# Holography from Singular Supertranslations on a Black Hole Horizon

Ratindranath Akhoury<EMAIL_ADDRESS>Leinweber Center for Theoretical
Physics, Randall Laboratory of Physics, Department of Physics, University of
Michigan, Ann Arbor, MI 48109, USA.  Sangmin Choi
<EMAIL_ADDRESS>CPHT, CNRS, École Polytechnique, Institut
Polytechnique de Paris, F-91128 Palaiseau, France.  Malcolm J. Perry
<EMAIL_ADDRESS>School of Physics and Astronomy, Queen Mary University
of London, Mile End Road, London E1 4NS, UK.  DAMTP, Centre for Mathematical
Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 OWA, UK.
Trinity College, Cambridge, CB2 1TQ, UK.

###### Abstract

We investigate the standard and dual Bondi-Metzner-Sachs (BMS)
supertranslation generators on a black hole horizon and draw some conclusions
about black hole physics. Recently, it has been shown that in addition to
conventional BMS supertranslation symmetries, there exists an additional
infinite set of magnetic asymptotic symmetries, dual BMS supertranslations,
again parametrized by a function on the two-sphere. We show that the Dirac
bracket between these generators exhibits an anomalous central term when one
parameter function exhibits a singularity in the complex stereographical
coordinates on the sphere. In order to preserve general coordinate invariance,
we demonstrate that this central term can be removed by postulating a
holographic gravitational Chern-Simons theory on the horizon. This indicates
that for an anomaly-free theory of quantum gravity in the presence of a black
hole, one should include a boundary theory on the horizon.

CPHT-RR034.052022

##  I Introduction

The recent interest in asymptotic symmetries [1, 2] is motivated in large part
by the applications to black hole physics. The asymptotic symmetries of
particular interest in this Letter are the Bondi-Metzner-Sachs (BMS)
supertranslations and the dual supertranslations. While the supertranslations
are associated with a certain class of diffeomorphisms, parametrized by a
function on a two-sphere, no such association is known for the dual ones.
However, in refs. [3, 4], it was noted that the dual symmetries are contained
in a complexification of the usual supertranslation. The action of the
complexified charge on phase space was also investigated there, indicating
that this situation is completely analogous to what happens in
electromagnetism, [5]. This analogy with electromagnetism is further explored
in this Letter by considering the algebra of the dual and standard
supertranslation charges (magnetic and electric charges in electromagnetism)
for a parameter function with singularities in terms of complex
stereographical coordinates on the sphere. There are several physical
motivations for considering singular parameter functions. In electromagnetism,
Dirac-string-like configurations in the bulk correspond to singular gauge
transformations, and thus in gravity it is natural to expect that singular
supertranslations correspond to the gravitational analog of such
configurations. Moreover, singular supertranslations arise naturally in the
BMS algebra of supertranslation and superrotation charges [6]. While a proper
understanding of BMS superrotations is a subject of ongoing research,
considering singularities in the supertranslation parameter would be one of
the first steps in this direction. Singular gauge transformations in
electromagnetism were considered in [7, 8], and it was found that the algebra
of the electric and magnetic charges at null infinity contained a central
term. Since asymptotic symmetry charges map one physical configuration to
another, one expects the charge algebra to be closed. Thus, a central term
appearing in the algebra signals the existence of an anomaly.

In this Letter, we consider pure gravity without matter and show that an
analogous central term arises in the algebra of standard and dual
supertranslation charges at the future black hole horizon even if there are no
bulk Newman-Unti-Tamburino charges. We further find that this anomaly in the
charge algebra may be canceled by including a holographic Chern-Simons theory
on the horizon of the black hole. We give an outline discussion of the
electromagnetic case first and construct the Chern-Simons theory here as well,
so as to bring out the analogies and differences between the two theories. We
find the Chern-Simons theory for electromagnetism to have gauge group
$\mathrm{U}(1)\otimes\mathrm{U}(1)$, in agreement with a similar result on
future null infinity in [8]. For the case of gravitation, we find the Chern-
Simons theory to have gauge group $\mathrm{SL}(2,\mathbb{C})$ on the horizon.
The implications of this for a quantum theory of gravity, in particular, the
relevance of the states of the Chern-Simons theory to black hole physics, is
beyond the scope of this Letter but will be treated elsewhere.

In section II, we review the analog of both the standard and dual
supertranslations on the black hole horizon in Bondi gauge. In section III, we
introduce the charges associated with the diffeomorphism symmetries. In
parallel, we also discuss the dual, magnetic, counterpart of the
diffeomorphism symmetries. In section IV, we continue the discussion of
charges and allow for the possibility that there could be singularities in the
supertranslations. We examine in detail the case of the supertranslation
generator having a simple pole when expressed in the usual complex coordinates
on the two-sphere of the horizon and show that the algebra of electric
(standard) and magnetic (dual) supertranslation charges is anomalous. The
central charge is explicitly calculated here. In section V, we examine
electromagnetic soft hair on the black hole horizon and show that in this
case, also, there is an anomaly in the charge algebra when one has both
electric and magnetic transformations. We show that this anomaly can be
canceled by supposing that the horizon has a Chern-Simons theory living on it.
In section VI, we repeat this analysis for the gravitational case and find
that the Chern-Simons theory that does the job here was formulated earlier in
[9]. We conclude with a brief discussion of our results.

##  II BMS transformations

We begin by reviewing the notion of supertranslations on the future horizon of
the Schwarzschild black hole. The Schwarzschild metric in terms of the
advanced Eddington-Finkelstein coordinate $v$, and the stereographic
coordinates $z^{A}$ ($A=1,2$) which parametrize a unit 2-sphere with metric
$\gamma_{AB}$ is given by,

$\displaystyle
ds^{2}=g_{ab}dx^{a}dx^{b}=-\left(1-\frac{2M}{r}\right)dv^{2}+2dvdr+r^{2}\gamma_{AB}dz^{A}dz^{B}.$
(1)

We will work in the Bondi gauge, where $g_{rr}=0$, $g_{rA}=0$,
${\partial}_{r}\det\frac{g_{AB}}{r^{2}}=0$ where $g_{AB}$ is the metric on the
two-sphere. The diffeomorphisms that respect the Bondi gauge conditions are
generated by the vector field [1, 2],

$\displaystyle\xi^{a}{\partial}_{a}=X{\partial}_{v}-\frac{1}{2}\left(rD_{A}X^{A}+D^{2}X\right){\partial}_{r}+\left(X^{A}+\frac{1}{r}D^{A}X\right){\partial}_{A},$
(2)

where, $D_{A}$ is the covariant derivative with respect to $\gamma_{AB}$ and
the $A,B,\ldots$ indices are raised and lowered using the metric on the unit
two-sphere $\gamma_{AB}$. Supertranslations are generated by $X=f$ and
$X^{A}=0$, while superrotations are generated by $X=\frac{u}{2}D_{A}Y^{A}$,
$X^{A}=Y^{A}$. Here $f$ parametrizes the diffeomorphism on a two-sphere
(either at null infinity or the horizon). In this Letter, we will be
interested in the charge associated with supertranslations and the related
dual supertranslations.

The dual gravitational charges are given by twisted fields [4, 10] defined
through the Levi-Civita tensor $\epsilon_{AB}$ on the two-sphere. Unlike the
supertranslation charge, they do not generate diffeomorphisms and may be
derived, as recently shown in [11, 12], by introducing the topological Holst
term to the Einstein action. The most convenient way to view the dual
symmetries is through a complexification of supertranslation charge [4]. It
was shown in this last reference that the complexified charge acts on the
phase space in a manner consistent with what one would expect for the
analogous case of electric and magnetic charges in electromagnetism [5].
Specifically, the complex charge on future null infinity is given by [4]

$\displaystyle\textbf{Q}^{+}_{f}=-\frac{1}{16\pi}\int_{S}d\Omega(-4fm_{B}+fD^{2}_{z}C^{zz}+D_{{\bar{z}}}fD_{{\bar{z}}}C^{{\bar{z}}{\bar{z}}}),$
(3)

where, $S$ is the two-sphere at future null infinity, $rC_{AB}=\delta g_{AB}$,
$\delta g_{AB}$ is the variation of the metric $g_{AB}$ and $m_{B}$ the Bondi
mass aspect. They further find that its action on phase space may be
represented by,

$\displaystyle\\{\textbf{Q}^{+}_{f},C_{zz}(u,z,{\bar{z}})\\}$
$\displaystyle=f\partial_{u}C_{zz}$ (4)
$\displaystyle\\{\textbf{Q}^{+}_{f},C_{{\bar{z}}{\bar{z}}}(u,z,{\bar{z}})\\}$
$\displaystyle=f\partial_{u}C_{{\bar{z}}{\bar{z}}}-2D^{2}_{{\bar{z}}}f.$ (5)

Thus, $\textbf{Q}^{+}_{f}$ generates time translation on the mode, $C_{zz}$
and a supertranslation on the other mode $C_{{\bar{z}}{\bar{z}}}$ in complete
analogy with the electromagnetic case discussed in [5].

##  III Horizon charges

Let us now construct charges that generate the standard and dual BMS
supertranslations on the horizon. Consider a spacelike hypersurface $\Sigma$
that extends from a section of $\mathcal{I}^{+}$ to a section of
$\mathcal{H}^{+}$. A charge $Q_{f}^{\Sigma}$ associated to $\Sigma$ then
splits into two parts,

$\displaystyle
Q_{f}^{\Sigma}=Q_{f}^{{\mathcal{H}^{+}}}+Q_{f}^{\mathcal{I}^{+}},$ (6)

where $Q_{f}^{{\mathcal{H}^{+}}}$ is defined on a section on the future
horizon $\mathcal{H}^{+}$ and $Q_{f}^{\mathcal{I}^{+}}$ is defined on a
section on the future null infinity $\mathcal{I}^{+}$. We refer to the horizon
contribution $Q_{f}^{\mathcal{H}^{+}}$ as the horizon charge.

The horizon charge can be written as an integral on ${\mathcal{H}^{+}}$ so
long as the contribution from the future boundary $\mathcal{H}^{+}_{+}$
vanishes. In this Letter, we take the viewpoint that the black hole ultimately
evaporates, and hence this contribution is negligible. If a horizon has a
future end-point, in classical general relativity this point must be singular.
Following common practice, we assume that this is not an issue and that the
quantum theory will ultimately resolve this problem. The horizon charge can
therefore be written either as a 3-dimensional integral of a total
$v$-derivative over ${\mathcal{H}^{+}}$, or a 2-dimensional integral over
$\mathcal{H}^{+}_{-}$

To each vector field $\xi$ that does not vanish on $\Sigma$, there exists a
diffeomorphism (electric) horizon charge $Q_{f}^{\mathcal{H}^{+}}$ associated
to the diffeomorphism generated by $\xi$, as well as a dual (magnetic) horizon
charge $\widetilde{Q}_{f}^{\mathcal{H}^{+}}$ that originates from the Holst
action [11]. Variations of such charges can be computed using the covariant
phase space formalism in the first-order formulation of gravity [11, 12].
Taking the vector field $\xi$ to be the supertranslation generator (2), we
obtain the standard and dual supertranslation horizon charges,

$\displaystyle\not{\delta}Q^{\mathcal{H}^{+}}_{f}$
$\displaystyle=\frac{1}{16\pi
M}\int_{\mathcal{H}^{+}}dv\,d^{2}\Theta\sqrt{\gamma}f(\Theta)D^{A}D^{B}\sigma_{AB},$
(7) $\displaystyle\not{\delta}\widetilde{Q}^{\mathcal{H}^{+}}_{f}$
$\displaystyle=\frac{-1}{32\pi
M}\int_{{\partial}{\mathcal{H}^{+}}}d^{2}\Theta\sqrt{\gamma}(D^{B}f)\epsilon_{A}{}^{C}D^{A}h_{BC},$
(8)

where $h_{AB}=\delta g_{AB}$ is the variation of the metric, and
$\sigma_{AB}=\frac{1}{2}{\partial}_{v}h_{AB}$ is its conjugate momentum. It is
possible that these variations are non-integrable, and we emphasize this point
with the notation $\not{\delta}$, in contrast to integrable variations denoted
by $\delta$.

##  IV Charge algebra

In this section, we demonstrate that the Dirac bracket algebra between the
standard and dual supertranslation charges on the horizon exhibits a central
term when the parameter function exhibits a singularity. To demonstrate this,
let us consider a supertranslation with a pole $f(z,{\bar{z}})=\frac{1}{z-w}$
and a dual supertranslation with a smooth function $f^{\prime}$.111We work
with a simple pole for explicit computation, but a similar line of argument
can be carried out for other functions with singularities, such as logarithms.
Then, the dual charge for smooth $f^{\prime}$ becomes integrable
${\not{\delta}{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}={\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}$,
but the supertranslation charge acquires a non-integrable piece,

$\displaystyle{\not{\delta}{Q_{f=\frac{1}{z-w}}^{\mathcal{H}^{+}}}}$
$\displaystyle={\delta{Q_{f=\frac{1}{z-w}}^{\mathcal{H}^{+}}}}-\frac{1}{4M}\int_{-\infty}^{\infty}dvD_{z}[D^{2}-1]^{-1}D^{B}D^{A}\sigma_{AB}\bigg{|}_{z=w}.$
(9)

It is straightforward to check that the non-integrable piece has vanishing
Dirac brackets with all charges, and therefore can be ignored for our
computation of brackets. Natural definitions of the integrable variations are
given by

$\displaystyle{\delta{Q_{f}^{\mathcal{H}^{+}}}}$
$\displaystyle\equiv\frac{1}{16\pi
M}\int_{\mathcal{H}^{+}}dv\,d^{2}\Theta\sqrt{\gamma}\,(D^{B}D^{A}f)\sigma_{AB},$
(10) $\displaystyle{\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}$
$\displaystyle\equiv\frac{-1}{32\pi
M}\int_{\mathcal{H}^{+}_{-}}d^{2}\Theta\sqrt{\gamma}\,(D^{B}D^{A}f^{\prime})\epsilon_{A}{}^{C}h_{BC}.$
(11)

It is worth noting that ${\delta{\widetilde{Q}_{f}^{\mathcal{H}^{+}}}}$ is
related to ${\delta{Q_{f}^{\mathcal{H}^{+}}}}$ by the twisting
$h_{AB}\to\epsilon_{A}{}^{C}h_{CB}$ [13, 4, 10]. These variations are first-
order perturbations around the Schwarzschild background. Thus, one can imagine
that a full (integrated) charges $Q$ has the expansion

$\displaystyle Q$ $\displaystyle=Q_{0}+\delta Q+O(h^{2}).$ (12)

Here $Q_{0}$ is the charge evaluated on the Schwarzschild metric, which is a
constant and hence does not carry degrees of freedom. This implies that at
leading order in $h$, the Dirac bracket between the full charges
${Q_{f}^{\mathcal{H}^{+}}}$ and
${\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}$ is

$\displaystyle\\{{Q_{f}^{\mathcal{H}^{+}}},{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}\\}$
$\displaystyle=\\{{\delta{Q_{f}^{\mathcal{H}^{+}}}},{\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}\\}+O(h).$
(13)

Since both ${\delta{Q_{f}^{\mathcal{H}^{+}}}}$ and
${\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}$ are linear in
$h_{AB}$ and $\sigma_{AB}$ respectively, their Dirac bracket is a constant.
This implies that the bracket
$\\{{\delta{Q_{f}^{\mathcal{H}^{+}}}},{\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}\\}$
contains information about the central term of the full charge algebra. By
straightforward computation, one finds that

$\displaystyle\left\\{{\delta{Q_{f=\frac{1}{z-w}}^{\mathcal{H}^{+}}}},{\delta{\widetilde{Q}_{f^{\prime}}^{\mathcal{H}^{+}}}}\right\\}$
$\displaystyle=\frac{-i}{4}(D^{z}D_{z}^{2}f^{\prime})\bigg{|}_{z=w}.$ (14)

One observes the appearance of a central term in the algebra.

##  V Electromagnetism

In the previous section, we have shown that the presence of a singularity in
the supertranslation parameter function leads to a central term in the charge
algebra. In this section, we review similar results for electromagnetism, and
show that such central term can be removed by adding a Chern-Simons theory on
the horizon. We refer the reader to [7, 8] for related discussions on
$\mathcal{I}^{+}$.

In electromagnetism, the asymptotic symmetries are the electric and magnetic
large gauge transformations. The soft horizon charges are given by the
expressions

$\displaystyle{\mathcal{Q}_{\psi}^{\mathcal{H}^{+}}}$
$\displaystyle=\int_{{\mathcal{H}^{+}}}d\psi\wedge*F,\qquad{\widetilde{\mathcal{Q}}_{\sigma}^{\mathcal{H}^{+}}}=\int_{{\mathcal{H}^{+}}}d\sigma\wedge
F,$ (15)

where $\psi=\psi(z,{\bar{z}})$ and $\sigma=\sigma(z,{\bar{z}})$ are functions
on the sphere. It is straightforward to see that they satisfy the algebra

$\displaystyle\\{{\mathcal{Q}_{\psi}^{\mathcal{H}^{+}}},{\widetilde{\mathcal{Q}}_{\sigma}^{\mathcal{H}^{+}}}\\}$
$\displaystyle=\int_{{\partial}{\mathcal{H}^{+}}}d\psi\wedge d\sigma,$ (16)
$\displaystyle\\{{\mathcal{Q}_{\psi}^{\mathcal{H}^{+}}},{\mathcal{Q}_{\sigma}^{\mathcal{H}^{+}}}\\}$
$\displaystyle=0,\quad\\{{\widetilde{\mathcal{Q}}_{\psi}^{\mathcal{H}^{+}}},{\widetilde{\mathcal{Q}}_{\sigma}^{\mathcal{H}^{+}}}\\}=0.$
(17)

One observes that singularities in $\psi$ (or $\sigma$) lead to central terms
in the algebra.

Central terms in a symmetry algebra imply the existence of anomalies. One way
to remove such a term is to add a boundary theory on ${\mathcal{H}^{+}}$ that
has an asymptotic charge algebra with a central term canceling it. For this
purpose, let us consider a $\mathrm{U}(1)\otimes\mathrm{U}(1)$ Chern-Simons
theory on ${\mathcal{H}^{+}}$,

$\displaystyle S=\int_{{\mathcal{H}^{+}}}a\wedge d\widetilde{a},$ (18)

where $a$ and $\widetilde{a}$ are the electric and magnetic $\mathrm{U}(1)$
gauge fields. Under an electric large gauge transformation (LGT), $\delta
a=d\phi$ and $\delta\widetilde{a}=0$, and under a magnetic one $\delta a=0$
and $\delta\widetilde{a}=d\widetilde{\phi}$. They are generated by the charges

$\displaystyle\delta\mathcal{Q}_{\phi}$
$\displaystyle=-\int_{{\partial}{\mathcal{H}^{+}}}d\phi\wedge\delta\widetilde{a},$
(19) $\displaystyle\delta\widetilde{\mathcal{Q}}_{\phi}$
$\displaystyle=\int_{{\partial}{\mathcal{H}^{+}}}\delta a\wedge d\phi,$ (20)

respectively. The charge algebra is

$\displaystyle\\{\mathcal{Q}_{\phi},\widetilde{\mathcal{Q}}_{\varphi}\\}$
$\displaystyle=-\int_{{\partial}{\mathcal{H}^{+}}}d\phi\wedge d\varphi,$ (21)
$\displaystyle\qquad\\{\mathcal{Q}_{\phi},\mathcal{Q}_{\varphi}\\}$
$\displaystyle=\\{\widetilde{\mathcal{Q}}_{\phi},\widetilde{\mathcal{Q}}_{\varphi}\\}=0.$
(22)

Therefore, one finds the algebra to be exactly parallel to that of standard
and dual LGT charges on the horizon. This tells us that putting a
$\mathrm{U}(1)\otimes\mathrm{U}(1)$ Chern-Simons theory on the horizon, we can
get rid of the central term in the standard and dual LGT algebra.

##  VI Gravitational Chern-Simons theory

In this section, we will follow Witten’s approach to gravity in three
dimensions [9] and consider a gravitational Chern-Simons theory on the
horizon. The algebra of the charges in this theory is calculated and the
parameters chosen such that the central term here cancels the one found in
section IV. In three dimensions, we use $i,j,\ldots$ for spacetime indices and
$a,b,\ldots$ for tangent space. The action is given in the canonical form by,

$I_{CS}=\frac{k}{4\pi}\int\operatorname{tr}\ \bigl{(}A\wedge
dA+\tfrac{2}{3}A\wedge A\wedge A\bigr{)}.$ (23)

Equations of motion fix the gauge group to be ${\bf
G}=\mathrm{SL}(2,\mathbb{C})$. The gauge field decomposes as
$A_{i}=e^{a}_{i}P_{a}+{\omega}^{a}_{i}J_{a}$, where the generators $P_{a}$ and
$J_{a}$ satisfy the commutators

$\displaystyle[J_{a},J_{b}]=\epsilon_{abc}J^{c},\quad[J_{a},P_{b}]=\epsilon_{abc}P^{c},\quad[P_{a},P_{b}]=\lambda\epsilon_{abc}J^{c},$
(24)

where the Cartan metric $\eta_{ab}=\operatorname{diag}(-++)$ is used to lower
and raise tangent space indices, and $\lambda=-\frac{1}{4M^{2}}$ is a negative
constant. As the names suggest, Witten found that this Chern-Simons theory is
equivalent to a first-order Einstein theory after identifying $e^{a}_{i}$ and
${\omega}^{a}_{i}$ as the vielbein and the spin connection.

There exist two Killing forms and therefore one can write two actions in terms
of $e^{a}$ and ${\omega}^{a}$. The first Killing form is $\langle
J_{a},P_{b}\rangle=\eta_{ab}$, with all other components vanishing. It leads
to the electric action

$I_{\text{electric}}=\frac{k}{2\pi}\int_{{\mathcal{H}^{+}}}2e^{a}\wedge
d\omega_{a}+\epsilon_{abc}e^{a}\wedge\omega^{a}\wedge\omega^{c}+\,\tfrac{1}{3}\lambda\epsilon_{abc}e^{a}\wedge
e^{b}\wedge e^{c}.$ (25)

The second Killing form has non-vanishing components $\langle
J_{a},J_{b}\rangle=\eta_{ab}$, $\langle P_{a},P_{b}\rangle=\lambda\eta_{ab}$,
and it leads to the magnetic action,

$I_{\text{magnetic}}=\frac{\widetilde{k}}{\pi}\int_{\mathcal{H}^{+}}\omega^{a}\wedge
d\omega_{a}+\tfrac{1}{3}\epsilon_{abc}\omega^{a}\wedge\omega^{b}\wedge\omega^{c}+\lambda
e^{a}\wedge de_{a}+\lambda\epsilon_{abc}\omega^{a}\wedge e^{b}\wedge e^{c}.$
(26)

$\widetilde{k}$ is arbitrary here but we will see later that it can be
expressed in terms of $k$ by demanding that the complexified charges form a
closed algebra in the absence of singularities in the function parametrizing
the diffeomorphisms. Drawing analogy to four-dimensional gravity, the electric
action corresponds to the Einstein-Hilbert action whereas the magnetic action
corresponds to the Holst action [11]. As in four dimensions, we take the
electric action to be our gravitational Chern-Simons theory, and use the
magnetic action only to derive the dual charge.

There are two types of gauge transformations, one labeled by a tangent-space
vector $\rho^{a}$ and the other by a second vector $\tau^{a}$. The fields
transform as $\delta
e_{i}^{a}=-\partial_{i}\rho^{a}-\epsilon^{abc}\omega_{ib}\rho_{c}$ and
$\delta\omega_{i}^{a}=-\lambda\epsilon^{abc}e_{i\,b}\rho_{c}$ under $\rho$,
and $\delta e_{i}^{a}=-\epsilon^{abc}e_{ib}\tau_{c}$ and
$\delta\omega_{i}^{a}=-\partial_{i}\tau^{a}-\epsilon^{abc}\omega_{ib}\tau_{c}$
under $\tau$. The $\tau$-transformations correspond to Lorentz transformations
on the tangent space, whereas the $\rho$-transformations correspond to
diffeomorphisms $\rho^{a}=\imath_{v}e^{a}$ generated by the spacetime vector
field $v$ up to a compensating Lorentz transformation.

Using the covariant phase space formalism, one finds the electric and magnetic
charges associated with the gauge transformations to be [14]

$\displaystyle Q^{E}_{\rho,\tau}$
$\displaystyle=-\frac{k}{\pi}\int_{\partial\Sigma}\
\tau_{a}e^{a}+\rho_{a}\omega^{a},$ (27) $\displaystyle Q^{M}_{\rho,\tau}$
$\displaystyle=-\frac{2\widetilde{k}}{\pi}\int_{\partial\Sigma}\tau_{a}\omega^{a}+\lambda\rho_{a}e^{a}.$
(28)

The domain of integration $\partial\Sigma$ is the collection of closed loops
around the singularities.222 This implies that the charges vanish for smooth
parameter functions. However, we are interested in the variation of these
charges under the action of another charge with singularities, which may not
be zero.  Having computed the magnetic charge, we dispose of the magnetic
action and work solely with the electric action. The symplectic structure of
the electric theory dictates that the only non-vanishing Dirac bracket of the
theory is

$\displaystyle\\{e^{a}_{z}(z,{\bar{z}}),w^{b}_{\bar{z}}(z^{\prime},{\bar{z}}^{\prime})\\}=-\frac{i\pi}{k}\eta^{ab}\delta^{2}(z-z^{\prime}),$
(29)

where $z,{\bar{z}}$ are stereographic coordinates of the unit two-sphere.
Using the brackets, one finds the following algebra of electric and magnetic
charges [14],

$\displaystyle\\{Q^{E}_{\tau,\rho},Q^{E}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=Q^{E}_{\tau^{\prime\prime},\rho^{\prime\prime}}-\frac{k}{\pi}\int_{{\partial}\Sigma}\left(\rho^{a}d\tau^{\prime}_{a}+\tau^{a}d\rho^{\prime}_{a}\right),$
(30)
$\displaystyle\\{Q^{E}_{\tau,\rho},Q^{M}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=Q^{M}_{\tau^{\prime\prime},\rho^{\prime\prime}}-\frac{2\widetilde{k}}{\pi}\int_{{\partial}\Sigma}\left(\tau_{a}d\tau^{\prime
a}+\lambda\rho_{a}d\rho^{\prime a}\right),$ (31)
$\displaystyle\\{Q^{M}_{\tau,\rho},Q^{M}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=4\lambda\frac{\widetilde{k}^{2}}{k^{2}}Q^{E}_{\tau^{\prime\prime},\rho^{\prime\prime}}-\frac{4\lambda\widetilde{k}^{2}}{\pi
k}\int_{{\partial}\Sigma}\left(\rho^{a}d\tau^{\prime}_{a}+\tau^{a}d\rho^{\prime}_{a}\right).$
(32)

The composition is given by $\tau^{\prime\prime
a}=\epsilon^{abc}(\tau^{\prime}_{b}\tau_{c}+\lambda\rho^{\prime}_{b}\rho_{c})$
and $\rho^{\prime\prime
a}=\epsilon^{abc}(\tau^{\prime}_{b}\rho_{c}-\tau_{b}\rho^{\prime}_{c})$. We
demand that the central term of this algebra cancels the central term observed
in the supertranslation algebra on the Schwarzschild horizon which we computed
for $f=\frac{1}{z-w}$. Since supertranslation is a diffeomorphism, it acts as
a gauge transformation $\rho^{a}=\iota_{v}e^{a}$ on the horizon Chern-Simons
theory with $v=f{\partial}_{v}+\frac{1}{2M}D^{A}f{\partial}_{A}$, up to a
Lorentz transformation. We define the compensating Lorentz transformation of
supertranslation to be

$\displaystyle\tau^{0}=\frac{(D^{2}+2)f}{8(2\widetilde{k})^{1/2}},\quad\tau^{1}=i\sqrt{\lambda}\rho^{2},\quad\tau^{2}=-i\sqrt{\lambda}\rho^{1}.$
(33)

Note that all components are real since $\lambda$ is negative. Then algebra of
charges with $f=\frac{1}{z-w}$, becomes

$\displaystyle\\{Q^{E}_{\tau,\rho},Q^{E}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=Q^{E}_{\tau^{\prime\prime},\rho^{\prime\prime}},$ (34)
$\displaystyle\\{Q^{E}_{\tau,\rho},Q^{M}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=Q^{M}_{\tau^{\prime\prime},\rho^{\prime\prime}}+\left.\frac{i}{4}(D^{z}D_{z}^{2}f^{\prime})\right|_{z=w},$
(35)
$\displaystyle\\{Q^{M}_{\tau,\rho},Q^{M}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=4\lambda\frac{\widetilde{k}^{2}}{k^{2}}Q^{E}_{\tau^{\prime\prime},\rho^{\prime\prime}}.$
(36)

The central term obtained here is exactly of the form to cancel the one
obtained in section IV. Note that $f^{\prime}$ in the above is an arbitrary
smooth function. It corresponds to the function $g$ introduced in section IV.

One can find the relation between $k$ and $\widetilde{k}$ by considering the
complexified charge
$\textbf{Q}_{\tau,\rho}=Q^{E}_{\tau,\rho}+iQ^{M}_{\tau,\rho}$. Demanding that
the charges $\textbf{Q}_{\tau,\rho}$ form a closed algebra in the absence of
singularities in the parameters fixes $\widetilde{k}$ in terms of $k$ to be
$\widetilde{k}^{2}=-\frac{k^{2}}{4\lambda}=k^{2}M^{2}$. The algebra of
complexified charge with a pole $f=\frac{1}{z-w}$ reads

$\displaystyle\\{\textbf{Q}_{\tau,\rho},\textbf{Q}_{\tau^{\prime},\rho^{\prime}}\\}$
$\displaystyle=\textbf{Q}_{2\tau^{\prime\prime},2\rho^{\prime\prime}}+\left.\frac{1}{2}(D^{z}D_{z}^{2}f^{\prime})\right|_{z=w}.$
(37)

##  VII Discussions

We have explored the consequences of allowing the function parametrizing
standard and dual supertranslations on the black hole horizon to have
singularities. The chief outcome is that the algebra of charges develops a
central term. What is novel is that we then showed that this anomaly can be
cancelled by postulating a holographic Chern-Simons theory on the horizon. It
was noted that both the gravitational case and the electromagnetic one are
analogous in this regard. It is fortunate that the Chern-Simons is a
topological theory as it is metric independent. There are two nice properties
that follow. The first is that since the horizon is a null surface, the metric
is degenerate there. One cannot invert the metric. Had the theory been metric
dependent, as most are, it would have been impossible to formulate a theory
that is restricted to the null surface. The second also follows from being
metric independent. The energy-momentum tensor of a theory is given by varying
the action with respect to the metric. Therefore, in the Chern-Simons case,
the energy-momentum tensor vanishes and the holographic theory does not
disturb the black hole geometry.

While we have shown that an $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory
cancels the central term, we have not shown that this is the unique theory
capable of this cancellation. There may exist other (arguably more obscure)
topological field theories that could accomplish this. If they exist, then it
would be very interesting to see explicit examples of such theories, as they
would share properties that are inherent to the structure of the black hole
horizon.

It is worth noting that, in the complexified Chern-Simons theory which
incorporates both the electric and magnetic actions, the level $\widetilde{k}$
of the magnetic action becomes quantized [15, 16]. For us this is not
relevant, since our Chern-Simons theory is just that of the electric action;
the magnetic action is only used to compute the dual charge.

The addition of a holographic Chern-Simons theory on the horizon in
conjunction with the soft hair makes the structure at the horizon much more
complex. The implications for the nature of black hole microstates will be
explored in a future publication.

###### Acknowledgements.

MJP acknowledges funding from the Science and Technology Facilities Council
(STFC) Consolidated Grant ST/T000686/1 “Amplitudes, Strings and duality”. MJP
would also like to thank the UK STFC for financial support under grant
ST/L000415/1. No new data were generated or analysed during this study. The
work of SC is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 852386). SC also acknowledges financial support from the Samsung
Scholarship.

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