $\omega_{\xi}^{u}$ does not fall off fast enough for large $r$, then the
integral over $\mathcal{C}$ of $k_{\xi}^{ur}$ will diverge in the limit
$r\to\infty$, and the charge is not defined. 444One might argue that this
means the boundary conditions must be tightened, so that the fields and their
variations in $\omega_{\xi}^{u}$ fall off more rapidly with $r$. Our
assumption is that the boundary conditions necessary for the theory to admit a
sufficiently wide array of solutions have already been determined, and faster
fall off in $r$ would be too restrictive. Working at large but finite $r$, one
typically finds that $k_{\xi}^{ur}$ can be written as a part that diverges, a
part that remains finite, and terms that vanish in this limit. Even if one
could simply discard the divergent part, the finite piece that remains may not
describe the expected charge [18, 37].

To address these problems we take advantage of a natural ambiguity in the
Iyer-Wald formalism to identify a choice of presymplectic current such that
(2.2) is well-defined. This choice is not unique; a residual ambiguity will
remain. Once the boundary conditions are set, the behavior of the fields is
known to some order in a large-r expansion near $\mathcal{B}$. Then the
components of the presymplectic potential along $\mathcal{B}$ can be
integrated, allowing us to write

$\Theta^{a}(\Psi;\delta\Psi)=\partial_{r}\int
dr\,\Theta^{a}(\Psi;\delta\Psi)~{},$ (2.4)

with $x^{a}=(u,x^{A})$ coordinates on $\mathcal{B}$. Here, the integral is
meant to capture the parts of $\Theta^{a}$ common to all field configurations
– there may also be subleading terms associated with degrees of freedom not
fixed by the boundary conditions. Expressing the components $\Theta^{a}$ as
total $r$-derivatives, the bulk divergence of the presymplectic potential can
now be written

$\displaystyle\partial_{\mu}\Theta^{\mu}(\Psi;\delta\Psi)=\partial_{r}\left(\Theta^{r}(\Psi;\delta\Psi)+\partial_{a}\int
dr\,\Theta^{a}(\Psi;\delta\Psi)\right)$
$\displaystyle=:\partial_{r}\Theta^{r}_{\text{ren.}}(\Psi;\delta\Psi)$ (2.5)
$\displaystyle\text{ with }\Theta^{a}_{\text{ren.}}(\Psi;\delta\Psi)$
$\displaystyle\simeq 0~{}$ (2.6)

Essentially, we shift components of the presymplectic potential along
$\mathcal{B}$ to zero (with $\simeq$ indicating the possibility of subleading
terms) while preserving their contributions to $\partial_{\mu}\Theta^{\mu}$ as
“corner terms” in a new $r$-component. This shift in the potential changes
both the presymplectic current and the associated codimension-2 form. For the
lower-dimensional examples we consider in this paper, $\Theta^{u}$ is the same
for all field configurations and $\Theta_{\text{ren}}^{u}=0$ with no
subleading terms. Then $\omega_{\text{ren}}^{u}$ is exactly zero and the
integral of (2.3) over $\mathcal{C}$ gives

$\partial_{r}\int_{\mathcal{C}}k^{ur}_{\xi}+\int_{\mathcal{C}}\partial_{A}k^{uA}_{\xi}=0~{}.$
(2.7)

The surface $\mathcal{C}$ is closed, so the second term vanishes, and hence
the $r\to\infty$ limit exists in the definition of the charge (2.2). The
codimension-2 form obtained from the new presymplectic current satisfies

$\displaystyle\partial_{r}\partial_{a}k_{\xi}^{ra}\simeq 0~{},$ (2.8)

such that $\partial_{a}k_{\xi}^{ra}$ is independent of $r$. For theories with
dynamical degrees of freedom the statements above may apply only up to sub-
leading terms in a large-$r$ expansion near $\mathcal{B}$. This is discussed
in the conclusion.

This is the main result of this paper: We give a prescription to obtain finite
charges based solely on corner terms identified in the bulk presymplectic
potential. The result does not rely on a particular choice of gauge, or the
presence of a particular set of boundary terms in the action.

Note that the conditions above do not completely fix the ambiguity in the
definition of the presymplectic potential in the Iyer-Wald formalism. In
passing from (2.4) to (2.5), one could still shift $\Theta^{r}_{\text{ren.}}$
by an $r$-independent piece. However, for the theories we consider, which do
not have local degrees of freedom in the bulk, a suitable choice of gauge
naturally gives the relevant components of $\Theta^{a}$ as total
$r$-derivatives of the fields and their variations. These expressions give
corner terms in $\Theta^{r}_{\text{ren.}}$ with both $r$-dependent and
$r$-independent parts, and result in integrable charges $\delta Q_{\xi}$.
Thus, for these theories, corner contributions extracted from the bulk
presymplectic potential are all that is needed to ensure that the construction
in (2.2) both exists and gives integrable charges.

###  2.2 Proof of finite charges at the boundary

Let us now give a more detailed derivation of the results sketched out above.
The setting is a diffeomorphism-invariant theory on an $n+1$-dimensional
spacetime $M$, describing a collection of tensor fields $\Psi_{i}$ that
includes the metric $g$. The definition of the theory specifies boundary
conditions for the fields, expressed as conditions on their behavior with
respect to some coordinate $r$ as $r\to\infty$. These conditions may refer to
quantities that vanish or diverge in this limit, so we work on a spacetime $M$
with a regulating boundary $\partial M$ that includes a component
$\mathcal{B}$ which is an isosurface of $r$. Calculations are carried out at
large but finite $r$, with the limit $r\to\infty$ taken at the end. (This
final step may be implicit in some our results, and we routinely use
$\mathcal{B}$ to refer to both the surface at finite $r$ and the component of
the boundary at $r\to\infty$.) The boundary may have other components, as
well, which possibly intersect $\mathcal{B}$ at corners that comprise
$\partial\mathcal{B}$. The fact that $\mathcal{B}$ may have a boundary
$\partial\mathcal{B}$ requires careful attention later on when dealing with
total derivatives appearing in boundary integrals.

The theory is described by a variational principle that considers all field
configurations satisfying the boundary conditions, on manifolds $M$ with a
boundary that includes $\mathcal{B}$ with corners $\partial\mathcal{B}$. These
field configurations are weighted by an action functional of the form

$\displaystyle\Gamma=\int_{M}\\!\\!\mathrm{d}^{n+1}x\,L_{M}+\int_{\partial
M}\\!\\!\\!\mathrm{d}^{n}x\,L_{\partial M}~{}.$ (2.9)

The boundary Lagrangian $L_{\partial M}$, which may have support on some
components of $\partial M$ and not others, will not be relevant for our
results on charges. Everything that is needed comes from the bulk part of the
action, which changes under an infinitesimal variation of the fields as

$\displaystyle\delta
L_{M}=E^{i}\,\delta\Psi_{i}+\partial_{\mu}\Theta^{\mu}(\Psi;\delta\Psi)~{}.$
(2.10)

The tensor densities $E^{i}$ are the equations of motion and $\Theta^{\mu}$ is
the presymplectic potential. Now consider a second variation of the fields.
Without loss of generality these field variations can be taken to be
independent, so they commute. It follows that the presymplectic current
$\omega^{\mu}$, defined as

$\displaystyle\omega^{\mu}(\Psi;\delta_{1}\Psi,\delta_{2}\Psi)=\delta_{2}\Theta^{\mu}(\Psi;\delta_{1}\Psi)-\delta_{1}\Theta^{\mu}(\Psi;\delta_{2}\Psi)~{},$
(2.11)

satisfies

$\displaystyle\partial_{\mu}\omega^{\mu}+\delta_{2}E^{i}\,\delta_{1}\Psi_{i}-\delta_{1}E^{i}\,\delta_{2}\Psi_{i}=0~{}.$
(2.12)

This is just a consequence of $[\delta_{2},\delta_{1}]=0$ applied to (2.10);
it holds independent of the equations of motion. The presymplectic current is
conserved, $\partial_{\mu}\omega^{\mu}=0$, when we restrict to field
variations that satisfy the equation of motion.

Under a diffeomorphism $x^{\mu}\to x^{\mu}+\xi^{\mu}$ the fields $\Psi_{i}$
change by their Lie derivative along $\xi$. The response of the bulk
Lagrangian is

$\displaystyle\delta_{\xi}L_{M}=E^{i}\,\delta_{\xi}\Psi_{i}+\partial_{\mu}\Theta^{\mu}(\Psi;\delta_{\xi}\Psi).$
(2.13)

After integration-by-parts this can be written as

$\displaystyle\delta_{\xi}L_{M}=\xi^{\mu}N_{\mu}+\partial_{\mu}\Big{(}\Theta^{\mu}(\Psi;\delta_{\xi}\Psi)+S^{\mu}_{\xi}\Big{)}.$
(2.14)

The quantity $N_{\mu}$ – a combination of the equations of motion, the fields,
and their derivatives – vanishes as a consequence of diffeomorphism
invariance. These Noether identities take the form

$\displaystyle
N_{\mu}=E^{i}\partial_{\mu}\Psi_{i}-\partial_{\nu}\big{(}E^{i}\Psi_{i}\big{)}^{\nu}{}_{\mu}=0~{},$
(2.15)

where $\big{(}E^{i}\Psi_{i}\big{)}^{\nu}{}_{\mu}$ denotes the contractions of
the tensor densities $E^{i}$ and tensors $\Psi_{i}$ appearing in

$\displaystyle
E^{i}\,\delta_{\xi}\Psi_{i}=E^{i}\,\xi^{\mu}\partial_{\mu}\Psi_{i}+\big{(}E^{i}\Psi_{i}\big{)}^{\mu}{}_{\nu}\,\partial_{\mu}\xi^{\nu}~{}.$
(2.16)

The current $S^{\mu}_{\xi}$ appearing as a divergence in (2.14) is then

$\displaystyle S^{\mu}_{\xi}=\xi^{\nu}\,(E^{i}\,\Psi_{i})^{\mu}{}_{\nu}~{}.$
(2.17)

This is the “weakly vanishing Noether current”. It is zero when the fields are
fully on-shell – when all the equations of motion $E^{i}=0$ are satisfied –
but may have non-zero components if one or more of the equations of motion are
not enforced.

Diffeomorphism invariance implies $N_{\mu}=0$ for all field configurations,
independent of whether the equations of motion hold. Thus, given two sets of
fields that differ by infinitesimal $\delta\Psi_{i}$, the difference $\delta
N_{\mu}=N_{\mu}(\Psi+\delta\Psi)-N_{\mu}(\Psi)$ must also vanish. Linearizing
(2.15), we have

$\displaystyle\delta N_{\mu}=\delta
E^{i}\,\partial_{\mu}\Psi_{i}+E^{i}\partial_{\mu}\delta\Psi_{i}-\partial_{\nu}\delta\big{(}E^{i}\Psi_{i}\big{)}^{\nu}{}_{\mu}=0~{}.$
(2.18)

Contracting this with $\xi^{\mu}$, and recalling that the $E^{i}$ transform as
tensor densities, leads to the important identity

$\displaystyle\delta
E^{i}\,\delta_{\xi}\Psi_{i}-\delta_{\xi}E^{i}\,\delta\Psi_{i}=\partial_{\mu}\Big{(}\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i}\big{)}^{\mu}{}_{\nu}-\xi^{\mu}\,E^{i}\,\delta\Psi_{i}\Big{)}~{}.$
(2.19)

The left-hand side takes the same form as the terms appearing in (2.12), for a
generic field variation $\delta_{2}=\delta$ and a field variation
$\delta_{1}=\delta_{\xi}$ with the same form as a diffeomorphism. Combining
these results gives

$\displaystyle\partial_{\mu}\Big{(}\omega^{\mu}_{\xi}+\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i})^{\mu}{}_{\nu}-\xi^{\mu}\,E^{i}\,\delta\Psi_{i}\Big{)}=0~{},$
(2.20)

where $\omega^{\mu}_{\xi}=\omega^{\mu}(\Psi;\delta_{\xi}\Psi;\delta\Psi)$.
This equation holds off-shell, like the other results above, and is
straightforward to verify for specific models. Since the divergence (2.20)
vanishes, it is always possible to find a (non-unique) codimension-2 form with
components $k^{\mu\nu}_{\xi}$ that satisfy

$\displaystyle\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}_{\xi}+\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i})^{\mu}{}_{\nu}-\xi^{\mu}\,E^{i}\,\delta\Psi_{i}~{}.$
(2.21)

This is a consequence of diffeomorphism invariance that holds for any field
configuration, not just solutions of the equations of motion. 555See the
discussion around equation (2.2.67) in [9] for a similar result. When the
fields are fully on-shell it reduces to the Iyer-Wald definition.

Covariant phase space methods impose the equations of motion $E^{i}=0$ for the
fields. Here, we take a different approach and enforce the slightly weaker
condition $E^{i}\,\delta\Psi_{i}=0$. This can be accomplished by gauge fixing
components of some of the fields, restricting to field variations that
preserve this gauge, and enforcing only some of the equations of motion. For
example, the Bondi gauge used in the next few sections fixes a particular
component $g_{rr}$ of the spacetime metric. Field variations that preserve
this gauge satisfy $\delta g_{rr}=0$. As a result, the term $E^{rr}\,\delta
g_{rr}$ appearing in $E^{i}\,\delta\Psi_{i}$ will vanish even if the component
of the equation of motion conjugate to $g_{rr}$ (which would normally be
enforced as a constraint) does not vanish: $E^{rr}\neq 0$. Field
configurations that satisfy

$\displaystyle E^{i}\,\delta\Psi_{i}=0$ (2.22)

in this way, through a combination of equations of motion and gauge
conditions, will be referred to as “partially on-shell.” For partially on-
shell fields, the defining equation (2.21) for the codimension-2 form reduces
to

$\displaystyle\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}_{\xi}+\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i})^{\mu}{}_{\nu}~{}.$
(2.23)

The motivation for imposing (2.22) rather than the stronger condition
$E^{i}=0$ comes from the last term in this equation. Now the variation of the
weakly vanishing Noether current may have non-zero components that contribute
to the codimension-2 form, supporting charges that are not present when
$E^{i}=0$. Such charges appear, for instance, in gravitational theories dual
to quantum mechanical models that exhibit a larger symmetry algebra when off-
shell, which is then broken to something simpler when the fields are fully on-
shell [42, 57, 60, 61].

There are natural ambiguities in the definitions of both $\Theta^{\mu}$ and
$\omega^{\mu}$ which may or may not affect the defining property (2.23) of the
codimension-2 form. For example, boundary terms in the action can be thought
of as shifting the presymplectic potential by a $\delta$-exact contribution;
i.e. $\Theta\to\Theta+\delta L_{\partial M}$. However, such a shift has no
effect on $\omega^{\mu}$, and a boundary term in the action does not change
the equations of motion, so the result (2.21) is not sensitive to this sort of
ambiguity in the definition of $\Theta$. 666A shift of this sort is relevant
for the variational problem, as we will discuss below. On the other hand, it
is clear from (2.10) that the presymplectic potential is only defined up to a
shift $\Theta^{\mu}\to\Theta^{\mu}+\partial_{\nu}\Upsilon^{\mu\nu}$ for an
anti-symmetric $\Upsilon^{\mu\nu}(\Psi;\delta\Psi)$ that is linear in the
field variations. This _does_ change $\omega^{\mu}$, by a term

$\displaystyle\omega^{\mu}(\Psi;\delta_{1}\Psi;\delta_{2}\Psi)\to\omega^{\mu}(\Psi;\delta_{1}\Psi;\delta_{2}\Psi)+\partial_{\nu}\mathcal{Y}^{\mu\nu}(\Psi;\delta_{1}\Psi,\delta_{2}\Psi)~{},$
(2.24)

where

$\displaystyle\mathcal{Y}^{\mu\nu}(\Psi;\delta_{1}\Psi,\delta_{2}\Psi)=\delta_{2}\Upsilon^{\mu\nu}(\Psi;\delta_{1}\Psi)-\delta_{1}\Upsilon^{\mu\nu}(\Psi;\delta_{2}\Psi)~{}.$
(2.25)

The shift in $\omega^{\mu}$ is consistent with both (2.12) and (2.20), but it
contributes a new term to (2.21) that changes $k^{\mu\nu}_{\xi}$ by

$\displaystyle k^{\mu\nu}_{\xi}\to
k^{\mu\nu}_{\xi}+\mathcal{Y}^{\mu\nu}(\Psi;\delta_{\xi}\Psi,\delta\Psi)~{}.$
(2.26)

For the models we consider, there is a natural prescription for
$\Upsilon^{\mu\nu}$, or equivalently $\mathcal{Y}^{\mu\nu}$, such that the
resulting codimension-2 form gives finite charges at the boundary.

Consider now the component $\mathcal{B}$ of $\partial M$, defined as a surface
of constant $r$. Coordinates along this surface are denoted by $x^{a}$. Since
we know the fall-offs of the fields close to the boundary 777 For instance in
Bondi gauge these can be obtained once we partially solve the equations of
motion., the components of the presymplectic potential along $\mathcal{B}$ can
be written as total derivatives with respect to $r$:

$\displaystyle\Theta^{a}\simeq\partial_{r}Y^{ar}~{}.$ (2.27)

This is trivially true once the $r$-dependence of the fields is known; simply
integrating $\Theta^{a}$ gives $Y^{ar}$ up to an $r$-independent ambiguity.
However, for the theories we consider, some of the components can already be
put in the form (2.27) simply by evaluating $\Theta^{a}$ in Bondi gauge and
applying integration-by-parts. For those components, the resulting $Y^{ar}$
are functions of the spacetime fields and their variations, and give specific
$r$-dependent and $r$-independent parts when the fields are partially on-
shell. Making a shift
$\Theta^{\mu}\to\Theta^{\mu}+\partial_{\nu}\Upsilon^{\mu\nu}$ and choosing
$\Upsilon^{\mu\nu}$ to set the components $\Theta^{a}$ to zero, gives
$\Upsilon^{ar}=-Y^{ar}$ up to the residual ambiguity mentioned above. As a
result, we have

$\displaystyle\Theta^{r}\to\Theta^{r}+\partial_{a}Y^{ar}\vphantom{\Big{|}}$
(2.28) $\displaystyle\omega^{a}(\Psi;\delta_{1}\Psi;\delta_{2}\Psi)\to
0\vphantom{\Big{|}}$ (2.29)
$\displaystyle\omega^{r}(\Psi;\delta_{1}\Psi;\delta_{2}\Psi)\to\omega^{r}(\Psi;\delta_{1}\Psi;\delta_{2}\Psi)+\partial_{a}\Big{(}\delta_{2}Y^{ar}(\Psi;\delta_{1}\Psi)-\delta_{1}Y^{ar}(\Psi;\delta_{2}\Psi)\Big{)}~{},$
(2.30)

along with a corresponding shift (2.26) to the codimension-2 form. Note that
both $\Theta^{r}$ and $\omega^{r}$, which naturally appear in integrals over
the constant $r$ surface $\mathcal{B}$, acquire corner terms (total
derivatives on $\mathcal{B}$) with support on $\partial\mathcal{B}$.

There is one additional modification to the codimension-2 form coming from the
variation of the weakly vanishing Noether current in (2.23). For the theories
we consider, its components along $\mathcal{B}$ for an asymptotic symmetry
$\xi$ can also be written as total $r$-derivatives

$\displaystyle\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i}\big{)}^{a}{}_{\nu}=\partial_{r}\mathcal{Z}^{ar}_{\xi}~{}.$
(2.31)

As a result, (2.20) can be expressed entirely as a total $r$-derivative

$\displaystyle\partial_{r}\Big{(}\omega^{r}_{\xi}+\partial_{a}\mathcal{Y}^{ar}_{\xi}+\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i}\big{)}^{r}{}_{\nu}+\partial_{a}\mathcal{Z}^{ar}_{\xi}\Big{)}=0~{},$
(2.32)

where $\mathcal{Y}^{ar}_{\xi}$ denotes the shift in (2.26). This gives a
family of codimension-2 forms $k^{\mu\nu}_{\xi}$ that satisfy

$\displaystyle\partial_{a}k^{ra}_{\xi}=\omega^{r}_{\xi}+\partial_{a}\mathcal{Y}^{ar}_{\xi}+\xi^{\nu}\,\delta\big{(}E^{i}\Psi_{i}\big{)}^{r}{}_{\nu}+\partial_{a}\mathcal{Z}^{ar}_{\xi}~{}.$
(2.33)

The quantity $\partial_{a}k^{ra}_{\xi}$ is independent of $r$, by virtue of
(2.32). Compared to the codimension-2 form obtained from (2.21), it is shifted
by

$\displaystyle k^{ra}_{\xi}\to
k^{ra}_{\xi}+\mathcal{Y}^{ar}_{\xi}+\mathcal{Z}^{ar}_{\xi}~{},$ (2.34)

up to a residual ambiguity which is independent of $r$.

In the lower-dimensional theories we investigate, setting
$\Theta^{u}\to\Theta_{\text{ren}}^{u}=0$ via the Iyer-Wald ambiguity described
above implies $\omega^{u}\to\omega_{\text{ren}}^{u}=0$. The codimension-2 form
then satisfies

$\displaystyle\partial_{r}k^{ur}_{\xi}+\partial_{A}k^{uA}_{\xi}=0~{},$ (2.35)

so that the $r\to\infty$ limit of $k_{\xi}^{ur}$ integrated over $\mathcal{C}$
exists. Alternately, one can begin with the right-hand side of (2.33), which
is explicitly independent of $r$ by virtue of (2.32), and use integration by
parts to obtain $k_{\xi}^{ur}$ up to total derivatives on $\mathcal{C}$. The
result is not simply the finite part of the problematic codimension-2 form
from our earlier statement of the problem; the new terms in (2.34) have also
contributed terms independent of $r$. Accounting for these finite shifts, we
will find that the charges are now integrable once proper field-dependence is
assigned to the generators of the asymptotic symmetries. Thus, we have
obtained a codimension-2 form such that the construction (2.2) of the charge
associated with the symmetry $\xi$ exists and is integrable, using only
quantities arising in the variation of the bulk Lagrangian.

This result can be generalized to anomalous Lagrangians, and to state
dependent asymptotic symmetries following the mathematical framework developed
for instance in [24, 27]. It can also be generalized for fields transforming
with higher derivatives of the symmetry parameters, see for instance [62].

###  2.3 A comment on boundary terms and variational principles

The construction detailed above makes no reference to specific boundary terms
in the action (2.9). But it is relevant for identifying boundary terms that
must be added to the bulk action to obtain a well-defined variational
principle for a given set of boundary conditions, including cases where
$\mathcal{B}$ is taken to have a boundary $\partial\mathcal{B}$.

When the fields are partially on-shell ($E^{i}\delta\Psi_{i}=0$) or fully on-
shell ($E^{i}=0$), the divergence of the presymplectic potential is the bulk
contribution to the variation of the action (2.10). It has support on
$\partial M$, including terms on $\mathcal{B}$ that typically do not vanish as
$r\to\infty$ for the full range of field variations that preserve the boundary
conditions. This is addressed by adding boundary terms to the action, a
procedure that will be discussed in more detail for specific models later in
this paper. The point we wish to make here is that the structure of the
shifted presymplectic potential – not just $\Theta^{r}$ on $\mathcal{B}$, but
also the corner terms obtained from the components $\Theta^{a}$, which
contribute at $\partial\mathcal{B}$ – will determine some of the boundary
terms which must be included in the action.

In the following sections we construct the codimension-2 form and identify the
charges for partially on-shell fields in lower-dimensional gravitational
theories. Then we apply what was learned about the structure of the
presymplectic potential to obtain an action with a well defined variational
principle for (Dirichlet) boundary conditions on the fields at $r\to\infty$.
In each example the charges are calculated first, to emphasize that our
prescription is not sensitive to boundary terms in the action, or how one
carries out “holographic renormalization” or related procedures.

##  3 2d Dilaton gravity

In this section we apply our prescription for the charges to the $UV$-family
of dilaton gravity theories in 2d dimensions, for a review see [41]. The bulk
Lagrangian is

$\displaystyle
L_{M}=\frac{1}{2\,\kappa^{2}}\,\sqrt{-g}\,\Big{(}X\,R-U(X)\,(\nabla
X)^{2}-2\,V(X)\Big{)}~{}.$ (3.1)

where $g_{\mu\nu}$ is the two-dimensional metric, $X$ is the dilaton, and $U$
and $V$ are two arbitrary functions of the dilaton.

We first present the solution space in linear dilaton Bondi gauge, and explain
which equations of motion are enforced for the partially on-shell fields. The
most general symmetries and transformation laws that preserve the form of the
fields are determined, with special care given to an on-shell Killing vector
that is present in all 2D dilaton gravity theories. When the theory is not
fully on-shell, as is the case here, this diffeomorphism acts on the metric at
the boundary. Incorporating it into the symmetries of the theory identifies a
specific field-dependence of the symmetry parameters which naturally leads to
integrable charges in the presence of corners. The adapted covariant phase
space construction outlined in the previous section ensures that our results
are finite as $r\to\infty$ . Carrying out this calculation before discussing
the variational principle emphasizes two points. First, the procedure is
independent of how “holographic renormalization” is implemented in the action.
And second, the terms needed to cancel divergences in the codimension-2 form
$k^{\mu\nu}_{\xi}$ are obtained directly from the variation of the bulk
Lagrangian, at the level of the presymplectic potential – there is no need for
_ad hoc_ subtractions. We also note that our prescription selects a specific
finite part of the codimension-2 form, and comment on its relation to the
charges that were considered in [36]. As pointed out in the previous section,
the construction of the charges takes place without the fields being fully on-
shell.

Next, we revisit the first variation of the action and discuss the variational
principle in the presence of corners at spatial infinity. This leads to a
discussion of holographic renormalization that generalizes the results of [53,
61] when spatial infinity is not obtained as the limit of an isocurve of the
dilaton. In the case of models with no kinetic term ($U=0$) the boundary terms
required for a well-defined variational principle (and a finite on-shell
action) are similar to the ones derived in [36]. However, the origin of these
terms is more clear in the present construction.

###  3.1 Solution space and symmetries in linear Bondi gauge

The first variation of the bulk Lagrangian (3.1) takes the form

$\displaystyle\delta L_{M}=(E^{\mu\nu}\,\delta
g_{\mu\nu}+E_{X}\,\delta{X})+\partial_{\mu}\Theta^{\mu}~{},$ (3.2)

where the equations of motion are

$\displaystyle E^{\mu\nu}$
$\displaystyle=\frac{\sqrt{-g}}{2\kappa^{2}}\left(\nabla^{\mu}\nabla^{\nu}X-g^{\mu\nu}\,\nabla^{2}X-g^{\mu\nu}\,V+U\,\Big{(}\nabla^{\mu}X\nabla^{\nu}X-\frac{1}{2}g^{\mu\nu}(\nabla
X)^{2}\Big{)}\right)$ (3.3) $\displaystyle E_{X}$
$\displaystyle=\frac{\sqrt{-g}}{2\kappa^{2}}\left(R+\partial_{X}U\,(\nabla
X)^{2}+2\,U\,\nabla^{2}X-2\partial_{X}V\right)~{}.$ (3.4)

The presymplectic potential receives contributions from both the metric and
the dilaton

$\displaystyle\Theta^{\mu}=\frac{\sqrt{-g}}{2\,\kappa^{2}}\,\Big{(}-2U\,\nabla^{\mu}X\,\delta
X+(g^{\mu\lambda}g^{\nu\sigma}-g^{\mu\nu}g^{\lambda\sigma})\big{(}X\,\nabla_{\nu}\delta
g_{\lambda\sigma}-{\delta}g_{\lambda\sigma}\nabla_{\nu}X\big{)}\Big{)}~{}.$
(3.5)

##### Linear dilaton Bondi gauge

The coordinate gauge is partially fixed by choosing a spacelike coordinate $r$
such that the other coordinate $u$ is null, which implies $g_{rr}=0$, and the
dilaton is linear in $r$.

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=2B(u,r)\,\mathrm{d}u^{2}-2\,e^{A(u,r)}\,\mathrm{d}u\,\mathrm{d}r$
(3.6) $\displaystyle X$ $\displaystyle=r\,e^{-q(u)}+\varphi(u)~{}.$ (3.7)

The condition on the dilaton is the 2D equivalent of specifying the transverse
metric in higher dimensional Bondi gauge. The constraint associated with
fixing $g_{rr}$, which would normally be enforced by setting the $E^{rr}$
component of the metric equation of motion to zero, is relaxed. The remaining
equations of motion are sufficient to completely determine the $r$-dependence
of the other fields. Solving the $E^{uu}$ and $E^{ur}$ components of the
metric equations of motion, the functions $A(u,r)$ and $B(u,r)$ in the metric
take the form

$\displaystyle A(u,r)=$ $\displaystyle\,\,a(u)+Q(X)$ (3.8a) $\displaystyle
B(u,r)=$
$\displaystyle\,\,-\frac{1}{2}\,e^{Q(X)}e^{2(a(u)+q(u))}\Big{(}w(X)+2\,e^{-(a(u)+q(u))}\,\partial_{u}X-2\,\mathcal{M}(u)\Big{)}~{}.$
(3.8b)

Here $Q(X)$ and $w(X)$ are familiar functions constructed from $U(X)$ and
$V(X)$ as

$\displaystyle Q(X)$
$\displaystyle=\int^{X}\negthickspace\negthickspace\negthickspace
dY\,U(Y)\qquad\qquad w(X)=-2\,\int^{X}dYe^{Q(Y)}\,V(Y)~{}$ (3.9)

The final term in $B(u,r)$ involves the Casimir 888When the dilaton gravity
admits a Poisson Sigma model formulation, $\mathcal{M}(u)$ is typically a
Casimir function when the fields are fully on-shell. $\mathcal{M}(u)$. Thus,
the fields in linear dilaton Bondi gauge are described by a set of four
functions $q(u)$, $\varphi(u)$, $a(u)$, and $\mathcal{M}(u)$. This matches the
results obtained in appendix A of [36].

With the fields parameterized as above, the $E^{rr}$ component of the metric
equation of motion becomes

$\displaystyle
E^{rr}=\frac{1}{2\kappa^{2}}\,e^{a(u)+q(u)}e^{-A(u,r)}\,\partial_{u}\mathcal{M}(u)~{}.$
(3.10)

Hence, the constraint $E^{rr}=0$ associated with the choice of coordinate
gauge $g_{rr}=0$ would imply constant $\mathcal{M}$. As stated earlier, we
relax this condition and consider the partially on-shell theory where
$\mathcal{M}$ is instead an arbitrary function of $u$. In the static case
$\partial_{u}\mathcal{M}=0$ the conventions above reproduce (after
transforming to Schwarzschild gauge and defining a new coordinate $\tilde{r}$)
the conventions of [53].

##### Symmetries and transformation laws

A diffeomorphism $\xi^{\mu}$ that preserves the gauge condition $g_{rr}=0$
requires $\xi^{u}$ independent of $r$, while preserving the form of the
dilaton fixes $\xi^{r}$ to also be linear in $r$,

$\displaystyle\xi=\epsilon(u)\,\partial_{u}+\Big{(}r\,\chi(u)+\eta(u)\Big{)}\,\partial_{r}~{}.$
(3.11)

Under this diffeomorphism the fields $A$ and $B$ appearing in the metric
transform as

$\displaystyle\delta_{\xi}A$
$\displaystyle=\partial_{u}\epsilon+\epsilon\,\partial_{u}A+\eta\partial_{r}A+\chi(1+r\partial_{r}A)$
(3.12) $\displaystyle\delta_{\xi}B$
$\displaystyle=2B\partial_{u}\epsilon+\epsilon\,\partial_{u}B-e^{A}\partial_{u}\big{(}r\,\chi(u)+\eta(u)\big{)}+\big{(}r\,\chi(u)+\eta(u)\big{)}\partial_{r}B~{}.$
(3.13)

In terms of the integration functions $\epsilon$, $\chi$, and $\eta$, the set
of functions of $u$ parameterizing the field configurations transform as

$\displaystyle\delta_{\xi}q=\epsilon\,\partial_{u}q-\chi\qquad\delta_{\xi}\varphi=\epsilon\,\partial_{u}\varphi+\eta\,e^{-q}$
(3.14)
$\displaystyle\delta_{\xi}a=\partial_{u}\epsilon+\epsilon\,\partial_{u}a+\chi\qquad\delta_{\xi}\mathcal{M}=\epsilon\,\partial_{u}\mathcal{M}=\epsilon\,e^{-(a+q)}\,e^{A}\,2\kappa^{2}\,E^{rr}~{}.$
(3.15)

There is one diffeomorphism of particular interest among these
transformations. In two dimensions, any solution of a dilaton gravity theory
possesses a Killing vector that also leaves the dilaton invariant [63, 64].
For the bulk Lagrangian (3.1) this vector takes the form

$\displaystyle\xi_{X}=e^{Q(X)}\,\epsilon^{\mu\nu}\,\partial_{\mu}X\,\partial_{\nu}~{},$
(3.16)

with $Q$ defined in (3.9) and $\epsilon^{\mu\nu}$ related to the Levi-Civita
alternating symbol by $\epsilon^{\mu\nu}=\varepsilon^{\mu\nu}/\sqrt{-g}$. By
construction, this diffeomorphism leaves the dilaton invariant. However, since
we are not fully on-shell, it acts non-trivially on the metric. In Bondi gauge
$\sqrt{-g}=e^{A}$, so using (3.6) and the fields (3.8a),(3.8b) this vector is

$\displaystyle\xi_{X}=e^{-(a+q)}\partial_{u}+\Big{(}e^{-(a+q)}\,r\,\partial_{u}q-e^{-a}\,\partial_{u}\varphi\Big{)}\partial_{r}~{}.$
(3.17)

This has the same general form as (3.11), and from (3.14)-(3.15) we see that
only $\mathcal{M}(u)$ transforms

$\displaystyle\delta_{\xi_{X}}\mathcal{M}=e^{-(a+q)}\,\partial_{u}\mathcal{M}~{}.$
(3.18)

Requiring that $\xi_{X}$ is among the asymptotic symmetries generated by
(3.11) suggests a specific field-dependence of the functions $\epsilon$,
$\chi$, and $\eta$. If we rewrite these functions as

$\displaystyle\epsilon=$
$\displaystyle\,\,\bar{\epsilon}\,e^{-(a+q)}\,,\qquad\chi=\,\,\bar{\chi}+\bar{\epsilon}\,e^{-(a+q)}\,\partial_{u}q\,,\qquad\eta=\,\,\bar{\eta}-\bar{\epsilon}\,e^{-a}\,\partial_{u}\varphi~{}$
(3.19)

where $\bar{\epsilon},\bar{\chi},\bar{\eta}$ are functions of $u$, then the
asymptotic symmetries are generated by vectors

$\displaystyle\xi=\bar{\epsilon}\,\xi_{X}+\bar{\chi}\,r\,\partial_{r}+\bar{\eta}\,\partial_{r}~{}.$
(3.20)

For instance the fields transform under this parameterization as

$\displaystyle\delta_{\xi}\big{(}e^{a+q}\big{)}=\partial_{u}\bar{\epsilon}\qquad\delta_{\xi}q=-\bar{\chi}\qquad\delta_{\xi}\varphi=\bar{\eta}\,e^{-q}\qquad\delta_{\xi}\mathcal{M}=e^{-(a+q)}\,\bar{\epsilon}\,\partial_{u}\mathcal{M}~{}.$
(3.21)

As we will show in the next section, there is a natural choice for the
codimension-2 form $k^{ur}_{\xi}$ such that the infinitesimal charges are
integrable when the symmetry parameters $\bar{\epsilon}$, $\bar{\eta}$, and
$\bar{\chi}$ are taken to be field-independent.

Using the modified Lie bracket [65], the algebra of the asymptotic Killing
vector is given by

$\displaystyle[\xi_{1},\xi_{2}]_{\star}=\left(\bar{\eta}_{1}\,\bar{\chi}_{2}-\bar{\chi}_{1}\bar{\eta}_{2}\right)\partial_{r}-\delta_{1}\xi_{2}+\delta_{2}\xi_{1}\,,$
(3.22)

and for the slicing

$\delta\bar{\epsilon}=\delta\bar{\eta}=\delta\bar{\chi}=0$ (3.23)

the algebra is three abelian Lie algebras of smooth functions with a
Heisenberg central extension. Note that if we take the functions appearing in
(3.16) to be state independent ($\delta\epsilon=\delta\chi=\delta\eta=0$) the
algebra is the diffeomorphism along the $u$-direction in semi-direct sum with
an Heisenberg algebra, see eq. (2.24) of [36].

###  3.2 Covariant phase space for two dimensional dilaton gravity

First, let us review the construction of section 2.2 applied to dilaton
gravity in two dimensions [4, 36, 45]. The first variation of the bulk
Lagrangian in (3.1) is

$\displaystyle\delta L_{M}=E^{\mu\nu}\,\delta g_{\mu\nu}+E_{X}\,\delta
X+\partial_{\mu}\Theta^{\mu}~{},$ (3.24)

with equations of motion given in (3.3) and the presymplectic potential given
by (3.5). When the fields variations are given by a diffeomorphism,
integration by parts of the bulk term in (3.24) gives

$\displaystyle
E^{\mu\nu}\,\delta_{\xi}g_{\mu\nu}+E_{X}\,\delta_{\xi}X=\xi^{\nu}N_{\nu}+\partial_{\mu}S^{\mu}_{\xi}~{}.$
(3.25)

The first term on the right-hand side, which vanishes by Noether’s second
theorem, takes the form

$\displaystyle
N_{\nu}=E_{X}\partial_{\nu}X+E^{\mu\lambda}\partial_{\nu}g_{\mu\lambda}-2\partial_{\mu}E^{\mu}{}_{\nu}=0~{},$
(3.26)

The second term in (3.25) is the weakly vanishing Noether current
$S^{\mu}_{\xi}=\xi^{\nu}(E^{i}\,\Psi_{i})^{\mu}{}_{\nu}$, which in this case
involves only

$\displaystyle(E^{i}\Psi_{i})^{\mu}{}_{\nu}=2\,E^{\mu\lambda}\,g_{\lambda\nu}~{}.$
(3.27)

Since we have relaxed the constraint involving $E^{rr}$, this quantity and its
variation have non-vanishing components which will contribute to the
construction of the charges.

The presymplectic current $\omega^{\mu}$ is obtained from the total derivative
term in the anti-symmetrized second variation of the bulk Lagrangian.
Calculating $\omega^{\mu}$ from (3.5) is straightforward, and gives

$\displaystyle\frac{2\kappa^{2}}{\sqrt{-g}}\,\omega^{\mu}=$
$\displaystyle\,\,U\,\nabla^{\mu}X\,\delta_{1}g\,\delta_{2}X+2\,U\,\delta_{2}X\,\nabla^{\mu}\delta_{1}X+2\,U\,\nabla_{\nu}X\,\delta_{1}X(\delta_{2}g)^{\mu\nu}+\delta_{1}X\,\nabla^{\mu}\delta_{2}g$
$\displaystyle+\delta_{1}g\,\nabla^{\mu}\delta_{2}X+\delta_{2}X\,\nabla_{\nu}(\delta_{1}g)^{\mu\nu}+(\delta_{2}g)^{\mu\nu}\nabla_{\nu}\delta_{1}X+\frac{1}{2}\,(\delta_{1}g)^{\mu\nu}\,\delta_{2}g\,\nabla_{\nu}X$
(3.28)
$\displaystyle+\frac{1}{2}\,X\,\Big{[}\delta_{2}g_{\nu\lambda}\nabla^{\mu}(\delta_{1}g)^{\nu\lambda}+\delta_{1}g\,\nabla^{\mu}\delta_{2}g+(\delta_{2}g)^{\mu\nu}\nabla_{\nu}\delta_{1}g-2\,\delta_{2}g_{\nu\lambda}\nabla^{\lambda}(\delta_{1}g)^{\mu\nu}$
$\displaystyle\qquad\qquad+\delta_{2}g\nabla_{\nu}(\delta_{1}g)^{\mu\nu}\Big{]}-\big{(}1\leftrightarrow
2\big{)}~{}.$

In this expression, $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$ and $(\delta
g)^{\mu\nu}=g^{\mu\lambda}\,g^{\nu\sigma}\,\delta
g_{\lambda\sigma}=-\delta(g^{\mu\nu})$. For the special case of a field
variation $\delta_{1}$ that takes the same form as a diffeomorphism
$\delta_{\xi}$, with $\delta_{2}$ arbitrary, the result (2.20) from the
introduction implies the existence of a (not unique) codimension-2 form
$k^{\mu\nu}_{\xi}$ such that

$\displaystyle\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}(\Psi;\delta_{\xi}\Psi,\delta\Psi)+2\,\delta(E^{\mu\lambda}g_{\lambda\nu})\,\xi^{\nu}-\xi^{\mu}\,\big{(}E^{\nu\lambda}\,\delta
g_{\nu\lambda}+E_{X}\,\delta X\big{)}~{}.$ (3.29)

This result holds off-shell as explained in section 2. When the fields are
fully on-shell, the presymplectic current is conserved and (3.29) reduces to
$\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}(\Psi;\delta_{\xi}\Psi,\delta\Psi)$.
This is not the case for the field configurations described in the previous
section, which instead satisfy $E^{i}\delta\Psi_{i}=0$ through a combination
of a subset of the equations of motion ($E^{uu}=E^{ur}=E_{X}=0$) and a
coordinate gauge condition ($\delta g_{rr}=0$). In this case $\omega^{\mu}$
and $\delta(E^{\mu\lambda}g_{\lambda\nu})\xi^{\nu}$ are independently
conserved. However, it is their combination appearing in (3.29) that will be
relevant for the calculation of the charges.

For dilaton gravity described by (3.1), we find $k^{\mu\nu}_{\xi}$ satisfying
(3.29) is given by

$\displaystyle\frac{\kappa^{2}}{\sqrt{-g}}\,k^{\mu\nu}_{\xi}=$
$\displaystyle\,\,U\,\delta
X\,\big{(}\xi^{\nu}\nabla^{\mu}X-\xi^{\mu}\,\nabla^{\nu}X\big{)}+\xi^{\nu}\nabla^{\mu}\delta
X-\xi^{\mu}\nabla^{\nu}\delta X$ (3.30) $\displaystyle\,\,+\frac{1}{2}\,\delta
X\,\big{(}\nabla^{\nu}\xi^{\mu}-\nabla^{\mu}\xi^{\nu}\big{)}+\frac{1}{2}\big{(}\xi^{\mu}(\delta
g)^{\lambda\nu}-(\delta g)^{\mu\lambda}\xi^{\nu}\big{)}\nabla_{\lambda}X$
$\displaystyle\,\,+\frac{1}{4}\,X\,\Big{(}(\delta
g)^{\mu\lambda}\,\big{(}\nabla_{\lambda}\xi^{\nu}+\nabla^{\nu}\xi_{\lambda}\big{)}-(\delta
g)^{\lambda\nu}\,\big{(}\nabla_{\lambda}\xi^{\mu}+\nabla^{\mu}\xi_{\lambda}\big{)}\Big{)}~{}.$

This is the same expression obtained in [36] via the Barnich-Brandt procedure
[6, 7]. Like that earlier result, this codimension-2 form is independent of
the potential $V(X)$ appearing in the bulk Lagrangian. And while (3.30)
includes explicit factors of the dilaton kinetic term $U(X)$, this dependence
drops out when it is evaluated in linear dilaton Bondi gauge. A subtle point
related to the last line of (3.30) is discussed in Appendix A.

As explained in section 2, the charges are not obtained directly from the $ur$
component of (3.30). It is instructive to see why this does not work. First,
evaluating (3.30) for the fields (3.6)-(3.8b), we find a term that is linear
in $r$

$\displaystyle k^{ur}_{\xi}=\frac{1}{2}\,e^{-q}\,r\,\Big{(}\chi\,(\delta
a+\delta q)+\epsilon\,(\partial_{u}a\,\delta q-\partial_{u}q\,\delta a)+\delta
q\,\partial_{u}\epsilon\Big{)}+\ldots~{},$ (3.31)

where $\ldots$ denotes terms that are finite as $r\to\infty$. This
$r$-dependence (which comes from terms in $\omega^{\mu}$ that are linear in
$r$) is the obstruction to taking the $r\to\infty$ limit. A second issue, not
obviously related to the first, is that the part of (3.31) that remains finite
as $r\to\infty$ is not integrable for diffeomorphisms with field dependence
(3.19). This is not a problem per se, since a different choice of field
dependence renders the finite part integrable [36]. However, since $\xi_{X}$
is always a Killing vector when the fields are fully on-shell, one would
prefer that it appears naturally among the asymptotic symmetries for the
partially on-shell field configurations. As we will show in the next section,
both issues are resolved by accounting for corner contributions coming from
the $\omega^{u}$ component of the presymplectic current. 999Corner
contributions from components of the weakly vanishing Noether current
$\xi^{\nu}\,\delta(E^{i}\,\Psi_{i})^{\mu}{}_{\nu}$ will be relevant for 3D
gravity, but do not contribute here since $E^{uu}=E^{ur}=0$.

###  3.3 Charges

We begin this section by implementing the prescription outlined in section 2,
and showing that the result is an $r$-independent codimension-2 form that
leads to integrable charges. Nevertheless, there remain ambiguities in this
procedure, as well as ambiguities in the slicing – defined as the choice of
field dependence of the symmetries generators – which we examine in the second
part.

The coordinate gauge condition $g_{rr}=\delta g_{rr}=0$ leads to a $u$
component of the presymplectic potential (3.5) given by

$\displaystyle\kappa^{2}\,\Theta^{u}=-\frac{1}{2}\,\delta
A\,\partial_{r}X+\delta Q\,\partial_{r}X+\frac{1}{2}\,X\,\partial_{r}\delta
A~{}.$ (3.32)

Since the $r$-dependence of the fields is completely determined by the
equations of motion and linear condition on the dilaton, one could simply
impose them and integrate the result to express $\Theta^{u}$ as a total
$r$-derivative. However, we note that already in (3.32), integration-by-parts
gives

$\displaystyle\kappa^{2}\,\Theta^{u}=\partial_{r}\Big{(}\frac{1}{2}\,X\,\delta
A+X\,(\delta Q-\delta A)\Big{)}+X\,\partial_{r}\big{(}\delta A-\delta
Q\big{)}~{}.$ (3.33)

The result $A(u,r)=a(u)+Q(X)$, which follows from $E^{uu}=0$ and the condition
that the dilaton is linear in $r$, reduces this to a total derivative
$\partial_{r}Y^{ur}$ with

$\displaystyle\kappa^{2}\,Y^{ur}=\frac{1}{2}\,\delta
Q(X)\,X-\frac{1}{2}\,\delta a\,X~{}.$ (3.34)

The first term is $\delta$-exact and will not contribute to $\omega^{\mu}$ or
$k^{\mu\nu}_{\xi}$, though it will be important when we return to the
variational problem. The second term in (3.34) is not $\delta$-exact and will
therefore impact the charges. Note that $Y^{ur}$ is defined only up to an
$r$-independent piece. Here we choose to identify $Y^{ur}$ as the combination
of $A$, $X$, and their variations that naturally arises from integration by
parts in $\Theta^{u}$, before imposing any equations of motion. This leads to
integrable charges for the slicing (3.19), (3.23) associated with the Killing
vector $\xi_{X}$. Another choice for $Y^{ur}$, which was used in [36], keeps
only the r-dependent part of (3.34). An integrable slicing also exists in that
case (though it is more complicated) and yields the same charge algebra.

The prescription described in section 2 is implemented at the level of the
presymplectic potential by taking
$\Theta^{\mu}\to\Theta^{\mu}+\partial_{\nu}\Upsilon^{\mu\nu}$, with

$\displaystyle\kappa^{2}\,\Upsilon^{ur}=-\kappa^{2}\,Y^{ur}=-\frac{1}{2}\,\delta
Q(X)\,X+\frac{1}{2}\,\delta a\,X~{}.$ (3.35)

This shifts the components of $\Theta^{\mu}$ as

$\displaystyle\Theta^{u}\to 0$ (3.36)
$\displaystyle\Theta^{r}\to\Theta^{r}+\frac{1}{\kappa^{2}}\,\partial_{u}\Big{(}\frac{1}{2}\,\delta
Q(X)\,X-\frac{1}{2}\,\delta a\,X\Big{)}~{}.$ (3.37)

The effect on the presymplectic current is to set the component $\omega^{u}$
to zero and shift the $\omega^{r}$ component by a corner term

$\displaystyle\omega^{r}\to\omega^{r}+\frac{1}{2\kappa^{2}}\,\partial_{u}\Big{(}\delta_{1}X\,\delta_{2}a-\delta_{2}X\,\delta_{1}a\Big{)}~{}.$
(3.38)

This in turn changes the $ur$ component of the codimension-2 form as

$\displaystyle k^{ur}_{\xi}\to
k^{ur}_{\xi}-\frac{1}{2\kappa^{2}}\,\Big{(}\delta_{\xi}X\,\delta a-\delta
X\,\delta_{\xi}a\Big{)}~{},$ (3.39)

with the minus sign relative to (3.38) coming from considering $k^{ur}_{\xi}$
rather than $k^{ru}_{\xi}$. The shift in $k^{ur}_{\xi}$ includes a piece that
is linear in $r$, from the leading part of the dilaton, which precisely
cancels the terms shown in (3.31). The result for $k^{ur}_{\xi}$ is now
independent of $r$ and given by

$\displaystyle\kappa^{2}\,k^{ur}_{\xi}=$
$\displaystyle\,\,\epsilon(u)\,e^{a(u)+q(u)}\,\delta\mathcal{M}(u)+\delta\varphi(u)\,\Big{(}\chi(u)-\epsilon(u)\,\partial_{u}q(u)\Big{)}$
(3.40) $\displaystyle\quad+\delta
q(u)\,\Big{(}\eta(u)\,e^{-q(u)}+\epsilon(u)\,\partial_{u}\varphi(u)\Big{)}~{}.$

Incorporating the field dependence (3.19) of the symmetry parameters, this
reduces to

$\displaystyle\kappa^{2}\,k^{ur}_{\xi}=\bar{\epsilon}(u)\,\delta\mathcal{M}(u)+\bar{\chi}(u)\,\delta\varphi(u)+\bar{\eta}(u)\,e^{-q(u)}\,\delta
q(u)~{}.$ (3.41)

In the slicing (3.23) where
$\delta\bar{\epsilon}=\delta\bar{\eta}=\delta\bar{\chi}=0$ this is integrable,
$k^{ur}_{\xi}=\delta Q_{\xi}$. The charge is then

$\displaystyle
Q_{\xi}=\frac{1}{\kappa^{2}}\left(\bar{\epsilon}(u)\,\mathcal{M}(u)+\bar{\chi}(u)\,\varphi(u)-\bar{\eta}(u)\,e^{-q(u)}\right)~{}.$
(3.42)

Since the charges are integrable, they form a representation of the asymptotic
symmetry algebra up to a possible central extension[38, 66] (see also [67, 6,
8] for the covariant formulation of this result). Thus,

$\delta_{\xi_{2}}(Q_{\xi_{1}})=:[(Q_{\xi_{1}}),(Q_{\xi_{2}})]=Q_{\xi_{12}}~{},$
(3.43)

where ${\xi_{12}}$ is given by (3.22), and the charge algebra is given by
three abelian Lie algebras with a Heisenberg central extension.

A few comments about the charges are in order. First, note that (3.42) does
not depend on the function $a(u)$. This is not surprising, since the fields in
linear dilaton Bondi gauge involve four functions of $u$, but only three
field-dependent symmetry parameters appear in the diffeomorphisms $\xi$
(3.20). Second, the charges are not sensitive to the specific potential $V(X)$
or kinetic term $U(X)$ appearing in the bulk Lagrangian. There is explicit
dependence on $U(X)$ in the first few terms of (3.30), but this is canceled by
other terms when the full expression is evaluated in linear dilaton Bondi
gauge. This is not changed by the shift in the presymplectic potential, which
depends on $U(X)$ only through a $\delta$-exact term that does not affect
$\omega^{r}$ or $k^{ur}_{\xi}$.

Finally, for the partially on-shell fields in linear Bondi gauge, the
codimension-2 form (3.40) is obtained from (3.29) by evaluating
$\omega^{r}+2\,\xi^{\nu}\,\delta(E^{r\lambda}g_{\lambda\nu})$ and writing the
result as a total $u$-derivative. As we saw above, when $\Theta^{u}$ is set to
zero there is a corresponding change in $\omega^{r}$ which leads to the shift
(3.39). This shift cancels the problematic term in (3.31) linear in $r$, and
also contributes a finite piece. But the contribution from
$2\,\xi^{\nu}\,\delta(E^{r\lambda}g_{\lambda\nu})$ also plays an important
role. It evaluates to

$\displaystyle
2\,\xi^{\nu}\,\delta(E^{r\lambda}g_{\lambda\nu})=-\frac{1}{\kappa^{2}}\,e^{a+q}\,\big{(}\delta\mathcal{M}^{\prime}+\delta(a+q)\,\mathcal{M}^{\prime}\big{)}\,\epsilon~{}.$
(3.44)

This combines with a term in $\omega^{r}$ to produce the total derivative
$\partial_{u}\big{(}-\epsilon\,e^{a+q}\,\delta\mathcal{M}\big{)}$, which is
the first term in (3.40). Without the contribution from the variation of the
weakly vanishing Noether current, we would not obtain $\mathcal{M}(u)$ as a
charge with the slicing (3.19). In fact, an integrable charge would only be
obtained in that case by requiring
$\partial_{u}\mathcal{M}=\partial_{u}\delta\mathcal{M}=0$, which is equivalent
to enforcing $E^{rr}=0$ and placing the fields fully on-shell. The
contribution from the weakly vanishing Noether current is essential for
capturing the full tower of charges $\mathcal{M}(u)$, as opposed to just the
zero mode. This is important, for example, when reproducing the off-shell
symmetry of the SYK dual of the Jackiw-Teitelboim model of dilaton gravity
[61].

###  3.4 First variation of the action and holographic renormalization

In the previous sections we obtained a finite and integrable codimension-2
form $k_{\xi}^{\mu\nu}$ via a shift in the presymplectic potential that set
$\Theta^{u}$ to zero. Now we turn to an analysis of the variational principle,
where this shift is associated with terms in the first variation of the action
that have support on the corners $\partial\mathcal{B}$.

The goal of this section is to construct an action for general models of
dilaton gravity that admits a well-defined variational problem when the fields
satisfy Dirichlet boundary conditions at a spatial infinity that includes
corners. We work in linear dilaton Bondi gauge, on a “regulated” spacetime $M$
with a boundary $\partial M$. The boundary includes a timelike component
$\mathcal{B}$ and null components $\mathcal{N}_{\pm}$ that intersect
$\mathcal{B}$ at corners
$\partial\mathcal{B}_{\pm}=\mathcal{B}\cap\mathcal{N}_{\pm}$. This is shown
schematically in the figure below. The component $\mathcal{B}$ is a curve of
constant $r=r_{c}$ which becomes spatial infinity when the cut-off is removed
by taking $r_{c}\to\infty$. Calculations involving the action and its first
variation are performed on this regulated spacetime, with $r_{c}$ large but
finite, and the $r_{c}\to\infty$ limit taken as the final step.

$\mathcal{N}_{+}$$\mathcal{N}_{-}$$\mathcal{B}$ Figure 1:

Before jumping into the construction, let us quickly review the variational
principle given in [36]. That analysis, tailored to dilaton gravity models
with $U=0$, accounts for corners and arrives at the action

$\displaystyle\Gamma^{{\tiny\text{\cite[cite]{[\@@bibref{}{Ruzziconi:2020wrb}{}{}]}}}}=$
$\displaystyle\,\,\frac{1}{2\kappa^{2}}\int_{M}\negthickspace
d^{2}x\sqrt{-g}\,\Big{(}X\,R-2\,V(X)\Big{)}$
$\displaystyle+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{1}x\sqrt{-\gamma}\big{(}X\,K+\mathcal{L}_{w}+\mathcal{L}_{c}+\mathcal{L}_{n}\big{)}~{}.$
(3.45)

Here $K$ is the extrinsic curvature of a constant $r$ curve embedded in $M$
with outward pointing unit normal $n^{\mu}$, and

$\displaystyle\mathcal{L}_{w}=-\sqrt{w+2e^{-(a+q)}\partial_{u}X},\quad\mathcal{L}_{c}=D_{u}(v^{u}\,X),\quad\mathcal{L}_{n}=-\sqrt{\gamma}Xv^{\mu}a_{\mu}~{}.$
(3.46)

In the last two functions, $v$ is a unit vector field
($v^{a}v_{a}=\gamma_{uu}v^{u}v^{u}=-1$) tangent to $r$ = constant leaves, $D$
is the covariant derivative compatible with the induced metric $\gamma_{uu}$
on $\mathcal{B}$, and $a_{\mu}$ is the “acceleration”
$n^{\nu}\nabla_{\nu}n_{\mu}$. This action is finite on-shell, has a well-
defined flat limit, and is stationary for Dirichlet boundary conditions. But
since it assumes $U=0$ it is not suitable for the full range of dilaton
gravity models we consider. Furthermore, the motivation for some of the
boundary terms is _ad hoc_ , so even though a number of generalizations to
$U\neq 0$ suggest themselves, it is not immediately clear if they are all on
equal footing. Rather than starting from this action and trying to modify it,
we will instead use variational arguments and holographic renormalization
techniques to build an appropriate action from the ground up, paying careful
attention to contributions at the corners and on components of $\partial M$
other than $\mathcal{B}$. The resulting action will reduce (in the
$r_{c}\to\infty$ limit) to (3.4) for models with $U=0$.

For our purposes, a theory described by an action $\Gamma$ and boundary
conditions on the fields has a well-defined variational principle if field
configurations satisfying $E^{i}\,\delta\Psi_{i}=0$ are local extrema of
$\Gamma$ among all nearby field configurations with the same boundary
conditions. An essential point is that $\delta\Gamma=0$ for any field
variation consistent with the boundary conditions, and not just field
variations that satisfy additional and more restrictive conditions. 101010This
is a common problem when constructing an action for fields that satisfy
conditions at spatial infinity. An example is four-dimensional gravity with
asymptotically flat boundary conditions. A meaningful variational principle
should distinguish the Schwarzschild solution from competing field
configurations with similar asymptotic fall-off; i.e., metrics whose $g_{tt}$
components differ from Schwarzschild at $\mathcal{O}(1/r)$ as $r\to\infty$.
But the variation of the Einstein-Hilbert action plus the Gibbons-Hawking-York
boundary term does not vanish unless the metric variation $\delta g_{tt}$
falls off faster than $1/r$ [68]. To address this problem, one must add
additional boundary terms to the action [69, 70, 71].  For the variational
principle to have any meaning, it must be able to distinguish solutions of the
equations of motion from all competing field configurations with similar
asymptotic behavior. This means that field variations with any asymptotic
behavior that preserves the boundary conditions must be allowed.

The focus here will be on Dirichlet boundary conditions for the fields at
$\mathcal{B}$. In that case $\delta\Gamma$ evaluated on
$E^{i}\,\delta\Psi_{i}=0$ should be a functional of the fields $\Psi_{i}$ and
their variations $\delta\Psi_{i}$ on $\mathcal{B}$, as well as derivatives
$\partial_{u}\delta\Psi_{i}$ of the field variations tangent to $\mathcal{B}$.
But it should not depend on the derivatives $\partial_{r}\delta\Psi_{i}$ of
the field variations in the direction normal to $\mathcal{B}$. Working in
linear dilaton Bondi gauge, with fields $\Psi_{i}=A$, $B$, and $X$, we take
Dirichlet boundary conditions to fix the boundary data $a(u)$, $q(u)$, and
$\varphi(u)$ in (3.7)-(3.8). Thus, when $E^{i}\delta\Psi_{i}=0$, a well-
defined variational principle should have $\delta\Gamma=0$ for all field
variations that preserve $g_{rr}=0$ and do not change this boundary data. An
immediate and useful consequence is that changing the boundary data should
produce a finite (as $r_{c}\to\infty$) change in the action. 111111If a change
in the boundary data caused $\delta\Gamma$ to diverge as $r_{c}\to\infty$,
then a suitably rescaled field variation would give a finite and non-zero
$\delta\Gamma$, while also falling off fast enough to preserve the boundary
data. This would contradict the defining property of the variational
principle, that $\delta\Gamma=0$ for any field variation consistent with the
boundary conditions. It will be convenient in the following section to use
this fact when analyzing the properties of the action, since changes in the
boundary data $\delta a$, $\delta q$, and $\delta\varphi$ producing a finite
response $\delta\Gamma$ is a necessary (though not sufficient) condition for a
well-defined variational principle.

It is easy to see that an action based only on the bulk Lagrangian (3.1) does
not lead to a well-defined variational principle. The variation of the bulk
Lagrangian, evaluated on $E^{i}\delta\Psi_{i}=0$, is just the total derivative
of the presymplectic potential. Integrated over $M$, it gives terms on
$\mathcal{B}$ (and $\partial\mathcal{B}$) that include factors which diverge
as $r_{c}\to\infty$. As a result, the variation of the bulk action on its own
does not vanish for all field variations preserving the boundary data. For
example, in the Jackiw-Teitelboim model [54, 55] ($U=0$, $V=-X$) there are
terms in $\Theta^{r}$ proportional to $r^{2}\,\delta A$ and $r\,\delta A$.
These vanish only if $\delta A$ falls off faster than $1/r^{2}$, which is much
more restrictive than the condition $\delta a=0$. This can be addressed by
supplementing the action with boundary contributions whose variations cancel
problematic terms on $\partial M$ coming from $\partial_{\mu}\Theta^{\mu}$ in
the bulk. As long as the problematic terms on $\partial M$ are $\delta$-exact
(total variations of some combination of the fields) it will be possible to
cancel them against the variation $\delta L_{\partial M}$ of appropriate
boundary terms.

In the following, we restrict our attention to models with potentials $U(X)$
and $V(X)$ such that $w(X)\to\infty$ as $X\to\infty$. This includes, for
example, models with AdS2 and Minkowski asymptotics. We also assume that
$\partial_{u}X/w\sim X/w\to 0$ as $X\to\infty$. Both conditions can be
relaxed, but doing so would require a qualitatively different construction
than the one we present here. Additional details, and comments on the regime
of applicability of some of our results, can be found a the end of appendix B.

#### Corner contribution from the bulk symplectic potential

A complication related to this last point arises when $\partial M$ includes
corners. Consider the $\partial_{r}\Theta^{r}$ part of $\delta L_{M}$. When
integrated over $M$ it produces terms on $\mathcal{B}$. In Bondi gauge they
are

$\displaystyle\kappa^{2}\,\Theta^{r}=$
$\displaystyle\,\,\delta\big{(}X\,e^{-A}\,\partial_{r}B\big{)}+\big{(}2\,e^{-A}\,B\,U\,\partial_{r}X-e^{-A}\,\partial_{r}B\big{)}\,\delta
X-e^{-A}\,\partial_{r}X\,\delta B$ (3.47)
$\displaystyle\quad+\partial_{u}X\,\delta\big{(}Q-A\big{)}+\partial_{u}\Big{(}\frac{1}{2}\,X\,\delta
A\Big{)}~{}.$

The first term includes a $\partial_{r}\delta B$ contribution, which is not
consistent with Dirichlet boundary conditions, while the other terms do not
vanish for all field variations that preserve the boundary conditions due to
various factors that diverge as $r_{c}\to\infty$. For fields of the form
(3.7)-(3.8) these problematic terms reduce to $\delta$-exact expressions that
will be addressed with appropriate boundary terms. However, the final term in
(3.47) – a total $u$-derivative with support on $\partial\mathcal{B}$ –
includes an $r_{c}\to\infty$ divergence proportional to $X\,\delta a$. This
corner term is _not_ $\delta$-exact and therefore cannot be canceled by adding
a boundary term to the action. Instead, the resolution comes from the
$\Theta^{u}$ component of the presymplectic potential. The integral over $M$
of $\partial_{u}\Theta^{u}$ gives terms on $\mathcal{N}_{\pm}$. As we saw
previously, $\Theta^{u}$ in Bondi gauge is a total $r$-derivative. When
integrated over $\mathcal{N}$ it gives contributions at the corners that can
be written as $\partial_{u}Y^{ur}$ on $\mathcal{B}$ with

$\displaystyle\kappa^{2}\,Y^{ur}=$ $\displaystyle\,\,\frac{1}{2}\,X\,\delta
A+X\,\delta(Q-A)=X\,\delta Q-\frac{1}{2}\,X\,\delta A$ (3.48)

Thus, once the full total derivative $\partial_{\mu}\Theta^{\mu}$ in the
variation of the bulk Lagrangian is accounted for, the terms on $\mathcal{B}$
are

$\displaystyle\kappa^{2}\,(\Theta^{r}+\partial_{u}Y^{ur})=$
$\displaystyle\,\,\delta\big{(}X\,e^{-A}\,\partial_{r}B\big{)}+\big{(}2\,e^{-A}\,B\,U\,\partial_{r}X-e^{-A}\,\partial_{r}B\big{)}\,\delta
X$ (3.49) $\displaystyle\quad-e^{-A}\,\partial_{r}X\,\delta
B+\partial_{u}X\,\delta\big{(}Q-A\big{)}+\partial_{u}\big{(}X\,\delta
Q\big{)}~{}.$

The parts of this expression which cause problems in the $r_{c}\to\infty$
limit are $\delta$-exact and will now be addressed with appropriate boundary
terms.

#### Gibbons-Hawking-York term

The first term in (3.49) is incompatible with Dirichlet boundary conditions
because it includes a contribution proportional to $\partial_{r}\delta B$.
This is remedied by adding the dilaton gravity version of the Gibbons-Hawking-
York (GHY) term [72, 73] on $\mathcal{B}$.

$\displaystyle\kappa^{2}\,L_{\text{GHY}}=\sqrt{-\gamma}\,X\,K=-X\,e^{-A}\,\partial_{r}B-X\,\Big{(}\partial_{u}A-\frac{\partial_{u}B}{2B}\Big{)}~{}.$
(3.50)

The dependence on the normal derivative of the metric eliminates the
$\partial_{r}\delta B$ term in (3.49). Combining all terms with support on
$\mathcal{B}$ up to this point, we have

$\displaystyle\kappa^{2}\,(\Theta^{r}+\partial_{u}Y^{ur}+\delta L_{GHY})=$
$\displaystyle\,\,\big{(}2\,e^{-A}\,B\,U\,\partial_{r}X-e^{-A}\,\partial_{r}B\big{)}\,\delta
X-e^{-A}\,\partial_{r}X\,\delta B$ (3.51)
$\displaystyle\quad+\partial_{u}X\,\delta\big{(}Q-A\big{)}-\delta\Big{(}X\,\Big{(}\partial_{u}A-\partial_{u}Q-\frac{\partial_{u}B}{2B}\Big{)}\Big{)}~{},$

Evaluating this for the solution (3.8) and using $\partial_{u}\big{(}X\,\delta
Q(X)\big{)}=\delta\big{(}X\,\partial_{u}Q(X)\big{)}$

$\displaystyle\kappa^{2}\,(\Theta^{r}+\partial_{u}Y^{ur}+\delta L_{GHY})=$
$\displaystyle\,\,\delta\big{(}e^{a+q}\,w\big{)}+\partial_{u}\big{(}\delta
X\big{)}-\delta\Big{(}X\,\partial_{u}\log\Big{(}\frac{e^{A-Q}}{\sqrt{-2\,B}}\Big{)}\Big{)}$
(3.52) $\displaystyle\quad+\partial_{u}\varphi\,\delta
q-\partial_{u}q\,\delta\varphi-e^{-(a+q)}\,\delta\big{(}e^{2(a+q)}\,\mathcal{M}\big{)}$

The terms in the first line on the RHS diverge as $r_{c}\to\infty$ for
variations of the boundary data, and hence do not vanish for all field
variations consistent with our boundary conditions. However, these terms are
all $\delta$-exact and can be addressed by additional boundary contributions
to the action. The terms in the second line are zero for variations that
preserve the boundary conditions, except for a non-zero term proportional to
$\delta\mathcal{M}$.

#### Boundary terms in the presence of corners

To understand the remaining boundary terms, let us now write an action with
the form suggested by [74] for spacetimes with a boundary that includes both
timelike and null parts, generalized to dilaton gravity in two dimensions. It
consists of a bulk integral over $M$ as well as contributions on the boundary
components $\mathcal{N}_{\pm}$ and $\mathcal{B}$.

$\displaystyle\Gamma=$
$\displaystyle\,\,\frac{1}{2\kappa^{2}}\int_{M}\negthickspace
d^{2}x\sqrt{-g}\,\Big{(}X\,R-U(X)\,\big{(}\nabla
X\big{)}^{2}-2\,V(X)\Big{)}+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{1}x\sqrt{-\gamma}\,X\,K$
(3.53)
$\displaystyle\qquad+\frac{1}{\kappa^{2}}\int_{\mathcal{N}_{\pm}}d\lambda\,X\,\mathcal{K}+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{1}x\,\Big{(}L_{X\partial\mathcal{B}}+L_{\text{{CT}}}\Big{)}~{}.$

As before, $\gamma$ is the induced metric on $\mathcal{B}$ and
$K=\nabla_{\mu}n^{\mu}$ is the extrinsic curvature formed from the outward-
pointing spacelike unit vector $n^{\mu}$. The integrals on the null components
of the boundary involve a parameter $\lambda$ along a null generator
$k_{\mu}$, while $\mathcal{K}$ measures the failure of this parameter to be
affine according to

$\displaystyle k^{\nu}\nabla_{\nu}k^{\mu}=\mathcal{K}\,k^{\mu}~{},$ (3.54)

or $\mathcal{K}=\nabla_{\mu}k^{\mu}$. The last set of terms in (3.53) are a
boundary term $L_{X\partial\mathcal{B}}$ that is needed when $\mathcal{B}$ has
corners, along with a “boundary counterterm” $L_{\text{CT}}$ that implements a
holographic renormalization prescription explained in the next section. Note
that $L_{X\partial\mathcal{B}}$ is not necessarily a pure corner term. Rather,
it represents terms which may be needed (in addition to the other boundary
terms) when there is a corner $\mathcal{B}\cap\mathcal{N}$.

An ambiguity in the action (3.53) is that it depends on the definition of the
null generator $k_{\mu}$, which is not unique. We fix part of this ambiguity
as in [74] by demanding affine $\lambda$, so that $\mathcal{K}=0$ and the
integrals over $\mathcal{N}_{\pm}$ vanish. Then motivated by the construction
in [74], the term $L_{X\partial\mathcal{B}}$ on $\mathcal{B}$ takes the form

$\displaystyle\kappa^{2}\,L_{X\partial\mathcal{B}}=\sqrt{-\gamma}\,X\,v^{u}\partial_{u}\log\big{(}n^{\mu}\,k_{\mu}\big{)}~{},$
(3.55)

where $v^{u}=1/\sqrt{-\gamma}$ is the timelike unit vector tangent to
$\mathcal{B}$. Requiring affine $\lambda$ gives
$k_{\mu}=-\beta(u)\,\delta_{\mu}{}^{u}$ with arbitrary $\beta(u)$. Contracting
this with the unit vector $n^{\mu}$ normal to a curve of constant $r$, we have

$\displaystyle\kappa^{2}\,L_{X\partial\mathcal{B}}=X\,\partial_{u}\log\Big{(}\frac{\beta(u)}{\sqrt{-2B}}\Big{)}~{}.$
(3.56)

The remaining ambiguity in the choice of $k_{\mu}$ is fixed by comparing this
boundary term with the third term on the first line of (3.52). An
$r_{c}\to\infty$ divergence cancels if $\beta(u)=e^{A-Q}$, which fixes (up to
an irrelevant constant factor) the choice of null generator

$\displaystyle k_{\mu}=-e^{A(u,r)-Q(X)}\,\delta_{\mu}{}^{u}~{}.$ (3.57)

The final form of $L_{X\partial\mathcal{B}}$ is then

$\displaystyle\kappa^{2}\,L_{X\partial\mathcal{B}}=\sqrt{-\gamma}X\,v^{u}\,\partial_{u}\log\big{(}n^{\mu}\,k_{\mu}\big{)}=X\,\partial_{u}\log\left(\frac{e^{A(u,r)-Q(X)}}{\sqrt{-2B(u,r)}}\right)~{}.$
(3.58)

Schematically, this can be thought of as the dimensional reduction of a higher
dimensional corner term, with the dilaton identified with the transverse part
of the metric.

#### Boundary counterterms for dilaton gravity models

The derivation of the boundary counterterms $L_{\text{CT}}$ is described in
Appendix B. Following the Hamilton-Jacobi approach used in [53], the
counterterm Lagrangian $L_{\text{CT}}$ is obtained as a solution of the radial
Hamiltonian constraint in a derivative expansion on $\mathcal{B}$. There are
no curvature invariants built from $\gamma$ and its derivatives along
$\mathcal{B}$, so this amounts to an expansion in derivatives of $X$ which
takes the form

$\displaystyle
L_{\text{{CT}}}(\gamma,X)=\frac{1}{\kappa^{2}}\sqrt{-\gamma}\,\Big{(}F_{0}(X)+F_{2}(X)\,\gamma^{uu}(\partial_{u}X)^{2}+\ldots\Big{)}~{}.$
(3.59)

The number of terms in this expansion needed for the renormalization of a
particular model will depend on the $r\to\infty$ behavior of the functions
$Q(X)$ and $w(X)$ constructed from $U(X)$ and $V(X)$. The derivation in
Appendix B gives the first two terms in the expansion as

$\displaystyle\kappa^{2}\,L_{\text{{CT}}}(\gamma,X)=-\sqrt{-\gamma}\,\Big{(}\sqrt{e^{-Q(X)}w(X)}+\frac{1}{2\sqrt{e^{-Q(X)}w(X)}}\,D^{u}X\,D_{u}X\Big{)}+\ldots~{},$
(3.60)

where $D$ is the derivative along $\mathcal{B}$ compatible with the induced
metric $\gamma$. In the analysis below, we will focus on the wide class of
models that require only these first two terms; i.e., models for which the
subsequent terms in the boundary derivative expansion of $L_{\text{CT}}$
approach zero as $r_{c}\to\infty$. It should be emphasized that, while the
boundary counterterms are normally presented as a way of systematically
addressing $r_{c}\to\infty$ problems with the variational principle,
$L_{\text{CT}}$ also makes a finite contribution to both the action and its
first variation. Evaluating (3.60) for the partially on-shell fields, the
relevant terms are

$\displaystyle\kappa^{2}L_{\text{CT}}\sim-w(X)e^{a+q}-\partial_{u}X+e^{a+q}\mathcal{M}+\frac{(\partial_{u}X)^{2}}{e^{a+q}w(X)}+\ldots~{}.$
(3.61)

The roles played by finite parts of this expression are explained below.

#### Summary: Renormalized action and first variation of the action

To summarize, the following action gives a well-defined variational principle
for general dilaton gravity models with fields satisfying Dirichlet boundary
conditions at a spatial infinity with corners

$\displaystyle\Gamma=$
$\displaystyle\,\,\frac{1}{2\kappa^{2}}\int_{M}\negthickspace
d^{2}x\sqrt{-g}\,\Big{(}X\,R-U(X)\,\big{(}\nabla
X\big{)}^{2}-2\,V(X)\Big{)}+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{1}x\sqrt{-\gamma}\,X\,K$
(3.62)
$\displaystyle\qquad+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{1}x\,\Big{(}L_{X\partial\mathcal{B}}+L_{\text{{CT}}}\Big{)}~{}.$

The boundary term $L_{X\partial\mathcal{B}}$ is given in (3.58), and
$L_{\text{{CT}}}$ is given in (3.60).

Using the results of the previous sections to evaluate the first variation of
(3.62) in linear dilaton Bondi gauge, the terms with support on $\mathcal{B}$
are

$\displaystyle\delta\Gamma\Big{|}_{\mathcal{B}}=\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}\negthickspace
du\,\left(\delta
q\,\partial_{u}\varphi-\delta\varphi\,\partial_{u}q-\delta(a+q)\,e^{a+q}\,\mathcal{M}+\delta\Big{(}e^{-(a+q)}\frac{(\partial_{u}X)^{2}}{w}\Big{)}\right)~{}.$
(3.63)

With our assumptions about the falloff of $w$ this vanishes for variations
that preserve the Dirichlet boundary data ($a$, $q$, and $\varphi$), and it is
finite for variations of the boundary data. A term in (3.52) proportional to
$\delta\mathcal{M}$ has canceled against a similar finite term coming from
$L_{\text{CT}}$. This is important, since $\mathcal{M}$ is not part of the
boundary data and hence $\delta\Gamma$ must vanish for a variation
$\delta\mathcal{M}$. Note that the last term in (3.63), which is the only part
of the expression that explicitly depends on (integrals of) the potentials
$V(X)$ and $U(X)$, is delta-exact and hence does not contribute to the
symplectic current. In the case of Jackiw-Teitelboim gravity and related
models, this term is needed to obtain the expected Schwarzian action for the
dual theory [61].

Finally we can also show that the limit $r_{c}\to\infty$ does not produce any
divergences in the (partially) on-shell action. This gives 121212In [36],
there was a finite term $-e^{a+q}\mathcal{M}$ missing in the computation of
the on-shell action.

$\displaystyle\Gamma=\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}\negthickspace
du\,\left(-e^{a+q}\,\mathcal{M}+e^{-(a+q)}\,\frac{(\partial_{u}X)^{2}}{w}\right)+\ldots~{},$
(3.64)

where “$\ldots$” refers to contributions to the action from integration-by-
parts terms evaluated in the interior of $M$; i.e. total $r$ derivatives that
may also give a finite contribution at $r=0$, the horizon of a black hole, or
similar. When considering $\delta\Gamma$ we are only interested in terms with
support on $\mathcal{B}$, since the fields satisfy boundary conditions at
spatial infinity. For applications that make use of the on-shell action, the
“$\ldots$” terms in (3.64) are just as important as the contributions on
$\mathcal{B}$. Those terms cannot be evaluated without having a specific model
and solution in mind, as they depend on what happens in the interior of $M$.
The point here is simply that there are no terms in the action that diverge as
$r_{c}\to\infty$.

##  4 3d Einstein gravity

In this section we revisit the results of [37] where a new gauge for three
dimensional Einstein gravity, dubbed the Bondi-Weyl gauge, was
introduced.131313One can show that this is precisely the three dimensional
analog of the partial Bondi gauge introduced later in four dimensions [75].
This is an enhancement of the Bondi gauge where the condition on the
determinant of the transverse metric is relaxed. The space of partially on-
shell field configurations in Bondi-Weyl gauge is characterized by four
charges, which is two more than in the usual Bondi gauge [36]. Previous
efforts to calculate the charges in this gauge encountered two problems: the
“obvious” choice of codimension-2 form diverges as $r\to\infty$, and its
finite part is not integrable. These issues were addressed using the intrinsic
corner ambiguities of the covariant phase space _à la_ Iyer-Wald, along with a
change in the field dependence of the symmetries generators. We show in this
section that this is not necessary when corner contributions in the symplectic
potential are taken into account, as described in section 2. Moreover, the
finite corner term introduced in [75] to restore integrability appears
naturally in our construction, in the same way as the dilaton gravity example
of section 3.

We begin by reviewing the solution space of the Bondi-Weyl gauge. This is
followed by a quick review of the covariant phase space in Einstein gravity.
Applying the results of section 2 we recover the same charges as in [37].
Finally, we present a new variational principle for three-dimensional gravity
with a negative cosmological that accounts for corners, and also admits a
straightforward flat-space limit.

The discussion in this section is carried out in Bondi-Weyl gauge, but many of
our results hold for other choices of gauge. In appendix D we consider the
variational properties of the action in the more familiar Fefferman–Graham
gauge [20], and show that corner contributions from the symplectic potential
cancel divergent total derivative terms on the boundary which cannot be
discarded in the presence of corners.

###  4.1 Solution space and symmetries in Bondi-Weyl gauge

We consider Einstein gravity in three dimensions with a negative cosmological
constant. The bulk Lagrangian is

$L_{M}=\frac{1}{2\kappa^{2}}\sqrt{-g}\,\left(R+\frac{2}{\ell^{2}}\right)\,,$
(4.1)

where $\ell$ is the AdS radius. 141414The results of this section assume a
negative cosmological constant, and also hold in the flat limit
$\ell\to\infty$. A similar construction can be carried out in the case of a
positive cosmological constant. Its variation

$\displaystyle\delta L_{M}=E^{\mu\nu}\,\delta
g_{\mu\nu}+\partial_{\mu}\Theta^{\mu}~{},$ (4.2)

gives the equations of motions

$E^{\mu\nu}:=\frac{\delta L_{M}}{\delta
g_{\mu\nu}}=-\frac{\sqrt{-g}}{2\kappa^{2}}\left(R^{\mu\nu}-\frac{1}{2}\,g^{\mu\nu}R-\frac{1}{\ell^{2}}g^{\mu\nu}\right)\,,$
(4.3)

and the symplectic potential

$\displaystyle\Theta^{\mu}=\frac{\sqrt{-g}}{2\kappa^{2}}\big{(}\nabla_{\nu}(\delta
g)^{\mu\nu}-\nabla^{\mu}(\delta g)^{\nu}{}_{\nu}\big{)}\,.$ (4.4)

##### Bondi-Weyl gauge

The line element in the Bondi-Weyl gauge is

$\displaystyle\mathrm{d}s^{2}=-2e^{2\beta(u,\phi)}\,\mathrm{d}u\,\mathrm{d}r+\mathcal{V}(u,r,\phi)\,e^{2\beta(u,\phi)}\,\mathrm{d}u^{2}+{\mathcal{W}}^{2}(u,r,\phi)\Big{(}\mathrm{d}\phi-\mathcal{U}(u,r,\phi)\,\mathrm{d}u\Big{)}^{2}\,,$
(4.5)

with $g_{rr}=g_{r\phi}=\partial_{r}g_{ur}=0$ imposed as gauge conditions. The
condition $\partial_{r}g_{ur}=0$ can be relaxed, but this will not enlarge the
solution space and many of the calculations that follow are simpler with this
condition in place. The coordinate $\phi$ in (4.5) is taken to be periodic:
$\phi\sim\phi+2\pi$. Later on we will use this to discard total
$\phi$-derivatives appearing in integrals, under the assumption that the
fields are single-valued. Solving the components of the equations of motion
associated with the non-zero components of the metric gives151515The notation
here is different than in [37], with
$C=H^{\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}$,
$\mathcal{V}=V^{\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}/r$,
and
$\mathcal{W}^{2}=r^{2}\mathcal{W}^{\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}$.
The functions $M,N$ have been shifted as
$N=N^{\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}+2\,C\,\beta^{\prime}\,,\qquad
M=M^{\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}+\frac{e^{2\beta}C^{2}}{2\ell^{2}}-C\,\big{(}U^{\prime}+U\varphi^{\prime}+\partial_{u}\varphi\big{)}~{}.$

$\displaystyle E^{uu}=0\ $ $\displaystyle\Rightarrow\
{\mathcal{W}}=e^{\varphi}\left(r-C\right),$ (4.6a) $\displaystyle E^{u\phi}=0\
$ $\displaystyle\Rightarrow\
\mathcal{U}=U+\frac{e^{2(\beta-\varphi)}}{(r-C)}\,\left(2\,\beta^{\prime}-\frac{N}{(r-C)}\right)$
(4.6b) $\displaystyle E^{\phi\phi}=E^{ur}=0\ $ $\displaystyle\Rightarrow\
\mathcal{V}=-\frac{e^{2\beta}}{\ell^{2}}(r-C)^{2}-2\,(U^{\prime}+U\varphi^{\prime}+\partial_{u}\varphi)(r-C)$
$\displaystyle\phantom{\ \Rightarrow\
V=}+2\,\big{(}M+2\,e^{2(\beta-\varphi)}(\beta^{\prime})^{2}\big{)}-e^{2(\beta-\varphi)}\left(2\,\beta^{\prime}-\frac{N}{(r-C)}\right)^{2}$
(4.6c)

where $C,U,\beta,\varphi$ are functions of $(u,\phi)$, and a prime ′ denotes a
partial derivative with respect to $\phi$. The two remaining components,
$E^{rr}$ and $E^{r\phi}$, would typically be enforced as constraints on the
functions $M$ and $N$, associated with the choice of coordinate gauge.
However, we do not impose these constraints. This leaves a total of six
arbitrary functions of $(u,\phi)$ parameterizing the partially on-shell fields
in Bondi-Weyl gauge.

##### Symmetries and transformation laws

The asymptotic Killing vectors $\xi^{\mu}$ of the metric (4.5) take the form

$\displaystyle\xi^{u}$ $\displaystyle=f,$ (4.7a) $\displaystyle\xi^{\phi}$
$\displaystyle=g-\frac{e^{2(\beta-\varphi)}}{r-C}f^{\prime},$ (4.7b)
$\displaystyle\xi^{r}$
$\displaystyle=(r-C)(h+Uf^{\prime}-g^{\prime}-g\varphi^{\prime}-f\partial_{u}\varphi)-k$
$\displaystyle\phantom{=\
}+e^{2(\beta-\varphi)}\left(f^{\prime\prime}+f^{\prime}(4\beta-\varphi)^{\prime}-\frac{N}{r-C}f^{\prime}\right).$
(4.7c)

They depend on four arbitrary functions of $(u,\phi)$ – denoted $f$, $g$, $h$
and $k$ – which may be field-dependent. Under these diffeomorphisms, the
fields transform as

$\displaystyle\delta_{\xi}\beta$
$\displaystyle=f\,\partial_{u}\beta+g\,\beta^{\prime}+Uf^{\prime}+\frac{1}{2}\big{(}\partial_{u}f-f\partial_{u}\varphi-g^{\prime}-g\varphi^{\prime}+h\big{)},$
(4.8a) $\displaystyle\delta_{\xi}\varphi$ $\displaystyle=h,$ (4.8b)
$\displaystyle\delta_{\xi}U$
$\displaystyle=f\,\partial_{u}U+g\,U^{\prime}+U\big{(}\partial_{u}f-g^{\prime}+Uf^{\prime}\big{)}-\partial_{u}g+\frac{e^{2(2\beta-\varphi)}}{\ell^{2}}f^{\prime},$
(4.8c) $\displaystyle\delta_{\xi}C$
$\displaystyle=f\,\partial_{u}C+g\,C^{\prime}+k.$ (4.8d)

The transformations $\delta_{\xi}M$ and $\delta_{\xi}N$ are more involved, but
can be easily derived.

Using the modified Lie bracket [65] these vector fields satisfy the
commutation relations

$\displaystyle\big{[}\xi(f_{1},g_{1},h_{1},k_{1}),\xi(f_{2},g_{2},h_{2},k_{2})\big{]}_{\star}$
$\displaystyle=\big{[}\xi(f_{1},g_{1},h_{1},k_{1}),\xi(f_{2},g_{2},h_{2},k_{2})\big{]}-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1}$
(4.9) $\displaystyle=\xi(f_{12},g_{12},h_{12},k_{12}),$ (4.10)

where

$\displaystyle f_{12}$
$\displaystyle=f_{1}\partial_{u}f_{2}+g_{1}f_{2}^{\prime}-\delta_{\xi_{1}}f_{2}-(1\leftrightarrow
2),$ (4.11a) $\displaystyle g_{12}$
$\displaystyle=f_{1}\partial_{u}g_{2}+g_{1}g^{\prime}_{2}-\delta_{\xi_{1}}g_{2}-(1\leftrightarrow
2),$ (4.11b) $\displaystyle h_{12}$
$\displaystyle=-\delta_{\xi_{1}}h_{2}-(1\leftrightarrow 2),$ (4.11c)
$\displaystyle k_{12}$
$\displaystyle=f_{1}\partial_{u}k_{2}+g_{1}k_{2}^{\prime}-\delta_{\xi_{1}}k_{2}-(1\leftrightarrow
2).$ (4.11d)

When the functions parameterizing the diffeomorphism (4.7) are taken to be
independent of the fields, $\delta f=\delta g=\delta h=\delta k=0$, the
algebra is
$\left(\text{Diff}(C^{2})^{(f,g)}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}}(\mathcal{C}^{\infty})^{(k)}(C^{2})\right)\oplus(\mathcal{C}^{\infty})^{(h)}(C^{2})$,
where $C^{2}$ is the cylinder spanned by $(u,\phi)$ and
$(\mathcal{C}^{\infty})(C^{2})$ denotes the smooth functions over $C^{2}$.

###  4.2 Covariant phase space for three dimensional gravity

We consider the covariant phase space, reviewed in section 2.2, for Einstein
gravity in three dimensions. 161616In fact, the expressions in this subsection
are valid for pure gravity in any spacetime dimension.

The presymplectic potential for the Einstein-Hilbert action was given in
(4.4). The antisymmetrized second variation of the action then gives the
presymplectic current

$\displaystyle\omega^{\mu}(g;\delta_{1}g,\delta_{2}g)=\frac{\sqrt{-g}}{2\kappa^{2}}\,\left[\vphantom{\frac{1}{2}}\right.$
$\displaystyle\,\,\frac{1}{2}\,g^{\mu\lambda}\Big{(}g^{\alpha\beta}g^{\sigma\nu}-g^{\alpha\sigma}g^{\beta\nu}\Big{)}+\frac{1}{2}\,g^{\beta\nu}\Big{(}g^{\alpha\lambda}g^{\sigma\mu}-g^{\alpha\sigma}g^{\lambda\mu}\Big{)}$
(4.12)
$\displaystyle\,\,+\left.\frac{1}{2}\,g^{\lambda\nu}\Big{(}g^{\mu\alpha}g^{\beta\sigma}-g^{\alpha\beta}g^{\sigma\mu}\Big{)}\right]\,\Big{(}\delta_{2}g_{\alpha\beta}\,\nabla_{\sigma}\delta_{1}g_{\lambda\nu}-(1\leftrightarrow
2)\Big{)}~{}.$

The other ingredient needed for the construction of the codimension-2 form is
the weakly vanishing Noether current. Contracting the bulk term in the
variation of the Lagrangian (4.1) with a diffeomorphism yields

$\displaystyle
E^{\mu\nu}\,\delta_{\xi}g_{\mu\nu}=\xi^{\nu}N_{\nu}+\partial_{\mu}S^{\mu}_{\xi}~{},$
(4.13)

which gives the Noether identities and the weakly vanishing Noether current as

$\displaystyle N_{\nu}$
$\displaystyle=-2\,\partial_{\mu}E^{\mu}{}_{\nu}+E^{\mu\lambda}\,\partial_{\nu}g_{\mu\lambda}=0~{}\,,\qquad
S_{\xi}^{\mu}=2\,E^{\mu\lambda}\,g_{\lambda\nu}\,\xi^{\nu}~{}.$ (4.14)

Then (2.20) implies the existence of a non-unique codimension-2 form
$k^{\mu\nu}_{\xi}$ such that

$\displaystyle\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}(g;\delta_{\xi}g,\delta
g)+2\,\xi^{\nu}\delta(E^{\mu\lambda}g_{\lambda\nu})-\xi^{\mu}\,E^{\nu\lambda}\,\delta
g_{\nu\lambda}~{}.$ (4.15)

Using the above results for the terms on the right-hand side, integration by
parts leads to $k^{\mu\nu}_{\xi}$ given by

$\displaystyle
k^{\mu\nu}_{\xi}=\frac{\sqrt{-g}}{2\,\kappa^{2}}\,\left(\vphantom{\frac{1}{2}}\right.$
$\displaystyle\xi^{\mu}\nabla_{\lambda}(\delta
g)^{\nu\lambda}-\xi^{\mu}\nabla^{\nu}(\delta
g)^{\lambda}{}_{\lambda}+\xi_{\lambda}\nabla^{\nu}(\delta g)^{\mu\lambda}$
$\displaystyle\left.+\frac{1}{2}(\delta
g)^{\lambda}{}_{\lambda}\nabla^{\nu}\xi^{\mu}-\delta
g^{\lambda\nu}\nabla_{\lambda}\xi^{\mu}-(\mu\leftrightarrow\nu)\right)~{}.$
(4.16)

Note that (4.15) is satisfied by (4.2) for all field configurations,
independent of whether the equations of motion hold. When the fields are fully
on-shell, so that the presymplectic current is conserved, (4.15) reduces to
$\partial_{\nu}k^{\mu\nu}_{\xi}=\omega^{\mu}(g;\delta_{\xi}g,\delta g)$.
However, for partially on-shell field configurations only the last term on the
right-hand side of (4.15) vanishes. In that case, some components of
$\xi^{\nu}\delta(E^{\mu\lambda}g_{\lambda\nu})$ are non-zero and contribute to
$k^{\mu\nu}_{\xi}$.

Evaluating the $k^{ur}_{\xi}$ component of (4.2) leads to the same problem
encountered in section 3: it diverges in the limit $r\to\infty$, and the
finite part is not integrable. We address this with the prescription outlined
in section 2 for constructing a finite ($r$-independent) codimension-2 form,
and determine the field dependence of the functions appearing in (4.7) that
leads to integrable charges.

###  4.3 Charges

The construction in section 2.2 shifts the components of the codimension-2
form by corner contributions $\mathcal{Y}^{ar}$ from the presymplectic current
and $\mathcal{Z}^{ar}$ from the weakly vanishing Noether current, as in
(2.34). This defines (up to residual ambiguities) a codimension-2 form such
that $\partial_{a}k^{ra}_{\xi}$ is independent of $r$. The charges are
obtained from the $ur$ component, which may depend on $r$ through terms whose
$u$-derivative cancels against similar terms in
$\partial_{\phi}k^{r\phi}_{\xi}$. In other words, any $r$-dependence in
$k^{ur}_{\xi}$ is a total $\phi$-derivative and therefore can be discarded in
integrals over $\mathcal{B}$ or $\partial\mathcal{B}$. Thus, to determine the
charges it is sufficient to calculate only the shift in the $ur$ component of
(4.2)

$\displaystyle k^{ru}_{\xi}\to
k^{ru}_{\xi}+\mathcal{Y}^{ur}_{\xi}+\mathcal{Z}^{ur}_{\xi}~{}.$ (4.17)

However, since we consider field configurations that satisfy the $E^{uu}$,
$E^{u\phi}$, and $E^{ur}$ components of the equation of motion,
$\mathcal{Z}^{ur}_{\xi}=0$ and the only shift to the $ur$ component of (4.2)
comes from $\mathcal{Y}^{ur}$. This is obtained from the anti-symmetrized
variation of $\Theta^{u}$ written as a total $r$-derivative.

Recall from section 2 that the contributions from the $\Theta^{a}$ components
to the divergence of the presymplectic current can be folded into the
$r$-derivative to define a $\Theta^{r}_{\text{ren.}}$ that includes corner
contributions

$\partial_{\mu}\Theta^{\mu}=\partial_{r}\left(\Theta^{r}+\partial_{u}\int
dr\Theta^{u}+\partial_{\phi}\int
dr\Theta^{\phi}\right):=\partial_{r}\Theta^{r}_{\text{{ren}}}~{}.$ (4.18)

Evaluating the symplectic potential (4.4) on the line element (4.5) one has

$\displaystyle\kappa^{2}\,\Theta^{u}$
$\displaystyle=\partial_{r}(\delta\mathcal{W}-\delta\beta\,\,\mathcal{W})+2\mathcal{W}\partial_{r}\delta\beta$
(4.19) $\displaystyle\kappa^{2}\Theta^{\phi}$
$\displaystyle=\delta\left(-\frac{1}{{\mathcal{W}}}\partial_{\phi}e^{2\beta}+\partial_{r}\mathcal{W}\,\mathcal{U}\right)+\partial_{r}\left(-\mathcal{W}\,\mathcal{U}\,\delta\beta+\frac{1}{2}\mathcal{W}\,\delta\mathcal{U}\right)$
(4.20)
$\displaystyle+\frac{2e^{2\beta}}{\mathcal{W}}\delta\beta\partial_{\phi}\beta+\mathcal{W}\,\partial_{r}\mathcal{U}\,\delta\beta+2\mathcal{W}\,\mathcal{U}\,\partial_{r}\delta\beta$
(4.21)

Since $\partial_{r}\beta=\partial_{r}\delta\beta=0$, $\Theta^{u}$ is
automatically a total $r$-derivative in Bondi-Weyl gauge. The $\Theta^{\phi}$
component, on the other hand, must be integrated with respect to $r$. However,
as explained above, only the shift coming from the
$\partial_{u}\Theta^{u}=\partial_{r}\partial_{u}Y^{ur}$ corner contribution
will be relevant for the charges.

Incorporating the corner contribution from $\Theta^{u}$ in the $r$ component
of the presymplectic potential, and ignoring both total $\phi$-derivatives
(including the contribution from $\Theta^{\phi}$) and $\delta$-exact terms, we
have

$\displaystyle\kappa^{2}\,\Theta^{r}_{\text{ren}}$
$\displaystyle=\kappa^{2}\,\Theta^{r}+\partial_{u}\left(\delta\mathcal{W}-\mathcal{W}\,\delta\beta\right)$
$\displaystyle=-e^{\varphi}(M-\partial_{u}C-U\,C^{\prime}+2e^{2\beta-\varphi}\beta^{\prime
2})\delta(2\beta-\varphi)$ (4.22)
$\displaystyle\quad+e^{\varphi}(N-C^{\prime})\delta
U-\partial_{u}(e^{\varphi}\delta C)~{}.$

The only $r$-dependence takes the form of a $\delta$-exact term, which will
not contribute to $\omega^{r}$ and hence has been dropped. Then the $r$
component of the symplectic current is

$\displaystyle\kappa^{2}\,\omega^{r}_{\text{ren}}$
$\displaystyle=\kappa^{2}\,\omega^{r}+\partial_{u}\big{(}\delta_{1}\mathcal{W}\,\delta_{2}\beta-\delta_{2}\mathcal{W}\,\delta_{1}\beta\big{)}$
$\displaystyle=-\delta_{1}\left(e^{\varphi}(M-\partial_{u}C-U\,C^{\prime})+2e^{2\beta-\varphi}\beta^{\prime
2}\right)\delta_{2}(2\beta-\varphi)$ (4.23)
$\displaystyle+\delta_{1}\left(e^{\varphi}(N-C^{\prime})\right)\delta_{2}U-\partial_{u}\left(\delta_{1}e^{\varphi}\delta_{2}C\right)-(1\leftrightarrow
2)$ (4.24)

The total $u$-derivative on the right-hand side of the first line gives
$\mathcal{Y}^{ur}_{\xi}$ when $\delta_{2}$ is a generic field variation, and
$\delta_{1}=\delta_{\xi}$ is a field variation with the same form as a
diffeomorphism.

Before giving the $ur$ component of the shifted codimension-2 form, let us
briefly compare the results above with the two corner terms introduced in [37]
to ‘renormalize’ the presymplectic potential. The presymplectic potential in
that reference is related to the $\Theta^{r}$ component of (4.4) by

$\displaystyle\Theta^{r\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}_{\text{ren}}$
$\displaystyle=\Theta^{r}-\frac{r}{\kappa^{2}}\partial_{u}(e^{\varphi}\delta\beta-\delta
e^{\varphi})+\frac{1}{\kappa^{2}}\partial_{u}(e^{\varphi}C\delta\beta)~{}.$
(4.25)

The first of the two terms canceled the $r$-dependent part of $\Theta^{r}$
that diverges in the $r\to\infty$ limit. The second corner term, which is
finite in this limit, led to an integrable codimension-2 form when the
functions parameterizing the diffeomorphism (4.7) have a specific field
dependence. This is equivalent to (4.3), up to total $\phi$-derivatives and
$\delta$-exact terms

$\displaystyle\Theta^{r}_{\text{ren}}$
$\displaystyle=\Theta^{r\text{\cite[cite]{[\@@bibref{}{Geiller:2021vpg}{}{}]}}}_{\text{ren}}+(\ldots)^{\prime}+\delta(\ldots)~{}.$
(4.26)

Thus, the _ad hoc_ corner terms introduced in [37] are in fact explained by
corner contributions from the component $\Theta^{u}$ of the symplectic
potential. Note the second corner term introduced in [37], which is necessary
for integrability of the charges, appears as the $r$-independent part of
$Y^{ur}$ when we use integration-by-parts to express
$\Theta^{u}=\partial_{r}Y^{ur}$ as a total $r$-derivative.

Incorporating the $\mathcal{Y}^{ur}_{\xi}$ shift to the $ur$ component of
(4.2), and dropping total $\phi$-derivatives, $k^{ur}_{\xi}$ is given by

$\displaystyle k^{ur}_{\xi}=$
$\displaystyle\,\,f\,\Big{[}e^{2\beta}\,\delta\Big{(}-2e^{-2\beta}\,\big{(}e^{2\beta-\varphi}\beta^{\prime}\big{)}^{\prime}\Big{)}+e^{2\beta-\varphi}\delta\big{(}-M\,e^{2\varphi-2\beta}\big{)}$
(4.27)
$\displaystyle\qquad+e^{2\beta}\,\delta\Big{(}U\,e^{\varphi-2\beta}C^{\prime}\Big{)}+e^{2\beta}\,\delta\Big{(}e^{\varphi-2\beta}\partial_{u}C\Big{)}\Big{]}$
$\displaystyle\,\,+\Big{(}-k+e^{2\beta-2\varphi}\,\big{(}f^{\prime\prime}+f^{\prime}\,(4\beta-\varphi)^{\prime}\big{)}\Big{)}\delta(e^{\varphi})$
$\displaystyle\,\,+\Big{(}h+U\,f^{\prime}-g^{\prime}-g\,\varphi^{\prime}-f\,\partial_{u}\varphi\Big{)}\,e^{\varphi}\,\delta
C$
$\displaystyle\,\,+\big{(}f\,U-g\big{)}\,\delta\big{(}e^{\varphi}\,N\big{)}+f\,\delta
C\,\Big{(}\partial_{u}e^{\varphi}+\big{(}U\,e^{\varphi}\big{)}^{\prime}\Big{)}~{}.$

This becomes integrable when the field dependence of the functions in the
asymptotic Killing vectors is

$\displaystyle f$ $\displaystyle=\bar{f}e^{\varphi-2\beta},\qquad
h=-\big{(}\bar{h}+(\bar{g}e^{\varphi})^{\prime}\big{)}e^{-\varphi}+\bar{g}^{\prime}+\bar{g}\varphi^{\prime},$
(4.28) $\displaystyle g$
$\displaystyle=\bar{g}+\bar{f}e^{\varphi-2\beta}U,\qquad
k=\bar{k}-\bar{g}C^{\prime}-\bar{f}e^{\varphi-2\beta}(UC^{\prime}+\partial_{u}C)\,.$
(4.29)

The functions $\bar{f}$, $\bar{g}$, $\bar{h}$, and $\bar{k}$ are taken to be
independent of the fields, but are otherwise arbitrary functions of
$(u,\phi)$. With this assignment, the charges are then precisely the ones
presented in [37]:

$\displaystyle\delta
Q_{\xi}=\int_{0}^{2\pi}\\!\\!\\!\\!\mathrm{d}\phi\,\Big{(}\bar{f}\,\delta\bar{M}+\bar{g}\,\delta\bar{N}+\bar{h}\,\delta
C+\bar{k}\,\delta e^{\varphi}\Big{)}\,,$ (4.30)

with

$\displaystyle\bar{M}=4(\beta^{\prime})^{2}-2\beta^{\prime}\varphi^{\prime}+\frac{1}{2}(\varphi^{\prime})^{2}+(2\beta-\varphi)^{\prime\prime}+e^{2(\varphi-\beta)}\big{(}M-\partial_{u}C{-U\,C^{\prime}}\big{)}$
(4.31) $\displaystyle\bar{N}=e^{\varphi}\big{(}N-C^{\prime}\big{)}.$ (4.32)

When $\delta\bar{f}=\delta\bar{g}=\delta\bar{h}=\delta\bar{h}=0$, the algebra
of charges is

$[Q_{\bar{\xi}_{1}},Q_{\bar{\xi}_{2}}]=Q_{[\bar{\xi}_{1},\bar{\xi}_{2}]_{\star}}+\int_{0}^{2\pi}\\!\\!\\!\\!\mathrm{d}\phi\Big{(}\bar{f}_{1}\bar{g}_{2}^{\prime\prime\prime}-\bar{f}_{2}\bar{g}_{1}^{\prime\prime\prime}+\bar{h}_{1}\bar{k}_{2}-\bar{h}_{2}\bar{k}_{1}\Big{)},$
(4.33)

where
$[\bar{\xi}_{1},\bar{\xi}_{2}]_{\star}=\xi(\bar{f}_{12},\bar{g}_{12},\bar{h}_{12},\bar{k}_{12})$
is the vector with

$\displaystyle\bar{f}_{12}$
$\displaystyle=\bar{f}_{1}\bar{g}_{2}^{\prime}+\bar{g}_{1}\bar{f}_{2}^{\prime}-(1\leftrightarrow
2),\qquad\bar{h}_{12}=0$ (4.34a) $\displaystyle\bar{g}_{12}$
$\displaystyle=\bar{g}_{1}\bar{g}^{\prime}_{2}+\frac{1}{\ell^{2}}\bar{f}_{1}\bar{f}_{2}^{\prime}-(1\leftrightarrow
2)\,,\qquad\bar{k}_{12}=0\,.$ (4.34b)

Explicitly this is an algebroid whose base space is parametrized by $u$ and at
each $u$ we have the conformal algebra in two dimensions in direct sum with
the Heisenberg algebra. In the flat limit the conformal algebra becomes the
BMS algebra in three dimensions.

###  4.4 First variation of the action and holographic renormalization

The goal of this section is to use what we learned about the structure of the
presymplectic form in the previous sections to construct an action with a
well-defined variational principle, for fields that satisfy Dirichlet boundary
conditions at $r\to\infty$. In Bondi-Weyl gauge (4.5) with the partially on-
shell fields (4.6a)-(4.6c), we take Dirichlet boundary conditions to fix the
functions $\beta$, $\varphi$, $U$, and $C$ characterizing the large-$r$
behavior of the metric. The first variation of the action should vanish for
variations of the fields that preserve this boundary data. The functions $M$
and $N$, which are sub-leading in the large $r$ expansion of the metric, are
not fixed.

Since $E^{\mu\nu}\,\delta g_{\mu\nu}=0$ for these field configurations, the
variation of the bulk Lagrangian is just the total derivative $\delta
L_{M}=\partial_{\mu}\Theta^{\mu}$ with support on $\partial M$. Our focus is
the contribution on the constant-$r$ surface $\mathcal{B}$ that becomes
spatial infinity in the limit $r\to\infty$. The $\Theta^{r}$ component of the
presymplectic potential contains factors that diverge as $r\to\infty$. As a
result, it is non-zero for some field variations that preserve the boundary
conditions. This is most easily seen by considering field variations that
change the boundary data; i.e. field variations $\delta\beta$,
$\delta\varphi$, $\delta U$, and $\delta C$. Such variations produce terms at
$\mathcal{O}(r^{2})$ and $\mathcal{O}(r)$ in $\Theta^{r}$. The
$\mathcal{O}(r^{2})$ term is $\delta$-exact, so it could in principle be
canceled by a suitable boundary term added to the action. But the
$\mathcal{O}(r)$ term is _not_ $\delta$-exact. So the terms in $\Theta^{r}$
that interfere with a well-defined variational principle cannot be entirely
removed by boundary terms in the action.

However, as we saw in the previous section, contributions from the
$\Theta^{a}$ components must be accounted for. They can be expressed as total
$r$-derivatives, $\Theta^{a}=\partial_{r}Y^{ar}$, which in turn give corner
contributions on $\mathcal{B}$. If we consider the shifted quantity
$\Theta^{r}+\partial_{a}Y^{ar}$, which represents all contributions on
$\mathcal{B}$ (including $\partial\mathcal{B}$) from the variation of the bulk
Lagrangian, then field variations which shift the boundary data produce
$\delta$-exact terms at both $\mathcal{O}(r^{2})$ and $\mathcal{O}(r)$.

$\displaystyle\Theta^{r}+\partial_{a}Y^{ar}=-\frac{1}{\ell^{2}}\,\delta\Big{(}e^{2\beta-\varphi}\,\mathcal{W}^{2}\Big{)}+\big{(}\text{terms
independent of $r$})~{}.$ (4.35)

The obstruction to a well-defined variational principle is now a
$\delta$-exact term which can be addressed by the variation of boundary terms;
this is the correct starting point for constructing an action with a well-
defined variational principle. All that remains is to identify suitable
boundary terms such that $\delta\Gamma=0$ for field variations that preserve
the Dirichlet boundary data.

Before presenting the action and verifying its properties, we comment on two
ambiguities present in these constructions. First, as we saw with the two-
dimensional dilaton gravity models of section 3, it is natural to consider
terms in the action with support on the null components of the boundary
$\mathcal{N}_{\pm}$. We always work with an affine parameterization of the
null generators on $\mathcal{N}_{\pm}$, so that these terms and their first
variation are zero except for contributions that arise at the corners
$\mathcal{B}\cap\mathcal{N}_{\pm}$ [76, 74]. Second, given an action with a
well-defined variational principle, one can define another such action by
adding an intrinsic functional of the boundary data on $\mathcal{B}$ that
remains finite as $r\to\infty$ (assuming such a functional of the fields
exists). Both actions have the same bulk equations of motion. And since the
new term vanishes for field variations which preserve the boundary data there
is no impact on the variational principle. Here, we fix (or at least reduce)
this ambiguity by demanding that the action have a finite flat-space limit
$\ell\to\infty$ for partially on-shell fields in Bondi-Weyl gauge, without any
additional rescalings or redefinitions of the fields.

Given the considerations outlined above, we arrive at an action that admits a
well-defined variational principle with Dirichlet boundary conditions, is
finite in the $r\to\infty$ limit that removes $\mathcal{B}$ to spatial
infinity, and has a sensible flat space limit $\ell\to\infty$. As expected, it
is similar in form to the usual action for AAdS3 gravity, with additional
terms needed in the presence of the corners $\partial\mathcal{B}$ and to
accommodate the $\ell\to\infty$ limit:

$\displaystyle\Gamma=$
$\displaystyle\,\,\frac{1}{2\kappa^{2}}\int_{M}\negthickspace
d^{3}x\sqrt{-g}\,\left(R+\frac{2}{\ell^{2}}\right)+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{2}x\sqrt{-\gamma}\left(K-\frac{1}{\ell}\right)$
(4.36)
$\displaystyle+\frac{1}{\kappa^{2}}\,\int_{\mathcal{B}}d^{2}x\sqrt{-\gamma}\left(\frac{\ell}{2}\,\,K_{c}^{2}\right)+\frac{1}{\kappa^{2}}\,\int_{\partial\mathcal{B}}\\!\\!d^{1}x\,\sqrt{\sigma}\,\big{(}-\ell\,K_{c}-1\big{)}~{}.$
(4.37)

Here $K=\nabla_{\mu}n^{\mu}$ is the trace of the extrinsic curvature of the
constant-$r$ surface $\mathcal{B}$ embedded in $M$ with outward-pointing,
spacelike unit normal $n^{\mu}$. The coordinates on $\mathcal{B}$ are
$x^{a}=(u,\phi)$, and $\gamma_{ab}$ is the induced metric. In addition,
$\mathcal{B}$ is foliated into curves of constant $u$, with forward-pointing
timelike unit normal $\rho^{a}$. The extrinsic curvature of the leaves in this
foliation is $K_{c}=D_{a}\rho^{a}$, where $D_{a}$ is the covariant derivative
on $\mathcal{B}$ compatible with the metric $\gamma_{ab}$. The one-dimensional
metric along each curve in the foliation is $\sigma\,\mathrm{d}\phi^{2}$.

The terms (4.36) in the first line of the action are the familiar bulk and
boundary terms for Einstein-Hilbert gravity in three dimensions, with
asymptotically anti-de Sitter boundary conditions. They include the Gibbons-
Hawking-York (GHY) term and a boundary counterterm [77, 78, 79]. The second
line (4.37) includes new terms that are needed to implement the variational
principle when $\mathcal{B}$ has corners, and to ensure a finite
$\ell\to\infty$ limit.

##### First variation of the action

The variational properties of the terms in (4.36) are well established when
$\partial M$ does not have corners. To verify our claims about the full action
it is sufficient to show that the contributions at the corners
$\partial\mathcal{B}$ do not interfere with the variational principle, and
that the $\ell\to\infty$ limit is well defined.

First, consider the terms in $\delta\Gamma$ with support on $\mathcal{B}$
coming from the first line (4.36).

$\displaystyle\delta\Gamma=\ldots+\int_{\mathcal{B}}\\!\\!\mathrm{d}^{2}x\,\left[\Theta^{r}+\partial_{a}Y^{ar}+\frac{1}{\kappa^{2}}\,\delta\,\left(\sqrt{-\gamma}\,K-\sqrt{-\gamma}\,\frac{1}{\ell}\right)\right]$
(4.38)

The variation of the GHY term and boundary counterterm gives

$\displaystyle\delta\Big{(}L_{\text{GHY}}+L_{\text{CT}}\Big{)}=$
$\displaystyle\,\,-\Theta^{r}+\frac{1}{2\,\kappa^{2}}\sqrt{-\gamma}\,\left(\gamma^{ab}\Big{(}K-\frac{1}{\ell}\Big{)}-K^{ab}\right)\delta\gamma_{ab}$
(4.39)
$\displaystyle\quad+\frac{1}{\kappa^{2}}\,\sqrt{-\gamma}\,D_{a}c^{a}~{}.$

This includes a commonly neglected total derivative term arising from the
variation of the extrinsic curvature [80], which cannot be ignored in the
presence of corners $\partial\mathcal{B}$. As above, $D$ is the two-
dimensional covariant derivative on $\mathcal{B}$ compatible with $\gamma$,
and the vector $c^{a}$ is given by

$\displaystyle c^{a}=-\frac{1}{2}\,\gamma^{ab}\,\delta
g_{b\lambda}\,n^{\lambda}~{}.$ (4.40)

Together, the terms in (4.38) reduce to a familiar contribution on
$\mathcal{B}$ and a corner term with support on $\partial\mathcal{B}$

$\displaystyle\delta\Gamma=\ldots+\int_{\mathcal{B}}\\!\\!\mathrm{d}^{2}x\,\left[\frac{\sqrt{-\gamma}}{2\,\kappa^{2}}\,\left(\gamma^{ab}\Big{(}K-\frac{1}{\ell}\Big{)}-K^{ab}\right)\delta\gamma_{ab}+\partial_{a}\Big{(}Y^{ar}+\frac{1}{\kappa^{2}}\,\sqrt{-\gamma}\,c^{a}\Big{)}\right]~{}.$
(4.41)

The first set of terms, involving the extrinsic curvature and the variation of
the induced metric on $\mathcal{B}$, vanishes for field variations that
preserve Dirichlet boundary conditions at $r\to\infty$ [81]. But this is not
the case for the corner term. From (4.19) we have
$Y^{ur}=\delta\mathcal{W}-\mathcal{W}\,\delta\beta$, and the definition (4.40)
gives $\sqrt{-\gamma}\,c^{u}=\mathcal{W}\,\delta\beta$. Thus, the variation of
the first line of the action (4.36) includes a corner term that interferes
with the definition of the variational principle. For field variations that
change the boundary data it takes the form

$\displaystyle\frac{1}{\kappa^{2}}\int_{\partial\mathcal{B}}\\!\\!d\phi\,\delta\mathcal{W}~{}.$
(4.42)

This diverges as $r\to\infty$, since
$\mathcal{W}=e^{\varphi}\,\big{(}r-C\big{)}$. Therefore, there will be field
variations that preserve the boundary data for which the corner terms in
(4.41) are non-zero. This is at odds with our requirements for the variational
principle.

The corner term described above, which is $\delta$-exact, is addressed by
adding the very last term in (4.37) to the action. It takes the form

$\displaystyle-\frac{1}{\kappa^{2}}\int_{\partial\mathcal{B}}\\!\\!d^{1}x\,\sqrt{\sigma}~{}.$
(4.43)

A leaf of the foliation of $\mathcal{B}$ by curves of constant $u$ has induced
metric $\sigma\,\mathrm{d}\phi^{2}$, so $\sqrt{\sigma}=\mathcal{W}$ for the
fields (4.6a). Thus, the variation of this term in the action immediately
cancels the problematic term (4.42).

All that remains in our proposed action are the terms in (4.37) involving the
extrinsic curvature $K_{c}$ of the constant-$u$ curves foliating
$\mathcal{B}$. It is straightforward to see that these terms are a finite (as
$r\to\infty$) functional of the Dirichlet boundary data on $\mathcal{B}$. The
future-pointing timelike unit vector normal to curves of constant $u$ on
$\mathcal{B}$ is

$\displaystyle\rho^{a}=\frac{1}{e^{\beta}\,\sqrt{-\mathcal{V}}}\,\Big{(}\delta^{a}{}_{u}+\mathcal{U}\,\delta^{a}{}_{\phi}\Big{)}~{},$
(4.44)

so the extrinsic curvature of these curves is

$\displaystyle
K_{c}=D_{a}\rho^{a}=\frac{1}{\sqrt{-\gamma}}\,\Big{(}\partial_{u}\mathcal{W}+\partial_{\phi}\big{(}\mathcal{W}\,\mathcal{U}\big{)}\Big{)}~{}.$
(4.45)

From the large-$r$ behavior of the fields (4.6a)-(4.6c) it is apparent that
the quantities $\sqrt{-\gamma}\,K_{c}^{2}$ and $\sqrt{\sigma}\,K_{c}$ in
(4.37) are finite functions of the Dirichlet boundary data as $r\to\infty$,
and hence do not interfere with the variational principle. An alternate
presentation of these terms is given in Appendix C.

Including the contributions from all terms in (4.36)-(4.37), and evaluating on
fields that satisfy $E^{\mu\nu}\,\delta g_{\mu\nu}=0$, we have
$\delta\Gamma=0$ for all field variations that preserve the Dirichlet boundary
data at $r\to\infty$. For field variations that change this boundary data, the
non-zero terms in $\delta\Gamma$ are

$\displaystyle\delta\Gamma=\frac{1}{\kappa^{2}}\int_{\mathcal{B}}\negthickspace
dud\phi\bigg{[}$ $\displaystyle\partial_{u}(-e^{\varphi}\delta
C)+e^{\varphi}\delta\left(2\beta-\varphi\right)\left(-M+\partial_{u}C+U\,C^{\prime}\right)$
(4.46) $\displaystyle+e^{\varphi}\delta
U\left(N-C^{\prime}\right)-\delta\beta\left(4e^{2\beta-\varphi}\beta^{\prime}\right)^{\prime}\bigg{]}~{}.$

Total derivatives with respect to $\phi$ have been dropped to obtain this form
of $\delta\Gamma$. It is interesting to note that this can also be written as

$\displaystyle\delta\Gamma=\frac{1}{\kappa^{2}}\int_{\mathcal{B}}\negthickspace
dud\phi\bigg{[}$ $\displaystyle\partial_{u}(-e^{\varphi}\delta
C)-\bar{M}\,\delta(e^{2\beta-\varphi})+\bar{N}\,\delta U$ (4.47)
$\displaystyle+\delta\left(-2\left(e^{2\beta-\frac{1}{2}\varphi}\right)^{\prime}\left(e^{-\frac{1}{2}\varphi}\right)^{\prime}\right)\bigg{]}~{},$

where $\bar{M}$ and $\bar{N}$ are the charges defined in (4.31)-(4.32).

##### On-shell action

The value of the action for a specific field configuration will depend on
details in the interior of $M$, like the presence of a horizon. Evaluating
(4.36)-(4.37) for the fields (4.6a)-(4.6c) on a semi-infinite interval
$r_{+}\leq r<\infty$, gives

$\displaystyle\Gamma=$
$\displaystyle\frac{1}{\kappa^{2}}\int_{\mathcal{B}}\mathrm{d}u\mathrm{d}\phi\,\left(-e^{\varphi}\,\big{(}M-\partial_{u}C-U\,C^{\prime}\big{)}+\frac{1}{\ell^{2}}\,e^{2\beta+\varphi}\big{(}r_{+}-C\big{)}^{2}\right)~{}.$
(4.48)

Total $\phi$-derivatives have been discarded, and the last term captures
contributions from the lower limit of the integral over $r$. This result makes
two things apparent. First, the boundary terms needed for a well-defined
variational principle also render the action finite as the regulator is
removed via the $r\to\infty$ limit. And second, the action remains finite in
the flat space $\ell\to\infty$ limit.

##  5 Conclusion

In this paper we have outlined a procedure for obtaining finite charges at an
asymptotic boundary, and applied it to two lower-dimensional examples where
the resulting charges are integrable. The existence of the limiting procedure
that defines the charges is ensured by appropriate corner terms in the
presymplectic current, obtained directly from the variation of the bulk
Lagrangian. This method of identifying the corner terms gives a systematic
accounting of the _ad hoc_ shifts and subtractions used in some previous
calculations [36, 37, 35, 52]. In these lower-dimensional examples, the
charges are not just finite at the asymptotic boundary, but fully independent
of the coordinate $r$ used to implement the limiting procedure. One major
advantage of this approach is that it reduces ambiguities in Iyer-Wald
covariant phase space methods. A (finite) ambiguity remains, which is to be
expected in any formulation that accommodates an open system where the
boundary dynamics is not specified. We also show that the corner terms in the
presymplectic potential provide important information needed to construct a
well-defined variational principle on spacetimes where components of the
boundary intersect at corners.

A notable feature of our analysis is that the fields are not fully on shell.
Some equations of motion are enforced, while others – constraints conjugate to
gauge-fixed quantities – are relaxed. Nevertheless, even with the fields only
“partially on-shell,” diffeomorphism invariance implies the existence of a
codimension-2 form which receives contributions from both the presymplectic
current and the weakly vanishing Noether current. When the fields are fully
on-shell it reduces to the usual result, but with our weaker conditions it
captures charges which are not otherwise accessible. This formalizes some of
the assumptions in the analysis of [61], where on-shell conservation of the
Casimir must be relaxed before the off-shell conformal symmetry is recovered
in the dilaton gravity dual of the SYK model.

The examples in sections 3 and 4 are lower-dimensional theories without local
degrees of freedom in the bulk. A natural next step is to apply the procedure
to higher dimensional theories with dynamical degrees of freedom that might
radiate across the asymptotic boundary. The proof outlined in section 2 will
still apply if the fields fall off sufficiently fast as $r\to\infty$. The
question now is whether those fall-off conditions are physically acceptable.
One starting point is four-dimensional gravity in the partial Bondi gauge
studied in [75], which is closely related to the three-dimensional Bondi gauge
used in section 4. The expectation is that the components $\Theta^{a}$ at
large $r$ will then consist of a leading part fixed by the boundary conditions
and kinematics, along with a sub-leading part where the dynamical degrees of
freedom appear. The Iyer-Wald ambiguity would presumably be used to shift the
former into corner contributions appearing in $\Theta_{\text{ren}}^{r}$. This
may be sufficient to define charges which satisfy an appropriate flux-balance
law [10, 16, 17, 82]. However, it is also possible that the remaining,
dynamical part of $\omega_{\text{ren}}^{u}$ does not fall off fast enough at
large $r$ for the arguments of section 2 to apply. In that case, one would
have to either relax the boundary conditions, or else modify our prescription.
Since a number of possible complications are expected to appear in four
dimensions, it might instead be helpful to start by coupling additional fields
to the simple models treated in this paper. One could then focus on just the
issue of extracting corner terms when dynamical degrees of freedom appear in
the relevant components of the presymplectic potential.

Even without adding new fields, there are still important results to recover
in the lower-dimensional models we consider. For three dimensional gravity,
one expects as many as six charges [83, 84]. The Bondi-Weyl gauge employed in
section 4 leads to four charges. Relaxing the $g_{r\phi}=0$ condition would
bridge the Bondi-Weyl [37] and covariant Bondi [35] gauges, enlarging the
space of field configurations and possibly increasing the number of charges.
As shown in sections 3 and 4, contributions from the variation of the weakly
vanishing Noether current in (2.23) are essential for obtaining the charges
when the fields are only partially on-shell. Removing the gauge condition
$g_{r\phi}=0$ would allow variations $\delta g_{r\phi}$, and therefore the
$E^{r\phi}=0$ component of the equation of motion would presumably need to be
enforced to completely fix the $r$-dependence of the fields. This will change
how the weakly vanishing Noether current contributes to the charges.

There are other extensions and refinements of this work that we would like to
explore. The focus here has been on asymptotic symmetries generated by
diffeomorphisms. It would be useful to generalize our analysis to incorporate
other gauge symmetries [85, 86, 62, 87, 88, 89]. Even the addition of a simple
$U(1)$ gauge field would be interesting, since unlike a scalar it would
contribute to the weakly vanishing Noether current. Another avenue of
investigation involves the choice of spacelike coordinate $r$ that is singled
out when specifying the boundary conditions and constructing the charges. For
some theories, such as asymptotically AdS gravity, the boundary conditions can
be formulated in a completely covariant manner [90, 91]. It seems reasonable
that the procedure for extracting the corner terms in the presymplectic
potential could also be specified in a covariant manner for such theories.

## Acknowledgments

The authors thank Adrien Fiorucci for helpful discussions and comments on a
draft of this paper. We also thank Daniel Grumiller and Florian Ecker for
stimulating conversations, and Technical University of Vienna for hospitality
during the early stages of this project. Some of the calculations presented in
this paper were performed with the xAct package for Mathematica [92]. Research
at Perimeter Institute is supported in part by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of Colleges and
Universities.

##  Appendix A “Einstein-Hilbert” contribution to $k_{\xi}^{\mu\nu}$ in 2d
dilaton gravity

The last line in (3.30) has been simplified. It is proportional to the product
of the dilaton $X$ and the codimension-2 form $k^{\mu\nu}_{\xi,EH}$ for
Einstein-Hilbert gravity, which is given by

$\displaystyle\frac{2\,\kappa^{2}}{\sqrt{-g}}\,k_{\xi,EH}^{\mu\nu}=$
$\displaystyle\,\,\xi^{\nu}\,\nabla^{\mu}\delta
g-\xi^{\mu}\,\nabla^{\nu}\delta g+\xi^{\mu}\,\nabla_{\lambda}(\delta
g)^{\lambda\nu}-\xi^{\nu}\,\nabla_{\lambda}(\delta g)^{\mu\lambda}$ (A.1)
$\displaystyle+\xi^{\lambda}\,\big{(}\nabla^{\nu}(\delta
g)^{\mu}{}_{\lambda}-\nabla^{\mu}(\delta
g)_{\lambda}{}^{\nu}\big{)}+\frac{1}{2}\,\delta
g\,\big{(}\nabla^{\nu}\xi^{\mu}-\nabla^{\mu}\xi^{\nu}\big{)}$
$\displaystyle+(\delta g)^{\mu\lambda}\,\nabla_{\lambda}\xi^{\nu}-(\delta
g)^{\lambda\nu}\,\nabla_{\lambda}\xi^{\mu}~{}.$

The terms in the last line of (3.30) (with a factor of $\sqrt{-g}/\kappa^{2}$)
are equal to $X\,k^{\mu\nu}_{\xi,EH}$ in two dimensions. This is not
immediately apparent when the two expressions are compared in covariant form,
and the condition (3.29) satisfied by $k^{\mu\nu}_{\xi}$ is most easily
verified by replacing the last line of (3.30) with $X\,k^{\mu\nu}_{\xi,EH}$.
We include it here for completeness. In any case, this part of the
codimension-2 form does not contribute to our charge calculation. Evaluating
(A.1) in linear dilaton Bondi gauge gives

$\displaystyle
k_{\xi,EH}^{ur}=\frac{1}{2\kappa^{2}}\,\delta\big{(}\sqrt{-g}\,g^{rr}\big{)}\,\partial_{r}\xi^{u}~{}.$
(A.2)

This vanishes for diffeomorphisms that preserve the gauge condition
$g_{rr}=0$, which requires $\partial_{r}\xi^{u}=0$.

##  Appendix B Counterterms for dilaton gravity

In this appendix we generalize the results of [53] to dilaton gravity models
where spatial infinity is the $r_{c}\to\infty$ limit of a regulating boundary
$\mathcal{B}$ that is not an isosurface of the dilaton. Under certain
assumptions the resulting terms can be organized in a derivative expansion on
$\mathcal{B}$, with the zeroth-order term reproducing the boundary counterterm
given in [53]. Depending on the potentials $U$ and $V$, additional terms in
this expansion may be necessary for a well-defined variational problem and a
finite action.

The approach of [53] identified a boundary counterterm for the dilaton gravity
action by solving the radial Hamiltonian constraint. That analysis assumed the
fields were fully on-shell, and used coordinates $\tilde{r}$ and $t$ adapted
to the Killing vector $\xi_{X}$. Curves of constant $\tilde{r}$, including
spatial infinity at $\tilde{r}\to\infty$, are then isocurves of the dilaton
$X(\tilde{r})$. Here we relax this condition and repeat the analysis for a
more general foliation. The starting point is a “radial ADM” decomposition of
the 2D metric 171717The linear Bondi gauge is obtained by taking
$\gamma_{uu}=2B,N^{u}=-e^{A}/(2B),N=e^{A}/\sqrt{-2B}$

$\displaystyle
ds^{2}=N^{2}\,dr^{2}+\gamma_{uu}\,\big{(}du+N^{u}\,dr\big{)}^{2}~{}.$ (B.1)

The bulk Lagrangian (3.1) can be written as

$\displaystyle\kappa^{2}\,L_{M}=$
$\displaystyle\,\,\sqrt{-\gamma}\,\frac{1}{2N}\,\gamma^{uu}\big{(}\partial_{r}\gamma_{uu}-2\gamma_{uu}D_{u}N^{u}\big{)}\big{(}\partial_{r}X-N^{u}D_{u}X\big{)}$
(B.2)
$\displaystyle\quad-\sqrt{-\gamma}\,\frac{1}{2N}\,U\,\big{(}\partial_{r}X-N^{u}D_{u}X\big{)}^{2}-\sqrt{-\gamma}\,N\,\Big{(}\frac{1}{2}\,U\,(DX)^{2}+D^{2}X+V\Big{)}$
$\displaystyle\quad+\partial_{u}\Big{(}-\sqrt{-g}\,n^{u}\,X\,K+\sqrt{-\gamma}\,\gamma^{uu}\,\big{(}N\,\partial_{u}X-X\,\partial_{u}N\big{)}\Big{)}$
$\displaystyle\quad+\partial_{r}\Big{(}-\sqrt{-g}\,n^{r}\,X\,K\Big{)}~{},$

where $K$ is the extrinsic curvature of a constant $r$ curve, $\gamma_{uu}$ is
the induced metric on that curve, and $D_{u}$ is the (1D) covariant derivative
compatible with that metric. The last term cancels the GHY term on
$\mathcal{B}$ so we drop it henceforth. 181818That term potentially gives
another contribution if the boundary has a component at some interior value of
$r$, like a horizon. But that is not relevant here, as we are only interested
in terms on $\mathcal{B}$. The momenta conjugate to $\partial_{r}\gamma_{uu}$
and $\partial_{r}X$ are

$\displaystyle\pi^{uu}=$ $\displaystyle\,\,\frac{\partial
L_{M}}{\partial(\partial_{r}\gamma_{uu})}=\frac{\sqrt{-\gamma}}{2\,\kappa^{2}N}\,\gamma^{uu}\big{(}\partial_{r}X-N^{u}D_{u}X\big{)}$
(B.3) $\displaystyle\pi_{X}=$ $\displaystyle\,\,\frac{\partial
L_{M}}{\partial(\partial_{r}X)}=\frac{\sqrt{-\gamma}}{2\,\kappa^{2}N}\,\gamma^{uu}\big{(}\partial_{r}\gamma_{uu}-2\,\gamma_{uu}D_{u}N^{u}\big{)}-2\,U\,\pi^{u}{}_{u}~{}.$
(B.4)

Now we construct the Hamiltonian
$H_{M}=\pi^{uu}\,\partial_{r}\gamma_{uu}+\pi_{X}\,\partial_{r}X-L_{M}$. Using
the above expressions, this is

$\displaystyle
H_{M}=N\,\mathcal{H}+N^{u}\,\mathcal{H}_{u}+\partial_{u}T^{u}~{},$ (B.5)

with

$\displaystyle\mathcal{H}=$