B_{\lambda}\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}\bigoplus_{r=\frac{|\lambda|}{2}}^{\infty}\bigoplus_{\begin{subarray}{c}s\in\frac{1}{r}\mathbb{Z}\\\
-1<s\leq 1\end{subarray}}c^{\lambda}_{(r,s)}W_{(r,s)}\ .$ (B.1)

where we omit the dependence on $L=\infty$. For example,
$B_{[3,1]}\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,\pm\frac{1}{2})}\oplus
W_{(2,1)}\oplus\ldots$, where we introduce the notation

$\displaystyle W_{(r,\pm s)}=W_{(r,s)}\oplus W_{(r,-s)}\ .$ (B.2)

#### Algorithm

We first construct a basis for $B^{(L)}_{\lambda}$. A suitable linear operator
${\cal O}^{(L)}$ belonging to $u\mathcal{J\\!T\\!\\!L}_{L}$ is then
numerically diagonalised within the corresponding space, giving rise to a
spectrum $\Lambda_{\cal O}(L,\lambda)$. Diagonalising next the same operator
${\cal O}^{(L)}$ within each of the representations $W^{(L)}_{(r,s)}$ with
$2r\leq L$ and any allowed $s$ gives rise to the spectra $\Lambda_{\cal
O}(L,r,s)$. It is then found that $\Lambda_{\cal O}(L,\lambda)$ is the union
of all the $\Lambda_{\cal O}(L,r,s)$, and the multiplicity of each eigenvalue
in $\Lambda_{\cal O}(L,\lambda)$ reveals the corresponding integer coefficient
$c_{(r,s)}^{\lambda}$ on the right-hand side of the branching rule.

For this strategy to be fully successful we should choose ${\cal O}^{(L)}$
such that the spectrum $\Lambda_{\cal O}(L,r,s)$ is simple for any $(r,s)$ and
such that the different spectra $\Lambda_{\cal O}(L,r,s)$ have no coinciding
eigenvalues. It turns out a good choice to take ${\cal O}^{(L)}$ as the row-
to-row transfer matrix of the $u\mathcal{J\\!T\\!\\!L}(n)$ loop model on the
square lattice. For a generic value $n>0$ of the loop weight, it turns out
that the spectra $\Lambda_{\cal O}(L,r,s)$ are real, simple and non-
overlapping, except for the symmetry $\Lambda_{\cal O}(L,r,s)=\Lambda_{\cal
O}(L,r,-s)$. This is however not a problem, since the branching coefficients
themselves also do not depend on the sign of $s$.

We now give a few more details. Each element of the basis for
$B^{(L)}_{\lambda}$ consists of a link pattern in which crossings are allowed,
plus some extra information that will be constructed to impose that the
irreducible representation $\lambda$ lives on the $2r=|\lambda|$ through-
lines. We first construct the set of link patterns consisting of $2r$ through-
lines and a perfect matching (which may include crossings) of the remaining
$L-2r$ points. This vector space of link patterns is denoted ${\cal S}_{0}$
and has dimension $\binom{L}{L-|\lambda|}(L-|\lambda|-1)!!$. We may endow each
of the link patterns in ${\cal S}_{0}$ with an extra label, which can take
$f_{\lambda}$ possible values, in order to have a space of total dimension
(A.9). We call this the vector space of labelled link patterns and denote it
${\cal S}_{1}$.

To diagonalise ${\cal O}^{(L)}$ within ${\cal S}_{1}$, we use the Arnoldi
method for non-symmetric matrices, which is an iterative eigenvalue method
that only relies on providing a method (or function, in algorithmic parlance)
that computes the vector $v_{2}={\cal O}^{(L)}v_{1}$ from any given vector
$v_{1}\in{\cal S}_{1}$. To provide this method we shall need to introduce a
larger space, the space of link patterns with distinguishable through-lines,
denoted ${\cal S}_{2}$, which will be used in intermediary stages of the
computation of $v_{2}$. The idea is that each state in ${\cal S}_{2}$ is a
link pattern in which any one of its through-lines carries a distinct “mark”
from the set $\\{1,2,\ldots,2r\\}$. Here is an illustration of two such link
patterns from $B^{(10)}_{\lambda}$ with $|\lambda|=4$, differing only by the
marks on the through-lines:

$1$$2$$3$$4$

$3$$2$$4$$1$

(B.3)

In this way, each link pattern in ${\cal S}_{2}$ is endowed with an element of
the symmetric group $S_{2r}$, enabling us to follow how the (now
distinguishable) through-lines are permuted by the action of ${\cal O}^{(L)}$.
In the space of permutations of $\\{1,2,\ldots,2r\\}$ we finally construct a
$(2r)!$-dimensional matrix representation $\Pi_{\lambda}$ of the Young
projector on $\lambda$. Choosing an overall normalising factor of
$\frac{f_{\lambda}}{|\lambda|!}$ we find that $\Pi_{\lambda}$ is an idempotent
of rank $f_{\lambda}$ (it has $f_{\lambda}$ unit eigenvalues). Let
$\\{w^{\Pi_{\lambda}}_{i}\\}_{i=1}^{f_{\lambda}}$ be the corresponding set of
orthonormalised eigenvectors.

The multiplication method of the Arnoldi algorithm is now decomposed into the
following steps. Given the vector $v_{1}\in{\cal S}_{1}$, we map each of the
corresponding basis states with label $i$ into a linear combination of $(2r)!$
basis states in ${\cal S}_{2}$, using the coefficients coming from the
eigenvector $w^{\Pi_{\lambda}}_{i}$. In this way $v_{1}$ has been “unfolded”
into the larger space ${\cal S}_{2}$. We then multiply this vector by ${\cal
O}^{(L)}$, using the hash-table data structures and the sparse-matrix
factorisation techniques which have been extensively described in [34]
(Appendix A). In this process the through-lines carrying their respective
marks will undergo permutations. The result after multiplication by ${\cal
O}^{(L)}$ is the vector $v_{2}$, but living in the enlarged space ${\cal
S}_{2}$. To complete the computation, we “refold” $v_{2}$ into the smaller
space ${\cal S}_{1}$ by projecting on the eigenvectors
$\\{w^{\Pi_{\lambda}}_{i}\\}$.

The result of this diagonalisation of ${\cal O}^{(L)}$ within ${\cal S}_{1}$
is the spectrum $\Lambda_{\cal O}(L,\lambda)$ corresponding to the
representation $B^{(L)}_{\lambda}$. The spectra $\Lambda_{\cal O}(L,r,s)$
corresponding to each representation $W^{(L)}_{(r,s)}$ are obtained by more
elementary means, which have been well described in [34] (Appendix A).
Comparing the two sets of spectra we finally deduce the branching rules.

#### Results

We have computed the branching rules of $B^{(L)}_{\lambda}$ up to size $L=10$
and for all representations with $|\lambda|\leq 4$, with the results:

$\displaystyle B_{[]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(0,0)}\oplus
W_{(2,0)}\oplus 2W_{(3,0)}\oplus W_{(3,\pm\frac{2}{3})}$ (B.4a)
$\displaystyle\oplus\ 7W_{(4,0)}\oplus 2W_{(4,\pm\frac{1}{4})}\oplus
5W_{(4,\pm\frac{1}{2})}\oplus 2W_{(4,\pm\frac{3}{4})}\oplus 6W_{(4,1)}$
$\displaystyle\oplus\ 36W_{(5,0)}\oplus 24W_{(5,\pm\frac{1}{5})}\oplus
34W_{(5,\pm\frac{2}{5})}\oplus 24W_{(5,\pm\frac{3}{5})}\oplus
34W_{(5,\pm\frac{4}{5})}\oplus 25W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[1]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(\frac{1}{2},0)}\oplus
W_{(\frac{5}{2},0)}\oplus W_{(\frac{5}{2},\pm\frac{2}{5})}\oplus
W_{(\frac{5}{2},\pm\frac{4}{5})}$ (B.4b) $\displaystyle\oplus\
5W_{(\frac{7}{2},0)}\oplus 5W_{(\frac{7}{2},\pm\frac{2}{7})}\oplus
5W_{(\frac{7}{2},\pm\frac{4}{7})}\oplus 5W_{(\frac{7}{2},\oplus\frac{6}{7})}$
$\displaystyle\oplus\ 36W_{(\frac{9}{2},0)}\oplus
36W_{(\frac{9}{2},\pm\frac{2}{9})}\oplus
36W_{(\frac{9}{2},\pm\frac{4}{9})}\oplus
36W_{(\frac{9}{2},\pm\frac{2}{3})}\oplus
36W_{(\frac{9}{2},\pm\frac{8}{9})}\oplus\ldots$ $\displaystyle B_{[2]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(1,0)}\oplus
W_{(2,0)}\oplus W_{(2,1)}\oplus 4W_{(3,0)}\oplus 2W_{(3,\pm\frac{1}{3})}\oplus
4W_{(3,\pm\frac{2}{3})}\oplus 2W_{(3,1)}$ (B.4c) $\displaystyle\oplus\
23W_{(4,0)}\oplus 18W_{(4,\pm\frac{1}{4})}\oplus
23W_{(4,\pm\frac{1}{2})}\oplus 18W_{(4,\pm\frac{3}{4})}\oplus 23W_{(4,1)}$
$\displaystyle\oplus\ 191W_{(5,0)}\oplus 174W_{(5,\pm\frac{1}{5})}\oplus
191W_{(5,\pm\frac{2}{5})}\oplus 174W_{(5,\pm\frac{3}{5})}\oplus
191W_{(5,\pm\frac{4}{5})}\oplus 174W_{5,1}\oplus\ldots$ $\displaystyle
B_{[1^{2}]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(1,1)}\oplus
W_{(2,\pm\frac{1}{2})}\oplus 2W_{(3,0)}\oplus 4W_{(3,\pm\frac{1}{3})}\oplus
2W_{(3,\pm\frac{2}{3})}\oplus 4W_{(3,1)}$ (B.4d) $\displaystyle\oplus\
18W_{(4,0)}\oplus 23W_{(4,\pm\frac{1}{4})}\oplus
18W_{(4,\pm\frac{1}{2})}\oplus 23W_{(4,\pm\frac{3}{4})}\oplus 18W_{(4,1)}$
$\displaystyle\oplus\ 174W_{(5,0)}\oplus 191W_{(5,\pm\frac{1}{5})}\oplus
174W_{(5,\pm\frac{2}{5})}\oplus 191W_{(5,\pm\frac{3}{5})}\oplus
174W_{(5,\pm\frac{4}{5})}\oplus 191W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[3]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(\frac{3}{2},0)}\oplus
W_{(\frac{5}{2},0)}\oplus W_{(\frac{5}{2},\pm\frac{2}{5})}\oplus
W_{(\frac{5}{2},\pm\frac{4}{5})}$ (B.4e) $\displaystyle\oplus\
7W_{(\frac{7}{2},0)}\oplus 7W_{(\frac{7}{2},\pm\frac{2}{7})}\oplus
7W_{(\frac{7}{2},\pm\frac{4}{7})}\oplus 7W_{(\frac{7}{2},\pm\frac{6}{7})}$
$\displaystyle\oplus\ 63W_{(\frac{9}{2},0)}\oplus
61W_{(\frac{9}{2},\pm\frac{2}{9})}\oplus
61W_{(\frac{9}{2},\pm\frac{4}{9})}\oplus
63W_{(\frac{9}{2},\pm\frac{2}{3})}\oplus
61W_{(\frac{9}{2},\pm\frac{8}{9})}\oplus\ldots$ $\displaystyle B_{[21]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(\frac{3}{2},\pm\frac{2}{3})}\oplus
2W_{(\frac{5}{2},0)}\oplus 2W_{(\frac{5}{2},\pm\frac{2}{5})}\oplus
2W_{(\frac{5}{2},\pm\frac{4}{5})}$ (B.4f) $\displaystyle\oplus\
14W_{(\frac{7}{2},0)}\oplus 14W_{(\frac{7}{2},\pm\frac{2}{7})}\oplus
14W_{(\frac{7}{2},\pm\frac{4}{7})}\oplus 14W_{(\frac{7}{2},\pm\frac{6}{7})}$
$\displaystyle\oplus\ 122W_{(\frac{9}{2},0)}\oplus
124W_{(\frac{9}{2},\pm\frac{2}{9})}\oplus
124W_{(\frac{9}{2},\pm\frac{4}{9})}\oplus
122W_{(\frac{9}{2},\pm\frac{2}{3})}\oplus
124W_{(\frac{9}{2},\pm\frac{8}{9})}\oplus\ldots$ $\displaystyle B_{[1^{3}]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(\frac{3}{2},0)}\oplus
W_{(\frac{5}{2},0)}\oplus W_{(\frac{5}{2},\pm\frac{2}{5})}\oplus
W_{(\frac{5}{2},\pm\frac{4}{5})}$ (B.4g) $\displaystyle\oplus\
7W_{(\frac{7}{2},0)}\oplus 7W_{(\frac{7}{2},\pm\frac{2}{7})}\oplus
7W_{(\frac{7}{2},\pm\frac{4}{7})}\oplus 7W_{(\frac{7}{2},\pm\frac{6}{7})}$
$\displaystyle\oplus\ 63W_{(\frac{9}{2},0)}\oplus
61W_{(\frac{9}{2},\pm\frac{2}{9})}\oplus
61W_{(\frac{9}{2},\pm\frac{4}{9})}\oplus
63W_{(\frac{9}{2},\pm\frac{2}{3})}\oplus
61W_{(\frac{9}{2},\pm\frac{8}{9})}\oplus\ldots$ $\displaystyle B_{[4]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,0)}\oplus
2W_{(3,0)}\oplus W_{(3,\pm\frac{1}{3})}\oplus 2W_{(3,\pm\frac{2}{3})}\oplus
W_{(3,1)}$ (B.4h) $\displaystyle\oplus\ 16W_{(4,0)}\oplus
12W_{(4,\pm\frac{1}{4})}\oplus 15W_{(4,\pm\frac{1}{2})}\oplus
12W_{(4,\pm\frac{3}{4})}\oplus 16W_{(4,1)}$ $\displaystyle\oplus\
158W_{(5,0)}\oplus 147W_{(5,\pm\frac{1}{5})}\oplus
158W_{(5,\pm\frac{2}{5})}\oplus 147W_{(5,\pm\frac{3}{5})}\oplus
158W_{(5,\pm\frac{4}{5})}\oplus 147W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[31]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,\pm\frac{1}{2})}\oplus
W_{(2,1)}\oplus 4W_{(3,0)}\oplus 5W_{(3,\pm\frac{1}{3})}\oplus
4W_{(3,\pm\frac{2}{3})}\oplus 5W_{(3,1)}$ (B.4i) $\displaystyle\oplus\
39W_{(4,0)}\oplus 43W_{(4,\pm\frac{1}{4})}\oplus
40W_{(4,\pm\frac{1}{2})}\oplus 43W_{(4,\pm\frac{3}{4})}\oplus 39W_{(4,1)}$
$\displaystyle\oplus\ 452W_{(5,0)}\oplus 463W_{(5,\pm\frac{1}{5})}\oplus
452W_{(5,\pm\frac{2}{5})}\oplus 463W_{(5,\pm\frac{3}{5})}\oplus
452W_{(5,\pm\frac{4}{5})}\oplus 463W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[2^{2}]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,0)}\oplus
W_{(2,1)}\oplus 4W_{(3,0)}\oplus 2W_{(3,\pm\frac{1}{3})}\oplus
4W_{(3,\pm\frac{2}{3})}\oplus 2W_{(3,1)}$ (B.4j) $\displaystyle\oplus\
31W_{(4,0)}\oplus 24W_{(4,\pm\frac{1}{4})}\oplus
31W_{(4,\pm\frac{1}{2})}\oplus 24W_{(4,\pm\frac{3}{4})}\oplus 31W_{(4,1)}$
$\displaystyle\oplus\ 316W_{(5,0)}\oplus 294W_{(5,\pm\frac{1}{5})}\oplus
316W_{(5,\pm\frac{2}{5})}\oplus 294W_{(5,\pm\frac{3}{5})}\oplus
316W_{(5,\pm\frac{4}{5})}\oplus 294W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[21^{2}]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,0)}\oplus
W_{(2,\pm\frac{1}{2})}\oplus 4W_{(3,0)}\oplus 5W_{(3,\pm\frac{1}{3})}\oplus
4W_{(3,\pm\frac{2}{3})}\oplus 5W_{(3,1)}$ (B.4k) $\displaystyle\oplus\
40W_{(4,0)}\oplus 43W_{(4,\pm\frac{1}{4})}\oplus
39W_{(4,\pm\frac{1}{2})}\oplus 43W_{(4,\pm\frac{3}{4})}\oplus 40W_{(4,1)}$
$\displaystyle\oplus\ 452W_{(5,0)}\oplus 463W_{(5,\pm\frac{1}{5})}\oplus
452W_{(5,\pm\frac{2}{5})}\oplus 463W_{(5,\pm\frac{3}{5})}\oplus
452W_{(5,\pm\frac{4}{5})}\oplus 463W_{(5,1)}\oplus\ldots$ $\displaystyle
B_{[1^{4}]}$
$\displaystyle\underset{u\mathcal{J\\!T\\!\\!L}(n)}{=}W_{(2,1)}\oplus
2W_{(3,0)}\oplus W_{(3,\pm\frac{1}{3})}\oplus 2W_{(3,\pm\frac{2}{3})}\oplus
W_{(3,1)}$ (B.4l) $\displaystyle\oplus\ 15W_{(4,0)}\oplus
12W_{(4,\pm\frac{1}{4})}\oplus 16W_{(4,\pm\frac{1}{2})}\oplus
12W_{(4,\pm\frac{3}{4})}\oplus 15W_{(4,1)}$ $\displaystyle\oplus\
158W_{(5,0)}\oplus 147W_{(5,\pm\frac{1}{5})}\oplus
158W_{(5,\pm\frac{2}{5})}\oplus 147W_{(5,\pm\frac{3}{5})}\oplus
158W_{(5,\pm\frac{4}{5})}\oplus 147W_{(5,1)}\oplus\ldots$

#### Observations

We recognise here the coefficients $c^{\lambda}_{(r,s)}$ computed in section
4.2. In addition we make a number of observations:

  * •

The results for $B_{[3]}$ and $B_{[1^{3}]}$ are identical. This is because the
even and odd permutations have the same sign in the Young projector
$\Pi_{[3]}$ and opposite signs in $\Pi_{[1^{3}]}$, and therefore cannot mix
under the action of the cyclic group.

  * •

The results for $B_{[4]}$ and $B_{[1^{4}]}$ coincide for $r$ odd. The results
for $B_{[2^{2}]}$ and $r$ odd have coefficients that are larger by a factor of
${\rm dim}\,[2^{2}]=2$.

  * •

The results for $B_{[31]}$ and $B_{[21^{2}]}$ coincide for $r$ odd.

###  B.2 Branching rules $\mathcal{P}_{L}(Q)\downarrow
P\\!\mathcal{T\\!\\!L}_{2L}(\sqrt{Q})$

Let us briefly describe our algorithm in this case, while focussing on the
differences with respect to the case of $\mathcal{B}_{L}(n)\downarrow
u\mathcal{J\\!T\\!\\!L}_{L}(n)$ in Appendix B.1.

We consider a system of $L$ sites (or Potts spins). Partition-algebra modules
$P^{(L)}_{\lambda}$ have a basis of set partitions of the $L$ sites, with
$r\in\mathbb{N}$ propagating blocks. The propagating blocks carry an
irreducible representation of the symmetric group $S_{|\lambda|}$ with
$|\lambda|=r$. We want to decompose $P^{(L)}_{\lambda}$ into representations
$W^{(2L)}_{(r,s)}$ of $P\\!\mathcal{T\\!\\!L}_{2L}(\sqrt{Q})$, with
$|\lambda|\leq r\leq L$. We consider the limit $L\to\infty$ and omit the
dependence on $L$. We have $r\geq 2$ except in the cases
$P_{[]}\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(0,0)}\oplus\cdots$
and
$P_{[1]}\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(0,\frac{1}{2})}\oplus\cdots$.

#### Algorithm

We first construct a basis of set partitions of $\\{1,2,\ldots,L\\}$ with
$|\lambda|$ propagating (but indistinguishable) blocks, defining the vector
space ${\cal S}_{0}$. Each of the set partitions in ${\cal S}_{0}$ is then
endowed with an extra label which can take $f_{\lambda}$ possible values,
defining the vector space of labelled set partitions ${\cal S}_{1}$. This
${\cal S}_{1}$ has the same dimension (A.7) as the module $P^{(L)}_{\lambda}$.
We then wish to diagonalise a suitable linear operator ${\cal O}^{(L)}$ within
${\cal S}_{1}$, in order to obtain the spectrum $\Lambda_{\cal O}(L,\lambda)$
corresponding to the representation $P^{(L)}_{\lambda}$. This is done, as in
Appendix B.1, by providing the multiplication method of the Arnoldi algorithm.
In this method, ${\cal S}_{1}$ is first unfolded—by means of the orthonormal
eigenvectors $\\{w^{\Pi_{\lambda}}_{i}\\}_{i=1}^{f_{\lambda}}$ of the Young
projector $\Pi_{\lambda}$—into an enlarged space ${\cal S}_{2}$ in which each
of the propagating blocks carries a distinct mark (each of the sites in a
given block carries the same mark). After the multiplication by ${\cal
O}^{(L)}$, acting within ${\cal S}_{2}$, the result is folded back into ${\cal
S}_{1}$ by projecting on the eigenvectors $w^{\Pi_{\lambda}}_{i}$.

We find that an appropriate choice of the linear operator ${\cal O}^{(L)}$ is
that of the row-to-row transfer matrix of the $Q$-state cluster model on an
axially oriented square lattice with periodic boundary conditions. Notice that
this is a product of $L$ operators adding the horizontal edges and another $L$
operators adding the vertical ones, each edge operator being a sparse matrix
expressed in terms of join and detach operators [34]. For generic values of
$Q>0$, the operator ${\cal O}^{(L)}$ has the advantage that the spectra do not
contain any accidental degeneracies. The branching rules can therefore be
unambiguously read off from the multiplicities in $\Lambda_{\cal
O}(L,\lambda)$ and by comparison with the spectra $\Lambda_{\cal O}(2L,r,s)$
of the Potts model.

#### Results

For the fully symmetric representations $P_{[p]}$ we find then the branching
rules

$\displaystyle P_{[]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(0,0)}\oplus
W_{(4,0)}\oplus 3W_{(6,0)}\oplus W_{(6,\pm\frac{1}{3})}\oplus\cdots$ (B.5a)
$\displaystyle P_{[1]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(0,\frac{1}{2})}\oplus
W_{(4,0)}\oplus W_{(4,\frac{1}{2})}\oplus W_{(5,0)}\oplus
W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}$ (B.5b)
$\displaystyle\oplus\ 5W_{(6,0)}\oplus 2W_{(6,\pm\frac{1}{6})}\oplus
4W_{(6,\pm\frac{1}{3})}\oplus 3W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[2]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(2,0)}\oplus
2W_{(4,0)}\oplus W_{(4,\frac{1}{2})}\oplus 2W_{(5,0)}\oplus
2W_{(5,\pm\frac{1}{5})}\oplus 2W_{(5,\pm\frac{2}{5})}\oplus$ (B.5c)
$\displaystyle\oplus\ 10W_{(6,0)}\oplus 5W_{(6,\pm\frac{1}{6})}\oplus
9W_{(6,\pm\frac{1}{3})}\oplus 5W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[3]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(3,0)}\oplus
W_{(4,0)}\oplus W_{(4,\frac{1}{2})}\oplus 2W_{(5,0)}\oplus
2W_{(5,\pm\frac{1}{5})}\oplus 2W_{(5,\pm\frac{2}{5})}$ (B.5d)
$\displaystyle\oplus\ 11W_{(6,0)}\oplus 6W_{(6,\pm\frac{1}{6})}\oplus
9W_{(6,\pm\frac{1}{3})}\oplus 7W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[4]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(4,0)}\oplus
W_{(5,0)}\oplus W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}$ (B.5e)
$\displaystyle\oplus\ 7W_{(6,0)}\oplus 3W_{(6,\pm\frac{1}{6})}\oplus
6W_{(6,\pm\frac{1}{3})}\oplus 4W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[5]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,0)}\oplus
2W_{(6,0)}\oplus W_{(6,\pm\frac{1}{6})}\oplus 2W_{(6,\pm\frac{1}{3})}\oplus
W_{(6,\frac{1}{2})}\oplus\cdots$ (B.5f)

For the fully antisymmetric representations $P_{[1^{p}]}$ we find

$\displaystyle P_{[1^{2}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(2,\frac{1}{2})}\oplus
W_{(4,\pm\frac{1}{4})}\oplus W_{(4,\frac{1}{2})}\oplus 2W_{(5,0)}\oplus
2W_{(5,\pm\frac{1}{5})}\oplus 2W_{(5,\pm\frac{2}{5})}$ (B.6a)
$\displaystyle\oplus\ 6W_{(6,0)}\oplus 8W_{(6,\pm\frac{1}{6})}\oplus
6W_{(6,\pm\frac{1}{3})}\oplus 9W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[1^{3}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(3,0)}\oplus
W_{(4,\pm\frac{1}{4})}\oplus 2W_{(5,0)}\oplus 2W_{(5,\pm\frac{1}{5})}\oplus
2W_{(5,\pm\frac{2}{5})}$ (B.6b) $\displaystyle\oplus\ 8W_{(6,0)}\oplus
9W_{(6,\pm\frac{1}{6})}\oplus 6W_{(6,\pm\frac{1}{3})}\oplus
10W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle P_{[1^{4}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(4,\frac{1}{2})}\oplus
W_{(5,0)}\oplus W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}$ (B.6c)
$\displaystyle\oplus\ 6W_{(6,0)}\oplus 4W_{(6,\pm\frac{1}{6})}\oplus
5W_{(6,\pm\frac{1}{3})}\oplus 5W_{(6,\frac{1}{2})}\oplus\cdots$ $\displaystyle
P_{[1^{5}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,0)}\oplus
2W_{(6,0)}\oplus W_{(6,\pm\frac{1}{6})}\oplus 2W_{(6,\pm\frac{1}{3})}\oplus
W_{(6,\frac{1}{2})}\oplus\cdots$ (B.6d)

We have also computed some, but not all, of the cases with $f_{\lambda}>1$ and
$|\lambda|\leq 6$:

$\displaystyle P_{[21]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(3,\pm\frac{1}{3})}\oplus
W_{(4,0)}\oplus W_{(4,\pm\frac{1}{4})}\oplus W_{(4,\frac{1}{2})}\oplus
4W_{(5,0)}\oplus 4W_{(5,\pm\frac{1}{5})}\oplus 4V_{(5,\pm\frac{2}{5})}$ (B.7a)
$\displaystyle\oplus\ 15W_{(6,0)}\oplus 16W_{(6,\pm\frac{1}{6})}\oplus
17W_{(6,\pm\frac{1}{3})}\oplus 15W_{(6,\frac{1}{2})}\oplus\cdots$
$\displaystyle P_{[31]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(4,\pm\frac{1}{4})}\oplus
W_{(4,\frac{1}{2})}\oplus
3W_{(5,0)}+3W_{(5,\pm\frac{1}{5})}+3V_{(5,\pm\frac{2}{5})}$ (B.7b)
$\displaystyle\oplus\ 14W_{(6,0)}\oplus 15W_{(6,\pm\frac{1}{6})}\oplus
14W_{(6,\pm\frac{1}{3})}\oplus 15W_{(6,\frac{1}{2})}\oplus\cdots$
$\displaystyle P_{[2^{2}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(4,0)}\oplus
W_{(4,\frac{1}{2})}\oplus
2W_{(5,0)}+2W_{(5,\pm\frac{1}{5})}+2W_{(5,\pm\frac{2}{5})}$ (B.7c)
$\displaystyle\oplus\ 11W_{(6,0)}\oplus 8W_{(6,\pm\frac{1}{6})}\oplus
12W_{(6,\pm\frac{1}{3})}\oplus 7W_{(6,\frac{1}{2})}\oplus\cdots$
$\displaystyle P_{[21^{2}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(4,0)}\oplus
W_{(4,\pm\frac{1}{4})}\oplus
3W_{(5,0)}+3W_{(5,\pm\frac{1}{5})}+3V_{(5,\pm\frac{2}{5})}$ (B.7d)
$\displaystyle\oplus\ 13W_{(6,0)}\oplus 16W_{(6,\pm\frac{1}{6})}\oplus
13W_{(6,\pm\frac{1}{3})}\oplus 16W_{(6,\frac{1}{2})}\oplus\cdots$
$\displaystyle P_{[41]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,\pm\frac{1}{5})}\oplus
W_{(5,\pm\frac{2}{5})}\oplus 6W_{(6,0)}\oplus 6W_{(6,\pm\frac{1}{6}})\oplus
6W_{(6,\pm\frac{1}{3})}\oplus 6W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7e)
$\displaystyle P_{[3,2]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,0)}\oplus
W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}\oplus 8W_{(6,0)}\oplus
7W_{(6,\pm\frac{1}{6})}\oplus 8W_{(6,\pm\frac{1}{3})}\oplus
7W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7f) $\displaystyle P_{[31^{2}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}2W_{(5,0)}\oplus
W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}\oplus 8W_{(6,0)}\oplus
10W_{(6,\pm\frac{1}{6})}\oplus 8W_{(6,\pm\frac{1}{3})}\oplus
10W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7g) $\displaystyle P_{[2^{2}1]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,0)}\oplus
W_{(5,\pm\frac{1}{5})}\oplus W_{(5,\pm\frac{2}{5})}\oplus 8W_{(6,0)}\oplus
7W_{(6,\pm\frac{1}{6})}\oplus 8W_{(6,\pm\frac{1}{3})}\oplus
7W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7h) $\displaystyle P_{[21^{3}]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(5,\pm\frac{1}{5})}\oplus
W_{(5,\pm\frac{2}{5})}\oplus 6W_{(6,0)}\oplus 6W_{(6,\pm\frac{1}{6})}\oplus
6W_{(6,\pm\frac{1}{3})}\oplus 6W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7i)
$\displaystyle P_{[51]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}W_{(6,\pm\frac{1}{6})}\oplus
W_{(6,\pm\frac{1}{3})}\oplus W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7j)
$\displaystyle P_{[42]}$
$\displaystyle\underset{P\\!\mathcal{T\\!\\!L}(\sqrt{Q})}{=}2W_{(6,0)}\oplus
W_{(6,\pm\frac{1}{6})}\oplus 2W_{(6,\pm\frac{1}{3})}\oplus
W_{(6,\frac{1}{2})}\oplus\cdots$ (B.7k)

#### Observations

We recognise here the coefficients $d^{\lambda}_{(r,s)}$ computed in Section
5.2. In particular, combining (B.5)–(B.7) with a dimensional count is enough
to uniquely determine the $\Lambda_{(r,s)}$ for all $r\leq 6$, thus recovering
all of the results displayed in (3.20).

Concerning the terms on the right-hand side with the minimal value of $r$,
that is $r=|\lambda|$, we can go a bit further. In that case we can infer some
families of branching coefficients, in which $\lambda$ consists of one long
row followed by one or two short rows. This proceeds by studying carefully the
corresponding Young projector $\Pi_{\lambda}$, either by hand or aided by
Mathematica for the more complicated cases, but without employing the above
method of numerically diagonalizing the transfer matrix.

We first find that

$d^{[L-1,1]}_{(L,\frac{k}{L})}=\begin{cases}0&\mbox{for }k=0\,,\\\
1&\mbox{otherwise}\,.\end{cases}$ (B.8)

The sum of multiplicities is of course
$\sum_{k=0}^{L-1}d^{[L-1,1]}_{(L,\frac{k}{L})}=L-1$, the dimension of the
representation $\lambda=[L-1,1]$ on the left-hand side. Similarly we find for
$P_{[L-2,2]}$ the multiplicities

$d^{[L-2,2]}_{(L,\frac{k}{L})}=\begin{cases}\left\lfloor\frac{L-2}{2}\right\rfloor&\mbox{for
}k\mbox{ even}\,,\\\ \left\lfloor\frac{L-3}{2}\right\rfloor&\mbox{for }k\mbox{
odd}\,,\end{cases}$ (B.9)

where $\lfloor x\rfloor$ denotes the integer part of $x$. Their sum is
$\frac{1}{2}L(L-3)$, as it should be by the hook length formula (4.10). The
result (B.9) has been checked up to $L=8$. For $P_{[L-2,1,1]}$ we find

$d^{[L-2,1,1]}_{(L,\frac{k}{L})}=\begin{cases}1+\left\lfloor\frac{L-3}{2}\right\rfloor&\mbox{for
}k=0\,,\\\ \left\lfloor\frac{L-3}{2}\right\rfloor&\mbox{for }k\neq 0\mbox{
even}\,,\\\ \left\lfloor\frac{L-2}{2}\right\rfloor&\mbox{for }k\mbox{
odd}\,,\end{cases}$ (B.10)

and the sum of multiplicities is $\frac{1}{2}(L-1)(L-2)$. This result has been
checked up to $L=7$. And finally we obtain

$d^{[L-3,2,1]}_{(L,\frac{k}{L})}=\begin{cases}\left\lfloor\frac{(L-3)^{2}}{3}\right\rfloor&\mbox{for
}3\nmid k\,,\\\\[6.0pt] \left\lfloor\frac{(L-3)^{2}}{3}\right\rfloor&\mbox{for
}3\mid k\mbox{ and }3\nmid L\,,\\\\[6.0pt]
\left\lfloor\frac{(L-3)^{2}}{3}-1\right\rfloor&\mbox{for }3\mid k\mbox{ and
}3\mid L\,,\end{cases}$ (B.11)

with total multiplicity $\frac{1}{3}L(L-2)(L-4)$. This has been checked up to
$L=7$.

## References

  * [1] R. J. Baxter,  _Potts model at the critical temperature_ ,  J. Phys. C: Solid State Phys. 6, L445 (1973).
  * [2] B. Nienhuis,  _Exact critical point and critical exponents of $\mathrm{O}(n)$ models in two dimensions_,  Phys. Rev. Lett. 49, 1062 (1982),  doi:10.1103/PhysRevLett.49.1062.
  * [3] P. Di Francesco, H. Saleur and J.-B. Zuber,  _Relations between the coulomb gas picture and conformal invariance of two-dimensional critical models_ ,  Journal of statistical physics 49(1-2), 57 (1987).
  * [4] N. Read and H. Saleur,  _Enlarged symmetry algebras of spin chains, loop models, and S-matrices_ ,  Nucl. Phys. B 777, 263 (2007),  doi:10.1016/j.nuclphysb.2007.03.007,  cond-mat/0701259.
  * [5] L. Grans-Samuelsson, R. Nivesvivat, J. L. Jacobsen, S. Ribault and H. Saleur,  _Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model_,  SciPost Phys. 12, 147 (2022),  doi:10.21468/SciPostPhys.12.5.147,  2111.01106.
  * [6] R. Nivesvivat,  _Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model_ (2022),  2205.09349.
  * [7] J. Böhm, J. L. Jacobsen, Y. Jiang and Y. Zhang,  _Geometric Algebra and Algebraic Geometry of Loop and Potts Models_ (2022),  2202.02986.
  * [8] B. Nienhuis, A. N. Berker, E. K. Riedel and M. Schick,  _First- and second-order phase transitions in potts models: Renormalization-group solution_ ,  Phys. Rev. Lett. 43, 737 (1979),  doi:10.1103/PhysRevLett.43.737.
  * [9] M. Picco, S. Ribault and R. Santachiara,  _A conformal bootstrap approach to critical percolation in two dimensions_ ,  SciPost Phys. 1(1), 009 (2016),  doi:10.21468/SciPostPhys.1.1.009,  1607.07224.
  * [10] Y. He, J. L. Jacobsen and H. Saleur,  _Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap_ (2020),  doi:10.1007/JHEP12(2020)019,  2005.07258.
  * [11] R. Nivesvivat and S. Ribault,  _Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model_,  SciPost Phys. 10(1), 021 (2021),  doi:10.21468/SciPostPhys.10.1.021,  2007.04190.
  * [12] L. Grans-Samuelsson, L. Liu, Y. He, J. L. Jacobsen and H. Saleur,  _The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$_ (2020),  doi:10.1007/JHEP10(2020)109,  2007.11539.
  * [13] W. Koo and H. Saleur,  _Representations of the virasoro algebra from lattice models_ ,  Nuclear Physics B 426(3), 459 (1994),  doi:https://doi.org/10.1016/0550-3213(94)90018-3.
  * [14] M. Zini and Z. Wang,  _Conformal field theories as scaling limit of anyonic chains_ ,  Communications in Mathematical Physics 363 (2018),  doi:https://doi.org/10.1007/s00220-018-3254-1.
  * [15] A. M. Gainutdinov, N. Read and H. Saleur,  _Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the ${\mathfrak{gl}(1|1)}$ periodic spin chain, Howe duality and the interchiral algebra_,  Commun. Math. Phys. 341(1), 35 (2016),  doi:10.1007/s00220-015-2483-9,  1207.6334.
  * [16] S. Ribault, M. Zabrocki,  _Representations of $o(n)$ and $s_{q}$ with generic parameters_ (2022).
  * [17] A. M. Vershik and A. Y. Okounkov,  _A New Approach to the Representation Theory of the Symmetric Groups. 2_ (1996),  math/0503040.
  * [18] G. D. James,  _The Representation Theory of the Symmetric Groups_ ,  Springer,  doi:10.1007/BFb0067708 (1978).
  * [19] R. E. Howe, E. C. Tan and J. F. Willenbring,  _Stable branching rules for classical symmetric pairs_ (2005),  math/0311159.
  * [20] V. Gorbenko, S. Rychkov and B. Zan,  _Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$_,  SciPost Phys. 5(5), 050 (2018),  doi:10.21468/SciPostPhys.5.5.050,  1808.04380.
  * [21] D. J. Binder and S. Rychkov,  _Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$_ (2019),  1911.07895.
  * [22] S. Migliaccio and S. Ribault,  _The analytic bootstrap equations of non-diagonal two-dimensional CFT_ ,  JHEP 05, 169 (2018),  doi:10.1007/JHEP05(2018)169,  1711.08916.
  * [23] R. Couvreur, J. L. Jacobsen and R. Vasseur,  _Non-scalar operators for the Potts model in arbitrary dimension_ ,  J. Phys. A 50(47), 474001 (2017),  doi:10.1088/1751-8121/aa7f32,  1704.02186.
  * [24] R. Orellana and M. Zabrocki,  _Products of characters of the symmetric group_ (2017),  1709.08098.
  * [25] S. H. Assaf and D. E. Speyer,  _Specht modules decompose as alternating sums of restrictions of schur modules_ ,  Proceedings of the American Mathematical Society 148(3), 1015 (2019),  doi:10.1090/proc/14815,  1809.10125.
  * [26] R. Orellana and M. Zabrocki,  _Symmetric group characters as symmetric functions (extended abstract)_ (2015),  1510.00438.
  * [27] R. Orellana and M. Zabrocki,  _Symmetric group characters as symmetric functions_ (2018),  1605.06672.
  * [28] J. L. Jacobsen,  _Loop models and boundary CFT_ ,  Lect. Notes Phys. 853, 141 (2012),  doi:10.1007/978-3-642-27934-8\\_4.
  * [29] H. Blöte and B. Nienhuis,  _Critical behaviour and conformal anomaly of the $O(n)$ model on the square lattice_,  J. Phys. A: Math. Gen. 22, 1415 (1989).
  * [30] R. Brauer,  _On algebras which are connected with the semi-simple continuous groups_ ,  Annals of Mathematics 38, 857 (1937),  doi:10.2307/1968843.
  * [31] G. Benkart and D. Moon,  _Tensor product representations of temperley-lieb algebras and chebyshev polynomials_ Representations of Algebras and Related Topics (2005),  doi:10.1090/fic/045/05.
  * [32] A. M. Gainutdinov, J. L. Jacobsen and H. Saleur,  _A fusion for the periodic Temperley-Lieb algebra and its continuum limit_ ,  J. High Energy Phys. 11, 117 (2018),  doi:10.1007/JHEP11(2018)117,  1712.07076.
  * [33] J. R. Stembridge,  _On the eigenvalues of representations of reflection groups and wreath products._ ,  Pacific Journal of Mathematics 140(2), 353 (1989),  doi:pjm/1102646930.
  * [34] J. L. Jacobsen and H. Saleur,  _Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra_,  JHEP 01, 084 (2019),  doi:10.1007/JHEP01(2019)084,  1809.02191.
  * [35] T. Halverson and A. Ram,  _Partition algebras_ ,  Eur. J. Comb. 26, 869 (2005),  doi:10.1016/j.ejc.2004.06.005,  0401314.
  * [36] Y. Ikhlef and A. Morin-Duchesne,  _Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models_ (2021),  2105.10240.
  * [37] G. Delfino and J. Viti,  _Potts q-color field theory and scaling random cluster model_ ,  Nucl. Phys. B852, 149 (2011),  doi:10.1016/j.nuclphysb.2011.06.012,  1104.4323.
  * [38] J. Björnberg, H. Rosengren and K. Ryan,  _Heisenberg models and schur–weyl duality_ (2022),  2201.10209.
  * [39] T. Halverson and E. delMas,  _Representations of the Rook-Brauer Algebra_ 1206.4576.
  * [40] G. Benkart and T. Halverson,  _Motzkin Algebras_ 1106.5277.
  * [41] J. Belletête and Y. Saint-Aubin,  _The principal indecomposable modules of the dilute Temperley-Lieb algebra_ ,  J. Math. Phys. 55, 111706 (2014),  doi:10.1063/1.4901546,  1310.4791.
  * [42] P. Martin and V. Mazorchuk,  _On the representation theory of partial Brauer algebras_ 1205.0464.
  * [43] U. Grimm,  _Dilute Algebras and Solvable Lattice Models_ q-alg/9511020.
  * [44] E. Vernier, J. L. Jacobsen and H. Saleur,  _Dilute oriented loop models_ ,  J. Phys. A: Math. Theor. 49, 064002 (2016),  doi:10.1088/1751-8113/49/6/064002,  1509.07768.
  * [45] J. L. Jacobsen,  _Critical points of Potts and $O(N)$ models from eigenvalue identities in periodic Temperley–Lieb algebras_,  J. Phys. A: Math. Theor. 48, 454003 (2015),  doi:10.1088/1751-8113/48/45/454003,  1507.03027.


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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.7664pt\raise-39.62746pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{X\smallsetminus
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
66.58575pt\raise-39.62746pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\hbox{{}{\hbox{\kern
5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
76.33995pt\raise-32.3576pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.32541pt\hbox{$\textstyle{j}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
94.28464pt\raise-39.62746pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

where $\pi$ is a projective morphism, $X^{\prime}$ is smooth, $Y^{\prime}$ is
a normal crossing hypersurface,
$Z^{*}_{\mathscr{M}}:=Z_{\mathscr{M}}\smallsetminus Y$ and $\pi_{Z}^{*}$ is
biholomorphic. Note that the assumption on $F$ entails that

$\pi^{-1}F=\pi^{-1}F\otimes\mathbb{C}_{(X^{\prime}\smallsetminus
Y^{\prime})\times S}\quad\text{and}\quad\pi_{*}\pi^{-1}F=F,$

while
${}_{\scriptscriptstyle\mathrm{D}}\pi^{*}\mathscr{M}[d_{X^{\prime}}-d_{X}]$ is
concentrated in degree zero and is of D-type along $Y^{\prime}$. Moreover,
$\mathscr{M}\simeq{}_{\scriptscriptstyle\mathrm{D}}\pi_{*}{}_{\scriptscriptstyle\mathrm{D}}\pi^{*}\mathscr{M}[d_{X^{\prime}}-d_{X}]$.
According to Proposition 2.9 and Lemma 2.14,

$R\pi_{*}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X^{\prime}\times
S/S}}({}_{\scriptscriptstyle\mathrm{D}}\pi^{*}\mathscr{M},{}_{\scriptscriptstyle\mathrm{D}}\pi^{*}\operatorname{RH}^{S}_{X}(F))\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))$

is an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$,
which ends the proof of Theorem 3. ∎

###  3.e. Proof of Corollary 4

For any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
we have functorial isomorphisms

$\operatorname{{}^{\mathrm{p}}Sol}_{X}\boldsymbol{D}(\operatorname{RH}^{S}_{X}(F))\simeq\boldsymbol{D}\operatorname{{}^{\mathrm{p}}Sol}_{X}(\operatorname{RH}^{S}_{X}(F))\simeq\boldsymbol{D}F\simeq\operatorname{{}^{\mathrm{p}}Sol}_{X}\operatorname{RH}^{S}_{X}(\boldsymbol{D}F).$

Corollary 4 then follows by the full faithfulness of the functor
$\operatorname{{}^{\mathrm{p}}Sol}_{X}$.∎

## References

  * [1] P. Deligne, _Équations différentielles à points singuliers réguliers_ , Lect. Notes in Math., vol. 163, Springer-Verlag, 1970.
  * [2] M. J. Edmundo and L. Prelli, _Sheaves on $\mathscr{T}$-topologies_, J. Math. Soc. Japan 68 (2016), no. 1, 347–381.
  * [3] L. Fiorot and T. Monteiro Fernandes, _$t$ -structures for relative $\mathscr{D}$-modules and $t$-exactness of the de Rham functor_, J. Algebra 509 (2018), 419–444.
  * [4] M. Kashiwara, _On the holonomic systems of differential equations II_ , Invent. Math. 49 (1978), 121–135.
  * [5] by same author, _The Riemann-Hilbert problem for holonomic systems_ , Publ. RIMS, Kyoto Univ. 20 (1984), 319–365.
  * [6] by same author, _$D$ -modules and microlocal calculus_, Translations of Mathematical Monographs, vol. 217, American Mathematical Society, Providence, R.I., 2003.
  * [7] M. Kashiwara and P. Schapira, _Sheaves on manifolds_ , Grundlehren Math. Wiss., vol. 292, Springer-Verlag, Berlin, Heidelberg, 1990.
  * [8] by same author, _Moderate and formal cohomology associated with constructible sheaves_ , Mém. Soc. Math. France (N.S.), vol. 64, Société Mathématique de France, Paris, 1996.
  * [9] by same author, _Ind-sheaves_ , Astérisque, vol. 271, Société Mathématique de France, Paris, 2001.
  * [10] by same author, _Categories and sheaves_ , Grundlehren Math. Wiss., vol. 332, Springer-Verlag, Berlin, Heidelberg, 2006.
  * [11] by same author, _Regular and irregular holonomic D-modules_ , London Mathematical Society Lecture Notes Series, vol. 433, Cambridge University Press, 2016.
  * [12] Ph. Maisonobe, _Filtration relative, l’idéal de Bernstein et ses pentes_ , 2016, hal-01285562v2.
  * [13] T. Mochizuki, _Mixed twistor D-Modules_ , Lect. Notes in Math., vol. 2125, Springer, Heidelberg, New York, 2015.
  * [14] T. Monteiro Fernandes and L. Prelli, _Relative subanalytic sheaves_ , Fund. Math. 226 (2014), no. 1, 79–100.
  * [15] T. Monteiro Fernandes and C. Sabbah, _On the de Rham complex of mixed twistor $\mathscr{D}$-modules_, Internat. Math. Res. Notices (2013), no. 21, 4961–4984.
  * [16] by same author, _Relative Riemann-Hilbert correspondence in dimension one_ , Portugal. Math. 74 (2017), no. 2, 149–159.
  * [17] by same author, _Riemann-Hilbert correspondence for mixed twistor $\mathscr{D}$-modules_, J. Inst. Math. Jussieu 18 (2019), no. 3, 629–672.
  * [18] C. Sabbah, _Polarizable twistor $\mathscr{D}$-modules_, Astérisque, vol. 300, Société Mathématique de France, Paris, 2005.
  * [19] P. Schapira and J.-P. Schneiders, _Index theorem for elliptic pairs_ , Astérisque, vol. 224, Société Mathématique de France, Paris, 1994\.
  * [20] Y.-T. Siu and G. Trautmann, _Gap-sheaves and extension of coherent analytic subsheaves_ , Lect. Notes in Math., vol. 172, Springer-Verlag, Berlin-New York, 1971.


# Massive Scalar Perturbation of Extremal Rotating Braneworld Black hole:
Superradiant Stability Analysis

Shauvik<EMAIL_ADDRESS><EMAIL_ADDRESS>
School of Physical Sciences
Indian Association for the Cultivation of Science, Kolkata-700032, India

###### Abstract

We analyse the superradiant stability of braneworld extremal Kerr and Kerr-
Newman black holes under massive scalar perturbation. These black hole
solutions differ from their four dimensional counterpart by the presence of a
tidal charge, which unlike electric charge, can take both positive and
negative values and carries the signature of extra dimensions. We consider the
perturbation of the brane geometry by a massive scalar field. From the radial
equation of motion of the scalar field, we have found the effective potential
felt by the field. For superradiant stability, there should not be any
trapping well of the effective potential outside of the horizon. Using this
condition we have specified the parameter space of superradiant stability
which depends on parameters of both black hole and perturbation. In the case
of Kerr-like extremal braneworld black holes, we have obtained a bound on
tidal charge for superradiant stability under a massive scalar perturbation.
Similarly, for slowly rotating Kerr-Newman like extremal braneworld black
holes, we have obtained a lower bound of the product of mass to charge ratio
of the black hole and scalar field, for which the system is superradiantly
stable.

##  1 Introduction

Black holes are solutions to Einstein’s equation in different scenarios[1, 2,
3]. These solutions play important role in high energy physics and
astrophysics. The fundamental role of these solutions makes it highly
important to study the nature of their stability under linear perturbations of
different spins. In this context one considers a test wave impinging on a
black hole, a part of which is reflected by the effective potential (felt by
it) and scattered to infinity. The transmitted part is swallowed by the black
hole. Thus, it is expected that the amplitude of the outgoing wave is smaller
than that of the incident wave. It was shown that the Schwarzschild black hole
is stable under all kinds of massless perturbation [4, 5, 6, 7]. On the other
hand, the stability issue of the rotating black holes is more complicated due
to the presence of ergoregion outside the event horizon. In this region the
conserved energy of causal matter becomes negative. One can utilize this fact
to extract the rest energy and rotational energy of the black hole[8, 9, 10].
It is shown [11, 12] that when a bosonic wave impinges on a Kerr-like black
hole, the reflected wave can be amplified if the so-called superradiance
condition [1] is satisfied. In this case, the wave near the horizon becomes
outgoing [12] and extracts the rotational energy of the black hole from the
ergosphere. If this process continues to occur then a Kerr-black hole finally
becomes a Schwarzschild black hole. Due to the Pauli exclusion principle, such
phenomena are absent in the case of fermionic scattering[11].

Press and Teukolsky [12] had considered massless scalar perturbation of Kerr
black hole and proposed a feedback mechanism of the superradiantly amplified
scattered wave by surrounding the black hole with a reflecting mirror. Then
the amplified scattered wave can not escape to infinity and is reflected by
the mirror. This wave is then amplified again. In this way, the total
extracted energy should grow indefinitely until radiation pressure destroys
the system[12, 13]. This is called a ”black hole bomb”. It was also shown [14]
that such a mirror can be realized by using a charged massive scalar field
propagating in the Kerr-Newman background. In this case, the effective
potential of the massive scalar field has a local minima [14, 15], that is
outside the horizon we have trapping well and the scalar field can be trapped
inside this well instead of letting it escape to infinity[10, 16]. Due to the
above mentioned feedback mechanism, the scalar field plus black hole system
become superradiantly unstable. Various work on instability growth rate had
been done in the range $M\mu\ll 1$ [15, 17] and $M\mu\gg 1$ [18], where M and
$\mu$ are the mass of the black hole and scalar field respectively. It is
proposed [16] that this instability growth rate is maximum for extremal black
holes, for which the inner and outer horizons coincide.

In this paper, we are going to study the above mentioned massive scalar
perturbation in the context of rotating braneworld black holes. Braneworld
black holes are natural candidates to study quantum black holes [19, 20, 21,
22]. Recently these solutions are extensively used in ECO
phenomenology333Recently an interesting work[23]analytically shows that highly
compact rotating ECO with reflective boundary condition will be marginally
stable under static massless scalar perturbation only for discrete spectra of
critical radii. [20], tidal heating of black holes [24], optical properties of
black holes [25, 26] and in some other interesting cases [27, 28, 29, 30] .
These black hole solutions differ from their counterpart in general relativity
by the presence of a tidal charge, unlike electric charge tidal charge can
take negative values, which has no analogue in GR. This charge depends on the
electric part of the bulk Weyl tensor and carries the effect of extra
dimensions. An immediate consequence of tidal charge is that in the case of
extremal Kerr-like black holes we do not have a=M, where a is the black hole
spin parameter and M is the mass parameter. Moreover, $a>M$ is allowed, which
is also absent in GR. As mentioned before, some earlier works [18, 17, 16]
have shown that the superradiant instability growth rate of massive scalar
perturbation of rotating black holes is maximum for extremal black holes. Here
we will do a superradiant stability analysis for rotating braneworld black
holes under massive scalar perturbation. That is, we will try to determine a
parameter space where braneworld Kerr-like (or Kerr-Newman like) black holes
are superradiantly stable under massive scalar (or charged massive scalar)
perturbations. To do the analysis we will follow the formalism presented in
[31, 32, 33, 34]. As stated before, superradiant instability happens when
there exists a local minima (that is trapping well) of effective potential
outside the horizon. So for stability under superradiance there should exist
only one maxima of effective potential outside the horizon. This is the key
argument of our analysis. In this analytic treatment, we will consider only
the real frequencies of the scalar wave. But the conclusions can be extended
to nearly real frequencies as in [14, 35, 18]. Like [14] we will assume that
both the superradiance condition and the bound state condition hold
simultaneously. In the case of Kerr-like extremal braneworld black holes, we
will obtain a bound on tidal charge for superradiant stability under a massive
scalar perturbation. Similarly, for Kerr-Newman like slowly rotating extremal
braneworld black holes, we will obtain a lower bound of the product of mass to
charge ratio of the black hole and scalar field for which the system is
superradiantly stable.

This paper is organized as follows: In 2 a brief review of the braneworld
scenario and the solution of the effective gravitational equation on brane
will be given. In 3 we will demonstrate the massive scalar perturbation of
braneworld extremal Kerr-like black holes. Then taking both the bound state
condition and superradiance condition we will find a parameter region where
the system is superradiantly stable. In 4 we will do the same analysis for
charged massive scalar perturbation of Kerr-Newman like extremal braneworld
black hole. In 5 a discussion of this formalism will be given.
_Notations and Conventions:_ In our calculations we will set $c=G=1$. The
lowercase Roman indices $a,b,c,\ldots$ will denote the four dimensional
braneworld spacetime coordinates and run over 0,1,2,3. On the other hand
uppercase Roman indices $A,B,C,\ldots$ will denote the five dimensional bulk
spacetime coordinates and run over 0,1,2,3,4. In both the cases 0 denotes the
time coordinate. We will follow the positive signature convention for our
metric, for which the Minkowski metric in four dimension takes the form
$\textrm{diag}(-1,+1,+1,+1)$.

##  2 A brief introduction to the braneworld scenario and solution of the
effective gravitational equation on the brane

In the braneworld model, we describe four dimensional spacetime as embedded in
five dimensional spacetime, with a spacelike extra dimension. The non-
gravitational physics of matter fields are confined in the four dimensional
space time; it is called brane. On the other hand, only gravity can probe the
entire five dimensional space-time, known as bulk. People would like to know
whether one can have a brane localized black hole. To answer this one has to
solve the effective gravitational equation on the brane, which can be done by
the following steps: i) we start by assuming that bulk spacetime satisfies
five dimensional Einstein’s equation. ii) then we introduce projector
$h^{A}_{B}=\delta^{A}_{B}-n^{A}n_{B}$ to the brane, where $n^{A}$ is normal to
brane hypersurface, which satisfies $n^{A}n_{A}=1$. iii) Ultimately we use
this projector and Gauss-Codazzi relations to derive the Einstein tensor on
the brane and thereby determining the effective equation. Considering a vacuum
brane, one can get[36, 37]

$\displaystyle{}^{(4)}G_{ab}+E_{ab}=0\quad.$ (1)

In this equation ${}^{(4)}G_{ab}$ is the induced Einstein tensor on the brane
and $E_{ab}\equiv C_{PQRS}n^{P}e^{Q}_{a}n^{R}e^{S}_{b}$ is the electric part
of the bulk Weyl tensor $C_{PQRS}$. Also $e^{A}_{a}\equiv(\partial
x^{A}/\partial y^{a})$, where $x^{A}$ and $y^{a}$ are the brane and bulk the
coordinates respectively. $e^{A}_{a}$’s are tangent to brane hypersurface
i.e.; $e^{A}_{a}n_{A}=0$.
The static spherically symmetric solution of the above equation was first
derived in [37], then using Kerr-Schild ansatz [38, 39, 40] it was generalized
to the rotating case [41]. Using the Boyer-Lindquist coordinate system, the
metric which describes a rotating braneworld black hole with mass $M$ and
angular momentum $J\equiv aM$ can be written as,

$\displaystyle ds^{2}=-\frac{\Delta}{\rho^{2}}(dt-a{}\sin^{2}\theta
d\phi)^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[(r^{2}+a^{2})d\phi-
adt\right]^{2}~{},$ (2)

where $\Delta\equiv r^{2}+a^{2}-2Mr-Q$ ; $\rho^{2}\equiv
r^{2}+a^{2}\cos^{2}\theta~{}.$

The tidal charge parameter $Q$ depends on the bulk Weyl tensor. It can take
both positive and negative values. When $Q$ is negative it is like the case of
the Kerr-Newman black hole in GR, on the other hand, if $Q$ is positive, it
carries the effect of extra dimensions. The horizons associated with metric in
2 can be obtained by setting $\Delta=0$, this yields,

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-a^{2}+Q}~{}.$ (3)

Brane localized observers will anticipate $r_{+}=M+\sqrt{M^{2}-a^{2}+Q}$ as
the horizon of the black hole. Now in case of extremal black holes the two
horizon coincides, that is,

$\displaystyle r_{+}=r_{-}=M\quad\text{with}\quad M^{2}=a^{2}-Q~{}.$ (4)

If the brane is not vacuum, that is we have a non-vanishing energy momentum
tensor $T_{ab}$ of matter fields, then the effective gravitational equation 1
on brane modifies as[36, 41],

$\displaystyle{}^{(4)}G_{ab}+E_{ab}=8\pi T_{ab}+\pi_{ab}\quad.$ (5)

In this equation $\pi_{ab}$ is called ”squared” energy momentum tensor; which
contains quadratic combinations of $T_{ab}$(for explicit expressions see[41,
36]). If a Maxwell field is confined on the brane, then $T_{ab}$ is given
by[42, 1, 2],

$\displaystyle
T_{ab}=\frac{1}{4\pi}\left(F_{ac}F^{c}_{b}-\frac{1}{4}h_{ab}F_{cd}F^{cd}\right)~{},$
(6)

where $F_{ab}$ is the electromagnetic field tensor. Using this energy momentum
tensor in 5 and then taking Kerr-Schild ansatz[38, 39, 40] one can derive a
metric describing rotating black hole on brane having both tidal charge and
electric charge 444Note that our notation of electric charge and tidal charge
is different from that of the reference given. [41], which in Boyer-Lindquist
coordinate system reads,

$\displaystyle
ds^{2}=-(1-H)dt^{2}+\frac{\rho^{2}}{\Delta_{l}}(1+\delta)dr^{2}+\rho^{2}d\theta^{2}+(r^{2}+a^{2}+Ha^{2}\sin^{2}\theta)\sin^{2}\theta
d\phi^{2}~{}$ (7) $\displaystyle-2\delta(dt-a\sin^{2}\theta
d\phi)dr-2aH\sin^{2}\theta dtd\phi~{},$

where

$\displaystyle
H=\frac{2Mr+Q-\beta^{2}}{\rho^{2}}-\frac{l^{2}\beta^{4}}{\rho^{2}}h~{},$ (8)
$\displaystyle\Delta_{l}=r^{2}+a^{2}+\beta^{2}-2Mr-Q+l^{2}\beta^{4}h_{0}\quad;\quad\delta=-\frac{l^{2}\beta^{4}}{\Delta_{l}}(h-h_{0})~{},$
(9) $\displaystyle
h=\frac{1}{8a^{4}\cos^{4}\theta}\left[2+\frac{r^{2}}{\rho^{2}}+\frac{3r}{a\cos\theta}tan^{-1}\left(\frac{r}{a\cos\theta}\right)\right]~{}.\quad$
(10)

Here $l$ is the AdS curvature radius and $h_{0}$ is the value of $h$ at fixed
angle $\theta_{0}$. One can expand $H$ and $h$ in powers of $a$ and keep
corrections up to second order to yield,

$\displaystyle\quad
h-h_{0}=-\frac{a^{2}}{14r^{6}}(\cos^{2}\theta-\cos^{2}\theta_{0})~{},$ (11)
$\displaystyle
H=\frac{2M}{r}-\frac{\beta^{2}-Q}{r^{2}}-\frac{l^{2}\beta^{4}}{20r^{6}}-\frac{a^{2}\cos^{2}\theta}{r^{2}}\left(\frac{2M}{r}-\frac{\beta^{2}-Q}{r^{2}}-\frac{17}{140}\frac{l^{2}\beta^{4}}{r^{6}}\right)~{}.$
(12)

Now since the terms of $\mathcal{O}\left(\frac{1}{r^{6}}\right)$ are highly
suppressed than the other terms, so we can approximate the metric in the form
of 2 with $\Delta\equiv r^{2}+a^{2}+\beta^{2}-2Mr-Q$, where $\beta$ is the
electric charge. In this case, the extremal black hole is characterized by the
condition:

$\displaystyle r_{+}=r_{-}=M\quad\text{with}\quad M^{2}=a^{2}+\beta^{2}-Q~{}.$
(13)

In the next section, we are going to study the massive scalar perturbation of
the brane geometry, where the extra dimension will be kept unperturbed.

##  3 Massive Scalar perturbation of Braneworld Extremal Kerr Black Hole

The dynamics of a massive scalar perturbation $\Psi$ in the Kerr background 2
is described by the covariant Klein-Gordon equation[18, 17, 43],

$\displaystyle(\nabla^{a}\nabla_{a}-\mu^{2})\Psi=0~{},$ (14)

where $\mu$ is the mass of the scalar field and $\nabla_{a}$ represents the
covariant derivative with respect to the background brane metric.

Since the background spacetime exhibits two killing vectors field,
$(\partial/\partial t)^{a}$ and $(\partial/\partial\phi)^{a}$, so we can
decompose the solution of the above differential equation as,

$\displaystyle\Psi(t,r,\theta,\phi)=\sum_{lm}R_{lm}(r)S_{lm}(\theta)e^{im\phi}e^{-i\omega
t}~{},$ (15)

here $l$ and $m$ respectively represent the spheroidal harmonics index and
azimuthal harmonics index of the mode. $R_{lm}(r)$ is the radial wave function
and $S_{lm}(\theta)$ is the angular wave function. $S_{lm}(\theta)$ satisfies
the spheroidal harmonic equation[44, 45, 46, 47, 48]:

$\displaystyle\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{dS_{lm}}{d\theta}\right)+\left[K_{lm}+(\mu^{2}-\omega^{2})a^{2}\sin^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}\right]S_{lm}=0~{},$
(16)

here $K_{lm}$ is the eigenvalue of the differential operator with
$S_{lm}(\theta)$ as eigenfunction. Lower bound of $K_{lm}$ is given by [48,
49, 50]:

$\displaystyle K_{lm}>m^{2}-a^{2}(\mu^{2}-\omega^{2}).$ (17)

On the other hand $R_{lm}(r)$ satisfies the differential equation[51, 44, 45,
46, 47],

$\displaystyle\Delta\frac{d}{dr}\left(\Delta\frac{dR_{lm}}{dr}\right)+U_{K}R_{lm}=0~{},$
(18)

where

$\displaystyle
U_{K}=\\{\omega(r^{2}+a^{2})-am\\}^{2}+\Delta\\{2am\omega-\mu^{2}(r^{2}+a^{2})-K_{lm}\\}.$
(19)

We define tortoise coordinate as

$\displaystyle\frac{dr_{*}}{dr}=\frac{r^{2}+a^{2}}{\Delta}~{}.$ (20)

By defining a new radial function [4]
$\tilde{R}_{lm}\equiv\sqrt{r^{2}+a^{2}}R_{lm}$ we can bring 18 in a form

$\displaystyle\frac{d^{2}\tilde{R}_{lm}}{dr_{*}^{2}}+\tilde{U}_{K}\tilde{R}_{lm}=0~{},$
(21)

where

$\displaystyle\tilde{U}_{K}=\frac{1}{(r^{2}+a^{2})^{2}}\left[(r^{2}+a^{2})^{2}\left(\omega-\frac{ma}{(r^{2}+a^{2})}\right)^{2}+\Delta(r^{2}+a^{2})\left(-\mu^{2}-\frac{K_{lm}}{(r^{2}+a^{2})}+\frac{2am\omega}{(r^{2}+a^{2})}\right)\right]-\frac{\Delta
P(r)}{(r^{2}+a^{2})^{2}}$ (22)

with
$P(r)\equiv\frac{[\\{r\Delta^{\prime}+\Delta\\}(r^{2}+a^{2})^{\frac{3}{2}}-3r^{2}\Delta\sqrt{r^{2}+a^{2}}]}{(r^{2}+a^{2})^{\frac{5}{2}}}$.

Considering asymptotic and near horizon behaviour of $\tilde{U}_{K}$ and
demanding exponentially decaying bound state solution at spatial infinity we
impose the following boundary condition on $\tilde{R}_{lm}$

$\displaystyle\tilde{R}_{\ell
m}(r)\sim\begin{cases}e^{-\sqrt{\mu^{2}-\omega^{2}}r_{*}}&\text{for}\ \quad
r_{*}\to\infty\\\ \\\ e^{-i(\omega-m\Omega_{H})r_{*}}&\text{for}\quad
r_{*}\to-\infty~{},\end{cases}$ (23)

where $\Omega_{H}=\frac{a}{r^{2}_{+}+a^{2}}=\frac{a}{2a^{2}-Q}$ is the angular
velocity of the horizon. Here we have used the fact, that near the horizon we
should have only ingoing wave (as seen by the co-moving observers).

From the above, it is clear that to get exponentially decaying bound state at
spatial infinity we need

$\displaystyle\mu^{2}>\omega^{2}.$ (24)

In case of Kerr-like braneworld black holes the superadiance condition[11, 52,
20]is given by

$\displaystyle 0<\omega<m\Omega_{H}\quad(m>0).$ (25)

###  3.1 Superradiant Stability Analysis

In order to find superradiantly stable parameter space of the braneworld
extermal Kerr black hole, we define a new radial function [31]
$\psi_{lm}=\sqrt{\Delta}R_{lm}$. Using this radial function in 18 we get

$\displaystyle\frac{d^{2}\psi_{lm}}{dr^{2}}+(\omega^{2}-V_{K})\psi_{lm}=0~{},$
(26)

where

$\displaystyle V_{K}$ $\displaystyle=\omega^{2}-\frac{U_{K}}{\Delta^{2}}$ (27)
$\displaystyle=\omega^{2}-\frac{\\{\omega(r^{2}+a^{2})-am\\}^{2}}{\Delta^{2}}-\frac{\\{2am\omega-\mu^{2}(r^{2}+a^{2})-K_{lm}\\}}{\Delta}~{}.$
(28)

Here we have used the fact, that for extremal black holes $M^{2}=a^{2}-Q$.

The asymptotic and near horizon behaviour of the above potential is given by

$\displaystyle
V_{K}\sim\begin{cases}\mu^{2}+\frac{2M}{r}(\mu^{2}-2\omega^{2})+\mathcal{O}(\frac{1}{r^{2}})&\text{for}\
\quad r\to\infty\\\ \\\ -\infty&\text{for}\quad r\to r_{+}.\end{cases}$ (29)

On the other hand
$V^{\prime}_{K}(r\rightarrow\infty)\rightarrow\frac{2M}{r^{2}}(-\mu^{2}+2\omega^{2})+\mathcal{O}(\frac{1}{r^{3}})$.
As explained in 1, for superradiant stability under massive scalar
perturbation, there should not be any trapping well outside the horizon for
the effective potential $V_{K}$[31, 32, 33]. That is, it can have only one
maxima outside the horizon and no local minima. To do such analysis, we
compute the derivative of effective potential $V_{K}$ using a new radial
coordinate $z=r-M$. In terms of this new radial coordinate, one can write the
derivative of $V_{K}$ as,

$\displaystyle V^{\prime}_{K}(z)$
$\displaystyle=\frac{z(Az^{3}+Bz^{2}+Cz+D)}{-z^{6}}\equiv\frac{zf(z)}{-z^{6}}~{},$
(30)

where

$\displaystyle A$ $\displaystyle=2M(\mu^{2}-2\omega^{2})~{},$ (31)
$\displaystyle B$
$\displaystyle=2K_{lm}+2\mu^{2}a^{2}+2\mu^{2}M^{2}-12\omega^{2}M^{2}-4\omega^{2}a^{2}~{},$
(32) $\displaystyle C$ $\displaystyle=12M\omega
ma-12M\omega^{2}(M^{2}+a^{2})~{},$ (33) $\displaystyle D$
$\displaystyle=-4[\omega(M^{2}+a^{2})-ma]^{2}~{}.$ (34)

Here we have used the fact, that for extremal black holes $\Delta=z^{2}$. In
the above expressions $M^{2}=a^{2}-Q$.

At the position of maxima or minima $V^{\prime}_{K}(r)=V^{\prime}_{K}(z)=0$.
Then we can avoid to have any trapping well (local minima) outside the horizon
if the equation $V^{\prime}_{K}(z)=0$ has only one positive root and
$V^{\prime\prime}_{K}(z)<0$ at that point. That is, one can remove the
possibility of getting the local minima outside the horizon by requiring that
$V_{K}(r)$ can have only one maxima outside the horizon. To specify the
parameter region consistent with this requirement, we analyze the sign of the
roots of the $f(z)=0$ equation. This can be done using Vieta’s theorem[53]. If
$z_{1}$, $z_{2}$ and $z_{3}$ are the three roots of the cubic equation
$f(z)=0$, then according to this theorem:

$\displaystyle z_{1}+z_{2}+z_{3}=-\frac{B}{A}~{},$ (35) $\displaystyle
z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}=\frac{C}{A}~{},$ (36) $\displaystyle
z_{1}z_{2}z_{3}=-\frac{D}{A}~{}.$ (37)

From the expression of the coefficients of $f(z)$ it is clear that $D<0$. Now
suppose $A<0$, then from 37 we can infer that the product of the roots is
negative. In this case either we will have two positive root and one negative
root or the three roots will be negative. Now using superradiance condition 25
we can see that $C>0$. Then from 36 it is clear that we can never have three
negative roots if $A<0$. In this case we are left with the possibility of
having two positive roots and one negative root. That is, with the condition
$A<0$ one can not avoid the possibility of forming trapping well outside the
horizon. On the other hand, if we let $A>0$, then the product of the roots
become positive. In this case, we can choose our parameters to have only one
positive (say $z_{1}$) and two negative roots (say $z_{2}$ and $z_{3}$). So
the necessary condition to avoid trapping well is $A>0$(see also [54]). In
terms of perturbation parameters this gives,

$\displaystyle\omega^{2}<\frac{\mu^{2}}{2}~{}.$ (38)

According to the asymptotic behaviour of the effective potential, this implies
$V^{\prime}_{K}(r\rightarrow\infty)<0$. So we have reduced our job to a
problem of finding the parameter region where $z_{2}<0$ and $z_{3}<0$. A
sufficient condition for the negativity of both $z_{2}$ and $z_{3}$ is (see
35)

$\displaystyle B>0~{}.$ (39)

Now using superradiance condition 25 and lower bound of $K_{lm}$ 17in the
inequality 39 we get

$\displaystyle\mu$ $\displaystyle>\omega\sqrt{3+\frac{Q}{a^{2}}}~{}.$ (40)

It only remains to confirm that $z_{1}(>0)$ is the maxima of $V_{K}(z)$ in
this parameter region; that is $V^{\prime\prime}_{K}(z_{1})<0$. The second
derivative of the potential $V_{K}(z)$ can be written in a form

$\displaystyle
V^{\prime\prime}_{K}(z)=\frac{5f(z)-(3Az^{3}+2Bz^{2}+Cz)}{z^{6}}~{}.$ (41)

Since $V^{\prime}_{K}(z_{1})=0$, so that $f(z_{1})=0$. It is easy to see that
when suprradience condition is satisfied i.e.,
$m>\frac{\omega(M^{2}+a^{2})}{a}$, then $C>0$. Also by our demand $B>0$. So
$V^{\prime\prime}_{K}(z_{1})<0$ in this parameter region.

###  3.2 Bound On Tidal Charge

To hold the inequality 38 (otherwise we can not ensure $A>0$) and 40
simultaneously, we must have

$\displaystyle Q\geq-a^{2}.$ (42)

Also we must have $M^{2}=a^{2}-Q>0$. Combining these two inequalities we get

$\displaystyle-a^{2}\leq Q<a^{2}.$ (43)

This serves as a bound on tidal charge. When the inequalities 25, 40 and 43
are satisfied, then we can ensure that there is no trapping potential well
outside the horizon and the system of scalar perturbation and braneworld
extremal Kerr black hole is superradiantly stable. Note that when these three
inequalities hold simultaneously, the bound state condition 24 and the
inequality 38 hold automatically. Therefore these three inequalities are
sufficient to specify superradiantly stable parameter space.

###  3.3 Behaviour of Effective Potential Outside the Horizon

In this section, we will demonstrate the behaviour of effective potential
$V_{K}(r)$ as a function of r for two sets of parameters. This will give a
justification of our results.

  1. 1.

Case-1 . In this case, we choose our parameters such that inequalities 25, 40
and 43 are satisfied simultaneously. For example we take $a=2$, $m=1$,
$\omega=0.250$, $\mu=0.500$, $M=1$, $Q=3$. As required by our result, there is
no potential well outside the horizon 1 and the system is superradiantly
stable.

Figure 1: The effective potential $V_{K}(r)$ has no trapping well outside the
horizon. The system is superradiantly stable. Here the parameters of
perturbation are $\omega=0.250$, $\mu=0.500$ and $m=1$. On the other hand, the
parameters of the black hole are chosen as $M=1$, $a=2$ and $Q=3$. Figure 2:
The effective potential $V_{K}(r)$ has a trapping well outside the horizon.
The system is superradiantly unstable. In this case parameters of the
perturbation are chosen as $m=2$, $\omega=0.250$ and $\mu=0.300$. On the hand,
the black hole parameters are $M=3$, $Q=-5$ and $a=2$.

  2. 2.

Case-2. In this case, we choose our parameters such that they do not satisfy
all the above mentioned inequalities. For example we take $a=2$, $m=2$,
$\omega=0.250$, $\mu=0.300$, $M=3$, $Q=-5$. In this case both bound state
condition 24 and superradiance condition 25 hold but inequality 43 does not
hold. As evident from the 2, we have a potential well outside the horizon and
the system is superradiantly unstable.

##  4 Charged Massive Scalar perturbation of Braneworld Extremal Kerr-Newman
Black Hole

As stated in 2 in this case the brane geometry is approximately described by
the metric 2 with $\Delta\equiv r^{2}+a^{2}+\beta^{2}-2Mr-Q$. The dynamics of
a charged massive scalar perturbation $\Psi$ of mass $\mu$ and charge $q$ in
the braneworld Kerr-Newman background is given by [44, 45, 55],

$\displaystyle(\nabla^{a}-iqA^{a})(\nabla_{a}-iqA_{a})\Psi=\mu^{2}\Psi~{},$
(44)

here $\nabla_{a}$ represents covariant derivative in the background geometry
and the background electromagnetic vector potential is given by

$\displaystyle A_{a}=\left(\frac{-r\beta}{\rho^{2}},0,0,\frac{ar\beta
sin^{2}\theta}{\rho^{2}}\right).$ (45)

Using the same argument presented in 3 the solution of the above differential
equation can be decomposed as

$\displaystyle\Psi(t,r,\theta,\phi)=\sum_{lm}R_{lm}(r)S_{lm}(\theta)e^{im\phi}e^{-i\omega
t}~{},$ (46)

here $l$ and $m$ receptively represent the spheroidal harmonics index and
azimuthal harmonics index of the mode. $R_{lm}(r)$ is the radial wave function
and $S_{lm}(\theta)$ is the angular wave function. $S_{lm}(\theta)$ satisfies
same the spheroidal harmonic 16. So $K_{lm}$ has the same lower bound 17.

$R_{lm}(r)$ satisfies a differential equation of the form 18 but with
different potential [55],

$\displaystyle\Delta\frac{d}{dr}\left(\Delta\frac{dR_{lm}}{dr}\right)+U_{KN}R_{lm}=0~{},$
(47)

where

$\displaystyle U_{KN}=\\{\omega(r^{2}+a^{2})-am-q\beta
r\\}^{2}+\Delta\\{2am\omega-\mu^{2}(r^{2}+a^{2})-K_{lm}\\}~{}.$ (48)

One can do a similar analysis as we have done in 3 to find the asymptotic and
near horizon behaviour of $R_{lm}(r)$. Demanding exponentially decaying bound
state solution at spatial infinity and purely ingoing wave solution near the
horizon, we can get the following behaviour of $R_{lm}$,

$\displaystyle R_{\ell
m}(r)\sim\begin{cases}\frac{e^{-\sqrt{\mu^{2}-\omega^{2}}r_{*}}}{r}&\text{for}\
\quad r_{*}\to\infty\\\ \\\ e^{-i(\omega-\omega_{c})r_{*}}&\text{for}\quad
r_{*}\to-\infty~{},\end{cases}$ (49)

where $\omega_{c}=m\Omega_{H}+q\Phi_{H}$. Here
$\Omega_{H}=\frac{a}{M^{2}+a^{2}}$ is the angular velocity of the horizon and
$\Phi_{H}=\frac{M\beta}{M^{2}+a^{2}}$ is the electric potential of the
horizon. From the above, it is clear that to get an exponentially decaying
bound state at spatial infinity we need

$\displaystyle\mu^{2}>\omega^{2}.$ (50)

In the case of charged rotating black holes the superradiance condition[11,
52] is,

$\displaystyle 0<\omega<\omega_{c}~{}.$ (51)

###  4.1 Superradiant Stability Analysis

To analyze the superradiant stability of the braneworld extremal Kerr-Newman
black hole we follow the same steps presented in 3.1. Using a new radial
function [31] $\psi_{lm}=\sqrt{\Delta}R_{lm}$, one can put the radial equation
47 in the form,

$\displaystyle\frac{d^{2}\psi_{lm}}{dr^{2}}+(\omega^{2}-V_{KN})\psi_{lm}=0~{},$
(52)

where

$\displaystyle V_{KN}$ $\displaystyle=\omega^{2}-\frac{U_{KN}}{\Delta^{2}}$
(53) $\displaystyle=\omega^{2}-\frac{\\{\omega(r^{2}+a^{2})-am-
rq\beta\\}^{2}}{\Delta^{2}}-\frac{\\{2am\omega-\mu^{2}(r^{2}+a^{2})-K_{lm}\\}}{\Delta}~{}.$
(54)

While deriving 52 we have used the fact, that for charged extremal black holes
$M^{2}=a^{2}+\beta^{2}-Q$.
The asymptotic and near horizon behaviour of the effective potential
$V_{KN}(r)$ is,

$\displaystyle
V_{KN}\sim\begin{cases}\mu^{2}+\frac{2M(\mu^{2}-2\omega^{2})+2q\beta\omega}{r}+\mathcal{O}(\frac{1}{r^{2}})&\text{for}\
\quad r\to\infty\\\ \\\ -\infty&\text{for}\quad r\to r_{+}.\end{cases}$ (55)

On the other hand
$V^{\prime}_{KN}(r\rightarrow\infty)\rightarrow\frac{-2M(\mu^{2}-\omega^{2})-2\omega(q\beta-M\omega)}{r^{2}}+\mathcal{O}(\frac{1}{r^{3}})$.
For superradiant stability under charged massive scalar perturbation, there
should not be any trapping well outside the horizon for the effective
potential $V_{KN}$ [31, 32, 33]. As in 3.1, we can implement this condition by
demanding

  1. 1.

$V^{\prime}_{KN}(r\rightarrow\infty)<0$ ,

  2. 2.

$V_{KN}(r)$ should have only one maxima outside the horizon.

If the bound state condition 50 holds then the first condition always holds
when

$\displaystyle\omega<\frac{q\beta}{M}\quad(q\beta\neq 0)~{}.$ (56)

To fulfill the second condition, we compute the derivative of the effective
potential $V_{KN}(r)$ using new radial coordinate $z=r-M$, which yields,

$\displaystyle V^{\prime}_{KN}(z)$
$\displaystyle=\frac{z(Ez^{3}+Fz^{2}+Gz+H)}{-z^{6}}\equiv\frac{zh(z)}{-z^{6}}~{},$
(57)

here

$\displaystyle E$
$\displaystyle=2[\omega(q\beta-M\omega)+M(\mu^{2}-\omega^{2})]~{},$ (58)
$\displaystyle F$
$\displaystyle=2[\mu^{2}(M^{2}+a^{2})-2\omega\\{\omega(M^{2}+a^{2})-qM\beta\\}+K_{lm}-(q\beta-2M\omega)^{2}]~{},$
(59) $\displaystyle G$
$\displaystyle=-6(2M\omega-q\beta)\\{\omega(M^{2}+a^{2})-am-qM\beta\\}~{},$
(60) $\displaystyle H$
$\displaystyle=-4(M^{2}+a^{2})^{2}(\omega-\omega_{c})^{2}~{}.$ (61)

In the above expressions $M^{2}=a^{2}+\beta^{2}-Q$. As argued in 3.1, the
effective potential $V_{KN}(r)$ will have only one maxima outside horizon if
the equation $h(z)=0$ has only one positive root. If $z_{1}$, $z_{2}$ and
$z_{3}$ are the three roots of the cubic equation $h(z)=0$, then according to
Vieta’s theorem[53]

$\displaystyle z_{1}+z_{2}+z_{3}=-\frac{F}{E}~{},$ (62) $\displaystyle
z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}=\frac{G}{E}~{},$ (63) $\displaystyle
z_{1}z_{2}z_{3}=-\frac{H}{E}~{}.$ (64)

From the asymptotic behaviour of the effective potential $V_{KN}(r)$ it is
clear that there exists at least one positive root (say $z_{1}$)outside the
horizon, that is

$\displaystyle z_{1}>0~{}.$ (65)

From 58 and using 56 and 50 we can see that $E>0$. It is also clear from 61
that $H<0$. So from 64 we see that the product of the roots is positive. Then
a sufficient condition for the negativity of both $z_{2}$ and $z_{3}$ (see 62)
is,

$\displaystyle F>0~{}.$ (66)

Now using the lower bound of $K_{lm}$ 17 in the inequality 66 we get,

$\displaystyle\mu^{2}>\frac{\omega^{2}(6M^{2}+a^{2})-6\omega\beta
qM+(q^{2}\beta^{2}-m^{2})}{M^{2}}\equiv\frac{y(\omega)}{M^{2}}~{}.$ (67)

The function $y(\omega)$ represents a parabola of the form
$(\omega-\omega_{A})^{2}=4a(y-b)$, where

$\displaystyle\omega_{A}=\frac{3q\beta M}{6M^{2}+a^{2}}~{},$ $\displaystyle
4a=\frac{1}{6M^{2}+a^{2}}~{},$ $\displaystyle
b=\frac{q^{2}\beta^{2}(a^{2}-3M^{2})}{6M^{2}+a^{2}}-m^{2}~{},$

here $\omega_{A}$ represents the axis of symmetry of the parabola. Using the
inequality $\frac{1}{6M^{2}+a^{2}}<\frac{1}{6M^{2}}$, one can show that,

$\displaystyle 2\omega_{A}<\frac{q\beta}{M}~{}.$ (68)

From this inequality, it is clear that if $0<\omega<\frac{q\beta}{M}$, the
maximum value of $y(\omega)$ is given by $y(\frac{q\beta}{M})$. So the
sufficient condition to hold the inequality 67 is

$\displaystyle\mu^{2}$
$\displaystyle>\frac{y(\frac{q\beta}{M})+m^{2}}{M^{2}}=\left(\frac{q\beta}{M}\right)^{2}(k^{2}+1)\quad\text{where}\quad
k=\frac{a}{M}~{}.$ (69)

In terms of tidal charge, the above inequality becomes

$\displaystyle\frac{\mu}{q}>\frac{\beta}{\sqrt{a^{2}+\beta^{2}-Q}}\sqrt{\frac{a^{2}}{a^{2}+\beta^{2}-Q}+1}~{}.$
(70)

When the above inequality holds together with inequalities 51 and 56, there is
no trapping well outside the horizon and our system of charged massive scalar
field and braneworld extremal Kerr-Newman black is superradiantly stable. When
these inequalities are not satisfied, we can not assure the superradiant
stability of the system.

###  4.2 Behaviour of Effective Potential Outside the Horizon

In this section, we will demonstrate the behaviour of effective potential
$V_{KN}(r)$ as a function of r for two sets of parameters. This will give a
justification of our results.

  1. 1.

Case-1. In this case we choose our parameters such that inequalities 69, 56
and 51 are satisfied simultaneously. For example we take $a=0.5$, $m=1$,
$\omega=0.280$, $\mu=0.350$, $M=2.961$, $Q=-2.27$, $q=0.4$, $\beta=2.5$. In
this case $\frac{M\mu}{q\beta}=1.036>\sqrt{k^{2}+1}=1.014$ and
$\omega=0.280<\frac{q\beta}{M}=0.337$. As required by our results, there is no
potential well outside the horizon (see 3) and the system is superradiantly
stable.

Figure 3: The effective potential $V_{KN}(r)$ has no trapping well outside the
horizon. Therefore system is superradiantly stable. The parameters of the
perturbation are taken as $m=1$, $\omega=0.280$, $\mu=0.350$, $q=0.4$. The
parameters of the black hole are $a=0.5$, $\beta=2.5$, $M=2.961$, $Q=-2.27$.
Figure 4: The effective potential $V_{KN}(r)$ has a trapping well outside the
horizon. So the system is superradiantly unstable. The parameters of the
perturbation are taken as $m=1$, $\omega=0.286$, $\mu=0.300$ and $q=0.300$.
The parameters of the black hole are $a=2$, $M=3$, $Q=-1$, $\beta=2.000$.

  2. 2.

Case-2. In this case we choose our parameters such that they do not satisfy
all the (above mentioned) inequalities. For example we take $a=2$, $m=1$,
$\omega=0.286$, $\mu=0.300$, $M=3$, $Q=-1$, $q=0.300$, $\beta=2.000$. In this
case both bound state condition 50 and superradiance condition 51 hold, but
inequality 56 does not hold. As evident from the 4, in this case we have a
potential well outside the horizon and the system is superradiantly unstable.

##  5 Discussions

In the above analysis, we have found superradiantly stable parameter regime of
braneworld extremal Kerr and Kerr-Newman black holes under massive scalar
perturbation. In our calculations, we have assumed that both the superradiance
condition and the bound state condition hold simultaneously. For
superradiantly stable parameter space, there should not be any trapping well
of the effective potential outside the horizon. Otherwise, the trapped modes
will be superradiantly amplified, this leads to the instability of the system.
It is the crucial point of our analysis. The radial part of the Klein-Gordon
equation is the main object of our concern. After choosing a suitable variable
we have reduced the radial equation in the Schrödinger form. From this
equation, we have determined the effective potential felt by the scalar field.
Next, one should choose the parameters in a way such that there remain only
one maxima (and no local minima) of the effective potential outside the
horizon. This requirement is implemented by demanding that asymptotically the
effective potential should behave like decreasing function and there should
remain only one positive root of derivative of effective potential outside the
horizon. Using this argument and Vieta’s theorem, we have obtained some
inequalities involving both black hole parameters and perturbation parameters.
These inequalities together with the superradiance condition and bound state
condition specify the parameter space of stability. For massive scalar
perturbation of braneworld Kerr-black holes, we have obtained a bound on tidal
charge. On the other hand for charged massive scalar perturbation of slowly
rotating braneworld extremal Kerr-Newman black holes, we have obtained a lower
bound of the product of mass to charge ratio of black hole and scalar field.

## Acknowledegment

I thank Sumanta Chakraborty for giving me the correct suggestion about the
metric used in 4. I also thank the unknown reviewer for making some valuable
comments about this. My research is supported by the graduate fellowship of
IACS. I thank Ramit Dey for a little help in latex.

## References

  * [1] T. Padmanabhan.  Gravitation: Foundations and Frontiers.  Cambridge University Press, 2010.
  * [2] Ray A d’Inverno.  Introducing Einstein’s relativity.  Clarendon Press, 1992.
  * [3] S. Chandrasekhar and S. Chandrasekhar.  The Mathematical Theory of Black Holes.  International series of monographs on physics. Clarendon Press, 1998.
  * [4] William H. Press and Saul A. Teukolsky.  Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric.  Astrophys. J., 185:649–674, 1973.
  * [5] Tullio Regge and John A. Wheeler.  Stability of a Schwarzschild singularity.  Phys. Rev., 108:1063–1069, 1957.
  * [6] C. V. Vishveshwara.  Stability of the schwarzschild metric.  Phys. Rev. D, 1:2870–2879, 1970.
  * [7] F. J. Zerilli.  Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics.  Phys. Rev. D, 2:2141–2160, 1970.
  * [8] Demetrios Christodoulou.  Reversible and irreversible transformations in black-hole physics.  Phys. Rev. Lett., 25:1596–1597, Nov 1970.
  * [9] Roger Penrose and RM Floyd.  Extraction of rotational energy from a black hole.  Nature Physical Science, 229(6):177–179, 1971.
  * [10] Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Nemanja Kaloper, and John March-Russell.  String Axiverse.  Phys. Rev. D, 81:123530, 2010.
  * [11] Richard Brito, Vitor Cardoso, and Paolo Pani.  Superradiance: New Frontiers in Black Hole Physics.  Lect. Notes Phys., 906:pp.1–237, 2015.
  * [12] William H. Press and Saul A. Teukolsky.  Floating Orbits, Superradiant Scattering and the Black-hole Bomb.  Nature, 238:211–212, 1972.
  * [13] Vitor Cardoso, Oscar J. C. Dias, Jose P. S. Lemos, and Shijun Yoshida.  The Black hole bomb and superradiant instabilities.  Phys. Rev. D, 70:044039, 2004.  [Erratum: Phys.Rev.D 70, 049903 (2004)].
  * [14] T. Damour, N. Deruelle, and R. Ruffini.  On Quantum Resonances in Stationary Geometries.  Lett. Nuovo Cim., 15:257–262, 1976.
  * [15] Hironobu Furuhashi and Yasusada Nambu.  Instability of massive scalar fields in Kerr-Newman space-time.  Prog. Theor. Phys., 112:983–995, 2004.
  * [16] Shahar Hod and Oded Hod.  Analytic treatment of the black-hole bomb.  Phys. Rev. D, 81:061502, 2010.
  * [17] Steven L. Detweiler.  KLEIN-GORDON EQUATION AND ROTATING BLACK HOLES.  Phys. Rev. D, 22:2323–2326, 1980.
  * [18] T. J. M. Zouros and D. M. Eardley.  INSTABILITIES OF MASSIVE SCALAR PERTURBATIONS OF A ROTATING BLACK HOLE.  Annals Phys., 118:139–155, 1979.
  * [19] Ramit Dey, Sumanta Chakraborty, and Niayesh Afshordi.  Echoes from braneworld black holes.  Phys. Rev. D, 101(10):104014, 2020.
  * [20] Ramit Dey, Shauvik Biswas, and Sumanta Chakraborty.  Ergoregion instability and echoes for braneworld black holes: Scalar, electromagnetic, and gravitational perturbations.  Phys. Rev. D, 103(8):084019, 2021.
  * [21] M. J. Soleimani, N. Abbasvandi, Shahidan Radiman, and W. A. T. Wan Abdullah.  Quasinormal Modes of Charged Black Holes Localized in the Randall-Sundrum Brane World.  10 2016.
  * [22] Roberto Emparan, Gary T Horowitz, and Robert C Myers.  Exact description of black holes on branes.  Journal of High Energy Physics, 2000(01):007–007, jan 2000.
  * [23] Shahar Hod.  Onset of superradiant instabilities in rotating spacetimes of exotic compact objects.  JHEP, 06:132, 2017.
  * [24] Sumanta Chakraborty, Sayak Datta, and Subhadip Sau.  Tidal heating of black holes and exotic compact objects on the brane.  arXiv preprint arXiv:2103.12430, 2021.
  * [25] He-Xu Zhang, Cong Li, Peng-Zhang He, Qi-Qi Fan, and Jian-Bo Deng.  Optical properties of a brane-world black hole as photons couple to the weyl tensor.  The European Physical Journal C, 80:1–11, 2020.
  * [26] Ahmadjon Abdujabbarov, Bobomurat Ahmedov, Naresh Dadhich, and Farruh Atamurotov.  Optical properties of a braneworld black hole: Gravitational lensing and retrolensing.  Phys. Rev. D, 96:084017, Oct 2017.
  * [27] Kirill A Bronnikov and Roman A Konoplya.  Echoes in brane worlds: Ringing at a black hole-wormhole transition.  Physical Review D, 101(6):064004, 2020.
  * [28] Juliano C. S. Neves.  Constraining the tidal charge of brane black holes using their shadows.  Eur. Phys. J. C, 80(8):717, 2020.
  * [29] Akash K Mishra, Abhirup Ghosh, and Sumanta Chakraborty.  Constraining extra dimensions using observations of black hole quasi-normal modes.  arXiv preprint arXiv:2106.05558, 2021.
  * [30] Indrani Banerjee, Sumanta Chakraborty, and Soumitra SenGupta.  Silhouette of M87*: A New Window to Peek into the World of Hidden Dimensions.  Phys. Rev. D, 101(4):041301, 2020.
  * [31] Shahar Hod.  On the instability regime of the rotating Kerr spacetime to massive scalar perturbations.  Phys. Lett. B, 708:320–323, 2012.
  * [32] Jia-Mao Lin, Ming-Jian Luo, Zi-Han Zheng, Lei Yin, and Jia-Hui Huang.  Extremal rotating black holes, scalar perturbation and superradiant stability.  Phys. Lett. B, 819:136392, 2021.
  * [33] Jun-Huai Xu, Zi-Han Zheng, Ming-Jian Luo, and Jia-Hui Huang.  Analytic study of superradiant stability of Kerr–Newman black holes under charged massive scalar perturbation.  Eur. Phys. J. C, 81(5):402, 2021.
  * [34] Jia-Hui Huang, Wen-Xiang Chen, Zi-Yang Huang, and Zhan-Feng Mai.  Superradiant stability of the Kerr black holes.  Phys. Lett. B, 798:135026, 2019.
  * [35] Sam R. Dolan.  Instability of the massive Klein-Gordon field on the Kerr spacetime.  Phys. Rev. D, 76:084001, 2007.
  * [36] Tetsuya Shiromizu, Kei-ichi Maeda, and Misao Sasaki.  The Einstein equation on the 3-brane world.  Phys. Rev. D, 62:024012, 2000.
  * [37] Naresh Dadhich, Roy Maartens, Philippos Papadopoulos, and Vahid Rezania.  Black holes on the brane.  Phys. Lett. B, 487:1–6, 2000.
  * [38] Donato Bini, Andrea Geralico, and Roy P. Kerr.  The Kerr-Schild ansatz revised.  Int. J. Geom. Meth. Mod. Phys., 7:693, 2010.
  * [39] Roy Patrick Kerr.  Rotating black holes and the Kerr metric.  AIP Conf. Proc., 1059(1):9–12, 2008.
  * [40] Robert H. Boyer and Richard W. Lindquist.  Maximal analytic extension of the Kerr metric.  J. Math. Phys., 8:265, 1967.
  * [41] A. N. Aliev and A. E. Gumrukcuoglu.  Charged rotating black holes on a 3-brane.  Phys. Rev. D, 71:104027, 2005.
  * [42] Lev Davidovich Landau.  The classical theory of fields, volume 2.  Elsevier, 2013.
  * [43] D. J. Rowan and G. Stephenson.  The Klein-Gordon Equation in a Kerr-Newman Background Space.  J. Phys. A, 10:15–23, 1977.
  * [44] Saul A. Teukolsky.  Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations.  Phys. Rev. Lett., 29:1114–1118, Oct 1972.
  * [45] Thomas Hartman, Wei Song, and Andrew Strominger.  Holographic Derivation of Kerr-Newman Scattering Amplitudes for General Charge and Spin.  JHEP, 03:118, 2010.
  * [46] Shahar Hod.  Weak cosmic censorship: As strong as ever.  Phys. Rev. Lett., 100:121101, Mar 2008.
  * [47] P. P. Fiziev.  Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations.  2 2009.
  * [48] Shahar Hod.  Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis.  Phys. Lett. B, 746:365–367, 2015.
  * [49] James M. Bardeen, William H. Press, and Saul A Teukolsky.  Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation.  Astrophys. J., 178:347, 1972.
  * [50] Shahar Hod.  The superradiant instability regime of the spinning Kerr black hole.  Phys. Lett. B, 758:181–185, 2016.
  * [51] B. Carter.  Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations.  Commun. Math. Phys., 10(4):280–310, 1968.
  * [52] Jacob D. Bekenstein.  Extraction of energy and charge from a black hole.  Phys. Rev. D, 7:949–953, Feb 1973.
  * [53] S. Barnard and J.M. Child.  Higher Algebra, by S. Barnard and J.M. Child.  Macmillan and Company, Limited, 1959.
  * [54] Shahar Hod.  No-bomb theorem for charged reissner–nordström black holes.  Physics Letters B, 718(4):1489–1492, 2013.
  * [55] Shahar Hod.  Analytic treatment of the system of a kerr-newman black hole and a charged massive scalar field.  Phys. Rev. D, 94:044036, Aug 2016.