aainstitutetext: DAMTP, Centre for Mathematical Sciences, University of
Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.

# Magnetic charges in Supergravity

Bilyana L. Tomova<EMAIL_ADDRESS>
###### Abstract

In this paper we study the dual charges of $\mathcal{N}=1$ supergravity in
asymptotically flat space-time. The action considered is the usual
supergravity action with a topological contribution. This is the Nieh-Yan term
and the magnetic term of the free Rarita-Schwinger field. Through methods of
the covariant phase space formalism we construct the charges conjugate to
supersymmetry, diffeomorphism and Lorentz transformations. The additional term
in the action will lead to new, non-vanishing contributions to these charges.
The magnetic diffeomorphism charges are equivalent to the ones previously
found for gravity, while the dual supersymmetric charges are new and do not
appear for the free Rarita-Schwinger field. The dual Lorentz charges serve to
regularize the previous two. We find that the asymptotic symmetry group for
supergravity can only include globally well-defined super-rotations.

##  1 Introduction

The usual Maxwell equations that govern the behavior of the electric and
magnetic fields, seemingly forbid the existence of magnetic monopoles. This
conclusion was first challenged by Dirac. His insight was that by choosing a
special couple of gauge potentials, that are not everywhere continuous, and
removing the origin of the coordinates, one can construct a magnetic field
with a monopole.

Later, in the study of Yang-Mills theory it was shown that one can add an
additional gauge and Lorentz invariant term to the action, with a coupling
constant $\theta$ (hence the name theta term). In special cases where the
coupling constant varies with the background, one can have magnetic field,
seemingly caused by monopole. This happens, for example, when one has
electromagnetic radiation propagating inside of a topological insulator
($\theta=\pi$) and outside ($\theta=0$).

There are a lot of parallels between Yang-Mills theory and the Einstein theory
of gravity. The study of dual charges is no exception. In the fifties and
sixties, a very peculiar solution to the vacuum Einstein equations was
discovered - the Taub-NUT metric 12 , 13 . This solution is characterized by
the NUT parameter, which can be viewed as a gravitomagnetic monopole 25  \-
the dual charge of the gravitational mass. In recent years it has been shown
27 , that the presence of the NUT charge requires that the subleading (with
respect to the radial coordinate) components of the angular metric (written in
the Bondi gauge 1 ), is not continuous on the sphere. What is more, this part
of the metric transforms as a gauge field under the action of the asymptotic
symmetry group 14 . If one chooses this field to be the gauge potential of the
Dirac monopole, one obtains the the Taub-NUT metric 14 .

On the other hand, the study of asymptotic symmetries of asymptotically flat
space-times lead to the discovery of an infinite dimensional group of
asymptotic charges 20 . They represent different modes of the mass and angular
momentum flux. This begs the question whether an infinite number of
gravitational magnetic charges exist. Indeed these charges were found 14 , 15
. Subsequently, their existence was better understood in the Hamiltonian
formalism of gravity 16 . Adding an additional term to the Einstein-Palatini
action, one can derive these charges via the covariant phase space formalism.
This additional term is the Holst term. It is similar to the theta term in
Yang-Mills theory and does not change the equations of motion. Indeed one can
derive the magnetic Yang-Mills charges in a similar fashion 39 . If one
couples gravity to fermions, the Holst term needs to be replaced by the Nieh-
Yan term 16 .

In this paper we extend these efforts to supergravity. The topological term in
the action will have two contributions - the Nieh-Yan term and the magnetic
term from the free Rarita-Schwinger field 24 . We choose the metric to be
asymptotically flat and we write it in Bondi coordinates. The usual asymptotic
symmetry group on this metric is the BMS. Here it would be enlarged to in two
ways. Firstly, because of supersymmerty it will be extended to the super-BMS
group 26 . Furthermore, it will have magnetic charges as well as the usual
super-BMS charges.

##  2 Symplectic structure

In order to study the normal and dual Hamiltonian charges for supergravity we
first need to set the tools of this analysis. Therefore, in this section, we
briefly show how to construct the phase space for a gauge theory. The phase
space for a field is an infinite-dimensional symplectic manifold, where every
point is a specific field configuration. These configurations are constrained
by the equations of motion and the boundary conditions. The conserved charges
of the theory are functions on phase space. Mathematically they are are
expressed as functionals of the fields. Their commutator relationship and
infinitesimal variations are defined via the symplectic form. The steps for
defining this symplectic form are straightforward 3 , 17 .

First we vary the Lagrangian,

$\displaystyle\delta L(\phi,\delta\phi)\approx d\Theta(\phi,\delta\phi)$ (1)

Here and throughout this paper the sign $\approx$ means equality on shell. The
boundary term $\Theta$ is the pre-symplectic potential. The pre-symplectic
density and form are defined respectively as,

$\displaystyle
w(\phi,\delta_{1}\phi,\delta_{2}\phi)=\delta_{1}\Theta(\phi,\delta_{2}\phi)-\delta_{2}\Theta(\phi,\delta_{1}\phi)$
(2)
$\displaystyle\tilde{\Omega}(\phi,\delta_{1}\phi,\delta_{2}\phi)=\int_{\Sigma}w(\phi,\delta_{1}\phi,\delta_{2}\phi)$
(3)

where $\Sigma$ is a Cauchy surface in the space-time manifold. The pre-
symplectic form will be in general degenerate on the phase space. Its
degenerate directions correspond to transformations, that are not physical,
but are rather redundant descriptions of the same field configuration. If we
quotient the phase space by the degenerate directions, we get the pre-
symplectic form $\Omega$. The infinitesimal variation of a Hamiltonian charge,
conjugate to a field variation $\delta_{\chi}\phi$ is

$\displaystyle\cancel{\delta}H_{\chi}[\phi]=\int_{\Sigma}w(\phi,\delta_{\chi}\phi,\delta\phi)$
(4)

For gauge transformations this can be re-written as a boundary term 17 . This
variation is not in general an exact one form on phase space. This can be due
to two to reasons- the transformation is not canonical, or there is a charge
flux through the boundary of the space-time region. In the second case, one
can impose the condition that the variation of the field on the boundary is
zero and thus make the Hamiltonian charge integrable 3 .

The commutator of two integrable charges is defined as 17 ,

$\displaystyle[H_{\chi},H_{\xi}][\phi]=\delta_{\xi}H_{\chi}[\phi]=\int_{\Sigma}w(\phi,\delta_{\chi}\phi,\delta_{\xi}\phi)=H_{[\xi,\chi]}+K_{\xi,\chi}[\bar{\phi}]$
(5) $\displaystyle
K_{\xi,\chi}=\int_{\Sigma}w(\bar{\phi},\delta_{\chi}\bar{\phi},\delta_{\xi}\bar{\phi})+N_{[\xi,\chi]}[\bar{\phi}]$
(6)

Where $K$ is the central extension and $\bar{\phi}$ is a reference field
configuration, which is the starting point of integration of the charges on
phase space.The term $N$ is a reference charge.

##  3 The action

The action of gravity with magnetic term, coupled to fermions has been derived
in 16 . It is the usual Einstein-Palatini action plus the Nieh-Yan term. The
magnetic charges of the free massless Rarita-Schwinger field were studied in
24 . Here, we will combine these efforts and study the magnetic charges of the
$\mathcal{N}=1$ supergravity theory.

We work in the 1.5 order formalism 22 . Initially, we consider the that the
frame field, spin connection and gravitino field are all independent
variables. By varying the action with respect to the spin connection, we can
determine its expression in terms of the two other fields. The presence of
spinor field, gives rise to a non-vanishing torsion. Once the expression for
the spin connection is determined, we put it back in the action and proceed as
if the only independent variables are the frame and gravitino field. For more
details of this formalism see 22 .

The usual action for $\mathcal{N}=1$ supergravity in the Einstein-Palatini
formalism is the following,

$\displaystyle
S_{SG}=\frac{1}{2\kappa^{2}}\int_{\mathcal{M}}\,\varepsilon_{abcd}\,e^{a}\wedge
e^{b}\wedge R^{cd}-\bar{\psi}\wedge\gamma_{5}\gamma\wedge\nabla\psi$ (7)

Here and throughout this paper, Greek letters $\mu,\nu$ are space-time
indices, and Latin letters $a,b$ are tangent space indices.

The topological term, that will give raise to magnetic charges is,

$\displaystyle S_{\ast SG}=-\frac{i\lambda}{2\kappa^{2}}\int_{\mathcal{M}}\
e_{a}\wedge e_{b}\wedge R^{ab}-T^{a}\wedge
T_{a}-d\left(\bar{\psi}\wedge\gamma\wedge\psi\right)$ (8)

where $T^{a}$ is the torsion, defined as $de^{a}+e^{b}\wedge\omega_{b}^{\
a}=T^{a}$. The first part is the topological term for gravity coupled to
fermions 16 . It is equall to $e_{a}\wedge e_{b}\wedge R^{ab}-T^{a}\wedge
T_{a}=d(e_{a}\wedge T^{a})$. Re-writing it like this however, hides away
important terms, that are technically zero, but whose contribution to the pre-
symplectic potential is not. The action is invariant under supersymmetry,
diffeomorphism and local Lorentz transformations 22 . These transformations
act respectively on the frame and gravitino field in the following way,

$\displaystyle\delta_{\epsilon}e^{a}_{\mu}=\frac{1}{2}\bar{\epsilon}\gamma^{a}\psi_{\mu}\quad\delta_{\epsilon}\psi_{\mu}=\nabla_{\mu}\epsilon$
(9)
$\displaystyle\delta_{\xi}e^{a}_{\mu}=\mathcal{L}_{\xi}e^{a}_{\mu}=\xi^{\nu}\partial_{\nu}e^{a}_{\mu}+e^{a}_{\nu}\partial_{\mu}\xi^{\nu}\quad\delta_{\xi}\psi_{\mu}=\mathcal{L}_{\xi}\psi_{\mu}=\xi^{\nu}\nabla_{\nu}\psi_{\mu}+\psi_{\nu}\nabla_{\mu}\xi^{\nu}-\frac{1}{4}\nabla_{\rho}\xi_{\nu}\gamma^{\rho}\gamma^{\nu}\psi_{\mu}$
(10) $\displaystyle\delta_{\Lambda}e^{a}=\Lambda^{a}_{\
b}e^{b}\quad\delta_{\Lambda}\psi=-\frac{1}{4}\Lambda_{ab}\gamma^{ab}\psi$ (11)

where $\epsilon$ is the gauge spinor, $\xi$ is the vector field, generating
diffeomorphism, and $\Lambda$ generates a local Lorentz transformation.

The variation of the action is,

$\displaystyle\delta S=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{M}}\left(2\varepsilon_{abcd}e^{b}\wedge
R^{cd}-\bar{\psi}\wedge\gamma_{5}\gamma_{a}\nabla\psi\right)\wedge\delta
e^{a}$ (12) $\displaystyle+\left(2\varepsilon_{abfd}\,de^{a}\wedge
e^{b}+2\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge\omega_{\
f}^{c}-\frac{1}{4}\bar{\psi}\wedge\gamma_{5}\gamma\gamma_{fd}\wedge\psi\right)\wedge\delta\omega^{fd}$
(13)
$\displaystyle-\left(\nabla\bar{\psi}\gamma_{5}\wedge\gamma\right)\wedge\delta\psi$
(14) $\displaystyle+d\left(e_{abcd}e^{a}\wedge
e^{b}\wedge\delta\omega^{cd}-\bar{\psi}\gamma_{5}\wedge\gamma\wedge\delta\psi\right)$
(15) $\displaystyle-i\lambda\left(e_{a}\wedge
e_{b}\wedge\delta\omega^{ab}-2\delta e^{a}\wedge
T_{a}-\delta\left(\bar{\psi}\wedge\gamma\wedge\psi\right)\right)$ (16)

In the first line we expressed the variation of the action with respect to the
frame field - this gives the Einstein equation. The second line is variation
with respect to the spin connection and it gives the equation for the torsion
and contorsion tensor,

$\displaystyle\omega_{\mu ab}=\omega_{\mu ab}(e)+K_{\mu ab}(\psi)\quad
K_{\mu[ab]}(\psi)=\frac{1}{4}\left(\bar{\psi}_{b}\gamma_{a}\psi_{\mu}-\bar{\psi}_{a}\gamma_{b}\psi_{\mu}-\bar{\psi}_{a}\gamma_{\mu}\psi_{b}\right)$
(17) $\displaystyle
T_{[\mu\nu]b}(\psi)=\frac{1}{2}\bar{\psi}_{\mu}\gamma_{b}\psi_{\nu}$ (18)

The third line is the equation of motion for the gravitino field. The two
final lines are pre-symplectic potential for $\mathcal{N}=1$ supergravity with
magnetic term,

$\displaystyle\Theta=$
$\displaystyle\frac{1}{2\kappa^{2}}\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\delta\omega^{cd}-\bar{\psi}\wedge\gamma_{5}e^{a}\gamma_{a}\wedge\delta\psi\right)$
(19) $\displaystyle-\frac{i\lambda}{\kappa^{2}}\underbrace{\left(e_{a}\wedge
e_{b}\wedge\delta\omega^{ab}-2\delta e^{a}\wedge
T_{a}-\delta(\bar{\psi}\wedge\gamma\wedge\psi)\right)}_{\Theta^{\ast}}$ (20)

Before we proceed with studying the charges we note something useful about the
magnetic piece in the symplectic density.

$\displaystyle\Theta^{\ast}(\delta)=\delta(e_{a}\wedge
e_{b}\wedge\omega^{ab})-2\delta e^{a}\wedge
de_{a}-\delta(\bar{\psi}\wedge\gamma\wedge\psi)$ (21)
$\displaystyle\Omega^{\ast}(\delta_{1},\delta_{2})=2d(\delta_{1}e^{a}\wedge\delta_{2}e_{a})=d\delta_{1}(e^{a}\wedge\delta_{2}e_{a})-d\delta_{2}(e^{a}\wedge\delta_{1}e_{a})$
(22)

Because $\delta$ is the exterior derivative on phase space.

###  3.1 Asymptotic conditions

The equations of motion, that the fields $e^{a}$ and $\psi$ need to satisfy
are already given in section 3. In this section, we introduce the boundary
conditions of the fields. Firstly, we demand that the metric is asymptotically
flat. In the Bondi gauge 1 , 6 , 33  an asymptotically flat metric takes the
following form,

$\displaystyle
ds^{2}=-e^{2\beta}f^{2}du^{2}-2e^{2\beta}dudr+r^{2}h_{AB}(dx^{A}-U^{A}du)(dx^{B}-U^{B}du)$
(23) $\displaystyle
h_{AB}=\gamma_{AB}+\frac{C_{AB}}{r}+\frac{d_{AB}}{r^{2}}+O(r^{-3})\quad
f^{2}(u,r,x^{A})=1-\frac{2M(u,x^{A})}{r}+O(r^{-1})$ (24) $\displaystyle
e^{2\beta}=1+O(r^{-2})\quad
g_{uA}=\frac{1}{2}D_{B}C^{AB}+\frac{1}{r}\left(\frac{1}{4}C_{BC}D_{A}C^{CB}+N_{A}\right)\quad
d_{AB}=\frac{1}{4}\gamma_{AB}C_{CD}C^{CD}$ (25)

Here $D_{A}$ denotes the covariant derivative with resepect to the metric of
the unit two-sphere $\gamma_{AB}$, $M(u,x^{A})$ is the Bondi mass, that can
vary with time and $N_{A}$ is the angular momentum. The subleading tensor
$C_{AB}$ characterizes a gravitational wave and is related to the time
derivative of the mass. Residual gauge freedom can be used to make it
traceless and to set the determinant $\det(g_{AB})$ to be
$r^{2}\det(\gamma_{AB})$.

Since we work in the Einstein-Palatini formalism, the metric is not the
gravitational field of interest. Rather we need the frame fields
$e^{a}_{\mu}$, which satisfy $e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}=g_{\mu\nu}$,
where $\eta_{ab}$ is the Minkowski metric. The frame fields are defined up to
a local Lorentz transformation. Our choice for them is,

$\displaystyle e^{0}=\frac{e^{\beta}}{f}dr+e^{\beta}fdu\quad
e^{1}=\frac{e^{\beta}}{f}dr$ (26) $\displaystyle
e^{i}=rE^{i}_{A}\left(dx^{A}-U^{A}du\right)\quad
E^{i}_{A}E^{j}_{B}\delta_{ij}=h_{AB}\quad i,j\in\\{2,3\\}$ (27)

Any variation should preserve the leading terms of these fields.

The gauge condition for the gravitino field is,

$\displaystyle\gamma^{\mu}\psi_{\mu}=0$ (28)

This simplifies the equations of motion for $\psi$. The boundary conditions on
$\psi$, will be dictated by the fact that supersymmetric transformations must
preserve the gauge and asymptotic behavior of the metric,

$\displaystyle\delta_{\epsilon}g_{rr}=\bar{\epsilon}\gamma_{r}\psi_{r}=0\quad\delta_{\epsilon}g_{ur}=\frac{1}{2}\bar{\epsilon}\gamma_{(u}\psi_{r)}=O(r^{-2})\quad\delta_{\epsilon}g_{rA}=\frac{1}{2}\bar{\epsilon}\gamma_{(r}\psi_{A)}=0$
(29)

To leading order this and the fermionic gauge condition are satisfied when,

$\displaystyle\epsilon\sim O(1)\quad\psi_{A}\sim O(1)\quad\psi_{u}\sim
O(r^{-1})\quad\psi_{r}\sim O(r^{-2})$ (30)
$\displaystyle\gamma^{u}\psi_{r}=0\quad\gamma^{u(0)}\psi_{A}^{0}=0\quad\gamma^{u(0)}\psi_{u}^{(-1)}=0\quad\gamma^{A(-1)}\psi_{A}=0$
(31)

Notice that, because the matrix $\gamma^{u}$ is nilpotent the equations of the
second line do not imply that $\psi_{r}$, $\psi_{A}^{(0)}$ or
$\psi_{u}^{(-1)}$ are 0. The justification for these boundary conditions are
the following. Firstly, $\epsilon\sim O(1)$ comes from the fact that the
commutator of the supersymmetric transformations of the metric, should be a
diffeomorphism. This is better explained in the next section. Next, because
$\gamma_{AB}$ is fixed $\psi_{A}$ should be $O(1)$. It cannot be more sub
leading, because the subleading tensors of $h_{AB}$ depend on $C_{AB}$. The
trace-free condition requires that $\gamma^{A(-1)}\psi_{A}^{(0)}=0$. Combining
this with $\delta_{\epsilon}g_{rA}$, we can reach the conclusion that
$\psi_{r}^{(-1)}=0$. Last but not least, remembering that the supersymmetric
variation, should also respect the linearized equations of motion for the
metric, we must also have to the following constraint on $\psi_{A}^{(-1)}$,

$\displaystyle\gamma_{(A}^{(0)}\psi_{B)}^{(0)}+\gamma_{(A}^{(1)}\psi_{B)}^{(-1)}=\frac{1}{2}\gamma_{AB}C^{DC}\gamma_{(D}^{(1)}\psi_{C)}^{0}\
\Rightarrow\ \gamma^{A(-1)}\psi_{A}^{(-1)}=0$ (32)

Finally it is important to discuss what variations are allowed on the
gravitino field. Super-translations preserve the leading order components of
the field, while super-rotations do not. From super-symmetric transformation
we require that $\delta_{\epsilon}\psi_{A}=O(r^{-1})$ and
$\delta_{\epsilon}\psi_{u}=O(r^{-2})$. This is discussed in detail in section
4.3. Finally, for integrability of the supersymmetric charges we require that
any other non-specific variation $\delta\psi_{A}$ is subleading. The fact that
super-rotations violate this condition will result in a contribution to the
central charge.

##  4 Charges

###  4.1 Diffeomorphism charges

The diffeomorphisms that respect the boundary and gauge conditions of the
fields form the BMS group. The BMS group is generated by vector fields, that
take the following form 1 , 6 ,

$\displaystyle\xi^{u}=f(x^{A})+\frac{u}{2}D_{B}f^{B}(x^{A})\quad\xi^{A}=f^{A}-\partial_{B}\xi^{u}\int\frac{1}{r^{2}}e^{2\beta}h^{AB}dr$
(33)
$\displaystyle\xi^{r}=-r\partial_{u}\xi^{u}+\frac{1}{2}\nabla^{r}\xi^{u}-\partial_{B}\xi^{u}\int
U^{B}dr\quad\partial_{u}\xi^{u}=D_{A}f^{A}$ (34)

These generators can be divided in two subcategories - super-translations and
super-rotations. The super-translation generators depend only on an arbitrary
function on the sphere $f(x^{A})$. When this function is taken to be the
lowest order spherical harmonics, the transformation reduces to ordinary
translation. Similarly, the super-rotation generators depend only on a vector
field on the two sphere $f^{B}(x^{A})$ and includes the Lorentz subgroup.

In order to calculate the diffeomorphism charge, we use the fact that the
variation of any quantity is just its Lie derivative, and we can use Cartan’s
magic formula,

$\displaystyle\Omega(\delta,\delta_{\xi})\approx\,$
$\displaystyle\delta\Theta(\delta_{\xi})-d\iota_{\xi}\Theta(\delta)-\iota_{\xi}\delta
L=\delta\left(\Theta(\delta_{\xi})-\iota_{\xi}L\right)-d\iota_{\xi}\Theta(\delta)$
(35)
$\displaystyle\cancel{\delta}H_{\xi}=\int_{\Sigma}\delta\left(\Theta(\delta_{\xi})-\iota_{\xi}L\right)-\int_{\partial\Sigma}\iota_{\xi}\Theta(\delta)$
(36)

We see that the Hamiltonian charge, naturally splits into integrable and non-
integrable part. The latter quantifies the flux of the given charge through
the Cauchy surface, which here is chosen to be future null-infinity. At the
past boundary of $\mathcal{I}^{+}$ this flux is set to be zero, by choosing
appropriate $u$ fall-off conditions for gravitational radiation.

The calculations for the Hamiltonian charge density are in appendix D. The
formula for the charge is,

$\displaystyle\cancel{\delta}H_{\xi}+i\lambda\cancel{\delta}\tilde{H}=\frac{1}{2\kappa^{2}}\int_{\partial\Sigma}$
$\displaystyle\delta\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\,\iota_{\xi}\omega^{cd}-\bar{\psi}\wedge\gamma_{5}\gamma\,\iota_{\xi}\psi\right)$
(37) $\displaystyle-\iota_{\xi}\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\delta\omega^{cd}-\bar{\psi}\wedge\gamma_{5}\gamma\wedge\delta\psi\right)+i\lambda(\mathcal{L}_{\xi}e^{a}\wedge\delta
e_{a})$ (38)

First we note, that due to the boundary conditions on the fermionic field 30,
the part of the charge, that depends on $\psi$, is finite and it vanishes for
super-translations. It is integrable for both super-translations and super-
rotations. The integrability crucially relies on
$\gamma^{A(-1)}\psi_{A}^{0}=\gamma^{u(0)}\psi_{A}^{(0)}=0$. We see that in the
context of supergravity, the finiteness of the global Lorentz generators of
the gravitino field is a direct consequence of the asymptotic flatness of the
metric. One does not need to make additional assumptions for the asymptotic
behavior of the field, as is the case for the free Rarita-Schwinger field 34 .

The rest of the above expression is the usual diffeomorphism charge of pure
gravity. The charge has normal and magnetic part. More specifically, comparing
with the results from 6 , 16  and 32 , the super-translation and super-
rotation charges are respectively,

$\displaystyle H_{ST}+i\lambda\tilde{H}_{ST}=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{I}^{+}_{-}}\sqrt{\gamma}\,fM+i\lambda\int_{\mathcal{I}^{+}_{-}}-C_{B[D}D_{A]}D^{B}\xi^{u}$
(39) $\displaystyle H_{SR}+i\lambda\tilde{H}_{SR}=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{I}^{+}_{-}}\sqrt{\gamma}\,-\frac{1}{4}D_{A}f^{A}C^{2}+f^{A}N_{A}+\bar{\psi}_{[A}\gamma_{5}\gamma_{B]}\psi_{C}f^{C}$
(40) $\displaystyle+$ $\displaystyle
i\lambda\int_{\mathcal{I}^{+}_{-}}-C_{B[D}D_{A]}D^{B}\xi^{u}+\frac{1}{4}f^{B}C_{C[A}D_{|B|}C^{C}_{D]}-\frac{1}{4}f_{[A}D_{D]}C^{2}$
(41)

where $C^{2}=C_{AB}C^{AB}$.

###  4.2 Lorentz charges

In addition to diffeomorphisms, we also need to consider local Lorentz
transformations. They do not affect the metric and therefore one expects that
they are unphysical. The frame field transforms under diffeomorphisms and
local Lorentz transformations as,

$\displaystyle\delta
e^{a}_{\mu}=\xi^{\nu}\partial_{\nu}e^{a}_{\mu}+e^{a}_{\nu}\partial_{\mu}\xi^{\nu}+\Lambda^{a}_{\
b}e^{b}_{\mu}$ (42)

If we take into consideration only the diffeomorphisms, the gauge and boundary
conditions are not satisfied. We need local Lorentz transformations to
regularize this. Their expressions up to leading order are,

$\displaystyle\Lambda^{0}_{\
i}=-e^{0}_{\mu}e^{A}_{i}\partial_{A}\xi^{\mu}\quad\Lambda^{1}_{\
i}=-e^{1}_{r}e^{A}_{i}\partial_{A}\xi^{r}$ (43) $\displaystyle\Lambda^{1}_{\
0}=-\frac{1}{f^{2}}\partial_{r}\xi^{r}\quad\Lambda^{i}_{\
j}=-e^{A}_{j}\mathcal{L}_{f^{B}}e^{i}_{A}+\delta_{\ j}^{i}D_{A}f^{A}$ (44)

Notice that in the third line, $\delta_{i}^{j}D_{A}f^{A}$ is the symmetric
part of the term $-e^{A}_{j}\mathcal{L}_{f^{B}}e^{i}_{A}$, making
$\Lambda_{ij}$ asymmetric as it should be. Furthemore, we observe that these
transformations are needed to preserve the boundary conditions, only if the
BMS transformation in question is a super-rotation. Intuitively, this can be
understood as the fact that super-rotations are a generalization of Lorentz
transformations. Therefore, part of their effect on the frame field needs to
be undone by a local Lorentz transformation.

We now proceed to study the charges related to these transformations. We know
from 16  that $\delta_{\Lambda}\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\delta\omega^{cd}\right)=0$. We also have,

$\displaystyle\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\delta_{\Lambda}\omega^{cd}=d\mathcal{Q}_{\Lambda}+2\varepsilon_{abcd}T^{a}\wedge
e^{b}\Lambda^{cd}=d\mathcal{Q}_{\Lambda}+2\psi\gamma_{5}\gamma_{bcd}\wedge\psi\wedge
e^{b}\Lambda^{cd}$ (45)

We know that the hamiltonian charge from $\mathcal{Q}$ in this setup is zero
16 . From the fermionic part we have:

$\displaystyle\delta_{\Lambda}\psi=-\frac{1}{4}\Lambda_{ab}(x)\gamma^{ab}\psi\quad\delta_{\Lambda}e^{b}=\Lambda(x)_{\
a}^{b}e^{a}$ (46)
$\displaystyle\bar{\psi}\wedge\gamma_{5}\gamma\wedge\delta_{\Lambda}\psi=\bar{\psi}\wedge\gamma_{5}\gamma\gamma_{cd}\wedge\psi\Lambda^{cd}$
(47)

Which simplifies the second term in 45. Furthermore, thanks to the identity 92
(which works in the same way if one replaces $\omega_{ab}$ by $\Lambda_{ab}$),
we can see that
$\delta_{\Lambda}\left(\bar{\psi}\wedge\gamma_{5}\gamma\wedge\delta\psi\right)=0$.
We conclude that the normal Lorentz charge vanishes.

On the other hand, the magnetic term from the Lorentz charges does not vanish.
This seems bizarre at a first glance. However, this is actually useful as it
serves to regularize the diffeomorphism magnetic charge.

$\displaystyle\left(\mathcal{L}_{\xi}e^{a}+\Lambda^{a}_{\
b}e^{b}\right)\wedge\delta e_{a}\sim O(1)$ (48)

###  4.3 SUSY charges

The charges conjugate to supersymmetry transformations are more complicated.
We will study their expression in this section. First we work out the
transformations of the fields and the spin connection.

$\displaystyle\delta_{\epsilon}e_{\mu}^{a}=$
$\displaystyle\frac{1}{2}\bar{\epsilon}\gamma^{a}\psi_{\mu}\quad\delta_{\epsilon}\psi_{\mu}=\nabla_{\mu}\epsilon$
(49) $\displaystyle\delta_{\epsilon}\omega_{\mu ab}(e)=$
$\displaystyle\frac{1}{2}\partial_{\mu}\left(\bar{\epsilon}\gamma_{[a}\psi_{b]}\right)+e^{\rho}_{a}\partial_{\mu}\left(\bar{\epsilon}\gamma_{b}\psi_{\rho}\right)+\frac{1}{2}\bar{\epsilon}\gamma_{[a}\psi^{[\rho}e^{\sigma]}_{b]}\partial_{\sigma}g_{\rho\mu}+\frac{1}{2}e^{\rho}_{[a}e^{\sigma}_{b]}\partial_{\rho}\left(\bar{\epsilon}\gamma_{(\sigma}\psi_{\mu)}\right)$
(50) $\displaystyle\begin{split}4\delta_{\epsilon}K_{\mu
ab}=&\nabla_{b}\bar{\epsilon}\gamma_{a}\psi_{\mu}+\frac{1}{2}\bar{\epsilon}\gamma_{b}\psi^{\nu}\bar{\psi}_{\nu}\gamma_{a}\psi_{\mu}+\bar{\psi}_{b}\gamma_{a}\nabla_{\mu}\epsilon\\\
&-\frac{1}{2}\bar{\psi}_{a}\gamma^{c}\psi_{b}\bar{\epsilon}\gamma_{c}\psi_{\mu}+\nabla_{a}\bar{\epsilon}\gamma_{\mu}\psi_{b}+\frac{1}{2}\bar{\epsilon}\gamma_{a}\psi^{\nu}\bar{\psi}_{\nu}\gamma_{\mu}\psi_{b}-(a\leftrightarrow
b)\end{split}$ (51)

Where $\epsilon$ is the gauge spinor. These transformations need to preserve
the boundary and gauge conditions of the frame and gravitino field. This leads
to various constraints. We first look at the gauge spinor,

$\displaystyle[\delta_{1},\delta_{2}]g_{\mu\nu}=\nabla_{(\mu}\xi_{\nu)}\quad\xi_{\nu}=\frac{1}{2}\bar{\epsilon}_{1}\gamma_{\nu}\epsilon_{2}$
(52)

This looks like the transformation of the metric under diffeomorphism. We
already know what conditions should be satisfied by the vector field
$\xi^{\mu}$ \- it should generate a member of the BMS group. In particular
$\xi^{u}$ should not be a function of $r$. This is achieved if the following
property is satisfied by the gauge spinor $\epsilon$,

$\displaystyle\frac{e^{\beta}}{f}\bar{\epsilon}_{1}\left(\gamma_{0}+\gamma_{1}\right)\epsilon_{2}\neq
g(r)\ \Rightarrow\
\epsilon_{i}=\sqrt{\frac{f}{e^{\beta}}}f_{i}(\theta,\phi)+\gamma_{r}^{(0)}\rho_{i}$
(53)

From the above we see that
$\epsilon^{-1}=-\frac{M}{r}\epsilon^{0}+\frac{\gamma^{u(0)}}{r}\rho^{(-1)}$,
(because $\gamma^{u(0)}\times\gamma^{u(0)}=0$). From the gauge condition 28 at
first and second order in $r$ the equations of motion for the gauge spinor
are,

$\displaystyle O(1):$ $\displaystyle\gamma^{u0}\partial_{u}\epsilon^{0}=0$
(54) $\displaystyle O(r^{-1}):$
$\displaystyle\gamma^{A(-1)}\partial_{A}\epsilon^{0}+\frac{1}{2}\cot\theta\gamma^{2}\epsilon^{0}+\gamma^{1}\epsilon^{0}+\gamma^{u0}\partial_{u}M\epsilon^{0}+\gamma^{u0}\partial_{u}\epsilon^{-1}=0$
(55)
$\displaystyle\gamma^{A(-1)}\partial_{A}\epsilon^{0}+\frac{1}{2}\cot\theta\gamma^{2}\epsilon^{0}+\gamma^{1}\epsilon^{0}=0$
(56)

This is the equation for covariantly constant spinor and it agrees with the
result from 26 . It’s expression is given in the appendix. This tells us that
the function $\xi^{u}=\frac{1}{2}\bar{\epsilon}_{1}\gamma^{u}\epsilon_{2}$ is
some linear combination of the lowest order spherical harmonics. Thus we
recognize that the vector field $\xi^{\mu}$ from 52 generates only ordinary
translations. The same conclusion was reached in 35 . Notice that in order to
obtain higher order spin $1/2$ spherical harmonics, 4 , and consequently more
interesting super-translations, we would need a different constant multiplying
$\gamma^{1}$. This would mean setting $U^{0}=\lambda\neq 1$. However Einstein
equations tells us that $U^{0}=\frac{1}{2}R[\gamma]$. Knowing what the Ricci
tensor is for the metric on the sphere, we can infer that this re-scaling
would also re-scale the coefficients of $\gamma_{AB}$, so as to leave the
above equation invariant. Therefore the equations of motion and the asymptotic
flatness impose, that the commutator of two supersymmetric transformations can
only be an ordinary translation. It would be interesting to see whether more
interesting results can be obtained for other metrics. Finally, a quick
calculation can show that by choosing $\rho^{(-1))}=-\gamma^{0}M\epsilon^{0}$,
we can set $\delta_{\epsilon}\psi_{u}\sim O(r^{-2})$, which is going to be
useful.

With these conditions in mind, we now proceed to the calculations of the
fermionic charges, arising from supersymmetry. The calculations for the part
of the charge, arising from the gravitino term in the action, are almost
identical to the calculation for gauge charges of the free, massless Rarita-
Schwinger field 24 ,

$\displaystyle\delta\Theta(\delta_{\epsilon})-\delta_{\epsilon}\Theta(\delta)\approx\delta\left(\Theta(\delta_{\epsilon})-I_{\epsilon}L\right)-dI_{\epsilon}\Theta(\delta)$
(57)
$\displaystyle\delta\left(\Theta(\delta_{\epsilon})-I_{\epsilon}L\right)\approx\delta\left(\Theta(\delta_{\epsilon})\right)=\delta\left(\bar{\psi}\wedge\gamma_{5}\gamma\wedge\nabla\epsilon\right)$
(58) $\displaystyle\approx
d\left(\delta(\bar{\psi}\wedge\gamma_{5}\gamma\epsilon)+\bar{\psi}\wedge\gamma_{5}T^{a}\gamma_{a}\epsilon\right)$
(59) $\displaystyle
I_{\epsilon}\Theta(\delta)=\bar{\epsilon}\gamma_{5}\gamma\wedge\delta\psi$
(60)

where the operator $I_{\epsilon}$ is defined as
$I_{\epsilon}=\epsilon\cdot\frac{\delta}{\delta\psi}$. In the expression for
$\delta_{\epsilon}\Theta$ I have ignored the term
$\bar{\psi}\wedge\gamma_{5}\delta_{\epsilon}\gamma\wedge\psi$ because this is
a four fermion term that vanishes thanks to the Fierz identity 93. Simiralry,
$\bar{\psi}\wedge\gamma_{5}T^{a}\gamma_{a}\epsilon$ vanishes thanks to the
cyclic identity for spinors 95.

We now turn to the gravitational part of the symplectic form.

$\displaystyle\Omega(\delta,\delta_{\epsilon})=\varepsilon_{abcd}\int_{\Sigma}\delta(e^{a}\wedge
e^{b}\wedge\delta_{\epsilon}\omega^{cd})-\delta_{\epsilon}(e^{a}\wedge
e^{b}\wedge\delta\omega^{cd})$ (61)

First we note two things - $\int_{\Sigma}\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge B^{cd}=\int_{\Sigma}B_{\mu}^{\ \mu\nu}d\Sigma_{\nu}$, for any one-
form with two antisymmetrized tangent space indices. Also, the connection form
$\omega$ has two parts $\omega=\omega(e)+K(\psi)$. From the expression of the
supersymmetric variation of the contorsion tensor $K_{\mu ab}$ 51 and the
gauge condition 28, we can conclude that $\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\delta_{\epsilon}K^{cd}=0$. Any variation $\delta$ should preserve
this gauge condition. Furthermore, most of the terms of
$\delta_{\epsilon}\left(\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge\delta
K^{cd}\right)$ again vanish by the gauge condition. The remaining ones are of
the form
$n_{\nu}\delta_{\epsilon}\bar{\psi}^{\nu}\gamma^{\mu}\delta\psi_{\mu}$. This
will be sub-leading because we have set $\nabla_{u}\epsilon\sim O(r^{-2})$. We
are left with the usual symplectic form of four-dimensional pure gravity. Its
expression is,

$\displaystyle\Omega(\delta_{1},\delta_{2})=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{I}^{+}}\sqrt{\gamma}\,dud\theta
d\phi\,\varepsilon^{AB}\,\varepsilon_{CD}\,\delta_{1}\left(C_{A}^{D}\right)\,\delta_{2}\left(\partial_{u}C^{C}_{B}\right)-(1\leftrightarrow
2)$ (62) $\displaystyle\Omega(\delta,\delta_{\epsilon})=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\mathcal{I}^{+}}\sqrt{\gamma}dud\theta
d\phi\,\varepsilon^{AB}\,\varepsilon_{CD}\,\bar{\epsilon}\gamma^{D}\psi_{A}\,\delta\left(\partial_{u}C^{C}_{B}\right)-(\delta\leftrightarrow\delta_{\epsilon})$
(63) $\displaystyle=$
$\displaystyle-\frac{1}{2\kappa^{2}}\int_{\mathcal{I}^{+}}dud\theta
d\phi\,\partial_{u}\left(\bar{\epsilon}\gamma_{5}\gamma_{D}\psi_{A}\,\delta
C^{D}_{B}\,\varepsilon^{AB}\right)$ (64)

where we have used the fact that
$\gamma_{5}=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$ and
$\gamma_{0}\gamma_{1}\psi_{A}^{(0)}=-\psi_{A}^{(0)}$, which is a consequence
of $\gamma^{u(0)}\psi_{A}^{(0)}=0$. Furthermore, writing this expression as a
total derivative is possible because the leading order terms of the spinors
and frame field do not depend on time. At subleading order this will no longer
be true.

The overall the charge is,

$\displaystyle\delta
H_{\epsilon}+i\lambda\cancel{\delta}\tilde{H}_{\epsilon}=-\frac{1}{2\kappa^{2}}\int_{\partial\Sigma}\,\delta\left(\bar{\epsilon}\gamma_{5}\gamma\wedge\psi\right)+\bar{\epsilon}\gamma_{5}\delta\gamma\wedge\
\psi-\bar{\epsilon}\gamma_{5}\gamma\wedge\delta\psi-i\lambda\
\bar{\epsilon}\,\delta\gamma\wedge\psi$ (65)

We see that unlike the Rarita-Schwinger case, there is contribution to the
magnetic charge. Furthermore because the boundary metric is fixed and because
of 30 both charges are finite. We can make both charges integrable by
requiring that any non-specific variation (that is not BMS or susy) of
$\psi_{A}^{0}$ vanishes to leading order. In the case where the variation is
supersymmetry or super-translation this still holds. When the variation is a
super-rotation one may naively think that the charge is not-integrable.
However, this is not the case. A simple way to understand why this is false is
by noting that $\delta_{\xi}{H}_{\epsilon}=-\delta_{\epsilon}{H}_{\xi}$.
However, we already know that ${H}_{\xi}$ is integrable up to some flux term
27 . Therefore, the obstruction that one gets can only be interpreted as a
central charge. This is studied in detail in the last section.

It is worth noting that the integrable part of the normal charge is the same
as the gauge charge of the Rarita-Schwinger field 24  up to an overall
numerical constant. In terms of the metric and gravitino components, the
integrable charges are,

$\displaystyle
H_{\epsilon}+i\lambda\tilde{H}_{\epsilon}=\frac{1}{2\kappa^{2}}\int_{\partial\Sigma}\,2\bar{\epsilon}^{(0)}\gamma_{5}\gamma_{D}^{(1)}\psi_{[A}^{(0)}\,C^{D}_{\
B]}-i\lambda\bar{\epsilon}^{(0)}\gamma_{D}^{(1)}\psi_{[A}^{(0)}\,C^{D}_{\ B]}$
(66)

We see that both charges depend on the tensor $\tilde{C}_{AB}=C^{C}_{\
(B}\varepsilon_{A)C}$, which also enters in the expression for the dual
diffeomorphism charges.

##  5 Algebra of integrable charges

Before studying the algebra of the charges, we need to specify the algebra of
the generators $\xi$ and $\epsilon$. The algebra for the BMS generators is
well-know 17 ,

$\displaystyle\xi_{f,f^{A}}=\\{\xi_{f_{1}.f_{1}^{A}},\,\xi_{f_{2},f^{A}_{2}}\\}$
(67) $\displaystyle
f=f^{A}_{1}D_{A}f_{2}-\frac{1}{2}f_{1}D_{A}f^{A}_{2}-(1\leftrightarrow 2)\quad
f^{A}=f^{B}_{1}D_{B}f^{A}_{2}-(1\leftrightarrow 2)$ (68)

Furthermore, we already know that the anti-commutator of two supersymmetric
parameters is a translation, characterized by the function
$\bar{\epsilon}_{1}\gamma^{u(0)}\epsilon_{2}$. The problem now is to work the
bracket of $\epsilon$ with $\xi$. We define the bracket between a BMS vector
and the gauge spinor to be,

$\displaystyle[\epsilon,\xi_{R}]=\frac{1}{2}\mathcal{L}_{\xi}\epsilon=f^{A}\partial_{A}\epsilon-\frac{1}{4}D_{A}f^{A}\left(\mathbb{1}+\gamma_{0}\gamma_{1}\right)\epsilon+O(r^{-1})$
(69)

We see that to leading order, this vanishes for a super-translations.
Asymptotically the generators of super-translation and supersymmetric
transformations commute, as they should. Using the fact that
$\gamma^{u(0)}\left(\mathbb{1}+\gamma_{0}\gamma_{1}\right)=2\gamma^{u(0)}$,
one can verify that with this bracket the following Jacobi identity is
satisfied.

$\displaystyle\\{\epsilon_{1},[\epsilon_{2},\xi_{SR}]\\}-\\{\epsilon_{2},[\epsilon_{1},\xi_{SR}]\\}=[\xi_{R},\\{\epsilon_{1},\epsilon_{2}\\}]$
(70)

This result agrees with that in 38 , where the same bracket was established in
the super-symmetric extension of BMS.

We will now discuss an interesting result from this bracket. The fact that
commutator of gauge spinors can only give an ordinary translation implies that
only Lorentz transformations are allowed in the super BMS group. In super
gravity we cannot extend the Lorentz group to include arbitrary conformal
transformations of the celestial sphere. In order to understand why this is
the case, using the notation of 37 , we write the vector field on the sphere
$f^{A}$, the spherical harmonics $f_{m.n}$ and the covariantly constant
spinors $\epsilon$ as,

$\displaystyle
l_{n}=f^{A}_{n}=-z^{n+1}\partial_{z}\quad\bar{l}_{n}=\bar{f}_{n}^{A}=-\bar{z}^{n+1}\partial_{\bar{z}}\quad
f_{m,n}=\frac{1}{1+z\bar{z}}z^{n}\bar{z}^{m}$ (71)
$\displaystyle\epsilon^{T}_{+}=e^{i\frac{\phi}{2}}\begin{pmatrix}ae^{\frac{i\theta}{2}},&ae^{-\frac{i\theta}{2}},&be^{\frac{i\theta}{2}},&be^{-\frac{i\theta}{2}}\end{pmatrix}$
(72)
$\displaystyle\epsilon^{T}_{-}=e^{i\frac{-\phi}{2}}\begin{pmatrix}ce^{\frac{i\theta}{2}},&-ce^{-\frac{i\theta}{2}},&de^{\frac{i\theta}{2}},&-de^{-\frac{i\theta}{2}}\end{pmatrix}$
(73) $\displaystyle
f_{0,0}=\bar{\epsilon}_{+}\gamma^{u(0)}\epsilon_{+}+\bar{\epsilon}_{-}\gamma^{u(0)}\epsilon_{-}\quad
f_{1,1}=\bar{\epsilon}_{+}\gamma^{u(0)}\epsilon_{+}-\bar{\epsilon}_{-}\gamma^{u(0)}\epsilon_{-}$
(74) $\displaystyle f_{1,0}=\bar{\epsilon}_{-}\gamma^{u(0)}\epsilon_{+}\quad
f_{0,1}=\bar{\epsilon}_{+}\gamma^{u(0)}\epsilon_{-}$ (75)

Where $z=e^{i\phi}\cot\frac{\theta}{2}$ is the standard stereographic
projection, complex coordinate. the We can re-write the BMS algebra in the
following way 37 ,

$\displaystyle[l_{n},l_{m}]=(n-m)l_{n+m}\quad[\bar{l}_{n},\bar{l}_{m}]=(n-m)\bar{l}_{n+m}\quad[\bar{l}_{n},l_{m}]=0$
(76)
$\displaystyle[l_{k},f_{n.m}]=\left(\frac{k+1}{2}-m\right)f_{m+k,n}\quad[\bar{l}_{k},f_{n.m}]=\left(\frac{k+1}{2}-n\right)f_{m+k,n}$
(77)

From these equations and the Jacobi identity (70) we see that whenever $n>1$
the commutators of $[l_{n},\epsilon]$ and $[\bar{l}_{n},\epsilon]$ are equal
to spinors, whose components must depend on higher order spin $1/2$ spherical
harmonics. This is not allowed by the equations of motion for the gauge
spinor. Therefore the allowed asymptotic symmetry group is the super-Poincaré
group, plus the infinite dimensional super-translations.

###  5.1 Normal charges

In this section we study the algebra of the normal Hamiltonian charges,
conjugate to diffeomorphic and supersymmetric transformations. They are a
representation of an extension of the BMS group - the super-BMS group 26 . It
contains a copy of the super-Poincaré group. The commutators of the BMS group
are already well understood 17 . Curiously the presence of the gravitino field
does not modify the central charge. Here, we will look only at the
commutators, involving supersymmetric charges.

We first look at the commutator of two supersymmetric charges. This is the
same as the commutator for gauge charges of the free Rarita-Schwinger field 31
.

$\displaystyle[H_{\epsilon 1},H_{\epsilon
2}]=\frac{1}{\kappa^{2}}\int_{\partial\Sigma}\nabla_{[\mu}\bar{\epsilon}_{1}\gamma_{\sigma]}\gamma_{5}\epsilon_{2}-(\epsilon_{1}\leftrightarrow\epsilon_{2})$
(78)

In 31  it is described in detail how the above expression is a BMS charge,
generated by $\xi^{\mu}=\bar{\epsilon}_{1}\gamma^{\mu}\epsilon_{2}$. This can
only be a super-translation, so the commutator is equal to,

$\displaystyle[H_{\epsilon 1},H_{\epsilon 2}]=$
$\displaystyle\frac{1}{2\kappa^{2}}\int_{\partial\Sigma}\sqrt{\gamma}\,\bar{\epsilon}_{1}^{0}\gamma^{u(0)}\epsilon_{2}^{0}\,M-(\epsilon_{1}\leftrightarrow\epsilon_{2})$
(79)

Notice that there is no central charge. What used to be the central charge for
this commutator in the free Rarita-Schwinger theory is now the super-
translation charge 24 , 31 . As already explained,
$\bar{\epsilon}_{1}^{0}\gamma^{u(0)}\epsilon_{2}^{0}$ is just a lowest order
spherical harmonic, and therefore the charge above is a global translation.
Again, we note that other supertranslations can not be obtained through the
commutator of fermionic charges.

Now we proceed to study the commutator of a BMS and a supersymmetric charge.
Looking at the expressions from subection 4.3 and section 2 we can identify
the commutator and the central charge as,

$\displaystyle[H_{\epsilon},H_{\xi}]=\frac{1}{\kappa^{2}}\int_{\partial\Sigma}\,(\mathcal{L}_{\xi}\bar{\epsilon})\gamma_{5}\gamma\wedge\psi+\,\bar{\epsilon}^{(0)}\gamma_{5}\gamma_{D}^{(1)}\psi_{[A}^{(0)}\,\mathcal{L}_{\xi}C^{D}_{\
B]}=H_{[\epsilon,\xi]}+K_{\epsilon,\xi}$ (80)

Notice that we obtain the same result if replace $\delta$, by $\delta_{\xi}$
in 65 and use the act that
$\int_{\mathcal{I}^{+}_{-}}\delta_{\xi}\left(\bar{\epsilon}\gamma_{5}\gamma\wedge\psi\right)=0$.

For super-translations $\mathcal{L}_{\xi}\epsilon=O(r^{-1})$, so
$H_{[\epsilon,\xi]}=0$ in agreement with the Poincarré algebra. The central
charge is,

$\displaystyle
K_{\epsilon,\xi_{ST}}=\frac{i\lambda}{\kappa^{2}}\int_{\partial\Sigma}\bar{\epsilon}\gamma_{5}\gamma^{B(-1)}\psi_{[A}^{(0)}\,D_{D]}D_{B}f$
(81)

It is worth nothing that for $l=0,1$ the spherical harmonics satisfy
$D_{A}D_{B}f=\gamma_{AB}f$ and the integral will vanish. Curiously, for higher
order spherical harmonics it will not necessarily vanish. This is also due to
the fact that, as shown in appendix C, $\psi_{A}$ can depend on arbitrarily
high spherical modes. This can be restricted in order to make the central
charge vanish. This expression also have a term involving
$\partial_{u}C_{AB}$. This is the non-integrable part and it vanishes at the
past boundary of $\mathcal{I}^{+}$. The integrable central charge for the
super-rotations is,

$\displaystyle K_{\epsilon,\xi_{SR}}=$
$\displaystyle\frac{i\lambda}{\kappa^{2}}\int_{\partial\Sigma}D_{C}f^{C}\,\bar{\epsilon}\gamma_{5}\gamma^{B}\psi_{[A}\,C_{D]B}+\bar{\epsilon}\gamma_{5}\gamma^{B(-1)}\psi_{[A}^{(0)}D_{|C|}f_{D]}C_{B}^{C}+\bar{\epsilon}\gamma_{5}\gamma^{B(-1)}\psi_{[A}^{(0)}C_{D]}^{C}D_{[B}f_{C]}$
(82)

We notice strong a parallel between this central charge and the one for super-
rotations, given in equation 7.11 of 16 .

###  5.2 Dual charges

The algebra of the dual charges is simpler to study. The commutator of two
dual diffeomorphsim charges is already studied in 16 . It was established that
the magnetic BMS charges satisfy the same algebra as the normal ones, but with
slightly different central extension. Here we will look at commutators,
involving dual fermioinic charges,

$\displaystyle\\{\tilde{H}_{\epsilon 1},\tilde{H}_{\epsilon
2}\\}=\frac{i\lambda}{2\kappa^{2}}\int_{\partial\Sigma}\nabla_{[\mu}\bar{\epsilon}_{1}\gamma_{\sigma]}\epsilon_{2}-(\epsilon_{1}\leftrightarrow\epsilon_{2})=0$
(83)

This seems to defer from the usual super-Poincaré algebra. However it is worth
nothing, that the dual global translation charge also vanishes 15 .
Furthermore, the above expression will not vanish if $C_{AB}$ is not a
continuous function on the sphere. This is precisely the condition we need for
the existence of BMS magnetic charge as well. We now proceed to compute the
commutator of supersymmetric and diffeomorphism charge.

$\displaystyle\\{\tilde{H}_{\epsilon},\tilde{H}_{\xi}\\}=\frac{i\lambda}{4\kappa^{2}}\int_{\partial\Sigma}\delta_{\epsilon}\left(e^{a}\wedge\mathcal{L}_{\xi}e_{a}\right)-\mathcal{L}_{\xi}\left(e^{a}\wedge\delta_{\epsilon}e_{a}\right)=$
(84)
$\displaystyle\frac{i\lambda}{2\kappa^{2}}\int_{\partial\Sigma}\,e^{a}\wedge\left(\mathcal{L}_{\xi}\epsilon\right)\gamma_{a}\psi+\delta_{\epsilon}e^{a}\wedge\mathcal{L}_{\xi}e_{a}=\tilde{H}_{[\epsilon,\xi]}+\tilde{K}_{\epsilon,\xi}$
(85)

The first term is the magnetic supersymmetric charge, defined from a gauge
spinor $\mathcal{L}_{\xi}\epsilon$. The second one is the central charge. Its
expression is almost identical to the one for the normal charges,

$\displaystyle\tilde{K}_{\epsilon,\xi_{ST}}=\frac{i\lambda}{2\kappa^{2}}\int_{\partial\Sigma}\bar{\epsilon}\gamma^{B(-1)}\psi_{[A}^{(0)}\,D_{D]}D_{B}f$
(86)
$\displaystyle\tilde{K}_{\epsilon,\xi_{SR}}=\frac{i\lambda}{2\kappa^{2}}\int_{\partial\Sigma}D_{C}f^{C}\,\bar{\epsilon}\gamma^{B}\psi_{[A}\,C_{D]B}+\bar{\epsilon}\gamma^{B(-1)}\psi_{[A}^{(0)}D_{|C|}f_{D]}C_{B}^{C}+\bar{\epsilon}\gamma^{B(-1)}\psi_{[A}^{(0)}C_{D]}^{C}D_{[B}f_{C]}$
(87)

##  6 Conclusion

In this paper, a new type of charges was discovered for $\mathcal{N}=1$
supergravity. They are dual to the usual diffeomorphisms and supersymmetric
charges. The diffeomorphism dual charges are the same as the ones studied
before 15 , 16 . The supersymmetric magnetic charges are new and do not appear
for the free Rarita-Schwinger field 24 . We saw that their integrability and
finiteness crucially depend on the conditions on the gravitino field, imposed
by the asymptotically flat metric. It would be interesting to study the dual
charges for more complicated supergravity theories, or for more exotic
metrics. The magnetic charges in supergravity can have potential interesting
applications for black holes in supergravity like the extreme Reissner-
Nordstr$\ddot{o}$m black hole.

##  Appendix A Conventions and useful identities

The choice of $\gamma$ matrices is,

$\displaystyle\gamma^{0}=-i\begin{pmatrix}0&\mathbb{1}\\\
\mathbb{1}&0\end{pmatrix}\quad\gamma^{j}=-i\begin{pmatrix}0&\sigma^{j}\\\
-\sigma^{j}&0\end{pmatrix}\quad\gamma^{5}=-i\begin{pmatrix}\mathbb{1}&0\\\
0&-\mathbb{1}\end{pmatrix}\quad C=\begin{pmatrix}0&\mathbb{1}\\\
-\mathbb{1}&0\end{pmatrix}$ (88)

The covariant and Lie derivative of a spinor are defined as,

$\displaystyle\nabla_{\mu}\psi_{\nu}=$
$\displaystyle\partial_{\mu}\psi_{\mu}-\Gamma^{\lambda}_{\mu\nu}\psi_{\lambda}+\frac{1}{4}\omega_{\mu
ab}\gamma^{ab}\psi_{\nu}$ (89) $\displaystyle\mathcal{L}_{\xi}\psi_{\nu}=$
$\displaystyle\xi^{\mu}\nabla_{\mu}\psi_{\nu}+\psi_{\mu}\nabla_{\nu}\xi^{\mu}-\frac{1}{4}\nabla_{\mu}\xi_{\rho}\gamma^{\mu}\gamma^{\rho}\psi_{\nu}$
(90) $\displaystyle=$
$\displaystyle\xi^{\mu}\partial_{\mu}\psi_{\nu}+\psi_{\mu}\partial_{\nu}\xi^{\mu}-\frac{1}{4}e_{a[\mu}\mathcal{L}_{\xi}e^{a}_{\nu]}\gamma^{\mu\nu}\psi_{\nu}$
(91)

Up to leading order the last term is simplified by local a Lorentz
transformation.

Simple, but useful commutation property,

$\displaystyle-\lambda_{ab}\gamma^{a}\gamma^{b}\gamma_{5}e_{c\nu}\gamma^{c}+\lambda_{ab}\gamma_{5}e_{c\nu}\gamma^{c}\gamma^{a}\gamma^{b}+4\lambda_{ab}e^{a}_{\nu}\gamma^{b}=0$
(92)

for any $\lambda_{ab}$ that is antisymmetric in $a$ and $b$.

Fierz identity in 4 dimensions 29 ,

$\displaystyle(\bar{\lambda}_{1}\lambda_{2})\lambda_{3\,a}=$
$\displaystyle-\frac{1}{4}(\bar{\lambda}_{1}\lambda_{3})\lambda_{2\,a}-\frac{1}{4}(\bar{\lambda}_{1}\gamma_{5}\lambda_{3})\gamma_{5}\lambda_{2\
a}-\frac{1}{4}(\bar{\lambda}_{1}\gamma_{\mu}\lambda_{3})(\gamma^{\mu}\lambda_{2})_{a}$
(93)
$\displaystyle+\frac{1}{4}(\bar{\lambda}_{1}\gamma_{\mu}\gamma_{5}\lambda_{3})(\gamma^{\mu}\gamma_{5}\lambda_{2})_{a}+\frac{1}{8}(\bar{\lambda}_{1}\gamma_{\mu\nu}\lambda_{3})(\gamma^{\mu\nu}\lambda_{2})_{a}$
(94)

Cyclic identity 22 ,

$\displaystyle\bar{\lambda}_{[1|}\gamma_{a}\lambda_{|2|}\bar{\lambda}_{|3]}\gamma^{a}$
(95)

Where $\lambda_{i}$ are arbitrary spinors and the letter $a$ is a spinor
index.

##  Appendix B Covariantly constant spinor

The differential equation, satisfied by a covariantly constant spinor $\chi$,
is,

$\displaystyle\gamma^{A}\partial_{A}\chi+\gamma^{1}\chi+\frac{1}{2}\cot\theta\gamma^{2}\chi=0$
(96)

With the particular choice we have made for the gamma matrices, the equations
for the first and second components are coupled, and are the same as the
equations for the third and forth. In particular we have,

$\displaystyle
i\partial_{\theta}\chi_{2}-\frac{1}{\sin\theta}\partial_{\phi}\chi_{1}+\frac{i}{2}\cot\theta\chi_{2}=\chi_{2}$
(97) $\displaystyle
i\partial_{\theta}\chi_{1}-\frac{1}{\sin\theta}\partial_{\phi}\chi_{2}+\frac{i}{2}\cot\theta\chi_{1}=-\chi_{1}$
(98)

We assume separation of variables and impose,
$\chi_{1}=e^{-i\frac{\theta}{2}}f(\phi)$ and
$\chi_{2}=e^{i\frac{\theta}{2}}h(\phi)$,

$\displaystyle
e^{-i\frac{\theta}{2}}\partial_{\phi}f-\frac{i}{2}e^{-i\frac{\theta}{2}}h=0\quad
e^{i\frac{\theta}{2}}\partial_{\phi}h-\frac{i}{2}e^{i\frac{\theta}{2}}f=0$
(99)
$\displaystyle\Rightarrow\partial_{\phi}h=\frac{i}{2}f\quad\partial_{\phi}f=\frac{i}{2}h$
(100)

Setting $h=ae^{i\frac{\phi}{2}}+be^{-i\frac{\phi}{2}}$ and
$f=ce^{i\frac{\phi}{2}}+de^{-i\frac{\phi}{2}}$ we have $a=c$ and $b=-d$.

##  Appendix C Equations of motion for $\psi$

In order to study the the charges and their properties, we need first to solve
(to some extent) the equations of motion.Since the metric is already fixed, in
this section we need to study only the gravitino field. The equations for
motion for it, subject to the gauge condition 28, are
$\gamma^{\mu}\nabla_{\mu}\psi_{\nu}=0$. Taking into consideration 30, at
different orders and for the different components of the gravitino, the exact
equations are,

  * •

$\psi_{A}$:

$\displaystyle O(1):$ $\displaystyle\gamma^{u(0)}\partial_{u}\psi_{A}=0$ (101)
$\displaystyle O(r^{-1}):$
$\displaystyle\gamma^{B}\partial_{B}\psi_{A}+\frac{1}{2r}\cot\theta\gamma^{2}\psi_{A}-\gamma^{C}\Gamma^{B}_{CA}\psi_{B}+h_{AB}\gamma^{B}\psi_{u}+\gamma^{u(0)}\partial_{u}\psi_{A}^{-1}=0$
(102)

  * •

$\psi_{u}$:

$\displaystyle O(r^{-1}):$ $\displaystyle\gamma^{u}\partial_{u}\psi_{u}=0$
(103) $\displaystyle O(r^{-2}):$ $\displaystyle
r\gamma^{A}\partial_{A}\psi_{u}+\frac{1}{2}\cot\theta\gamma^{2}\psi_{u}+\gamma^{u}\partial_{u}\psi_{u}^{-2}-r\gamma^{A}\partial_{u}C_{A}^{B}\psi_{B}=0$
(104)

  * •

$\psi_{r}$:

$\displaystyle O(r^{-3}):$
$\displaystyle\gamma^{A(-1)}\partial_{A}\psi_{r}^{(-2)}+\frac{1}{2}\cot\theta\gamma^{2}\psi_{r}^{(-2)}-\gamma^{1}\psi_{r}-\gamma^{A(-1)}\psi_{A}^{(-1)}$
(105) $\displaystyle+\gamma^{C(-1)}\frac{1}{2}C^{\
A}_{C}\psi_{A}^{(0)}-\gamma^{A(-2)}\psi_{A}^{(0)}=0$ (106)

We note that $K_{\mu\nu\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\sim
O(r^{-2})$ and that’s why it does not enter these equations. Using the gauge
condition 28 at second order the last equation can be re-written as,

$\displaystyle\gamma^{A(-1)}\partial_{A}\psi_{r}^{(-2)}+\frac{1}{2}\cot\theta\gamma^{2}\psi_{r}^{(-2)}+\gamma^{C(-1)}\frac{1}{2}C^{\
A}_{C}\psi_{A}^{(0)}+\gamma^{u(0)}\psi_{u}^{(-2)}=0$ (107)

We would need to components of the gravitino field to study the commutators of
the charges. This is an integral evaluated at $u\rightarrow-\infty$. We would
like the field to be finite in this limit, so we assume it can be decomposed
as $\psi_{\mu}=\varphi_{\mu}(x^{A})+\phi_{\mu}(u,x^{A})$ with
$\lim_{u\to-\infty}\phi_{\mu}=0$. Let’s look at the equation for
$\varphi_{u}^{(-1)}$.

$\displaystyle\gamma^{A}\partial_{A}\varphi_{u}+\frac{1}{2}\cot\theta\gamma^{2}\varphi_{u}^{(-1)}=0$
(108)

Remembering that $\gamma^{u(0}\varphi_{u}^{(-1)}=0$, this equation is solved
by the spinor
$\sqrt{\sin\theta}\,\varphi_{u}^{(-1)}=\begin{pmatrix}az^{-m},&az^{-m},&-b\bar{z}^{-m},&b\bar{z}^{-m}\end{pmatrix}^{T}$,
where $z$ is the stereographic coordinate $z=e^{i\phi}\tan\frac{\theta}{2}$,
and $a$ and $b$ are arbitrary constants. Taking into consideration that
$\gamma^{A(-1)}\psi_{A}^{(0)}$, the equation for $\varphi_{\phi}^{(0)}$ is,

$\displaystyle\gamma^{A}\partial_{A}\varphi_{\phi}+\frac{1}{2}\cot\theta\gamma^{2}\varphi_{\phi}^{(-1)}+\gamma_{\phi}\varphi_{u}^{(-1)}=0$
(109)

If we write $\varphi_{\phi}=\begin{pmatrix}\iota&\lambda\end{pmatrix}$, the
equations for the components of $\lambda$ are,

$\displaystyle\partial_{\theta}(\lambda_{1}+\lambda_{2})-\frac{i}{\sin\theta}\partial_{\phi}(\lambda_{1}+\lambda_{2})+\frac{1}{2}\cot\theta(\lambda_{1}+\lambda_{2})=0$
(110)
$\displaystyle\partial_{\theta}(\lambda_{1}-\lambda_{2})+\frac{i}{\sin\theta}\partial_{\phi}(\lambda_{1}-\lambda_{2})+\frac{1}{2}\cot\theta(\lambda_{1}-\lambda_{2})=\frac{2b\sin\theta}{\sqrt{\sin\theta}}e^{im\phi}\cot^{m}\frac{\theta}{2}$
(111)

We remember that because $\gamma^{u(0)}\varphi_{A}^{(0)}=0$,
$\lambda_{1}+\lambda_{2}=0$. This means that
$\lambda_{1}=-\lambda_{2}=\frac{1}{2}\left(\lambda_{1}-\lambda_{2}\right)=-b\frac{\cos\theta}{\sqrt{\sin\theta}}z^{m}$.
Similralrly,
$\iota_{1}=\iota_{2}=\frac{1}{2}\left(\iota_{1}+\iota_{2}\right)\frac{\cos\theta}{\sqrt{\sin\theta}}\bar{z}^{m}$.
From the expression of $\psi_{\phi}$ one can easily work out $\psi_{\theta}$.
Curiously they take a form, very similar to the generators of the super-
rotations.

$\displaystyle\psi^{\phi}_{m}=\frac{\cos\theta}{\sqrt{\sin\theta}}\begin{pmatrix}az^{-m},&az^{-m},&-b\bar{z}^{-m},&b\bar{z}^{-m}\end{pmatrix}\quad$
(112)
$\displaystyle\psi^{\theta}_{m}=i\sin\theta\frac{\cos\theta}{\sqrt{\sin\theta}}\begin{pmatrix}az^{-m},&az^{-m},&b\bar{z}^{-m},&-b\bar{z}^{-m}\end{pmatrix}$
(113)

Because the algebra of super-rotations has already been established, one can
easily work the Lie derivative of this siponor, along a generator of the BMS
group (after one corrects with Local Lorentz transformations,

$\displaystyle\mathcal{L}_{l_{n}+\bar{l}_{n}}\psi^{A}_{m}=\left(n-m-\frac{1+\cos^{2}\theta}{2\cos\theta}\right)\psi^{A}_{n+m}$
(114)

It may seem odd that the spinor change under the BMS group, even though the
components of the metric, on which its equation of motion depend do not.
However, this linear variation can be compensated by $\delta_{\xi}\psi_{u}$
and $\delta_{\xi}\psi_{A}^{(-1)}$ in the linearized equation of moiton.

From 107, we see that the leading order $u$ \- independent part of of
$\psi_{r}$ depends on $\psi_{A}$ and the tensor $C_{AB}$. For Minkowksi,
$\psi_{r}$ will have the same solution as $\psi_{u}$. Lastly, $m$ should be a
half-integer because $\psi$ is a spinor.

##  Appendix D  Diffeomorphism variation of the pre-symplectic potential

In this appendix we provide in detail the calculations necessary for the
deriving the expression for the diffeomorphism charge.

$\displaystyle 2\kappa^{2}\left(\Theta(\delta_{\xi})-\iota_{\xi}L\right)=$
(115) $\displaystyle\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\mathcal{L}_{\xi}\omega^{cd}-\bar{\psi}\wedge\gamma_{5}e^{a}\gamma_{a}\wedge\mathcal{L}_{\xi}\psi$
(116) $\displaystyle-\iota_{\xi}\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge R^{cd}-\bar{\psi}\wedge\gamma_{5}\gamma\wedge\nabla\psi\right)$
(117) $\displaystyle\approx\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\mathcal{L}_{\xi}\omega^{cd}-\bar{\psi}\wedge\gamma_{5}e^{a}\gamma_{a}\wedge\mathcal{L}_{\xi}\psi$
(118) $\displaystyle\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\mathcal{L}_{\xi}\omega^{cd}=\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge\left(d\iota_{\xi}\omega^{cd}+\iota_{\xi}d\omega^{cd}\right)$ (119)
$\displaystyle\approx d\left(\varepsilon_{abcd}e^{a}\wedge
e^{b}\iota_{\xi}\omega^{cd}\right)-2\varepsilon_{abcd}\left(T^{a}\wedge
e^{b}\iota_{\xi}\omega^{cd}\right)+2\varepsilon_{abcd}\xi^{a}e^{b}\wedge
R^{cd}$ (120)

We obtain in the end three terms. One is a total derivative and the other two
will be simplified.

$\displaystyle\bar{\psi}\wedge\gamma_{5}e^{a}\gamma_{a}\wedge\mathcal{L}_{\xi}\psi=\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\left(\iota_{\xi}d\psi+d\iota_{\xi}\psi\right)$
(121)
$\displaystyle\approx\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\iota_{\xi}d\psi+d\left(\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\iota_{\xi}\psi\right)-\frac{1}{4}\bar{\psi}\gamma_{5}\wedge\omega^{cd}\gamma_{cd}\wedge
e^{a}\gamma_{a}\iota_{\xi}\psi+\bar{\psi}\gamma_{5}e_{b}\wedge\omega^{ab}\gamma_{a}\iota_{\xi}\psi$
(122) $\displaystyle-\bar{\psi}\gamma_{5}T^{a}\gamma_{a}\iota_{\xi}\psi$ (123)
$\displaystyle\approx\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\iota_{\xi}d\psi+d\left(\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\iota_{\xi}\psi\right)+\frac{1}{4}\bar{\psi}\gamma_{5}\wedge\gamma\wedge\gamma_{cd}\omega^{cd}\iota_{\xi}\psi-\bar{\psi}\gamma_{5}T^{a}\gamma_{a}\iota_{\xi}\psi$
(124)

Notice that we have omitted the quantity
$\bar{\psi}\wedge\gamma_{5}e^{a}\gamma_{a}\wedge\psi\,\left(\omega_{ab}\gamma^{ab}-\nabla_{a}\xi_{b}\gamma^{a}\gamma^{b}\right)$
in the Lie derivative of the gravitino, because it is compensated by the
Lorentz transformation. As we already saw, the charge from this transformation
is 0.

$\displaystyle 2\varepsilon_{abcd}\xi^{a}e^{b}\wedge
R^{cd}=\bar{\psi}\wedge\gamma_{5}\iota_{\xi}\gamma\,\nabla\psi\approx\bar{\psi}\wedge\gamma_{5}\gamma\,\iota_{\xi}\nabla\psi$
(126) $\displaystyle 2\varepsilon_{abcd}\left(T^{a}\wedge
e^{b}\iota_{\xi}\omega^{cd}\right)=\frac{1}{4}\bar{\psi}\wedge\gamma_{5}\gamma\gamma_{cd}\iota_{\xi}\omega^{cd}\wedge\psi$
(127)
$\displaystyle\bar{\psi}\wedge\gamma_{5}\gamma\,\iota_{\xi}\nabla\psi-\frac{1}{4}\bar{\psi}\wedge\gamma_{5}\gamma\gamma_{cd}\iota_{\xi}\omega^{cd}\wedge\psi-\frac{1}{4}\bar{\psi}\gamma_{5}\wedge\gamma\wedge\gamma_{cd}\omega^{cd}\iota_{\xi}\psi=\bar{\psi}\gamma_{5}e^{a}\gamma_{a}\iota_{\xi}d\psi$
(128)

We are left with $\bar{\psi}\gamma_{5}T^{a}\gamma_{a}\iota_{\xi}\psi$ that
disappears thanks to the cyclic identity 95.

###### Acknowledgements.

The author thanks the contribution of her supervisor Malcolm Perry, in
motivating, discussing and reviewing the contents of this paper and Jonathan
Crabbé and Filipe Miguel for useful feedback. The author is jointly funded by
the University of Cambridge, the Cambridge Trust and King’s College.

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