aainstitutetext: Institute of Theoretical Physics,
China West Normal University,
Nanchong, 637009, Chinabbinstitutetext: Center for Theoretical Physics,
College of Physical Science and Technology,
Sichuan University,
Chengdu, 610051, China

# Remnants, fermions’ tunnelling and effects of quantum gravity

D.Y. Chen a Q.Q. Jiang b P. Wang b and H. Yang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The remnants are investigated by fermions’ tunnelling from a 4-dimensional
charged dilatonic black hole and a 5-dimensional black string. Based on the
generalized uncertainty principle, effects of quantum gravity are taken into
account. The quantum numbers of the emitted fermions affects the Hawking
temperatures. For the black hole, the quantum gravity correction slows down
the increase of the temperature, which leads to the remnant left in the
evaporation. For the black string, the existence of the remnant is determined
by the quantum gravity correction and effects of the extra compact dimension.

##  1 Introduction

The standard Hawking formula predicts the complete evaporation of black holes.
In the original research SWH , the formula was gotten in the frame of quantum
field theory on curved spacetime. It is based on the Heisenberg uncertainty
principle (HUP). Therefore, it is natural to find that the complete
evaporation is a direct consequence of the HUP. The semi-classical tunnelling
method put forward by Parikh and Wilczek is an effective way to research on
Hawking radiation PW . With the consideration of the variable background
spacetime, the tunnelling behavior of photons across the horizons was
described veritably. The corrected temperatures were gotten and higher than
the standard Hawking temperatures SWH . This result indicates that the
variable spacetimes speed up the increases of the temperatures and the black
holes evaporate completely. The extension of this method to the tunnelling
radiation of massive scalar particles was found in the subsequent work ZZ ;
JWC . The Hamilton-Jacobi ansatz is another version of the tunnelling method
ANVZ ; KM1 . Adopting this ansatz, the standard Hawking temperatures were
recovered by fermions’ tunnelling across the horizons of the black holes KM2 .
All of these results lead to that black holes evaporate completely and there
are no remnants left AAS .

On the other hand, various theories of quantum gravity, such as string theory,
loop quantum gravity and quantum geometry, predict the existence of the
minimal observable length PKT ; ACV ; KPP ; LJG ; GAC ; NIC2 . This view is
supported by the Gedanken experiment FS . An effective model to realize this
minimal length is the generalized uncertainty principle (GUP),

$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\left[1+\beta(\Delta
p)^{2}\right],$ (1)

which is derived by the modified fundamental commutation relations.
$\beta=\beta_{0}\frac{l^{2}_{p}}{\hbar^{2}}$ is a small value,
$\beta_{0}<10^{34}$ is a dimensionless parameter and $l_{p}$ is the Planck
length. Kempf et. al. first modified the commutation relations and got
$\left[x_{i},p_{j}\right]=i\hbar\delta_{ij}\left[1+\beta p^{2}\right]$, where
$x_{i}$ and $p_{i}$ are position and momentum operators defined respectively
as KMM

$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle x_{0i},$ $\displaystyle
p_{i}$ $\displaystyle=$ $\displaystyle p_{0i}(1+\beta p^{2}),$ (2)

$x_{0i}$ and $p_{0j}$ satisfy the canonical commutation relations
$\left[x_{0i},p_{0j}\right]=i\hbar\delta_{ij}$. The modification is not
unique. Other modifications are referred to AK ; FB ; ADV ; DV ; NIC3 .

These modifications play an important role on the black hole physics. Based on
the GUP, it was found that there is no existence of black holes at LHC in AFA
. The black hole thermodynamics was discussed in XW ; BJM ; KP ; ZDM ,
respectively. The relation between the area and entropy and the corrected
Hawking temperatures were gotten. An interested result is that the remnants
exist in black holes’ evaporation ACS ; SGC ; BG ; LX ; NS ; NIC1 .
Incorporating the GUP into the tunnelling radiation in scalar fields, the
corrected Hawking temperatures in the Schwarzschild and the noncommutative
spacetimes were obtained NS ; NM . Using the modified commutation relation
between the radial coordinate and the conjugate momentum and considering the
natural cutoffs as minimal and maximal momentum, the tunnelling rates were
derived in NS . The interesting result is that the minimal mass and the
maximum temperature in the scalar field were found.

In this paper, taking into account effects of quantum gravity, we investigate
the tunnelling radiation of fermions from a 4-dimensional charged dilatonic
black hole and a 5-dimensional black string. The remnants are discussed by the
corrected Hawking temperatures. The temperatures are affected by the quantum
numbers (mass, charge and energy) of the emitted fermions. For the dilatonic
black hole, the quantum gravity correction slows down the increase of the
Hawking temperature. It is natural to lead to the remnant existed in the
evaporation. In the black string spacetime, the quantum gravity correction and
the effect of the extra compact dimension affect the evaporation.

The rest is outlined as follows. In the next section, based on the modified
commutation relations put forward in KMM , we modify the Dirac equation in
curved spacetime. In section 3, with the consideration of effects of quantum
gravity, the fermion’s tunnelling from the charged dilatonic black hole is
investigated and the remnant is derived. In section 4, we investigate the
fermion’s tunnelling from the black string. The evaporation of the string is
discussed. Section 5 is devoted to our conclusion.

##  2 Generalized Dirac equation

In this section, we adopt the modified operators of position and momentum in
eqn. (2) to modify the Dirac equation in curved spacetime. To achieve this
purpose, we first introduce the GUP into Dirac equation. We then generalize
Dirac equation to curved background by standard process. Respecting covariance
is certainly of importance during the derivations. Under this constraint, the
modification of Dirac equation in flat spacetime based on GUP can be uniquely
determined NK ; HBH ; OS ; MK . The square of momentum operators is

$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle
p_{i}p^{i}=-\hbar^{2}\left[{1-\beta\hbar^{2}\left({\partial_{j}\partial^{j}}\right)}\right]\partial_{i}\cdot\left[{1-\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)}\right]\partial^{i}$
(3) $\displaystyle\simeq$
$\displaystyle-\hbar^{2}\left[{\partial_{i}\partial^{i}-2\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)\left({\partial^{i}\partial_{i}}\right)}\right].$

In the last step, the higher order terms of $\beta$ are neglected. In the
theory of quantum gravity, the generalized frequency takes on the form as WG

$\displaystyle\tilde{\omega}=E(1-\beta E^{2}),$ (4)

with the definition of energy operator $E=i\partial_{t}$. Using the energy
mass shell condition $p^{2}+m^{2}=E^{2}$, we get the generalized expression of
energy NS ; WG ; NK ; HBH

$\displaystyle\tilde{E}=E[1-\beta(p^{2}+m^{2})].$ (5)

Then, in flat background, the modified Dirac equation based on GUP follows
straightforwardly as in NK . In curved spacetime, the Dirac equation with an
electromagnetic field takes on the form as

$\displaystyle
i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}+\frac{i}{\hbar}eA_{\mu}\right)\psi+\frac{m}{\hbar}\psi=0,$
(6)

where $\Omega_{\mu}\equiv\frac{i}{2}\omega_{\mu}\,^{ab}\Sigma_{ab}$,
$\Sigma_{ab}=\frac{i}{4}\left[{\gamma^{a},\gamma^{b}}\right]$,
$\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}$, $\omega_{\mu}\,^{ab}$ is the spin
connection defined by
$\omega_{\mu}\,^{a}\,{}_{b}=e_{\nu}\,^{a}e^{\lambda}\,_{b}\Gamma^{\nu}_{\mu\lambda}-e^{\lambda}\,_{b}\partial_{\mu}e_{\lambda}\,^{a}$,
$\Gamma^{\nu}_{\mu\lambda}$ is the ordinary connection and $e^{\lambda}\,_{b}$
is the tetrad. The Greek indices are raised and lowered by the curved metric
$g_{\mu\nu}$, while the Latin indices are governed by the flat metric
$\eta_{ab}$. The construction of a tetrad satisfies the following relations

$g_{\mu\nu}=e_{\mu}\,^{a}e_{\nu}\,^{b}\eta_{ab},\hskip
14.22636pt\eta_{ab}=g_{\mu\nu}e^{\mu}\,_{a}e^{\nu}\,_{b},\hskip
14.22636pte^{\mu}\,_{a}e_{\nu}\,^{a}=\delta^{\mu}_{\nu},\hskip
14.22636pte^{\mu}\,_{a}e_{\mu}\,^{b}=\delta_{a}^{b}.$ (7)

Therefore, it is readily to construct the $\gamma^{\mu}$’s in curved spacetime
as

$\gamma^{\mu}=e^{\mu}\,_{a}\gamma^{a},\hskip
19.91692pt\left\\{{\gamma^{\mu},\gamma^{\nu}}\right\\}=2g^{\mu\nu}.$ (8)

To modify the Dirac equation, we rewrite eqn. (6) as

$\displaystyle-i\gamma^{0}\partial_{0}\psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\psi,$
(9)

namely,

$\displaystyle
i\partial_{0}\psi=-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\psi.$
(10)

The left hand side of the above equation is related to the energy. Using the
generalized expression of energy (5), we get the modified Dirac equation as
follows NK ; HBH

$\displaystyle i\partial_{0}\Psi$ $\displaystyle=$
$\displaystyle-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1-\beta
p^{2}-\beta m^{2}\right)\Psi$ (11) $\displaystyle=$
$\displaystyle-\gamma_{0}\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\Psi.$

The last equal sign was derived by the expression of the square of momentum
operators in eqn. (3) and the neglect of the higher order terms of $\beta$. In
this equation, $\Psi$ is the generalized Dirac field. Thus the modified Dirac
equation in curved spacetime is

$\displaystyle-i\gamma^{0}\partial_{0}\Psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\Psi,$ (12)

which is rewritten as

$\displaystyle\left[i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta
m^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{m}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right.$
$\displaystyle\left.+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right]\Psi=0.$ (13)

When $eA_{\mu}=0$, it describes the Dirac equation without the electromagnetic
field. In the following sections, we adopt eqn. (13) to investigate fermions’
tunnelling across the horizons of the 4-dimensional and the 5-dimensional
spacetimes. If one only considers the modification of momenta in the Dirac
equation. It is not covariant. From covariance, therefore, we modified the
momenta and energy in the Dirac equation.

##  3 The remnant in the 4-dimensional dilatonic black hole

The general solution of dilatonic black holes GHSHH was derived from the
action

$\displaystyle
S=\int{dx^{4}\sqrt{-g}\left[-R+2(\Delta\Phi)^{2}+e^{-2\alpha\Phi}F^{2}\right]},$
(14)

which describes the standard matter, gravity coupled to a Maxwell field and a
dilaton. $\alpha$ is a parameter expressed the strength of coupling of the
dilation field $\Phi$ to the Maxwell field $F$. It reduces to the usual
Einstein-Maxwell scalar theory when $\alpha=0$, while it is part of the low
energy action of string theory when $\alpha=1$. The metric of the spherically
symmetric charged dilatonic black hole is given by

$\displaystyle
ds^{2}=-f\left(r\right)dt^{2}+\frac{1}{f\left(r\right)}dr^{2}+R^{2}(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (15)

with the electromagnetic potential
$A_{\mu}=\left(A_{t},0,0,0\right)=\left(\frac{Q}{r},0,0,0\right)$, where

$\displaystyle R\left(r\right)$ $\displaystyle=$ $\displaystyle
r\left(1-\frac{r_{-}}{r}\right)^{\frac{\alpha^{2}}{1+\alpha^{2}}},$
$\displaystyle f\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}.$
(16)

The event horizon is located at $r=r_{+}$ for all $\alpha$ and $r_{+}>r_{-}$.
The mass and charge of the black hole are represented by
$M=\frac{r_{+}}{2}+\frac{r_{-}}{2}\left(\frac{1-\alpha^{2}}{1+\alpha^{2}}\right)$
and $Q=\sqrt{\frac{r_{+}r_{-}}{1+\alpha^{2}}}$, respectively.

For a spin-1/2 fermion, there are two states corresponding to spin up and spin
down. In this paper, we only consider the state with spin up without loss of
generality. The investigation of the state with spin down is parallel. The
motion of a fermion in the dilaton black hole obeys the generalized Dirac
equation (13). To describe the motion, we first suppose that the wave function
of the fermion with spin up state takes on the form as

$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\theta,\phi\right)\right),$
(21)

where $I$ is the action of the fermion with spin up state and $A$ and $B$ are
functions of $(t,r,\theta,\phi)$. To solve the equation (13), one should
construct gamma matrices. The construction of gamma matrices is relied on a
tetrad. It is straightforward to get the tetrad from the metric (15) as

$\displaystyle
e_{\mu}^{a}=\rm{diag}\left(\sqrt{f},1/\sqrt{f},R,R\sin\theta\right).$ (22)

Then the gamma matrices is constructed as

$\displaystyle\gamma^{t}=\frac{1}{\sqrt{f\left(r\right)}}\left(\begin{array}[]{cc}0&\rm
I\\\ -\rm I&0\end{array}\right),$
$\displaystyle\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (27)
$\displaystyle\gamma^{r}=\sqrt{f\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right).$ (32)

In the above equations, $\sqrt{g^{\theta\theta}}=R^{-1}$,
$\sqrt{g^{\phi\phi}}=(R\sin\theta)^{-1}$, $\rm I$ is the unit matrix,
$\sigma^{i}$ are the Pauli matrices,

$\displaystyle\sigma^{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma^{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma^{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (39)

We insert the gamma matrices and the wave function into the equation (13) and
divide by the exponential term. Applying the WKB approximation, we get the
resulting equations to leading order in $\hbar$. They are decoupled into four
equations

$\displaystyle-B\frac{1}{\sqrt{f}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}I-Am\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]$
$\displaystyle+B\beta\sqrt{f}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]+Am\left(1-\beta
m^{2}\right)$ $\displaystyle-B\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\beta\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right)\right]=0,$
(40)

$\displaystyle A\frac{1}{\sqrt{f}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}I-Bm\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]$
$\displaystyle+A\beta\sqrt{f}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]+Bm\left(1-\beta
m^{2}\right)$ $\displaystyle+A\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\beta\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right)\right]=0,$
(41)

$\displaystyle A\left\\{-(1-\beta
m^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right.$
$\displaystyle\left.-i(1-\beta
m^{2})\sqrt{g^{\phi\phi}}\partial_{\phi}I+i\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right\\}=0,$
(42)

$\displaystyle B\left\\{-(1-\beta
m^{2})\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right.$
$\displaystyle\left.-i(1-\beta
m^{2})\sqrt{g^{\phi\phi}}\partial_{\phi}I+i\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}\right]\right\\}=0.$
(43)

It is difficult to solve the action from the above equations. Considering
properties of the dilatonic spacetime and the above equations, we carry out
separation of variables on the action and get

$I=-\omega t+W(r)+\Xi(\theta,\phi),$ (44)

where $\omega$ is the energy of the emitted fermion.

From eqns. (40)-(43), it is found that eqn. (42) and eqn. (43) are irrelevant
to $A$ and $B$ and can be reduced into the same equation. Then inserting eqn.
(44) into eqn. (42) and eqn. (43) and rewriting the equation yield

$\displaystyle\left(\sqrt{g^{\theta\theta}}\partial_{\theta}\Xi+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Xi\right)\times$
$\displaystyle\left[\beta g^{rr}(\partial_{r}W)^{2}+\beta
g^{\theta\theta}(\partial_{\theta}\Xi)^{2}+\beta
g^{\phi\phi}(\partial_{\phi}\Xi)^{2}+\beta m^{2}-1\right]=0,$ (45)

which implies

$\displaystyle\sqrt{g^{\theta\theta}}\partial_{\theta}\Xi+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Xi=0.$
(46)

Thus the solution of $\Xi$ can be gotten. It is a complex function other than
the trivial solution of constant. This complex function produces a
contribution on the action. However, it has no contribution on the tunelling
rate since the contributions of the outgoing and ingoing solutions are
canceled in the calculation. From the above equation, it is easily to derive
the relation

$\displaystyle
g^{\theta\theta}\left(\partial_{\theta}\Xi\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}\Xi\right)^{2}=0.$
(47)

Now we focus our attention on the radial action. Inserting eqn. (44) into
eqns. (40) and (41) and using eqn. (47), we get

$\displaystyle B\frac{\omega}{\sqrt{f}}-B\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}W-Am\beta
g^{rr}\left(\partial_{r}W\right)^{2}+B\beta\sqrt{f}g^{rr}\left(\partial_{r}W\right)^{3}$
$\displaystyle-B\frac{eA_{t}}{\sqrt{f}}\left(1-\beta m^{2}-\beta
g^{rr}\left(\partial_{r}W\right)^{2}\right)+Am\left(1-\beta m^{2}\right)=0,$
(48)

$\displaystyle-A\frac{\omega}{\sqrt{f}}-A\left(1-\beta
m^{2}\right)\sqrt{f}\partial_{r}W-Bm\beta
g^{rr}\left(\partial_{r}W\right)^{2}+B\beta\sqrt{f}g^{rr}\left(\partial_{r}W\right)^{3}$
$\displaystyle+A\frac{eA_{t}}{\sqrt{f}}\left(1-\beta m^{2}-\beta
g^{rr}\left(\partial_{r}W\right)^{2}\right)+Bm\left(1-\beta m^{2}\right)=0.$
(49)

In the above equations, $A$ and $B$ are irrelevant to the result. Eliminating
them yields

$C_{6}\left(\partial_{r}W\right)^{6}+C_{4}\left(\partial_{r}W\right)^{4}+C_{2}\left(\partial_{r}W\right)^{2}+C_{0}=0,$
(50)

where

$\displaystyle C_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}f^{4},$
$\displaystyle C_{4}$ $\displaystyle=$ $\displaystyle\beta
f^{3}\left(m^{2}\beta-2\right)-\beta^{2}f^{2}e^{2}A_{t}^{2},$ $\displaystyle
C_{2}$ $\displaystyle=$ $\displaystyle
f^{2}\left(1-\beta^{2}m^{4}\right)+2\beta feA_{t}\left[-\omega+eA_{t}(1-\beta
m^{2})\right],$ $\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle-m^{2}f\left(1-\beta m^{2}\right)^{2}-\left[\omega-
eA_{t}\left(1-\beta m^{2}\right)\right]^{2}.$ (51)

Keeping the leading order terms of $\beta$ and solving $W$ at the event
horizon, we derive the imaginary part of the radial action

$\displaystyle ImW_{\pm}$ $\displaystyle=$ $\displaystyle\pm\int
dr\frac{1}{f}\sqrt{m^{2}f+\left[\omega-eA_{t}(1-\beta
m^{2})\right]^{2}}\left(1+\beta
m^{2}+\beta\frac{\tilde{\omega}^{2}}{f}-\frac{\beta
eA_{t}\tilde{\omega}}{f}\right)$ (52) $\displaystyle=$
$\displaystyle\pm\pi\frac{r_{+}}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}(\omega-
eA_{t+})\times\left(1+\beta\Pi\right),$

where

$\displaystyle\tilde{\omega}$ $\displaystyle=$ $\displaystyle\omega-eA_{t},$
$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle
m^{2}+\frac{m^{2}}{\omega_{0}}eA_{t+}+\frac{1}{2}\frac{m^{2}\left(\omega_{0}-eA_{t+}\right)}{\omega-
eA_{t+}(1-\beta
m^{2})}-\frac{eA_{t+}\left(\omega_{0}-eA_{t+}\right)}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}$
(53)
$\displaystyle+\frac{\omega_{0}}{\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}\left[2\omega_{0}+eA_{t+}-\frac{r_{-}(1-\alpha^{2})\left(2\omega_{0}-eA_{t+}\right)}{(r_{+}-r_{-})(1+\alpha^{2})}-\frac{2e^{2}Q^{2}}{\omega_{0}r_{+}}\right],$

$+(-)$ denotes the outgoing(ingoing) solutions, $\omega_{0}=\omega-eA_{t+}$,
$A_{t}=\frac{Q}{r}$, $A_{t+}=\frac{Q}{r_{+}}$ is the electromagnetic potential
at the event horizon. Using the relations between $M$, $Q$ and $r_{\pm}$, it
is found that $\Pi>0$. Thus the tunnelling rate of the fermion with spin up
state at the event horizon is

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{P_{(emission)}}{P_{(absorption)}}=\frac{\exp{\left(-\frac{2}{\hbar}ImI_{+}\right)}}{\exp{\left(-\frac{2}{\hbar}ImI_{-}\right)}}=\frac{exp\left(-\frac{2}{\hbar}ImW_{+}-\frac{2}{\hbar}Im{\Xi}\right)}{exp\left(-\frac{2}{\hbar}ImW_{-}-\frac{2}{\hbar}Im{\Xi}\right)}$
(54) $\displaystyle=$ $\displaystyle
exp\left[-4\pi\frac{r_{+}}{\hbar\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}(\omega-
eA_{t+})\times\left(1+\beta\Pi\right)\right].$

This is the Boltzmann factor with the Hawking temperature at the event horizon
of the dilatonic black hole taking

$\displaystyle
T=\frac{\hbar\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}}{4\pi
r_{+}\left(1+\beta\Pi\right)}=T_{0}\left(1-\beta\Pi\right),$ (55)

where $T_{0}=\frac{\hbar}{4\pi
r_{+}}\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}$
is the standard Hawking temperature. It is evidently that the corrected
Hawking temperature appears and is lower than the standard one. This
temperature is also lower than that derived by the semi-classical tunnelling
method PW . The correction value is determined not only by the mass and charge
of the black hole but also by the quantum number (energy, mass and charge) of
the emitted fermion. Due to the radiation, the temperature increases. Eqn.
(55) shows that the quantum gravity correction slows down the increase of the
temperature during the radiation. This correction therefore causes the
radiation ceased at some particular temperature, leaving the remnant mass.

Eqn. (55) describes the temperature of the Reissner-Nordstrom black hole when
$\alpha=0$ and that of the Schwarzschild black hole when $\alpha=0$ and $Q=0$.
Using an assumption that the emitted particle is massless, we estimate the
remnant of the Schwarzschild black hole. The evaporation stops when
$\left(M-dM\right)\left(1+\beta\omega^{2}\right)\simeq M$. With the
observation $dM=\omega$ and $\beta=\frac{\beta_{0}}{M_{p}^{2}}$, we get the
the remnant as
$M_{R}\simeq\frac{M_{p}^{2}}{\beta_{0}\omega}\geq\frac{M_{p}}{\beta_{0}}$,
where we assumed that the maximal energy of the emitted particle is the Planck
mass $M_{p}$ .

##  4 The remnant in the 5-dimensional black string

In this section, we investigate the remnant by the fermion’s tunnelling from a
5-dimensional spacetime. The emitted fermion is supposed to be uncharged.
Therefore, the electromagnetic effect in eqn. (13) is not taken into account
here. The 4-dimensional Schwarzschild metric is a static spherically symmetric
solution to the vacuum Einstein equations. When an extra compact spatial
dimension $z$ is added, the metric becomes

$\displaystyle
ds^{2}=-F\left(r\right)dt^{2}+\frac{1}{F\left(r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)+dz^{2},$ (56)

where $F=1-\frac{r_{h}}{r}$, $r_{h}=2M$ is the location of the event horizon
and $M$ is proportional to the black hole mass. The metric (56) describes a
neutral uniform black string. Here we investigate a fermion tunnelling from
this string.

We still only consider the spin up state. The wave function of the fermion
with spin up state is now assumed as

$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\theta,\phi,z\right)\right),$
(61)

$A$ and $B$ are functions of $(t,r,\theta,\phi,z)$. The fermion’s motion
satisfies the generalized Dirac equation. Now the tetrad is different from
that in the above section. It is
$e_{\mu}^{a}=\rm{diag}\left(\sqrt{F},1/\sqrt{F},r,r\sin\theta,1\right)$. Then
gamma matrices are constructed as follows

$\displaystyle\gamma^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{F\left(r\right)}}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),\hskip
19.91692pt\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (66) $\displaystyle\gamma^{r}$
$\displaystyle=$
$\displaystyle\sqrt{F\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),\hskip
19.91692pt\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right),$ (71) $\displaystyle\gamma^{z}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right).$ (74)

To apply the WKB approximation, we insert the wave function and the gamma
matrices into the Dirac equation and divide by the exponential term.
Multiplying by $\hbar$, the equations to leading order in $\hbar$ are obtained
and decoupled into four equations

$\displaystyle-iA\frac{1}{\sqrt{F}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{F}\partial_{r}I+Am\left(1-\beta m^{2}\right)+A\left(1-\beta
m^{2}\right)\partial_{z}I$
$\displaystyle+\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+\left(\partial_{z}I\right)^{2}\right]\left(B\sqrt{F}\partial_{r}I-A\partial_{z}I-Am\right)=0,$
(75)

$\displaystyle iB\frac{1}{\sqrt{F}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{F}\partial_{r}I+Bm\left(1-\beta m^{2}\right)-B\left(1-\beta
m^{2}\right)\partial_{z}I$
$\displaystyle+\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+\left(\partial_{z}I\right)^{2}\right]\left(A\sqrt{F}\partial_{r}I+B\partial_{z}I-Bm\right)=0,$
(76)

$\displaystyle
A\left(\sqrt{g^{\theta\theta}}\partial_{\theta}I+i\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)\times$
$\displaystyle\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}+(\partial_{z}I)^{2}+\beta
m^{2}-1\right]=0.$ (77)

$\displaystyle
B\left(\sqrt{g^{\theta\theta}}\partial_{\theta}I+i\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)\times$
$\displaystyle\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{\phi\phi}(\partial_{\phi}I)^{2}+(\partial_{z}I)^{2}+\beta
m^{2}-1\right]=0.$ (78)

It is also difficult to solve the action from the above equations. Considering
the property of the black string spacetime, we carry out separation of
variables as

$\displaystyle I=-\omega t+W(r)+\Theta(\theta,\phi)+Jz,$ (79)

where $\omega$ is the energy, $J$ is a conserved momentum and describes a
constant of motion corresponding to the compact dimension $z$.

We first focus our attention on the last two equations. They are irrelevant to
$A$ and $B$ and can be reduced to the same equation. Inserting eqn. (79) into
them, we get
$\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+i\sqrt{g^{\phi\phi}}\partial_{\phi}\Theta=0$
since the summation of factors in the square brackets in eqn. (77) and (78)
can not be zero. Thus the solution of $\Theta$ is a complex function (other
than the constant solution). The following relation,

$\displaystyle
g^{\theta\theta}\left(\partial_{\theta}\Theta\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}\Theta\right)^{2}=0,$
(80)

is easily obtained. Return to eqns. (76) and (77). Inserting eqns. (79) and
(80) into them and eliminating $A$ and $B$, we get the equation of the radial
action

$\displaystyle
D_{6}\left(\partial_{r}W\right)^{6}+D_{4}\left(\partial_{r}W\right)^{4}+D_{2}\left(\partial_{r}W\right)^{2}+D_{0}=0$
(81)

where

$\displaystyle D_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}F^{4},$
$\displaystyle D_{4}$ $\displaystyle=$ $\displaystyle-2\beta
xF^{3}-\left(m^{2}-J\right)\beta^{2}F^{3},$ $\displaystyle D_{2}$
$\displaystyle=$ $\displaystyle x^{2}F^{2}+2\beta
xF^{2}\left(m^{2}-J\right)-i2\beta J\omega F^{3/2},$ $\displaystyle D_{0}$
$\displaystyle=$
$\displaystyle-\left(m^{2}-J\right)x^{2}F-\omega^{2}+i2J\omega x\sqrt{F},$
$\displaystyle x$ $\displaystyle=$ $\displaystyle 1-\beta m^{2}-\beta J^{2}.$
(82)

Neglect higher order terms of $\beta$ and solve the equation (81) at the event
horizon. We only interest the imaginary part of the action because the
tunnelling rate is determined by it. The imaginary part is

$\displaystyle ImW_{\pm}$ $\displaystyle=$ $\displaystyle\pm
Im\int{dr\frac{\sqrt{\omega^{2}+\left(m^{2}-J^{2}\right)F-i2\sqrt{F}J\omega}}{F}\left[1+\beta\left(m^{2}-J^{2}+\frac{\omega^{2}-2iJ\omega\sqrt{F}}{F}\right)\right]}$
(83) $\displaystyle=$ $\displaystyle\pm\pi\omega
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right],$

where $+(-)$ are the outgoing(ingoing) solutions. Thus the tunnelling rate of
the uncharged fermion across the event horizon of the 5-dimensional black
string is

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\exp{\left(-\frac{2}{\hbar}ImI_{+}\right)}}{\exp{\left(-\frac{2}{\hbar}ImI_{-}\right)}}=\frac{exp\left(-\frac{2}{\hbar}ImW_{+}-\frac{2}{\hbar}Im{\Theta}\right)}{exp\left(-\frac{2}{\hbar}ImW_{-}-\frac{2}{\hbar}Im{\Theta}\right)}$
(84) $\displaystyle=$ $\displaystyle exp\left\\{-\frac{1}{\hbar}4\pi\omega
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right]\right\\},$

which shows that the Hawking temperature is

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{\hbar}{4\pi
r_{h}\left[1+\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right]}$
(85) $\displaystyle=$ $\displaystyle
T_{0}\left[1-\frac{1}{2}\beta\left(3m^{2}+4\omega^{2}-2J^{2}\right)\right].$

In the above equation, $T_{0}=\frac{\hbar}{4\pi r_{h}}$ is the standard
Hawking temperature. It shows that the corrected Hawking temperature is not
only determined by the quantum number (energy and mass) of the emitted fermion
but also affected by the effect of the extra compact dimension.

It is of interest to discuss the value of $3m^{2}+4\omega^{2}-2J^{2}$. When
$3m^{2}+4\omega^{2}>2J^{2}$, it is easily found that the corrected temperature
is lower than the standard Hawking temperature. This implies that the
combination of the quantum gravity correction and the effect of the extra
compact dimension slows down the increase of the temperature caused by the
radiation. Finally, the black string should be in a stable balanced state. At
this state, the remnant is left. The special case is $J=0$. In this case, the
fermion’s motion is limited in the 4-dimensional spacetime. Thus eqn. (85)
reduces to the temperature of the 4-dimensional Schwarzschild black hole.

When $3m^{2}+4\omega^{2}<2J^{2}$, the corrected temperature is higher than the
standard one. It shows that the black string accelerates the evaporation and
there is no remnant left. If $3m^{2}+4\omega^{2}=2J^{2}$, the effect of the
quantum gravity correction and that of the extra dimension are canceled. Then
the standard Hawking temperature appears and results in the complete
evaporation. Therefore, the evaporation of the black string is affected by the
quantum gravity correction and the effect of the extra compact dimension.

##  5 Conclusion

In this paper, based on the modified fundamental commutation relation, we
modified the HUP and investigated the fermions’ tunnelling across the horizons
of the 4-dimensional charged dilatonic black hole and the 5-dimensional
neutral black string. The corrected Hawking temperatures were gotten. The
remnants were discussed by the temperatures. For the dilatonic black hole, the
correction is determined not only by the mass and charge of the black hole but
also by the quantum number (mass, charge and energy) of the emitted fermion.
The interesting point is that the quantum gravity correction slows down the
increase of the Hawking temperature. It is natural to lead to the remnant left
in the evaporation. For the black string, the temperature is affected by the
quantum number (mass and energy) of the emitted fermion and the effect of
extra compact dimension. The existence of the remnant is determined by the
combined effect of the quantum gravity correction and the compact dimension.
In NIC4 , noncommutative black holes was discussed. In RMS ; SVZR , the
quantum tunnelling radiation were researched beyond the semiclassical
approximation. The corrected Hawking temperatures were derived and also lower
than the standard semiclassical Hawking temperatures.

###### Acknowledgements.

This work is supported in part by the NSFC (Grant Nos. 11205125, 11005016,
11005086, 11175039 and 11375121), the Innovative Research Team in College of
Sichuan Province (Grant No. 13TD0003) and SiChuan Province Science Foundation
for Youths (Grant No. 2012JQ0039).

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&-2{u^{\lambda}}_{\mu}{u^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}-4{u^{\prime\lambda}}_{\mu}{v^{\mu}}_{\nu}{t^{\nu}}_{\lambda}-4{u^{\lambda}}_{\mu}{v^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}+2\bar{\psi}^{+\,\lambda}\nabla\psi^{+\,\lambda}+2\bar{\psi}^{-\,\lambda}\nabla\psi^{-\,\lambda}\bigg{]}\\\
&-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}+\alpha_{1}{t^{\lambda}}_{\mu}{s^{\mu}}_{\lambda}+\alpha_{1}{t^{\prime\lambda}}_{\mu}{s^{\prime\mu}}_{\lambda}+2\alpha_{1}{u^{\lambda}}_{\mu}{v^{\prime\mu}}_{\lambda}\right)\Bigg{\\}}\,,\end{split}$
(6.17)

which is written in terms of the super-Carroll curvatures appearing in (6.16)
(we emphasize that the latter must not be confused with the super-AdS Carroll
ones given in (4.17)). Note that the exotic term in (6.17), that is the one
proportional to $\alpha_{0}$, is now given by the Lorentz Lagrangian plus
additional terms involving the $1$-form fields $t^{\lambda\mu}$,
$t^{\prime\lambda\mu}$, and $u^{\lambda\mu}$. Let us further mention that the
action (6.17) can also be derived by using the non-vanishing components of the
invariant tensor obtained by taking the limit $\ell\rightarrow\infty$ of
(4.20) together with the connection $1$-form for the
$\mathcal{N}=(\mathcal{N},0)$ (flat) Carroll superalgebra (6.15) in the
general expression (2.11). In particular, taking the $\ell\rightarrow\infty$
limit of (4.20) we are left with the non-vanishing components

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\
&\langle{T}^{\lambda\mu}{T}^{\nu\rho}\rangle=\langle{T}^{\prime\lambda\mu}{T}^{\prime\nu\rho}\rangle=2\alpha_{0}\left(\delta^{\lambda\rho}\delta^{\nu\mu}-\delta^{\lambda\nu}\delta^{\rho\mu}\right)\,,\\\
&\langle{U}^{\lambda\mu}{U}^{\nu\rho}\rangle=-2\alpha_{0}\delta^{\lambda\nu}\delta^{\rho\mu}\,,\\\
&\langle{T}^{\lambda\mu}{S}^{\nu\rho}\rangle=\langle{T}^{\prime\lambda\mu}{S}^{\prime\nu\rho}\rangle=-2\alpha_{1}\left(\delta^{\lambda\rho}\delta^{\nu\mu}-\delta^{\lambda\nu}\delta^{\rho\mu}\right)\,,\\\
&\langle{U}^{\lambda\mu}{V}^{\nu\rho}\rangle=2\alpha_{1}\delta^{\lambda\nu}\delta^{\rho\mu}\,,\\\
&\langle Q^{+\,\lambda}_{\alpha}Q^{+\,\mu}_{\beta}\rangle=\langle
Q^{-\,\lambda}_{\alpha}Q^{-\,\mu}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{\lambda\mu}\,.\end{split}$
(6.18)

The action (6.17) is invariant by construction under the super-Carroll group
associated with (6.15); in particular, concerning the $\ell\rightarrow\infty$
limit of the local gauge transformations (4.24), we obtain

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}\,,\\\ \delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\delta
t^{\lambda\mu}&=d\varrho^{\lambda\mu}-2{\varrho^{[\lambda}}_{\nu}t^{\nu|\mu]}-2{\varphi^{\prime[\lambda}}_{\nu}u^{\nu|\mu]}\,,\\\
\delta
t^{\prime\lambda\mu}&=d\varrho^{\prime\lambda\mu}-2{\varrho^{\prime[\lambda}}_{\nu}t^{\prime\nu|\mu]}-2{\varphi^{[\lambda}}_{\nu}u^{\prime\nu|\mu]}\,,\\\
\delta
u^{\lambda\mu}&=d\varphi^{\lambda\mu}-{\varphi^{\lambda}}_{\nu}t^{\nu\mu}+{u^{\lambda}}_{\nu}\varrho^{\nu\mu}-{\varrho^{\prime\lambda}}_{\nu}u^{\nu\mu}+{t^{\prime\lambda}}_{\nu}\varphi^{\nu\mu}\,,\\\
\delta
s^{\lambda\mu}&=d\vartheta^{\lambda\mu}-2{\varrho^{[\lambda}}_{\nu}s^{\nu|\mu]}+2{t^{[\lambda}}_{\nu}\vartheta^{\nu|\mu]}-2{\varphi^{\prime[\lambda}}_{\nu}v^{\nu|\mu]}+2{u^{\prime[\lambda}}_{\nu}\varsigma^{\nu|\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}\\\
&+\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
s^{\prime\lambda\mu}&=d\vartheta^{\prime\lambda\mu}-2{\varrho^{\prime[\lambda}}_{\nu}s^{\prime\nu|\mu]}+2{t^{\prime[\lambda}}_{\nu}\vartheta^{\prime\nu|\mu]}-2{\varphi^{[\lambda}}_{\nu}v^{\prime\nu|\mu]}+2{u^{[\lambda}}_{\nu}\varsigma^{\prime\nu|\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}\\\
&-\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
v^{\lambda\mu}&=d\varsigma^{\lambda\mu}-{\varsigma^{\lambda}}_{\nu}t^{\nu\mu}+{v^{\lambda}}_{\nu}\varrho^{\nu\mu}-{\varrho^{\prime\lambda}}_{\nu}v^{\nu\mu}+{t^{\prime\lambda}}_{\nu}\varsigma^{\nu\mu}-{\varphi^{\lambda}}_{\nu}s^{\nu\mu}+{u^{\lambda}}_{\nu}\vartheta^{\nu\mu}-{\vartheta^{\prime\lambda}}_{\nu}u^{\nu\mu}+{s^{\prime\lambda}}_{\nu}\varphi^{\nu\mu}\\\
&-\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}-\bar{\varepsilon}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\Gamma^{0}\psi^{+\,\mu]}\,,\\\
\delta\psi^{+\,\lambda}&=d\varepsilon^{+\,\lambda}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+\,\lambda}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}-\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{+}_{\mu}\\\
&+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}-\varphi^{(\lambda\mu)}\Gamma_{0}\psi^{+}_{\mu}+u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{+}_{\mu}-\varphi^{[\lambda\mu]}\Gamma_{0}\psi^{-}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{-}_{\mu}\,,\\\
\delta\psi^{-\,\lambda}&=d\varepsilon^{-\,\lambda}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-\,\lambda}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}+\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{+}_{\mu}\\\
&-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}+\varphi^{(\lambda\mu)}\Gamma_{0}\psi^{-}_{\mu}-u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{-}_{\mu}+\varphi^{[\lambda\mu]}\Gamma_{0}\psi^{+}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{+}_{\mu}\,.\end{split}$
(6.19)

Then, restricting ourselves to the supersymmetry transformations in the limit
$\ell\rightarrow\infty$, we find

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta k^{a}&=0\,,\\\ \delta
V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\delta t^{\lambda\mu}&=0\,,\\\ \delta t^{\prime\lambda\mu}&=0\,,\\\ \delta
u^{\lambda\mu}&=0\,,\\\ \delta
s^{\lambda\mu}&=\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
s^{\prime\lambda\mu}&=\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
v^{\lambda\mu}&=-\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}-\bar{\varepsilon}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\Gamma^{0}\psi^{+\,\mu]}\,,\\\
\delta\psi^{+\,\lambda}&=d\varepsilon^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+\,\lambda}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}+u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{+}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{-}_{\mu}\,,\\\
\delta\psi^{-\,\lambda}&=d\varepsilon^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-\,\lambda}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}-u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{-}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{+}_{\mu}\,.\end{split}$
(6.20)

Finally, we find that the equations of motion for the action (6.17) (flat
limit of the equations of motion given in (4.27)) read

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\ \delta
h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t^{\lambda\mu}&:\quad\alpha_{0}\mathcal{T}^{\lambda\mu}+\alpha_{1}\mathcal{S}^{\lambda\mu}=0\,,\\\
\delta
t^{\prime\lambda\mu}&:\quad\alpha_{0}\mathcal{T}^{\prime\lambda\mu}+\alpha_{1}\mathcal{S}^{\prime\lambda\mu}=0\,,\\\
\delta
u^{\lambda\mu}&:\quad-\alpha_{0}\mathcal{U}^{\lambda\mu}+\alpha_{1}\mathcal{V}^{\lambda\mu}=0\,,\\\
\delta s^{\lambda\mu}&:\quad\alpha_{1}\mathcal{T}^{\lambda\mu}=0\,,\\\ \delta
s^{\prime\lambda\mu}&:\quad\alpha_{1}\mathcal{T}^{\prime\lambda\mu}=0\,,\\\
\delta v^{\lambda\mu}&:\quad\alpha_{1}\mathcal{U}^{\lambda\mu}=0\,,\\\
\delta\psi^{+\,\lambda}&:\quad\alpha_{1}\nabla\psi^{+\,\lambda}=0\,,\\\
\delta\psi^{-\,\lambda}&:\quad\alpha_{1}\nabla\psi^{-\,\lambda}=0\,.\end{split}$
(6.21)

We can see that when $\alpha_{1}\neq 0$, the eqs. (6.21) reduce to the
vanishing of the curvature $2$-forms given in (6.16) ($\alpha_{1}\neq 0$ is a
sufficient condition to recover the vanishing of the curvature $2$-forms in
(6.16); the coefficient $\alpha_{0}$ can be consistently set to zero, making
the exotic term disappear from the action (6.17).

Let us observe that, restricting ourselves to the purely bosonic theory, we
end up with the $\mathcal{N}=(\mathcal{N},0)$ Carroll gravity theories (with
$\mathcal{N}$ even) in three dimensions, invariant under the
$\mathcal{N}=(\mathcal{N},0)$ Carroll algebra. At the purely bosonic level,
the fields $t^{\lambda\mu}$, $t^{\prime\lambda\mu}$, $u^{\lambda\mu}$,
$s^{\lambda\mu}$, $s^{\prime\lambda\mu}$, and $v^{\lambda\mu}$, and the
corresponding terms in the action, can also be consistently discarded by
performing an IW contraction.

On the other hand, considering the special case $\mathcal{N}=(2,0)$, that is
$x=1$, after some algebraic calculations, we can prove that the $(2,0)$ theory
in the flat limit previously discussed in this section is exactly reproduced.

###  6.4 $(p,q)$ Carroll supergravity theories from the
$\ell\rightarrow\infty$ limit

Applying the flat limit $\ell\rightarrow\infty$ to the (anti)commutation
relations given in (5.6), we get the following non-vanishing ones:

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[{T}^{ij},{T}^{kl}\right]=\delta^{jk}{T}^{il}-\delta^{ik}{T}^{jl}-\delta^{jl}{T}^{ik}+\delta^{il}{T}^{jk}\,,\\\
&\left[{T}^{IJ},{T}^{KL}\right]=\delta^{JK}{T}^{IL}-\delta^{IK}{T}^{JL}-\delta^{JL}{T}^{IK}+\delta^{IL}{T}^{JK}\,,\\\
&\left[{T}^{ij},{S}^{kl}\right]=\delta^{jk}{S}^{il}-\delta^{ik}{S}^{jl}-\delta^{jl}{S}^{ik}+\delta^{il}{S}^{jk}\,,\\\
&\left[{T}^{IJ},{S}^{KL}\right]=\delta^{JK}{S}^{IL}-\delta^{IK}{S}^{JL}-\delta^{JL}{S}^{IK}+\delta^{IL}{S}^{JK}\,,\\\
&\left[J_{ab},Q^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{i}\right)_{\alpha}\,,\\\
&\left[J_{ab},Q^{I}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{I}\right)_{\alpha}\,,\\\
&\left[{T}^{ij},{Q}^{k}_{\alpha}\right]=\delta^{jk}{Q}^{i}_{\alpha}-\delta^{ik}{Q}^{j}_{\alpha}\,,\\\
&\left[{T}^{IJ},{Q}^{K}_{\alpha}\right]=\delta^{JK}{Q}^{I}_{\alpha}-\delta^{IK}{Q}^{J}_{\alpha}\,,\\\
&\left\\{Q^{i}_{\alpha},Q^{j}_{\beta}\right\\}=\delta^{ij}\left(\Gamma^{0}C\right)_{\alpha\beta}H+C_{\alpha\beta}S^{ij}\,,\\\
&\left\\{Q^{I}_{\alpha},Q^{J}_{\beta}\right\\}=\delta^{ij}\left(\Gamma^{0}C\right)_{\alpha\beta}H-C_{\alpha\beta}S^{IJ}\,.\end{split}$
(6.22)

These are the (anti)commutation relations of the $\mathcal{N}=(p,q)$, $D=3$
super-Carroll algebra, and we could also have obtained the latter by applying
the Carroll contraction to the semi-direct extension of the
$\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ automorphism algebra by the
$\mathcal{N}=(p,q)$, $D=3$ Poincaré superalgebra (see Ref. [40]).

Then, as $\ell\rightarrow\infty$, the $2$-form curvatures (5.9) become

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}=R^{ab}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}=\mathfrak{K}^{a}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{i}\Gamma^{0}\psi^{i}-\frac{1}{2}\bar{\psi}^{I}\Gamma^{0}\psi^{I}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{i}\Gamma^{0}\psi^{i}-\frac{1}{2}\bar{\psi}^{I}\Gamma^{0}\psi^{I}\,,\\\
\mathcal{T}^{ij}&=dt^{ij}+{t^{i}}_{k}t^{kj}\,,\\\
\mathcal{T}^{IJ}&=dt^{IJ}+{t^{I}}_{K}t^{KJ}\,,\\\
\mathcal{S}^{ij}&=ds^{ij}+2{t^{i}}_{k}s^{kj}-\bar{\psi}^{i}\psi^{j}\,,\\\
\mathcal{S}^{IJ}&=ds^{IJ}+2{t^{I}}_{K}s^{KJ}+\bar{\psi}^{I}\psi^{J}\,,\\\
\nabla\psi^{i}&=d\psi^{i}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{i}+t^{ij}\psi_{j}\,,\\\
\nabla\psi^{I}&=d\psi^{I}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{I}+t^{IJ}\psi_{J}\,,\end{split}$
(6.23)

and by applying the $\ell\rightarrow\infty$ limit to the CS action (5.11), we
get

$\begin{split}I^{(p,q)}_{CS}|_{\ell\rightarrow\infty}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+2{t^{i}}_{j}{dt^{j}}_{i}+\frac{4}{3}{t^{i}}_{j}{t^{j}}_{k}{t^{k}}_{i}+2{t^{I}}_{J}{dt^{J}}_{I}+\frac{4}{3}{t^{I}}_{J}{t^{J}}_{K}{t^{K}}_{I}\right)\\\
&+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}-2{t^{i}}_{j}\left({ds^{j}}_{i}+{t^{j}}_{k}{s^{k}}_{i}\right)+2{t^{I}}_{J}\left({ds^{J}}_{I}+{t^{J}}_{K}{s^{K}}_{I}\right)\\\
&+2\bar{\psi}^{i}\nabla\psi^{i}+2\bar{\psi}^{I}\nabla\psi^{I}\bigg{]}-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}+\alpha_{1}{t^{i}}_{j}{s^{j}}_{i}-\alpha_{1}{t^{I}}_{J}{s^{J}}_{I}\right)\Bigg{\\}}\,,\end{split}$
(6.24)

which is written in terms of the super-Carroll curvatures appearing in (6.23)
(let us stress that the latter must not be confused with the super-AdS Carroll
ones given in (5.9)). We can see that the exotic term appearing in (6.24),
namely the one proportional to $\alpha_{0}$, now is given by the Lorentz
Lagrangian plus additional terms involving the $1$-form fields $t^{ij}$ and
$t^{IJ}$.

Notice that the action (6.24) can also be derived by using the non-vanishing
components of the invariant tensor obtained by taking the limit
$\ell\rightarrow\infty$ of (5.10) together with the connection $1$-form for
the $\mathcal{N}=(p,q)$ (flat) Carroll superalgebra (6.22) in the general
expression (2.11). Specifically, the limit $\ell\rightarrow\infty$ of (5.10)
gives us the following non-vanishing components:

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\
&\langle{T}^{ij}{T}^{kl}\rangle=2\alpha_{0}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle{T}^{IJ}{T}^{KL}\rangle=2\alpha_{0}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle{T}^{ij}{S}^{kl}\rangle=-2\alpha_{1}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle{T}^{IJ}{S}^{KL}\rangle=2\alpha_{1}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle
Q^{i}_{\alpha}Q^{j}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{ij}\,,\\\
&\langle
Q^{I}_{\alpha}Q^{J}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{IJ}\,.\end{split}$
(6.25)

The CS action (6.24) is invariant by construction under the super-Carroll
group associated with (6.22), and, in particular, concerning the
$\ell\rightarrow\infty$ limit of the local gauge transformations (5.14), we
get

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}\,,\\\ \delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{i}\Gamma^{0}\psi^{i}+\bar{\varepsilon}^{I}\Gamma^{0}\psi^{I}\,,\\\
\delta t^{ij}&=d\varphi^{ij}-2{\varphi^{[i}}_{k}t^{k|j]}\,,\\\ \delta
t^{IJ}&=d\varphi^{IJ}-2{\varphi^{[I}}_{K}t^{K|J]}\,,\\\ \delta
s^{ij}&=d\varsigma^{ij}-2{\varphi^{[i}}_{k}s^{k|j]}+2{t^{[i}}_{k}\varsigma^{k|j]}+2\bar{\varepsilon}^{[i}\psi^{j]}\,,\\\
\delta
s^{IJ}&=d\varsigma^{IJ}+2{\varphi^{[I}}_{K}s^{K|J]}+2{t^{[I}}_{K}\varsigma^{K|J]}-2\bar{\varepsilon}^{[I}\psi^{J]}\,,\\\
\delta\psi^{i}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{i}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{i}-\varphi^{ij}\psi_{j}+t^{ij}\varepsilon_{j}\,,\\\
\delta\psi^{I}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{I}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{I}-\varphi^{IJ}\psi_{J}+t^{IJ}\varepsilon_{J}\,.\end{split}$
(6.26)

Thus, restricting ourselves to the supersymmetry transformations in the limit
$\ell\rightarrow\infty$, we are left with

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta k^{a}&=0\,,\\\ \delta
V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{i}\Gamma^{0}\psi^{i}+\bar{\varepsilon}^{I}\Gamma^{0}\psi^{I}\,,\\\
\delta t^{ij}&=0\,,\\\ \delta t^{IJ}&=0\,,\\\ \delta
s^{ij}&=2\bar{\varepsilon}^{[i}\psi^{j]}\,,\\\ \delta
s^{IJ}&=-2\bar{\varepsilon}^{[I}\psi^{J]}\,,\\\
\delta\psi^{i}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{i}+t^{ij}\varepsilon_{j}\,,\\\
\delta\psi^{I}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{I}+t^{IJ}\varepsilon_{J}\,.\end{split}$
(6.27)

Concluding, the equations of motion for the action (6.24) (flat limit of the
equations of motion given in (5.16)) read as follows:

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta V^{a}&:\quad
2\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\ \delta
h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t^{ij}&:\quad-\alpha_{0}\mathcal{T}^{ij}+\alpha_{1}\mathcal{S}^{ij}=0\,,\\\
\delta
t^{IJ}&:\quad-\alpha_{0}\mathcal{T}^{IJ}-\alpha_{1}\mathcal{S}^{IJ}=0\,,\\\
\delta s^{ij}&:\quad\alpha_{1}\mathcal{T}^{ij}=0\,,\\\ \delta
s^{IJ}&:\quad\alpha_{1}\mathcal{T}^{IJ}=0\,,\\\
\delta\psi^{i}&:\quad\alpha_{1}\nabla\psi^{i}=0\,,\\\
\delta\psi^{I}&:\quad\alpha_{1}\nabla\psi^{I}=0\,.\end{split}$ (6.28)

When $\alpha_{1}\neq 0$, the eqs. (6.28) exactly reduce to the vanishing of
the curvature $2$-forms given in (6.23) ($\alpha_{1}\neq 0$ is a sufficient
condition to recover the vanishing of the curvature $2$-forms (6.23), and the
coefficient $\alpha_{0}$ can also be consistently set to zero, making the
exotic term disappear from the action (6.24)).

Restricting ourselves to the purely bosonic theory, we end up with the
$\mathcal{N}=(p,q)$ Carroll gravity theories in three dimensions, invariant
under the $\mathcal{N}=(p,q)$ Carroll algebra. At the purely bosonic level,
the fields $t^{ij}$, $t^{IJ}$, $s^{ij}$, and $s^{IJ}$, and the corresponding
terms in the action, can also be consistently discarded by performing an IW
contraction.

On the other hand, let us finally mention that, if we now consider the
particular case $p=q=1$, we exactly reproduce the results previously obtained
in this section for the $(1,1)$ theory in the flat limit.

All the studies of the flat limit presented in this section represent a new
development and generalization of the previous works concerning Carroll
superalgebras in three dimensions, in particular in the context of three-
dimensional CS supergravity theories.

##  7 Conclusions

Motivated by the recent development of applications of Carroll symmetries (in
particular, by their prominent role in the context of holography), and by the
fact that, nevertheless, the study of their supersymmetric extensions in the
context of supergravity models still remains poorly explored, in this paper we
have presented, in a systematic fashion, the ultra-relativistic
$\mathcal{N}$-extended AdS CS supergravity theories in three ($2+1$) spacetime
dimensions, which are invariant under $\mathcal{N}$-extended AdS Carroll
superalgebras, extending the results recently presented in [35] (where the
construction of the three-dimensional $\mathcal{N}=1$ CS supergravity theory
invariant under the so-called AdS Carroll superalgebra, ultra-relativistic
contraction of the $\mathcal{N}=1$ AdS superalgebra [14], together with the
study of its flat limit, has been presented for the first time). In
particular, we have applied the method introduced in [36] with the
improvements of [35] to construct the aforesaid ultra-relativistic
$\mathcal{N}$-extended AdS CS supergravity theories.

We have first considered the $(2,0)$ and $(1,1)$ cases, and subsequently
generalized our analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$
even integer, and to $\mathcal{N}=(p,q)$, that is $\mathcal{N}=p+q$, with
$p,q>0$. The $\mathcal{N}$-extended AdS Carroll superalgebras have been
obtained through the Carrollian (i.e., ultra-relativistic) contraction applied
to an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$, to
$\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$, to an
$\mathfrak{so}(\mathcal{N})$ extension of
$\mathfrak{osp}(2|\mathcal{N})\otimes\mathfrak{sp}(2)$, and to the direct sum
of an $\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ algebra and
$\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$, respectively.

An $\mathcal{N}=(2,0)$ AdS Carroll superalgebra in three dimensions was
previously introduced in [14]. Nevertheless, the latter does not allow for a
non-degenerate invariant tensor, meaning that one cannot construct a well-
defined CS action based on this superalgebra. To overcome this point, we have
considered an $so(2)$ extension of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$ and performed the ultra-
relativistic contraction on it, ending up with a new $\mathcal{N}=(2,0)$ AdS
Carroll superalgebra endowed with a non-degenerate invariant tensor. This has
allowed us to develop the three-dimensional CS supergravity action invariant
under this $\mathcal{N}=(2,0)$ AdS Carroll superalgebra. We have called this
action the $(2,0)$ AdS Carroll CS supergravity action. We have done an
analogous analysis in the $(1,1)$ case, and subsequently generalized our study
to $\mathcal{N}=(\mathcal{N},0)$ (with $\mathcal{N}$ even) and to
$\mathcal{N}=(p,q)$ (with $p,q>0$). In particular, after having introduced the
ultra-relativistic superalgebras, we have constructed the respective CS
supergravity theories in three-dimensions by exploiting the non-vanishing
components of the corresponding invariant tensor. The aforementioned actions
are all based on a non-degenerate, invariant bilinear form (i.e., an invariant
metric), and each of them is characterized by two coupling constants and
involve an exotic contribution. The results presented in this paper were also
open problems suggested in Ref. [14], and they represent the
$\mathcal{N}$-extended generalization of [35]. Interestingly, one can observe
that the CS formulation in the $\mathcal{N}$-extended cases
$\mathcal{N}=(\mathcal{N},0)$ and $\mathcal{N}=(p,q)$ requires the presence of
$\mathfrak{so}(\mathcal{N})$ and $\mathfrak{so}(p)\oplus\mathfrak{sp}(q)$
generators, respectively, also at the ultra-relativistic level, that is in the
Carroll limit; thus, what happens at the relativistic level for three-
dimensional $\mathcal{N}$-extended CS Poincaré and AdS supergravity theories
(see [40]), that is the need to introduce the aforementioned extra generators
(together with their dual $1$-form fields) in the theory in order to obtain a
non-degenerate invariant tensor, has repercussions also on (and still holds
at) the ultra-relativistic level.

We have also analyzed the flat limit $\ell\rightarrow\infty$ of the
aforementioned models, in which we have recovered the ultra-relativistic
$\mathcal{N}$-extended (flat) CS supergravity theories invariant under
$\mathcal{N}$-extended super-Carroll algebras. The flat limit has been applied
at the level of the superalgebras, CS actions, supersymmetry transformation
laws, and field equations. Also all the studies of the flat limit presented in
Section 6 represent a new development and generalization of the previous works
presented in the literature concerning Carroll (super)algebras in three
dimensions, in particular in the context of three-dimensional CS
(super)gravity theories.

The recently discovered relations among the Carrollian world and flat
holography suggest that this work might represents a starting point to go
further in the analysis of supersymmetry invariance of flat supergravity in
the presence of a non-trivial boundary, along the lines of [55]. Besides, now,
having well-defined three-dimensional CS (super)gravity theories respectively
invariant under the $\mathcal{N}$-extended AdS-Carroll and Carroll
(super)algebras, it would be intriguing to go beyond and study the asymptotic
symmetry of these models, following, for instance, the prescription given in
Ref. [56]. It would also be interesting to further extend our analysis to more
general amount of supersymmetry, involving also odd $\mathcal{N}$ cases, and
to higher-dimensional models (recently, a study exploring the Carroll limit
corresponding to M2- as well as M3-branes propagating over $D=11$ supergravity
backgrounds in M-theory has been presented [10]), where Carrollian
(super)gravity theories still remain poorly explored. Finally, all these
ultra-relativistic theories constructed à la CS could have some applications
in the context of Carrollian fluids (and their relations with flat holography,
see Refs. [31, 32, 33, 34]).

##  8 Acknowledgments

L.R. is grateful to L. Andrianopoli and B.L. Cerchiai for useful discussions
and support.

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