KIAS-P23007
SNUTP23-001

Towards quantum black hole microstates

Sunjin Choi1, Seok Kim2, Eunwoo Lee2, Siyul Lee3 and Jaemo Park4

1School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea.

2Department of Physics and Astronomy & Center for Theoretical Physics,
Seoul National University, Seoul 08826, Korea.

3Leinweber Center for Theoretical Physics, University of Michigan,
Ann Arbor, MI 48109, USA.

4 Department of Physics, Pohang University of Science and Technology (POSTECH)
Pohang 37673, Korea.

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
We study the cohomology of local BPS operators in $\mathcal{N}=4$ Yang-Mills
theory. The finite $N$ cohomologies consist of the graviton part (subject to
the stringy exclusion principle) and the rest which may describe black hole
microstates in quantum AdS/CFT. We construct an infinite tower of non-graviton
cohomologies in the $SU(2)$ theory and study to what extent they simulate
quantum black holes. We find signals for partial no-hair behaviors by showing
that certain gravitons are forbidden to dress these cohomologies. This is in
qualitative agreement with the perturbative hairs allowed around black holes,
which also leads us to a natural setup to construct hairy BPS black holes. The
cohomologies are simpler to study in the BMN matrix model truncation of the
classical field theory.

###### Contents

  1. 1 Introduction
  2. 2 The cohomology problem
    1. 2.1 The finite $N$ gravitons
  3. 3 The black hole cohomologies for $SU(2)$
    1. 3.1 The BMN sector
    2. 3.2 General sector and partial no-hair theorem
  4. 4 Comments on hairy black holes
  5. 5 Conclusion and remarks
  6. A Supersymmetry and supergravitons
  7. B Representations of $PSU(2,2|4)$ and $\frac{1}{16}$-BPS states
  8. C Counting BMN gravitons for $SU(2)$
  9. D $Q$-exactness of some graviton hairs

##  1 Introduction

Black holes exhibit extreme properties with the information they carry. For
instance, the Bekenstein-Hawking entropy [1, 2] of black holes is supposed to
represent the maximal amount of information allowed per size [3]. So regarding
this entropy as the most fine-grained information of Nature, we can learn from
it the fundamental degrees of freedom of quantum gravity [4, 5, 6]. To explore
other extreme behaviors of black holes, we want access to the individual black
hole microstates which account for this entropy. In this paper, we ‘construct’
(in certain sense) and study the local BPS operators which can describe the
microstates of BPS black holes in quantum AdS/CFT [7]. The entropies of BPS
black holes in AdSD>3 were recently computed from the dual field theories: see
[8, 9, 10] and references thereof. We shall construct representations of some
of these microstates and study their properties, hoping that we can better
address and exactly solve interesting black hole information problems with
this knowledge.

We shall study the 4d $\mathcal{N}=4$ Yang-Mills theory dual to type IIB
string theory in $AdS_{5}\times S^{5}$. We are interested in BPS states that
preserve $2$ of the $32$ supersymmetries [11, 12], called $\frac{1}{16}$-BPS
states. The spectrum of these BPS states may change as the coupling constant
$g_{\rm YM}$ is varied, and in fact they are different between the free and
1-loop levels. The 1-loop BPS states of the $SU(N)$ theory were studied in the
past [12, 13, 14, 15, 16]. They can be reformulated as classical cohomologies
with respect to a nilpotent supercharge $Q$. It has been conjectured that the
spectrum of 1-loop BPS states remains unchanged at general nonzero coupling
[12, 13, 17]. This conjecture is partly proved in perturbation theory [18]
with certain assumptions. The discussions in this paper will often assume this
conjecture. Although we shall construct representatives of new cohomologies,
they are generally not equal to the BPS operators. The two in general differ
by certain $Q$-exact terms. So what we achieve is weaker than ‘constructing’
the black hole microstates. It remains to be seen what kind of questions can
be addressed just from cohomologies. (See section 3 for an example.)

Our precise goal is to construct cohomologies at finite $N$ which are not of
the graviton type. One may wonder what gravitons mean at finite $N$. First,
single-particle graviton cohomologies are constructed with single trace
operators only. Multi-gravitons are simply defined as products of the single-
particle gravitons.111While products of BPS operators are generally not BPS,
cohomologies multiply to yield cohomologies. At large $N$, all these operators
are independent because there are no trace relations of matrix fields. More
precisely, an operator is free of trace relations when the number of fields
appearing in the operator is no greater than $N$. When the number of fields is
larger than $N$, trace relations may apply. On the gravity side, trace
relations are realized by gravitons polarizing into D3-brane giant gravitons
[19, 20, 21], after which fewer states are allowed than the naive estimate.
This is called the ‘stringy exclusion principle.’ The physical mechanism of
this principle is same at all $N\geq 2$. So it makes physical sense to define
finite $N$ graviton cohomologies as multiplications of single-trace
cohomologies modulo trace relations. These cohomologies are fully understood.

Unless there are exotic microstates that neither qualify to be called
gravitons nor black holes, we expect the remaining cohomologies to describe
the black hole microstates in AdS. We shall therefore call them black hole
cohomologies for simplicity of nomenclature, but also having in mind that
‘black hole’ could broadly mean all possible novelties beyond gravitons. There
may be two viewpoints on these cohomologies at finite $N$. First, they are
intermediate steps to the cohomologies at parametrically large $N$. Second,
more progressively, one may regard the finite $N$ Yang-Mills theory as a model
of quantum AdS/CFT at finite Newton constant. The ‘black hole cohomologies’ at
finite $N$ may simulate quantum black holes therein. Not all interesting
questions on semiclassical black holes survive in these finite $N$ models, but
some questions do. For instance, the Cardy limit [9, 22, 23, 24, 25] exhibits
universal deconfining behaviors at large charges, naturally generalizing the
large black hole physics to all finite $N$.

The progress in this paper is all with the $SU(2)$ theory, the most quantum
version of AdS/CFT. Already in this model, the new cohomologies exhibit some
qualitative features reminiscent of black holes. Although this problem has
been discussed since 2005, not a single black hole cohomology was found until
last year. In [18], it was shown that the $SU(2)$ theory has the lowest black
hole cohomology at energy $E=\frac{19}{2}$, R-charges
$R_{1}=R_{2}=R_{3}=\frac{3}{2}$ and angular momenta $J_{1}=J_{2}=\frac{5}{2}$.
A simple representative of this cohomology was constructed in [26]. This is
the primary of a superconformal representation of $PSU(1,2|3)$ $\subset
PSU(2,2|4)$. There are infinitely many descendants that one can trivially
construct from this primary.

We construct an infinite tower of new black hole primaries in the $SU(2)$
theory. For technical reasons, we often focus on the ‘BMN sector’ of the Yang-
Mills theory on $S^{3}\times\mathbb{R}$ [27, 28]. This is a consistent
truncation of the classical theory, also yielding a consistent truncation of
our cohomology problem. The superconformal index [11, 12] of the Yang-Mills
theory can also be restricted to the BMN sector. For the $SU(2)$ theory in the
BMN sector, we find infinitely many new cohomologies which saturate the index.
More concretely, the energy $E_{(n)}$, R-charge $R_{(n)}(=R_{1}=R_{2}=R_{3})$
and the angular momentum $J_{(n)}(=J_{1}=J_{2})$ of our $n$’th ‘core’ black
hole primary $O_{n}$ are given by

$(E_{(n)},R_{(n)},J_{(n)})=({\textstyle\frac{19}{2}+4n,\frac{3}{2},\frac{5}{2}+2n})\
,\ \ n=0,1,2,\cdots\ .$ (1.1)

The operator found in [18] corresponds to $n=0$. One can exactly compute the
full index $Z$ and the graviton index $Z_{\rm grav}$ in the BMN sector. Their
difference is given by

$Z-Z_{\rm
grav}=\left[-\frac{e^{-4(\Delta_{1}+\Delta_{2}+\Delta_{3})}}{1-e^{-2(\Delta_{1}+\Delta_{2}+\Delta_{3})}}\right]\cdot\left[\prod_{I=1}^{3}(1-e^{-\Delta_{I}})\right]\cdot\left[\prod_{I=1}^{3}\frac{1}{1-e^{-\Delta_{I}}e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\right]\
,$ (1.2)

where $\Delta_{I}$ is the chemical potential conjugate to the charge
$R_{I}+J$. The three factors in the square brackets come respectively from:
(left) the tower of core black hole primaries $O_{n}$, (middle) the Fock space
of their superconformal descendants, (right) the hairs by a limited subset of
gravitons. The BMN sector in the $SU(2)$ theory does not show large enough
entropy of BPS states even at large charges. However, we expect that the BPS
entropy in the BMN sector should exhibit a black hole like growth at large
$N$.

Since cohomologies are multiplicative, we can consider the product
cohomologies of gravitons and core black hole cohomologies. We find that a
surprisingly large set of them does not appear in the index. The most natural
interpretation of this phenomenon is that these product operators are
$Q$-exact (i.e. absent in the BPS Hilbert space), which we prove explicitly at
some low orders. We view this as a partial no-hair theorem of black holes in
the $SU(2)$ model, in that our black hole cohomologies abhor the dressing by
some gravitons. This phenomenon is most clearly visible for the conformal
primary states of gravitons. In (1.2), the index only captures $3$ out of $17$
species of BMN graviton particles dressing our black holes. We also studied
the general $SU(2)$ index for black holes up to $40$’th order in the charge
$j=6(R+J)$. Till this order, the index captures only $3$ out of $32$ conformal
primary gravitons dressing the core black holes. (See sections 3.2 and 4 for
the conformal descendants.) Since AdS black holes are expected to allow
certain graviton hairs [29, 30, 31, 32], partial no-hair phenomenon seems to
be the right behavior in AdS/CFT models. In section 4 we perturbatively study
the black hole hairs in the BPS limit to clarify the similar behaviors in the
gravity dual.

We also estimate when new black hole cohomologies should appear in the $SU(2)$
theory beyond $O_{n}$. By studying the index and all possibilities of finite
$N$ graviton hairs, we show that new black hole core primaries should appear
at or before the $j=39$ order.

Note that black holes in the BMN model were studied recently [33, 34, 35]. As
emphasized in [34, 35], microscopic detection of the quasinormal modes is a
signature of seeing black holes since these modes are falling into the
horizon. In our stationary (BPS) setup, a direct consequence is the no-hair
theorem for the corresponding modes: we try to perturb a black hole by
multiplying certain gravitons and find that they do not exist in the BPS
sector after a long time.

The remaining part of this paper is organized as follows. In section 2, we
explain the cohomology problem and the graviton cohomologies at finite $N$. In
section 3.1, we study the BMN sector and find an infinite class of new black
hole cohomologies for $SU(2)$. In section 3.2, we study the general $SU(2)$
cohomologies up to $j\leq 40$ and find no-hair behaviors. In section 4, we
study how perturbative black hole hairs behave in the BPS limit and comment on
similarities with the results in section 3. In section 5, we conclude with
remarks.

##  2 The cohomology problem

We review the problem of local BPS operators in $\mathcal{N}=4$ Yang-Mills
theory on $\mathbb{R}^{4}$ and its cohomological formulation. By operator-
state map, they map to BPS states of the Yang-Mills theory on
$S^{3}\times\mathbb{R}$. We shall discuss the $SU(N)$ theory. We sketch the
problem briefly before explaining the details, to emphasize the nature of the
problem. We start by selecting two supercharges among $32$ of them, which will
annihilate our BPS operators. We first consider all gauge-invariant local BPS
operators in the free limit. Then we turn on small coupling $g_{\rm YM}\neq 0$
and see how many of the free BPS operators remain BPS, order by order in
perturbation theory.

The $\mathcal{N}=4$ Yang-Mills theory consists of the following fields in the
adjoint representation of $SU(N)$ ($N\times N$ traceless matrices):

vector $\displaystyle:$ $\displaystyle A_{\mu}\sim A_{\alpha\dot{\beta}}\ ,\
\mu=1,2,3,4\ ,\ \alpha=\pm\ ,\ \dot{\beta}=\dot{\pm}$ scalar $\displaystyle:$
$\displaystyle\Phi_{ij}(=-\Phi_{ji})\ ,\
\overline{\Phi}^{ij}\sim{\textstyle\frac{1}{2}}\epsilon^{ijkl}\Phi_{kl}\ ,\ \
i,j,k,l=1,2,3,4$ fermion $\displaystyle:$ $\displaystyle\Psi_{i\alpha}\ ,\
\overline{\Psi}^{i}_{\dot{\alpha}}\ .$ (2.1)

We consider the Euclidean CFT on $\mathbb{R}^{4}$, which is related to the CFT
on $S^{3}\times\mathbb{R}$ by radial quantization. $\alpha,\dot{\alpha}$ are
the doublet indices of $SU(2)_{L}\times SU(2)_{R}\sim SO(4)$ which rotate the
$S^{3}$, and $\mu$ is the vector index. Uppercase $i,j$ are the indices for
the fundamental representation of $SU(4)$ R-symmetry, while the lowercase
indices are for the anti-fundamental representation. Hermitian conjugations of
these fields should be understood with a bit care in radial quantization. See,
for instance, [15] for the details. It will be useful to decompose these
fields in the $\mathcal{N}=1$ language as follows, with manifest $SU(3)\times
U(1)\subset SU(4)$ symmetry:

vector multiplet $\displaystyle:$ $\displaystyle A_{\alpha\dot{\beta}}\ ,\
\lambda_{\alpha}=\Psi_{4\alpha}\ ,\
\bar{\lambda}_{\dot{\alpha}}=\overline{\Psi}^{4}_{\dot{\alpha}}$ (2.2) chiral
multiplets $\displaystyle:$ $\displaystyle\phi_{m}=\Phi_{4m}\ ,\
\bar{\phi}^{m}=\overline{\Phi}^{4m}\ ,\ \psi_{m\alpha}=-i\Psi_{m\alpha}\ ,\
\bar{\psi}^{m}_{\dot{\alpha}}=i\overline{\Psi}^{m}_{\dot{\alpha}}\ .$

$m=1,2,3$ is the $SU(3)$ index, either for the fundamental/anti-fundamental
representation. The system carries a continuous real coupling $g_{\rm YM}$,
and enjoys $\mathcal{N}=4$ superconformal symmetry at any value of $g_{\rm
YM}$. (The theta angle will not be relevant in our discussions.) The
supercharges consist of $16$ Poincare supercharges $Q^{i}_{\alpha}$,
$\overline{Q}_{i\dot{\alpha}}$ and $16$ conformal supercharges $S_{i\alpha}$,
$\overline{S}^{i}_{\dot{\alpha}}$. In radial quantization, $Q$ and $S$ are
Hermitian conjugate to each other:
$(Q^{i}_{\alpha})^{\dagger}=S_{i}^{\alpha}$,
$(\overline{Q}_{i\dot{\alpha}})^{\dagger}=\overline{S}^{i\dot{\alpha}}$. The
quantum supercharges may depend on the coupling $g_{\rm YM}$, which is hard to
write down in general. The supercharges of the interacting classical field
theory contain terms up to half-loops, i.e. $\mathcal{O}(g_{\rm YM}^{1})$
order. The transformations for the classical Poincare supercharges are listed
in Appendix A.

The BPS operators of our interest are defined with $Q\equiv Q^{4}_{-}$ and
$S\equiv S_{4}^{-}=Q^{\dagger}$. We are interested in gauge-invariant local
operators $O$ located at the origin $x^{\mu}=0$ of $\mathbb{R}^{4}$,
satisfying

$[Q,O\\}=0\ ,\ [Q^{\dagger},O\\}=0\ .$ (2.3)

The part of the superconformal algebra which is important to us is

$Q^{2}=0\ ,\ (Q^{\dagger})^{2}=0\ ,\ \\{Q,Q^{\dagger}\\}\sim
H-\sum_{I=1}^{3}R_{I}-\sum_{i=1}^{2}J_{i}\ .$ (2.4)

$H$ is the scaling dimension of the local operators, or the energy of the
corresponding states times the radius of $S^{3}$. $R_{I}$ are the three Cartan
charges of $SO(6)\sim SU(4)$ which rotate three orthogonal 2-planes in
$\mathbb{R}^{6}$. $J_{i}$ are the two angular momenta on $\mathbb{R}^{4}$
rotating two orthogonal 2-planes. The eigenvalues of $R_{I}$ and $J_{i}$ are
integers for bosons and half of odd integers for fermions. The BPS operators
$O$ annihilated by $Q$ and $Q^{\dagger}$ commute with $\\{Q,Q^{\dagger}\\}$.
They can be arranged to be the eigenstates of $H,R_{I},J_{i}$, whose
eigenvalues $E,R_{I},J_{i}$ satisfy

$E=\sum_{I}R_{I}+\sum_{i}J_{i}\ .$ (2.5)

field | $E$ | ($R_{1}$, $R_{2}$, $R_{3}$) | ($J_{1}$, $J_{2}$)
---|---|---|---
$\phi_{m}$ | $1$ | ($-1,0,0$), ($0,-1,0$), ($0,0,-1$) | ($0,0$)
$\bar{\phi}^{m}$ | $1$ | ($1,0,0$), ($0,1,0$), ($0,0,1$) | ($0,0$)
$\psi_{m\pm}$ | $\frac{3}{2}$ | ($-\frac{1}{2},\frac{1}{2},\frac{1}{2}$), ($\frac{1}{2},-\frac{1}{2},\frac{1}{2}$), ($\frac{1}{2},\frac{1}{2},-\frac{1}{2}$) | ($\pm\frac{1}{2},\pm\frac{1}{2}$)
$\bar{\psi}^{m}_{\dot{\pm}}$ | $\frac{3}{2}$ | ($\frac{1}{2},-\frac{1}{2},-\frac{1}{2}$), ($-\frac{1}{2},\frac{1}{2},-\frac{1}{2}$), ($-\frac{1}{2},-\frac{1}{2},\frac{1}{2}$) | ($\pm\frac{1}{2},\mp\frac{1}{2}$)
$A_{+\dot{\pm}}$ | $1$ | $(0,0,0)$ |  $(1,0)$, $(0,1)$
$A_{-\dot{\pm}}$ | $1$ | $(0,0,0)$ |  $(-1,0)$, $(0,-1)$
$f_{++},f_{+-},f_{--}$ | $2$ | ($0,0,0$) | ($1,1$), ($0,0$), ($-1,-1$)
$f_{\dot{+}\dot{+}},f_{\dot{+}\dot{-}},f_{\dot{-}\dot{-}}$ | $2$ | ($0,0,0$) | ($1,-1$), ($0,0$), ($-1,1$)
$\lambda_{\pm}$ | $\frac{3}{2}$ | $(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$ | ($\pm\frac{1}{2},\pm\frac{1}{2}$)
$\bar{\lambda}_{\dot{\pm}}$ | $\frac{3}{2}$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | ($\pm\frac{1}{2},\mp\frac{1}{2}$)
$\partial_{+\dot{\alpha}}$ | $1$ | ($0,0,0$) | ($1,0$), ($0,1$)
$\partial_{-\dot{\alpha}}$ | $1$ | ($0,0,0$) | ($-1,0$), ($0,-1$)
Table 1: The charges of elementary fields. Charges in the parenthesis are
listed in the order of $m=1,2,3$ or $\dot{\alpha}=\dot{+},\dot{-}$ or in the
order of the fields listed in the first column.

In the free limit, the operators satisfying the BPS relation (2.5) can be
easily constructed using the BPS elementary fields satisfying the same
relation. The charges for the fields and the derivative
$\partial_{\mu}\sim\partial_{\alpha\dot{\beta}}$ are listed in Table 1. The
field strengths in the bispinor basis are defined by
$f_{\alpha\beta}\sim(\sigma^{\mu\nu})_{\alpha\beta}F_{\mu\nu}$ and
$f_{\dot{\alpha}\dot{\beta}}\sim(\bar{\sigma}^{\mu\nu})_{\dot{\alpha}\dot{\beta}}F_{\mu\nu}$.
Among them, the gauge-covariant BPS fields and derivatives which satisfy the
BPS relation are given by

$\bar{\phi}^{m}\ ,\ \ \psi_{m+}\ ,\ \ f_{++}\ ,\ \
\bar{\lambda}_{\dot{\alpha}}\ ,\ \ \partial_{+\dot{\alpha}}\ .$ (2.6)

With these, we construct independent ‘letters’ for the gauge invariant
operators. Basically, acting derivatives $\partial_{+\dot{\alpha}}$ to a BPS
field forms a letter. However, the equation of motion operator is zero and
should not be included. The only equation of motion constructed using (2.6) is

$\partial_{+\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}=0\ \Leftrightarrow\
\partial_{+[\dot{\alpha}}\bar{\lambda}_{\dot{\beta}]}=0\ .$ (2.7)

So the $SU(2)_{R}$ indices carried by the derivatives and the gaugino have to
be symmetrized in a letter. Also, in the free theory, the derivatives
$\partial_{+\dot{\alpha}}$ acting on the same field commute. Therefore, all
$SU(2)_{R}$ indices appearing in a letter should be symmetrized. So we find
the following independent letters (all at $x^{\mu}=0$),

$\displaystyle\partial_{+(\dot{\alpha}_{1}}\cdots\partial_{+\dot{\alpha}_{n})}\varphi$
$\displaystyle\sim$
$\displaystyle(\partial_{+\dot{+}})^{n_{1}}(\partial_{+\dot{-}})^{n_{2}}\varphi\
\ \ (n_{1},n_{2}\geq 0,n\geq 0)$ (2.8)
$\displaystyle\partial_{+(\dot{\alpha}_{1}}\cdots\partial_{+\dot{\alpha}_{n-1}}\bar{\lambda}_{\dot{\alpha}_{n})}$
$\displaystyle\sim$ $\displaystyle
n_{1}(\partial_{+\dot{+}})^{n_{1}-1}(\partial_{+\dot{-}})^{n_{2}}\bar{\lambda}_{\dot{+}}+n_{2}(\partial_{+\dot{+}})^{n_{1}}(\partial_{+\dot{-}})^{n_{2}-1}\bar{\lambda}_{\dot{-}}\
\ \ (n_{1},n_{2}\geq 0,n\geq 1)$

where $n_{1}+n_{2}=n$. $\varphi$ denotes any field except
$\bar{\lambda}_{\dot{\alpha}}$, i.e. chosen among $\bar{\phi}^{m}$,
$\psi_{m+}$, $f_{++}$. Multiplying these letters and pairwise contracting the
$SU(N)$ indices, one can construct general gauge-invariant BPS operators in
the free theory.

Now we consider the 1-loop BPS operators, i.e. operators which diagonalize $H$
and saturate the BPS relation till $O(g_{\rm YM}^{2})$ order. For very small
$g_{\rm YM}$, we can start from the free BPS operators explained in the
previous paragraph and study which of them remain BPS at the 1-loop level. The
only modification needed is to replace $\partial_{+\dot{\alpha}}$ and $f_{++}$
by the covariant expressions by adding $\mathcal{O}(g_{\rm YM}^{1})$
corrections. The covariant letters used in the free theory should be modified
as follows:

$D_{+(\dot{\alpha}_{1}}\cdots D_{+\dot{\alpha}_{n})}\varphi\ ,\ \
D_{+(\dot{\alpha}_{1}}\cdots
D_{+\dot{\alpha}_{n-1}}\bar{\lambda}_{\dot{\alpha}_{n})}\ .$ (2.9)

Here $D_{+\dot{\alpha}}$ is the covariant derivative defined by
$D_{+\dot{\alpha}}=\partial_{+\dot{\alpha}}-ig_{\rm YM}[A_{+\dot{\alpha}},\ \
]$, related to $f_{++}$ by $[D_{+\dot{\alpha}},D_{+}^{\
\dot{\alpha}}]\sim[D_{+[\dot{\alpha}},D_{+\dot{\beta}]}]\sim g_{\rm
YM}f_{++}$. Although two different $D_{+\dot{\alpha}}$’s do not commute
anymore, we still completely symmetrize the derivatives in a letter. This is
to avoid introducing redundant letters: if some $D$’s are antisymmetrized in
(2.9), it can be rewritten as a sum of products of letters involving
$f_{++}$’s. We also symmetrize the $SU(2)_{R}$ indices in $D_{+\dot{\alpha}}$
and $\bar{\lambda}_{\dot{\alpha}}$ to avoid redundant letters, because the
gaugino equation is given by $g_{\rm YM}[\bar{\phi}^{m},\psi_{m+}]\sim
D_{+\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}\sim
D_{+[\dot{\alpha}}\bar{\lambda}_{\dot{\beta}]}$. Among all possible gauge-
invariant operators $O$ constructed from (2.9), we would like to find the
combinations satisfying

$0=\left[\frac{}{}\\!\\!\right.H-\sum_{I}R_{I}-\sum_{i}J_{i},O\left.\\!\\!\frac{}{}\right]\sim\left[\\{Q,Q^{\dagger}\\},O\right]\
.$ (2.10)

The last equation holds if and only if $[Q,O\\}=0$ and $[Q^{\dagger},O\\}=0$.
Note that the coupling dependence is only in $H,Q,Q^{\dagger}$, since the
quantized non-Abelian charges $R_{I}$, $J_{i}$ cannot depend on $g_{\rm YM}$.
At the 1-loop level, $H-\sum_{I}R_{I}-\sum_{i}J_{i}$ acting on the BPS letters
(2.9) is at $\mathcal{O}(g_{\rm YM}^{2})$ order, while $Q,Q^{\dagger}$ acting
on (2.9) are at $\mathcal{O}(g_{\rm YM}^{1})$ order. Their complete forms are
given in [36]. In particular, $Q$ at the half-loop order is simply that of the
classical field theory at nonzero $g_{\rm YM}$, which transform the BPS fields
by

$\displaystyle[Q,\bar{\phi}^{m}]=0\ ,\ \\{Q,\bar{\lambda}_{\dot{\alpha}}\\}=0\
,\ \\{Q,\psi_{m+}\\}=-ig_{\rm
YM}\epsilon_{mnp}[\bar{\phi}^{n},\bar{\phi}^{p}]$ (2.11)
$\displaystyle[Q,f_{++}]=-ig_{\rm YM}[\psi_{m+},\bar{\phi}^{m}]\ ,\
[Q,D_{+\dot{\alpha}}](\cdots)=-ig_{\rm
YM}[\bar{\lambda}_{\dot{\alpha}},(\cdots)\\}\ .$

$S=Q^{\dagger}$ acts on pairs of letters [36]. From here, we absorb $g_{\rm
YM}$ into the normalization of fields.

Instead of directly constructing the BPS operators, we study the cohomology
classes which are in 1-to-1 map to the BPS operators. The cohomology class is
defined by the set of operators $O$ made of (2.9) that are closed under the
action of $Q$, i.e. $[Q,O\\}=0$, with the equivalence relation $O\sim
O+[Q,\Lambda\\}$. Here, $\Lambda$ is also an operator constructed with (2.9).
We can call this a cohomology because of the nilpotency $Q^{2}=0$. This
cohomology class is in 1-to-1 map to the BPS operators $O_{\rm BPS}$
satisfying $[Q,O_{\rm BPS}\\}=0$ and $[Q^{\dagger},O_{\rm BPS}\\}=0$, because
the latter is basically a harmonic form in this language [15]. The cohomology
problem is defined using $Q$ without referring to $S=Q^{\dagger}$. Note also
that this problem is completely classical, since the transformation (2.11) is
that of the classical field theory.

It has been conjectured implicitly/explicitly (especially in [17]) that the
1-loop BPS states remain BPS at general nonzero coupling. However, even if
this conjecture is true, it does not mean that the form of the BPS operator
$O_{\rm BPS}$ takes the same form at different values of $g_{\rm YM}$. The
conjecture just claims that the operators will survive to be BPS with $g_{\rm
YM}$-dependent deformations, being in 1-to-1 map with the 1-loop BPS states
without any pairwise jumps. Recently, the validity of this conjecture was
argued at all orders in perturbation theory [18].

We shall construct the representatives of new cohomology classes. Our interest
is those which have chances to describe black holes rather than BPS gravitons.
The BPS graviton cohomologies are completely understood, even at finite $N$
subject to the stringy exclusion principle. We shall explain them in section
2.1. However, enumerating them without overcounting is technically quite
tricky, due to the trace relations. We shall present a strategy which allows
us to solve the trace relations in terms of diagonal matrix fields. The
strategy applies to arbitrary $N$, but is particularly useful for $N=2$. (At
$N=2$, this idea was employed in [15] in special subsectors.) We shall
analytically/numerically implement this idea in section 3 (and appendix C) to
efficiently count graviton-type cohomologies.

###  2.1 The finite $N$ gravitons

We first explain the cohomologies for the BPS gravitons in the large $N$
limit, and then explain how to define and understand the finite $N$ graviton
cohomologies.

We first take $N\gg 1$ and the number of fields in the cohomologies much
smaller than $N$ (say at order $1$). In this limit, the mixing of single- and
multi-trace operators by the dilatation operator $H$ is suppressed by
$\frac{1}{N}$. So one can first construct the BPS operators using single trace
operators only, and then multiply them to obtain multi-trace BPS operators.
The former and latter are the single- and multi-particle graviton states,
respectively.

$(-1)^{F}E^{\prime}$ | $J^{\prime}$ | $R_{1}^{\prime}$ | $R_{2}^{\prime}$ | construction
---|---|---|---|---
$n$ | $0$ | $n$ | $0$ | $|n\rangle$
$-(n+\frac{1}{2})$ | $\frac{1}{2}$ | $n-1$ | $0$ | $\overline{Q}_{m\dot{\alpha}}|n\rangle$
$n+1$ | $0$ | $n-2$ | $0$ | $\overline{Q}_{m\dot{+}}\overline{Q}_{n\dot{-}}|n\rangle$
$-(n+1)$ | $0$ | $n-1$ | $1$ | $Q^{m}_{+}|n\rangle$
$n+\frac{3}{2}$ | $\frac{1}{2}$ | $n-2$ | $1$ | $Q^{m}_{+}\overline{Q}_{n\dot{\alpha}}|n\rangle$
$-(n+2)$ | $0$ | $n-3$ | $1$ | $Q^{m}_{+}\overline{Q}_{n\dot{+}}\overline{Q}_{p\dot{-}}|n\rangle$
$n+2$ | $0$ | $n-1$ | $0$ | $Q^{m}_{+}Q^{n}_{+}|n\rangle$
$-(n+\frac{5}{2})$ | $\frac{1}{2}$ | $n-2$ | $0$ | $Q^{m}_{+}Q^{n}_{+}\overline{Q}_{p\dot{\alpha}}|n\rangle$
$n+3$ | $0$ | $n-3$ | $0$ | $Q^{m}_{+}Q^{n}_{+}\overline{Q}_{p\dot{+}}\overline{Q}_{q\dot{-}}|n\rangle$
Table 2: The state contents of the $PSU(1,2|3)$ multiplet $S_{n}$. For low
$n$’s, the rows with negative $R_{1}^{\prime}$ are absent. $|n\rangle$
schematically denotes the superconformal primaries.

We consider the single-trace BPS operators from the viewpoint of the
$Q$-cohomology problem. The cohomology problem is well-defined in the single
trace sector, since the action of $Q$ does not change trace numbers. This
problem is completely solved [14, 16]. The single trace cohomologies can be
arranged into the supermultiplets of $PSU(1,2|3)\subset PSU(2,2|4)$ which
commute with our $Q,Q^{\dagger}$. In [12], the short multiplets for the single
trace cohomologies are called $S_{n}$, with $n=2,3,\cdots$.222The $S_{1}$
multiplet comes from the overall $U(1)$ mode of the $U(N)$ theory, which is
absent for $SU(N)$. In this paper, we shall call the
multiplets/representations of $PSU(1,2|3)$ ‘$\frac{1}{16}$-BPS
multiplets/representations’ and those of $PSU(2,2|4)$ ‘$\mathcal{N}=4$
multiplets/representations,’ respectively. The ways in which
$\frac{1}{16}$-BPS representations sit in the $\mathcal{N}=4$ representations
are explained in the appendix B.3 of [12], which we review in appendix B. The
$\frac{1}{16}$-BPS multiplet $S_{n}$ is contained in the short $\mathcal{N}=4$
multiplet $B\overline{B}[0;0]^{[0,n,0]}_{n}$ in the notation of [37], where
the superscripts/subscript are the $SU(4)$ Dynkin labels and the scaling
dimension of the primaries, respectively, and
$[J_{1}+J_{2},J_{1}-J_{2}]=[0,0]$ denotes the spins of the primaries. (This is
a multiplet of ${\bf bb}$ type in the notation of [12].) Following [12], we
define $E^{\prime}=E+\frac{J_{1}+J_{2}}{2}$,
$J^{\prime}=\frac{J_{1}-J_{2}}{2}$, and $[R_{1}^{\prime},R_{2}^{\prime}]$ is
the $SU(3)$ highest weight. The states in $S_{n}$ obtained by acting
$Q^{m}_{+}$ and $\overline{Q}_{m\dot{\alpha}}$ generators of $PSU(1,2|3)$ are
listed in Table 2. These are obtained by assuming that all supercharges
anticommute. In fact some of the anticommutators yield
$\\{Q^{m}_{+},\overline{Q}_{n\dot{\alpha}}\\}\sim P_{+\dot{\alpha}}$, but they
generate the conformal descendants can be generated later. In the gravity dual
language, the table lists the Kaluza-Klein field contents in AdS5. The
conformal descendants are obtained by acting $P_{+\dot{\alpha}}$ on the
‘fields’ in the table, which generate the wavefunctions in $AdS_{5}$.

The superconformal primaries $|n\rangle$ of Table 2 are given by the following
operators,

${\rm tr}[\bar{\phi}^{(m_{1}}\cdots\bar{\phi}^{m_{n})}]\ .$ (2.12)

From the cohomology viewpoint, (2.12) can be understood as follows. All gauge-
invariant operators made of BPS scalars are $Q$-closed since
$Q\bar{\phi}^{m}=0$. However, the presence of antisymmetrized pairs of scalars
will make the operator $Q$-exact because
$[\bar{\phi}^{m},\bar{\phi}^{n}]\sim\epsilon^{mnp}Q\psi_{p+}$ from
(2.11).333$\bar{\phi}^{m_{1}}\bar{\phi}^{n_{1}}\cdots\bar{\phi}^{n_{i}}\bar{\phi}^{m_{2}}-\bar{\phi}^{m_{2}}\bar{\phi}^{n_{1}}\cdots\bar{\phi}^{n_{i}}\bar{\phi}^{m_{1}}$
is also a linear combination of terms involving commutators. So among the
single-trace operators made of $n$ scalars, only (2.12) represent nontrivial
cohomology classes. All the other states in the table are obtained by acting
$Q^{m}_{+},\overline{Q}_{m\dot{\alpha}}$’s on (2.12). Multiplying them yields
independent multi-particle cohomologies at large $N$. Cohomologies in $S_{2}$
are given in Appendix A.

Now we discuss the finite $N$ cohomologies of graviton type. Finite $N$
gravitons are nothing but the ‘graviton cohomologies’ that we have explained
so far. Various steps that we explained above go through for finite $N$
cohomologies. For instance, constructing cohomologies within single trace
operators is consistent because $Q$ does not change the trace number. Also,
despite the existence of trace relations at finite $N$ which may relate some
single trace cohomologies to multi-trace ones, it is just a matter of using
redundant basis and we can eliminate them later. (Of course, it will be
convenient later to use only a subset of a such single trace cohomologies, as
we will explain.) Next we consider the step of multiplying them to make multi-
trace operators. Due to the Leibniz rule of $Q$, cohomologies multiply to
yield a new cohomology. Therefore, even at finite $N$, multiplying the single
trace cohomologies yields multi-trace cohomologies. Again due to trace
relations, there might be some multiplications which yield a trivial
cohomology, i.e. $Q$-exact operator. However, this is again a question of
redundancy.

The only new issue to consider is that the number of independent cohomologies
reduce because of the trace relation. Even at large $N$, one should take into
account the trace relations if the number of fields becomes larger than $N$.
This is the regime in which semi-classical giant gravitons can account for the
trace relations as the stringy exclusion principle. Even at finite $N$, it
makes good sense to regard these cohomologies as the graviton cohomologies in
quantum AdS gravity. In other words, these BPS states can be completely
understood by knowing the (giant) graviton physics.444Historically, giant
gravitons are discovered as objects realizing the trace relations, reducing
the number of states than the naive estimates. This is the context in which we
quote giant gravitons here. However, note that it has been shown [40] recently
that more complicated giant graviton states (with brane intersections and
light open strings stretched between the branes) represent black hole
microstates.

On the other hand, these operators cannot account for certain finite $N$
physics. For instance, in the high-temperature Cardy limit [9], charges/energy
are taken to be $N^{2}$ times a large number independent of $N$. The entropy
in this limit scales like $N^{2}$ times a universal large number for the whole
sequence of $U(N)$ gauge theories, from small to large $N$. Although this is
not large $N$ at all, such a deconfining behavior $\propto N^{2}$ is universal
at large energy/temperature. This generalizes the large black hole physics
straightforwardly to finite $N$. (The black hole like growth of the entropy at
finite $N$ was also confirmed by numerical studies [38, 39].) Such behaviors
do not happen with the finite $N$ graviton cohomologies. All these make it
natural to study novel finite $N$ cohomologies which are not of the graviton
type as defined above.

In section 3, an important technical step of identifying the black hole
cohomologies will be counting finite $N$ gravitons and subtracting them from
the full degeneracy (index). Once we know the overcomplete set for gravitons
as described above, it is in principle straightforward to identify the
redundant operators due to trace relation. In practice, even if one uses a
computer, this becomes quickly cumbersome as the charges and $N$ grow. In the
remaining part of this section, we will explain some structures which make
this estimate easier. We explain two tricks, both related to the fact that the
primaries (2.12) are given by symmetrized scalars.

The single-trace cohomologies in the $S_{n}$ multiplets are the generators of
the graviton cohomologies, i.e. we multiply them to generate multi-trace
graviton cohomologies. If some of them decompose to products of other
generators in $S_{n}$’s, clearly they need not be used as generators. For
$SU(N)$, one can show that all elements of $S_{n\geq N+1}$ decompose to the
those of $S_{n\leq N}$. To see this, we first show that any primary of the
form (2.12) at $n\geq N+1$ decomposes to those at $n\leq N$. This can be shown
using the Cayley-Hamilton equation,

$M^{N}+c_{N-1}(M)M^{N-1}+\cdots c_{1}(M)M+c_{0}(M){\bf 1}_{N\times N}=0\ ,$
(2.13)

where $M$ is an $N\times N$ matrix, and the coefficients $c_{n}(M)$’s are
explicitly known in terms of ${\rm tr}(M^{k})$ with $k\leq N-n$. (In
particular, $c_{0}(M)=(-1)^{N}\det(M)$.) Using this identity, one can express
the matrix $M^{N}$ (the first term) as a linear combination of various multi-
trace operators times the matrix $M^{n}$, where $0\leq n\leq N-1$. Inserting
$M=M_{1}+M_{2}$, $M=M_{1}+M_{2}+M_{3}$, $\cdots$, $M=M_{1}+\cdots+M_{N}$ with
independent matrices and applying the Cayley-Hamilton equations repeatedly to
various parts, one obtains a more generalized identity for the symmetrized
product $M_{(1}\cdots M_{N)}$ of $N$ matrices $M_{i}$. For traceless $M_{i}$,
some examples at low $N$ are

$\displaystyle N=2$ $\displaystyle:$ $\displaystyle
M_{(1}M_{2)}={\textstyle\frac{1}{2}}{\rm tr}(M_{1}M_{2}){\bf 1}_{2\times 2}$
(2.14) $\displaystyle N=3$ $\displaystyle:$ $\displaystyle
M_{(1}M_{2}M_{3)}={\textstyle\frac{1}{2}}{\rm
tr}(M_{(1}M_{2})M_{3)}+{\textstyle\frac{1}{3}}{\rm tr}(M_{(1}M_{2}M_{3)}){\bf
1}_{3\times 3}$ $\displaystyle N=4$ $\displaystyle:$ $\displaystyle
M_{(1}M_{2}M_{3}M_{4)}={\textstyle\frac{1}{2}}{\rm
tr}(M_{(1}M_{2})M_{3}M_{4)}+{\textstyle\frac{1}{3}}{\rm
tr}(M_{(1}M_{2}M_{3})M_{4)}$ $\displaystyle\hskip
85.35826pt+{\textstyle\frac{1}{4}}{\rm tr}(M_{(1}M_{2}M_{3}M_{4)}){\bf
1}_{4\times 4}-{\textstyle\frac{1}{2\cdot 2\cdot 2!}}{\rm tr}(M_{(1}M_{2}){\rm
tr}(M_{3}M_{4)}){\bf 1}_{4\times 4}$ $\displaystyle N=5$ $\displaystyle:$
$\displaystyle M_{(1}\cdots M_{5)}=-{\textstyle\frac{1}{2\cdot 2\cdot 2!}}{\rm
tr}(M_{(1}M_{2}){\rm tr}(M_{3}M_{4})M_{5)}-{\textstyle\frac{1}{2\cdot 3}}{\rm
tr}(M_{(1}M_{2}){\rm tr}(M_{3}M_{4}M_{5)}){\bf 1}_{5\times 5}$
$\displaystyle\hskip 73.97733pt+{\textstyle\frac{1}{2}}{\rm
tr}(M_{(1}M_{2})M_{3}M_{4}M_{5)}+{\textstyle\frac{1}{3}}{\rm
tr}(M_{(1}M_{2}M_{3})M_{4}M_{5)}$ $\displaystyle\hskip
73.97733pt+{\textstyle\frac{1}{4}}{\rm
tr}(M_{(1}M_{2}M_{3}M_{4})M_{5)}+{\textstyle\frac{1}{5}}{\rm tr}(M_{(1}\cdots
M_{5)}){\bf 1}_{5\times 5}\ .$

The factors on the right-hand sides are the symmetry factors for the cyclicity
of matrices inside a trace and also for exchanging identical single-trace
operators. The sign is $-1$ if the term involves even number of single-trace
operators. Inserting this expression into (2.12) with $n\geq N+1$, many times
if necessary, the single trace operators can be written as sums of products of
operators of the form (2.12) with $n\leq N$. In other words, the chiral
primaries of $S_{n\geq N+1}$ can be written in terms of those of $S_{n\leq
N}$. Acting $Q^{m}_{+}$, $\overline{Q}_{m\dot{\alpha}}$, $P_{+\dot{\alpha}}$
to these expressions, all cohomologies in $S_{n\geq N+1}$ can be written in
terms of those in $S_{n\leq N}$. So finite $N$ gravitons can be constructed
with the single-trace generators in $S_{n\leq N}$.

Even if the single-trace generators $S_{n\leq N}$ are independent in the sense
that none of them can be written as multi-trace operators, still there are new
trace relations among multi-traces operators after multiplying them. For
example, consider the $SU(2)$ theory and the following primaries of the
$S_{2}$ multiplet:

${\rm tr}(\bar{\phi}^{(m}\bar{\phi}^{n)})\sim{\rm tr}(X^{2})\ ,\ {\rm
tr}(Y^{2})\ ,\ {\rm tr}(Z^{2})\ ,\ {\rm tr}(XY)\ ,\ {\rm tr}(YZ)\ ,\ {\rm
tr}(ZX)\ .$ (2.15)

Here and below, we shall often use the notation
$(X,Y,Z)\equiv(\bar{\phi}^{1},\bar{\phi}^{2},\bar{\phi}^{3})$. Since we
consider cohomologies, we study the trace relations up to $Q$-exact terms. One
finds the following relations at the double trace order,

$\displaystyle{\rm tr}(X^{2}){\rm tr}(Y^{2})-[{\rm tr}(XY)]^{2}\sim{\rm
tr}([X,Y][X,Y])\sim Q{\rm tr}(\psi_{3}[X,Y])$ (2.16) $\displaystyle{\rm
tr}(X^{2}){\rm tr}(YZ)-{\rm tr}(XY){\rm tr}(XZ)\sim{\rm tr}([X,Y][X,Z])\sim
Q{\rm tr}(\psi_{3}[X,Z]-[X,Y]\psi_{2})\ ,$

and $4$ more relations obtained by permuting $X,Y,Z$. More covariantly, one
can write them as

$\epsilon_{mab}\epsilon_{ncd}{\rm tr}(\bar{\phi}^{(a}\bar{\phi}^{c)}){\rm
tr}(\bar{\phi}^{(b}\bar{\phi}^{d)})=Q\textrm{-exact}\ ,$ (2.17)

in the representation $\bar{\bf 6}$, or $[0,2]$, of $SU(3)$. They are the
primaries of a constraint supermultiplet of $PSU(1,2|3)$, contained in
$B\overline{B}[0;0]_{4}^{[2,0,2]}$ of $PSU(2,2|4)$. These constraints are not
independent in the sense that there are ‘relations of relations’ at higher
orders. See, for instance, [15] on these relations of relations in special
sectors. Unfortunately, we do not know a simple algorithm to identify all such
trace relations in terms of the generators.

The strategy we explained so far, using single-trace generators, is analogous
or dual to the counting of BPS states from giant gravitons (D3-branes expanded
in $S^{5}$) . There is another way of counting graviton-like BPS states using
dual giant gravitons, which are D3-branes expanded in $AdS_{5}$. In the field
theory, this is dual to counting operators in terms of the eigenvalues of
diagonal matrices, as we shall explain in a moment. For the chiral primaries
(2.12), the two complementary methods are fully explored and yield the same
result [41, 42]. For general graviton-like BPS states, neither method has been
fully developed. In a sense, we shall now provide a hybrid method of the two.

The multiplet $B\overline{B}[0;0]^{[0,n,0]}_{n}$ for single-particle gravitons
is absolutely protected, meaning that it cannot combine with other short
multiplets to form a long multiplet. BPS operators in this multiplet never
acquire anomalous dimensions as the coupling changes. Therefore, one may count
the graviton cohomologies in the free limit $g_{\rm YM}\rightarrow 0$.

We start by considering the chiral primaries (2.12). Since all the scalars are
symmetrized in the trace, we can regard all the fields as diagonal matrices
for the purpose of enumerating cohomologies. Each scalar contains $N-1$
eigenvalues, or $N$ eigenvalues whose sum is zero. The generators (2.12) are
Weyl-invariant polynomials of the eigenvalues. Then we consider the
superconformal descendants of (2.12) in the free theory limit. In free theory,
supersymmetry transformation of diagonal fields only involves diagonal
components of the superpartner fields. So we can restrict all the elementary
fields appearing in $S_{n\leq N}$ to be diagonal. The covariant derivatives on
the fields also reduce to ordinary derivatives since $g_{\rm YM}=0$. In
particular, trace relations are now relations of eigenvalue polynomials,
including the ordinary derivatives acting on the eigenvalues. So we should
count the polynomials which can be written as sums of products of the
generators in $S_{n\leq N}$. The reduction to the polynomial counting will
ease numerical studies, and sometimes will admit analytic counting.

This procedure becomes simpler in the $SU(2)$ theory, because each elementary
field is represented by one eigenvalue. We denote by
$x,y,z,\psi_{m},\lambda_{\alpha},f$ the eigenvalues of the fields
$X,Y,Z,\psi_{m+},\bar{\lambda}_{\dot{\alpha}},f_{++}$, respectively, e.g.
$X={\rm diag}(x,-x)$. The BPS derivatives $D_{+\dot{\alpha}}$ can also be
replaced by ordinary derivatives, which we call $\partial_{\alpha}$. We should
consider polynomials of these eigenvalues and derivatives acting on them,

$\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(x,y,z,\psi_{m},f)\ \ ,\ \
n_{1}\partial_{+}^{n_{1}-1}\partial_{-}^{n_{2}}\lambda_{+}+n_{2}\partial_{+}^{n_{1}}\partial_{-}^{n_{2}-1}\lambda_{-}$
(2.18)

with $n_{1},n_{2}\geq 0$. In order for a polynomial to be a BPS graviton, it
must be a sum of products of the generators in $S_{2}$. In terms of
eigenvalues, those generators are given by

$\displaystyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(x^{2},\ y^{2},\ z^{2},\
xy,\ yz,\ zx)$ (2.19)
$\displaystyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(y\psi_{1},\ z\psi_{1},\
x\psi_{2},\ z\psi_{2},\ x\psi_{3},\ y\psi_{3},\ x\psi_{1}-y\psi_{2},\
y\psi_{2}-z\psi_{3})\ $
$\displaystyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(x\lambda_{\pm},\
y\lambda_{\pm},\ z\lambda_{\pm})\ \ ,\ \
{\textstyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(xf-\frac{1}{2}\psi_{2}\psi_{3},yf-\frac{1}{2}\psi_{3}\psi_{1},zf-\frac{1}{2}\psi_{1}\psi_{2})}$
$\displaystyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(\psi_{1}\lambda_{\pm}+2y\partial_{\pm}z,\psi_{2}\lambda_{\pm}+2z\partial_{\pm}x,\psi_{3}\lambda_{\pm}+2x\partial_{\pm}y)$
$\displaystyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(\lambda_{+}\lambda_{-})\
,\ \
{\textstyle\partial_{+}^{n_{1}}\partial_{-}^{n_{2}}(\lambda_{\pm}f-\frac{2}{3}\psi_{m}\partial_{\pm}\phi^{m}+\frac{1}{3}\phi^{m}\partial_{\pm}\psi_{m})}\
.$

If a polynomial is expressed as a sum of the products of these generators, it
must be invariant under the Weyl reflection of the eigenvalues:

$(x,y,z,\psi_{m},\lambda_{\alpha},f)\rightarrow-(x,y,z,\psi_{m},\lambda_{\alpha},f)\
.$ (2.20)

In other words it must be an even polynomial. We used this approach to
numerically count the $SU(2)$ graviton operators till certain order in the
charge expansion. The studies of section 3.2 will be based on this
calculation. In a simple subsector of the $SU(2)$ theory, we can analytically
count independent graviton cohomologies using this approach as explained in
Appendix C.

##  3 The black hole cohomologies for $SU(2)$

[18] systematically constructed all cohomologies till certain order. We shall
employ a more streamlined approach in this section. We will first compute the
index over the black hole cohomologies by subtracting the contributions from
finite $N$ gravitons. Then with some guesses if necessary, we shall explicitly
construct the black hole cohomologies which account for this index. We find
that this approach is much more efficient in detecting new cohomologies.

The index of the $\mathcal{N}=4$ Yang-Mills theory is defined as [11, 12]

$Z(\Delta_{I},\omega_{i})={\rm
Tr}\left[(-1)^{F}e^{-\sum_{I=1}^{3}\Delta_{I}R_{I}}e^{-\sum_{i=1}^{2}\omega_{i}J_{i}}\right]$
(3.1)

where the chemical potentials should satisfy
$\sum_{I}\Delta_{I}-\sum_{i}\omega_{i}=0$ for this quantity to be an index.
For the $SU(N)$ theory, this is given by the following integral

$Z(\Delta_{I},\omega_{i})=\frac{1}{N!}\int_{0}^{2\pi}\prod_{a=1}^{N}\frac{d\alpha_{a}}{2\pi}\exp\left[\sum_{a\neq
b}\sum_{n=1}^{\infty}\frac{f(n\Delta_{I},n\omega_{i})-1}{n}e^{in(\alpha_{a}-\alpha_{b})}\right]\exp\left[(N\\!-\\!1)\sum_{n=1}^{\infty}\frac{f(n\Delta_{I},n\omega_{i})}{n}\right]\
,$ (3.2)

where $f(\Delta_{I},\omega_{i})\equiv
1-\frac{(1-e^{-\Delta_{1}})(1-e^{-\Delta_{2}})(1-e^{-\Delta_{3}})}{(1-e^{-\omega_{1}})(1-e^{-\omega_{2}})}$
is the single letter index. Since this index is independent of the coupling
$g_{\rm YM}$, we can regard it as the index over our 1-loop cohomologies.

###  3.1 The BMN sector

The radially quantized QFT lives on $S^{3}\times\mathbb{R}$. The fields are
expanded in spherical harmonics of $SO(4)$. It was shown in [28] that the
classical $\mathcal{N}=4$ Yang-Mills theory has a consistent truncation which
keeps finite degrees of freedom, described by the BMN matrix model [27]. The
modes kept after the truncation are given by: (1) s-wave modes $\phi_{m}(t)$,
$\bar{\phi}^{m}(t)$ of the scalars, (2) lowest spinor harmonics modes
$\psi_{m\alpha}(t)$, $\lambda_{\alpha}(t)$ (the spinor indices are defined
using the labels of Killing spinor fields [28]), (3) vector potential 1-form
restricted to $A=A_{0}(t)dt+A_{i}(t)\sigma_{i}$ where $\sigma_{i}$ with
$i=1,2,3$ are the right-invariant 1-forms on $S^{3}$ in our convention. This
is a consistent truncation of the nonlinear equations of motion, and not a
quantum reduction in any sense. So the full quantum BMN theory is a priori
unrelated to the 4d Yang-Mills theory. However, since our 1-loop cohomology
problem uses classical supercharge $Q$ only, it can be truncated to the BMN
model. If the conjecture of [17] is true, the whole BPS cohomology problem
would have a quantum truncation to this model.

In general, the BMN theory and the full Yang-Mills theory behave differently
in many ways. The difference starts from the number of ground states. The
Yang-Mills theory on $S^{3}\times\mathbb{R}$ has a unique vacuum, while the
BMN model has many ground states labeled semiclassically by the discrete
values of $A_{i}$. In the quantum BMN theory, viewed as M-theory in the plane
wave background, these ground states describe various M2/M5-brane
configurations with zero lightcone energies [43]. In the Yang-Mills theory,
however, there are large gauge transformations on $S^{3}$ which can gauge away
these ground states to $A_{i}=0$. So if one wishes to study the Yang-Mills
theory using this matrix model, it suffices to consider the physics around
$A_{i}=0$.

Recall that our cohomology problem is completely classical, using the
classical supercharge $Q$ at half-loop order. Therefore, this problem should
have a truncation to the BMN matrix model. This turns out to be the cohomology
problem defined using

$\bar{\phi}^{m}\ ,\ \ \psi_{m+}\ ,\ \ f_{++}\ ,$ (3.3)

without using any gauginos $\bar{\lambda}_{\dot{\alpha}}$ or derivatives
$D_{+\dot{\alpha}}$. These operators close by the action of $Q$:
$[Q,\bar{\phi}^{m}]=0$,
$\\{Q,\psi_{m+}\\}=-i\epsilon_{mnp}[\bar{\phi}^{m},\bar{\phi}^{n}]$,
$[Q,f_{++}]=-i[\psi_{m+},\bar{\phi}^{m}]$. So it is possible to restrict the
cohomology problem by using operators constructed using the letters (3.3).
Note that the truncation is also applied to the operator $\Lambda$ when one
identifies two operators $O_{1}$ and $O_{2}$ related as
$O_{2}-O_{1}=[Q,\Lambda\\}$. This is why the gauginos
$\bar{\lambda}_{\dot{\alpha}}$ cannot be included in this truncation. Although
it is $Q$-closed by itself, $\bar{\lambda}_{\dot{\alpha}}$ can be obtained by
acting $Q$ on the covariant derivative,
$[Q,D_{+\dot{\alpha}}]=-i[\bar{\lambda}_{\dot{\alpha}},\ \\}$. So if one had
tried to include $\bar{\lambda}_{\dot{\alpha}}$ into the truncation and
construct operators like $O_{1}$, $O_{2}$, $\Lambda$, one may incorrectly
conclude that certain $O_{1}$ and $O_{2}$ are different by not including
derivatives in $\Lambda$. This truncation of the cohomology problem was known
in [18, 26], although the relation to the BMN truncation was not explicitly
addressed.555We thank Nakwoo Kim for first pointing this out to us. Notice
also that this truncation is not kinematic, i.e. cannot be inferred without
knowing the dynamical information of the classical theory.

The BMN truncation is the $SU(2)_{R}$ invariant truncation. In our cohomology
problem, this means that no ingredients include the $\dot{\alpha}$ indices for
$SU(2)_{R}$. This is why $\bar{\lambda}_{\dot{\alpha}}$ and
$D_{+\dot{\alpha}}$ are excluded. Similarly, in the representation theory,
only a small subset of $PSU(1,2|3)$ generators can be used to generate a
multiplet. Among the $PSU(1,2|3)$ generators $Q^{m}_{+}$,
$\overline{Q}_{m\dot{\alpha}}$ and $P_{+\dot{\alpha}}$, only the three
supercharges $Q^{m}_{+}$ which belong to $SU(1|3)$ act within BMN
cohomologies.

We also consider the index over the BMN cohomologies. Keeping the letters
(3.3) only, the letter index is given by

$\tilde{f}=\sum_{I=1}^{3}e^{-\Delta_{I}}-\sum_{I<J}e^{-\Delta_{I}-\Delta_{J}}+e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}=1-(1-e^{-\Delta_{1}})(1-e^{-\Delta_{2}})(1-e^{-\Delta_{3}})\
.$ (3.4)

The first three terms come from $\bar{\phi}^{m}$, the next three terms from
$\psi_{m+}$, and the last term from $f_{++}$. $\Delta_{I}$ is the chemical
potential conjugate to $R_{I}+\frac{J_{1}+J_{2}}{2}\equiv R_{I}+J$. The fourth
chemical potential $\omega_{1}-\omega_{2}$ of (3.1) does not appear since BMN
truncation is $SU(2)_{R}$ invariant. The matrix integral expression for the
BMN index is

$\displaystyle Z(\Delta_{I})$ $\displaystyle=$
$\displaystyle\frac{1}{N!}\int_{0}^{2\pi}\prod_{a=1}^{N}\frac{d\alpha_{a}}{2\pi}\exp\left[\sum_{a\neq
b}\sum_{n=1}^{\infty}\frac{\tilde{f}(n\Delta_{I})-1}{n}e^{in(\alpha_{a}-\alpha_{b})}\right]\exp\left[(N\\!-\\!1)\sum_{n=1}^{\infty}\frac{\tilde{f}(n\Delta_{I})}{n}\right]$
(3.5) $\displaystyle=$
$\displaystyle\frac{1}{N!}\int_{0}^{2\pi}\prod_{a=1}^{N}\frac{d\alpha_{a}}{2\pi}\cdot\prod_{a\neq
b}\frac{(1-e^{i\alpha_{ab}})\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}}e^{i\alpha_{ab}})}{(1-e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}e^{i\alpha_{ab}})\prod_{I=1}^{3}(1-e^{-\Delta_{I}}e^{i\alpha_{ab}})}$
$\displaystyle\hskip
85.35826pt\cdot\left[\frac{\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}})}{(1-e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}})\prod_{I=1}^{3}(1-e^{-\Delta_{I}})}\right]^{N-1}$

where $\alpha_{ab}\equiv\alpha_{a}-\alpha_{b}$. This expression cannot be
obtained from the original index (3.2) by taking limits of the chemical
potentials, because the BMN truncation is not kinematic. To ease the
calculations, one may replace the Haar measure part of the integrand by half
of it [44],

$\frac{1}{N!}\prod_{a\neq b}(1-e^{i\alpha_{ab}})\ \rightarrow\
\prod_{a<b}(1-e^{i\alpha_{ab}})\ ,$ (3.6)

without changing the integral.

(3.5) can be computed easily for low $N$’s because the integrand is a finite
product. One can evaluate it by a residue sum. For instance, for the $SU(2)$
index, the only nontrivial integral variable is $e^{i\alpha_{12}}$ along the
unit circle. We evaluate the integral by the residue sum by shrinking this
unit circle. After the replacement (3.6) in the integrand, we should sum over
four residues at the following poles:

$e^{i\alpha_{12}}=e^{-\Delta_{1}}\ ,\ \ e^{-\Delta_{2}}\ ,\ \ e^{-\Delta_{3}}\
,\ \ e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}\ .$ (3.7)

We have a general result, but just to illustrate it at
$e^{-\Delta_{1}}=e^{-\Delta_{2}}=e^{-\Delta_{3}}\equiv t^{2}$, one obtains

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\left[\frac{}{}\\!\\!\right.1+\\!3t^{2}+\\!12t^{4}+\\!20t^{6}+\\!42t^{8}+\\!48t^{10}+\\!75t^{12}+\\!66t^{14}+\\!81t^{16}$
$\displaystyle\hskip
8.5359pt+\\!55t^{18}+\\!54t^{20}+\\!27t^{22}+\\!19t^{24}+\\!6t^{26}+\\!3t^{28}\left.\\!\\!\frac{}{}\right]\frac{(1-t^{2})^{3}}{(1-t^{12})(1-t^{8})^{3}}\
.$

This index contains both black hole and graviton cohomologies made with (3.3).

To compute the index over black hole cohomologies in this sector, we need to
enumerate the graviton cohomologies and subtract their index from the full
index. This can be done analytically for the $SU(2)$ gravitons in the BMN
sector, employing the eigenvalue counting explained in section 2. We need to
consider even polynomials of the following seven eigenvalues:

$x\ ,\ y\ ,\ z\ ,\ f\ ,\ \psi_{1}\ ,\ \psi_{2}\ ,\ \psi_{3}\ ,$ (3.9)

where one should remember that the last three are Grassmannian numbers. We can
write all possible even monomials of these eigenvalues, and find the
combinations which can be written as sums of products of the following BMN
generators in $S_{2}$:

$\displaystyle{\bf 6}$ $\displaystyle:$ $\displaystyle x^{2}\ ,\ y^{2}\ ,\
z^{2}\ ,\ xy\ ,\ yz\ ,\ zx$ (3.10) $\displaystyle{\bf 8}$ $\displaystyle:$
$\displaystyle\psi_{1}\cdot(y,z)\ ,\ \psi_{2}\cdot(z,x)\ ,\
\psi_{3}\cdot(x,y)\ ,\ \psi_{1}x-\psi_{2}y\ ,\ \psi_{2}y-\psi_{3}z$
$\displaystyle{\bf 3}$ $\displaystyle:$ $\displaystyle
xf-\textstyle{\frac{1}{2}}\psi_{2}\psi_{3}\ ,\
yf-\textstyle{\frac{1}{2}}\psi_{3}\psi_{1}\ ,\
zf-\textstyle{\frac{1}{2}}\psi_{1}\psi_{2}\ .$

Counting these polynomials in the right order, one can count them basically at
the monomial levels. The procedure is explained in Appendix B. We can compute
the full partition function for these cohomologies. Giving $-1$ weights to
fermions and unrefining some fugacities, we can compute the index $Z_{\rm
grav}(\Delta_{I})$ over gravitons.

Subtracting this from the full index, the difference between $Z$ and $Z_{\rm
grav}$ is given by

$Z-Z_{\rm
grav}=-\frac{e^{-4(\Delta_{1}+\Delta_{2}+\Delta_{3})}}{1-e^{-2(\Delta_{1}+\Delta_{2}+\Delta_{3})}}\cdot\prod_{I=1}^{3}(1-e^{-\Delta_{I}})\cdot\prod_{I=1}^{3}\frac{1}{(1-e^{-\Delta_{I}}e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}})}\
.$ (3.11)

Unrefining as $e^{-\Delta_{I}}=t^{2}$, one obtains a series in $t$ given by

$Z-Z_{\rm grav}=-t^{24}+3t^{26}-3t^{28}+t^{30}-3t^{32}+\cdots\ .$ (3.12)

$t$ is conjugate to $j\equiv 6(R+J)$, where
$R\equiv\frac{R_{1}+R_{2}+R_{3}}{3}$, $J\equiv\frac{J_{1}+J_{2}}{2}$. From
this formula, one finds the first black hole cohomology at $j=24$. This
‘threshold’ black hole cohomology was already identified in [18, 26], as we
shall review and rewrite in a more compact form below. It may look like there
are many black hole states beyond this threshold, but most of them are rather
trivial. To make this point clear, we would like to first interpret various
factors of (3.11), which will be extensively justified later.

(3.11) is a multiplication of three factors, divided by the $\cdot$ products.
We interpret the first factor as the ‘core’ black hole primary operators.
Constructing this part of the cohomologies will be the goal of this
subsection. The second factor comes from the $SU(1|3)$ descendants obtained
from the first factor by acting $Q^{m}_{+}$. The supercharge $Q^{m}_{+}$
carries charges $R_{I}=\delta_{I,m}-\frac{1}{2}$ and $J=\frac{1}{2}$, so is
weighted by $e^{-\Delta_{I}}$. So the second factor comes from the Fock space
obtained by acting three $Q^{m}_{+}$’s. Finally, the third factor comes from
the multiplications of certain multi-gravitons to the core black hole
cohomologies. Among the $17$ graviton states listed in (3.10), only $3$ types
on the third line can contribute. The remaining $14$ gravitons of (3.10)
multiplying the core black hole operators do not appear in the index. This
aspect will be discussed further in section 3.2.

For $SU(2)$, the BMN index (3.11) does not show very large entropy even at
large charges. From an inspection of the integral (3.5) and having in mind
applying the techniques developed in [45], we expect that at large enough $N$
and charges one will be able to show that the entropies will scale like those
for black holes.

Now we construct the cohomologies that will account for the first factor of
(3.11),

$-\frac{t^{24}}{1-t^{12}}=-t^{24}-t^{36}-t^{48}-t^{60}-\cdots\ ,$ (3.13)

where $t^{6}=e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}$. The index predicts unique
fermionic cohomology at $j=24+12n$ ($n=0,1,2,\cdots$), all singlets of
$SU(3)\subset SU(4)$. For $SU(2)$ gauge group, we use the 3-dimensional vector
notation for the adjoint fields. In the remaining part of this subsection,
$\phi^{m}=(X,Y,Z)$, $\psi_{m},f$ would mean 3 dimensional vectors, and
inner/outer products will replace the trace/commutators. The
$Q$-transformations of these $3$-vectors are given by

$Q\phi^{m}=0\ ,\ \
Q\psi_{m}={\textstyle\frac{1}{2}}\epsilon_{mnp}\phi^{n}\times\phi^{p}\ ,\
Qf=\phi^{m}\times\psi_{m}\ .$ (3.14)

See [26] and appendix A for our normalization.

$O_{0}$ operator at $t^{24}$

This operator has charges $E=\frac{19}{2}$, $R_{1}=R_{2}=R_{3}=\frac{3}{2}$,
$J_{1}=J_{2}=\frac{5}{2}$. A representative of this cohomology [26] is given
by

$\displaystyle O_{0}^{\prime}$ $\displaystyle=$
$\displaystyle(X\cdot\psi_{1}-Y\cdot\psi_{2})(X\cdot\psi_{3})(\psi_{2}\cdot\psi_{1}\times\psi_{1})+(Y\cdot\psi_{2}-Z\cdot\psi_{3})(Y\cdot\psi_{1})(\psi_{3}\cdot\psi_{2}\times\psi_{2})$
(3.15)
$\displaystyle+(Z\cdot\psi_{3}-X\cdot\psi_{1})(Z\cdot\psi_{2})(\psi_{1}\cdot\psi_{3}\times\psi_{3})\
.$

Note that the second and third terms are obtained by making cyclic
permutations of $(X,\psi_{1})$, $(Y,\psi_{2})$, $(Z,\psi_{3})$ on the first
term. The cyclic permutations are part of the $SU(3)$ symmetry, thus
symmetries of the cohomology problem, On the other hand, odd permutations
accompanied by the sign flips of all $\psi_{m}$’s and $\phi^{m}$’s are part of
$SU(4)\times SU(2)_{L}$ symmetry which leave $Q$ invariant, thus being
symmetries of the cohomology problem. To construct a better representative of
this cohomology, consider the following operator obtained by permuting
$(X,\psi_{1})\leftrightarrow(Y,\psi_{2})$ and flipping signs of all
$\phi^{m},\psi_{m}$ on (3.15):

$\displaystyle O_{0}^{\prime\prime}$ $\displaystyle=$
$\displaystyle(X\cdot\psi_{1}-Y\cdot\psi_{2})(Y\cdot\psi_{3})(\psi_{1}\cdot\psi_{2}\times\psi_{2})+(Y\cdot\psi_{2}-Z\cdot\psi_{3})(Z\cdot\psi_{1})(\psi_{2}\cdot\psi_{3}\times\psi_{3})$
(3.16)
$\displaystyle+(Z\cdot\psi_{3}-X\cdot\psi_{1})(X\cdot\psi_{2})(\psi_{3}\cdot\psi_{1}\times\psi_{1})\
.$

One can show

$\displaystyle O_{0}^{\prime}-O_{0}^{\prime\prime}$ $\displaystyle=$
$\displaystyle-2Q[(\psi_{1}\cdot\psi_{2})(\psi_{2}\cdot\psi_{3})(\psi_{3}\cdot\psi_{1})]\
,$ (3.17) $\displaystyle O_{0}$ $\displaystyle\equiv$
$\displaystyle-5(O_{0}^{\prime}+O_{0}^{\prime\prime})=\epsilon^{p_{1}p_{2}p_{3}}v^{m}_{\
\ p_{1}}v^{n}_{\ \ p_{2}}(\psi_{m}\cdot\psi_{n}\times\psi_{p_{3}})\ ,$

where

$v^{m}_{\ \
n}\equiv(\phi^{m}\cdot\psi_{n})-{\textstyle\frac{1}{3}}\delta^{m}_{n}(\phi^{p}\cdot\psi_{p})$
(3.18)

are graviton cohomologies in the $S_{2}$ multiplet. $O_{0}$ is manifestly an
$SU(3)$ singlet. Note that the second term of $v$ proportional to
$\delta^{m}_{n}$ drops out when $v$ is inserted into (3.17), because of the
symmetry of $\psi_{m}\cdot\psi_{n}\times\psi_{p_{3}}$ and the antisymmetry of
$\epsilon^{p_{1}p_{2}p_{3}}$. So we can write

$O_{0}=\epsilon^{p_{1}p_{2}p_{3}}(\phi^{m}\cdot\psi_{p_{1}})(\phi^{n}\cdot\psi_{p_{2}})(\psi_{m}\cdot\psi_{n}\times\psi_{p_{3}})\
.$ (3.19)

To show that $O_{0}$ is a black hole cohomology, one should check that it is
$Q$-closed, not $Q$-exact, and not of graviton type. The first and third are
trivial. $O_{0}$ is not graviton-like because it consists of seven (odd)
letters: since $SU(2)$ gravitons are made of operators in $S_{2}$, they always
have an even number of letters. To check $Q$-closedness, first note that $Q$
acts only on $\psi_{m}\cdot\psi_{n}\times\psi_{p_{3}}$ because $v^{m}_{\ \ n}$
are $Q$-closed. One finds

$Q(\psi_{m}\cdot\psi_{n}\times\psi_{p})={\textstyle\frac{3}{2}}\epsilon_{(m|qr}(\phi^{q}\times\phi^{r})\cdot(\psi_{|n}\times\psi_{p)})=3\epsilon_{(m|qr}(\phi^{q}\cdot\psi_{|n})(\phi^{r}\cdot\psi_{p)})=3\epsilon_{(m|qr}v^{q}_{\
\ |n}v^{r}_{\ \ p)}\ .$ (3.20)

At the last step, the second term of $\phi^{q}\cdot\psi_{n}=v^{q}_{\ \
n}+\delta^{q}_{n}(\cdots)$ etc. does not survive after the index contractions.
Inserting it to $QO_{0}$ and replacing the product of two $\epsilon$’s by
three $\delta$’s, $QO_{0}$ is given by various row/column contractions of four
$3\times 3$ traceless matrices $v^{m}_{\ \ n}$. Possible terms are ${\rm
tr}(v^{4})$ and ${\rm tr}(v^{2}){\rm tr}(v^{2})$, but the fermionic nature of
$v$ and the cyclicity of trace ensure that they are all zero. So $QO_{0}$ is
zero because there are no nonzero terms that can contribute.

The non-$Q$-exactness was originally shown after a calculation using computer
[18, 26]. Here we provide an analytic argument. We assume $Q$-exactness,
narrow down the possible $Q$-exact terms and then show that no combination of
them works. $O_{0}$ is at the $\mathcal{O}(\phi^{2}\psi^{5})$ order. If this
is $Q$-exact, the schematic structure should be as follows:

$\phi^{2}\psi^{5}=Q_{f}[f\phi\psi^{4}]+Q[\psi^{6}]\ .$ (3.21)

$Q_{f}$ means the part of $Q$ acting on $f$. $Q$ may also act on $\psi$ in
this term to produce a term at $\mathcal{O}(f\phi^{3}\psi^{3})$ order, and if
$O_{0}$ is completely $Q$-exact, $Q_{\psi}(f\phi\psi^{4})$ should cancel
$Q_{f}[f^{2}\phi^{2}\psi^{2}]$. We shall only consider the $Q$-exactness of
$O_{0}$ within the $\phi^{2}\psi^{5}$ order and find a contradiction. The
terms on the right hand side should respect all the $SU(3)\times SU(2)$ tensor
structures of the left hand side. There might be terms violating some of these
structures separately on the first and second terms, but they should cancel by
themselves and we do not care about this part. We consider the terms which
respect them and the equation should hold within this sector separately if
(3.21) is generally true. (3.19) is given by multiplying the following scalar
and fermion factors,

$\displaystyle\phi^{(m}_{(i}\phi^{n)}_{j)}$ $\displaystyle:$
$\displaystyle({\bf 6},{\bf 5}+{\bf 1})\in SU(3)\times SU(2)$ (3.22)
$\displaystyle\psi_{p_{1}(i|}\psi_{p_{2}|j)}(\psi_{m}\cdot\psi_{n}\times\psi_{p_{3}})$
$\displaystyle:$ $\displaystyle({\bf 3},{\bf 5}+{\bf 1})\otimes(\overline{\bf
10},{\bf 1})\ ,$

where $i,j=1,2,3$ are $SU(2)\sim SO(3)$ indices. The operators on the right
hand side of (3.21) should respect these structures.

We shall first write down all possible terms on the right hand side satisfying
several consequences of (3.22) after contracting all the indices, obtaining
only a small number of terms. Some useful requirements are: (1) $SU(3)$
singlet condition of $O_{0}$, (2) exchange symmetry of the two $SU(3)$ indices
carried by the scalars. We first consider the term $Q[\psi^{6}]$. $Q$ acting
on any $\psi$ produces a term of the form $\phi^{m}\times\phi^{n}$, violating
the condition (2). So there are no terms of the form $Q[\psi^{6}]$ that we can
write down. Now we try to write down all the gauge-invariant operators at
$f\phi\psi^{4}$ order which can appear inside $Q_{f}$ in (3.21). Since it
consists of six letters, we take three pairwise inner products. (Contractions
by two $\epsilon$ tensors can also be written as three inner products.) The
possible terms are

$(f\cdot\phi^{m})(\psi_{[n_{1}}\cdot\psi_{n_{2}]})(\psi_{[p_{1}}\cdot\psi_{p_{2}]})\
\ ,\ \
(\phi^{m}\cdot\psi_{n_{1}})(f\cdot\psi_{n_{2}})(\psi_{[p_{1}}\cdot\psi_{p_{2}]})\
.$ (3.23)

$Q_{f}$ transformation of the first term violates the condition (2) since
$Q[f\cdot\phi^{m}]=(\phi^{n}\times\psi_{n})\cdot\phi^{m}=(\phi^{m}\times\phi^{n})\cdot\psi_{n}$.
Now imposing the condition (1) on the second term, one should contract the
$SU(3)$ indices to form singlets. One finds

$({\bf 8}\oplus{\bf 1})\otimes{\bf 3}\otimes\overline{\bf 3}\rightarrow{\bf
27}\oplus{\bf 10}\oplus\overline{\bf 10}\oplus{\bf 8}\oplus{\bf 8}\oplus{\bf
8}\oplus{\bf 8}\oplus{\bf 1}\oplus{\bf 1}$

so there are two possible singlets. They are

$(\phi^{m}\cdot\psi_{m})\epsilon^{npq}(f\cdot\psi_{n})(\psi_{p}\cdot\psi_{q})\
\ ,\ \
(\phi^{m}\cdot\psi_{n})(f\cdot\psi_{m})\epsilon^{npq}(\psi_{p}\cdot\psi_{q})\
.$ (3.24)

Acting $Q_{f}$ on them and separating the $\phi^{2}$ and $\psi^{5}$ parts as
we did in (3.22), we obtain a part consistent with (3.22) and the rest.
Focusing on the former part, they are given by $\phi^{(m}_{(i}\phi^{r)}_{j)}$
times

$\psi_{(m}^{(i}(\psi_{r)}\times\psi_{n})^{j)}\epsilon^{npq}(\psi_{p}\cdot\psi_{q})\
\ ,\ \
\psi_{n}^{(i}(\psi_{r}\times\psi_{m})^{j)}\epsilon^{npq}(\psi_{p}\cdot\psi_{q})$
(3.25)

respectively. If $O_{0}$ is $Q$-exact, a suitable linear combination of these
two terms should yield $O_{0}$. The agreement should happen for every
coefficient of $\phi^{(m}_{(i}\phi^{r)}_{j)}$ separately, demanding

$\epsilon^{pqr}\psi_{p}^{(i}\psi_{q}^{j)}(\psi_{m}\cdot\psi_{n}\times\psi_{r})=A\psi_{(m}^{(i}(\psi_{n)}\times\psi_{r})^{j)}\epsilon^{pqr}(\psi_{p}\cdot\psi_{q})+B\psi_{r}^{(i}(\psi_{n}\times\psi_{m})^{j)}\epsilon^{pqr}(\psi_{p}\cdot\psi_{q})$
(3.26)

for suitable $A,B$. Inserting two different sets of $m,r,i,j$, we found that
there are no solutions for $A$ and $B$. This proves that $O_{0}$ is not
$Q$-exact.

One can also easily show the non-$Q$-exactness by studying the $SU(1|3)$
descendants obtained by acting $Q_{+}^{a}Q_{+}^{b}$. For instance, one obtains

$\displaystyle Q_{+}^{2}Q_{+}^{1}O_{0}^{\prime}=$ (3.27)
$\displaystyle-(Y\cdot
f+\psi_{3}\cdot\psi_{1})^{2}\psi_{3}\cdot(\psi_{2}\times\psi_{2})-(X\cdot
f+\psi_{2}\cdot\psi_{3})(Z\cdot
f+\psi_{1}\cdot\psi_{2})\psi_{1}\cdot(\psi_{3}\times\psi_{3})$
$\displaystyle-(X\cdot
f+\psi_{2}\cdot\psi_{3})(X\cdot\psi_{3})f\cdot(\psi_{1}\times\psi_{1})+2(Y\cdot\psi_{2}-Z\cdot\psi_{3})(Y\cdot
f+\psi_{3}\cdot\psi_{1})\psi_{3}\cdot(\psi_{2}\times f)$
$\displaystyle-2(Y\cdot
f+\psi_{3}\cdot\psi_{1})(X\cdot\psi_{3})\psi_{2}\cdot(\psi_{1}\times
f)-(Z\cdot\psi_{3}-X\cdot\psi_{1})(Z\cdot
f+\psi_{1}\cdot\psi_{2})f\cdot(\psi_{3}\times\psi_{3})$

which contains uncanceled $\phi^{0}\psi^{7}$ terms on the second line. Since
acting $Q$ always creates one or more $\phi$ factors, these terms cannot be
$Q$-exact. Since a descendant of $O_{0}^{\prime}$ is not $Q$-exact, $O_{0}$
cannot be $Q$-exact either, providing a simpler proof. Or alternatively, one
can prove non-$Q$-exactness by acting three $Q_{+}$’s to $O_{0}^{\prime}$ and
check that it contains nonzero term at $f\psi^{6}$ order,

$\displaystyle
Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{0}^{\prime}=Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{0}^{\prime\prime}=$
(3.28) $\displaystyle(X\cdot
f+\psi_{2}\cdot\psi_{3})^{2}f\cdot(\psi_{1}\times\psi_{1})+2(X\cdot
f+\psi_{2}\cdot\psi_{3})(Y\cdot
f+\psi_{3}\cdot\psi_{1})f\cdot(\psi_{1}\times\psi_{2})$
$\displaystyle+(1,2,3\rightarrow 2,3,1)+(1,2,3\rightarrow
3,1,2)=G^{m}G^{n}f\cdot(\psi_{m}\times\psi_{n})$

where $G^{m}\equiv\phi^{m}\cdot
f+\frac{1}{2}\epsilon^{mnp}\psi_{n}\cdot\psi_{p}$. Proof of this sort will
sometimes be useful later. For instance, one can show that $(Z\cdot
f+\psi_{1}\cdot\psi_{2})O_{0}^{\prime}$ is not $Q$-exact, since its descendant

$Q^{2}_{+}Q^{1}_{+}\left[(Z\cdot
f+\psi_{1}\cdot\psi_{2})O_{0}^{\prime}\right]=(Z\cdot
f+\psi_{1}\cdot\psi_{2})Q^{2}_{+}Q^{1}_{+}O_{0}^{\prime}$ (3.29)

contains a term at $\phi^{0}\psi^{9}$ order.

$O_{1}$ operator at $t^{36}$

Now we construct the cohomology which accounts for the $-t^{36}$ term of
(3.13). It should be fermionic, has charge $j=6(R+J)=36$, and should be an
$SU(2)_{R}\times SU(3)$ singlet because we expect unique cohomology (unless
there is a cancellation at this order which obscures the true degeneracy). We
call this operator $O_{1}$. From the last condition, we set three $R_{I}$
equal and two $J_{i}$ equal. Still, we do not know the individual $R$ and $J$
so we should make a guess. Our first guess was to add extra $\Delta J=2$ to
the charges $R=\frac{3}{2}$, $J=\frac{5}{2}$ of $O_{0}$. We listed all
operators in this sector and found the cohomology by computer. Then we made
several trials until we found the following $SU(3)$-invariant representative:

$\displaystyle O_{1}$ $\displaystyle=$ $\displaystyle(f\cdot
f)\epsilon^{c_{1}c_{2}c_{3}}(\phi^{a}\cdot\psi_{c_{1}})(\phi^{b}\cdot\psi_{c_{2}})(\psi_{a}\cdot\psi_{b}\times\psi_{c_{3}})$
$\displaystyle+\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}(f\cdot\psi_{b_{1}})(\phi^{a}\cdot\psi_{c_{1}})(\psi_{b_{2}}\cdot\psi_{c_{2}})(\psi_{a}\cdot\psi_{b_{3}}\times\psi_{c_{3}})$
$\displaystyle-{\textstyle\frac{1}{72}}\epsilon^{a_{1}a_{2}a_{3}}\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}(\psi_{a_{1}}\cdot\psi_{b_{1}}\times\psi_{c_{1}})(\psi_{a_{2}}\cdot\psi_{b_{2}}\times\psi_{c_{2}})(\psi_{a_{3}}\cdot\psi_{b_{3}}\times\psi_{c_{3}})\
.$

It is not graviton type since it is made of nine (odd) letters. One can also
easily check that it is not $Q$-exact. This is because the last term contains
no scalars. Since $Q$ transformations (3.14) always yield scalars, the last
term cannot be made $Q$-exact. So $O_{1}$ is not $Q$-exact.

Now we discuss the $Q$-closedness. $O_{1}$ takes the form of

$O_{1}=(f\cdot f)O_{0}+f\cdot\xi+\chi\ ,$ (3.31)

where the $SU(2)$ triplet $\vec{\xi}$ and the singlet $\chi$ are given by

$\displaystyle\vec{\xi}$ $\displaystyle=$
$\displaystyle\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}\vec{\psi}_{b_{1}}(\phi^{a}\cdot\psi_{c_{1}})(\psi_{b_{2}}\cdot\psi_{c_{2}})(\psi_{a}\cdot\psi_{b_{3}}\times\psi_{c_{3}})$
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle-{\textstyle\frac{1}{72}}\epsilon^{a_{1}a_{2}a_{3}}\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}(\psi_{a_{1}}\cdot\psi_{b_{1}}\times\psi_{c_{1}})(\psi_{a_{2}}\cdot\psi_{b_{2}}\times\psi_{c_{2}})(\psi_{a_{3}}\cdot\psi_{b_{3}}\times\psi_{c_{3}})$
(3.32) $\displaystyle=$
$\displaystyle-120\psi_{1}^{1}\psi_{1}^{2}\psi_{1}^{3}\psi_{2}^{1}\psi_{2}^{2}\psi_{2}^{3}\psi_{3}^{1}\psi_{3}^{2}\psi_{3}^{3}\
.$

$Q$-closedness is equivalent to the following equations:

$2(\vec{\phi}^{m}\times\vec{\psi}_{m})O_{0}+Q_{\psi}\vec{\xi}=0\ \ ,\ \
\vec{\phi}^{m}\cdot(\vec{\psi_{m}}\times\vec{\xi})+Q_{\psi}\chi=0\ .$ (3.33)

Note that $\vec{\xi}$ is related to $O_{0}$ by

$\vec{\xi}=-{\textstyle\frac{1}{2}}\epsilon^{mnp}\vec{\psi}_{m}\psi_{n}\cdot{\textstyle\frac{\partial}{\partial\phi^{p}}}O_{0}\
.$ (3.34)

So the first equation can be written as the following equations of $O_{0}$:

$4(\vec{\phi}^{m}\times\vec{\psi}_{m})O_{0}=\left[(\vec{\phi}^{a}\times\vec{\phi}^{b})\psi_{a}\cdot{\textstyle\frac{\partial}{\partial\phi^{b}}}+\vec{\psi}_{a}(\phi^{a}\times\phi^{b})\cdot{\textstyle\frac{\partial}{\partial\phi^{b}}}-\vec{\psi}_{a}(\psi_{b}\times\phi^{a})\cdot{\textstyle\frac{\partial}{\partial\psi_{b}}}+\vec{\psi}_{b}(\psi_{a}\times\phi^{a})\cdot{\textstyle\frac{\partial}{\partial\psi_{b}}}\right]O_{0}\
.$ (3.35)

This is a property of $O_{0}$. The second/third terms cancel due to
$\left(\phi^{b}\times\frac{\partial}{\partial\phi^{b}}+\psi_{b}\times\frac{\partial}{\partial\psi_{b}}\right)O_{0}=0$,
which holds because it is the $SU(2)$ gauge transformation on a gauge
invariant operator $O_{0}$. One can further simplify (3.35) using various
properties of $O_{0}$. Obvious ones are

$\displaystyle{\textstyle\phi^{m}\cdot\frac{\partial}{\partial\phi^{m}}O_{0}=n_{\rm
B}O_{0}\ \ ,\ \ \psi_{m}\cdot\frac{\partial}{\partial\psi_{m}}O_{0}=n_{\rm
F}O_{0}}\ \ \ (n_{\rm B},n_{\rm F})=(2,5)$
$\displaystyle\vec{\varepsilon}_{a}^{\
b}\left[{\textstyle\phi^{a}\cdot\frac{\partial}{\partial\phi^{b}}-\psi_{b}\cdot\frac{\partial}{\partial\psi_{a}}}\right]O_{0}=0\
\ \ (\vec{\varepsilon}_{a}^{\ a}=0)\ .$ (3.36)

The first two equations count the numbers of bosonic/fermionic fields in
$O_{0}$. The last equation is the $SU(3)$ invariance of $O_{0}$, which holds
for any $\varepsilon$. Equivalently, one obtains

${\textstyle\left[\phi^{a}\cdot\frac{\partial}{\partial\phi^{b}}-\psi_{b}\cdot\frac{\partial}{\partial\psi_{a}}\right]O_{0}=\frac{1}{3}(n_{\rm
B}-n_{\rm F})}\delta^{a}_{b}O_{0}\ .$ (3.37)

Finally, note that $\delta_{ij}$ contracts the $SU(2)$ gauge triplet indices
only between boson-fermion pairs in $O_{0}$, while fermion indices are
contracted only with $\epsilon_{ijk}$. This effectively promotes $SU(2)\sim
SO(3)$ to $SL(3)$ within $O_{0}$, where bosons/fermions transform in the
fundamental and anti-fundamental representations, respectively. This leads to
the following property:

${\textstyle\left[\phi^{a}_{i}\cdot\frac{\partial}{\partial\phi^{a}_{j}}-\psi_{a}^{j}\cdot\frac{\partial}{\partial\psi_{a}^{i}}\right]O_{0}=\frac{1}{3}(n_{\rm
B}-n_{\rm F})}\delta^{j}_{i}O_{0}\ .$ (3.38)

Using these properties, (3.35) can be written as

$(\vec{\psi}_{a}\times\vec{\phi}^{b})(\phi^{a}\cdot{\textstyle\frac{\partial}{\partial\phi^{b}}})O_{0}=({\textstyle
4-\frac{n_{\rm F}+2n_{\rm
B}}{3}})(\vec{\phi}^{m}\times\vec{\psi}_{m})O_{0}=(\vec{\phi}^{m}\times\vec{\psi}_{m})O_{0}\
.$ (3.39)

Both (3.39) and the second equation of (3.33) can be easily checked on a
computer. We have no extra analytic insights on why (3.39) this holds, except
that using complicated representation analysis of $SU(2)\times SU(3)$ should
provide the analytic proof. (We tried to simplify the equation for $O_{0}$ as
much as possible since they might provide insights on the generalization to
higher $N$’s in the future.) On the other hand, one can easily prove the
second equation of (3.33). First note that $\psi_{m}\times\xi$ is an $SU(2)$
vector involving $8$ $\psi$’s. There are nine independent operators involving
eight $\psi$’s, depending on which of the $9$ components is lacking. So it is
proportional to $\frac{\partial}{\partial\psi_{m}^{i}}\chi$. Since it has to
form a gauge-invariant by contracting with two scalars $\phi^{m}_{i}$,
$\phi^{a}_{j}$, one should be able to write
$\frac{\delta}{\delta\psi_{m}^{i}}\chi$ as an object with two $SU(3)$
antifundamental and two $SU(2)$ triplet indices by multiplying invariant
tensors. The only possible term is
$\epsilon_{man}\epsilon_{ijk}\frac{\partial}{\partial\psi_{n}^{k}}\chi$. One
can compute the proportionality constant by computing a term, e.g. at
$m=1,a=2,i=1,j=2$, finding $-\frac{1}{2}$. So one obtains

$\phi^{m}\cdot(\psi_{m}\times\xi)=-{\textstyle\frac{1}{2}\phi^{m}_{i}\phi^{a}_{j}\epsilon_{man}\epsilon_{ijk}\frac{\partial\chi}{\partial\psi_{n}^{k}}}=-{\textstyle\frac{1}{2}\epsilon_{man}(\phi^{m}\times\phi^{a})\cdot\frac{\partial\chi}{\partial\psi_{n}}}=-Q_{\psi}\chi\
,$ (3.40)

proving the second equation of (3.33).

One may wonder if $O_{1}$ is a descendant of $O_{0}$, or a lower black hole
operator times graviton operators appearing in (3.11). Since $O_{1}$ is at
$t^{36}$ order, the only possible way of getting operators at this order from
$O_{0}$ is $(Q^{m}_{+}Q^{n}_{+}O_{0})(\phi^{p}\cdot
f+\frac{1}{2}\epsilon^{pqr}\psi_{q}\cdot\psi_{r})$. However, during our
numerical construction of the cohomologies at this order, we separately
constructed the last operator which is not cohomologous to $O_{1}$. See also
the end of this subsection for an analytic proof (applicable to all $O_{n}$’s
with $n\geq 1$).

$O_{n}$ operator at $t^{24+12n}$ ($n\geq 2$)

We can use the structures of the operators $O_{0}$ and $O_{1}$ to analytically
construct an infinite tower of cohomologies $O_{n}$ accounting for (3.13).
Consider

$O_{n}\equiv(f\cdot f)^{n}O_{0}+n(f\cdot
f)^{n-1}f\cdot\xi+{\textstyle\frac{2n^{2}+n}{3}}(f\cdot f)^{n-1}\chi$ (3.41)

for $n\geq 2$. At $n=1$, this is just $O_{1}$ that we discussed above. We will
now show that these are new black hole like cohomologies at $t^{24+12n}$
order. It is again easy to show that these are not graviton type because they
are made of odd letters. It is not $Q$-exact because the last term does not
contain scalars.

Now we derive the $Q$-closedness. Its $Q$-action is given by

$\displaystyle QO_{n}$ $\displaystyle=$ $\displaystyle(f\cdot
f)^{n-1}\left[\vec{f}\cdot\left(2n(\vec{\phi}^{m}\times\vec{\psi}_{m})O_{0}+nQ_{\psi}\vec{\xi}\right)+n(\vec{\phi}^{m}\times\vec{\psi}_{m})\cdot\vec{\xi}+{\textstyle\frac{2n^{2}+n}{3}}Q_{\psi}\chi\right]$
$\displaystyle+2(n^{2}-n)(f\cdot
f)^{n-2}\vec{f}\cdot(\vec{\phi}^{m}\times\vec{\psi}_{m})(f\cdot\xi)+{\textstyle\frac{2n(n-1)(2n+1)}{3}}(f\cdot
f)^{n-2}\vec{f}\cdot(\vec{\phi}^{m}\times\vec{\psi}_{m})\chi\ .$

The first two terms on the first line cancel due to the first equation of
(3.33). The last term on the second line is zero because it includes $10$
fermions. Inserting the second equation of (3.33) to the last term on the
first line, one obtains

$QO_{n}={\textstyle\frac{2(n^{2}-n)}{3}}(f\cdot f)^{n-2}\left[-(f\cdot
f)(\phi^{m}\times\psi_{m})\cdot\xi+3(f\times\phi^{m})\cdot\psi_{m}(f\cdot\xi)\right]\
.$ (3.43)

The second term contains $8$ fermions, where the fermions carry $ma$ indices
for $SU(3)$ and three $SU(2)$ triplet indices to be contracted with
$(f\times\phi^{m})_{k}$, $f_{i}$, $\phi^{a}_{j}$. From the contraction
structures of $\xi$, one finds that $b_{1},c_{1}$ are antisymmetric so the
corresponding $i,j$ indices should be symmetric. The only possible $8$-fermion
terms satisfying these conditions are

$\epsilon_{man}\delta_{ij}\frac{\partial}{\partial\psi_{n}^{k}}\chi\ \ ,\ \
\epsilon_{man}\delta_{k(i}\frac{\partial}{\partial\psi_{n}^{j)}}\chi\ .$
(3.44)

Explicitly computing two components in the second term of (3.43), one finds
that the linear combination is

$\epsilon_{man}\left[\delta_{ij}{\textstyle\frac{\partial}{\partial\psi_{n}^{k}}}-{\textstyle\delta_{k(i}\frac{\partial}{\partial\psi_{n}^{j)}}}\right]\chi\
.$ (3.45)

Contracting this with $f_{i}$, $\phi^{a}_{j}$, $(f\times\phi^{m})_{k}$, one
obtains

$\displaystyle\epsilon_{man}\left[(f\cdot\phi^{a})(f\times\phi^{m})\cdot{\textstyle\frac{\partial}{\partial\psi_{n}}}-{\textstyle\frac{1}{2}}[(f\times\phi^{m})\cdot\phi^{a}]f\cdot{\textstyle\frac{\partial}{\partial\psi_{n}}}\right]\chi$
(3.46)
$\displaystyle={\textstyle\frac{1}{2}}\epsilon_{man}\left[\left[f\times(f\times(\phi^{m}\times\phi^{a}))\right]\cdot{\textstyle\frac{\partial}{\partial\psi_{n}}}-[(f\times\phi^{m})\cdot\phi^{a}]f\cdot{\textstyle\frac{\partial}{\partial\psi_{n}}}\right]\chi$
$\displaystyle=-{\textstyle\frac{1}{2}\epsilon_{man}(f\cdot
f)(\phi^{m}\times\phi^{a})\cdot{\textstyle\frac{\partial}{\partial\psi_{n}}}}\chi=-(f\cdot
f)Q_{\psi}\chi=(f\cdot f)(\phi^{m}\times\psi_{m})\cdot\xi\ .$

So the second term of (3.43) cancels the first term, ensuring that $O_{n}$ is
$Q$-closed. So we have shown that the operator

$\displaystyle O_{n}$ $\displaystyle=$ $\displaystyle(f\cdot
f)^{n}\epsilon^{c_{1}c_{2}c_{3}}(\phi^{a}\cdot\psi_{c_{1}})(\phi^{b}\cdot\psi_{c_{2}})(\psi_{a}\cdot\psi_{b}\times\psi_{c_{3}})$
$\displaystyle+n(f\cdot
f)^{n-1}\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}(f\cdot\psi_{b_{1}})(\phi^{a}\cdot\psi_{c_{1}})(\psi_{b_{2}}\cdot\psi_{c_{2}})(\psi_{a}\cdot\psi_{b_{3}}\times\psi_{c_{3}})$
$\displaystyle-{(\textstyle\frac{n}{72}+\frac{n^{2}-n}{108})}(f\cdot
f)^{n-1}\epsilon^{a_{1}a_{2}a_{3}}\epsilon^{b_{1}b_{2}b_{3}}\epsilon^{c_{1}c_{2}c_{3}}(\psi_{a_{1}}\cdot\psi_{b_{1}}\times\psi_{c_{1}})(\psi_{a_{2}}\cdot\psi_{b_{2}}\times\psi_{c_{2}})(\psi_{a_{3}}\cdot\psi_{b_{3}}\times\psi_{c_{3}})$

at $t^{24+12n}$ order is a black hole cohomology.

One may wonder if these are primaries captured in the first factor of (3.11),
or if they are related to other $O_{n^{\prime}}$ with $n^{\prime}<n$ by acting
some $Q^{m}_{+}$’s and/or gravitons on the third factor. One can show that the
latter possibilities are all impossible. Suppose $O_{n}$ is obtained by acting
acting $p$ $Q$’s on $O_{n^{\prime}}$ and multiplying $q$ gravitons. Then $p,q$
should satisfy

$2p+8q=12(n-n^{\prime})\ ,\ \ p=0,1,2,3\ ,\ q\geq 0\ .$ (3.48)

Possible solutions are

$(p,q,n-n^{\prime})=(2,1,1)\ ,\ (0,3,2)\ ,\ (2,4,3)\ ,\ (0,6,4)\ ,\ (2,7,5)\
,\ (0,9,6)\ ,\ \cdots\ .$ (3.49)

The cases with even $n-n^{\prime}$ and $p=0$ yield operators at $t^{24+12n}$
order obtained by multiplying $O_{n^{\prime}}$ and $\frac{3}{2}(n-n^{\prime})$
graviton operators of the form $\phi^{m}\cdot
f+\frac{1}{2}\epsilon^{mnp}\psi_{n}\psi_{p}$. However, these cannot be
cohomologous to $O_{n}$ because they do not have a term at
$\mathcal{O}(f^{2n-2}\phi^{0}\psi^{9})$ order that $O_{n}$ has, which cannot
be changed by adding $Q$-exact terms. Now we consider the cases with odd
$n-n^{\prime}$ and $p=2$, $q=\frac{3}{2}(n-n^{\prime})-\frac{1}{2}$, and again
consider whether the operator $(Q^{a}_{+}Q^{b}_{+}O_{n^{\prime}})(\phi\cdot
f+\psi\cdot\psi)^{q}$ has a term at $f^{2n-2}\phi^{0}\psi^{9}$ order. Let us
first study how the actions of $Q^{a}_{+}$ and $Q^{b}_{+}$ on $O_{n^{\prime}}$
can produce a term with no scalars. $Q^{a}_{+}$ either act as
$\phi\rightarrow\psi$ or $\psi\rightarrow f$, so there are following
possibilities:

$f^{2n^{\prime}}\phi^{2}\psi^{5}\rightarrow f^{2n^{\prime}}\psi^{7}\ ,\
f^{2n^{\prime}-1}\phi\psi^{7}\rightarrow f^{2n^{\prime}}\psi^{7}\ ,\ \
f^{2n^{\prime}-2}\psi^{9}\rightarrow f^{2n^{\prime}}\psi^{7}\ .$ (3.50)

In all three cases, we multiply gravitons of the form $(\phi\cdot
f+\psi\cdot\psi)^{q}$ and see whether there can be a term at
$f^{2n-2}\phi^{0}\psi^{9}$ order. This is possible only if $n=n^{\prime}+1$,
$p=2$, $q=1$. That is, the only possible relations between different $O_{n}$’s
are

$O_{n}\stackrel{{\scriptstyle?}}{{\sim}}\epsilon_{abc}(Q^{a}_{+}Q^{b}_{+}O_{n-1})(\phi^{c}\cdot
f+{\textstyle\frac{1}{2}}\epsilon^{cde}\psi_{d}\cdot\psi_{e})\ ,$ (3.51)

where $\sim$ means up to a multiplicative factor and addition of $Q$-exact
terms. We act three $Q^{a}_{+}$’s on (3.51) and show that this equation cannot
hold. Acting $Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}$ on the right hand side yields zero,
so if this equation is true, $Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{n}$ should be
$Q$-exact. However, this cannot be the case since it contains a term at
$f^{2n+1}\psi^{6}$ order, which does not contain scalars so cannot be
$Q$-exact. More concretely, one starts from

$\displaystyle O_{n}$ $\displaystyle=$ $\displaystyle(f\cdot
f)^{n}O_{0}+\frac{20n}{3}(f\cdot
f)^{n-1}\sum_{\textrm{cyclic}}(f\cdot\psi_{3})(\psi_{3}\cdot\psi_{2})(X\cdot\psi_{2})(\psi_{1}\cdot\psi_{1}\times\psi_{1})$
(3.52) $\displaystyle-\frac{10}{3}(\frac{n}{6}+\frac{n^{2}-n}{9})(f\cdot
f)^{n-1}(\psi_{1}\cdot\psi_{1}\times\psi_{1})(\psi_{2}\cdot\psi_{2}\times\psi_{2})(\psi_{3}\cdot\psi_{3}\times\psi_{3})$

where $\sum_{\textrm{cyclic}}$ means summation over the cyclic permutations of
$(X,\psi_{1})$, $(Y,\psi_{2})$, $(Z,\psi_{3})$. Acting
$Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}$, one obtains the following terms without
scalars,

$\displaystyle Q^{+}_{1}Q^{+}_{2}Q^{+}_{3}O_{0}$ (3.53)
$\displaystyle=-10(X\cdot
f+\psi_{2}\cdot\psi_{3})^{2}f\cdot(\psi_{1}\times\psi_{1})-20(X\cdot
f+\psi_{2}\cdot\psi_{3})(Y\cdot
f+\psi_{3}\cdot\psi_{1})f\cdot(\psi_{1}\times\psi_{2})+\textrm{cyclic}$
$\displaystyle\rightarrow-20(\psi_{2}\cdot\psi_{3})^{2}(f\cdot\psi_{1}\times\psi_{1})+\textrm{cyclic}$
$\displaystyle
Q^{+}_{1}Q^{+}_{2}Q^{+}_{3}(f\cdot\psi_{3})(\psi_{3}\cdot\psi_{2})(X\cdot\psi_{2})(\psi_{1}\cdot\psi_{1}\times\psi_{1})+\textrm{cyclic}$
$\displaystyle\rightarrow-3(f\cdot
f)(\psi_{2}\cdot\psi_{3})^{2}(f\cdot\psi_{1}\times\psi_{1})+6(f\cdot\psi_{2})(f\cdot\psi_{3})(\psi_{2}\cdot\psi_{3})(f\cdot\psi_{1}\times\psi_{1})+\textrm{cyclic}$
$\displaystyle
Q^{+}_{1}Q^{+}_{2}Q^{+}_{3}(\psi_{1}\cdot\psi_{1}\times\psi_{1})(\psi_{2}\cdot\psi_{2}\times\psi_{2})(\psi_{3}\cdot\psi_{3}\times\psi_{3})=-27(f\cdot\psi_{1}\times\psi_{1})(f\cdot\psi_{2}\times\psi_{2})(f\cdot\psi_{3}\times\psi_{3})$
$\displaystyle=18((f\cdot
f)(\psi_{2}\cdot\psi_{3})^{2}(f\cdot\psi_{1}\times\psi_{1}))-2(f\cdot\psi_{2})(f\cdot\psi_{3})(\psi_{2}\cdot\psi_{3})(f\cdot\psi_{1}\times\psi_{1})+\textrm{cyclic})\
.$

These terms at $f^{2n+1}\phi^{0}\psi^{6}$ order do not cancel, implying that
$Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{0}$ cannot be $Q$-exact. So at least among the
possibilities visible in the index (3.11), we have checked that different
$O_{n}$’s are not related in trivial manners.

Note also that the product of two $O_{n}$’s vanishes, $O_{m}O_{n}=0$. This is
because each operator includes $5$ or more $\psi$’s, so the product involves
$10$ or more $\psi$’s which vanishes by Fermi statistics.

###  3.2 General sector and partial no-hair theorem

Counting the graviton cohomologies with a computer using the eigenvalue setup
explained in section 2, we obtained its index $Z_{\rm grav}$ till $t^{40}$
order. The remaining $SU(2)$ index is given by

$\displaystyle Z-Z_{\rm grav}$ $\displaystyle=$
$\displaystyle\left[-t^{24}-\chi_{(1,3)}t^{32}-(\chi_{(1,\bar{3})}+\chi_{(3,6)})t^{34}-\chi_{(2,3)}t^{35}+(\chi_{(3,1)}+\chi_{(3,8)})t^{36}\right.$
(3.54)
$\displaystyle\left.-(\chi_{(2,\bar{3})}+\chi_{(4,6)})t^{37}+\chi_{(5,3)}t^{38}+(\chi_{(2,1)}+2\chi_{(4,1)}+\chi_{(4,8)})t^{39}\right.$
$\displaystyle\left.-(2\chi_{(1,6)}+\chi_{(3,\bar{3})}+\chi_{(5,\bar{3})}+\chi_{(5,6)})t^{40}\right]\chi_{\rm
D}+\mathcal{O}(t^{41})\ .$

The $SU(2)_{R}\times SU(3)$ characters and the factor $\chi_{D}$ are given by

$\displaystyle\chi_{(2J^{\prime}+1,{\bf{\rm R}})}\\!$
$\displaystyle\\!\equiv\\!$
$\displaystyle\\!\chi^{SU(2)_{R}}_{J^{\prime}}(p)\chi^{SU(3)}_{\bf\rm R}(x,y)$
(3.55) $\displaystyle\chi_{D}\\!$ $\displaystyle\\!\equiv\\!$
$\displaystyle\\!\frac{(1-\\!t^{2}z_{1})(1-\frac{t^{2}}{z_{2}})(1-\\!\frac{t^{2}z_{2}}{z_{1}})\cdot(1-\frac{tp}{z_{1}})(1-\\!\frac{t}{pz_{1}})(1-tz_{2}p)(1-\frac{tz_{2}}{p})(1-\frac{tz_{1}p}{z_{2}})(1-\frac{tz_{1}}{z_{2}p})}{(1-t^{3}p)(1-\frac{t^{3}}{p})}$

where
$t^{6}=e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}=e^{-\omega_{1}-\omega_{2}}$,
$z_{1}=e^{\frac{-2\Delta_{1}+\Delta_{2}+\Delta_{3}}{3}}$,
$z_{2}^{-1}=e^{\frac{\Delta_{1}-2\Delta_{2}+\Delta_{3}}{3}}$,
$p=e^{\frac{-\omega_{1}+\omega_{2}}{2}}$. The function $\chi_{D}$ is factored
out for later convenience (but we do not necessarily assume that only a
particular type of representation appears).

Since descendant operators of $PSU(1,2|3)$ are not really new, our interest is
the new possible $\frac{1}{16}$-BPS primaries contained in (3.54).
Constructing all the cohomologies order by order as done in [18] will clarify
this in principle. However, we shall not comprehensively do this job in this
paper. Rather, we shall study the possible superconformal representation
structures of the $\frac{1}{16}$-BPS states compatible with this index,
finding many illuminating structures. As emphasized, we may miss some BPS
states in case their multiplets completely cancel in the index.

The index can be written as a sum over the short $\mathcal{N}=4$
representations. Equivalently, it can be written as a sum over
$\frac{1}{16}$-BPS multiplets of $PSU(1,2|3)\subset PSU(2,2|4)$. The last
multiplets are embedded in the short representations of $PSU(2,2|4)$ in
canonical manners: see appendix B. Knowing this representation sum is knowing
the primary contents. We will study this expansion order by order in $t$. As
already mentioned in section 3.1 (and in the introduction), there are two
classes of black hole cohomologies: those which can be written as products of
other black hole cohomologies and gravitons which we call ‘hairy’ and the rest
which we call ‘core.’

We start by studying the black hole cohomologies that we identified in section
3.1. Among these, two of them $O_{0}$, $O_{1}$ appear within the $t^{40}$
order. We can show that all $O_{n}$’s are core black hole primaries of the
$\frac{1}{16}$-BPS multiplets. The coreness of $O_{n}$ is already shown in
section 3.1, at least within the states visible in the index (3.11), since it
suffices to show this within the BMN sector. We only need to show that they
are $\frac{1}{16}$-BPS primaries in their full $PSU(1,2|3)$ representations.
$O_{0}$ is clearly a $\frac{1}{16}$-BPS primary since it is the lowest black
hole cohomology. Since $j\equiv J_{1}+J_{2}=5$ is too large, $O_{0}$ can only
belong to the $\mathcal{N}=4$ multiplet
$A_{1}\overline{L}[4;0]^{[2,0,0]}_{9}$. The primary $O_{0}$ of the
$\frac{1}{16}$-BPS multiplet is obtained by acting $Q^{\prime}\equiv
Q^{4}_{+}$ on a primary of this $\mathcal{N}=4$ multiplet. The index over this
multiplet is

$\chi_{24}\equiv-t^{24}\chi_{D}(t,x,y,p)\ ,$ (3.56)

where $\chi_{D}$ is defined in (3.55) and in appendix B. So the first term
$-t^{24}$ in the square bracket of (3.54) corresponds to the contribution of
this multiplet. Next we consider other $O_{n}$’s. We can prove that they are
also primaries by showing that acting any of the nine $Q$’s in $PSU(1,2|3)$
yields nontrivial and independent cohomologies. (This is because $O_{n}$ does
not contain derivatives and cannot be a conformal descendant.) We have shown
in section 3.1 that the action of any $Q^{m}_{+}$ on $O_{n}$ is nontrivial and
independent because acting all three of them yields a nontrivial cohomology.
One can also show that $\overline{Q}_{m\dot{\alpha}}O_{n}$ are all nontrivial
and independent. It suffices to show that the six
$\overline{Q}_{m\dot{\alpha}}$’s acting on $Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{n}$
are independent. This is easily shown by studying the terms obtained by acting
$\overline{Q}_{m\dot{\alpha}}$ on the $\mathcal{O}(f^{2n+1}\phi^{0}\psi^{6})$
order terms of $Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{n}$ in (3.53). In particular,
one obtains terms at $f^{2n}\phi^{0}\psi^{6}D\psi$ by acting
$\overline{Q}_{m\dot{\alpha}}$ on $f$. These terms cannot be $Q$-exact since
it involves neither $\bar{\phi}^{m}$ or $\bar{\lambda}_{\dot{\alpha}}$. This
proves that all $6$ operators
$\overline{Q}_{m\dot{\alpha}}Q^{1}_{+}Q^{2}_{+}Q^{3}_{+}O_{n}$ are nontrivial.
They are also independent since their $SU(2)_{R}\times SU(3)$ quantum numbers
are different. This shows that $O_{n\geq 1}$ are $\frac{1}{16}$-BPS primaries.
$O_{n}$ belongs to the $\mathcal{N}=4$ multiplet
$A_{1}\overline{L}[4+4n;0]^{[2,0,0]}_{9+4n}$, which contributes to to the
index as $-t^{24+12n}\chi_{D}(t,x,y,p)$.

Now with the nature of $O_{n}$ understood, we come back to study the series
(3.54) till $t^{40}$ order, trying to better characterize other cohomologies
order by order in $t$. Once the lowest operator $O_{0}$ is identified, all the
states in its $\frac{1}{16}$-BPS multiplet are not really new operators. So we
subtract $\chi_{24}$ from $Z-Z_{\rm grav}$ and see what are left:

$\displaystyle Z-Z_{\rm gra}-\chi_{24}$ $\displaystyle=$
$\displaystyle\left[-\chi_{(1,3)}t^{32}-(\chi_{(1,\bar{3})}+\chi_{(3,6)})t^{34}-\chi_{(2,3)}t^{35}+(\chi_{(3,1)}+\chi_{(3,8)})t^{36}\right.$
(3.57)
$\displaystyle-(\chi_{(2,\bar{3})}+\chi_{(4,6)})t^{37}+\chi_{(5,3)}t^{38}+(\chi_{(2,1)}+2\chi_{(4,1)}+\chi_{(4,8)})t^{39}$
$\displaystyle\left.-(2\chi_{(1,6)}+\chi_{(3,\bar{3})}+\chi_{(5,\bar{3})}+\chi_{(5,6)})t^{40}\right]\chi_{D}+\mathcal{O}(t^{41})\
.$

Somewhat surprisingly, after subtracting the multiplet of $O_{0}$, one finds
that the remaining index starts from $t^{32}$ order. Namely, in the range
$t^{25}\sim t^{31}$, the index does not capture any new black hole
cohomologies except the trivial descendants of $O_{0}$. At first sight this
may look like a boring result, but the triviality of the index in this range
has a nontrivial implication.

Recall that cohomologies multiply to yield new cohomologies. This is because
of the Leibniz rule of the classical $Q$ acting on product operators. So
apparently, one can multiply light graviton cohomologies to $O_{0}$ or its
descendants to obtain many new cohomologies in the range $t^{25}\sim t^{31}$.
The possible product cohomologies of $O_{0}$ and gravitons below $t^{32}$
order are

$\displaystyle O_{0}(\bar{\phi}^{(m}\cdot\bar{\phi}^{n)})\ ,\ \
O_{0}(\bar{\phi}^{m}\cdot\bar{\lambda}_{\dot{\alpha}})\ ,\ \
O_{0}(\bar{\lambda}_{\dot{+}}\cdot\bar{\lambda}_{\dot{-}})\ ,$ $\displaystyle
O_{0}(\bar{\phi}^{m}\cdot\psi_{n+}-{\textstyle\frac{1}{3}}\delta^{m}_{n}\bar{\phi}^{p}\cdot\psi_{p+})\
,\ \
O_{0}(\bar{\lambda}_{\dot{\alpha}}\cdot\psi_{m+}-{\textstyle\frac{1}{2}}\epsilon_{mnp}\phi^{n}\cdot
D_{+\dot{\alpha}}\phi^{p})\ ,$ $\displaystyle
O_{0}\partial_{+\dot{\alpha}}(\bar{\phi}^{(m}\cdot\bar{\phi}^{n)})\ .$ (3.58)

Other possible products below $t^{32}$ involving the descendants of $O_{0}$
are

$\displaystyle\overline{Q}O_{0}$ $\displaystyle\times$
$\displaystyle(\bar{\phi}^{m}\cdot\bar{\phi}^{n}\ ,\
\bar{\phi}^{m}\cdot\bar{\lambda}_{\dot{\alpha}}\ ,\
\bar{\lambda}_{\dot{+}}\cdot\bar{\lambda}_{\dot{-}}\ ,\
\bar{\phi}^{m}\cdot\psi_{n+}-{\textstyle\frac{1}{3}}\delta^{m}_{n}\bar{\phi}^{p}\cdot\psi_{p+})\
,$ $\displaystyle(Q,\overline{Q}\overline{Q})O_{0}$ $\displaystyle\times$
$\displaystyle(\bar{\phi}^{m}\cdot\bar{\phi}^{n}\ ,\
\bar{\phi}^{m}\cdot\bar{\lambda}_{\dot{\alpha}})\ ,$
$\displaystyle(Q\overline{Q},\overline{Q}\overline{Q}\overline{Q},\partial)O_{0}$
$\displaystyle\times$ $\displaystyle(\bar{\phi}^{m}\cdot\bar{\phi}^{n})\ .$
(3.59)

The triviality of the index (3.57) in this range implies two possibilities for
these product cohomologies. The first possibility is that these product
cohomologies are $Q$-exact, i.e. absent in the BPS spectrum. Another
possibility is that these product cohomologies are nontrivial but there are
cancellations in the index, either among themselves or with new core black
hole cohomologies.666We have checked that cancellations cannot happen within
the product cohomologies listed above. It is logically possible (although a
bit unnatural) that some new core black hole primaries appear in this range,
precisely canceling with some of the product operators above if they are not
$Q$-exact. Although in different contexts, certain black holes are known not
to appear in the index. For instance, asymptotically flat multi-center BPS
black holes or BPS black rings are not captured by the index [46]. Among (3.2)
and (3.2), we have explicitly shown after very nontrivial numerical/analytic
calculations that

$O_{0}(\bar{\phi}^{m}\cdot\bar{\phi}^{n})\ ,\
O_{0}(\bar{\phi}^{m}\cdot\bar{\lambda}_{\dot{\alpha}})\ ,\
O_{0}(\bar{\phi}^{m}\cdot\psi_{n+}-{\textstyle\frac{1}{3}}\delta^{m}_{n}\bar{\phi}^{p}\cdot\psi_{p+})$
(3.60)

are all $Q$-exact. See Appendix D for some details. We did not manage to prove
the $Q$-exactness of other operators. Since these operators do not appear at
all in the index, all of them may be $Q$-exact till $t^{31}$ order. More
robustly/modestly, we can say that our index exhibits a no-hair behavior for
$O_{0}$ till $t^{31}$ order. It will be interesting to clarify this issue in
the future.

The $Q$-exactness of these product operators implies that $O_{0}$ abhors the
dressings by certain gravitons, reminiscent of the black hole no-hair theorem.
Especially, $(\bar{\phi}^{m}\cdot\bar{\phi}^{n})\sim{\rm
tr}(\bar{\phi}^{m}\bar{\phi}^{n})$ multiplied to $O_{0}$ are $Q$-exact. This
is interesting because these operators correspond to bulk scalar fields which
have been discussed in the context of hairy AdS5 black holes [30, 31, 32].
More precisely, it is the ‘s-wave’ modes of these scalars that have been used
to construct hairy black holes, precisely dual to the conformal primary
operator ${\rm tr}(\bar{\phi}^{m}\bar{\phi}^{n})$. Here, note that the BPS
limits of the hairy black holes constructed this way all exhibit substantial
back reactions to the core black holes, at least near the horizon, no matter
how small the hair parameter is [31, 32]. In section 4, we shall revisit these
aspects in the gravity dual and more carefully discuss the possible forms of
hairy BPS black holes.

Now we consider the lowest term $-\chi_{(1,3)}t^{32}$ of (3.57). In fact, this
term comes from the following product of $O_{0}$ and gravitons:

$O_{0}(\bar{\phi}^{m}\cdot
f_{++}+{\textstyle\frac{1}{2}}\epsilon^{mnp}\psi_{n+}\cdot\psi_{p+})\ .$
(3.61)

It is easy to show that this is not $Q$-exact, e.g. by acting two $Q^{m}_{+}$
as shown in (3.29). These operators contain terms at $f^{0}\phi^{0}\psi^{9}$
order, which cannot be $Q$-exact. So the operators (3.61) themselves are not
$Q$-exact either. Therefore, the no-hair interpretation that we made so far
holds only for certain low-lying gravitons, at best. See section 4 for more
detailed discussions on examples of ‘allowed’ hairs. Among the conformal
primaries of $S_{2}$ listed in Appendix A, these three gravitons are the only
ones which explicitly appear in the index when multiplied to $O_{0}$. At this
stage, it may seem that two more gravitons
$f_{++}\cdot\bar{\lambda}_{\dot{\alpha}}+{\textstyle\frac{2}{3}}\psi_{m+}\cdot
D_{+\dot{\alpha}}\bar{\phi}^{m}-{\textstyle\frac{1}{3}}\bar{\phi}^{m}\cdot
D_{+\dot{\alpha}}\psi_{m+}$ at $\mathcal{O}(t^{9})$ might multiply $O_{0}$ to
show up at $t^{33}$ order, but we will see below that the index does not
capture them. Therefore, out of the $32$ particle species of conformal primary
particles in the $S_{2}$ multiplet, $29$ gravitons except $\bar{\phi}^{m}\cdot
f_{++}+{\textstyle\frac{1}{2}}\epsilon^{mnp}\psi_{n+}\cdot\psi_{p+}$ do not
appear in the index when they multiply $O_{0}$. In the BMN sector, our studies
in section 3.1 imply a similar theorem for all $O_{n}$, at least as seen by
the index. Among the $17$ particle species of gravitons in the BMN sector, all
$14$ particles except $\bar{\phi}^{m}\cdot
f_{++}+{\textstyle\frac{1}{2}}\epsilon^{mnp}\psi_{n+}\cdot\psi_{p+}$ do not
appear in the index when they multiply any $O_{n}$.

The $3$ product coholomologies at $t^{32}$ order violating the no-hair theorem
should be the primaries of $PSU(1,2|3)$. This is again contained in a short
multiplet of $A_{1}\overline{L}$ type, whose contribution to the index is
given by $\chi_{32}=t^{32}\chi_{(1,3)}\chi_{D}$. We subtract this from
$Z-Z_{\rm grav}-\chi_{24}$, and study the remaining cohomologies. We can then
try to interpret the lowest order term of the remainder and judge whether it
comes from new core black hole primaries or products of already known core
primaries and gravitons. If one can clarify the nature of the cohomologies at
this lowest order, one can again subtract the characters of their
supermultiplets and keep exploring even higher orders. Since it becomes more
and more difficult to judge the $Q$-exactness of the possible product
operators, we shall only make much simpler and structural studies till the
$t^{40}$ order. Namely, we shall try to see if the surviving index can be
explained as the products of known gravitons and core primaries $O_{n}$,
without the need of any new core black hole primaries. Studies we made so far
showed that this is possible till $t^{32}$ order. Namely, the index till this
order is compatible with having no more new core primaries and only three more
product cohomologies (3.61). We shall show that the graviton spectrum is such
that new core black hole primaries should appear at $t^{39}$ order at the
latest. This not only proves from the index the existence of new core black
hole primaries, but will also show scenarios of possible hairy black holes (to
be further discussed in section 4).

After eliminating the contribution of the multiplet $\chi_{32}$ to the index,
the remaining index vanishes at $t^{33}$ order. In principle, there are two
possible product operators which completely cancel in the index if they are
not $Q$-exact, given by

$O_{0}\partial_{+\dot{\alpha}}(\bar{\lambda}_{\dot{\beta}}\cdot\bar{\lambda}^{\dot{\beta}})\
\ ,\ \
O_{0}\left(f_{++}\cdot\bar{\lambda}_{\dot{\alpha}}+{\textstyle\frac{2}{3}}\psi_{m+}\cdot
D_{+\dot{\alpha}}\bar{\phi}^{m}-{\textstyle\frac{1}{3}}\bar{\phi}^{m}\cdot
D_{+\dot{\alpha}}\psi_{m+}\right)\ .$ (3.62)

So these product operators, even if they exist, do not appear in the index.
The lowest nonzero term of $Z-Z_{\rm grav}-\chi_{24}-\chi_{32}$ is
$-(\chi_{(1,\bar{3})}+\chi_{(3,6)})t^{34}$. The only possible product
operators which may account for this term, if they are not $Q$-exact, are

$O_{0}{\partial_{+}}^{\dot{\alpha}}\left(\bar{\lambda}_{\dot{\alpha}}\cdot\psi_{m+}-\textstyle{\frac{1}{2}}\epsilon_{mnp}\bar{\phi}^{n}\cdot
D_{+\dot{\alpha}}\bar{\phi}^{p}\right)\ \ ,\ \
O_{0}\partial_{+\dot{\alpha}}\partial_{+{\dot{\beta}}}(\bar{\phi}^{m}\cdot\bar{\phi}^{n})\
.$ (3.63)

If they are nontrivial, they are in the $\mathcal{N}=4$ representations
$A_{1}\overline{L}[6;2]^{[2,2,0]}_{13}$ and
$A_{1}\overline{L}[6;0]^{[3,0,1]}_{13}$ representations, respectively.
Assuming that they are both non-$Q$-exact, this order can be accounted for by
hairy black hole operators. Their multiplets will contribute
$-(\chi_{(1,\bar{3})}+\chi_{(3,6)})t^{34}\chi_{D}$ to the index. Subtracting
them, the leading term is $-\chi_{(2,3)}t^{35}$. The only possible product
cohomologies which can account for this term are

$O_{0}\partial_{+\dot{\alpha}}(f_{++}\cdot\bar{\phi}^{m}+{\textstyle\frac{1}{2}}\epsilon^{mnp}\psi_{n+}\cdot\psi_{p+})\
,$ (3.64)

if they are not $Q$-exact. In this case, its multiplet is again
$A_{1}\overline{L}$ type and contributes $-\chi_{(2,3)}t^{35}\chi_{D}$ to the
index. Subtracting this, the lowest term is
$+(\chi_{(3,1)}+\chi_{(3,8)})t^{36}$. Since there is one fermionic black hole
primary $O_{1}$, we study whether the product cohomologies may account for
$+(1+\chi_{(3,1)}+\chi_{(3,8)})t^{36}$. The only possible set is

$O_{0}\partial_{+\dot{\alpha}}(f_{++}\cdot\bar{\lambda}_{\dot{\beta}}+{\textstyle\frac{2}{3}}\psi_{m+}\cdot
D_{+\dot{\beta}}\bar{\phi}^{m}-{\textstyle\frac{1}{3}}\bar{\phi}^{m}\cdot
D_{+\dot{\beta}}\psi_{m+})\ \ ,\ \
O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}(\bar{\phi}^{m}\cdot\psi_{n+}-{\textstyle\frac{1}{3}}\delta^{m}_{n}\bar{\phi}^{p}\cdot\psi_{p+})\
.$ (3.65)

Subtracting the contributions of these multiplets if not $Q$-exact, again in
the $A_{1}\overline{L}$ multiplets, the lowest term is
$-(\chi_{(2,\bar{3})}+\chi_{(4,6)})t^{37}$. The only possible product
cohomologies that can account for this term are

$O_{0}\partial_{+\dot{\alpha}}\partial_{+}^{\ \
\dot{\beta}}(\bar{\lambda}_{\dot{\beta}}\cdot\psi_{m+}-\textstyle{\frac{1}{2}}\epsilon_{mnp}\bar{\phi}^{n}\cdot
D_{+\dot{\beta}}\bar{\phi}^{p})\ \ ,\ \
O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}\partial_{+\dot{\gamma}}(\bar{\phi}^{m}\cdot\bar{\phi}^{n})\
.$ (3.66)

Further processing to subtract the contributions of their multiplets, the
lowest term is $+\chi_{(5,3)}t^{38}$. one possible set of product cohomologies
which can account for this is

$O_{0}\partial_{+(\dot{\alpha}}\partial_{+\dot{\beta}}\partial_{+\dot{\gamma}}(\bar{\lambda}_{\dot{\delta})}\cdot\bar{\phi}^{m})\
.$ (3.67)

Apart from these, the following two sets of product cohomologies

$O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}{\partial_{+}}^{\dot{\gamma}}(\bar{\lambda}_{\dot{\gamma}}\cdot\bar{\phi}^{m})\
\ ,\ \
O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}(f_{++}\cdot\bar{\phi}^{m}+{\textstyle\frac{1}{2}}\epsilon^{mnp}\psi_{n+}\cdot\psi_{p+})$
(3.68)

exactly cancel in the index, so there are two possible ways in which product
hairy cohomologies can account for this order. In either case, they are all in
the $A_{1}\overline{L}$ type multiplets.

Subtracting the last multiplets, the lowest term is
$+(\chi_{(2,1)}+2\chi_{(4,1)}+\chi_{(4,8)})t^{39}$. All possible product
cohomologies at this order are

$\displaystyle({\bf 4},{\bf 1})^{\textrm{F}}$ $\displaystyle:$ $\displaystyle
O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}\partial_{+\dot{\gamma}}(\bar{\lambda}_{\dot{\delta}}\cdot\bar{\lambda}^{\dot{\delta}})\
,$ (3.69) $\displaystyle({\bf 2},{\bf 1})^{\textrm{B}}$ $\displaystyle:$
$\displaystyle O_{0}\partial_{+\dot{\alpha}}\partial_{+}^{\ \
\dot{\beta}}(f_{++}\cdot\bar{\lambda}_{\dot{\beta}}+{\textstyle\frac{2}{3}}\psi_{m+}\cdot
D_{+\dot{\beta}}\bar{\phi}^{m}-{\textstyle\frac{1}{3}}\bar{\phi}^{m}\cdot
D_{+\dot{\beta}}\psi_{m+})\ ,$ $\displaystyle({\bf 4},{\bf 1})^{\textrm{B}}$
$\displaystyle:$ $\displaystyle
O_{0}\partial_{+(\dot{\alpha}}\partial_{+\dot{\beta}}(f_{++}\cdot\bar{\lambda}_{\dot{\gamma})}+{\textstyle\frac{2}{3}}\psi_{m+}\cdot
D_{+\dot{\gamma})}\bar{\phi}^{m}-{\textstyle\frac{1}{3}}\bar{\phi}^{m}\cdot
D_{+\dot{\gamma})}\bar{\psi}_{m})\ ,$ $\displaystyle({\bf 4},{\bf
8})^{\textrm{B}}$ $\displaystyle:$ $\displaystyle
O_{0}\partial_{+\dot{\alpha}}\partial_{+\dot{\beta}}\partial_{+\dot{\gamma}}(\bar{\phi}^{m}\cdot\psi_{n+}-{\textstyle\frac{1}{3}}\delta^{m}_{n}\bar{\phi}^{p}\cdot\psi_{p+})\
.$

We have shown the superscripts B/F for their bosonic/fermionic statistics,
respectively. With these candidates, we find that the closest one can get to
the index at this order is the case in which all three classes of bosonic
operators are nontrivial while the fermionic operators are $Q$-exact. In this
case, their contribution at this order is maximal and becomes
$+(\chi_{(2,1)}+\chi_{(4,1)}+\chi_{(4,8)})t^{39}$. Therefore, there should be
at least $4$ core black hole primaries to account for the remaining
$+\chi_{(4,1)}t^{39}$. Of course this is the latest order in which new core
black hole primaries should appear, and it may well appear at the lowest
orders by some non-$Q$-exactness assumptions we made for product cohomologies
being invalid.

So we have shown that, from the index data till $t^{40}$ order, there should
exist more core primary operators except $O_{n}$ in the BMN sector. This
conclusion is obtained by supposing otherwise, and trying to explain the index
as product cohomologies of $O_{n}$ and gravitons but finding a contradiction
at $t^{39}$. We should also emphasize that the structure of the index admits
natural explanations in terms of hairy product operators in a wide range
$t^{33}\sim t^{38}$. Note also that most of the gravitons appearing in this
range are conformal descendants in the $S_{2}$ multiplet. We shall find an
interpretation of this in the next section.

##  4 Comments on hairy black holes

In section 3.2, we explored possible hairy black hole states in the $SU(2)$
theory. To better address this concept, we should understand the ways in which
hairy BPS black holes can form in AdS. In this section, we study scalar hairs
of the BPS black holes in both wave and probe particle perspectives and
present a qualitative picture in the bulk gravity as to which scalar hairs may
dress the BPS black holes.

The simplest form of hairy black holes has been explored in [29, 30], called
the ‘non-interacting mix’ of small black holes and dilute graviton hairs. The
idea is that the back-reaction of the black hole to the hair is weak when the
black hole size $q$ is small, and vice versa when the hair density
$\varepsilon$ is low. In the limit of small $q$ and $\varepsilon$, the leading
hairy black hole solution is a superposition of the non-hairy black hole and
the graviton wavefunction in vacuum AdS. Systematic back-reactions to each
other can be computed as a series in small $q,\varepsilon$, rendering a
nonlinear superposition. Nonlinear superposition is naturally expected in the
BPS sector, even at finite $q,\varepsilon$, where the cancellation of mutual
forces is often ensured by supersymmetry. In fact, our product cohomologies
realize such superpositions of BPS states in some sense. Although the true BPS
states do not generally admit product forms, there are product representatives
of the cohomologies. Such superposition representations were crucial in [15]
for gaining a deeper understanding of multi-graviton states beyond the low-
energy limit.

Meanwhile, some numerical solutions for hairy BPS black holes are known from
[31, 32]. These solutions appear to be somewhat different from the non-
interacting mix picture of [29, 30] in the following sense. No matter how
small the hair density $\varepsilon$ is, it is shown that the hair back-reacts
substantially to the black hole near the horizon. In fact, the back-reaction
is such that a (mild) singularity is created at the horizon. On the other
hand, we would like to interpret our hairy black hole operators of section 3
as realizing the non-interacting mix picture of [29, 30], as superpositions
are realized in the cohomology setup. In this section, we will try to better
understand these two notions of hairy BPS black holes and explore a setup in
which BPS solutions along the line of [29, 30] may exist.

One no-hair theorem proved in appendix D is the absence of
$(\bar{\phi}^{m}\cdot\bar{\phi}^{n})O_{0}$. On the other hand, in [30, 31,
32], the bulk scalar dual to ${\rm tr}(\bar{\phi}^{m}\bar{\phi}^{n})$ was
discussed as a hair. We shall first review our current knowledge on hairy BPS
black holes in $AdS_{5}\times S^{5}$ with this scalar. We shall try to clarify
the relations of these results with our findings in section 3 along the way.

Attempts made in the literature were to find continuous deformations of non-
hairy black holes. The simplest non-hairy BPS black holes are those carrying
equal R-charges and angular momenta, $R\equiv R_{1}=R_{2}=R_{3}$, $J\equiv
J_{1}=J_{2}$ [47]. They carry one free parameter $q$. (This is related to the
three $\mu_{I}$ parameters of [48] by $\mu_{1}=\mu_{2}=\mu_{3}=q$.) We shall
follow the notation of [49] for later convenience. The metric and the gauge
field are given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f^{2}(dt+w\sigma_{3})^{2}+f^{-1}h_{mn}dx^{m}dx^{n}$
$\displaystyle=$
$\displaystyle-f^{2}\left(dt+w\sigma_{3}\right)^{2}+f^{-1}\left[\frac{dx^{2}}{4xh}+\frac{x}{4}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+h\sigma_{3}^{2}\right)\right]$
$\displaystyle A$ $\displaystyle=$
$\displaystyle-f(dt+w\sigma_{3})+U\sigma_{3}$ (4.1)

where the 1-forms $\sigma_{i}$ satisfying
$d\sigma_{1}=-\sigma_{2}\wedge\sigma_{3}$, etc. are given by

$\sigma_{1}=\cos\psi d\theta+\sin\psi\sin\theta d\phi\ ,\ \
\sigma_{2}=-\sin\psi d\theta+\cos\psi\sin\theta d\phi\ ,\ \
\sigma_{3}=d\psi+\cos\theta d\phi\ ,$ (4.2)

and various functions are given by

$H\equiv f^{-1}=1+\frac{q}{x}\ \ ,\ \ U=\frac{x}{2\ell}+\frac{q}{\ell}\ \ ,\ \
h=1+\frac{x+3q}{\ell^{2}}\ \ ,\ \ w=\frac{x+3q}{2\ell}+\frac{3q^{2}}{4\ell x}\
.$ (4.3)

$\ell$ is the radius of AdS5. The following 4d vierbeins

${\textstyle e^{\hat{x}}=\frac{dx}{2\sqrt{xh}}\ \ ,\ \
e^{\hat{1},\hat{2}}=\frac{\sqrt{x}}{2}\sigma_{1,2}\ \ ,\ \
e^{\hat{3}}=\frac{\sqrt{xh}}{2}\sigma_{3}\ .}$ (4.4)

for the metric $h_{mn}$ will be useful below. The boundary
$x\rightarrow\infty$ is asymptotically AdS5, whose canonical time and angle
coordinates $t^{\prime},\psi^{\prime}$ are related to those above by
$t^{\prime}=t$, $\psi^{\prime}=\psi+2t$. The unprimed coordinate system
rotates with the event horizon, located at $x=0$. The mass and charges are
given by

${\textstyle M=\frac{3R+2J}{\ell}\ \ ,\ \
R=\frac{N^{2}}{2}\left(\frac{q}{\ell^{2}}+\frac{q^{2}}{2\ell^{2}}\right)\ \ ,\
\
J=\frac{N^{2}}{2}\left(\frac{3q^{2}}{2\ell^{4}}+\frac{q^{3}}{\ell^{6}}\right)\
.}$ (4.5)

The solution has one free parameter $q$ but two distinct charges $R,J$. So
$R,J$ should satisfy a relation on the solution space, which is given by [9]

${\textstyle\mathcal{G}(R,J)\equiv
R^{3}+\frac{N^{2}}{2}J^{2}-\left(3R+\frac{N^{2}}{2}\right)\left(3R^{2}-N^{2}J\right)=0\
.}$ (4.6)

Hairy BPS black holes have been studied as a 1-parameter deformation of the
solution above. A real scalar field $\varphi$ was added, only depending on the
radial variable $x$ and preserving the $SU(2)_{R}\times U(1)\subset SO(4)$
isometry. The functions $f(x),U(x),h(x),w(x),\varphi(x)$ now satisfy coupled
ODE’s in $x$. [31, 32] solved these equations numerically first in the non-BPS
case and then reduced the mass to get close to the BPS limit. One important
feature reported in [31, 32] is that the solutions seem to exhibit a
singularity which replaces the event horizon. The limiting solution has an
extra continuous parameter apart from $q$. We would like to first see whether
BPS black holes in this setup can be understood from the non-interacting mix
picture, in the sense of superposing the solution. In particular, we set the
hair condensate $\varphi\sim\varepsilon$ to be small, keeping the black hole
size general. We shall try to keep $\varphi$ as a small perturbation around
fixed black hole background (4), trying to realize the spirit of [29, 30].
(The failure of this procedure will be related to the unique nature of the
solutions of [31, 32].)

We study the BPS equations for a real scalar $\varphi$ in the notation of
[49]. Their scalar is related to the scalar $\Phi$ of [30] by
$\Phi=2\sinh\varphi$. The functions $f(x),U(x),w(x),h(x),\varphi(x)$ satisfy
eqn.(4.7) of [49]. For small $\varphi\sim\varepsilon$, $f,U,w,h$ are given by
(4.3) at the leading $\mathcal{O}(\varepsilon^{0})$ order. At the next-to-
leading $\mathcal{O}(\varepsilon^{1})$ order, $\varphi$ should satisfy

$\varphi^{\prime}(x)=-\frac{2U(x)}{\ell xh(x)}\sinh\varphi(x)\ .$ (4.7)

At this order, we plug in the zeroth order solutions (4.3) for $U,h$. Since we
keep the hair small, $\varphi\sim\mathcal{\varepsilon}$, we take
$\sinh\varphi\approx\varphi$ and consider the linear differential equation for
$\varphi$. Normally, if we solve the general equation of motion in the AdS
black hole background, one studies the second order differential equation with
the normalizable boundary condition at the boundary $x\rightarrow\infty$ and
the infalling boundary condition at the horizon $x\rightarrow 0$. This yields
the quasinormal modes at discrete choices of frequency $\omega_{n}$. These
frequencies will typically have nonzero imaginary parts, either implying that
the modes fall into the horizon (${\rm Im}(\omega_{n})<0$) or that they
trigger superradiant instability (${\rm Im}(\omega_{n})>0$). However, in the
BPS equation, we only expect stationary modes with ${\rm Im}(\omega_{n})=0$.
(In the time coordinate chosen above, $\varphi$ will be $t$-independent and
real.) (4.7) can be integrated with the integration constant being its overall
amplitude $\varepsilon$. So one has no free parameter left to match either
normalizability at $x=\infty$ or the infalling (or in the BPS case, stationary
normalizable) condition at $x=0$. We just find either acceptable or
pathological solution.

Plugging in (4.3) to (4.7), the solution for $\varphi$ is given by

$\tanh\frac{\varphi(x)}{2}=\varepsilon
x^{-\frac{2q/\ell^{2}}{1+3q/\ell^{2}}}\left(1+\frac{3q}{\ell^{2}}+\frac{x}{\ell^{2}}\right)^{-\frac{1+q/\ell^{2}}{1+3q/\ell^{2}}}\
.$ (4.8)

The left hand side should be understood as $\approx\frac{\varphi(x)}{2}$ in
our small hair approximation. At the boundary $x\to\infty$, it behaves like
the proper normalizable mode $\sim x^{-1}$ for the scalar dual to the Yang-
Mills operator ${\rm tr}(X^{2}+Y^{2}+Z^{2})$ [30]. However, the solution
diverges at $x=0$ as

$\varphi(x)\sim\varepsilon x^{-\frac{2q/\ell^{2}}{1+3q/\ell^{2}}}\ .$ (4.9)

So our perturbative approach with small $\varphi$ breaks down. Also, for small
black hole size $q$, this solution provides the first correction to the
graviton wavefunction in vacuum AdS ($q=0$) due to the small black hole
($q\neq 0$), which again diverges. So the iteration procedure of [29, 30] for
the nonlinear superposition would not work even for small $\varepsilon$, $q$
(say, without finite temperature regularization). The exact numerical results
of [31, 32] report divergence not with the scalar but with certain tidal
forces (geometry), very different from what we find above. Since the nature of
the background black hole solution changes by turning on small $\varepsilon$,
the superposition picture of [29, 30] does not seem to apply straightforwardly
in this setup.

To improve the situation, we first show that $\varphi$ beyond the s-wave
ansatz may provide smooth solutions in the perturbative regime. To find these
solutions, we should keep a complex scalar including $\varphi$ and consider
the BPS equations given by PDE’s in AdS5. We study a small complex scalar
sitting in a hypermultiplet and identify the complex scalar which contains
$\varphi$. The general BPS equation for the scalars in the hypermultiplet is
given in [50], eqn. (3.12) and (3.27). (3.12) demands $t$-independence of the
scalar, while (3.27) is a spatial differential equation given by

$D_{m}q^{X}={(\Phi^{r})_{m}}^{n}D_{n}q^{Y}{(J^{r})_{Y}}^{X}\ .$ (4.10)

Here, $q^{X}$ is the scalars in the hypermultiplet, with $X,Y=1,\cdots,4$,
$(\Phi^{r})_{m}^{\ n}$ with $r=1,2,3$ are the three complex structures of the
base space and $(J^{r})_{X}^{\ Y}$ are the three complex structures of the
scalar target space. They satisfy

${(\Phi^{1})_{m}}^{n}{(\Phi^{2})_{n}}^{p}={(\Phi^{3})_{m}}^{p}\ ,\
{(\Phi^{2})_{m}}^{n}{(\Phi^{3})_{n}}^{p}={(\Phi^{1})_{m}}^{p}\ ,\
{(\Phi^{3})_{m}}^{n}{(\Phi^{1})_{n}}^{p}={(\Phi^{2})_{m}}^{p}\ ,\ $ (4.11)

and similar relations hold for $J^{r}$. $D_{m}$ is the covariant derivative
involving our $U(1)$ gauge field $A_{\mu}$. $\Phi^{r}$ can be taken to be

$\Phi^{1}=e^{\hat{x}}\wedge e^{\hat{2}}-e^{\hat{3}}\wedge e^{\hat{1}}\ ,\
\Phi^{2}=e^{\hat{2}}\wedge e^{\hat{3}}-e^{\hat{x}}\wedge e^{\hat{1}}\ ,\
\Phi^{3}=e^{\hat{x}}\wedge e^{\hat{3}}-e^{\hat{1}}\wedge e^{\hat{2}}$ (4.12)

on the black hole background (after some permutations of $r=1,2,3$: e.g.
$\Phi^{3}$ given above is called $J=\Phi^{2}$ in [49] and $J^{1}$ in [47]). In
this setup, introducing the following inverse-vierbeins
$E_{\hat{x}},E_{\hat{1},\hat{2},\hat{3}}$ for (4.4) on the 4d base space,

$\displaystyle{\textstyle E_{\hat{x}}=2\sqrt{xh}\frac{\partial}{\partial x}\ \
,\ \ E_{\hat{3}}=\frac{2}{\sqrt{xh}}\frac{\partial}{\partial\psi}}$ (4.13)
$\displaystyle{\textstyle
E_{\hat{1}}=\frac{2}{\sqrt{x}}(\cos\psi\frac{\partial}{\partial\theta}+\frac{\sin\psi}{\sin\theta}(\frac{\partial}{\partial\phi}-\cos\theta\frac{\partial}{\partial\psi}))\
,\ \
E_{\hat{2}}=\frac{2}{\sqrt{x}}(-\sin\psi\frac{\partial}{\partial\theta}+\frac{\cos\psi}{\sin\theta}(\frac{\partial}{\partial\phi}-\cos\theta\frac{\partial}{\partial\psi}))}\
,$

one can show

$\mathcal{E}_{i}^{m}{(\Phi^{r})_{m}}^{n}=-i{(\sigma^{r})_{i}}^{j}\mathcal{E}_{j}^{n}\
\ ,\ \ i,j=1,2$ (4.14)

with
$\mathcal{E}_{i}^{m}\equiv((E_{\hat{x}}+iE_{\hat{3}})^{m},(E_{\hat{1}}+iE_{\hat{2}})^{m})$,
where $\sigma^{r}$ are the Pauli matrices. Defining
$\tilde{\mathcal{E}}^{m}_{i}\equiv((E_{\hat{1}}-iE_{\hat{2}})^{m},-(E_{\hat{x}}-iE_{\hat{3}})^{m})$,
it also satisfies

$\tilde{\mathcal{E}}^{m}_{i}{(\Phi^{r})_{m}}^{n}=-i{(\sigma^{r})_{i}}^{j}\tilde{\mathcal{E}}_{j}^{n}\
.$ (4.15)

The $\Phi^{r}$ in (4.12) are chosen such that the right hand side is
proportional to $\sigma^{r}$. (The overall factor in front of $\sigma^{r}$
should be $-i$ for the relation (4.11) to hold.) One can similarly introduce a
complex basis in the field space. For small scalar fluctuations, we can
approximate the hypermultiplet target space to be flat. Combining four real
scalars $q^{X}$ to two complex scalars $Q^{a}=(Q,\tilde{Q}^{\ast})$, with
$a=1,2$, $q^{Y}{(J^{r})_{Y}}^{X}$ can be rewritten as
$-iQ^{b}{(\sigma^{r})_{b}}^{a}$. As we shall see below, with the gauging
$D_{\mu}Q^{a}\equiv(\partial_{\mu}-2iA_{\mu})Q^{a}$, $Q$ will become the
complex scalar which contains $\varphi$.

Now contracting the equation (4.10) with $\mathcal{E}_{i}^{m}$ or
$\tilde{\mathcal{E}}_{i}^{m}$, and changing the real index $X$ to the complex
index $a$, we study the BPS equation for $Q^{a}$. Using
${(\sigma^{r})_{i}}^{j}{(\sigma^{r})_{b}}^{a}=\frac{3}{2}\delta^{a}_{i}\delta_{b}^{j}-\frac{1}{2}{(\sigma^{s})_{i}}^{a}{(\sigma^{s})_{b}}^{j}$,
these equations reduce to

$\mathcal{E}_{i}^{m}D_{m}Q^{i}=0\ ,\ \tilde{\mathcal{E}}_{i}^{m}D_{m}Q^{i}=0\
.$ (4.16)

In terms of the component scalars, these equations are

$(E_{\hat{x}}+iE_{\hat{3}})^{m}D_{m}Q=-(E_{\hat{1}}+iE_{\hat{2}})^{m}D_{m}\tilde{Q}^{\ast}\
,\
(E_{\hat{1}}-iE_{\hat{2}})^{m}D_{m}Q=(E_{\hat{x}}-iE_{\hat{3}})^{m}D_{m}\tilde{Q}^{\ast}\
.$ (4.17)

Keeping only one complex scalar $Q$ while turning off the other $\tilde{Q}=0$,
one obtains

$(E_{\hat{x}}+iE_{\hat{3}})^{m}D_{m}Q=0\ \ ,\ \
(E_{\hat{1}}-iE_{\hat{2}})^{m}D_{m}Q=0\ .$ (4.18)

This is the BPS equation for the complex scalar $Q$ that we wish to
study.777On the other hand, turning off $Q=0$, $\tilde{Q}$ satisfies similar
equations with $D_{\mu}\tilde{Q}\equiv(\partial_{\mu}+2iA_{\mu})\tilde{Q}$.
Solving these equations following our studies in the next paragraph, one finds
that there are no normalizable solutions at $x\rightarrow\infty$. In general,
we think the normalizability of the solutions of (4.17) at
$x\rightarrow\infty$ should demand $\tilde{Q}=0$. We have checked that (4.18)
implies the following equation of motion

${\textstyle\left[\frac{1}{\sqrt{-g}}D_{\mu}(\sqrt{-g}g^{\mu\nu}D_{\nu})+\frac{4}{\ell^{2}}\right]Q=0}\
.$ (4.19)

Inserting the inverse-vierbeins, (4.18) can be written as

$\left({\textstyle xh\frac{\partial}{\partial
x}+i\frac{\partial}{\partial\psi}}\right)Q={\textstyle-\frac{2}{\ell}}UQ\ \ ,\
\
\left({\textstyle\sin\theta\frac{\partial}{\partial\theta}-i(\frac{\partial}{\partial\phi}-\cos\theta\frac{\partial}{\partial\psi})}\right)Q=0\
.$ (4.20)

We believe that $Q$ is the complex scalar of [30] appearing in their
consistent truncation. (It carries the same $U(1)$ charge as their scalar and
satisfies the same linearized equation of motion.) When $Q$ depends on $x$
only, the first equation reduces to (4.7) with $Q=2\sinh\varphi\approx
2\varphi$.

Now we study all smooth solutions of (4.20) at $\mathcal{O}(\varepsilon^{1})$.
We take

$Q(x,\theta,\phi,\psi)=\Phi(x)e^{im\psi}f(\theta,\phi)$ (4.21)

where $m$ is half-integral. The equations reduce to

$\Phi^{\prime}=\frac{m\ell^{2}-x-2q}{x(x+3q+\ell^{2})}\Phi\ \ \ ,\ \ \
z^{\ast}\frac{\partial f}{\partial z^{\ast}}=2m\frac{1-|z|^{2}}{1+|z|^{2}}f$
(4.22)

where $z\equiv\tan\frac{\theta}{2}e^{-i\phi}$. The solutions are given by

$\displaystyle\Phi(x)$ $\displaystyle=$ $\displaystyle{\textstyle\varepsilon
x^{\frac{m-2q/\ell^{2}}{1+3q/\ell^{2}}}\left(1+\frac{3q}{\ell^{2}}+\frac{x}{\ell^{2}}\right)^{-\frac{1+m+q/\ell^{2}}{1+3q/\ell^{2}}}}$
(4.23) $\displaystyle f(\theta,\phi)$ $\displaystyle=$
$\displaystyle{\textstyle\left(\frac{z^{\ast}}{(1+|z|^{2})^{2}}\right)^{\frac{m}{2}}z^{\alpha}=(\cos\frac{\theta}{2})^{\frac{3m}{2}-\alpha}(\sin\frac{\theta}{2})^{\frac{m}{2}+\alpha}}e^{i(\frac{m}{2}-\alpha)\phi}$

where the factor $z^{\alpha}$ represents the integration constant for
$z^{\ast}$ given by arbitrary functions of $z$. One should take
$\alpha=-\frac{m}{2},-\frac{m}{2}+1,\cdots,\frac{3m}{2}$ to have regular
solutions on $S^{2}(\theta,\phi)$,

$f(\theta,\phi)e^{im\psi}={\textstyle(\cos\frac{\theta}{2}e^{i\phi_{1}})^{m_{1}}(\sin\frac{\theta}{2}e^{i\phi_{2}})^{m_{2}}}\
\ ,\ \ (\psi,\phi)=\phi_{1}\pm\phi_{2}$ (4.24)

where $m_{1}+m_{2}=2m$ and $m_{1},m_{2}=0,1,2,\cdots$. The angle-dependent
part is the BPS spherical harmonics. It is just the usual ‘derivative’ factor
for the BPS graviton wavefunctions in AdS5, or in the dual QFT,
$(\partial_{+\dot{+}})^{m_{1}}(\partial_{+\dot{-}})^{m_{2}}$ acting on the
conformal primary operators. One finds the following behaviors at infinity and
on the horizon:

$\Phi(x)\sim\left\\{\begin{array}[]{ll}x^{-1}&\textrm{for
}x\rightarrow\infty\\\ x^{\frac{m-2q/\ell^{2}}{1+3q/\ell^{2}}}&\textrm{for
}x\rightarrow 0\end{array}\right.\ .$ (4.25)

We get the expected normalizable boundary condition at $x\rightarrow\infty$.
At the horizon $x\rightarrow 0$, the solution remains finite only for the
modes satisfying

$m=\frac{m_{1}+m_{2}}{2}\geq\frac{2q}{\ell^{2}}\ .$ (4.26)

This gives an intuitive understanding of the well-behaved perturbative BPS
hairs. If the scalar field ($\sim$ particle) rotates with large spherical
harmonics quantum numbers $m_{1},m_{2}\gg 1$ in vacuum AdS, its average radial
position will be far away from the center. On the other hand, the conformal
primary state with $m_{1},m_{2}=0$ prefers to be at the central region of AdS.
This can be seen from $\Phi(x)$ of (4.23) at $q=0$ (no black hole):

$\Phi(x)\rightarrow\frac{\varepsilon x^{m}}{(1+x/\ell^{2})^{m+1}}\ .$ (4.27)

The radial wavefunction $\Phi(x)$ is peaked at $x=x_{\star}\equiv m\ell^{2}$,
but the peak is not sharp. This is basically because the damping after the
peak $x_{\star}$ is slow, related to its slow asymptotic decay $\Phi\sim
x^{-1}$ for the mass $M^{2}\ell^{2}=-4$. Anyway, the average radial position
moves outward for larger $m=\frac{m_{1}+m_{2}}{2}$. Now imagine placing the
same field ($\sim$ particle) in the black hole background. The central region
of AdS is now occupied by the black hole, behind the event horizon. Therefore,
the fields ($\sim$ particles) with lower $m_{1},m_{2}$ quantum numbers will be
more likely to be swallowed by the black hole, or to back-react more strongly
to it. (4.23) shows that, below the threshold $\frac{2q}{\ell^{2}}$ for $m$,
this expectation is manifested by the exterior wavefunction exhibiting a
divergence at the event horizon. To have reasonable hairy black holes below
this threshold, the hair should give a large back-reaction to the background
black hole.

Our studies so far intrinsically used the wave picture. Let us also consider
the particle picture. Note that $\Phi(x)$ in (4.23) is never peaked around its
maximum at any choice of $m,q$, basically due to the slow fall-down $\sim
x^{-1}$ at large $x$. However, if one considers a semi-classical Kaluza-Klein
graviton with large mass and large orbital angular momentum quantum number
$m$, one can have a sharply peaked wavefunction in AdS. We shall reconsider
the perturbative hair as a particle probe in this case, assuming that it is
realized by a sharply peaked wavepacket of a field with large mass and angular
momenta. This viewpoint will provide new understandings of (4.26). For
instance, consider a spinless graviton particle dual to the chiral primary
(2.12) with $n\gg 1$. The scaling dimension $\Delta=n$ and the charge $\delta
R=\frac{n}{3}$ of this operator are dual to the following mass and charge for
the graviton:

$(M\ell)^{2}=n(n-4)\approx n^{2}\ ,\ \ \delta R={\textstyle\frac{n}{3}}\ .$
(4.28)

The bosonic probe particle action in the black hole background (4) is given by

$S={\textstyle\frac{n}{\ell}}\int
d\tau\left[-\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}-{\textstyle\frac{1}{3}}A_{\mu}\dot{x}^{\mu}\right]\
.$ (4.29)

We try to find the rotating BPS solutions for the time-like worldline with
$t(\tau)=\tau$, outside the horizon $x>0$. Denoting by $\delta J_{1},\delta
J_{2}$ the Noether charges of the particle for the rotations on
$\phi_{1},\phi_{2}$, respectively, one finds the following stationary radius
$x$ as a function of charges:

$x=\ell^{2}\left(\frac{\delta J_{1}+\delta J_{2}}{3\delta
R}-\frac{2q}{\ell^{2}}\right)\equiv\ell^{2}\left(\frac{2\delta J}{3\delta
R}-\frac{2q}{\ell^{2}}\right)>0\ \rightarrow\ \delta J>\frac{nq}{\ell^{2}}\ .$
(4.30)

The particle can orbit outside the event horizon when the particle is rotating
fast enough. This is morally in accordance with (4.26). In particular, blindly
inserting $n=2$ and $\delta J=m$ for the quantum particle of the field $\Phi$,
one happens to recover the bound (4.26).

When the bound (4.30) is saturated, the particle is precisely at the horizon.
Then it will be ambiguous to distinguish whether this is a hair or part of the
black hole, at least in the point particle approximation. Indeed one finds an
interesting signature of this ambiguity. To explain this, recall that the
background black hole satisfies a charge relation (4.6). Absorbing a probe
hair into the black hole will shift its charges by $R\to R+\delta R$ and $J\to
J+\delta J$, where $\delta R,\delta J$ from the probe should be much smaller
than the background charges. Since we take the background charges $R,J$ to be
at order $N^{2}$, the hair charges should also satisfy $\delta R\ll N^{2}$,