Cornering Quantum Gravity

Luca Ciambelli,a Alessandra D’Alise,b Vittorio D’Esposito,b Dušan Đorđevic,c

Diego Fernández-Silvestre,d Ludovic Varrine

a Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo
ON, Canada, N2L 2Y5

b Dipartimento di Fisica “E.Pancini”, Università di Napoli Federico II,
I-80125 Napoli, Italy and INFN, Sezione di Napoli, Complesso Universitario di
Monte S. Angelo, Via Cintia Edificio 6, 80126 Napoli, Italy

c Faculty of Physics, University of Belgrade, Studentski trg 12, Belgrade

d Departamento de Matemáticas y Computación, Universidad de Burgos, 09001
Burgos, Spain

e National Centre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

These lecture notes cover materials exposed by Luca Ciambelli at the 59 Winter
School of Theoretical Physics and third COST Action CA18108 Training School
"Gravity – Classical, Quantum and Phenomenology", held in Pałac Wojanów,
Poland, 12–21 Feb 2023.

After introducing the covariant phase space calculus, Noether’s theorems are
discussed, with particular emphasis on Noether’s second theorem and the role
of gauge symmetries. This is followed by the enunciation of the theory of
asymptotic symmetries, and later its application to gravity. Specifically, we
review how the BMS group arises as the asymptotic symmetry group of gravity at
null infinity. Symmetries are so powerful and constraining that memory effects
and soft theorems can be derived from them. The lectures end with more recent
developments in the field: the corner proposal as a unified paradigm for
symmetries in gravity, the extended phase space as a resolution to the problem
of charge integrability, and eventually the implications of the corner
proposal on quantum gravity.

###### Contents

  1. 1 Symmetries
    1. 1.1 Covariant phase space
      1. 1.1.1 Spacetime (De Rham) calculus
      2. 1.1.2 Field space (variational) calculus
      3. 1.1.3 Lagrangian approach
    2. 1.2 Noether’s theorems and charge algebra
      1. 1.2.1 Noether’s first theorem
      2. 1.2.2 Noether’s second theorem
      3. 1.2.3 Charge algebra
    3. 1.3 Application to gauge theories
    4. 1.4 Asymptotic symmetries
  2. 2 Gravity
    1. 2.1 What is wrong?
    2. 2.2 BMS
      1. 2.2.1 Dynamics
      2. 2.2.2 Boundary conditions
      3. 2.2.3 Gauge fixing
      4. 2.2.4 EOM
      5. 2.2.5 Symmetries
      6. 2.2.6 Algebra
      7. 2.2.7 Charges
      8. 2.2.8 Charge algebra
    3. 2.3 The power of symmetries: memory effects
    4. 2.4 The power of symmetries: soft theorems
  3. 3 The Corner Proposal
    1. 3.1 Universal algebra
    2. 3.2 Extended phase space
      1. 3.2.1 Keeping track of the embedding
      2. 3.2.2 Extended symplectic form
    3. 3.3 The corner proposal for quantum gravity
  4. A Proof of Important Relations

##### Technical Note:

Subsecs. 1.1, 1.2, 1.4, 2.1, and Section 3 follow closely reference [1]. We
refer to it for a detailed introduction on the topics exposed and a complete
list of references.

##  1 Symmetries

Symmetries are one of the most powerful tools in the description of physical
theories. From symmetry arguments physical laws can be derived and the
solution to complicated problems can be simplified by restricting the allowed
solutions to particular classes obeying a given symmetry. Symmetries can also
be used to constraint physical theories: for instance, the number of allowed
Lagrangians in relativistic quantum field theory is heavily limited by the
requirement of Lorentz invariance. In these notes, we will adopt a geometric
approach to the description of symmetries based on the so-called _covariant
phase space_ formalism.

The first naive distinction that can be found in textbooks is between _local_
and _global_ symmetries.

  * •

Local symmetries are described as transformations characterized by spacetime
dependent parameters. They are typically thought of as pure redundancies
without any physical content.

  * •

Global symmetries are described as transformations being specified by constant
parameters. These are typically regarded as physical transformations acting
non-trivially on physical systems.

The above distinction is somewhat confusing in that there are some contexts,
specifically in the presence of _boundaries_ , when local symmetries are
actually physical. The spacetime dependence of transformation parameters
should not be the discerning feature in the distinction between physical
symmetries and non-physical ones. The latter should be rather guided by the
presence (or absence) of non-vanishing associated charges. Therefore, the
fundamental tools in this distinction are Noether’s theorems [2].

There are two main frameworks in the study of Noether’s theorems and
symmetries:111As already mentioned, we refer to [1] for a complete list of
references.

  * •

In the Hamiltonian formalism, see [3, 4], the evolution of the system is
described by trajectories in phase space. The positive aspects of this
formalism are that it is a very robust and unambiguous framework and, more
importantly, it is the classical approach which is closest to quantum physics
(it is indeed the one used within the canonical quantization procedure).
However, there are also some important drawbacks, the most important of which
for our purposes is that it breaks the manifest spacetime covariance. This is
the main reason why we will not adopt this formalism.

  * •

In the Lagrangian approach [2], see also [5], the phase space is in one to one
correspondence with the set of solutions to the equations of motion. This is
why this approach is manifestly covariant, indeed there is no specification of
what time is but rather the physical system is described in terms of this
abstract set of solutions. However, this comes with some negative aspects,
mainly that this approach is somewhat more ambiguous and thus it is sometimes
less intuitive.

As anticipated, we will adopt the Lagrangian formalism in these notes in that
it allows us to apply the covariant phase space formalism which we will now
introduce.

###  1.1 Covariant phase space

The idea of this formalism is to describe spacetime calculus (see for instance
[6]) and field space calculus (see [5] for a recent review) on the same
footing.

####  1.1.1 Spacetime (De Rham) calculus

Given a manifold $M$, it is possible to introduce the De Rham calculus on the
space of forms. A $1$-form is a linear map from $TM$, the space of vector
fields over $M$, to $C^{\infty}(M)$, the space of infinitely differentiable
functions over $M$. The space of $1$-forms is dual to $TM$ and is therefore
denoted by $T^{*}M$. A generic $p$-form is a linear, skew-symmetric map, from
$p$ copies of $TM$ to $C^{\infty}(M)$. $p$ is also referred to as degree of
the $p$-form. Given the skew-symmetric nature of these objects, the space of
$p$-forms is denoted by
$\Omega^{p}(M,\mathbb{R})=\bigwedge\nolimits^{p}T^{*}M$, where $\bigwedge$
denotes the skew-symmetric product. If $\text{dim}\,M=d$, it is clear that the
maximum degree of a form is $d$. Such $d$-forms are called top forms.
Therefore, we can build the space of all forms, called the _De Rham complex_ ,
as

$\Omega^{\bullet}(M,\mathbb{R})\vcentcolon=\bigoplus_{p=0}^{d}\bigwedge\nolimits^{p}T^{*}M\;,$
(1)

where $0$-forms are scalar functions, namely
$\bigwedge\nolimits^{0}T^{*}M\vcentcolon=C^{\infty}(M)$. Observe that
$\text{dim}\,\Omega^{d}(M,\mathbb{R})=1$, namely there exist only one top
form, modulo function multiplication. On this complex, three derivations can
be defined, namely the interior product, the exterior derivative, and the Lie
derivative.

The interior product is defined as a contraction of a $p$-form with a vector
field and it is a (anti-)derivation of degree $-1$, namely it decreases by one
the degree of forms. If $X\in TM$, the interior product is a map

$\iota_{X}\vcentcolon\Omega^{p}(M,\mathbb{R})\mapsto\Omega^{p-1}(M,\mathbb{R})$
(2)

defined as

$\iota_{X}\omega\quantity(Y_{1},\ldots,Y_{p-1})\vcentcolon=\omega\quantity(X,Y_{1},\ldots,Y_{p-1})\;,$
(3)

where $\omega\in\Omega^{p}(M,\mathbb{R})$. If $\alpha$ is a $1$-form, then
$\iota_{X}\alpha=\alpha(X)$, while $\iota_{X}f=0\;\,\forall\,f\in
C^{\infty}(M)$. If $\alpha$ is a $p$-form and $\beta$ is a $q$-form, then the
interior product satisfies the graded Leibnitz rule

$\iota_{X}(\alpha\wedge\beta)=\iota_{X}(\alpha)\wedge\beta+(-1)^{p}\alpha\wedge\iota_{X}\beta\;,$
(4)

from which it is clear that this map is an anti-derivation. Another important
property of the interior product is that
$\iota_{X}\iota_{Y}=-\iota_{Y}\iota_{X}$ from which it follows that
$\iota_{X}\iota_{X}=0$.

The exterior derivative is an anti-derivation of degree $1$, namely it raises
by one the degree of forms. It is a map

$\text{d}\vcentcolon\Omega^{p}(M,\mathbb{R})\mapsto\Omega^{p+1}(M,\mathbb{R})$
(5)

satisfying the graded Leibnitz rule

$\text{d}(\alpha\wedge\beta)=\text{d}\alpha\wedge\beta+(-1)^{p}\alpha\wedge\text{d}\beta\;,$
(6)

where $\alpha$ and $\beta$ are a $p$-form and a $q$-form respectively. It is
assumed that the exterior derivative is a coboundary, namely a nilpotent
operator $\text{d}^{2}=0$.

The third derivation that can be defined on the De Rham complex is the Lie
derivative, which is a derivation of degree $0$, it does not change the degree
of forms. Given a vector field $X$, the Lie derivative along $X$ is given by
the anti-commutator of the interior product and the exterior derivative

${\mathcal{L}}_{X}\vcentcolon=\text{d}\iota_{X}+\iota_{X}\text{d}\;.$ (7)

This is typically referred to as _Cartan’s magic formula_. From this, we can
also derive the full algebra satisfied by the three derivations we have
defined, which turns out to be

$\commutator{{\mathcal{L}}_{X}}{\iota_{X}}=\commutator{{\mathcal{L}}_{X}}{\text{d}}=0\;,\;\commutator{{\mathcal{L}}_{X}}{\iota_{Y}}=\commutator{\iota_{X}}{{\mathcal{L}}_{Y}}=\iota_{\commutator{X}{Y}}\;.$
(8)

All in all, the Lie derivative defines the directional derivative along the
vector field $X$. Given a tensor $g$, for instance a metric defined on $M$, an
_isometry_ of $g$ is a vector field $X$ such that

${\mathcal{L}}_{X}g=0\;,$ (9)

namely the tensor $g$ does not change along integral curves of $X$.

Until now we have only used intrinsic geometric definitions, without referring
to any coordinate system. If $x^{\mu}$ are coordinates on $M$, then
$\text{d}x^{\mu}$ is a basis of $T^{*}M$ and $\partial_{\mu}$ is a basis of
$TM$. Therefore, a generic $1$-form and a generic vector field are given by
$\omega=\omega_{\mu}\text{d}x^{\mu}$ and $X=X^{\mu}\partial_{\mu}$
respectively (implicit sum on repeated indexes). Therefore, the interior
product yields

$\iota_{X}\omega=X^{\mu}\omega_{\mu}\;,$ (10)

i.e. the contraction. Analogously, the exterior derivative of a function
yields

$\text{d}f=\partial_{\mu}f\text{d}x^{\mu}\;,$ (11)

which is nothing else than the differential of the function. If $g$ is a
metric tensor defined on $M$, the top form, also called volume form, in
coordinates is given by

$\sqrt{\absolutevalue{\det{g}}}\text{d}x^{1}\wedge\ldots\wedge\text{d}x^{d}=\sqrt{\absolutevalue{\det{g}}}\differential[d]{x}=\star
1\;,$ (12)

where $\star$ is the Hodge star which, for a generic $p$-form $\omega$, is
defined as

$\star\vcentcolon\Omega^{p}(M,\mathbb{R})\mapsto\Omega^{d-p}(M,\mathbb{R})\;\;\Big{|}\;\;\star\omega\vcentcolon=\frac{\sqrt{\absolutevalue{\det{g}}}}{(d-p)!}\omega_{\mu_{1},\ldots,\mu_{p}}\tensor{\epsilon}{{}^{\mu_{1},\ldots,\mu_{p}}}{{}_{\nu_{1},\ldots,\nu_{d-p}}}\text{d}x^{\nu_{1}}\wedge\ldots\wedge\text{d}x^{\nu_{d-p}}\;,$
(13)

with $\epsilon$ being the Levi-Civita symbol.

To conclude this survey of spacetime calculus, we define the integration of
forms. A $p$-form can be integrated over $p$-dimensional sub-manifolds (i.e.
on sub-manifolds on which it is a top form). For instance, a one form can be
integrated over a curve $\gamma\vcentcolon[0,1]\mapsto M$ on the manifold $M$

$\int_{\gamma}\omega=\int_{\gamma}\omega_{\mu}\text{d}x^{\mu}\Big{|}_{\gamma}=\int_{\gamma(0)=a}^{\gamma(1)=b}\omega_{\mu}\derivative{\gamma^{\mu}}{s}\differential{s}\;,$
(14)

where $\text{d}x^{\mu}\big{|}_{\gamma}$ is the embedding of the coordinate
functions over the curve $\gamma$ and $\\{a,b\\}$ is the _boundary_ of
$\gamma$, i.e. $\partial\gamma=\\{a,b\\}$. In general, a $p$-form is
integrated over a $p$-chain, which is a map
$[0,1]\times\stackrel{{\scriptstyle p}}{{\ldots}}\times[0,1]\mapsto M$. The
set of all $p$-chains is a vector space, which we denote as $\mathcal{C}_{p}$,
and the boundary map is a linear map between these vector spaces, namely
$\partial\vcentcolon\mathcal{C}_{p}\mapsto\mathcal{C}_{p-1}$. This map allows
to define the _homology_ of $M$, i.e. the number of holes of the manifold. The
$k$-th homology group is defined as the vector space of $k$-chains with no
boundary (sometimes called $k$-cycles) which are not the boundary of a
$k+1$-chain, namely as the quotient

$H_{k}(M)\vcentcolon=\faktor{\left\\{c\in\mathcal{C}_{k}\,\mid\,\partial
c=0\right\\}}{\left\\{c\in\mathcal{C}_{k}\,\mid\,c=\partial
c^{\prime}\,,\,c^{\prime}\in\mathcal{C}_{k+1}\right\\}}\;.$ (15)

The dimension of $H_{k}(M)$ gives the number of $k$-dimensional holes of $M$.
The notion of boundary is somewhat the notion of exterior derivative. Indeed,
while the boundary decreases by one the degree of chains, the exterior
derivative goes in the opposite direction, increasing by one the dimension of
forms. The exterior derivative allows one to define the cohomology of a
manifold, indeed the $k$-cohomology group is defined as the vector space of
closed $p$-forms $(\text{d}\omega=0)$ which are not exact
($\omega\neq\text{d}\alpha$), namely it is the quotient

$H^{k}(M)\vcentcolon=\faktor{\left\\{\omega\in\Omega^{k}(M,\mathbb{R})\,\mid\,\text{d}\omega=0\right\\}}{\left\\{\omega\in\Omega^{k}(M,\mathbb{R}),\mid\,\omega=\text{d}\omega^{\prime}\,,\,\omega^{\prime}\in\Omega^{k-1}(M,\mathbb{R})\right\\}}\;.$
(16)

The De Rham theorem states that (15) and (16) are isomorphic, $H_{k}(M)\cong
H^{k}(M)$, therefore also the dimension of $H^{k}(M)$ gives the number of
$k$-dimensional holes of $M$. The isomorphism is given by the integral which
to a given $k$-chain, $c$, associates the unique $k$-form $\omega$ such that
the integral of $\omega$ over $c$ is $0$

$c\mapsto\omega\;\;\;\Big{|}\;\;\;\int_{c}\omega=0\;.$ (17)

The integral therefore defines a product between $k$-chains and $k$-forms,
$\int_{c}\omega\vcentcolon=(c,\omega)$. Finally, through the Stokes theorem, a
duality between the boundary and the exterior derivative is established

$(\text{d}\omega,c)=\int_{c}\text{d}\omega=\int_{\partial
c}\omega=(\omega,\partial c)\;,$ (18)

where $c\in\mathcal{C}_{k+1}$ and $\omega\in\Omega^{k}(M,\mathbb{R})$.

Having reviewed the relevant notions of spacetime calculus, we can now move to
field-space calculus.

####  1.1.2 Field space (variational) calculus

We can define analogous operations on the space of fields. The field space
$\Gamma$ is defined as the set of all possible fields configurations, assumed
to be a differentiable manifold. If this is the case, it is possible to
introduce a calculus on the space of forms on $\Gamma$. A $1$-form is an
element of $T^{*}\Gamma$ which maps vector fields in $T\Gamma$ to functionals
in $F=C^{\infty}(\Gamma)$. A $p$-form is an element of
$\bigwedge\nolimits^{p}T^{*}\Gamma$ and the space of all forms is given by

$\Omega^{\bullet}(\Gamma,F)\vcentcolon=\bigoplus_{p=0}^{\text{dim}\,\Gamma}\bigwedge\nolimits^{p}T^{*}\Gamma\;,$
(19)

which is called _variational complex_. $0$-forms are now functionals of the
fields. The exterior derivative and the interior product are maps

$\displaystyle\delta\vcentcolon\Omega^{p}(M,F)\mapsto\Omega^{p+1}(M,F)\;,$
(20) $\displaystyle
I_{\hat{V}}\vcentcolon\Omega^{p}(M,F)\mapsto\Omega^{p-1}(M,F)\;,$ (21)

where $\hat{V}\in T\Gamma$. The exterior derivative increases by one the
degree of a form and it is commonly referred to as field variation, while the
interior product decreases by one the degree of a form and it is usually
identified with a field contraction. The properties of these two maps are
similar to the ones we have in spacetime calculus: the exterior derivative is
a nilpotent operator, $\delta^{2}=0$, the interior product of a functional is
$0$ and two contractions anti-commute. It is also assumed, unless otherwise
stated, that vector fields in $TM$ are fields-independent, namely
$\mathfrak{L}_{\hat{V}}X=0\;\,\forall X\in TM,\hat{V}\in T\Gamma$, and
$\mathfrak{L}_{\hat{V}}$ the field space Lie derivative defined hereafter. The
exterior derivative and interior product define the field space Lie derivative
through the analogue of Cartan’s magic formula

$\mathfrak{L}_{\hat{V}}\vcentcolon=\delta I_{\hat{V}}+I_{\hat{V}}\delta\;.$
(22)

Of course, the Lie derivative does not change the degree of forms. The De Rham
complex and the variational complex can be put together yielding the
_variational bi-complex_ defined on $(M,\Gamma)$. The forms on the variational
bi-complex will be denoted as $(p,q)$-forms. A $(p,q)$-form is a $p$-form in
spacetime and a $q$-form in field space. The exterior derivative on the bi-
complex is given by $\text{d}+\delta$. There are mainly two conventions that
can be adopted for this derivative:

  * •

We can assume that $\text{d}+\delta$ is a coboundary, namely that
$(\text{d}+\delta)^{2}=0$, which yields $\text{d}\delta=-\delta\text{d}$;

  * •

We can assume that d and $\delta$ commute, namely
$\text{d}\delta=\delta\text{d}$.

The two conventions above yield the same results when equations are evaluated
on shell of the equations of motion. While the first convention has been
adopted in [1], we will adopt the second convention in these notes.

In field space it is possible to define coordinate functions as done in
spacetime. If $\phi^{i}(x)$ are coordinates on $\Gamma$, then
$\delta\phi^{i}(x)$ is a basis on $T^{*}\Gamma$ and
$\functionalderivative{\phi^{i}(x)}$, namely the functional derivatives with
respect to coordinate functions on field space, is a basis of $T\Gamma$. A
generic $1$-form on field space can therefore be written as

$\hat{\omega}=\int_{M}\hat{\omega}_{i}(x)\delta\phi^{i}(x)\differential[d]{x}\vcentcolon=\hat{\omega}_{i}\delta\phi^{i}\;,$
(23)

namely, the omitted sum in this case involves also an integration over
spacetime (we can think of $x$ as another index on which we have to sum). For
instance, a $(1,2)$-form is written as

$\omega=\omega_{\mu,ij}\,\text{d}x^{\mu}\otimes\delta\phi^{i}\wedge\delta\phi^{j}\;,$
(24)

and so on. In general, given a functional
$f[\varphi]=\int_{M}h(\varphi(x),x)\differential[d]{x}$, the functional
derivative is given by

$\functionalderivative{f[\varphi]}{\varphi}=\int_{M}\functionalderivative{f[\varphi]}{\varphi(x)}\differential[d]{x}=\int_{M}\int_{M}\functionalderivative{h(\varphi(y),y)}{\varphi(x)}\differential[d]{y}\differential[d]{x}=\int_{M}\functionalderivative{h(\varphi(x),x)}{\varphi(x)}\differential[d]{x}\;,$
(25)

since $\functionalderivative{\phi(y)}{\phi(x)}=\delta^{(d)}(x-y)$.

####  1.1.3 Lagrangian approach

We want now to apply the above formalism to a Lagrangian theory. The action of
a theory is given by

$S=\int_{M}L=\int_{M}\tilde{L}(\varphi,x)\sqrt{\absolutevalue{\det{g}}}\differential[d]{x}\;,$
(26)

where $L$ is a $(d,0)$-form and $\tilde{L}(\varphi,x)$ is a $(0,0)$-form
(namely the functional of fields that multiplies the volume form). If we vary
the Lagrangian with respect to the fields, it is always possible to rewrite
such a variation as a term linear in the variation and a total derivative,
namely

$\delta L=(\text{e.o.m.})\delta\varphi+\text{d}\theta\ \hat{=}\
\text{d}\theta\;,$ (27)

where e.o.m. stands for _equations of motion_ , $\ \hat{=}\ $ denotes an
equality that is valid on shell of the equations of motion and $\theta$ is a
$(d-1,1)$-form called _pre-symplectic_ potential. If $M$ has a boundary
$\partial M$, this means that

$\delta S=\int_{M}\Big{[}(\text{e.o.m.})\delta\varphi+\text{d}\theta\Big{]}\
\hat{=}\ \int_{\partial M}\theta\;.$ (28)

From the pre-symplectic potential we can define the pre-symplectic local
density

$\omega\vcentcolon=\delta\theta\;,$ (29)

which is a $(d-1,2)$-form, and from this, by integrating over a Cauchy slice
$\Sigma$ (a codimension-1 submanifold on $M$), we define

$\Omega\vcentcolon=\int_{\Sigma}\omega=\int_{\Sigma}\delta\theta\;,$ (30)

which is a $(0,2)$-form called _pre-symplectic $2$-form_. This is a key
ingredient of this formalism since, as we will see, it encodes the Poisson
brackets.

It is of course possible to define isometries on field space similarly to what
has been done in spacetime. An isometry of $\omega$, for instance, is a vector
field $\hat{V}$ such that

$\mathfrak{L}_{\hat{V}}\omega=0\;.$ (31)

Since $\omega=\delta\theta$ and $\delta^{2}=0$, assuming a trivial cohomology
on the space of $1$-forms over $\Gamma$ we have that

$\mathfrak{L}_{\hat{V}}\omega=0\Rightarrow\delta
I_{\hat{V}}\omega=0\Rightarrow I_{\hat{V}}\omega\vcentcolon=-\delta
J_{\hat{V}}\;,$ (32)

where $J_{\hat{V}}$ is a $(d-1,0)$-form. Analogously, an isometry of $\Omega$
is defined as

$\mathfrak{L}_{\hat{V}}\Omega=0\;.$ (33)

A vector field satisfying (33) is called _symplectomorphism_. If we do not
consider embedding fields in our theory, we can move the exterior derivative
$\delta$ inside the integral, therefore
$\delta\Omega=\delta\int_{\Sigma}\omega=\int_{\Sigma}\delta\omega=\int_{\Sigma}\delta^{2}\theta=0$.
Thus, given a symplectomorphism, we have

$0=\mathfrak{L}_{\hat{V}}\Omega=\delta I_{\hat{V}}\Omega\;,$ (34)

Since we have assumed trivial cohomology on the space of $1$-forms over
$\Gamma$, the above equation yields

$I_{\hat{V}}\Omega\vcentcolon=-\delta
H_{\hat{V}}\;\;,\;\;H_{\hat{V}}=\int_{\Sigma}J_{\hat{V}}\;,$ (35)

where $H_{\hat{V}}$ are called canonical charges and they are defined as the
contraction of the pre-symplectic $2$-form. A vector field $\hat{V}\in
T\Gamma$ satisfying (35) is called _Hamiltonian vector field_. For trivial
cohomology, Hamiltonian vector fields are symplectomorphisms and vice versa.

We are now ready to derive the Noether’s theorems in this formalism. Indeed,
in Noetherian jargon, $J_{\hat{V}}$ are called _Noether currents_ while
$H_{\hat{V}}$ are the _Noether charges_ and are the physical charges of the
theory.

###  1.2 Noether’s theorems and charge algebra

We have already introduced in the previous subsection the local Noether
current $J_{\hat{V}}$. Its expression is determined by the nature of the
symmetry. In the case of an internal symmetry, we have that

$\delta J_{\hat{V}}=-I_{\hat{V}}\omega\ ,$ (36)

and by taking into account that $\omega=\delta\theta$,

$\delta J_{\hat{V}}=-(\mathfrak{L}_{\hat{V}}\theta-\delta I_{\hat{V}}\theta)\
.$ (37)

Assuming that $\mathfrak{L}_{\hat{V}}\theta=0$, we just obtain

$J_{\hat{V}}=I_{\hat{V}}\theta\ ,$ (38)

modulo $\delta$-exact terms. This calculation is completely general for
internal symmetries, where the equations of motion have not been imposed at
all. Let us analyze the case of a spacetime symmetry. In this case, we will
consider that, associated to a vector field $\xi$ in spacetime, we have a
vector field in field space denoted by $\hat{\xi}$. Then,

$\delta J_{\hat{\xi}}=-(\mathfrak{L}_{\hat{\xi}}\theta-\delta
I_{\hat{\xi}}\theta)\ .$ (39)

As we are considering now spacetime transformations,
$\mathfrak{L}_{\hat{\xi}}\theta=\mathcal{L}_{\xi}\theta$,222This is background
independence, which is just $\delta_{\hat{\xi}}=\mathcal{L}_{\xi}$ in the
standard formalism. we get

$\delta
J_{\hat{\xi}}=-(\iota_{\xi}\mathrm{d}\theta+\mathrm{d}\iota_{\xi}\theta-\delta
I_{\hat{\xi}}\theta)\ .$ (40)

If we assume that $\mathrm{d}\iota_{\xi}\theta$ vanishes at the boundary under
consideration,333This assumption has to be reconsidered when there are fluxes,
and led to different appreciations of the concept of Noether current in the
literature. and we go on-shell, we obtain

$J_{\hat{\xi}}\mathrel{\hat{=}}I_{\hat{\xi}}\theta-\iota_{\xi}L\ ,$ (41)

modulo $\delta$-exact terms. Unlike the case of internal symmetries, the
equations of motion have been imposed. This current here derived is called the
weakly-vanishing Noether current, since it turns out to identically vanish on-
shell, modulo $\mathrm{d}$-exact terms.

####  1.2.1 Noether’s first theorem

This theorem concerns global symmetries. It states that there is an associated
codimension-1 conserved quantity for each global symmetry of the theory. This
can be shown straightforwardly by evaluating the field space Lie derivative of
the action, i.e

$\begin{split}\mathfrak{L}_{\hat{V}}S&=\int_{M}(I_{\hat{V}}\delta+\delta
I_{\hat{V}})L\\\ &=\int_{M}I_{\hat{V}}\delta L\\\
&=\int_{M}\left[I_{\hat{V}}(\mathrm{e.o.m})\delta\varphi+\mathrm{d}I_{\hat{V}}\theta\right]\
.\end{split}$ (42)

If $\hat{V}$ defines a global symmetry of the theory, then
$\mathfrak{L}_{\hat{V}}S=0$, and we further assume here
$\mathfrak{L}_{\hat{V}}L=0$. If we recall that
$J_{\hat{V}}=I_{\hat{V}}\theta$, we finally get

$\mathrm{d}J_{\hat{V}}=-I_{\hat{V}}(\mathrm{e.o.m})\delta\varphi\ ,$ (43)

hence

$\mathrm{d}J_{\hat{V}}\mathrel{\hat{=}}0\ .$ (44)

The result is that the local Noether current is conserved on-shell. As a
consequence, the global functional defined as

$H_{\hat{V}}\mathrel{\mathop{\ordinarycolon}}=\int_{\Sigma}J_{\hat{V}}$ (45)

is the integral of a codimension-1 conserved global charge on-shell. This is
the associated Noether charge.

####  1.2.2 Noether’s second theorem

This theorem concerns gauge symmetries. It states that there is an associated
codimension-2 conserved quantity for each gauge symmetry of the theory.
Explicitly, the local Noether current is such that

$J_{\hat{V}}\mathrel{\hat{=}}\mathrm{d}Q_{\hat{V}}\ ,$ (46)

so it is a $\mathrm{d}$-exact term on-shell.444We are here considering
$\hat{V}$ as a generic field space vector field. However, if $\hat{V}$ is
associated to a spacetime diffeomorphism $\xi$, then the notation $\hat{\xi}$
is used in these notes. Its conservation is directly inferred from
$\mathrm{d}^{2}=0$. This can also be shown by evaluating the field space Lie
derivative of the action, i.e

$\begin{split}\mathfrak{L}_{\hat{V}}S&=\int_{M}(I_{\hat{V}}\delta+\delta
I_{\hat{V}})L\\\
&=\int_{M}\left[I_{\hat{V}}(\mathrm{e.o.m})\delta\varphi+\mathrm{d}I_{\hat{V}}\theta\right]\\\
&=\int_{M}I_{\hat{V}}(\mathrm{e.o.m})\delta\varphi+\int_{\partial
M}I_{\hat{V}}\theta\ .\end{split}$ (47)

As before, $\mathfrak{L}_{\hat{V}}S=0$ if $\hat{V}$ defines a gauge symmetry
of the theory. Additionally, if $\hat{V}=\hat{\xi}$ defines a spacetime
symmetry, then $\mathfrak{L}_{\hat{\xi}}\theta=\mathcal{L}_{\xi}\theta$. Note
that

$\begin{split}\mathcal{L}_{\xi}S&=\int_{M}\mathcal{L}_{\xi}L\\\
&=\int_{M}\mathrm{d}\iota_{\xi}L\\\ &=\int_{\partial M}\iota_{\xi}L\
.\end{split}$ (48)

Finally,

$\int_{\partial M}I_{\hat{\xi}}\theta\mathrel{\hat{=}}\int_{\partial
M}\iota_{\xi}L\ ,$ (49)

and we obtain

$J_{\hat{\xi}}\mathrel{\hat{=}}\mathrm{d}Q_{\hat{\xi}}\ .$ (50)

Analogously, the global functional defined as

$H_{\hat{\xi}}\mathrel{\mathop{\ordinarycolon}}=\int_{\Sigma}J_{\hat{\xi}}$
(51)

is a conserved charge on-shell. This is the associated Noether charge when
considering gauge symmetries. Note that it is now a codimension-2 quantity, if
the hypersurface $\Sigma$ is such that its boundary $\partial\Sigma=S$ is a
codimension-2 surface. If so,

$H_{\hat{\xi}}\mathrel{\hat{=}}\int_{S}Q_{\hat{\xi}}\ .$ (52)

This is the reason why in the literature the name surface charge is typically
used for the Noether charge for gauge symmetries. This charge is also called
corner charge since the surface $S$ is generically called corner.

####  1.2.3 Charge algebra

Let us explain now why the adjective “pre”-symplectic is used for the 2-form
$\Omega$. Recall that

$\Omega=\int_{\Sigma}\omega=\int_{\Sigma}\delta\theta\ ,$ (53)

where $\omega$ and $\theta$ are the pre-symplectic local density and potential
respectively. A 2-form $\Omega^{\prime}$ is called symplectic if it is
characterized by the following properties:

  * •

$\delta\Omega^{\prime}=0$

  * •

$I_{\hat{V}}\Omega^{\prime}=0\Longleftrightarrow\hat{V}=0$

That is, the 2-form $\Omega^{\prime}$ is closed in field space and non-
degenerate. In this notes, the 2-form $\Omega$ is called pre-symplectic
because the non-degeneracy property does not hold in general. This means that
there are non-vanishing field space vector fields $\hat{V}$ (non-vanishing
symmetries) such that $I_{\hat{V}}\Omega=0$. If we quotient these symmetries
out the pre-symplectic 2-form $\Omega$ becomes symplectic and can define a
Poisson bracket of the charge algebra. This Poisson bracket is defined as

$\\{H_{\hat{V}},H_{\hat{W}}\\}=\mathfrak{L}_{\hat{W}}H_{\hat{V}}\ ,$ (54)

where $\hat{V},\hat{W}\in T\Gamma$ are symplectomorphisms. Indeed, one is able
to show that the Poisson bracket is skew-symmetric by using the properties of
symplectomorphisms. Explicitly,

$\begin{split}\\{H_{\hat{V}},H_{\hat{W}}\\}&=I_{\hat{W}}\delta H_{\hat{V}}\\\
&=-I_{\hat{W}}I_{\hat{V}}\Omega\\\ &=I_{\hat{V}}I_{\hat{W}}\Omega\\\
&=-I_{\hat{V}}\delta H_{\hat{W}}\\\ &=-\\{H_{\hat{W}},H_{\hat{V}}\\}\
.\end{split}$ (55)

Apart from proving its skew-symmetry, we also showed that the Poisson bracket
is determined by the symplectic 2-form $\Omega$, i.e

$\\{H_{\hat{V}},H_{\hat{W}}\\}=I_{\hat{V}}I_{\hat{W}}\Omega\ .$ (56)

At this point, we consider spacetime vector fields $\xi,\zeta\in TM$
associated with spacetime symmetries. The corresponding Poisson bracket reads

$\\{H_{\hat{\xi}},H_{\hat{\zeta}}\\}=\mathfrak{L}_{\hat{\zeta}}H_{\hat{\xi}}\
.$ (57)

In this case, we also have the Lie bracket vector fields in spacetime, which
reads

$[\xi,\zeta]=\mathcal{L}_{\xi}\zeta\ .$ (58)

One may now wonder how this Lie bracket of (spacetime) vector fields defining
the symmetry algebra is related to the Poisson bracket of the charges
associated to the corresponding (field space) vector fields defining the
charge algebra. This is a fundamental question that we will answer next. Let
us start by writing the identity

$I_{[\hat{\xi},\hat{\zeta}]}=[\mathfrak{L}_{\hat{\xi}},I_{\hat{\zeta}}]\ .$
(59)

If we apply it to the symplectic 2-form $\Omega$, we get

$I_{[\hat{\xi},\hat{\zeta}]}\Omega=\mathfrak{L}_{\hat{\xi}}I_{\hat{\zeta}}\Omega-
I_{\hat{\zeta}}\mathfrak{L}_{\hat{\xi}}\Omega\ .$ (60)

Since $\hat{\xi}$ is a symplectomorphism, $\mathfrak{L}_{\hat{\xi}}\Omega=0$,
and we have

$I_{[\hat{\xi},\hat{\zeta}]}\Omega=\delta
I_{\hat{\xi}}I_{\hat{\zeta}}\Omega+I_{\hat{\xi}}\delta I_{\hat{\zeta}}\Omega\
.$ (61)

If we recall that $\delta\Omega=0$, the second term of the above equation
gives $I_{\hat{\xi}}\delta
I_{\hat{\zeta}}\Omega=I_{\hat{\xi}}\mathfrak{L}_{\hat{\zeta}}\Omega=0$, since
also $\hat{\zeta}$ is a symplectomorphism. Therefore,

$I_{[\hat{\xi},\hat{\zeta}]}\Omega=\delta I_{\hat{\xi}}I_{\hat{\zeta}}\Omega\
,$ (62)

and thus, introducing the Poisson bracket,

$I_{[\hat{\xi},\hat{\zeta}]}\Omega=\delta\\{H_{\hat{\xi}},H_{\hat{\zeta}}\\}\
.$ (63)

Although we will not demonstrate it explicitly in these notes, one can see
using similar arguments that the Lie brackets Jacobi identity induces the
Poisson brackets Jacobi identity.

Finally, we can derive the correspondence between the charge algebra given by
the Poisson brackets and the symmetry algebra given by the Lie brackets.
First, note that the identity

$I_{\hat{\xi}}\Omega=-\delta H_{\hat{\xi}}$ (64)

relates the Noether charge $H_{\hat{\xi}}$ and the field space vector field
$\hat{\xi}$ associated to the spacetime symmetry $\xi$ via the symplectic
2-form $\Omega$. It we consider two charges, for instance $H_{\hat{\xi}}$ and
$H_{\hat{\xi}}+\kappa$, both charges correspond to the same vector field
$\hat{\xi}$ as long as $\delta\kappa=0$. This correspondence is said to be
cohomological. We can similarly conclude from Eq. (63) that both
$\\{H_{\hat{\xi}},H_{\hat{\zeta}}\\}$ and
$\\{H_{\hat{\xi}},H_{\hat{\zeta}}\\}+\kappa_{\hat{\xi},\hat{\zeta}}$
correspond to the same vector field $[\hat{\xi},\hat{\zeta}]$ as long as
$\delta\kappa_{\hat{\xi},\hat{\zeta}}=0$. Finally, we can write

$I_{[\hat{\xi},\hat{\zeta}]}\Omega=-\delta H_{[\hat{\xi},\hat{\zeta}]}\ ,$
(65)

to deduce from (63) and the above cohomological argument that

$\\{H_{\hat{\xi}},H_{\hat{\zeta}}\\}=-H_{[\hat{\xi},\hat{\zeta}]}+\kappa_{\hat{\xi},\hat{\zeta}}\
,$ (66)

with $\delta\kappa_{\hat{\xi},\hat{\zeta}}=0$. We can read from this result
that the Poisson bracket of charges represents the Lie bracket of symmetries
projectively. This is essentially the same as stating that the charge algebra
represents the symmetry algebra modulo central extensions. A central extension
is just a constant in field space appearing in the bracket structure of a
theory. It can be considered as a new algebra generator, with the word
“central" characterizing its commutativity with the rest of the algebra
generators. This is a general and highly important result, in particular for
the theory of asymptotic symmetries that we will introduce. We will do it in
section 1.4, but in section 1.3 we will first apply the formalism presented in
previous subsections to the computation of conserved charges in gauge
theories.

###  1.3 Application to gauge theories

Let us consider a classical gauge theory on a 4-dimensional flat spacetime. We
are generally interested in non-abelian gauge theories, with compact simple
gauge Lie group $G$ and corresponding Lie algebra $\mathfrak{g}$. The Lie
algebra generators $T_{a}$ satisfy

$\left[T_{a},T_{b}\right]=f_{ab}^{c}T_{c}\ ,$ (67)

where $f_{ab}^{c}$ are the structure constants.555Note that the structure
constants are antisymmetric in the indices $a$, $b$. It is possible to show
that, under certain conditions, the following orthogonality condition
holds:666The structure constants obtained as $f_{abc}=f_{ab}^{d}\delta_{dc}$
are completely antisymmetric in the indices $a$, $b$, $c$.

$\mathrm{Tr}(T_{a}T_{b})=\frac{\delta_{ab}}{2}\ .$ (68)

The gauge connection/field $A$ is a Lie algebra-valued 1-form
$A=A_{\mu}^{a}\,\mathrm{d}x^{\mu}\otimes T_{a}$, and the field strength $F$ is
a Lie algebra-valued 2-form
$F=\frac{1}{2}F_{\mu\nu}^{a}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\otimes
T_{a}$ given by

$F=\mathrm{d}A+A\wedge A\ .$ (69)

Recall that
$F_{\mu\nu}^{a}=\delta_{\mu\nu}^{\rho\sigma}\partial_{\rho}A_{\sigma}^{a}+f_{bc}^{a}A_{\mu}^{b}A_{\nu}^{c}$,
with $\delta$ the generalized Kronecker tensor. We also introduce the gauge-
covariant exterior derivative,

$\mathrm{D}\equiv\mathrm{d}+A\ ,$ (70)

as the generalization of the exterior derivative $\mathrm{d}$ with respect to
the gauge connection $A$.

The Lagrangian reads in this setting

$L=\frac{1}{2}\mathrm{Tr}\left(F^{2}\right)\,\mathrm{Vol}_{M}\ ,$ (71)

where $\mathrm{Vol}_{M}$ is the volume form of the manifold $M$. This volume
form $\mathrm{Vol}_{M}$ is, by definition, a top-form, then
$\mathrm{Vol}_{M}=\star 1$, with $\star$ denoting the Hodge duality operator.
If we consider its definition, we have

$\mathrm{Vol}_{M}=\frac{1}{4!}\epsilon_{\alpha\beta\gamma\delta}\,\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{\beta}\wedge\mathrm{d}x^{\gamma}\wedge\mathrm{d}x^{\delta}\
,$ (72)

where $\epsilon_{\alpha\beta\gamma\delta}$ is the covariant Levi-Civita tensor
in flat spacetime (which is equal to the Levi-Civita symbol). Its
contravariant version $\epsilon^{\alpha\beta\gamma\delta}$ is defined by means
of the metric as usual, so we can now use the identity

$\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{\beta}\wedge\mathrm{d}x^{\gamma}\wedge\mathrm{d}x^{\delta}=-\epsilon^{\alpha\beta\gamma\delta}\,\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}$
(73)

to write

$\mathrm{Vol}_{M}=\frac{1}{4!}\delta_{\alpha\beta\gamma\delta}^{\alpha\beta\gamma\delta}\,\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\
,$ (74)

but $\delta_{\alpha\beta\gamma\delta}^{\alpha\beta\gamma\delta}=4!$, so

$\mathrm{Vol}_{M}=\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\
.$ (75)

The above relation and the skew-symmetry of the strength tensor allow us to
write the Lagrangian as

$\begin{split}L&=\frac{1}{2}\mathrm{Tr}\left(F^{2}\right)\,\mathrm{Vol}_{M}\\\
&=-\frac{1}{4}F_{\mu\nu}^{a}F_{a}^{\mu\nu}\,\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\\\
&=-\frac{1}{8}F_{\mu\nu}^{a}\delta_{\rho\sigma}^{\mu\nu}F_{a}^{\rho\sigma}\,\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\\\
&=\frac{1}{16}F_{\mu\nu}^{a}\epsilon_{\alpha\beta\rho\sigma}\epsilon^{\alpha\beta\mu\nu}F_{a}^{\rho\sigma}\,\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\\\
&=-\frac{1}{16}F_{\mu\nu}^{a}\epsilon_{\alpha\beta\rho\sigma}F_{a}^{\rho\sigma}\,\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{\beta}\wedge\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\\\
&=-\frac{1}{4}\left(\frac{1}{2}F_{\mu\nu}^{a}\mathrm{d}x^{\mu}\,\wedge\mathrm{d}x^{\nu}\right)\wedge\left(\frac{1}{2}\epsilon_{\alpha\beta\rho\sigma}F_{a}^{\rho\sigma}\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{\beta}\right)\\\
&=-\frac{1}{2}\mathrm{Tr}(F\wedge\star F)\ .\end{split}$ (76)

The gauge connection is the dynamical field when considering gauge theories in
flat spacetime. The field space $\Gamma$ is then formed by $A$. Its equations
of motion and the pre-symplectic potential follow from taking the
corresponding field variation of the action,

$\delta S=\int_{M}\delta L\ ,$ (77)

giving

$\begin{split}\delta S&=-\frac{1}{2}\int_{M}\mathrm{Tr}\,\delta(F\wedge\star
F)\\\ &=-\int_{M}\mathrm{Tr}(\delta F\wedge\star F)\\\
&=-\int_{M}\mathrm{Tr}\left((\mathrm{d}\delta A+A\wedge\delta A+\delta A\wedge
A)\wedge\star F\right)\\\ &=\int_{M}\mathrm{Tr}\left((\mathrm{d}\star
F+[A,\star F])\wedge\delta A-\mathrm{d}(\delta A\wedge\star F)\right)\\\
&=\int_{M}\mathrm{Tr}\left(\mathrm{D}\star F\wedge\delta A-\mathrm{d}(\delta
A\wedge\star F)\right)\ .\end{split}$ (78)

where we have used $\delta\mathrm{d}=\mathrm{d}\delta$. We can now read off
the field equations for $A$,

$\mathrm{D}\star F=0\ ,$ (79)

which are the Yang-Mills equations (together with the Bianchi identity
$\mathrm{D}F=0$) as expected, as well as the pre-symplectic potential,

$\theta=-\,\mathrm{Tr}(\delta A\wedge\star F)\ .$ (80)

The corresponding gauge symmetry of the theory is encoded in the invariance
under the transformation
$\delta_{\lambda}A\mathrel{\mathop{\ordinarycolon}}=I_{\hat{\lambda}}\delta
A=\mathrm{D}\lambda=\differential\lambda+[A,\lambda]$, with
$[A,\lambda]=f_{ab}^{c}A^{a}\lambda^{b}T_{c}$ the Lie algebra commutator. Note
that

$\begin{split}\mathfrak{L}_{\hat{\lambda}}F\mathrel{\mathop{\ordinarycolon}}&=I_{\hat{\lambda}}\delta
F\\\ &=I_{\hat{\lambda}}(\mathrm{d}\delta A+A\wedge\delta A+\delta A\wedge
A)\\\ &=I_{\hat{\lambda}}\mathrm{D}\delta A\\\ &=\mathrm{D}^{2}\lambda\\\
&=(\mathrm{d}+A)(\mathrm{d}\lambda+[A,\lambda])\\\
&=\mathrm{d}[A,\lambda]+[A,\mathrm{d}\lambda]+[A,[A,\lambda]]\\\
&=[\mathrm{d}A+A\wedge A,\lambda]\\\ &=[F,\lambda]\ ,\end{split}$ (81)

hence

$\begin{split}\mathfrak{L}_{\hat{\lambda}}S=-\int_{M}\mathrm{Tr}([F,\lambda]\wedge\star
F)=0\;,\end{split}$ (82)

because, in components,

$\begin{split}\mathrm{Tr}([F,\lambda]\wedge\star
F)\;\propto\;\delta_{ab}f_{cd}^{a}F_{\mu\nu}^{c}\lambda^{d}F^{\mu\nu b}=0\
,\end{split}$ (83)

since $\delta_{ab}f_{cd}^{a}=f_{bcd}$ is antisymmetric in the indices $b,c$,
while $F_{\mu\nu}^{c}F^{\mu\nu b}$ is symmetric.

The associated Noether current follows from evaluating

$I_{\hat{\lambda}}\Omega=\int_{\Sigma}I_{\hat{\lambda}}\omega=\int_{\Sigma}I_{\hat{\lambda}}\delta\theta\
,$ (84)

giving

$\begin{split}I_{\hat{\lambda}}\Omega&=-\int_{\Sigma}\mathrm{Tr}[I_{\hat{\lambda}}(\delta
A\wedge\star\delta F)]\\\
&=-\int_{\Sigma}\mathrm{Tr}(\mathrm{D}\lambda\wedge\star\delta F-\delta
A\wedge[\star F,\lambda])\\\
&=-\int_{\Sigma}\mathrm{Tr}(\mathrm{d}\lambda\wedge\star\delta
F+[A,\lambda]\wedge\star\delta F-\delta A\wedge[\star F,\lambda])\\\
&=-\int_{\Sigma}\mathrm{Tr}\,\delta(\mathrm{d}\lambda\wedge\star
F+[A,\lambda]\wedge\star F)\;,\end{split}$ (85)

where we used $\delta\lambda=0$. The Noether current is then,

$J_{\hat{\lambda}}=-\,\mathrm{Tr}(\mathrm{d}\lambda\wedge\star
F+[A,\lambda]\wedge\star F)\ .$ (86)

It is possible to rewrite this Noether current as an on-shell total
derivative, due to Noether’s second theorem. Explicitly,

$\begin{split}J_{\hat{\lambda}}&=-\,\mathrm{Tr}(\mathrm{d}\lambda\wedge\star
F+[A,\lambda]\wedge\star F)\\\ &=-\,\mathrm{Tr}\left(\mathrm{d}(\lambda\star
F)-\lambda(\mathrm{d}\star F+[A,\star F])\right)\\\
&=-\,\mathrm{Tr}\left(\mathrm{d}(\lambda\star F)-\lambda\mathrm{D}\star
F\right)\\\ &\mathrel{\hat{=}}-\,\mathrm{Tr}\left(\mathrm{d}(\lambda\star
F)\right)\ .\end{split}$ (87)

Finally, the Noether charge is given by

$H_{\hat{\lambda}}=\int_{\Sigma}J_{\hat{\lambda}}\ ,$ (88)

hence

$\begin{split}H_{\hat{\lambda}}&\mathrel{\hat{=}}-\int_{\Sigma}\mathrm{Tr}\left(\mathrm{d}(\lambda\star
F)\right)\\\ &=-\int_{S}\mathrm{Tr}(\lambda\star F)\ ,\end{split}$ (89)

where $\Sigma$ is an arbitrary Cauchy surface with boundary $S$.

This is our final result, showing how Noether’s second theorem applies to
gauge theories and how to explicitly compute the conserved charge associated
to the local symmetry of the theory. The charge has support on a codimension-2
surface $S$, called the corner. We now turn our attention to the
aforementioned theory of asymptotic symmetries.

###  1.4 Asymptotic symmetries

What we have learned so far is that the basic ingredients needed to define a
classical physical theory are the spacetime manifold $M$ with boundary $B$ and
a field space $\Gamma$. These are the basic ingredients of the theory of
asymptotic symmetries.

A classical dynamical theory is specified by

  * (i)

The dynamics on $M$, i.e. the Lagrangian describing the system.

  * (ii)

Boundary conditions $\Gamma\rvert_{B}$ defining the asymptotic behaviour of
the bulk fields near the boundary, namely defining the so-called _falloffs_ of
the bulk fields.

  * (iii)

Gauge fixing conditions if the theory is a gauge theory.

At this point, in order to compute the Noether charges we need to identify the
real physical symmetries of the system. To this aim, let us introduce the
definitions of residual symmetries and trivial symmetries.

  * •

Residual symmetries: they are defined as symplectomorphisms preserving the
dynamics in the bulk and the boundary conditions. We stress that they are
usually known as _gauge transformations_.

  * •

Trivial symmetries: they are residual symmetries with vanishing Noether
charges and therefore they are true redundancies of the system. Strictly
speaking, these are the quantities that should be called gauge symmetries.

  * •

Asymptotic symmetries: they are residual symmetries with non-vanishing Noether
charges. They are, therefore, physical transformations acting non-trivially on
the field space, mapping the system into an inequivalent configuration.

Due to their redundancy nature, the trivial transformations represent the zero
modes of the pre-symplectic 2-form. Additionally, the group they form is an
ideal inside the residual symmetry group. Finally, the only symmetries to
focus on are the non-trivial residual symmetries that we have defined as
asymptotic symmetries. Precisely, the asymptotic symmetry group is defined as
the quotient

$\text{Asymptotic symmetries}=\faktor{\text{Residual
symmetries}}{\text{Trivial symmetries}}\;.$ (90)

Having restricted to the asymptotic symmetries, the pre-symplectic 2-form is
now invertible and this allows us to deal with a well-defined Poisson bracket
of physical charges. This defines the charge algebra of the theory and the
physical observables consequently.

Note that to be as general as possible, in presence of a gauge theory the
Noether charges _should be_ computed without any a priori restrictions given
by gauge fixing. In fact, only after the charges’ computation, the
transformation needed to perform the gauge fixing _should be_ shown to be a
trivial symmetry (leading to a vanishing charge). Typically, however, it is
not possible to carry on this procedure due to technical reasons.
Nevertheless, on the contrary, the gauge fixing procedure, as well as the
enforcing of the boundary conditions, may lead to pathologies for the well-
definiteness of physical charges. For this reason, the Noether charges
associated to asymptotic symmetries are canonical if they satisfy these three
conditions:

  * •

Integrable: $I_{\hat{V}}\Omega=-\delta\int_{S}Q_{\hat{V}}$.

  * •

Conserved:
$H_{\hat{V}}\rvert_{S_{2}}-H_{\hat{V}}\rvert_{S_{1}}=\int^{S_{2}}_{S_{1}}\differential
Q_{\hat{V}}\ \hat{=}\ 0$.

  * •

Finite: charges should not diverge approaching the codimension-2 surface $S$.

If these conditions are satisfied, points (ii) and (iii) are pathologies-free
and the theory is well defined.

The last thing to mention before going on is that the asymptotic symmetry
group can be (and typically is) larger than the bulk symmetry group. A
remarkable example is given by Minkowski in $4$ dimensions, whose bulk
symmetry group is given by the Poincaré group, while the asymptotic symmetry
group at null infinity is given by the infinite dimansional Bondi-Van der
Burg–Metzner–Sachs ($\mathrm{BMS_{4}}$) group that we will discuss in the next
section.

##  2 Gravity

###  2.1 What is wrong?

For the sake of simplicity, the previous discussion was done assuming a closed
system. Then, as expected, we obtained conserved charges. We elaborate here on
the problems one encounters when dealing with gravity. In this case, every one
of the three previously-mentioned properties (integrability, conservation and
finiteness) fails to be a priori true.

##### Integrability

It is not always guaranteed that $I_{\hat{\xi}}\Omega=-\delta H_{\hat{\xi}}$.
If this does not hold, then the charge algebra does not close and thus one
cannot derive the Poisson bracket of the theory. Generally, we have

$\displaystyle
I_{\hat{\xi}}\Omega=\int_{\Sigma}I_{\hat{\xi}}\delta\theta=\int_{\Sigma}\mathrm{d}(\iota_{\xi}\theta+Q_{\delta\hat{\xi}})-\delta\mathrm{d}Q_{\hat{\xi}}\;,$
(91)

where we added the term $Q_{\delta\hat{\xi}}$ in case $\xi$ is field
dependant, but we will assume it is zero from now on. The term
$\mathrm{d}(\iota_{\hat{\xi}}\theta)$ is not generically of the form
$\delta(...)$, and therefore the total expression cannot be equal to $-\delta
H_{\hat{\xi}}$. We can then define the symplectic flux as

$F_{\hat{\xi}}=\int_{S}\iota_{\xi}\theta,$ (92)

where $S=\partial\Sigma$. This term is responsible for dissipation in the
system.

There are various proposals to deal with non-integrability, and we will
mention two of them:

  1. 1.

Following [7] (see also [8]) we can split between integrable and non-
integrable parts introducing a modified Poisson bracket. There is no canonical
way to perform such splitting, and here we will use the Noetherian split
proposed in [8]. This means that we split

$I_{\hat{\xi}}\Omega=-\delta H_{\hat{\xi}}+F_{\hat{\xi}},$ (93)

such that $H_{\hat{\xi}}=\int_{S}Q_{\hat{\xi}}$, with
$\mathrm{d}Q_{\hat{\xi}}=I_{\hat{\xi}}\theta-\iota_{\xi}L$ being the
unmodified Noether charge, and $F_{\hat{\xi}}=\int_{S}\iota_{\xi}\theta$ the
Noetherian flux. It can then be proved that we have the modified bracket

$\poissonbracket{H_{\hat{\xi}}}{H_{\hat{\zeta}}}=\mathfrak{L}_{\hat{\zeta}}H_{\hat{\xi}}-I_{\hat{\zeta}}F_{\hat{\xi}}+\int_{S}i_{\xi}i_{\zeta}L\;.$
(94)

  2. 2.

We can consider an enlarged field space, and define another symplectic form
$\Omega^{\mathrm{ext}}$ such that

$I_{\hat{\xi}}\Omega^{\mathrm{ext}}=-\delta H_{\hat{\xi}},$ (95)

for every diffeomorphism $\xi$. This procedure goes under the name of
_extended phase space_ [9, 10, 11, 12], and we will have to say more about it
later. The system is still dissipative but is now integrable, thanks to the
introduction of edge modes.

##### Conservation

Charges are not always conserved. In general

$H_{\hat{\xi}}|_{S_{2}}-H_{\hat{\xi}}|_{S_{1}}=\int_{S_{1}}^{S_{2}}\mathrm{d}Q_{\hat{\xi}}=\int_{S_{1}}^{S_{2}}I_{\hat{\xi}}\theta-\iota_{\xi}L$
(96)

is not equal to zero. There are two main reasons why this might happen. One is
that there might be gravitational fluxes through the boundary under
consideration. In other words, the subregion we consider is not isolated from
its complement (there is a "leakage"). Another reason is that, for a
gravitational theory defined on an odd-dimensional bulk with cosmological
constant, a conformal anomaly exists, and this anomaly contributes to the non-
conservation of charges (see [13]). This breaking of conformal symmetry is
holographically understood as an anomalous Ward-Takahashy identity, as
discussed below.

##### Finiteness

The charges are guaranteed to be finite only if the boundary is at finite
distance in the bulk. In gravity, however, we often deal with asymptotic
boundaries. Consider the case of gravity with a negative cosmological constant
in three dimensions, with action (in suitable units) given by

$S=\frac{1}{16\pi G}\int\mathrm{d}^{3}x\;\sqrt{|\det{g}|}(R-2\Lambda)\;,$ (97)

where $\Lambda$ is a cosmological constant. Einstein equations imply
$R=6\Lambda$, and the on-shell action is thus proportional to the spacetime
volume, and therefore suffers from divergences.777We do not consider the
Gibbons-Hawing-York term in this lecture notes, but it also plays an important
role in the value of the on-shell action. It is then clear that the surface
charges, derived from this action, may also be infinite. We therefore have to
perform a suitable renormalization, called phase space renormalization. We add
a boundary term to the Lagrangian

$L_{R}=L+\mathrm{d}\ell_{\mathrm{div}}$ (98)

and tune it so to cure the divergences. Also, we add corner symplectic
potentials $\nu_{\mathrm{div}}$, such that

$\theta_{R}=\theta-\mathrm{d}\nu_{\mathrm{div}}-\delta\ell_{\mathrm{div}}.$
(99)

This implies that we have

$H_{\hat{\xi}}^{R}=H_{\hat{\xi}}-\int_{S}J_{\hat{\xi}}^{\mathrm{div}},$ (100)

where
$J_{\xi}^{\mathrm{div}}=I_{\xi}\nu_{\mathrm{div}}-\iota_{\xi}\ell_{\mathrm{div}}$.
It is not a priori obvious that $\ell_{\mathrm{div}}$ and $\nu_{\mathrm{div}}$
can always be selected to cure divergences, but this turns out to be true on a
case by case basis. The symplectic flux also gets renormalised, and the final
result is

$F_{\hat{\xi}}^{R}=F_{\hat{\xi}}+\int_{S}(\delta\iota_{\xi}\ell_{\mathrm{div}}-\mathfrak{L}_{\hat{\xi}}\;\nu_{\mathrm{div}}).$
(101)

This procedure is the rigorous generalization of the addition of counterterms
in AdS holographic renormalization.

Before providing a detailed gravitational example, we continue discussing
$3$-dimensional gravity further. We saw earlier that Poisson brackets are a
projective representation of Lie brackets. Brown and Henneaux showed in [14]
that the asymptotic symmetry algebra of $AdS_{3}$ is given by two copies of
the Virasoro algebra with equal central charges $c=\frac{3l}{2G}$, where $l$
is the $AdS_{3}$ radius, and $G$ the Newton constant. Nowadays, we understand
this result in the light of holography ($AdS/CFT$ correspondence), [15]. This
correspondence establishes a duality between a gravitational theory defined on
an (asymptotically) $AdS$ spacetime in the bulk and a conformal field theory
(CFT) living on the asymptotic boundary. This duality is best understood in a
regime where the gravity theory is classical and the dual field theory is
strongly coupling. It is a well-known fact that the stress-energy tensor in
CFT is traceless: $\langle T^{\mu}_{\;\;\mu}\rangle=0$. This property holds
both at the classical level and at the quantum level on flat spacetime, but
when the CFT is coupled to a curved background, a Weyl anomaly appears. In the
case of a $2d$ $\mathrm{CFT}$, we have $\langle
T^{\mu}_{\;\;\mu}\rangle=\frac{c}{24\pi}R$, where $R$ is the Ricci scalar, and
$c$ is the central charge. Using the holographic dictionary and holographic
renormalization, one can compute the expectation value of the holographic
stress-energy tensor purely from the classical gravity side and obtain the
result $\langle T^{\mu}_{\;\;\mu}\rangle=\frac{l}{16\pi G}R^{(0)}$, where
$R^{0}$ is Ricci scalar obtained from the boundary metric $g^{0}$, that can be
read from the asymptotic expansion (in Fefferman-Graham gauge) of the bulk
metric [16]. Comparing the last two formulas, we obtain a holographic
prediction: $3d$ gravity is dual to a $2d$ CFT with central charge given by
$c=\frac{3l}{2G}$. We refer the interested reader to [1], where one can find a
complete set of references on the topic and this example is proposed as a
solved exercise.

###  2.2 BMS

Time has come to study a gravitational example in detail. We will consider an
asymptotically flat spacetime in four dimensions and throughout this section
we will work with $8\pi G=1=c$ conventions with metric signature being
$(-,+,+,+)$. Referring to the previous section, we will follow the steps
(i)-(iii) to define our classical theory hence we will specify the dynamics,
the boundary conditions and the gauge fixing. Useful references for this
section are [17, 18, 19, 7, 20, 21, 8].

####  2.2.1 Dynamics

The bulk dynamics of the theory under exam is defined via the following
Einstein-Hilbert action

$S_{EH}=\frac{1}{2}\int R\sqrt{\absolutevalue{\det{g}}}\differential[4]{x}\;,$
(102)

with zero cosmological constant. The goal of this section is to extrapolate
the asymptotic charges at the future null infinity of the spacetime. We will
show that the symmetries in the bulk are those defined by the Poincaré group
which are finite-dimensional, while those defined on the boundary are
infinite-dimensional and form the so-called $\mathrm{BMS}$ group. The set of
coordinates we will use is $(u,r,\sigma^{A})$, where

  * •

$u$ is the null time that follows outgoing null geodesic congruences without
vorticity,

  * •

$r$ is the radial coordinate,

  * •

$\sigma^{A}$ are spatial coordinates on the codimension-2 surface (they can be
thought of as the angular coordinates $(\theta,\phi)$ on the sphere).

In this set of coordinates, the bulk metric is written as

$\differential s^{2}=-2e^{2\beta}\differential u\left(\differential
r+F\differential
u\right)+r^{2}q_{AB}\left(\differential\sigma^{A}-U^{A}\differential
u\right)\left(\differential\sigma^{B}-U^{B}\differential u\right)\;,$ (103)

where $q_{AB}$ is the codimension-2 metric and generally
$\beta,F,U^{A},q_{AB}$ are functions of the full coordinates
$(u,r,\sigma^{A})$ defined in the bulk.

####  2.2.2 Boundary conditions

We assume the following falloffs:

$\displaystyle g_{ur}$ $\displaystyle=-1+\order{\frac{1}{r^{2}}}\;,$ (104)
$\displaystyle g_{uA}$ $\displaystyle=\order{1}\;,$ (105) $\displaystyle
g_{uu}$ $\displaystyle=\order{1}\;,$ (106) $\displaystyle q_{AB}$
$\displaystyle=\order{1}\;.$ (107)

Given these falloffs, the causal structure of the spacetime is reported in the
figure below, fig. 1.

Figure 1: Conformal diagram of the bulk.

Firstly, the two black lines drawn at $45^{\circ}$ represent the future null
infinity and past null infinity, labelled with $\mathfrak{I}^{+}$ and
$\mathfrak{I}^{-}$ respectively. The points $i^{0}$, $i^{-}$ and $i^{+}$ are
spacelike infinity, timelike past infinity and timelike future infinity
respectively. The $u$-direction is parallel to the $\mathfrak{I}^{+}$-line
while the $r$-direction is radial. Each point in this diagram is of
codimension-2 which means that we can think of its structure as that of a
sphere. Therefore, the point following the radial direction can be pictured as
a _sphere_ having increasing radius as the point approaches null infinity.
Since a photon is a particle with zero mass, it moves on null geodesics by
definition, so it moves following straight $45^{\circ}$-lines, represented by
red lines in the figure. In terms of a photon path, $\mathfrak{I}^{-}$
represents the initial surface from which photons propagate into the bulk
while $\mathfrak{I}^{+}$ is the final surface reached asymptotically by
photons.
The bulk dynamics can be, for instance, that of a star forming a black hole.
Referring to fig. 2, at $\mathfrak{i}^{-}$ there is the star, our massive
object, which collapses into a black hole whose horizon is represented by the
blue line. during its collapse, the radiation emitted can be detected in the
vicinity of $\mathfrak{I}^{+}$. A practical example of this dynamics is given
by two infalling spiraling black holes whose radiation emitted in form of
gravitational waves is detected at (near) future null infinity (see fig. 2).

Figure 2: Conformal diagram of a collapsing star.

####  2.2.3 Gauge fixing

We consider the so-called _Bondi gauge_ which consists in enforcing the
following constraints ($\sqrt{q}=\sqrt{\det{q}}$)

$g_{rr}=0\ ,\quad g_{rA}=0\ ,\quad\partial_{r}\sqrt{q}=0\;,$ (108)

along with the following asymptotic expansions of the bulk fields

$\displaystyle F(u,r,\sigma^{A})$
$\displaystyle=\bar{F}(u,\sigma^{A})-\frac{M}{r}+\dots\;,$ (109)
$\displaystyle\beta(u,r,\sigma^{A})$
$\displaystyle=\frac{\bar{\beta}(u,\sigma^{A})}{r^{2}}+\dots\;,$ (110)
$\displaystyle q_{AB}(u,r,\sigma^{A})$
$\displaystyle=\bar{q}_{AB}(u,\sigma^{A})+\frac{c_{AB}}{r}+\dots\;,$ (111)
$\displaystyle U^{A}(u,r,\sigma^{A})$
$\displaystyle=\frac{\bar{U}^{A}(u,\sigma^{A})}{r^{2}}-\frac{2}{3r^{3}}\bar{p}^{A}+\dots\;,$
(112)

where the dots stand for subleading terms. The important thing to note is that
the $r$ dependency on the right hand side (RHS) of the previous expressions is
explicit. This means that the Bondi gauge completely decouples the radial
dependence. Additionally, $M$ will turn out to be (as we will later show) the
mass of the system that one would observe while $\bar{p}$ is its total angular
momentum. Moreover, $c_{AB}$ is called the _shear_ tensor and carries the
gravitational waves information of the system. This object is in fact a
$2\times 2$ traceless, symmetric matrix in terms of which we define the _news
tensor_ $N_{AB}$

$N_{AB}=\partial_{u}c_{AB}\ .$ (113)

Now that we have defined the theory we move on to the analysis of symmetries
which will lead us to the computation of charges.

####  2.2.4 EOM

Let us start by writing down the equations of motion (EOM) obtained by solving
Einstein’s equations order by order as $r$ goes to infinity.

They can be divided into two types of EOMs. Firstly, we have four constraints
which read

$\displaystyle\partial_{u}\bar{q}_{AB}=0\;,$ (114)
$\displaystyle\bar{\beta}+\frac{1}{32}c_{AB}c^{AB}=0\;,$ (115)
$\displaystyle\bar{R}-4\bar{F}=0\;,$ (116)
$\displaystyle\bar{U}^{A}+\frac{1}{2}\bar{D}_{B}c^{AB}=0\;,$ (117)

where $\bar{R}$ is the codimension-2 Ricci scalar of the boundary metric
$\bar{q}_{AB}$ and $\bar{D}$ is the covariant derivative with respect to the
metric $\bar{q}_{AB}$.

The other type are evolution equations since they describe the temporal
evolution of the system. For the sake of simplicity we focus our attention
only on the _Bondi mass-loss formula_

$\partial_{u}M=-\frac{1}{8}N_{AB}N^{AB}+\frac{1}{4}\bar{D}_{A}\bar{D}_{B}N^{AB}\;.$
(118)

This is called “mass-loss formula” since, as we will shortly show, $M$ is
indeed the charge associated to $\partial_{u}$, namely to time translations,
and is therefore the energy of the system. The equation above also helps us in
understanding why $N_{AB}$ is called “news tensor”: it is basically providing
us information about the source by telling us how the latter affects the mass
dynamics.
This EOM can be integrated on a sphere $S$ leading to

$\partial_{u}\int_{S}\sqrt{\bar{q}}M=-\frac{1}{8}\int_{S}\sqrt{\bar{q}}N_{AB}N^{AB}\;,$
(119)

where the covariant derivative term is dropped out since it is a total
derivative contribution on the sphere. Note that if the RHS is null then the
mass of the system is conserved while, on the contrary, its time evolution is
negative. This means that, given a finite mass at initial time, as time
evolves radiation is emitted resulting in a mass-decreasing mechanism. This is
exactly the reason why the Bondi equation is known as the _mass-loss formula_.
These are the EOMs that we will analyze later on. Note that of course there
are subleading EOMs, however we are only interested in the leading asymptotic
ones, since we will exclusively deal with the field space asymptotically to
the boundary.

####  2.2.5 Symmetries

By construction, a symmetry _must_ preserve the chosen gauge. Therefore, we
impose that the Bondi gauge in (108) is unaffected under diffeomorphisms,
namely when acting with the Lie derivative. Hence, we impose the following

$\mathcal{L}_{\xi}g_{rr}=0\ ,\quad\mathcal{L}_{\xi}g_{rA}=0\ ,\mbox{\quad
and\quad}\mathcal{L}_{\xi}\partial_{r}(\sqrt{q})=0\;,$ (120)

which can be solved leading to

$\displaystyle\xi^{u}$ $\displaystyle=\tau(u,\sigma)\;,$ (121)
$\displaystyle\xi^{A}$
$\displaystyle=Y^{A}(u,\sigma)-I^{AB}\partial_{B}\tau(u,\sigma)\;,$ (122)
$\displaystyle\xi^{r}$
$\displaystyle=-rw(u,\sigma)+\frac{r}{2}\left[D_{A}\left(I^{AB}\partial_{B}\tau(u,\sigma)+U^{A}\partial_{A}\tau(u,\sigma)\right)\right]\;,$
(123)

where

$I^{AB}=\int_{r}^{\infty}\frac{\differential r^{\prime}}{r^{\prime\ 2}}\
e^{2\beta}\ q^{AB}\ .$ (124)

The next step is to impose the falloffs as well. Again, looking at (104)-(107)
we immediately have that

$\displaystyle\mathcal{L}_{\xi}g_{ur}$
$\displaystyle=\order{\frac{1}{r^{2}}}\;,$ (125)
$\displaystyle\mathcal{L}_{\xi}g_{uu}$ $\displaystyle=\order{1}\;,$ (126)
$\displaystyle\mathcal{L}_{\xi}g_{uA}$ $\displaystyle=\order{1}\;,$ (127)
$\displaystyle\mathcal{L}_{\xi}g_{AB}$ $\displaystyle=\order{r^{2}}\;,$ (128)

which result in constraining the functions $Y^{A}(u,\sigma),w(u,\sigma)$ and
$\tau(u,\sigma)$ to be

$\displaystyle Y^{A}(u,\sigma)$ $\displaystyle=Y^{A}(\sigma)\;,$ (129)
$\displaystyle w(u,\sigma)$ $\displaystyle=w(\sigma)\;,$ (130)
$\displaystyle\tau(u,\sigma)$ $\displaystyle=T(\sigma)+uw(\sigma)\;,$ (131)

and, as a result, the vector field $\xi$ at first leading order in $r$ becomes

$\xi=T(\sigma)\partial_{u}+Y^{A}(\sigma)\partial_{A}+w(\sigma)(u\partial_{u}-r\partial_{r})+\dots\
.$ (132)

This vector field will give rise to conserved quantities. Moreover, since
$T(\sigma)$, $Y^{A}(\sigma)$ and $w(\sigma)$ parameterize $\xi$, they are
actually the parameters of the symmetries coming from the residual
diffeomorphisms. Having obtained $\xi$ we can move on to write down the
residual symmetry algebra.

####  2.2.6 Algebra

Using (132) the algebra of symmetries of the theory is

$\displaystyle\commutator{\xi_{1}}{\zeta_{2}}=T_{12}\partial_{u}+Y^{A}_{12}\partial_{A}+w_{12}(u\partial_{u}-r\partial_{r})\;,$
(133) $\displaystyle Y^{A}_{12}=\commutator{Y_{1}}{Y_{2}}^{A}\;,$ (134)
$\displaystyle w_{12}\vcentcolon=Y_{1}(w_{2})-Y_{2}(w_{1})\;,$ (135)
$\displaystyle
T_{12}\vcentcolon=Y_{1}(T_{2})-w_{1}(T_{2})-Y_{2}(T_{1})+w_{2}(T_{1})\;.$
(136)

Since we have not computed the charges yet, these symmetries are residual
symmetries. Therefore they can be trivial ones or physical ones. We will
determine it in a moment by computing the associated charges.

Let us now make some important observations. We know that the conformal group
on the $2$-sphere is $SO(1,3)$ which generalises to $SO(1,d+1)$ when
considering the $d$-sphere. As a result, if $Y^{A}$ were a conformal Killing
vector on the $2$-sphere, the group of symmetries that we would have obtained
from it is $SO(1,3)$, which is indeed the Lorenz group. Additionally, $\xi$
has the $T(\sigma)\partial_{u}$ contribution which is exactly the one defining
the so-called supertranslations. Therefore, focusing for a moment only on the
$T(\sigma)$ and $Y^{A}(\sigma)$ terms in $\xi$, we see an enhancement of the
Poincaré group. It is an enhancement because these parameters are a priori
arbitrary functions on the $2$-sphere hence they are infinitely many. If we
were restricted to globally well-defined conformal Killing vectors on the
$2$-sphere, and constant translations, we would get the usual finite Poincaré
group $SO(1,3)\ltimes\mathbb{R}^{4}$. Nevertheless, what we obtained is an
enhancement of it, namely the semi-direct product of diffeomorphisms of the
$2$-dimensional space and supertranslations.
To have a visual representation and intuition of the supertranslation
$T(\sigma)\partial_{u}$ (see fig. 3), just imagine a past light cone of an
observer $O$. A cut of this cone is the surface $S$ and a supertranslation is
an angle-dependent translation of this surface along the $u$-direction,
allowing its points to move with different velocities.

Figure 3: Sketch of the past light cone of an observer $O$ representing the
action of a supertranslation.

The algebra we obtained is known as the _generalised BMS algebra_ [22]

$\text{generalised BMS algebra}\;=\;\mathrm{diff}(S)\oleft\mathbb{R}^{S}\;.$
(137)

In our treatment, we also had another generator, called $w(\sigma)$, which
enhances the algebra to the $\mathrm{BMSW}$ algebra studied in [8]. We are
ready to finally compute the charges of the theory. We will show that the
$\mathrm{BMSW}$ algebra is indeed charged, and thus its generators are
asymptotic symmetries.

####  2.2.7 Charges

Let us recall that at leading order in $r$, the metric we are considering
contains $M,\bar{\beta},\bar{U}^{A},c_{AB}$ as parameters and the equation to
solve to obtain the charges is

$\differential Q_{\hat{\xi}}=I_{\widehat{\xi}}\,\theta_{EH}-i_{\xi}L_{EH}\;.$
(138)

As a consequence, this means that we need to know how our fields transform
under the symmetries. This latter task is solved by computing the Lie
derivative of the metric at leading orders, so that we can extrapolate
${\mathcal{L}}_{\xi}M$, etc.
Having setting the stage, we complete these preliminary information by
recalling that $L=\frac{1}{2}R\sqrt{|\det{g}|}\ \differential[4]{x}$ and using
$\delta L=G_{\mu\nu}\delta g^{\mu\nu}+\differential\theta$ we have

$\theta_{EH}=\frac{1}{2}\left(g^{\mu\nu}\delta\Gamma^{\alpha}_{\mu\nu}-g^{\alpha\mu}\delta\Gamma^{\nu}_{\mu\nu}\right)\epsilon_{\alpha}$
(139)

with $\epsilon_{\alpha}$ being the codimension-1 volume form.
We skip all the technicalities and give the final expressions of the charge
associated to $T$, $w$ and $Y$, respectively. For each of the charges
associated to a parameter, all the others are set to zero. In order:

  * •

$H_{\hat{T}}$
Setting $Y=0=w$ we have that the charge associated to supertranslations reads

$H_{\hat{T}}=\int_{S}Q_{\hat{\xi}}=\int_{S}\sqrt{\bar{q}}\,T(\sigma)\left(M-\frac{1}{2}\bar{D}_{A}\bar{U}^{A}\right)\;,$
(140)

from which it immediately follows that if $T$ is $\sigma$-independent, the
last term on the RHS would drop since it is a total derivative contribution
and the charge would be the total mass of the system. We have thus shown why
we were so confident in calling $M$ in eq. (109) the total mass of the system.
It is important to stress here that this result is finite, there is no need
for renormalization. On the contrary, the other two charges turn out to be
divergent and renormalization is needed. For the sake of simplicity we only
report the finite part, which is the endpoint of the renormalization
procedure, and refer to [8] for more details.

  * •

$H_{\hat{w}}$
After setting $T=0=Y$ the finite part is

$H_{\hat{w}}^{\text{finite}}=\int_{S}\sqrt{\bar{q}}\,w(\sigma)\left[4\bar{\beta}+u\left(M-\frac{1}{2}\bar{D}_{A}\bar{U}^{A}\right)\right]\;.$
(141)

  * •

$H_{\hat{Y}}$
After setting $T=0=w$, again the finite part reads

$H_{\hat{Y}}^{\text{finite}}=\int_{S}\sqrt{\bar{q}}\,Y^{A}(\sigma)\left(\bar{p}_{A}+2\bar{D}_{A}\bar{\beta}\right)\;,$
(142)

which, as for the charge associated to supertranslations, immediately tells us
that if $Y^{A}$ is $\sigma$-independent the last term on the RHS drops and the
charge associated to the diffeomorphisms on the $2$-dimensional space becomes
the total angular momentum. As before, this is the reason why in eq. (112) we
called $\bar{p}$ the total angular momentum of the system.

Let us pause for a moment to appreciate that we have just obtained the
observables of gravity and we have demonstrated that they arise as Noether
(corner) charges associated to diffeomorphisms.

####  2.2.8 Charge algebra

This will be nothing else than the representation of the $\mathrm{BMSW}$
algebra. The analysis of the previous section can be repeated here. In the
presence of symplectic fluxes as we already discussed the final result is
(94), that is,

$\poissonbracket{H_{\hat{\xi}}}{H_{\hat{\zeta}}}=\mathfrak{L}_{\hat{\zeta}}H_{\hat{\xi}}-I_{\hat{\zeta}}F_{\hat{\xi}}+\int_{S}i_{\xi}i_{\zeta}L\;.$
(143)

We are going to show that the previous expression contains the EOMs, so the
evolution of the system expressed via the Bondi mass loss formula emerges from
symmetries. To this aim, we consider two supertranslation generators, since
their Poisson brackets is zero by definition because the algebra of
(super)translations is abelian. We thus choose as generators the following two
vectors

$\xi=T\partial_{u}\mbox{\quad and\quad}\zeta=\partial_{u}\;.$ (144)

Note that $\zeta$ is the generator of the symmetries associated to the energy
of the system. Along with the Poisson brackets, also the last term on the RHS
of (143) is zero when considering two supertranslations because the two
contractions are along the same direction. Consequently, the final expression
is

$\mathfrak{L}_{\hat{\zeta}}H_{\hat{\xi}}=I_{\hat{\zeta}}F_{\hat{\xi}}\quad\implies\quad\mathfrak{L}_{\hat{\partial_{u}}}H_{\hat{T\partial_{u}}}=I_{\hat{\partial_{u}}}F_{\hat{T\partial_{u}}}\;,$
(145)

which, using the expression in (140) and in (139), reads

$\int_{S}\sqrt{\bar{q}}T(\sigma)\left(\dot{M}+\frac{1}{8}N^{AB}N_{AB}-\frac{1}{4}\bar{D}_{A}\bar{D}_{B}N^{AB}\right)=0\
,$ (146)

which is exactly the Bondi mass-loss formula.

###  2.3 The power of symmetries: memory effects

In this section we will introduce the concept of _memory effects_ in gravity.
We again refer to [1] for original references on the topic, while this section
is mainly based on [23], see also [20]. _Memory effects_ stands for permanent
relative displacement due to a burst of gravitational waves. More precisely,
if the system under consideration consists of a set of massive objects, they
will be subject to a permanent displacement after a gravitational wave passes
through. In this sense, the masses have memory of the passage of gravitational
waves. This effect can be computed from the displacement equation in general
relativity. Consider a particle moving on a worldline. Let $v^{a}$ denote the
vector tangent to the particle worldline and $\xi^{b}$ the deviation vector,
describing how a direction has changed infinitesimally. The displacement
equation reads

$\left(v^{a}\nabla_{a}\right)^{2}\xi^{b}=-\tensor{R}{{}_{a}{}_{c}{}_{d}}{{}^{b}}\
v^{a}v^{d}\xi^{c}\ .$ (147)

Our goal in this section will be to derive the vector $\xi$: we want to
compute the displacement of the particles. To achieve that, let us consider a
number of particles near asymptotic infinity such that at initial time
$v^{a}=\delta^{a}_{u}$. Therefore, to leading order in $r$, (147) gives
($x^{a}=(r,x^{\mu})$)

$\partialderivative[2]{\xi^{\mu}}{u}=-\tensor{R}{{}_{u\alpha u}}{{}^{\mu}}\
\xi^{\alpha}\;.$ (148)

Now, it can be shown that at leading order when $r\to\infty$ the Riemann
tensor, made up by a traceless contribution given by the Weyl tensor plus the
Ricci scalar and the Ricci tensor counterparts, simply becomes the Weyl
tensor. That is: $\tensor{R}{{}_{u\alpha
u}^{\mu}}\xrightarrow{r\to\infty}\tensor{\bar{C}}{{}_{u\alpha u}^{\mu}}$ at
leading order (the bar on the Weyl tensor stands for the leading order
contribution). Additionally, if $\xi^{\alpha}_{(0)}$ denotes the initial
displacement and if $\xi^{\alpha}-\xi^{\alpha}_{(0)}$ goes to zero fast enough
as $r\to\infty$, one can safely replace $\xi^{\alpha}$ with
$\xi^{\alpha}_{(0)}$ on the RHS of (148) so that

$\partialderivative[2]{\xi^{\mu}}{u}=-\tensor{\bar{C}}{{}_{u\alpha
u}}{{}^{\mu}}\xi^{\alpha}_{(0)}\;,$ (149)

which, after integrating in time twice, becomes

$\xi^{\mu}\rvert^{u=+\infty}_{u=-\infty}=-\int^{\infty}_{-\infty}\differential
u^{\prime}\int^{\infty}_{-\infty}\differential u\ \tensor{\bar{C}}{{}_{u\alpha
u}}{{}^{\mu}}\xi^{\alpha}_{(0)}\;.$ (150)

This can be recasted as follows:

$\xi^{\mu}\rvert^{u=+\infty}_{u=-\infty}=\tensor{\Delta}{{}^{\mu}}{{}_{\alpha}}\
\xi^{\alpha}_{(0)}\ ,$ (151)

where we have introduced the _memory tensor_
$\tensor{\Delta}{{}^{\mu}}{{}_{\alpha}}$ as

$\tensor{\Delta}{{}^{\mu}}{{}_{\alpha}}=-\int^{\infty}_{-\infty}\differential
u^{\prime}\int^{\infty}_{-\infty}\differential u\ \tensor{\bar{C}}{{}_{u\alpha
u}}{{}^{\mu}}\;.$ (152)

The memory tensor keeps track of the passage of the gravitational waves
originated by an unspecified black hole dynamics and the left hand side (LHS)
of (151) gives the evolution between the stationary state in the far past and
the stationary state in the far future.

From now on we will work in the Bondi gauge previously introduced. In this
gauge a straightforward computation leads us to the following crucial result:

$\bar{C}_{uAuB}=-\frac{1}{2}\partial_{u}N_{AB}\;,$ (153)

where $\\{A,B\\}$ are coordinates on the $2$-sphere matching the notation of
the previous section and $N_{AB}$ is the news tensor. The news tensor
introduced in (113) is such that

$\Delta_{AB}=\frac{1}{2}\int^{+\infty}_{-\infty}\differential u\
N_{AB}\overset{\eqref{news}}{=}\frac{1}{2}c_{AB}\rvert^{u=+\infty}_{u=-\infty}\;,$
(154)

from which it is now evident why the memory tensor has information about
gravitational waves, it is written in terms of the shear tensor.

Now that we have the memory tensor, we can further investigate its relation
with the symmetries. Suppose for instance that in the far past $c_{AB}=0$
corresponding to a _stationary era_. This would for example be the case in a
configuration in which two black holes are far away. A natural question is
then what is the fate of $c_{AB}$ in the far future. The latter can at best be
the BMS-transform of its initial value, that is:

$c_{AB}\rvert_{u=-\infty}=0\quad\implies\quad
c_{AB}\rvert^{u=+\infty}=\mathrm{BMS}(c_{AB}\rvert_{u=-\infty})\;.$ (155)

It can be shown that the variation of the shear tensor under a
supertranslation gives ($\bar{D}_{\langle A}\bar{D}_{B\rangle}$ is the
symmetric traceless part of $\bar{D}_{A}\bar{D}_{B}$):

$\mathfrak{L}_{\hat{T}}c_{AB}=T\partial_{u}c_{AB}-2\bar{D}_{\langle
A}\bar{D}_{B\rangle}T\;.$ (156)

Applying this result to $c_{AB}\rvert_{u=-\infty}$ with our initial hypothesis
of a stationary era we obtain

$c_{AB}\rvert^{u=+\infty}=\mathfrak{L}_{\hat{T}}c_{AB}\rvert_{u=-\infty}=T\partial_{u}c_{AB}\rvert_{u=-\infty}-2\bar{D}_{\langle
A}\bar{D}_{B\rangle}T\underbrace{=}_{c_{AB}\rvert_{u=-\infty}=0}-2\bar{D}_{\langle
A}\bar{D}_{B\rangle}T\;.$ (157)

The memory tensor (154) thus becomes

$\Delta_{AB}=-\bar{D}_{\langle A}\bar{D}_{B\rangle}T\;,$ (158)

from which we can express $T$ as a function of $\Delta_{AB}$. This
manipulation is essential when considering the Bondi mass loss formula.
Indeed, plugging it into $T(\Delta_{AB})$ we are able to have an expression of
the memory tensor in terms of measurable quantities. To have more insight, we
move to the wave forms and briefly illustrate the effect of this prediction
with a sketch. In fig. 4 the wave form of the gravitational wave as a function
of time is displayed.

Figure 4: Sketch of the metric perturbation as a function of time.

More precisely, on the left of fig. 4 the effect is represented with the
Poincaré group as a symmetry group while on the right, the group of symmetry
is the $\mathrm{BMS}_{4}$ symmetry group. There is a clear mismatch between
the two figures due to the presence of an offset in the second case. Let us
explain better what information is hidden in this fig. 4. Firstly, at some
time, the profile becomes constant. Suppose two black holes collide and emit
gravitational waves, their passage in our detector is highlighted by the
oscillating behaviour of the function in the figure. After this passage,
however, there is a remnant of the presence of the gravitational waves: the
memory displacement. Consequently, this displacement that we have studied in
this section culminates in the prediction of a constant shift in fig. 4.
Furthermore, we expect that the latter could be detected and measured by
upcoming experiments.

We conclude this section with a concept introduced in 2017 by Strominger [24],
the _infrared triangle_ that well summarises what we have learned so far. In
the triangle depicted in fig. 5, all the vertices are equivalent. Once one
vertex is known, the others can be attained by mathematical manipulations. We
however remark that we have followed here an approach based entirely on
symmetries. We have shown how symmetries are so powerful that memory effects
can be derived from them. This is the logic of the _corner proposal_ ,
enunciated in the next section, in which symmetries are exploited to infer new
constraints in quantum gravity. Before doing so, we fill the last tip of the
triangle and show how symmetries can be used to deduce the infrared soft
theorems.

Figure 5: The infrared triangle.

###  2.4 The power of symmetries: soft theorems

In this section we show how symmetries can be used to derive Weinberg’s soft
theorems [25]. The relationship between these two topics was observed in [26],
which we use here as main reference, and we again refer to [1] for more
references. Soft theorems establish a connection between the amplitude of a
scattering process involving one soft particle and the one involving only hard
particles, where the soft particle contribution factorises.

We start computing the asymptotic phase space Poisson bracket. In order to do
so, one can show that the renormalised symplectic two-form takes the form

$\Omega^{R}=\int_{\mathfrak{I}}\frac{1}{4}\delta
N_{AB}\wedge\delta\quantity(\sqrt{\bar{q}}c^{AB})+\dots\;.$ (159)

This is the famous Ashtekar-Streubel result [27]. Using this two-form, one can
compute the Poisson bracket between $N_{AB}$ and $c_{AB}$, and obtain

$\poissonbracket{N_{AB}(u,z,\bar{z})}{c_{CD}(u^{\prime},w,\bar{w})}=2\gamma_{<AC}\gamma_{BD>}\delta(u-u^{\prime})\delta^{2}(z-w),$
(160)

where $z$ and $\bar{z}$ are coordinates on the sphere $S^{2}$. As a side note,
we mention that Ashtekar promoted this Poisson bracket to a commutator, using
standard rules of canonical quantization
($\\{,\\}\rightarrow\frac{1}{i\hbar}[,]$), therefore quantizing gravity
without requiring the Newton constant to be small. This is possible only
asymptotically, where this quantization is still made on a classical geometric
background.

Let us now define the Fourier transformed field $c_{AB}(\omega,z,\bar{z})$,
for $\omega\geq 0$, as

$c_{AB}(\omega,z,\bar{z})=\int_{-\infty}^{+\infty}c_{AB}e^{i\omega
u}\mathrm{d}u\;.$ (161)

We can then define

$\displaystyle a_{+}(\omega,z,\bar{z})$ $\displaystyle\equiv 2\pi
ic_{zz}(\omega,z,\bar{z})\;,$ (162) $\displaystyle a_{-}(\omega,z,\bar{z})$
$\displaystyle\equiv 2\pi ic_{\bar{z}\bar{z}}(\omega,z,\bar{z})\;,$ (163)

such that $a_{\pm}^{*}$ become positive/negative-helicity graviton creation
operators. The field $c_{AB}$ is part of the metric, and therefore we can use
this field, previously shown to be related to gravitational waves, to
create/annihilate gravitons. Furthermore, we can compute

$\poissonbracket{a_{\pm}(\omega,z,\bar{z})}{a_{\pm}^{*}(\omega^{\prime},w,\bar{w})}=-2i\omega(2\pi)^{3}\delta(\omega-\omega^{\prime})\delta^{2}(z-w)\;,$
(164)

which proves that $a_{\pm}$ satisfy the canonical creation/annihilation
operators algebra.

For the rest of this section, we will have to slightly modify the definition
of charges and fluxes. We define888For simplicity we assume that the space is
the projective plane, with metric $\differential s^{2}=2\differential
z\differential\bar{z}$, such that $\sqrt{\bar{q}}=1$.

$\tilde{H}_{\hat{T}}=2\int_{S}TM\;;$ (165)

together with an appropriate expression for the flux. It can be proved that in
this case we have

$\tilde{H}_{\hat{T}}^{+}-\tilde{H}_{\hat{T}}^{-}=-\int_{\mathfrak{I}^{+}}T{\cal
F}+\frac{1}{2}\int_{S}T\bar{D}_{A}\bar{D}_{B}\Delta^{AB}\;,$ (166)

where we introduced the local Bondi hard flux, defined as

${\cal F}=\frac{1}{4}N_{AB}N^{AB}\;.$ (167)

For simplicity, we use the notation ${H}_{T}^{+}\equiv{H}_{T}(u=+\infty)$. We
have shown previously that $\Delta_{AB}=c_{AB}^{+}-c_{AB}^{-}$. We would like
to interpret the first term as the hard charge and the second term as the soft
charge. To see this, note that we have

$\displaystyle
c_{AB}^{+}-c_{AB}^{-}=\int_{-\infty}^{+\infty}\partial_{u}c_{AB}=\lim_{\omega\rightarrow
0}\int_{-\infty}^{+\infty}e^{i\omega
u}\partial_{u}c_{AB}=\lim_{\omega\rightarrow
0}\int_{-\infty}^{+\infty}-i\omega c_{AB}(\omega,z,\bar{z})\;,$ (168)

where we integrated by parts and used $\lim_{\omega\rightarrow 0}(e^{i\omega
u}c_{AB})^{+}_{-}=0$. We also impose one extra condition

${\bar{D}}_{z}{\bar{D}}_{z}(c_{+}^{zz}-c_{-}^{zz})={\bar{D}}_{\bar{z}}{\bar{D}}_{\bar{z}}(c_{+}^{\bar{z}\bar{z}}-c_{-}^{\bar{z}\bar{z}}),$
(169)

satisfied by the so called Christodoulou-Klainerman spacetimes [28, 29]. Using
this equation, we rewrite the second term in (166) as

$Q_{\mathrm{soft}}=\frac{1}{2}\int_{S}T\bar{D}_{A}\bar{D}_{B}\Delta^{AB}=\frac{1}{2\pi}\lim_{\omega\rightarrow
0}\omega\int_{S}T{\bar{D}}_{\bar{z}}{\bar{D}}_{\bar{z}}a_{+}(\omega,z,\bar{z}).$
(170)

For this reason we call this term a soft charge (word soft originating from
the limit $\omega\rightarrow 0$) and denote it $Q_{\mathrm{soft}}$. This term
corresponds to a conformally soft graviton insertion. Analogously, the first
term in (166) is called the hard charge, denoted $Q_{\mathrm{hard}}$.

For simplicity, we consider asymptotic $\ket{in}$ and $\ket{out}$ states
defined as a cloud of massless particles. Charge conservation implies

$\tilde{H}_{\hat{T}}^{+}-\tilde{H}_{\hat{T}}^{-}|_{\mathfrak{I}^{+}}=\tilde{H}_{\hat{T}}^{+}-\tilde{H}_{\hat{T}}^{-}|_{\mathfrak{I}^{-}}.$

At the quantum level, charge conservation is expressed as the fact that the
charge operator commutes with the $S$ matrix, that is,
$\bra{out}[Q,S]\ket{in}=0$. This expression is a Ward Identity for a given
symmetry. We thus have

$Q_{\mathrm{hard}}+Q_{\mathrm{soft}}|_{\mathfrak{I}^{+}}=Q_{\mathrm{hard}}+Q_{\mathrm{soft}}|_{\mathfrak{I}^{-}}\Rightarrow\bra{out}[Q_{\mathrm{soft}},S]\ket{in}+\bra{out}[Q_{\mathrm{hard}},S]\ket{in}=0.$
(171)

The last equation is a Ward identity for BMS, where here we only used
supertranslations. Now, since $Q_{\mathrm{hard}}$ is the energy operator for
incoming particles, we have

$Q_{\mathrm{hard}}\ket{in}=\sum_{k}E_{k}T(z_{k},\bar{z}_{k})\ket{in},$ (172)

together with a similar expression for $out$. In our case on the other hand we
have

$Q_{\mathrm{soft}}\ket{in}\sim a_{+}\ket{in}=0\;.$ (173)

Using this, we obtain

$\frac{1}{2\pi}\lim_{\omega_{g}\rightarrow
0}\int_{S}\mathrm{d}^{2}wT(w,\bar{w}){\bar{D}}_{\bar{w}}{\bar{D}}_{\bar{w}}\bra{out}a_{+}(\omega_{g},w,\bar{w})S\ket{in}=-\sum_{k}E_{k}T(z_{k},\bar{z}_{k})\bra{out}S\ket{in},$
(174)

where $\omega_{g}$ is the soft graviton frequency. This equation is true for
any supertranslation $T$. We can thus apply it to the special supertranslation

$T(z,\bar{z})=\frac{\bar{w}-\bar{z}}{w-z}\;.$ (175)

Using the following identity:

${\bar{D}}_{\bar{w}}{\bar{D}}_{\bar{w}}\Big{(}\frac{\bar{w}-\bar{z}}{w-z}\Big{)}=2\pi\delta^{2}(w-z)\;,$
(176)

we can finally write down

$\lim_{\omega_{g}\rightarrow
0}\omega_{g}\bra{out}a_{+}(\omega_{g},w,\bar{w})S\ket{in}=-\sum_{k}E_{k}\frac{\bar{w}-\bar{z}_{k}}{w-z_{k}}\bra{out}S\ket{in}.$
(177)

This is the desired identity, expressing Weinberg’s soft graviton theorem in
coordinates representation. It is often stated in momentum representation,
where it has the form

$-\sum_{k}E_{k}\frac{\bar{w}-\bar{z}_{k}}{w-z_{k}}=\sum_{k}\frac{p_{k}^{\mu}p_{k}^{\nu}\varepsilon_{\mu\nu}^{+}(q)}{2p_{k}\cdot
q}\;.$ (178)

Here, $q$ is the soft graviton four-momentum parametrised by (in cartesian
coordinates)

$q^{\mu}=(1+w\bar{w},w+\bar{w},-i(w-\bar{w}),1-w\bar{w})\;,$ (179)

while $p_{k}$ is the $k$-th particle momentum

$p^{\mu}=E(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z})\;.$ (180)

Moreover, we introduced the positive helicity graviton polarization tensor
given by

$\varepsilon^{+}_{\mu\nu}=\varepsilon^{+}_{\mu}\varepsilon^{+}_{\nu}\;,\qquad\varepsilon^{+\mu}=\partial_{w}q^{\mu}\;.$
(181)

Plugging all this together, one can verify (178).

Note that the procedure utilized here is fully general, soft theorems can be
formulated for particles of any spin. Also, one could work out the subleading
contributions in $\omega_{g}$, thus obtaining additional expressions that must
hold in a scattering process, known as subleading soft theorems, here derived
as Ward identities for the BMS symmetry.

##  3 The Corner Proposal

General considerations of symmetries in the previous sections gave us a top
down approach to associate to any classical theory a set of charges and their
associated algebras. Noether second’s theorem assures that charges associated
to local symmetries have support on codimension-2 surfaces, also called
corners. Therefore, it seems that the observables of the theory have to be
understood and measured on these corners, on which they are defined, rather
than in the entirety of spacetime. However, the main issue is that the algebra
seems to depend on the problem at hand and, in particular, on the boudary
conditions and gauge fixing (C.f section 2.2). The question that thus arises
is: "Is there a universal algebra for gravity?". If the answer is positive,
then any classical theory of gravity would have symmetries that fall in a
subset of the universal ones. In section 3.1 we answer this question by
considering embeddings of codimension-2 surfaces and in section 3.2 we tackle
the issue of non integrability of the charges by introducing the concept of
extended phase space. For more details on the corner proposal and corners in
general we refer the reader to [30, 31, 32, 33, 9, 10, 11, 34, 1].

###  3.1 Universal algebra

Let $M$ be a $d$-dimensional manifold and $S$ a codimension-2 manifold (the
corner). The way $S$ is embedded into $M$ is defined by the injective map

$\phi\mathrel{\mathop{\ordinarycolon}}S\longrightarrow M.$ (182)

Let
$\sigma^{\alpha}\mathrel{\mathop{\ordinarycolon}}S\longrightarrow\mathbb{R}^{d-2},\,(\alpha=1,...,d-2)$
be local coordinates on $S$. Let
$y^{M}\mathrel{\mathop{\ordinarycolon}}M\longrightarrow\mathbb{R}^{d}$ be
local coordinates on M. A choice of embedding (182), is then given by
$y^{M}(\sigma)$. Without loss of generality (and without choosing a specific
embedding) one can choose the $y^{M}$ coordinates such that

$y^{M}=(u^{a},x^{i}),$ (183)

where $a=1,...,d-2$ and $i=1,2$. Since we know from the covariant phase space
formalism that the spacetime Lie bracket represents the symmetry algebra
modulo central extensions (Equation (66)), one can ask what is the maximal
closed algebra by calculating the Lie bracket of two vector fields. However,
since the interest lies in the codimension-2 surface, one must only calculate
the bracket "close" to it. In order to do so, one can consider a particularly
simple form of the embedding (182). Once the coordinates (183) have been
chosen, this so-called "trivial" embedding is given by

$y^{M}_{0}(\sigma)=(u_{0}^{a}(\sigma),x_{0}^{i}(\sigma))=(0,\delta^{i}_{\alpha}\sigma^{\alpha}).$
(184)

Expanding "close to" this codimension-2 embedding thus corresponds to
expanding around $u^{a}=0$. Let us consider two vector fields $\xi,\zeta\in
TM$. In the coordinate system (183) they are expressed as

$\displaystyle\xi$
$\displaystyle=\xi^{a}(u,x)\partial_{a}+\xi^{i}(u,x)\partial_{i},$ (185)
$\displaystyle\zeta$
$\displaystyle=\zeta^{a}(u,x)\partial_{a}+\zeta^{i}(u,x)\partial_{i}.$

Expending close to the surface $u^{a}=0$, one gets

$\displaystyle\xi^{a}(u,x)$
$\displaystyle=\xi^{a}_{(0)}(x)+u^{b}\xi^{a}_{(1)b}(x)+u^{b}u^{c}\xi^{a}_{(2)bc}(x)+\mathcal{O}(u^{3}),$
(186) $\displaystyle\xi^{i}(u,x)$
$\displaystyle=\xi^{i}_{(0)}(x)+u^{b}\xi^{i}_{(1)b}(x)+u^{b}u^{c}\xi^{i}_{(2)bc}(x)+\mathcal{O}(u^{3}),$
$\displaystyle\zeta^{a}(u,x)$
$\displaystyle=\zeta^{a}_{(0)}(x)+u^{b}\zeta^{a}_{(1)b}(x)+u^{b}u^{c}\zeta^{a}_{(2)bc}(x)+\mathcal{O}(u^{3}),$
$\displaystyle\zeta^{i}(u,x)$
$\displaystyle=\zeta^{i}_{(0)}(x)+u^{b}\zeta^{i}_{(1)b}(x)+u^{b}u^{c}\zeta^{i}_{(2)bc}(x)+\mathcal{O}(u^{3}).$

Calculating their Lie bracket at second order reveals that the maximal closed
sub-algebra is generated by vectors of the type

$\displaystyle\xi$
$\displaystyle=\xi^{i}_{(0)}\partial_{i}+\quantity(\xi_{(0)}^{a}+u^{b}\xi_{(1)b}^{a})\partial_{a},$
(187) $\displaystyle\zeta$
$\displaystyle=\zeta^{i}_{(0)}\partial_{i}+\quantity(\zeta_{(0)}^{a}+u^{b}\zeta_{(1)b}^{a})\partial_{a}.$

Any additional term in the expansion (186) results in a non-closing algebra
and is thus discarded in this analysis. The Lie bracket of the vector fields
expressed in (187) gives

$\displaystyle\quantity[\xi,\zeta]$
$\displaystyle=\quantity[\hat{\xi}_{(0)},\hat{\zeta}_{(0)}]^{i}\partial_{i}$
(188)
$\displaystyle\qquad+\quantity[\xi_{(0)}^{k}\zeta_{(0),k}^{a}-\zeta_{(0)}^{k}\xi_{(0),k}^{a}+\zeta_{(1)b}^{a}\xi_{(0)}^{b}-\xi_{(1)b}^{a}\zeta_{(0)}^{b}]\partial_{a}$
$\displaystyle\qquad+u^{b}\quantity[-\quantity[\xi_{(1)},\zeta_{(1)}]^{a}{}_{\,b}+\xi_{(0)}^{k}\zeta_{(1)b,k}^{a}-\zeta_{(0)}^{k}\xi_{(1)b,k}^{a}]\partial_{a},$

where $\hat{\zeta}\equiv\zeta^{j}\partial_{j}$ and the comma denotes partial
differentiation. Remembering that the indices $i,j,k,...$ denote coordinates
on the surface and $a,b,c,...$ normal coordinates, it is now easy to interpret
this result. The first line simply corresponds to diffeomorphisms on the
surface. The first part of the second line corresponds to surface
diffeomorphisms acting on the normal translations and the second part to the
general linear group acting on normal translations. Finally, the first part of
the last line corresponds simply to the general linear algebra, and the second
part to the surface diffeomorphisms acting on this general linear algebra.
Thus the complete group corresponding to this maximally closed algebra is

$\mathrm{UCS}=\quantity(\mathrm{Diff}(S)\ltimes\mathrm{GL}(2,\mathbb{R})^{S})\ltimes\quantity(\mathbb{R}^{2})^{S}.$
(189)

UCS stands for the Universal Corner Symmetries. A few remarks on this result
are in order:

  * •

The semi-direct product emphasizes that, while the corner diffeomorphisms act
on the linear transformations and the translations, the converse is not true.
Similarly for the linear group acting on the translations.

  * •

The $S$ in the exponent of the linear group and the translations means that
there is a copy of the general linear group and the translation group at each
point of the corner. This is the origin of supertranslations in section 2.2.7.

  * •

This entire analysis was done without the need of introducing a metric on the
manifold. Mathematically this results is topological and not geometrical,
which translates to kinematical and not dynamical in the physical realm.
Therefore this algebra does not depend on the particular theory of gravity at
hand and is thus called universal.

It is believed in the corner community that this algebra is the most general
one that can have charges [9, 34], providing a bottom up approach to quantum
gravity as a condition that any theory should hold a representation of (189)
at some level. For example, the connection of this symmetry group to Loop
Quantum Gravity is discussed in [35].

###  3.2 Extended phase space

####  3.2.1 Keeping track of the embedding

Let us briefly recap the results up to this point. We know that the UCS Lie
bracket realises the algebra of the charges modulo central extensions. The
next step towards the quantization of the theory would be to promote the
charges to hermitian operators on some Hilbert space and replace the algebra
brackets by Dirac brackets. However there is another issue here. In general,
as we have seen in previous sections, there might be fluxes present.
Mathematically, this translates to the non-closure of the algebra brackets,
and thus we do not know how to quantize the theory. This issue arises from a
careless treatment of the embedding in the action. More precisely, we assumed
something of the following form in a number of computations:

$\mathfrak{L}_{\hat{\xi}}\int_{S}\mathcal{A}=\int_{S}\mathfrak{L}_{\hat{\xi}}\mathcal{A},$
(190)

for a general functional $\mathcal{A}$. However, the commutation of the
variation and the integral is only true in the case where the variation does
not move the surface we are integrating on. A way of being more careful about
it is to keep track of the embedding (182) when integrating a quantity on the
surface:

$\mathfrak{L}_{\hat{\xi}}\int_{S}\phi^{*}\quantity(\mathcal{A})=\int_{S}\phi^{*}\quantity(\mathfrak{L}_{\hat{\xi}}\mathcal{A})+\int_{S}\quantity(\mathfrak{L}_{\hat{\xi}}\phi^{*})\quantity(\mathcal{A}),$
(191)

where we see the appearance of a second term that was not present in equation
(190). The question now becomes: "How can we keep track of the variation of
the embedding at the phase space level?" This is done by the introduction of a
new field $\chi$ which is a vector in spacetime and a one-form on field space:

$\chi\in TM\otimes T^{*}\Gamma,$ (192)

defined through the relation

$I_{\hat{\xi}}\chi=-\xi,$ (193)

This field keeps track of the variation of the embedding in the following
sense: For any functional $\mathcal{F}$ on field space, we have

$\delta\phi^{*}(\mathcal{F})=\phi^{*}(\mathcal{L}_{\chi}\mathcal{F}).$ (194)

Note that this relation can be thought off as the equivalence between the
passive and the active interpretation of diffeomorphisms. The $\chi$ field,
which is sometimes called an edge mode because it lives on the corner, is then
added to the phase space of the theory.

####  3.2.2 Extended symplectic form

We will now calculate again the symplectic form but on the extended phase
space that includes the new field. In the following derivation we will use the
following relations that are proved in appendix A:

$\displaystyle\delta\chi$ $\displaystyle=-\frac{1}{2}[\chi,\chi],$ (195)
$\displaystyle\mathcal{L}_{\chi}\iota_{\chi}L$
$\displaystyle=\frac{1}{2}\iota_{\quantity[\chi,\chi]}L+\frac{1}{2}\differential(\iota_{\chi}i_{\chi}L),$
(196) $\displaystyle I_{\hat{\xi}}\iota_{\delta\chi}L$
$\displaystyle=\iota_{\quantity[\xi,\chi]}L.$ (197)

Consider the action defined in a subregion $R$ that allows us to keep track of
the embedding:

$S_{R}=\int_{R}\phi^{*}\quantity(L)\;,$ (198)

where $L$ is a functional of the dynamical fields $\varphi$ and their
derivatives. Taking its variation yields

$\displaystyle\delta S_{R}$
$\displaystyle=\int_{R}\quantity[\delta\phi^{*}(L)+\phi^{*}(\delta L)]$ (199)
$\displaystyle=\int_{R}\phi^{*}(\mathcal{L}_{\chi}L+\delta L)$
$\displaystyle\hat{=}\int_{R}\phi^{*}(\differential\iota_{\chi}L+\differential\theta)$
$\displaystyle=\int_{R}\phi^{*}(\differential\theta^{\mathrm{ext}})\;,$

where one should stress that the extended symplectic potential is now a
function of the dynamical fields $\varphi$, their variation $\delta\varphi$,
as well as the new field $\chi$, that is,
$\theta^{\mathrm{ext}}[\varphi,\delta\varphi,\chi]$. The extended symplectic
potential is now given by

$\Theta^{\mathrm{ext}}=\int_{\Sigma}\phi^{*}\theta^{\mathrm{ext}}=\int_{\Sigma}\phi^{*}(\iota_{\chi}L+\theta).$
(200)

The extended symplectic two-form can be calculated:

$\displaystyle\Omega^{\mathrm{ext}}$
$\displaystyle=\delta\Theta^{\mathrm{ext}}$ (201)
$\displaystyle=\int_{\Sigma}[\delta\phi^{*}(\iota_{\chi}L+\theta)+\phi^{*}_{(}\delta(\iota_{\chi}L+\theta))]$
$\displaystyle=\int_{\Sigma}\phi^{*}(\mathcal{L}_{\chi}\iota_{\chi}L+\mathcal{L}_{\chi}\theta+\iota_{\delta\chi}L-\iota_{\chi}\delta
L+\delta\theta)$ $\displaystyle\ \hat{=}\
\int_{\Sigma}\phi^{*}(\mathcal{L}_{\chi}\iota_{\chi}L+\differential\iota_{\chi}\theta+\delta\theta+\iota_{\delta\chi}L)$
$\displaystyle=\int_{\Sigma}\phi^{*}(\delta\theta+\frac{1}{2}\differential(\iota_{\chi}\iota_{\chi}L)+\differential\iota_{\chi}\theta)$
$\displaystyle=\int_{\Sigma}\phi^{*}\left(\delta\theta+\differential(\iota_{\chi}\theta+\frac{1}{2}\iota_{\chi}\iota_{\chi}L)\right),$

where we have used equations (195) and (196) to go from the fourth to the
fifth line. One can thus rewrite the extended symplectic form in the following
way:

$\Omega^{\mathrm{ext}}=\Omega+\int_{S}\phi^{*}(\iota_{\chi}\theta+\frac{1}{2}\iota_{\chi}\iota_{\chi}L).$
(202)

The additional term is thus a corner contribution, which will render all
charges integrable. Loosely speaking, the additional field $\chi$ renders the
flux part of the augmented phase space. Indeed, let us now calculate the
contraction (35) within this extended phase space formalism:

$\displaystyle I_{\hat{\xi}}\Omega^{\mathrm{ext}}$
$\displaystyle=I_{\hat{\xi}}\int_{\Sigma}\phi^{*}(\delta(\theta+\iota_{\chi}L)+\mathcal{L}_{\chi}(\theta+\iota_{\chi}L))$
(203)
$\displaystyle=\int_{\Sigma}\phi^{*}(I_{\hat{\xi}}\delta\theta+I_{\hat{\xi}}\mathcal{L}_{\chi}\theta+I_{\hat{\xi}}\mathcal{L}_{\chi}\iota_{\chi}L+I_{\hat{\xi}}\iota_{\delta\chi}L-I_{\hat{\xi}}\iota_{\chi}\delta
L)$ $\displaystyle=\int_{\Sigma}\phi^{*}(\mathcal{L}_{\xi}\theta-\delta
I_{\hat{\xi}}\theta+I_{\hat{\xi}}\mathcal{L}_{\chi}\theta+I_{\hat{\xi}}\mathcal{L}_{\chi}\iota_{\chi}L+\iota_{\quantity[\xi,\chi]}L-I_{\hat{\xi}}\iota_{\chi}\delta
L)$ $\displaystyle=\int_{\Sigma}\phi^{*}(-\delta
I_{\hat{\xi}}\theta-\mathcal{L}_{\chi}I_{\hat{\xi}}\theta+I_{\hat{\xi}}\mathcal{L}_{\chi}\iota_{\chi}L+\iota_{\quantity[\xi,\chi]}L-I_{\hat{\xi}}\iota_{\chi}\delta
L)$ $\displaystyle=\int_{\Sigma}\phi^{*}(-\delta
I_{\hat{\xi}}\theta-\mathcal{L}_{\chi}I_{\hat{\xi}}\theta-\mathcal{L}_{\xi}\iota_{\chi}L+\mathcal{L}_{\chi}\iota_{\xi}L+\iota_{\quantity[\xi,\chi]}L+\iota_{\xi}\delta
L+\iota_{\chi}I_{\hat{\xi}}\delta L)$
$\displaystyle=\int_{\Sigma}\phi^{*}(-\delta
I_{\hat{\xi}}\theta-\mathcal{L}_{\chi}I_{\hat{\xi}}\theta+\mathcal{L}_{\chi}\iota_{\xi}L+\iota_{\xi}\delta
L)$ $\displaystyle=\int_{\Sigma}\phi^{*}(-\delta
I_{\hat{\xi}}\theta-\mathcal{L}_{\chi}I_{\hat{\xi}}\theta+\mathcal{L}_{\chi}\iota_{\xi}L+\iota_{\xi}\delta
L)$
$\displaystyle=-\int_{\Sigma}\phi^{*}(\delta(I_{\hat{\xi}}\theta-\iota_{\xi}L)+\mathcal{L}_{\chi}(I_{\hat{\xi}}-\iota_{\xi}L)),$

where in the second line we have used (197) and in the third line we used that
the spacetime Lie derivative coincide with the field-space Lie derivative for
$L$ and $\theta$. Recalling the expression for the Noether current associated
to a diffeomorphism:

$J_{\hat{\xi}}=I_{\hat{\xi}}\theta-\iota_{\xi}L,$ (204)

we can rewrite the contraction as

$\displaystyle I_{\hat{\xi}}\Omega^{\mathrm{ext}}$
$\displaystyle=-\int_{\Sigma}\phi^{*}(\delta
J_{\hat{\xi}}+\mathcal{L}_{\chi}J_{\hat{\xi}})$ (205)
$\displaystyle=-\delta(\int_{\Sigma}\phi^{*}(J_{\hat{\xi}}))$
$\displaystyle=-\delta H_{\hat{\xi}}.$

where we have used (194) one last time and the charge $H_{\hat{\xi}}$ is the
same as in the non-extended phase space. We thus see that introducing this new
field takes care of the term that spoils integrability. This crucial result
established in [10] implies that the charge algebra is represented by the
standard Poisson bracket, see [11]. This is a remarkable feature, because the
system is still dissipative, albeit integrable. Rendering all diffeomosphisms
integrable is a turning point in the corner proposal because it prepares the
classical gravitational setup to finally address quantum gravity, as we will
discuss below.

###  3.3 The corner proposal for quantum gravity

We here enunciate the main idea of the corner proposal, and offer some
conclusive remarks, to summarize the journey taken.

We are finally ready to state the proposal. It is hard to formulate a
consistent theory of quantum gravity, and therefore it would be very useful to
have a bottom-up approach to answer some questions in this field. The main
idea of the corner proposal is the following:

_Gravity is described by a set of charges and their algebra at corners_.

Here, corners refer to any codimension-2 surface. This proposal instructs us
to loose ties with classical concepts and notions, that would not survive in
the quantum realm. For example, a full bulk metric is at best a semi-classical
notion. What we have shown to be more robust is the concept of corners, on
which charges are defined. Even more fundamental is the concept of symmetries
and algebras. In this regard, it was instrumental to enlarge the phase space,
such that all diffeomorphisms are canonically realized without fluxes. There,
the Universal Corner Symmetry algebra can be derived, and the corner proposal
posits that it survives in the quantum regime and becomes the algebra of
observables. We must therefore focus on charges and their Poisson brackets,
which are then promoted to operators in the quantum theory, with the usual
prescription
$\poissonbracket{Q_{\hat{\xi}}}{Q_{\hat{\zeta}}}\rightarrow\frac{1}{i\hbar}[Q_{\hat{\xi}},Q_{\hat{\zeta}}]$.
We can then study the representation theory of the charge algebra, and
investigate whether there are unitary representations that could be suitable
to describe the quantum geometry. The classical spacetime emerges from one
particular representation of this algebra, but there are other
representations, that are relevant for the quantum theory. Obviously, this is
not an easy task, but it is a promising and concrete program toward a better
understanding of quantum gravity.

From the algebraic perspective, the mission of the corner proposal is clear.
Starting from the Lie bracket of two symmetry generators, we have a way, using
the covariant phase space, to descend to the Poisson brackets of the system.
This is already a delicate and mathematically challenging step. The endpoint
is a projective representation of the symmetry algebra, at the level of
charges. From there, one can pursue a canonical quantization procedure to
derive the Dirac bracket. Which algebra should we focus on? If each
quantization procedure leads to a different algebra, than this proposal stops
being predictive. Thankfully, we have shown that the Universal Corner Symmetry
algebra is very robust and maximal, which therefore means that it is
independent of a particular setup and classical Lagrangian: the latter can
only further reduce the charged symmetries. Therefore, we have a narrow window
to probe quantum gravity: study the Universal Corner Symmetry algebra in
particular setups. Clearly, loosing the concept of metric and manifold opens
the door to the idea of quantum geometries, and hopefully this could represent
a new avenue of investigation for the quantum gravity community.

Let us quickly summarize the results exposed. We have shown that the covariant
phase space is the suitable arena to study Noether’s theorems. The nature of a
local symmetry is encoded in its corner charge. If the latter is non-zero,
then this is a physical symmetry. Gravity makes the study of the nature of the
symmetry hard, because it brings complications, such as integrability issues,
divergences, and dissipation. At the end, after curing those, we have shown
how symmetries are the crucial ingredients from which one can derive physical
effects, such as memory effects and soft theorems. In this regard, the
infrared triangle should rather be depicted as a symmetries pyramid, where the
concept of symmetries lies on the top of the pyramid, and all the rest can be
derived from it (see fig. 6). Using this as a guiding principle, we arrived to
the corner proposal, which is the idea that symmetries are the ultimate clues
toward a bottom-up understanding of quantum gravity. We exposed these ideas in
these lecture notes, and in particular we discussed how to prepare the
classical milieu using only tools that are expected to survive in quantum
gravity, such as algebras, charges, and symmetries. In this respect, the
derivation of the Universal Corner Symmetry algebra and the extension of the
phase space to make all diffeomorphisms integrable are important steps, to
close the classical gravity chapter and open the quantum gravity one.

Figure 6: Rather than an infrared triangle, our approach culminates in "the
symmetries pyramid". Indeed, the Corner Proposal puts emphasis on the
symmetries of the theory, from which everything can be derived.

##### Acknowledgements

We would like to thank the organisers and the participants of the 59 Winter
School of Theoretical Physics and third COST Action CA18108 Training School
for the very stimulating atmosphere. 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. This work has been
partially supported by the Regional Government of Castilla y León (Junta de
Castilla y León), by the Ministry of Science and Innovation MICIN and the
European Union NextGenerationEU (PRTR C17.I1) and by the Faculty of Physics,
University of Belgrade (grant number 451-03-47/2023-01/200162).

##  Appendix A Proof of Important Relations

The first relation we will look at is the variation of the field $\chi$.
Consider a general integrated functional:

$\mathcal{A}=\int_{R}\phi^{*}(\mathcal{F}),$ (206)

where $R$ denotes a subregion of spacetime which makes it simpler to keep
track of the embedding. We have

$\displaystyle\delta\mathcal{A}$
$\displaystyle=\int_{R}[\delta\phi^{*}(\mathcal{F})+\phi^{*}(\delta\mathcal{F})]$
(207)
$\displaystyle=\int_{R}\phi^{*}(\delta\mathcal{F}+\mathcal{L}_{\chi}\mathcal{F}).$

Taking the variation a second time yields

$\displaystyle 0=\delta^{2}\mathcal{A}$
$\displaystyle=\int_{R}[\delta\phi^{*}(\delta\mathcal{F}+\mathcal{L}_{\chi}\mathcal{F})+\phi^{*}(\delta(\mathcal{L}_{\chi}\mathcal{F}))]$
(208)
$\displaystyle=\int_{R}\phi^{*}[\mathcal{L}_{\chi}\delta\mathcal{F}+\mathcal{L}_{\chi}\mathcal{L}_{\chi}\mathcal{F}+\mathcal{L}_{\delta\chi}\mathcal{F}-\mathcal{L}_{\chi}\delta\mathcal{F}]$
$\displaystyle=\int_{R}\phi^{*}[(\mathcal{L}_{\chi}\mathcal{L}_{\chi}+\mathcal{L}_{\delta\chi})\mathcal{F}]$
$\displaystyle=\int_{R}\phi^{*}[(\mathcal{L}_{\frac{1}{2}[\chi,\chi]}+\mathcal{L}_{\delta\chi})\mathcal{F}].$

Note that the minus sign in front of the last term of the second line comes
from the fact that $\chi$ is a field space one-form. We thus have

$\delta\chi=-\frac{1}{2}[\chi,\chi].$ (209)

The second relation is the following

$\mathcal{L}_{\chi}\iota_{\chi}L=\frac{1}{2}\iota_{\quantity[\chi,\chi]}L+\frac{1}{2}\differential(\iota_{\chi}\iota_{\chi}L),$
(210)

which comes from the identity
$[\mathcal{L}_{\xi_{1}},i_{\xi_{2}}]=i_{[\xi_{1},\xi_{2}]}$:

$\displaystyle\mathcal{L}_{\chi}\iota_{\chi}L$
$\displaystyle=[\mathcal{L}_{\chi},\iota_{\chi}]L-\iota_{\chi}\mathcal{L}_{\chi}L$
(211)
$\displaystyle=\iota_{\quantity[\chi,\chi]}L-\iota_{\chi}\differential\iota_{\chi}L$
$\displaystyle=\iota_{\quantity[\chi,\chi]}L-\mathcal{L}_{\chi}(\iota_{\chi}L)+\differential(\iota_{\chi}\iota_{\chi}L).$

The third idendity that we will use is

$\displaystyle
I_{\hat{\xi}}\iota_{\delta\chi}L=-\frac{1}{2}I_{\hat{\xi}}\iota_{\quantity[\chi,\chi]}L=\iota_{\quantity[\xi,\chi]}L,$
(212)

where (193) and (209) were used.

## References

  * [1] L. Ciambelli, _From Asymptotic Symmetries to the Corner Proposal_ , _PoS_ Modave2022 (2023) 002 [2212.13644].
  * [2] E. Noether, _Invariant variation problems_ , _Transport Theory and Statistical Physics_ 1 (1971) 186.
  * [3] T. Regge and C. Teitelboim, _Role of surface integrals in the hamiltonian formulation of general relativity_ , _Annals of Physics_ 88 (1974) 286.
  * [4] R. Benguria, P. Cordero and C. Teitelboim, _Aspects of the hamiltonian dynamics of interacting gravitational gauge and higgs fields with applications to spherical symmetry_ , _Nuclear Physics B_ 122 (1977) 61.
  * [5] D. Harlow and J.-Q. Wu, _Covariant phase space with boundaries_ , _JHEP_ 10 (2020) 146 [1906.08616].
  * [6] M. Nakahara, _Geometry, topology and physics_ , CRC press (2003).
  * [7] G. Barnich and C. Troessaert, _BMS charge algebra_ , _JHEP_ 12 (2011) 105 [1106.0213].
  * [8] L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, _The Weyl BMS group and Einstein’s equations_ , _JHEP_ 07 (2021) 170 [2104.05793].
  * [9] L. Ciambelli and R.G. Leigh, _Isolated surfaces and symmetries of gravity_ , _Phys. Rev. D_ 104 (2021) 046005 [2104.07643].
  * [10] L. Ciambelli, R.G. Leigh and P.-C. Pai, _Embeddings and Integrable Charges for Extended Corner Symmetry_ , _Phys. Rev. Lett._ 128 (2022)  [2111.13181].
  * [11] L. Freidel, _A canonical bracket for open gravitational system_ , [2111.14747].
  * [12] A.J. Speranza, _Ambiguity resolution for integrable gravitational charges_ , _JHEP_ 07 (2022) 029 [2202.00133].
  * [13] F. Alessio, G. Barnich, L. Ciambelli, P. Mao and R. Ruzziconi, _Weyl charges in asymptotically locally AdS 3 spacetimes_, _Phys. Rev. D_ 103 (2021) 046003 [2010.15452].
  * [14] J.D. Brown and M. Henneaux, _Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity_ , _Commun. Math. Phys._ 104 (1986) 207.
  * [15] J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * [16] M. Henningson and K. Skenderis, _The Holographic Weyl anomaly_ , _JHEP_ 07 (1998) 023 [hep-th/9806087].
  * [17] H. Bondi, _Gravitational Waves in General Relativity_ , _Nature_ 186 (1960) 535.
  * [18] P.J. McCarthy, _Structure of the Bondi-Metzner-Sachs Group_ , _Journal of Mathematical Physics_ 13 (2003) 1837.
  * [19] G. Barnich and C. Troessaert, _Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited_ , _Phys. Rev. Lett._ 105 (2010) 111103 [0909.2617].
  * [20] A. Strominger and A. Zhiboedov, _Gravitational Memory, BMS Supertranslations and Soft Theorems_ , _JHEP_ 01 (2016) 086 [1411.5745].
  * [21] F. Alessio and G. Esposito, _On the structure and applications of the Bondi–Metzner–Sachs group_ , _Int. J. Geom. Meth. Mod. Phys._ 15 (2018) 1830002 [1709.05134].
  * [22] M. Campiglia and A. Laddha, _Asymptotic symmetries and subleading soft graviton theorem_ , _Phys. Rev. D_ 90 (2014) 124028 [1408.2228].
  * [23] G. Satishchandran and R.M. Wald, _Asymptotic behavior of massless fields and the memory effect_ , _Phys. Rev. D_ 99 (2019) 084007 [1901.05942].
  * [24] A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , [1703.05448].
  * [25] S. Weinberg, _Infrared photons and gravitons_ , _Phys. Rev._ 140 (1965) B516.
  * [26] T. He, V. Lysov, P. Mitra and A. Strominger, _BMS supertranslations and Weinberg’s soft graviton theorem_ , _JHEP_ 05 (2015) 151 [1401.7026].
  * [27] A. Ashtekar and M. Streubel, _Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity_ , _Proc. Roy. Soc. Lond. A_ 376 (1981) 585.
  * [28] D. Christodoulou and S. Klainerman, _The global nonlinear stability of the Minkowski space_ , _Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"_ (1989-1990) .
  * [29] A. Strominger, _On BMS Invariance of Gravitational Scattering_ , _JHEP_ 07 (2014) 152 [1312.2229].
  * [30] W. Donnelly and L. Freidel, _Local subsystems in gauge theory and gravity_ , _JHEP_ 09 (2016) 102 [1601.04744].
  * [31] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity. Part I. Corner potentials and charges_ , _JHEP_ 11 (2020) 026 [2006.12527].
  * [32] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity. Part II. Corner metric and Lorentz charges_ , _JHEP_ 11 (2020) 027 [2007.03563].
  * [33] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity. Part III. Corner simplicity constraints_ , _JHEP_ 01 (2021) 100 [2007.12635].
  * [34] L. Ciambelli and R.G. Leigh, _Universal corner symmetry and the orbit method for gravity_ , _Nucl. Phys. B_ 986 (2023) 116053 [2207.06441].
  * [35] L. Freidel, M. Geiller and W. Wieland, _Corner symmetry and quantum geometry_ , [2302.12799].