SNUTP23-002
KIAS-P23070
LCTP-23-20

Finite $N$ black hole cohomologies

Jaehyeok Choi1, Sunjin Choi2, Seok Kim1, Jehyun Lee1 and Siyul Lee3

1Department of Physics and Astronomy & Center for Theoretical Physics,
Seoul National University, Seoul 08826, Korea.

2School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea.

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

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We study new cohomologies for the BPS operators of the $\mathcal{N}=4$ Yang-
Mills theory with $SU(3)$ and $SU(4)$ gauge groups, to better understand the
black hole microstates. We first study the index of these black hole operators
and identify their apparent threshold levels. For $SU(3)$, we find many towers
of states and partial no-hair behaviors. We explicitly construct the threshold
cohomology in the $SU(3)$ theory. We study throughout this paper a subsector
of the field theory corresponding to the BMN matrix theory. We also argue that
the BMN sector exhibits a black hole like entropy growth at large $N$.

###### Contents

  1. 1 Introduction and summary
  2. 2 The cohomology problem
    1. 2.1 The BMN sector
    2. 2.2 The graviton cohomologies
  3. 3 The index
    1. 3.1 $SU(3)$
    2. 3.2 $SU(4)$
  4. 4 Constructing cohomologies
    1. 4.1 $SU(3)$ threshold cohomology from ansätze
    2. 4.2 $Q$-exactness checks and ansatz-independent studies
  5. 5 Future directions
  6. A Graviton trace relations

##  1 Introduction and summary

Microscopically counting the black hole entropy [1] and better characterizing
its microstates are one of the central problems in quantum gravity. These
questions were also naturally asked in AdS/CFT from its early days [2, 3, 4,
5]. The entropy of BPS black holes in AdS5 has been counted from the CFT dual
rather recently [6, 7, 8] by studying the index [9, 5]. Motivated by this
success, we wish to better characterize the BPS microstates which contribute
to this entropy. Explicitly constructing these quantum states at strong
coupling is difficult, even for BPS states. However, a modest version of this
program was posed in terms of the 1-loop BPS states and their cohomologies [5,
10, 11, 12, 13, 14]. In this program, the 1-loop BPS states annihilated by a
pair of classical supercharges $Q$, $Q^{\dagger}$ are studied from
$Q$-cohomologies, assuming that they remain BPS at strong coupling. See [15]
for a study of perturbative non-renormalization. There has been progress in
this program recently [15, 16, 17, 18, 19, 20]. In particular, new
cohomologies [15, 16, 17] and 1-loop BPS states [18, 19] beyond the familiar
‘multi-graviton type’ operators were constructed in the $SU(2)$ maximal super-
Yang-Mills theory.

Although the $SU(2)$ results already shed interesting light on the black hole
microstates, this theory is at best a highly quantum toy model of AdS/CFT. The
ultimate goal is to study the $SU(N)$ theory at parametrically large $N$. With
this in mind, in this paper, we find and study the new cohomologies for $N>2$.
We shall detect and construct new cohomologies for the $SU(3)$ and $SU(4)$
theories, and discuss their physics.

Identifying and constructing these operators are computationally very
demanding. To partly overcome this difficulty, we focus on a quantum
mechanical subsector corresponding to the BMN matrix model [21, 22]. Among the
full set of the BPS letters (elementary fields dressed by derivatives), this
subsector keeps three scalars $\bar{\phi}^{m}$ ($m=1,2,3$), three fermions
$\psi_{m+}$ and a component $f_{++}$ of the field strength only, without any
derivatives. The computations are easier in this sector, though we should
further pursue efficient methods. As we shall explain in section 2.1, this
sector exhibits a black hole like entropy growth at large $N$. Throughout this
paper, we shall only study cohomologies within this sector.

In the rest of this section, we sketch the main technical/conceptual advances
in this paper.

Completely finding all cohomologies is a very difficult calculation when the
charges or $N$ are large. So far the comprehensive studies are made only in
[15] for $N=2,3,4$ till not-too-large charges. At fixed charges, the procedure
consists of the following steps:

  1. 1.

Find all $Q$-closed operators.

  2. 2.

Find a subset of operators found in step 1 which are not $Q$-exact.

  3. 3.

Find a subset of cohomologies in step 2 which are not of multi-graviton type.

Each step demands substantial computational resources. One should also scan
through all the charges by repeating these procedures. In this paper, we
establish streamlined strategies, partly based on [17], which allow faster
detection and construction of new cohomologies.

Before going through the $3$ steps above, we compute the index over the non-
graviton cohomologies at finite $N$. With this index computed, one can easily
identify the charge sectors that contain non-graviton states and construct
their cohomologies. Of course one could miss pairs of non-graviton states
which cancel in the index: we give up finding all cohomologies and search for
those captured by the index. Exactly computing the full index in the BMN
sector is easy at not too large $N$. More difficult is to count the finite $N$
gravitons to be subtracted, taking into account the trace relations of finite
$N$ matrices. As we review in section 3, counting finite $N$ gravitons reduces
to counting certain class of polynomials, whose generators are subject to
certain relations. In principle, these relations can be systematically studied
using the Gröbner basis. In practice, finding Gröbner basis can be
computationally very difficult. We used a hybrid method of the Gröbner basis
(in a subsector in which this basis can be found easily) and a more brutal
counting of independent polynomials by computer, order by order in the
charges.

Even after identifying the charge sectors to study from the index, the steps
$1$, $2$, and $3$ are still quite hard. We have established the following
procedures which make steps 1 and 2 somewhat easier, and trivialize step 3
within our setup. See p.23 for how our implementation trivializes step 3.

As for step 1, we present a class of ansatz for the $Q$-closed operators. In
order for the final cohomology not to be of graviton type, $Q$ acting on the
operator should vanish by trace relations. Systematically finding useful trace
relations is difficult. We find a method of constructing a class of operators
which become $Q$-closed only after imposing trace relations. Our ansatz uses
the trace relations of the graviton cohomologies that we detected while
computing the index. Trace relations of gravitons mean that certain
polynomials of single-graviton cohomologies are $Q$-exact. These relations
satisfy ‘relations of relations’, i.e. certain linear combinations of trace
relations (with the coefficients being graviton cohomologies) are identically
zero. In other words, relations of relations are linear combinations of
$Q$-exact terms which vanish. So they provide operators which become
$Q$-closed thanks to the trace relations, which are our ansätze for the non-
graviton cohomologies. See section 4.1 for more details.

Some $Q$-closed operators mentioned in the previous paragraph are not
$Q$-exact, providing new cohomologies, while others are $Q$-exact. Determining
whether our ansätze are $Q$-exact or not, step 2, is very hard. We developed a
numerics-assisted approach to make this step affordable on the computer, by
ordering the Grassmann variables and then inserting many random integers to
the matrix elements. See section 4.2 for the details.

The new cohomology thus constructed, and often just the index for the non-
graviton cohomologies, provides insights on the structures of the BPS
spectrum.

First, for $SU(3)$ and $SU(4)$, we address the threshold (lightest) non-
graviton cohomologies as seen by the index. Let us denote the $SO(6)$
R-charges and spins by $R_{1},R_{2},R_{3}$ and $J_{1},J_{2}$ respectively. For
$SU(3)$, the index detects one fermionic threshold cohomology at
$R_{1}=R_{2}=R_{3}$, $J_{1}=J_{2}$ ($\equiv J$) and $j\equiv
2(R_{1}+R_{2}+R_{3})+6J=24$. ($j$ is a combination which can grade the index.)
For $SU(4)$, the index detects six fermionic threshold cohomologies at
$J_{1}=J_{2}\equiv J$ and $j=28$ in the $[2,0]$ representation of
$SU(3)\subset SO(6)$. Combining with the results known in the $SU(2)$ theory
[15, 16], the threshold levels for $j$ at $N=2,3,4$ form a non-decreasing
sequence $24,24,28$. These are apparent thresholds for $N=3,4$. They would be
true thresholds if there are no lower cohomologies canceling in the index or
outside the BMN sector.

For $SU(3)$, we construct the threshold cohomology. The threshold cohomology
is fermionic and has charges $R_{1}=R_{2}=R_{3}=\frac{5}{2}$ ($\equiv R$) and
$J=\frac{3}{2}$, thus the scaling dimension $E=3R+2J=\frac{21}{2}$. The
scaling dimension is larger than $E=\frac{19}{2}$ of the $SU(2)$ threshold
operator at $j=24$. So the energy threshold for non-graviton operators
increases in $N$. Such an increase is also expected at large $N$ because the
graviton description should be reliable till $E=N$.

The non-graviton index itself reveals various organized patterns which allow
us to guess some underlying structures of the spectrum. For $SU(4)$, we barely
managed to detect the threshold level and did not go to higher orders to find
richer structures. However, we computed the $SU(3)$ non-graviton index up to
$j\leq 54$, much higher than the threshold $j=24$, and observed various
intriguing structures. First of all, like the $SU(2)$ non-graviton index, we
find organized towers of states. In $SU(2)$ BMN case, we found just one tower
of states starting from the threshold operator. However, for $SU(3)$, there
are more than one towers starting at different levels. We also find that some
towers are related to others by developing hierarchies. We constrain their
charge structures and compare them with the expected asymptotic entropy of the
BMN sector. Possibly, these towers will continue to exist at larger $N$’s and
may be related to a particular subset of the giant graviton towers [23, 24,
25, 26].

Another important aspect of the $SU(3)$ non-graviton index is the partial no-
hair behavior. Namely, the index does not see many product cohomologies made
of core black hole cohomologies multiplied by graviton cohomologies. In
$SU(2)$, many of these product states not seen by the index were shown to be
$Q$-exact, thus absent in the BPS spectrum [17]. So the black hole
cohomologies do not want to be dressed by certain gravitons, reminiscent of
the black hole no-hair theorem. It was also pointed out, both from $SU(2)$ QFT
and the large $N$ gravity dual, that the BPS no-hair theorem holds only
partially. We find an empirical signal that the $SU(3)$ no-hair theorem may
also be partial, in that assuming particular graviton hairs makes the
remaining core spectrum better organized. See section 3.1 for the details.

The remaining part of this paper is organized as follows. In section 2, we
explain the cohomology problem, the BMN sector and its entropy, and the
graviton cohomologies. In section 3, we present our strategies for computing
the non-graviton index and present the results for $N=3,4$. Qualitative
discussions are also made for the $SU(3)$ index. In section 4, we present the
new $SU(3)$ cohomology at the threshold level. In section 5, we discuss future
directions. In the appendix, we list the graviton trace relations for the
$SU(3)$ theory.

##  2 The cohomology problem

The $\mathcal{N}=4$ Yang-Mills theory with $SU(N)$ gauge group has six real
scalars $\Phi_{ij}=-\Phi_{ji}$ (subject to
${\overline{\Phi}}^{ij}\sim\frac{1}{2}\epsilon^{ijkl}\Phi_{kl}$), fermions
$\Psi_{i\alpha},{\overline{\Psi}}^{i}_{\dot{\alpha}}$ and the gauge field
$A_{\mu}\sim A_{\alpha\dot{\beta}}$, all in the $SU(N)$ adjoint
representation. (Here, $i,j\cdots=1,\cdots,4$, $\alpha=\pm$,
$\dot{\beta}=\dot{\pm}$, $\mu=1,\cdots,4$.) For later convenience, we arrange
these fields into $\mathcal{N}=1$ supermultiplets as follows, with manifest
covariance only for the $SU(3)\subset SU(4)$ part of R-symmetry,

vector multiplet $\displaystyle:$ $\displaystyle A_{\alpha\dot{\beta}}\ ,\
\lambda_{\alpha}=\Psi_{4\alpha}\ ,\
\bar{\lambda}_{\dot{\alpha}}=\overline{\Psi}^{4}_{\dot{\alpha}}~{},$ (2.1)
$\displaystyle 3\textrm{ chiral multiplets}$ $\displaystyle:$
$\displaystyle\phi_{m}=\Phi_{4m}\ ,\ \bar{\phi}^{m}=\overline{\Phi}^{4m}\ ,\
\psi_{m\alpha}=-i\Psi_{m\alpha}\ ,\
\bar{\psi}^{m}_{\dot{\alpha}}=i\overline{\Psi}^{m}_{\dot{\alpha}}\ ,$

where $m=1,2,3$. In this paper, we consider the Euclidean CFT on
$\mathbb{R}^{4}$. This is related to the Lorentzian CFT on
$S^{3}\times\mathbb{R}$ by radial quantization, which regards the radius of
$\mathbb{R}^{4}$ as the exponential of the Euclidean time $\tau$ and makes a
Wick rotation $\tau=it$. Here we note the operator-state map, in which the
local operators at the origin of $\mathbb{R}^{4}$ map to the states
propagating in $S^{3}\times\mathbb{R}$. The CFT carries a marginal coupling
constant $g_{\rm YM}$.

The CFT is invariant under $32$ supersymmetries, represented by the $16$
Poincare supercharges $Q^{i}_{\alpha}$, $\overline{Q}_{i\dot{\alpha}}$ and the
$16$ conformal supercharges $S_{i\alpha}$, $\overline{S}^{i}_{\dot{\alpha}}$.
In the radially quantized theory, $S$’s are Hermitian conjugates of $Q$’s:
$S_{i}^{\alpha}=(Q^{i}_{\alpha})^{\dagger}$,
$\overline{S}^{i\dot{\alpha}}=(\overline{Q}_{i\dot{\alpha}})^{\dagger}$.
Together with other symmetry generators, these supercharges form the
$PSU(2,2|4)$ superconformal algebra. The most important part of the algebra
for this paper is [5]

$\\{Q^{i}_{\alpha},S_{j}^{\beta}\\}={\textstyle\frac{1}{2}}H\delta^{i}_{j}\delta_{\alpha}^{\beta}+{R^{i}}_{j}\delta_{\alpha}^{\beta}+{J_{\alpha}}^{\beta}\delta^{i}_{j}~{},$
(2.2)

where $H$ is the dilatation operator (or the Hamiltonian on
$S^{3}\times\mathbb{R}$ multiplied by the radius of $S^{3}$), ${R^{i}}_{j}$ is
the $SU(4)$ R-charges, and ${J_{\alpha}}^{\beta}$ is the left $SU(2)\subset
SO(4)$ angular momenta. The BPS states/operators of our interest preserve $2$
Hermitian supercharges $Q\equiv Q^{4}_{-}$ and $Q^{\dagger}\equiv S_{4}^{-}$,
thus called $\frac{1}{16}$-BPS states/operators. These two supercharges
satisfy $Q^{2}=0$, $(Q^{\dagger})^{2}=0$, and from (2.2) one obtains

$2\\{Q,Q^{\dagger}\\}=H-(R_{1}+R_{2}+R_{3}+J_{1}+J_{2})\ .$ (2.3)

On the right hand side, we expressed $2{R^{4}}_{4}=-R_{1}-R_{2}-R_{3}$ and
$2{J_{-}}^{-}=-J_{1}-J_{2}$ in terms of the five charges which rotate the
mutually orthogonal 2-planes on $\mathbb{R}^{6}\supset S^{5}$ and
$\mathbb{R}^{4}\supset S^{3}$, respectively, all normalized to have
$\pm\frac{1}{2}$ values for spinors. The BPS operators of our interest
saturate the bound $E\geq R_{1}+R_{2}+R_{3}+J_{1}+J_{2}$. The charges $R_{I}$,
$J_{i}$ on the right hand side are part of the non-Abelian charges and cannot
depend on the coupling $g_{\rm YM}$. However, $E$ is in general a function of
$g_{\rm YM}$, so that a BPS state may become anomalous as $g_{\rm YM}$
changes.

The gauge-invariant BPS operators are easily identified (and counted) in the
free theory limit, $g_{\rm YM}\rightarrow 0$. They are given by any gauge-
invariant operators made of the following fields

$\bar{\phi}^{m}\ ,\ \psi_{m+}\ ,\ f_{++}\ ,\ \bar{\lambda}_{\dot{\alpha}}$
(2.4)

as well as the derivatives $\partial_{+\dot{\alpha}}$ acting on these fields,
subject to the free equation of motion constraint
$\partial_{+\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}=0$. See, for instance,
[5, 17] for a more detailed explanation. We want to study how many of these
operators remain BPS at the 1-loop level. The dilatation operator $H(g_{\rm
YM})$ can be expanded in $g_{\rm YM}^{2}$, $H(g_{\rm
YM})=\sum_{L=0}^{\infty}g_{\rm YM}^{2L}H_{(L)}$. At least in perturbation
theory, this operator can be diagonalized within the subspace of free BPS
operators.111More precisely, for the gauge invariance in the interacting
theory, the subsector is defined at $g_{\rm YM}\neq 0$ by promoting the
derivatives $\partial_{+\dot{\alpha}}$ appearing in the operators to the
covariant derivatives
$D_{+\dot{\alpha}}\equiv\partial_{+\dot{\alpha}}-i[A_{+\dot{\alpha}},\ ]$.
Within this subspace, $H_{(0)}$ is zero. We want to find the subset of free
BPS operators which are annihilated by $H_{(1)}$. Within the free BPS sector,
one finds that

$\\{Q(g_{\rm YM}),Q^{\dagger}(g_{\rm YM})\\}=H(g_{\rm
YM})-\sum_{I}R_{I}-\sum_{i}J_{i}=\sum_{L=1}^{\infty}g_{\rm YM}^{2L}H_{(L)}\ .$
(2.5)

$Q$ and $Q^{\dagger}$ also depend on $g_{\rm YM}$. Since the free BPS fields
are annihilated by them at the leading $\mathcal{O}(g_{\rm YM}^{0})$ order,
their coupling expansions start from the $\mathcal{O}(g_{\rm YM}^{1})$ ‘half-
loop’ order. Therefore, the leading 1-loop Hamiltonian $H_{(1)}$ in (2.5) is
given by the anticommutator of $Q$ and $Q^{\dagger}$ at the half-loop order.
In particular, $Q_{(\frac{1}{2})}$ at $\mathcal{O}(g_{\rm YM}^{1})$ is
precisely the supercharge of the classical interacting field theory. So the
1-loop BPS operators should be annihilated by both $Q$ and $Q^{\dagger}$ at
the classical half-loop order.

Due to the nilpotency of $Q$ and $Q^{\dagger}$, the operators annihilated by
$Q$ and $Q^{\dagger}$ are in 1-1 map with the cohomologies of $Q$, which are
$Q$-closed operators with identifications of operators which differ by
$Q$-exact operators. We shall construct and study the representatives of the
cohomologies of the classical half-loop supercharge $Q$, which maps to the
1-loop BPS operators. The actions of classical (half-loop) $Q$ on the free BPS
fields are given by

$Q\bar{\phi}^{m}=0\ ,\ \ Q\bar{\lambda}_{\dot{\alpha}}=0\ ,\
Q\psi_{m+}=-{\textstyle\frac{i}{2}}\epsilon_{mnp}[\bar{\phi}^{n},\bar{\phi}^{p}]\
,\ Qf=-i[\bar{\phi}^{m},\psi_{m+}]\ ,\
[Q,D_{+\dot{\alpha}}]=-i[\lambda_{\dot{\alpha}},\ \ \\}\ ,$ (2.6)

where we absorbed the $g_{\rm YM}$ factors on the right hand sides into the
normalization of fields.

It is well known that there are fewer BPS states at the 1-loop level than in
the free theory. It has been conjectured (for instance, explicitly in [11])
that the 1-loop BPS states remain BPS at general non-zero coupling. Some
perturbative evidence of this conjecture was discussed in [15]. Various
discussions in this paper will assume this conjecture.

In our studies, the index over these cohomologies will be important. It is
defined by

$Z(\Delta_{I},\omega_{i})={\rm
Tr}\left[(-1)^{F}e^{-\Delta_{1}R_{1}-\Delta_{2}R_{2}-\Delta_{3}R_{3}-\omega_{1}J_{!}-\omega_{2}J_{2}}\right]$
(2.7)

with the constraint $\Delta_{1}+\Delta_{2}+\Delta_{3}=\omega_{1}+\omega_{2}$
(mod $4\pi i$). The trace is taken over the BPS states, or equivalently over
the cohomologies. We may regard it as the index over the cohomologies of $Q$
given by (2.6). A matrix integral formula for this index is given in [5].

###  2.1 The BMN sector

The cohomology problem has a consistent truncation to the so-called BMN matrix
model. This is the sector in which only the following three letters are used
to construct the operators,

$\bar{\phi}^{m}\ ,\ \ \psi_{m+}\ ,\ \ f_{++}\ ,$ (2.8)

without any derivatives acting on them. In this paper, to simplify the
calculations, we shall study cohomologies only in this sector. To simplify
notations, from now on we call them

$(\bar{\phi}^{m},\psi_{m+},f_{++})\ \rightarrow\ (\phi^{m},\psi_{m},f)\ .$
(2.9)

This truncation is consistent only in the classical interacting field theory,
i.e. till 1-loop BPS states. The truncation forbids all the modes in the
classical field theory with $J_{1}\neq J_{2}$. In general quantum theories,
the letters carrying nonzero $J_{1}-J_{2}$ charges may mix with the fields in
the BMN sector. So the 1-loop BPS operators in the BMN sector may mix with
non-BMN operators at higher orders in $g_{\rm YM}^{2}$. But if the conjecture
that the spectrum of 1-loop BPS states is isomorphic to the BPS states at
general coupling is true, the spectrum computed in the BMN sector will remain
unchanged.

Since the truncation is made for the modes of elementary fields, the same
truncation can be implemented to compute the index over the cohomologies in
the BMN sector,

$Z(\Delta_{I})={\rm Tr}_{\rm
BMN}[(-1)^{F}e^{-\sum_{I=1}^{3}\Delta_{I}(R_{I}+J)}]\ ,$ (2.10)

where $J=\frac{J_{1}+J_{2}}{2}$. Its matrix integral expression is given by
[17]

$\displaystyle Z(\Delta_{I})$ $\displaystyle=$
$\displaystyle\frac{1}{N!}\int_{0}^{2\pi}\frac{d^{N}\alpha}{(2\pi)^{N}}\frac{\prod_{a\neq
b}(1-e^{i\alpha_{ab}})\prod_{a,b=1}^{N}\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}}e^{i\alpha_{ab}})}{\prod_{a,b=1}^{N}\left[(1-e^{-(\Delta_{1}+\Delta_{2}+\Delta_{3})}e^{i\alpha_{ab}})\prod_{I=1}^{3}(1-e^{-\Delta_{I}}e^{i\alpha_{ab}})\right]}$
(2.11) $\displaystyle\hskip 76.82234pt\times\
\frac{(1-e^{-(\Delta_{1}+\Delta_{2}+\Delta_{3})})\prod_{I=1}^{3}(1-e^{-\Delta_{I}})}{\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}})}~{},$

where the second line (inverse of the $U(1)$ index) is multiplied to make it
an $SU(N)$ index rather than $U(N)$. This integral can be computed either
exactly using the residue sum or in a series expansion in $t$ defined by
$(e^{-\Delta_{1}},e^{-\Delta_{2}},e^{-\Delta_{3}})=t^{2}(x,y^{-1},x^{-1}y)$.

The entropy of BMN cohomologies will be smaller than the entropy of all
cohomologies. Despite, the large $N$ BMN entropy will still exhibit the black
hole like growth. Taking $j$ (schematically) to be the charges, the black hole
like entropy growth is

$S(j,N)=N^{2}f({\textstyle\frac{j}{N^{2}}})~{},$ (2.12)

for a function $f(x)$ which does not explicitly depend on $N$, where $N\gg 1$,
$j\gg 1$ and the ratio $\epsilon\equiv\frac{j}{N^{2}}$ does not scale in $N$.
Roughly, the scaled charge parameter $\epsilon$ measures the size of the black
hole in the AdS unit. In this subsection, we show that the BMN entropy scales
like (2.12) when $\epsilon$ is parametrically small (but not scaling in $N$).
We expect the same to be true at general $\epsilon$, although we do not prove
it.

We first compute the large $N$ entropy in the ‘small black hole’ regime: $j\gg
1$, $N\gg 1$ and $\epsilon\equiv\frac{j}{N^{2}}$ fixed and much smaller than
$1$ (but not scaling in $N$).222The term ‘small black hole’ has at least three
different meanings in the literature. It sometimes denotes string scale black
holes, for which 2-derivative gravity description breaks down near the
horizon. In our example, since $\epsilon$ does not scale in $N$, the
2-derivative gravity is reliable everywhere. Also, small black holes sometimes
mean AdS black holes with negative specific heat or susceptibility. What we
call ‘small black holes’ belong to this class, but are more specific. Our
notion is precisely the same as [27, 28]. This regime is reached by taking all
$\Delta_{I}$’s to be small. The approximate large $N$ calculation of the
entropy can be done by following all the calculations in section 5.3 of [28]
with minor changes in the setup. In particular, the calculations from (5.88)
to (5.91) there can be repeated by simply replacing all
$2-(-e^{\gamma})^{n}-(-e^{\gamma})^{-n}$ by $1$ (which are the denominators of
the letter indices in the two setups) and remembering that $\beta_{I}$ there
are $\frac{\Delta_{I}}{2}$ here. The resulting eigenvalue distribution is
along the interval $\alpha\in(-\pi,\pi)$ on the real axis (the gap closes in
the small black hole limit), with the distribution function

$\rho(\alpha)=\frac{3}{4\pi^{3}}(\pi^{2}-\alpha^{2})\ .$ (2.13)

The free energy $\log Z$ of this saddle point is given by

$\log Z=-\frac{3N^{2}}{\pi^{2}}\Delta_{1}\Delta_{2}\Delta_{3}\ .$ (2.14)

(For small black holes with negative susceptibility, the grand canonical index
is not well defined. Whenever we address $\log Z$, a Laplace transformation to
the micro-canonical ensemble is assumed.) The entropy at given charges
$q_{I}\equiv R_{I}+J$ is given by extremizing

$S_{\rm BMN}(q_{I};\Delta_{I})=\log Z+\sum_{I}q_{I}\Delta_{I}~{},$ (2.15)

in $\Delta_{I}$’s, which is given by

$S_{\rm BMN}(q_{I})=2\pi\sqrt{\frac{q_{1}q_{2}q_{3}}{3N^{2}}}\ .$ (2.16)

This expression is valid when $q_{I}=N^{2}\epsilon_{I}$ with $\epsilon_{I}\ll
1$. The entropy $\sim N^{2}\sqrt{\epsilon_{1}\epsilon_{2}\epsilon_{3}}\propto
N^{2}$ exhibits a black hole like scaling (2.12). $S_{\rm BMN}$ is smaller
than the full entropy $S(q_{I})=2\pi\sqrt{\frac{2q_{1}q_{2}q_{3}}{N^{2}}}$ in
the small black hole regime [28] by a factor of $\frac{1}{\sqrt{6}}$. This is
natural since the BMN truncation loses cohomologies. However, the fact that
$S_{\rm BMN}(q_{I})$ scales like $S(q_{I})$ implies that the truncation
provides a good simplified model for black holes, at least for
$\epsilon_{I}\ll 1$.

At large charges, we have not computed the asymptotic entropies at large $N$
and $q_{I}$. Instead, we have computed the large charge entropies at
$N=2,3,4$. This was done by first computing the exact index $Z(t)$ by a
residue sum, and then extracting the coefficient $\Omega_{j}$ of the expansion
$Z(t)=\sum_{j}\Omega_{j}t^{j}$ for large $j=2(q_{1}+q_{2}+q_{3})$. This
calculation was done both by saddle point approximation at large $j$, and also
by expanding $Z(t)$ in $t$ by computer until very high order. The asymptotic
BMN entropies $S_{\rm BMN}(j,N)$ at very large $j\gg 1$ are given by

$S_{\rm BMN}(j,2)\sim 3\log j\ ,\ S_{\rm BMN}(j,3)\sim 2\log j\ ,\ S_{\rm
BMN}(j,4)\sim 8\log j\ .$ (2.17)

Even at small $N$’s, these are much slower growths than the full entropy,
which is [7, 29, 30]

$S\propto(N^{2}\\!-\\!1)^{\frac{1}{3}}j^{\frac{2}{3}}\ .$ (2.18)

The large discrepancy $S_{\rm BMN}/S\ll 1$ is natural since truncating a 4d
QFT to quantum mechanics loses almost all cohomologies at higher energies.
This is because the dominant part of the entropy (2.18) is supposed to come
from the infinitely many letters dressed by derivatives. If $n$ unconstrained
derivatives can appear in the operators, one expects an entropy $S\propto
j^{\frac{n}{n+1}}$. The case with $n=2$ BPS derivatives in 4d indices yields
(2.18), while the quantum mechanical case with $n=0$ replaces
$j^{\frac{0}{0+1}}$ by $\log j$.

So we expect the BMN subsector to give us good lessons on the small black
holes for sufficiently large $N$. We hope that the studies at finite $N=3,4$
in this paper will start to unveil important structures which will continue at
larger $N$’s.

###  2.2 The graviton cohomologies

Some cohomologies are very well known, which are the multi-graviton
cohomologies. We want to exclude them in our discussions. In this subsection,
we review the notion of graviton cohomologies, especially at finite $N$, and
also explain how to list and count them.

The multi-graviton cohomologies are ‘defined’ to be the polynomials of single-
trace cohomologies. (This definition naturally yields the familiar large $N$
cohomologies for the supergravitons.) Single-trace cohomologies are completely
understood [5, 12, 14], as we shall review in a moment. Single-trace
cohomologies are nontrivial cohomologies at arbitrary $N$ by definition, since
no trace relations apply within the single-trace sector. Polynomials of these
single-trace cohomologies define the multi-trace cohomologies. Some
polynomials may be trivial, i.e. $Q$-exact at finite $N$. However, they are
$Q$-closed at arbitrary $N$ without using any trace relations. This will be in
contrast to the black hole cohomologies, which should become $Q$-closed only
after applying trace relations at particular $N$.

When $N$ is larger than the energy, the multi-graviton operators defined above
are all nontrivial cohomologies since no trace relations can be applied to
make them $Q$-exact. So in this setup, the ‘graviton cohomologies’ defined
abstractly in the previous paragraph actually map to the familiar
$\frac{1}{16}$-BPS graviton states in $AdS_{5}\times S^{5}$. Trace number of
the operator is regarded as the particle number.

At finite $N$, all the multi-trace operators mentioned in the previous
paragraphs are still $Q$-closed. However, some of their linear combinations
may be zero or $Q$-exact when their energies are larger than $N$, due to the
trace relations. So the independent graviton cohomologies reduce at finite
$N$. Such reductions of states are a well known finite $N$ effect in the
gravity dual. It is called the stringy exclusion principle [31], which happens
because gravitons polarize into D-brane giant gravitons [32, 33, 34]. The
reduction/exclusion mechanism is the same for any $N$ in QFT, making it
natural to call them ‘finite $N$ gravitons’ at general finite $N$.

Now we concretely explain the list of the graviton cohomologies. One starts by
listing the single-trace graviton cohomologies. These are completely found and
collected into supermultiplets. The relevant algebra for these multiplets is
the $PSU(1,2|3)$ subset of the superconformal symmetry $PSU(2,2|4)$ that
commutes with $Q,Q^{\dagger}$. The multiplets for single-trace graviton
cohomologies are called $S_{n}$ with $n=2,3,\cdots$ [5]. $S_{n}$ is obtained
by acting the Poincare supercharges $Q^{m}_{+}$,
$\overline{Q}_{m\dot{\alpha}}$ and the translations $P_{+\dot{\alpha}}$ in
$PSU(1,2|3)$ on the following primary operators

$u^{i_{1}i_{2}\cdots i_{n}}={\rm
tr}(\bar{\phi}^{(i_{1}}\bar{\phi}^{i_{2}}\cdots\bar{\phi}^{i_{n})})\ .$ (2.19)

See [5] for more details. At large $N$, multiplying the operators in $S_{n}$’s
yields independent multi-trace cohomologies. At finite $N$, trace relations
reduce the independent single-trace and multi-trace operators. Following [17],
we first identify the dependent single-trace operators as follows. Using the
Cayley-Hamilton identity, one can show that all single-trace operators in
$S_{n\geq N+1}$ can be expressed as polynomials of operators in $S_{n\leq N}$
[17]. So it suffices to use only the operators in $S_{n\leq N}$ to generate
graviton cohomologies. The remaining single-trace generators in $S_{n\leq N}$
are not independent when we multiply them. In other words, there are further
trace relations for gravitons within $S_{n\leq N}$. The last trace relations
are not systematically understood, to the best of our knowledge.

To simplify the discussions, let us consider the BMN sector only from now. The
subset of $PSU(2,2|4)$ that acts within the BMN sector is $SU(2|4)$. The
subset $SU(1|3)\subset SU(2|4)$ commutes with $Q,Q^{\dagger}$ and generates
the supermultiplets of BMN cohomologies. In each $S_{n}$, there is a finite
number of single-trace generators in the BMN sector. They are given by

$\displaystyle(u_{n})^{i_{1}\cdots i_{n}}$ $\displaystyle=$ $\displaystyle{\rm
tr}(\phi^{(i_{1}}\cdots\phi^{i_{n})})\ \ \ $ (2.20)
$\displaystyle{(v_{n})^{i_{1}\cdots i_{n\\!-\\!1}}}_{j}$ $\displaystyle=$
$\displaystyle{\rm
tr}(\phi^{(i_{1}}\cdots\phi^{i_{n\\!-\\!1})}\psi_{j})-\textrm{`trace'}\ \ \ \
$ $\displaystyle(w_{n})^{i_{1}\cdots i_{n\\!-\\!1}}$ $\displaystyle=$
$\displaystyle{\rm tr}(\phi^{(i_{1}\cdots
i_{n\\!-\\!1})}f+{\textstyle\frac{1}{2}}\epsilon^{jk(i_{p}}{\textstyle\sum_{p=1}^{n\\!-\\!1}}\phi^{i_{1}}\cdots\phi^{i_{p\\!-\\!1}}\psi_{j}\phi^{i_{p\\!+\\!1}}\cdots\phi^{i_{n\\!-\\!1})}\psi_{k})\
.\ \ \ \ $

Here, ‘trace’ denotes the terms to be subtracted to ensure that the
contractions of the upper/lower $SU(3)$ indices are zero. The BMN multi-
graviton cohomologies are polynomials of $u_{n}$, $v_{n}$, $w_{n}$. These
polynomials are subject to trace relations. These trace relations hold up to
$Q$-exact terms.333In principle there might be relations which hold without
any $Q$-exact terms. In practice, with extensive studies of the $SU(2)$ and
$SU(3)$ graviton operators in the BMN sector, all trace relations of this sort
that we found have nontrivial $Q$-exact terms. For instance, the lowest trace
relations for $N=2$ are

$R_{ij}\equiv\epsilon_{ikm}\epsilon_{jln}(u_{2})^{kl}(u_{2})^{mn}=Q\left[-i\epsilon_{a_{1}a_{2}(i}{\rm
tr}(\psi_{j)}\phi^{a_{1}}\phi^{a_{2}})\right]\ .$ (2.21)

More concretely, some components of these relations are

${\rm tr}(X^{2}){\rm tr}(Y^{2})-[{\rm tr}(XY)]^{2}\sim 0\ ,\ {\rm tr}(XY){\rm
tr}(XZ)-{\rm tr}(X^{2}){\rm tr}(YZ)\sim 0~{},$ (2.22)

where $\sim$ hold up to $Q$-exact terms. Such $Q$-exact combinations are zeros
in cohomology. Of course multiplying gravitons to such relations yields
further relations. Trace relations cannot be seen if one does not know that
the ‘meson’ or ‘glueball’ operators $u_{n},v_{n},w_{n}$ are made of ‘gluons’
$\phi,\psi,f$. To enumerate graviton cohomologies without overcounting, we
first consider the Fock space made by the operators $\\{u_{n},v_{n},w_{n}\\}$
with $n=2,\cdots,N$ and then take care of the trace relations to eliminate the
dependent states.

It is important to find all fundamental trace relations of the polynomials of
$u_{n},v_{n},w_{n}$, which cannot be decomposed into linear combinations of
smaller relations. Let us denote by $R_{a}(\\{u_{n},v_{n},w_{n}\\})$ the
fundamental trace relations, with $a$ being the label. Non-fundamental trace
relations are obtained by linear combinations of $R_{a}$’s,

$\sum_{a}f_{a}(\\{u_{n},v_{n},w_{n}\\})R_{a}(\\{u_{n},v_{n},w_{n}\\})\ .$
(2.23)

In general, (2.23) is nonzero and $Q$-exact. However, for some choices of
$f_{a}$’s, the combination (2.23) may be exactly zero. If (2.23) exactly
vanishes, this yields a ‘relation of relations.’ In terms of the mesonic
variables $u_{n},v_{n},w_{n}$, they are trivial expressions, meaning that
various terms just cancel to zero. They just represent the ways in which
fundamental relations $R_{a}$ can be redundant at higher orders. For example,
consider the relations $R_{ij}$ of (2.21) in the $SU(2)$ gauge theory. Some
relations of these relations are given by

$u^{ik}R_{jk}(u_{2})-{\textstyle\frac{1}{3}}\delta^{i}_{j}u^{kl}R_{kl}(u_{2})=0~{},$
(2.24)

in the $[1,1]$ representation. For instance, one can immediately see for
$i=1$, $j=2$ that

$u^{1i}R_{2i}=u^{11}[u^{23}u^{13}-u^{12}u^{33}]+u^{12}[u^{33}u^{11}-(u^{13})^{2}]+u^{13}[u^{12}u^{13}-u^{11}u^{23}]=0\
.$ (2.25)

This is a trivial identity if expanded in mesons. $u^{11}R_{21}$ and
$-u^{12}R_{22}-u^{13}R_{23}$ represent same constraint
$u^{11}(u^{23}u^{13}-u^{12}u^{33})=Q[\cdots]$, implying that $R_{ij}$’s are
not independent.

Interestingly, trace relations described so far will be used in section 4 to
construct the ansatz for the non-graviton cohomologies. In the meantime, we
shall exploit a more practical way of enumerating the graviton cohomologies,
as we explain in section 3.

##  3 The index

In this section, we explain how to enumerate the finite $N$ graviton
cohomologies described in section 2.2. Then it is straightforward to compute
the index over graviton cohomologies, and subtract from the full index to
obtain the index over non-graviton cohomologies. The results for $SU(3)$ and
$SU(4)$ are presented in respective subsections.

Recall that graviton operators in the BMN sector are the polynomials of mesons
(2.20). We wish to enumerate linearly independent operators among these, i.e.
we wish to mod out by linear relations between them. There are two main
strategies that we exploit to ease this computation: the eigenvalue counting
and the Gröbner basis.

Let us explain the first idea, the eigenvalue counting. We first review how
the multi-gravitons made of the chiral primaries $u_{n}$ of (2.19) are
enumerated. Based on rather physical arguments, [5] proposed to count them by
taking all three scalars $\phi^{m}$ to be diagonal matrices.444The argument is
often dubbed ‘quantizing the moduli space’ of the QFT. For exact quantum
states, it relies on the protection of the moduli space against quantum
corrections. At the level of classical cohomologies, its proof should be
elementary, although we do not pursue it here. With this restriction, the
problem of enumerating gauge-invariant operators reduces to enumerating Weyl-
invariant polynomials of the eigenvalues. Now our interest is in counting the
finite $N$ graviton cohomologies involving all the descendants (2.20) in
$S_{n}$, not only the chiral primaries $u_{n}$. The descendants are obtained
from $u_{n}$ by acting the supercharges in $PSU(1,2|3)$. Since the single-
graviton states belong to absolutely protected multiplets $S_{n}$, and since
their multiplications trivially remain in cohomology both for free and 1-loop
calculations, we can generate the descendants by acting the supercharges of
the strictly free theory [17]. The actions of free supercharges are linear so
that diagonal $\phi^{m}$’s transform to diagonal $\psi_{m}$ and $f$. (In the
BMN sector, the supercharges $Q^{m}_{+}$ in $SU(1|3)$ act linearly even in the
classical interacting theory.) Therefore, $\psi_{m}$ and $f$ that appear in
descendants can be taken to be diagonal matrices as well, for the purpose of
enumerating graviton operators.

So the counting of graviton operators is reduced to the counting of certain
polynomials of the eigenvalues. We have $N-1$ eigenvalues for each field
$\phi^{m}$, $\psi_{m}$ and $f$, so in total $7(N-1)$ variables are needed to
describe graviton operators in the BMN sector. Let us denote these eigenvalues
collectively as $\lambda_{I}$. Let us also denote the ‘mesonic generators’
$\\{u_{n},v_{n},w_{n}\\}$ (2.20) with $n=2,\cdots,N$, collectively as
$g_{i}$’s. These are now regarded as polynomials $g_{i}(\lambda_{I})$ of the
eigenvalues $\lambda_{I}$. Then, we want to count the polynomials $p(g_{i})$
of the mesons $g_{i}$, which can be written as polynomials
$p(g_{i}(\lambda_{I}))$ of eigenvalues $\lambda_{I}$.

These polynomials are not all independent because certain polynomials
$p(g_{i})$ of $g_{i}$’s may be zero when written as polynomials
$p(g_{i}(\lambda_{I}))$ of $\lambda_{I}$. Such polynomials can be thought of
as constraints on the space of polynomials. These are remnants of the trace
relations of the $N\times N$ matrices. Had we been keeping all the $N\times N$
matrix elements, trace relation would have been zero up to a $Q$-exact term.
Since the action of $Q$ yields a commutator, the $Q$-exact term vanishes when
the fields are diagonal. So general trace relations up to $Q$-exact terms
reduce to exact polynomial constraints.

Counting constrained polynomials is a classic mathematical problem, with known
solution. This brings us to the second strategy that we exploit: the Gröbner
basis. See e.g. [35]. Let us briefly explain a flavor of its properties and
how it is used to solve the enumeration problem.

Recall that the multi-graviton operators are given by the set of all
polynomials $p(g_{i})$ of $g_{i}$’s. However, this set is overcomplete and
therefore not suitable for the counting purpose, because of the constraints.
That is, some of the polynomials are zero and consequently some of the
polynomials are equivalent to each other.

We want to better understand the constraints, i.e. polynomials of $g_{i}$ that
are zero. The constraints appear because each meson $g_{i}$ is not an
independent variable but instead made of the gluons $\lambda_{I}$, i.e.
$g_{i}=g_{i}(\lambda_{I})$ where the right hand side is a polynomial of
$\lambda_{I}$ that corresponds to the meson $g_{i}$. All constraints are
derived from the fact that

$G_{i}(g_{i},\lambda_{I})\equiv g_{i}-g_{i}(\lambda_{I})=0\ ,$ (3.1)

for each meson labeled by $i$. Therefore, the set of all polynomials of the
mesons $g_{i}$ and the eigenvalues $\lambda_{I}$ that are zero (also known as
the ideal) is _generated_ by (3.1), in the sense that any element of this set
can be written as

$\sum_{i}q_{i}(g_{i},\lambda_{I})G_{i}(g_{i},\lambda_{I})~{},$ (3.2)

where $q_{i}(g_{i},\lambda)$ are polynomials of $g_{i}$ and $\lambda_{I}$. If
we restrict to elements of this ‘set of zeroes’ that only involve $g_{i}$ but
not $\lambda_{I}$, those will be precisely the constraints that mod out the
set of all polynomials $p(g_{i})$.

Although (3.1) is the most intuitive basis that generates the set of zeroes
like (3.2), it is often not the most convenient basis. The same set of zeroes
can be generated by many different choices of the basis, possibly with
different numbers of generators. Gröbner basis is one of these choices with
the following special property. Let $\\{G_{a}(g_{i},\lambda_{I})\\}$ be a
basis of the set of zero polynomials of $(g_{i},\lambda_{I})$. Then, for any
polynomial $p(g_{i},\lambda_{I})$, suppose one tries to ‘divide’ this
polynomial by the basis $\\{G_{a}(g_{i},\lambda_{I})\\}$. This is a process of
writing the polynomial as

$p(g_{i},\lambda_{I})=\sum_{a}q_{a}(g_{i},\lambda_{I})G_{a}(g_{i},\lambda_{I})+r(g_{i},\lambda_{I})\
,$ (3.3)

where $r(g_{i},\lambda_{I})$ can no longer be ‘divided by’
$\\{G_{a}(g_{i},\lambda_{I})\\}$, which can be well-defined by setting an
ordering scheme between variables and their monomials. Naturally,
$r(g_{i},\lambda_{I})$ can be thought of as the remainder of the division. In
general, there can be multiple ways — with different $q_{a}$ and $r$ — to
write $p(g_{i},\lambda_{I})$ as (3.3). The special property of the Gröbner
basis is that if $\\{G_{a}(g_{i},\lambda_{I})\\}$ were the Gröbner basis of
the set of zeroes, then the remainder $r(g_{i},\lambda_{I})$ is unique for
each given $p(g_{i},\lambda_{I})$. Note that since
$\\{G_{a}(g_{i},\lambda_{I})\\}$ generates the set of zeroes, (3.3) implies
that the polynomial $p(g_{i},\lambda_{I})$ is equivalent to its remainder
$r(g_{i},\lambda_{I})$. It follows that the set of all polynomials
$p(g_{i},\lambda_{I})$ is identical to the set of all possible remainders
$r(g_{i},\lambda_{I})$ under division by the Gröbner basis. However, unlike in
the set of all polynomials $p(g_{i},\lambda_{I})$, there are no polynomials in
the set of all remainders that are equivalent due to the constraints, because
otherwise one of them should have been divided once more to yield the other as
the remainder. Therefore, the set of remainders can be used to count the
number of independent polynomials of $(g_{i},\lambda_{I})$ under constraints.

There is a canonical procedure to find the Gröbner basis of the set of zeroes
given one choice of basis (3.1), known as _Buchberger’s algorithm_. Many
computer algebra softwares implement this algorithm or its improved versions.
The Gröbner basis depends wildly on the ordering scheme between variables and
monomials, so it is important to choose a nice ordering scheme which eases the
calculations. This ordering is difficult to know in advance, so some amount of
trials and errors is involved in finding the Gröbner basis.

By setting an appropriate ordering scheme, it is possible to consistently
truncate the Gröbner basis for zero polynomials of $(g_{i},\lambda_{I})$, into
that for zero polynomials of $g_{i}$ only. Then, the set of all possible
remainders $r(g_{i})$ under division by the truncated Gröbner basis form a
faithful — complete but not overcomplete — set of all independent polynomials
of $g_{i}$, and therefore the set of all independent graviton operators.
Moreover, one can easily construct a monomial basis for this set of
remainders, from which it is straightforward to compute both the partition
function and the index over graviton operators.

Employing the two strategies — the eigenvalue counting and the Gröbner basis —
explained so far, we have obtained a closed-form expression for the graviton
index for the $SU(2)$ theory,

$Z_{\rm
grav}^{SU(2)}=\frac{1\\!+\\!3t^{4}\\!-\\!8t^{6}\\!-\\!6t^{10}\\!+\\!10t^{12}\\!+\\!9t^{14}\\!-\\!9t^{16}\\!+\\!16t^{18}\\!-\\!18t^{20}\\!-\\!3t^{22}\\!+\\!t^{24}\\!-\\!3t^{26}\\!+\\!9t^{28}\\!-\\!2t^{30}\\!+\\!3t^{32}\\!-\\!3t^{34}}{(1-t^{4})^{3}(1-t^{8})^{3}}~{}.$
(3.4)

This result was obtained in [17] where the eigenvalue counting strategy was
used but the Gröbner basis was not. Here, we have reproduced this result by
finding a Gröbner basis of relations between $SU(2)$ BMN gravitons that
consists of 66 generators (after truncation), and counting the set of all
possible remainders under division by those.

Unfortunately, the computation of the Gröbner basis quickly becomes very
cumbersome if the generators of the constraints
$\\{g_{i}-g_{i}(\lambda_{I})\\}$ are numerous and complicated. For relations
between a subset of $SU(3)$ BMN gravitons that do not involve $f$, i.e.
$u_{n}$ and $v_{n}$ in (2.20), we found the Gröbner basis with $1170$
generators (after truncation) after several hours of computation on a
computer. For the complete set of $SU(3)$ BMN gravitons including $w_{n}$, we
were unable to find the Gröbner basis due to lack of computing resources: it
takes months at least and it is tricky to parallelize. Therefore, we have
devised a hybrid method to take maximal advantage of the Gröbner basis
obtained for the non-$f$ subsector as we now describe.

We first list the complete and independent monomial basis of graviton
operators, i.e. set of monomials of the mesons $g_{i}$, that consist of
$u_{n},v_{n}$ but not of $w_{n}$ ($n=2,3$), up to the charge order $j=54$.
This can be done for any order $j$ because the Gröbner basis for the non-$f$
subsector has been obtained. Then, one can construct an overcomplete set of
all graviton operators by multiplying each basis from the previous step by
arbitrary numbers of $w_{2}$ and $w_{3}$, again up to $j=54$. Note that
$w_{2}$ and $w_{3}$ include 3 and 6 different species of single-graviton
operators, respectively, so the size of the overcomplete set grows quickly.

It is helpful to fragment the problem by classifying the operators according
to their charges. Namely, each charge sector is specified by 4 non-negative
integers $2J$ and $q_{I}=R_{I}+J$ (where $I=1,2,3$). Note that the overall
order $j$ used for grading the operators is $j=2(q_{1}+q_{2}+q_{3})$ and
therefore it is always even in the BMN sector. This classification is useful
because all single-graviton operators $u_{n},v_{n},w_{n}$ and therefore all
multi-graviton operators have definite charges, and operators with different
sets of charges can never have a linear relation between them. Moreover,
different charge sectors with merely permuted charges $(q_{1},q_{2},q_{3})$
should contain the same number of independent graviton operators. Therefore,
we separately consider the overcomplete basis of gravitons in each charge
sector with $q_{1}\leq q_{2}\leq q_{3}$.

In order to count linearly independent operators among the overcomplete set in
any charge sector, we rewrite each operator as a polynomial of the
eigenvalues. This is done by substituting the mesons with corresponding
eigenvalue polynomials $u_{n}(\lambda_{I})$, $v_{n}(\lambda_{I})$ and
$w_{n}(\lambda_{I})$, which are obtained by writing the gluons in terms of
their eigenvalues. For the eigenvalues of the $SU(3)$ traceless elementary
fields, we use the convention

$\displaystyle f=\begin{pmatrix}f_{1}&0&0\\\ 0&f_{2}&0\\\
0&0&-f_{1}-f_{2}\end{pmatrix}~{},$ (3.5)

and likes.

The number of independent polynomials within each charge sector is determined
as the rank of their coefficient matrix. We have used the software Singular
[36] for finding the Gröbner basis, writing each operator as an eigenvalue
polynomial, and extracting the coefficient matrix within each charge sector,
and numpy for computing the rank of the matrix.

The computation of indices for the $SU(3)$ theory have been performed up to
$j=54$ on personal computers. For example, the computation for the charge
sector $(2J,q_{1},q_{2},q_{3})=(7,9,9,9)$, which turns out to be the largest,
the coefficient matrix was $31026\times 20940$ with rank 3242.

For the counting of $SU(4)$ BMN gravitons, we take a similar hybrid approach.
Separation into charge sectors works identically to the $SU(3)$ theory.
However, computation of the Gröbner basis is even more heavy, both time-wise
and memory-wise, so we were only able to obtain the Gröbner basis for a
subsector of $SU(4)$ BMN gravitons involving $u_{n}$ ($n=2,3,4$), i.e. the
chiral primaries. We first list the complete and independent monomial basis of
the chiral primaries $u_{n}$ using the Gröbner basis, up to the order $j=30$.
Then we construct an overcomplete set of all multi-graviton operators within
each charge sector by multiplying each independent basis by appropriate
numbers of $v_{2}$, $v_{3}$, $v_{4}$, $w_{2}$, $w_{3}$ and $w_{4}$, again up
to $j=30$.

We write each operator in the overcomplete basis as a polynomial of the
eigenvalues. For the traceless elementary fields in the $SU(4)$ theory, we
used the following convention for the diagonal entries:

$\displaystyle f=\begin{pmatrix}f_{1}&0&0&0\\\ 0&f_{2}&0&0\\\
0&0&f_{3}-f_{1}&0\\\ 0&0&0&-f_{2}-f_{3}\end{pmatrix}~{},$ (3.6)

which slightly simplifies the polynomials compared to the more canonical
convention $f={\rm diag}(f_{1},~{}f_{2},~{}f_{3},~{}-f_{1}-f_{2}-f_{3})$.

The computation of indices for the $SU(4)$ theory have been performed up to
$j=30$ on personal computers. For example, the computation for the charge
sector $(2J,q_{1},q_{2},q_{3})=(3,5,5,5)$, which turns out to be the largest,
the coefficient matrix was $12079\times 116042$ with rank 3788.

###  3.1 $SU(3)$

Following the computational procedures explained above, we have computed the
$SU(3)$ graviton index $Z_{\rm grav}$ until $t^{54}$ order. We write the
difference $Z-Z_{\rm grav}$ with the full index $Z$, which is the index over
non-graviton cohomologies or the ‘black hole’ cohomologies, as

$Z-Z_{\rm grav}=Z_{\rm
core}(\Delta_{I})\cdot\prod_{I=1}^{3}\frac{1}{1-e^{-\Delta_{I}}e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\cdot\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}})\
.$ (3.7)

The factors that dress the index over _core_ non-graviton cohomologies will be
explained shortly. $Z_{\rm core}(\Delta_{I})\equiv f(t,x,y)$ with
$e^{-\Delta_{1}}=t^{2}x$, $e^{-\Delta_{2}}=t^{2}y^{-1}$,
$e^{-\Delta_{3}}=t^{2}x^{-1}y$ can be expanded as

$f(t,x,y)=\sum_{j=0}^{54}\sum_{{\bf R}_{j}}(-1)^{F({\bf R}_{j})}\chi_{{\bf
R}_{j}}(x,y)t^{j}+\mathcal{O}(t^{56})\ ,$ (3.8)

where ${\bf R}_{j}$ runs over the $SU(3)$ irreducible representations which
appear at $t^{j}$ order ($j$ is even in the BMN sector), $\chi_{{\bf
R}_{j}}(x,y)$ is its character, and $F({\bf R}_{j})$ is its fermion number.
The representations ${\bf R}_{j}$ appearing in the expansion of $f$, together
with their bosonic/fermionic natures, are shown in Table 1. We have classified
the representations into several groups, i.e. what we suspect to be the
fermionic towers $F_{0},...,F_{4}$, the bosonic towers $B_{1},...,B_{3}$, and
the remainders $F_{\rm exc}$, $B_{\rm exc}$ for which we do not see particular
patterns (thus named ‘exceptional’).

$j$ | $F_{0}$ | $F_{1}$ | $F_{2}$ | $F_{3}$ | $F_{4}$ | $F_{\rm exc}$ | $B_{1}$ | $B_{2}$ | $B_{3}$ | $B_{\rm exc}$
---|---|---|---|---|---|---|---|---|---|---
$24$ | $[0,0]$ |  |  |  |  |  |  |  |  |
$26$ |  |  |  |  |  |  |  |  |  |
$28$ |  |  |  |  |  |  |  |  |  |
$30$ | $[0,0]$ | $[3,0]$ |  |  |  |  |  |  |  |
$32$ |  | $[4,0]$ |  |  |  |  |  |  |  |
$34$ |  | $[5,0]$ |  |  |  |  | $[3,1]$ |  |  |
$36$ | $[0,0]$ | $[6,0]$ |  |  |  |  | $[4,1]$ |  |  | $[3,0]$
$38$ |  | $[7,0]$ |  |  |  | $[1,0]$ | $[5,1]$ |  |  |
$40$ |  | $[8,0]$ | $\color[rgb]{0,1,1}[5,0]$ |  | $\color[rgb]{0,1,1}[3,1]$ |  | $[6,1]$ |  |  |
$42$ | $[0,0]$ | $[9,0]$ | $\color[rgb]{0,1,1}[6,0]$ |  | $\color[rgb]{0,1,1}[4,1]$ |  | $[7,1]$ |  |  | $[1,1]$
$44$ |  | $[10,0]$ | $\color[rgb]{0,1,1}[7,0]$ |  | $\color[rgb]{.75,.75,.75}[5,1]$ |  | $[8,1]$ | $\color[rgb]{.75,.75,.75}[5,1]$ |  |
$46$ |  | $[11,0]$ | $\color[rgb]{0,1,1}[8,0]$ |  | $\color[rgb]{.75,.75,.75}[6,1]$ | $[2,0]$ | $[9,1]$ | $\color[rgb]{.75,.75,.75}[6,1]$ |  | $[5,0]$
$48$ | $[0,0]$ | $[12,0]$ | $\color[rgb]{0,1,1}[9,0]$ |  | $\color[rgb]{.75,.75,.75}[7,1]$ | $[3,0]$ | $[10,1]$ | $\color[rgb]{.75,.75,.75}[7,1]$ |  | $[4,1]$
$50$ |  | $[13,0]$ | $\color[rgb]{0,1,1}[10,0]$ | $\color[rgb]{0,1,1}[7,0]$ | $\color[rgb]{.75,.75,.75}[8,1]$ |  | $[11,1]$ | $\color[rgb]{.75,.75,.75}[8,1]$ |  | $[4,0]$
$52$ |  | [14,0] | [11,0] | [8,0] | [9,1] | [2,0] | $[12,1]$ | [9,1] |  | [3,1]
$54$ |  | [15,0] | [12,0] | [9,0] | [10,1] | [4,1] | $[13,1]$ | [10,1] | [7,1] |
Table 1: $SU(3)$ Dynkin labels of fermionic/bosonic black hole cohomologies
after factoring out the descendants and the conjectured graviton hairs of
$w_{2}$. The towers in cyan/gray colors may be related to $F_{1}$ or $B_{1}$
tower by partial hairs of $w_{3}$ gravitons. The representations colored in
gray are not detected by the index, but may exist in cancelling pairs in
$F_{4}$/$B_{2}$.

We comment on the factors which we have taken out in (3.7). The factor
$\prod_{I<J}(1-e^{-\Delta_{1}-\Delta_{2}})$ accounts for $SU(1|3)$
descendants. For each non-graviton cohomology in ${\bf R}_{j}$ that
contributes to $Z_{\rm core}$, the entire $SU(1|3)$ multiplet obtained by
acting the three fermionic generators $Q_{+}^{m}$ must also be non-graviton
cohomologies. Every such multiplet is a long multiplet of the $SU(1|3)$, so
the corresponding character is simply the contribution from the primary times
the factor $\prod_{I<J}(1-e^{-\Delta_{1}-\Delta_{2}})$. This fact can be
argued using the embedding supergroup $PSU(2,2|4)$ of the 4d $\mathcal{N}=4$
theory. For any of the three generators $Q_{+}^{m}$ to annihilate the
$SU(1|3)$ primary, the primary of a bigger representation of $PSU(2,2|4)$ that
includes the $SU(1|3)$ multiplet must be annihilated by $Q_{+}^{4}$ _and_ by
the $SU(4)_{R}$ lowering operator that is not part of the $SU(3)\subset
SU(4)_{R}$. The only $PSU(2,2|4)$ representations that satisfy this property
are $B_{1}\bar{B}_{1}[0;0]^{(0,n,0)}$, namely the graviton operators, or the
identity. For details on the relevant representation theory, we refer to [37],
particularly its section 2.2.4, or to appendix B of [17].

The second factor of (3.7) was taken out for an empirical reason, with an
expectation that they come from the graviton hairs of $w_{2}$’s in (2.20).
Namely, we conjecture that $w_{2}$ gravitons multiplying the core black hole
cohomologies represented by $Z_{\rm core}$ provide nontrivial product
cohomologies. Although we have little logical justification of the last claim
(except that similar hairs are allowed in the $SU(2)$ theory), we think that
the phenomenological evidence of this claim is compelling since various simple
patterns in Table 1 are clear only after factoring it out.

Now we comment on various structures that we observe from Table 1.

We first discuss the possible product cohomologies obtained by multiplying
black hole cohomologies and gravitons. When a product is $Q$-exact, it is
interpreted [17] as a finite $N$ generalization of the black hole no-hair
theorem in the BPS sector.555This does not mean that the product states are
absent: the $Q$-exact products acquire anomalous dimensions and become non-
BPS. Similar structures are known in the gravity dual: a charged bulk field
develops hairs around AdS black holes in the non-BPS regime, which become
pathological/singular in the BPS limit [38, 39, 17]. We cannot conclude just
from the index whether these product states exist or not. However, if possible
product states do not appear in the index, it is suggestive that the
corresponding hair is not allowed in the cohomology. In the $SU(2)$ theory,
several simple product cohomologies which do not appear in the index were
explicitly shown to be $Q$-exact. Similar calculations are much more difficult
for $SU(3)$, which we did not attempt.

First of all, as explained above, all possible product cohomologies obtained
by multiplying $w_{2}$ are factored out. Mainly appealing to the simpler
spectral structures of the resulting $Z_{\rm core}$ as shown in Table 1, we
conjecture that the $w_{2}$ hairs are (at least mostly) allowed. Apart from
$w_{2}$, we discuss below the possibilities of other graviton hairs. We divide
the discussions into the possible hairs of $w_{3}$ (which are the only
gravitons heavier than $w_{2}$) and the rest.

Table 1 does not seem to signal hairs from gravitons which are no heavier than
$w_{2}$. The gravitons with $j$ charges no larger than $w_{2}$ are: $u_{2}$ in
$[2,0]_{4}$, $v_{2}$ in $[1,1]_{6}$, $u_{3}$ in $[3,0]_{6}$, $v_{3}$ in
$[2,1]_{8}$, where the subscript denotes $j$. Some products clearly do not
appear in Table 1. For instance, $[0,0]_{24}^{F_{0}}$ in Table 1 times $u_{2}$
does not appear in the index, because we find no states at $[2,0]_{28}$ with
fermionic statistics in the table. There are many more products which
similarly do not appear in the table. Of course, just by matching charges and
representations, there are several possibilities in which an entry of Table 1
can be accounted for by a tensor product of another entry and these gravitons.
For instance, $u_{2}$ may multiply the state $[p,q]_{j}$ in $F_{1},...,F_{4}$
or $B_{1},...,B_{3}$ (with either $q=0,1$) to yield $[p+2,q]_{j+4}$ in the
same tower. However, we feel that this seems quite unlikely. For instance, in
$F_{1}$, this may explain $[5,0]_{34},[7,0]_{38},\cdots$ from $[3,0]_{30}$,
but not $[4,0]_{32},[6,0]_{36},\cdots$. In other words, all these towers are
more naturally generated by adding one scalar at a time (as an adjoint letter)
rather than two scalars (in the gauge-invariant form of $u_{2}$). Similarly,
$u_{3}$ may multiply $[p,q]_{j}$ in a tower to yield $[p+3,q]_{j+6}$ in the
same tower, but again for the same reason we feel this unappealing. Apart from
these, sporadically, there exist several group theoretic possibilities of
light graviton hairs in Table 1, in particular accounting for some entries in
$F_{\rm exc}$ or $B_{\rm exc}$ as product cohomologies. We shall not list all
the possibilities here. It may be worthwhile to think about whether some of
the ‘exceptional’ cohomologies in $F_{\rm exc}$ or $B_{\rm exc}$ can be
accounted for by such sporadic graviton hairs.

Now we consider possible hairs obtained by multiplying $w_{3}$ of (2.20).
Again, this is just a possibility from matching the charges/representations.
It is group theoretically possible that the towers $F_{2}$, $F_{3}$ are the
hairy cohomologies obtained by multiplying $w_{3}$’s to the $F_{1}$ tower.
This is because the product of $[n,0]_{j}$ in $F_{1}$ and $[2,0]_{10}$
decomposes to

$[n,0]_{j}\otimes[2,0]_{10}=[n+2,0]_{j+10}\oplus[n,1]_{j+10}\oplus[n-2,2]_{j+10}\
,$ (3.9)

containing $[n+2,0]_{j+10}$ which is in $F_{2}$. Similarly, multiplying two
such gravitons, one obtains states in $[n+4,0]_{j+20}$ which is in $F_{3}$. If
$F_{4}$ also forms a tower, with partial cancellations with $B_{2}$, then
$F_{4}$ may also be product cohomologies of $F_{1}$ and these gravitons
because the right hand side of (3.9) contains $[n,1]_{j+10}$. Similarly, if
the tower $B_{2}$ exists (by cancelling partly with $F_{4}$), $B_{2}$ and
$B_{3}$ could be product cohomologies of $B_{1}$ and $w_{3}$’s because
$[n,1]_{j}\otimes[2,0]_{10}$ contains $[n+2,1]_{j+10}$. The possibilities
discussed above suggest that the colored towers (cyan or gray) in Table 1 may
be product cohomologies containing $w_{3}$. If these colored towers are indeed
obtained by multiplying $w_{3}$ to $F_{1}$ or $B_{1}$, note that the last
possible tower of $[n-2,2]_{j+10}$ in (3.9) does not exist in the index, which
could mean that this part of the product is $Q$-exact.

If the possibilities raised in the previous paragraph are indeed true, it
implies that the allowed hair structure of the $w_{3}$ gravitons in
$[2,0]_{10}$ is more delicate than that of $w_{2}$. For instance, our
conjecture on $w_{2}$ is that their hairs are universally allowed,
irrespective of the core black hole cohomology chosen. On the other hand, if
the scenario of the previous paragraph is true, the allowed/disallowed
combinations of the $w_{3}$ hair are determined after entangling the gravitons
with the core black hole states. This is not the familiar form of the no-hair
theorem of semiclassical black hole physics. That is, the no-hair theorem as
well as its violation is stated for a given black hole background, which
represents the whole ensemble of states. It will be interesting to see if the
allowed hairs exhibit subtle dependence on the fine-grained information within
the black hole ensemble or not, and if they do, whether it may have
implications to the fuzzball paradigm of black holes [40]. We emphasize that
all the scenarios discussed above can be straightforwardly confirmed/disproved
once the cohomologies of $F_{1},B_{1}$ are constructed, by checking whether
the product cohomologies discussed above are $Q$-exact or not.

Now we discuss the possible charge structures and the field contents of the
towers. We first make general considerations. Suppose that $n_{\phi}$ scalars,
$n_{\psi}$ fermions and $n_{f}$ field strengths are used to make the operator
in $SU(3)$ representation $[p,q]$. The operator is not necessarily made by
just one choice of $n_{\phi},n_{\psi},n_{f}$ but superposes many different
terms in general. So we are in fact studying the structures of a given term in
the operator. Let us also introduce the following non-negative integers: $l$
pairs of contractions are made between the $SU(3)$ indices of $\phi$’s and
$\psi$’s; $m_{\phi}$ pairs of $\phi$’s are contracted with $\epsilon_{abc}$ to
yield lower indices; $m_{\psi}$ pairs of $\psi$’s are contracted with
$\epsilon^{abc}$ to yield upper indices; $b_{\phi}$ threesomes of $\phi$’s are
contracted with $\epsilon_{abc}$; $b_{\psi}$ threesomes of $\psi$’s are
contracted with $\epsilon^{abc}$. Once the above contractions are made, the
remaining upper/lower indices are respectively symmetrized to yield the
$[p,q]$ representation (after subtracting certain terms to ensure that the
indices are traceless). So one obtains

$\displaystyle p$ $\displaystyle=$ $\displaystyle
n_{\phi}-l-2m_{\phi}-3b_{\phi}+m_{\psi}~{},$ (3.10) $\displaystyle q$
$\displaystyle=$ $\displaystyle n_{\psi}-l-2m_{\psi}-3b_{\psi}+m_{\phi}\ .$

Since $\phi$, $\psi$ and $f$ carry $(R,J)=(\frac{1}{3},0)$,
$(\frac{1}{6},\frac{1}{2})$ and $(0,1)$ respectively, the charges are given by

$\displaystyle R$ $\displaystyle=$
$\displaystyle{\textstyle\frac{n_{\phi}}{3}+\frac{n_{\psi}}{6}}={\textstyle\frac{p}{3}+\frac{q}{6}+\frac{l}{2}+\frac{m_{\phi}}{2}+b_{\phi}+\frac{b_{\psi}}{2}}~{},$
(3.11) $\displaystyle J$ $\displaystyle=$
$\displaystyle{\textstyle\frac{n_{\psi}}{2}}+n_{f}={\textstyle\frac{q}{2}+n_{f}+\frac{l}{2}+m_{\psi}+\frac{3b_{\psi}}{2}-\frac{m_{\phi}}{2}}~{},$
$\displaystyle j$ $\displaystyle=$ $\displaystyle
6(R+J)=2p+4q+6n_{f}+6l+6m_{\psi}+6b_{\phi}+12b_{\psi}\ .$

Now we apply these results to the tower $F_{1}$, for the states with
$[p,q]_{j}=[n+3,0]_{30+2n}$, where $n=0,1,2,\cdots$. (Similar studies can be
made to other towers in Table 1 except $F_{0}$.) Inserting these expressions
to (3.11), one obtains

$\displaystyle 4$ $\displaystyle=$ $\displaystyle
n_{f}+l+m_{\psi}+b_{\phi}+2b_{\psi}~{},$ (3.12) $\displaystyle J$
$\displaystyle=$ $\displaystyle
n_{f}+{\textstyle\frac{l}{2}+m_{\psi}+\frac{3b_{\psi}}{2}-\frac{m_{\phi}}{2}}~{},$
$\displaystyle R$ $\displaystyle=$
$\displaystyle{\textstyle\frac{n}{3}+1+\frac{l}{2}+\frac{m_{\phi}}{2}+b_{\phi}+\frac{b_{\psi}}{2}}\geq{\textstyle\frac{n}{3}+1}\
.$

The last inequality is saturated if and only if
$l=m_{\phi}=b_{\phi}=b_{\psi}=0$. On the other hand, from $j=2n+30$ and the
lower bound of $R$, one obtains

$J={\textstyle\frac{j}{6}-R}={\textstyle 5+\frac{n}{3}-R}\leq 4\ .$ (3.13)

This implies that $F_{1}$ is a tower in which $R$ increases indefinitely with
$J\leq 4$ bound. From the first equation of (3.12),
$n_{f},l,m_{\psi},b_{\phi},b_{\psi}$ are all bounded from above. Also, from
the second equation of (3.10), one finds
$n_{\psi}=l+2m_{\psi}+3b_{\psi}-m_{\phi}$. The first three terms on the right
hand side are bounded, so $n_{\psi}$ is bounded from above. Furthermore, for
the non-negativity of $n_{\psi}$, $m_{\phi}$ is also bounded from above. The
only unbound non-negative integer is $n_{\phi}$. So this tower has increasing
numbers of scalars. In other words, this is a Kaluza-Klein tower, carrying
increasing momentum charges on $S^{5}$, rather than a higher-spin tower.

We do not know which type of tower $F_{0}$ is, between higher-spin and Kaluza-
Klein. Apparently, the states in $F_{0}$ appear when $j$ increases by $6$,
except at $j=54$ where we found no states in $[0,0]$ representation. It is
unclear which of the following is true (if any): (i) the absence of the tower
$F_{0}$ beyond this charge; (ii) the multiple tower structure within $F_{0}$
with different periods; (iii) existence of an exceptional bosonic cohomology
at this order which cancels with the tower at $j=54$. Related to this tower,
we comment that the $SU(2)$ cohomologies in the BMN sector showed the
following index [17]:

$\displaystyle Z-Z_{\rm grav}$ $\displaystyle=$ $\displaystyle Z_{\rm
core}(\Delta_{I})\cdot\prod_{I=1}^{3}\frac{1}{1-e^{-\Delta_{I}}e^{-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\cdot\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}})~{},$
$\displaystyle Z_{\rm core}$ $\displaystyle=$
$\displaystyle=-\sum_{n=0}^{\infty}t^{24+12n}\ .$ (3.14)

The whole tower of cohomologies for $Z_{\rm core}$ was also constructed at
arbitrarily large $j$, with $R$ fixed and $J$ increasing. So this is a higher
spin tower of core black hole primaries. This suggests that the $SU(2)$ tower
may be protected by certain symmetries, perhaps of the sort discussed in [20].
It is not clear whether the $F_{0}$ tower in Table 1 can be understood in a
similar way.

The Kaluza-Klein towers are apparently not related to any symmetries and thus
may not be protected at large energies or large $j$. In particular, it will be
interesting to see if these towers are related to the towers of giant
gravitons appearing in the so-called ‘giant graviton expansion’ of the index
[23, 24, 25, 26]. This expansion recasts the index as an auxiliary summation
over a tower, with increasing giant graviton number. More precisely, the giant
graviton tower includes both finite $N$ gravitons (or the finite $N$ trace
relations to be subtracted) as well as new black hole states formed by bound
states of D-branes and open string excitations. In this framework, in certain
charge regimes, the black hole entropy is obtained by first computing the
entropy $S(j,n)$ with fixed giant graviton number $n$ and then maximizing
$S(j,n)$ in $n$ at fixed $j$ [5, 41]. (See [41, 42] for how one may generalize
this calculation.) Indeed there is a signal that the giant graviton tower
loses its meaning at higher $n\gg N$ [41]. With this interpretation in mind,
it will be interesting to compute $Z-Z_{\rm grav}$ either till higher orders
in $j$ or exactly, to see whether the KK towers survive at arbitrary high $j$
or not.

###  3.2 $SU(4)$

In the $SU(4)$ case, we computed $Z_{\rm grav}$ till $j=30$ level. The index
$Z-Z_{\rm grav}$ over non-graviton cohomologies is given by

$Z-Z_{\rm
grav}=\left[-\chi_{[2,0]}(x,y)t^{28}-\chi_{[3,0]}(x,y)t^{30}+\mathcal{O}(t^{32})\right]\cdot\prod_{I<J}(1-e^{-\Delta_{I}-\Delta_{J}})\
.$ (3.15)

The second factor generates the Fock space of each $SU(1|3)$ multiplet, while
the first factor in the square parenthesis represents the non-graviton
primaries. One finds that the BMN index predicts an apparent threshold of non-
graviton cohomologies at $j=6(R+J)=28$. Again, conservatively, this is an
upper bound for the threshold for two different reasons: first because the
index may miss a pair of canceling threshold cohomologies at lower charges,
and also because the true threshold might lie outside the BMN sector (carrying
nonzero $SU(2)_{r}$ spin $J_{1}-J_{2}$). Anyway, the above apparent threshold
is higher than the $SU(3)$ threshold. So it is natural to expect that it was
an exception that the $SU(2)$ and $SU(3)$ thresholds were the same: the
(apparent) thresholds for $j=6(R+J)$ are $24,24,28,\cdots$ for
$N=2,3,4\cdots$. To obtain the threshold level in terms of energy $E=3R+2J$,
one should construct the actual cohomologies which account for the $t^{28}$
term. This will not be done in this paper.

##  4 Constructing cohomologies

The cohomologies we would like to construct should be $Q$-closed and not
$Q$-exact. Unlike gravitons, the $Q$-closedness of the black hole cohomologies
should be ensured by the trace relations. (Otherwise, if it is a cohomology at
given energy and at arbitrary values of $N$, it is a graviton cohomology.) So
it is important to know what kind of nontrivial trace relations are available
for $N\times N$ matrices when the number of fields is larger than $N$.

It seems to be widely believed that all $SU(N)$ trace relations are derived
from the Cayley-Hamilton identity. For instance, see [43] (p.7, below
eqn.(19)) and [44]. But in practice it is inefficient to search for the trace
relations that we need just from this identity. Fortunately, we already know
many trace relations from the calculations reported in section 3. Namely, when
enumerating finite $N$ gravitons, we have counted them subject to various
trace relations of the generators $g_{i}$. So one can take advantage of these
trace relations to construct black hole cohomologies. This leads to our
‘ansatz’ for black hole cohomologies, which we explain now.

We can motivate the ideas with a simple example in the $SU(2)$ theory [15, 16,
17]. A representative of the threshold non-graviton cohomology in $SU(2)$ is
given by

$O_{0}\equiv\epsilon^{abc}{(v_{2})^{m}}_{a}{(v_{2})^{n}}_{b}{\rm
tr}(\psi_{(c}\psi_{m}\psi_{n)})$ (4.1)

where $v_{2}$ is the graviton operator in the $S_{2}$ multiplet. Let us see
how this operator becomes $Q$-closed. Acting $Q$ on $O_{0}$, $Q$ acts only on
${\rm tr}(\psi_{(c}\psi_{m}\psi_{n)})$ since $v_{2}$ is $Q$-closed. One
obtains

$Q{\rm
tr}(\psi_{(c}\psi_{m}\psi_{n)})\propto\epsilon_{ab(c}{(v_{2})^{a}}_{m}{(v_{2})^{b}}_{n)}\equiv
R(v_{2})_{cmn}$ (4.2)

after using $SU(2)$ trace relations. Plugging this into $QO_{0}$, one obtains

$QO_{0}\propto\epsilon^{abc}{(v_{2})^{m}}_{a}{(v_{2})^{n}}_{b}R(v_{2})_{cmn}=0\
.$ (4.3)

At the last step, one can show that the quartic mesonic polynomial
$\epsilon^{abc}{(v_{2})^{m}}_{a}{(v_{2})^{n}}_{b}R(v_{2})_{cmn}$ is
identically zero [17]. From the viewpoint of section 3, (4.2) are graviton
trace relations and the last step of (4.3) is a relation of relations. So the
operator $O_{0}$ is shown to be $Q$-closed by using the trace relations and a
relation of relations of the finite $N$ graviton operators.

This idea can be extended to construct operators which become $Q$-closed only
after using trace relations. Namely, for each relation of relations such as
(4.3), we can construct a $Q$-closed operator such as (4.1). We still need to
check that they are not $Q$-exact for them to represent nontrivial
$Q$-cohomologies. Also, there are non-graviton cohomologies which are not
constructed this way [17]. For these reasons, the $Q$-closed operators
constructed in this way are mere ansätze for the non-graviton cohomologies. In
the appendix, we have collected all $SU(3)$ fundamental trace relations
including $u_{n},v_{n}$ only, and manifestly wrote them in $Q$-exact forms. We
have also found trace relations involving $u_{n},v_{n},w_{n}$ till $j=20$
order. We have also found all relations between the fundamental graviton trace
relations at $j=24$ and some of them at $j=30$ orders in the $SU(3)\subset
SO(6)_{R}$ singlet sector where the index predicts non-graviton cohomologies
(see Table 1). In other charge sectors, one can immediately write down
$Q$-closed operators if one finds new relations of the fundamental trace
relations.

When we write a fundamental trace relation $R_{a}$ in a $Q$-exact form,
$R_{a}\sim Qr_{a}$, there is an ambiguity in $r_{a}$ by addition of arbitrary
$Q$-closed operators. We partly fix it so that $r_{a}$ vanishes when all the
letters are restricted to diagonal matrices. Since the $Q$-closed operators
constructed from relations of relations are linear combinations of $r_{a}$’s,
they vanish with diagonal letters. This makes it impossible for our ansatz to
be gravitons. So our ansatz either yields $Q$-exact operators or non-graviton
cohomologies.

Based on the ansatz, in subsection 4.1, we construct a number of gauge-
invariant $Q$-closed non-graviton operators at $j=24$ order that are singlets
of $SU(3)$ global symmetry. Only one of them is not $Q$-exact, representing
the non-graviton cohomology predicted by the index. In subsection 4.2, we
sketch how we checked the (non-)$Q$-exactness of these operators, while also
showing that there are no other non-graviton cohomologies in the $j=24$,
$SU(3)$ singlet sector.

###  4.1 $SU(3)$ threshold cohomology from ansätze

In this subsection, we present the explicit form of the black hole cohomology
at the threshold level $j=24$ which is singlet under $SU(3)\subset SU(4)_{R}$
global symmetry, in the BMN sector of the $SU(3)$ gauge theory. We first list
the non-graviton $Q$-closed operators from our ansatz. We find one
non-$Q$-exact operator among them, which is the threshold cohomology.

At $j\equiv 6(R+J)=24$, operators are further distinguished by R-charges
$R\equiv\frac{R_{1}+R_{2}+R_{3}}{3}$. The BMN operators which are
$SU(3)\subset SU(4)_{R}$ singlets satisfy $R_{1}=R_{2}=R_{3}$ and
$J_{1}=J_{2}$. Then the possible charges of the operators are
$(R,J)=(\frac{n}{2},\frac{8-n}{2})$ where $n=0,\cdots,8$. In each charge
sector, the number of letters is fixed to $n+4$. However, our ansatz further
restricts the charges since acting $Q$ on our ansatz should become a
polynomial of $u_{2,3},v_{2,3},w_{2,3}$. As a result, there exist total 7
possible charge sectors within our ansatz: $(R,J)=(\frac{n}{2},\frac{8-n}{2})$
where $n=1,\cdots,7$.

When $(R,J)=(\frac{1}{2},\frac{7}{2})$ or $(1,3)$, there are no $Q$-closed
operators within our ansatz using the trace relations in the appendix. One can
understand it heuristically as follows. At these charges, $R$ is so small that
only a small number of scalars is admitted. As the graviton generators contain
at least one scalar field, only few types of graviton polynomials exist in
these sectors, which are not enough to host relations of relations. Therefore,
these charge sectors are incompatible with our ansatz. The other $5$ charge
sectors host $Q$-closed operators in our ansatz, whose explicit forms will be
presented below.

We now present the $Q$-closed non-graviton operators in each of the five
charge sectors, $(R,J)=(\frac{n}{2},\frac{8-n}{2})$ where $n=3,\cdots,7$. For
convenience, we rewrite here the definition of the single-trace generators of
the $SU(3)$ BMN gravitons $u_{2,3},v_{2,3},w_{2,3}$:

$\displaystyle u^{ij}\equiv$
$\displaystyle\;\textrm{tr}\left(\phi^{(i}\phi^{j)}\right)\ ,\ \
u^{ijk}\equiv\;\textrm{tr}\left(\phi^{(i}\phi^{j}\phi^{k)}\right)\ ,$ (4.4)
$\displaystyle{v^{i}}_{j}\equiv$
$\displaystyle\;\textrm{tr}\left(\phi^{i}\psi_{j}\right)-{\textstyle\frac{1}{3}}\delta^{i}_{j}\textrm{tr}\left(\phi^{a}\psi_{a}\right)\
,\ \
{v^{ij}}_{k}\equiv\;\textrm{tr}\left(\phi^{(i}\phi^{j)}\psi_{k}\right)-{\textstyle\frac{1}{4}}\delta^{i}_{k}\textrm{tr}\left(\phi^{(j}\phi^{a)}\psi_{a}\right)-{\textstyle\frac{1}{4}}\delta^{j}_{k}\textrm{tr}\left(\phi^{(i}\phi^{a)}\psi_{a}\right)\
,$ $\displaystyle w^{i}\equiv$
$\displaystyle\;\textrm{tr}\left(f\phi^{i}+{\textstyle\frac{1}{2}}\epsilon^{ia_{1}a_{2}}\psi_{a_{1}}\psi_{a_{2}}\right)\
,\ \
w^{ij}\equiv\;\textrm{tr}\left(f\phi^{(i}\phi^{j)}+\epsilon^{a_{1}a_{2}(i}\phi^{j)}\psi_{a_{1}}\psi_{a_{2}}\right)\
.$

#### i) $(R,J)=(\frac{3}{2},\frac{5}{2})$

The operators in this sector are made of 7 letters. The possible numbers
$(n_{\phi},n_{\psi},n_{f})$ of scalars, fermions and $f$ in each term are
$(n_{\phi},n_{\psi},n_{f})=(4,1,2),(3,3,1),(2,5,0)$. We find 1 $Q$-closed
operator in this sector from the trace relations and a relation of relations
in appendix A. This $Q$-closed operator is given by

$O^{(2,1)}\equiv
65u^{ij}(r_{20}^{(2,1)})_{ij}-39w^{ij}(r_{14}^{(1,1)})_{ij}+5w^{i}(r_{16}^{(1,1)})_{i}+312{v^{jk}}_{i}(r_{16}^{(1,2)})^{i}_{jk}+26{v^{j}}_{i}(r_{18}^{(1,2)})^{i}_{j}+6w^{i}(r_{16}^{(0,3)})_{i}\
.$ (4.5)

The superscripts denote $(n_{f},n_{\psi})$ of the terms with maximal $n_{f}$
in the operator. $r_{j}^{(n_{f},n_{\psi})}$’s are given in (A), (A) where
$R_{j}^{(n_{f},n_{\psi}-1)}\equiv i\,Q\,r_{j}^{(n_{f},n_{\psi})}$’s are the
fundamental trace relations. The $Q$-closed operator (4.5) turns out to be
$Q$-exact. In fact, (4.5) is even under the parity transformation of [45]. It
is already known that all such even operators in this charge sector are
$Q$-exact for all $N\geq 3$ [19].

#### ii) $(R,J)=(2,2)$

The operators in this sector are made of 8 letters. Allowed
$(n_{\phi},n_{\psi},n_{f})$ are $(6,0,2),(5,2,1),(4,4,0)$. We find 4
$Q$-closed operators in this sector given by

$\displaystyle O_{1}^{(1,2)}\equiv$
$\displaystyle-3{v^{(j}}_{i}w^{k)}(r_{10}^{(0,1)})^{i}_{jk}-3u^{(ij}w^{k)}(r_{12}^{(0,2)})_{ijk}+\epsilon_{a_{1}a_{2}i}u^{a_{1}j}w^{a_{2}}(r_{12}^{(0,2)})^{i}_{j}\
,$ (4.6) $\displaystyle O_{2}^{(1,2)}\equiv$
$\displaystyle-9u^{a(i}{v^{j)}}_{a}(r_{14}^{(1,1)})_{ij}+10\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(r_{14}^{(1,1)})^{ij}_{k}+30{v^{(j}}_{i}w^{k)}(r_{10}^{(0,1)})^{i}_{jk}+60u^{(jk}{v^{l)}}_{i}(r_{14}^{(0,3)})^{i}_{jkl}\
,$ $\displaystyle O_{3}^{(1,2)}\equiv$
$\displaystyle-3u^{a(i}{v^{j)}}_{a}(r_{14}^{(1,1)})_{ij}+6\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(r_{14}^{(1,1)})^{ij}_{k}+4u^{ijk}(r_{18}^{(1,2)})_{ijk}+14{v^{(j}}_{i}w^{k)}(r_{10}^{(0,1)})^{i}_{jk}$
$\displaystyle-6w^{ij}(r_{14}^{(0,2)})_{ij}-12\epsilon^{a_{1}a_{2}(i}{v^{j}}_{a_{1}}{v^{k)}}_{a_{2}}(r_{12}^{(0,2)})_{ijk}-4{v^{j}}_{a}{v^{a}}_{i}(r_{12}^{(0,2)})^{i}_{j}\
,$ $\displaystyle O_{4}^{(1,2)}\equiv$
$\displaystyle-3u^{a(i}{v^{j)}}_{a}(r_{14}^{(1,1)})_{ij}+14\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(r_{14}^{(1,1)})^{ij}_{k}+8{v^{jk}}_{i}(r_{16}^{(1,1)})^{i}_{jk}+42{v^{(j}}_{i}w^{k)}(r_{10}^{(0,1)})^{i}_{jk}$
$\displaystyle+12u^{(ij}w^{k)}(r_{12}^{(0,2)})_{ijk}-24w^{ij}(r_{14}^{(0,2)})_{ij}-36\epsilon^{a_{1}a_{2}(i}{v^{j}}_{a_{1}}{v^{k)}}_{a_{2}}(r_{12}^{(0,2)})_{ijk}-8{v^{jk}}_{i}(r_{16}^{(0,3)})^{i}_{jk}\
.$

All operators in (4.6) are $Q$-exact.

#### iii) $(R,J)=(\frac{5}{2},\frac{3}{2})$

The operators in this sector are made of 9 letters. Allowed
$(n_{\phi},n_{\psi},n_{f})$ are $(7,1,1),(6,3,0)$. We find 13 $Q$-closed
operators in this sector given by

$\displaystyle
O_{1}^{(1,1)}\equiv\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}w^{k)a_{2}}(r_{10}^{(0,1)})^{i}_{jk}\
,$ (4.7) $\displaystyle
O_{2}^{(1,1)}\equiv\epsilon_{a_{1}a_{2}i}u^{a_{1}jk}w^{a_{2}}(r_{10}^{(0,1)})^{i}_{jk}\
,$ $\displaystyle
O_{3}^{(1,1)}\equiv\epsilon_{a_{1}a_{2}i}\epsilon_{b_{1}b_{2}j}u^{a_{1}b_{1}}u^{a_{2}b_{2}k}(r_{14}^{(1,1)})^{ij}_{k}+5{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}-2{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}\
,$ $\displaystyle
O_{1}^{(0,3)}=-\epsilon_{ia_{1}a_{2}}\left(4u^{a_{1}b}{v^{ja_{2}}}_{b}+3u^{ja_{1}b}{v^{a_{2}}}_{b}\right)(r_{12}^{(0,2)})^{i}_{j}=\frac{1}{2}i\,Q((r_{12}^{(0,2)})^{i}_{j}(r_{12}^{(0,2)})^{j}_{i})\
,$ $\displaystyle
O_{2}^{(0,3)}=-\epsilon_{a_{1}a_{2}(i}\left(u^{a_{1}(k}{v^{l)a_{2}}}_{j)}+u^{kla_{1}}{v^{a_{2}}}_{j)}\right)(r_{12}^{(0,2)})^{ij}_{kl}=\frac{1}{2}i\,Q((r_{12}^{(0,2)})^{kl}_{ij}(r_{12}^{(0,2)})^{ij}_{kl})\
,$ $\displaystyle
O_{3}^{(0,3)}\equiv-u^{a(i}{v^{jk)}}_{a}(r_{12}^{(0,2)})_{ijk}\ ,$
$\displaystyle
O_{4}^{(0,3)}\equiv-\epsilon_{a_{1}a_{2}i}u^{a_{1}b}{v^{a_{2}}}_{b}(r_{14}^{(0,2)})^{i}\
,$ $\displaystyle O_{5}^{(0,3)}\equiv
6{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}+6u^{a(ij}{v^{k)}}_{a}(r_{12}^{(0,2)})_{ijk}+\epsilon_{a_{1}a_{2}i}u^{a_{1}bj}{v^{a_{2}}}_{b}(r_{12}^{(0,2)})^{i}_{j}\
,$ $\displaystyle O_{6}^{(0,3)}\equiv
24{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}+6u^{a(i}{v^{j)}}_{a}(r_{14}^{(0,2)})_{ij}-\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(r_{14}^{(0,2)})^{ij}_{k}\
,$ $\displaystyle
O_{7}^{(0,3)}\equiv{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}-10{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}+6u^{a(ij}{v^{k)}}_{a}(r_{12}^{(0,2)})_{ijk}+10\epsilon_{a_{1}a_{2}(i}u^{a_{1}kl}{v^{a_{2}}}_{j)}(r_{12}^{(0,2)})^{ij}_{kl}\
,$ $\displaystyle O_{8}^{(0,3)}\equiv
5{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}-2{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}+9u^{a(ij}{v^{k)}}_{a}(r_{12}^{(0,2)})_{ijk}+6\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}u^{kl)a_{2}}(r_{14}^{(0,3)})^{i}_{jkl}\
,$ $\displaystyle O_{9}^{(0,3)}\equiv
6{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}+12{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}+18u^{a(ij}{v^{k)}}_{a}(r_{12}^{(0,2)})_{ijk}-\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(r_{14}^{(0,2)})^{ij}_{k}\
,$ $\displaystyle O_{10}^{(0,3)}\equiv
38{v^{a}}_{i}{v^{jk}}_{a}(r_{10}^{(0,1)})^{i}_{jk}+4{v^{(j}}_{a}{v^{k)a}}_{i}(r_{10}^{(0,1)})^{i}_{jk}+24u^{a(ij}{v^{k)}}_{a}(r_{12}^{(0,2)})_{ijk}+5u^{(jk}{v^{l)}}_{i}(r_{14}^{(0,2)})^{i}_{jkl}\
.$

All except for $O^{(0,3)}_{6}$ in (4.7) are $Q$-exact. Therefore, a
representative of the cohomology in this sector can be written as

$\displaystyle O$ $\displaystyle\equiv$ $\displaystyle-6O_{6}^{(0,3)}$
$\displaystyle=$
$\displaystyle\;288{v^{j}}_{a}{v^{ka}}_{i}\epsilon_{c_{1}c_{2}(j}\textrm{tr}\left(\phi^{c_{1}}\phi^{c_{2}}\phi^{i}\psi_{k)}\right)-72{v^{a}}_{b}{v^{bk}}_{a}\epsilon_{c_{1}c_{2}(k}\textrm{tr}\left(\phi^{c_{1}}\phi^{c_{2}}\phi^{d}\psi_{d)}\right)$
$\displaystyle+36\epsilon_{a_{1}a_{2}i}u^{a_{1}k}{v^{a_{2}}}_{j}\left[2\textrm{tr}\left(\phi^{(i}\phi^{c}\phi^{j)}\psi_{(c}\psi_{k)}\right)+2\textrm{tr}\left(\phi^{(i|}\phi^{c}\phi^{|j)}\psi_{(c}\psi_{k)}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.+9\textrm{tr}\left(\phi^{(i}\phi^{j}\psi_{(c}\phi^{c)}\psi_{k)}\right)-6\textrm{tr}\left(\phi^{(i}\phi^{j)}\psi_{(c}\phi^{c}\psi_{k)}\right)\right]$
$\displaystyle-9\epsilon_{a_{1}a_{2}j}u^{a_{1}b}{v^{a_{2}}}_{b}\left[2\textrm{tr}\left(\phi^{(j}\phi^{c}\phi^{d)}\psi_{(c}\psi_{d)}\right)+2\textrm{tr}\left(\phi^{(j|}\phi^{c}\phi^{|d)}\psi_{(c}\psi_{d)}\right)\right.$
$\displaystyle\qquad\qquad\qquad\quad\;\;\,\left.\,+9\textrm{tr}\left(\phi^{(j}\phi^{d}\psi_{(c}\phi^{c)}\psi_{d)}\right)-6\textrm{tr}\left(\phi^{(j}\phi^{d)}\psi_{(c}\phi^{c}\psi_{d)}\right)\right]$
$\displaystyle-20u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}b_{3}}\left[2\textrm{tr}\left(\psi_{(i}\psi_{j)}\phi^{b_{1}}\phi^{b_{2}}\phi^{b_{3}}\right)+\textrm{tr}\left(\psi_{(i}\phi^{b_{1}}\psi_{j)}\phi^{b_{2}}\phi^{b_{3}}\right)\right]$
$\displaystyle-36u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\psi_{c}\phi^{b_{1}}\phi^{b_{2}}\phi^{c}\right)+\textrm{tr}\left(\psi_{j)}\psi_{c}\phi^{b_{1}}\phi^{c}\phi^{b_{2}}\right)+\textrm{tr}\left(\psi_{j)}\psi_{c}\phi^{c}\phi^{b_{1}}\phi^{b_{2}}\right)\right]$
$\displaystyle-36u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\psi_{c}\phi^{b_{2}}\phi^{c}\right)+\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\psi_{c}\phi^{c}\phi^{b_{2}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{c}\psi_{c}\phi^{b_{1}}\phi^{b_{2}}\right)\right]$
$\displaystyle-36u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\phi^{b_{2}}\psi_{c}\phi^{c}\right)+\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\phi^{c}\psi_{c}\phi^{b_{2}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{c}\phi^{b_{1}}\psi_{c}\phi^{b_{2}}\right)\right]$
$\displaystyle-36u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\phi^{b_{2}}\phi^{c}\psi_{c}\right)+\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\phi^{c}\phi^{b_{2}}\psi_{c}\right)+\textrm{tr}\left(\psi_{j)}\phi^{c}\phi^{b_{1}}\phi^{b_{2}}\psi_{c}\right)\right]$
$\displaystyle+12u^{ai}{v^{j}}_{a}\epsilon_{b_{1}b_{2}(i}\left[5\textrm{tr}\left(\psi_{j)}\phi^{b_{1}}\phi^{b_{2}}\right)\textrm{tr}\left(\psi_{c}\phi^{c}\right)+2\textrm{tr}\left(\psi_{j)}\phi^{(b_{1}}\phi^{c)}\right)\textrm{tr}\left(\psi_{c}\phi^{b_{2}}\right)-2\textrm{tr}\left(\psi_{j)}\phi^{b_{2}}\right)\textrm{tr}\left(\psi_{c}\phi^{(b_{1}}\phi^{c)}\right)\right]\
.$

The scaling dimension of this cohomology $O$ is $E=3R+2J=\frac{21}{2}$. Note
that the representative found above does not contain the letter $f$.

#### iv) $(R,J)=(3,1)$

The operators in this sector are made of 10 letters. Allowed
$(n_{\phi},n_{\psi},n_{f})$ are $(9,0,1),(8,2,0)$. We find 6 $Q$-closed
operators in this sector given by

$\displaystyle
O^{(0,2)}_{1}\equiv-\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}+2\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{a_{2}(j}{v^{k)}}_{b}(r_{10}^{(0,1)})^{i}_{jk}\
,$ (4.9) $\displaystyle
O^{(0,2)}_{2}\equiv-6\epsilon_{a_{1}a_{2}i}u^{a_{1}b(j}{v^{k)a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}-\epsilon_{a_{1}a_{2}(i}u^{a_{1}(k}{v^{l)a_{2}}}_{j)}(r_{12}^{(0,1)})^{ij}_{kl}\
,$ $\displaystyle
O^{(0,2)}_{3}\equiv-\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}-\epsilon_{a_{1}a_{2}(i}u^{a_{1}kl}{v^{a_{2}}}_{j)}(r_{12}^{(0,1)})^{ij}_{kl}\
,$ $\displaystyle
O^{(0,2)}_{4}\equiv-\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}+\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{kl}(r_{12}^{(0,2)})^{ij}_{kl}\
,$ $\displaystyle
O^{(0,2)}_{5}\equiv-4\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}-24\epsilon_{a_{1}a_{2}i}u^{a_{1}b(j}{v^{k)a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}-\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}k}(r_{14}^{(0,2)})^{ij}_{k}\
,$ $\displaystyle
O^{(0,2)}_{6}\equiv-\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}+12\epsilon_{a_{1}a_{2}i}u^{a_{1}b(j}{v^{k)a_{2}}}_{b}(r_{10}^{(0,1)})^{i}_{jk}+3\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}u^{kl)a_{2}}(r_{14}^{(0,2)})^{i}_{jkl}\
.$

All the operators in (4.9) are $Q$-exact.

#### v) $(R,J)=(\frac{7}{2},\frac{1}{2})$

The operators in this sector are made of 11 letters. Allowed
$(n_{\phi},n_{\psi},n_{f})$ is $(9,1,0)$. We find 1 $Q$-closed operator in
this sector given by

$\displaystyle O^{(0,1)}$ $\displaystyle\equiv$ $\displaystyle
36\epsilon_{a_{1}a_{2}a_{3}}\epsilon_{b_{1}b_{2}i}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{a_{3}jk}(r_{10}^{(0,1)})^{i}_{jk}+5\epsilon_{a_{1}a_{2}a_{3}}\epsilon_{b_{1}b_{2}b_{3}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{a_{3}b_{3}}r_{12}^{(0,1)}$
(4.10)
$\displaystyle-6\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{kl}(r_{12}^{(0,1)})^{ij}_{kl}\
.$

The operator (4.10) is $Q$-exact.

In summary, we have found 1 fermionic black hole cohomology using our ansatz,
which is a singlet under $SU(3)\subset SU(4)_{R}$, at $j=24$ whose
representative is given by (4.1). Its charges and scaling dimension are given
by $(R,J,E)=\left(\frac{5}{2},\frac{3}{2},\frac{21}{2}\right)$.

###  4.2 $Q$-exactness checks and ansatz-independent studies

In this subsection, we sketch how to determine $Q$-exactness of various
$Q$-closed operators introduced in the previous subsection. We also show that
(4.1) is the only non-graviton cohomology at $j=24$ in the $SU(3)$ singlet
sector most generally, without imposing the ansatz.

To check whether a given operator is $Q$-exact or not, especially to check
non-$Q$-exactness, one has to rule out all possible ways of writing the
operator as $Q$ of ‘something’. That being said, one needs to construct all
possible operators that can participate in ‘something’ (the meaning of which
will be made clear shortly) and show that the _target_ operator is linearly
independent of $Q$-actions of them. More specifically, we divide the check of
$Q$-exactness into 4 steps, that we summarize as follows.

  1. 1.

Construct all gauge-invariant operators whose $Q$-action may participate in
reproducing the _target_.

  2. 2.

Count the number of linearly independent operators from step 1, and extract
the maximal subset of linearly independent operators. This is called the
_basis_.

  3. 3.

Act $Q$ on the basis operators, then again count and extract the maximal
subset of linearly independent ones between them.

  4. 4.

Check if the target is linearly independent of the result of step 3.

Now we explain what operators ‘may participate in reproducing the target’ in
step 1. This consists of two criteria: the charges and the parity under
permutation.

First, the charges of the target operator constrain the charges, thus the
letter contents of the basis operators. Note that the action of $Q$ increases
$R_{I=1,2,3}$ by $\frac{1}{2}$ and decreases $J=J_{1}=J_{2}$ by $\frac{1}{2}$.
Therefore, the basis operators must have the set of charges that differ by the
corresponding amount from the target, otherwise their $Q$-actions are disjoint
from the target. Note that all of our targets are $SU(3)$ singlets, so we
always have $R=R_{1}=R_{2}=R_{3}$.

Second, all of our targets being singlets under the $SU(3)$ subgroup of the
$SU(4)$ R-symmetry group, imposes a stronger constraint than just restricting
to the charge sectors with $R_{1}=R_{2}=R_{3}$. Each basis operator must be
invariant under cyclic permutation $\phi^{i}\to\phi^{i+1}$ and simultaneously
$\psi_{i}\to\psi_{i+1}$, where $i=1,2,3$ mod 3. Moreover, if there are
even/odd number of $\phi$’s and $\psi$’s combined, which carries one $SU(3)$
index each, it requires even/odd number of Levi-Civita symbols to write the
operator covariantly while contracting all indices. Therefore, we may restrict
to i) operators with even number of $\phi$’s and $\psi$’s combined, that are
even under all $3!$ permutations of $SU(3)$ indices, and ii) operators with
odd number of $\phi$’s and $\psi$’s combined, that are even under even
(cyclic) permutations of $SU(3)$ indices and odd under odd (swap) permutations
of $SU(3)$ indices. Also note that this permutation property commutes with the
action of $Q$, so that $Q$ of a non-trivial operator satisfies this property
if and only if the original operator does. This permutation property is
necessary but not sufficient for an operator to be an $SU(3)$ singlet.
However, we impose this property on the basis instead of requiring $SU(3)$
singlets, because the latter requires many sums over dummy indices and thus
the former is computationally more efficient. Our conclusions on the singlet
sector will be valid despite.

For example, suppose that the target operator is (4.1), which has charges
$(R,J)=(\frac{5}{2},\frac{3}{2})$. Operators whose $Q$-action may reproduce
this target operator must then have $(R,J)=(2,2)$. Possible choices of letter
contents are $(n_{\phi},n_{\psi},n_{f})=(6,0,2),(5,2,1)$, or $(4,4,0)$, and
numbers of $\phi^{i}$ minus numbers of $\psi_{i}$ must be equal between
$i=1,2,3$. Further taking into account the permutation property, the basis
operators whose $Q$-action ‘may participate in reproducing the target’ (4.1)
can be classified into the following 7 subsectors. ($(-1)^{\epsilon}$ in
subsectors 5 and 6 indicates minus sign for odd permutations, because there
are odd number of $\phi$’s and $\psi$’s in those subsectors.)

  * •

Subsector 1: $(\phi^{1})^{4}(\psi_{1})^{4}+$ (permutations)

  * •

Subsector 2: $(\phi^{1})^{3}(\phi^{2})^{1}(\psi_{1})^{3}(\psi_{2})^{1}+$
(permutations)

  * •

Subsector 3: $(\phi^{1})^{2}(\phi^{2})^{2}(\psi_{1})^{2}(\psi_{2})^{2}+$
(permutations)

  * •

Subsector 4:
$(\phi^{1})^{2}(\phi^{2})^{1}(\phi^{3})^{1}(\psi_{1})^{2}(\psi_{2})^{1}(\psi_{3})^{1}+$
(permutations)

  * •

Subsector 5:
$(\phi^{1})^{3}(\phi^{2})^{1}(\phi^{3})^{1}(\psi_{1})^{2}f^{1}+(-1)^{\epsilon}$
(permutations)

  * •

Subsector 6:
$(\phi^{1})^{2}(\phi^{2})^{2}(\phi^{3})^{1}(\psi_{1})^{1}(\psi_{2})^{1}f^{1}+(-1)^{\epsilon}$
(permutations)

  * •

Subsector 7: $(\phi^{1})^{2}(\phi^{2})^{2}(\phi^{3})^{2}f^{2}+$ (permutations)

Appropriate sums over permutations of single- and multi-trace operators in
each of these subsectors are the result of step 1, some of which we write down
below to help visualize:

$\displaystyle{\rm
tr}(\phi^{1}\phi^{1}\psi_{1}\phi^{1}\psi_{1}\psi_{1}\phi^{1}\psi_{1})+{\rm
tr}(\phi^{2}\phi^{2}\psi_{2}\phi^{2}\psi_{2}\psi_{2}\phi^{2}\psi_{2})+{\rm
tr}(\phi^{3}\phi^{3}\psi_{3}\phi^{3}\psi_{3}\psi_{3}\phi^{3}\psi_{3})~{},$
$\displaystyle{\rm tr}(\phi^{1}\phi^{1}\phi^{2}\psi_{2}){\rm
tr}(\psi_{1}\psi_{2}){\rm tr}(\phi^{2}\psi_{1})+{\rm
tr}(\phi^{2}\phi^{2}\phi^{3}\psi_{3}){\rm tr}(\psi_{2}\psi_{3}){\rm
tr}(\phi^{3}\psi_{2})+{\rm tr}(\phi^{3}\phi^{3}\phi^{1}\psi_{1}){\rm
tr}(\psi_{3}\psi_{1}){\rm tr}(\phi^{1}\psi_{3})$ $\displaystyle~{}~{}+{\rm
tr}(\phi^{3}\phi^{3}\phi^{2}\psi_{2}){\rm tr}(\psi_{3}\psi_{2}){\rm
tr}(\phi^{2}\psi_{3})+{\rm tr}(\phi^{1}\phi^{1}\phi^{3}\psi_{3}){\rm
tr}(\psi_{1}\psi_{3}){\rm tr}(\phi^{3}\psi_{1})+{\rm
tr}(\phi^{2}\phi^{2}\phi^{1}\psi_{1}){\rm tr}(\psi_{2}\psi_{1}){\rm
tr}(\phi^{1}\psi_{2})~{},$ $\displaystyle{\rm
tr}(\phi^{2}\phi^{2}\psi_{1}\psi_{2}\phi^{3}){\rm tr}(f\phi^{1}\phi^{1})+{\rm
tr}(\phi^{3}\phi^{3}\psi_{2}\psi_{3}\phi^{1}){\rm tr}(f\phi^{2}\phi^{2})+{\rm
tr}(\phi^{1}\phi^{1}\psi_{3}\psi_{1}\phi^{2}){\rm tr}(f\phi^{3}\phi^{3})$
$\displaystyle~{}~{}-{\rm tr}(\phi^{3}\phi^{3}\psi_{1}\psi_{3}\phi^{2}){\rm
tr}(f\phi^{1}\phi^{1})-{\rm tr}(\phi^{1}\phi^{1}\psi_{2}\psi_{1}\phi^{3}){\rm
tr}(f\phi^{2}\phi^{2})-{\rm tr}(\phi^{2}\phi^{2}\psi_{3}\psi_{2}\phi^{1}){\rm
tr}(f\phi^{3}\phi^{3})~{}.$ (4.11)

Given the operators from step 1, the rest is relatively straightforward, at
least conceptually. There are non-trivial trace relations between operators
from step 1, so in step 2 we extract linearly independent basis operators.
Then in step 3, we consider $Q$-actions of the basis operators, and again
count the number of linearly independent ones among them. These should form a
complete basis of all $Q$-exact operators in the target charge sector and with
the aforementioned permutation property. Therefore, the target operator is
$Q$-exact if and only if it is a linear combination of the $Q$-actions of the
basis operators. More generally, if there are multiple target operators, the
number of cohomologies among them would be equal to the number of linearly
independent ones among the basis _and_ all target operators, minus the number
of linearly independent ones among the basis only.

Each of step 2-4 involves counting and/or finding linearly independent
operators among a given set of gauge-invariant operators. Each operator is a
sum over single- and multi-trace operators written in terms of seven species
of fields $\phi^{m}$, $\psi_{m}$ and $f$. To completely account for trace
relations between them, we first convert the operators written in terms of
adjoint fields into polynomials of their matrix elements, by substituting

$\displaystyle f$ $\displaystyle=$
$\displaystyle\begin{pmatrix}f_{1}&f_{2}&f_{3}\\\ f_{4}&f_{5}&f_{6}\\\
f_{7}&f_{8}&-f_{1}-f_{5}\end{pmatrix}~{},$ (4.12)

and likes for 6 other fields. In this way, every operator is now written as a
polynomial of $8\times 7=56$ variables, 24 of which are Grassmannian. So the
problem boils down to finding linear dependence between a set of polynomials.
Although this is the same problem that was encountered while computing the
graviton index in section 3, the same method of extracting the coefficient
matrix is extremely unpractical here. It is because there are four times as
many variables (recall that for counting gravitons, we substituted each field
with a diagonal matrix), and therefore exponentially larger number of
monomials appear in polynomials. As a result, the coefficient matrix will have
a huge number of columns that is not viable for computers.

For this reason, we have devised a numerics-assisted approach to find linear
dependence between the polynomials with large number of variables. The
approach stems from the basic fact that if some linear combination of certain
polynomials vanishes, it must also be zero if we attribute any specific number
to each variable. So let us represent each polynomial by an array of numbers,
i.e. a row vector, by substituting each variable with a set of randomly chosen
integers. Then we examine the linear dependence between vectors, instead of
polynomials.

The substitution can be repeated for arbitrarily many sets of integers, so the
row vector can be made arbitrarily long. Obviously, the length of the row
vectors, i.e. number of columns, must be at least as many as there are
independent polynomials. Otherwise, it will be always possible to find a
relation between the row vectors even if the polynomials they represent are
independent. On the other hand, the length of the row vectors need not be much
more than the number of independent polynomials, as we will explain shortly.

This makes it clear why this method is efficient. It naturally realizes the
basic principle that in order to distinguish $n$ different entities, one needs
at least $n$ data for each entity, whereas extracting the coefficient matrix
for the polynomials with so many variables will equivalently convert each
polynomial into an unnecessarily long vector.

There are two issues with this approach that we need to address. The first is
that 24 of 56 variables are Grassmannian, which cannot be properly substituted
with c-numbers. The second is that randomness is involved in this approach,
and it may lead to errors albeit unlikely.

The issue with Grassmann variables can be easily addressed by ordering them in
a definite manner within each monomial. That is, we fully expand each
polynomial (which includes eliminating squares of Grassmann variables), and
let variables be multiplied only in a certain order within each monomial.
During this process the coefficients may flip signs, but the result of this
process is unique for each polynomial. Once we have done this, none of the
Grassmann properties will be used when finding linear relations between the
polynomials, because each monomial is now compared verbatim with monomials in
other polynomials. Therefore, it is now safe to substitute Grassmann variables
with c-numbers. This principle was also implied while extracting the
coefficient matrix of graviton operators in section 3.

As for the randomness, first note that substituting (sufficiently many sets
of) random integers never miss the true dependence between polynomials. If
there is a true linear dependence between polynomials, i.e. a linear
combination that vanishes, the same linear combination must be zero for
whatever numbers are put in, so the row vectors corresponding to the
polynomials must be linearly dependent. Note that all polynomials have
rational coefficients and we put in random integers, so there is no issue with
machine precision.

However, the converse is possible: this method may find false linear
dependence between polynomials. This is simply because a non-vanishing
polynomial may evaluate to zero when certain values are put into variables.
That is, the randomly chosen values could miraculously be the roots of the
polynomial. This type of error can be made arbitrarily more unlikely by
increasing the number of columns, i.e. number of sets of random integers that
are put in. Let us roughly estimate the unlikelihood.

Suppose that the number of columns is $m+n$ where $m$ is the true number of
independent polynomials. For this method to find a false dependence, both of
the followings must happen: i) there exists a non-trivial linear combination
of the polynomials that vanishes for the first $m$ sets of random integers,
and ii) this polynomial further vanishes for the additional $n$ sets of random
integers. The probability of i) is relatively difficult to estimate, since it
involves intricate tuning of $m-1$ coefficients in a linear combination of the
polynomials. Therefore we only estimate the probability of ii) as follows. A
typical basis polynomial such as $Q$-action of those in (4.2) 666These are
used in step 3 and 4 of determining $Q$-exactness of $Q$-closed operators in
the charge sector $(R,J)=(\frac{5}{2},\frac{3}{2})$, of which one is the non-
graviton cohomology (4.1) evaluates to $\sim 10^{28}$ when a random integer
between $1$ and $1000$ are substituted into each variable. (See Fig. 1. for an
example.) This is a natural scale considering that the typical polynomial is a
sum over $\sim 10^{6}$ monomials (with both signs) that each consists of 9
letters, so for example $10^{6}\times(10^{2.5})^{9}\sim 10^{28}$. This value
is far smaller than the number of all possible random choices — which is
$(10^{3})^{56}$ if all 7 gluons, thus $7\times(3^{2}-1)$ variables, are
involved — so each integer value within magnitude $\sim 10^{28}$ will be
sufficiently populated. Furthermore, since a typical polynomial consists of
many$\sim 10^{6}$ monomials, we assume that the evaluation of the polynomial
is like a random walk with sufficient iterations, and thus the factorization
property of integers is blurred. For these reasons, let us assume that the
distribution of the evaluated values is continuous. Then, the probability that
this value falls within $O(1)$ is estimated to be $\sim 10^{-28}$, even
accounting for the shape of the distribution. For ii), this must happen for
$n$ independent sets of random variables, so the probability of ii) is
estimated to be $10^{-28n}$. In step 3, $n$ was taken to be 175, so the
estimated probability of ii) is $10^{-4900}$.

Figure 1: An example distribution of 1.5$\times 10^{6}$ evaluated values of
$Q$-action of basis polynomials in paragraph iii). Width of each bin is
$10^{27}$.

This method of detecting linear dependence was used between numerous sets of
polynomials while determining $Q$-exactness of various operators in different
charge sectors. Numbers that appeared in the previous paragraph slightly
differ between occasions. Typical values of the polynomials differ because
they consist of different numbers of letters, and $n$ is inevitably different
because the number of columns are set before we know the number of true
independent polynomials. However, in any case, we use at least $n\geq 30$ and
the estimated probability of ii) has order of magnitude of a few negative
hundreds at the worst. Furthermore, when a $Q$-closed operator is determined
to be $Q$-exact, we checked analytically the relation between the target and
basis polynomials to further eradicate the margin for error.

Employing the method explained so far, we have constructed the basis operators
in each and all charge sectors with $R_{1}=R_{2}=R_{3}$ at $j=24$ order, with
the aforementioned permutation property. We have also evaluated the
$Q$-actions of the bases, that should form the basis of $Q$-exact operators.
Then we have determined $Q$-exactness of all $Q$-closed non-graviton operators
obtained from our ansatz in the previous subsection. The result is that all
operators except for the fermionic (4.1) are $Q$-exact.

$R$ | $J$ | #letters | #basis | #closed | #exact | #coh. | #gravitons | #BH coh.
---|---|---|---|---|---|---|---|---
0 | 4 | 4 | 1 | 0 | 0 | 0 | 0 | 0
$\frac{1}{2}$ | $\frac{7}{2}$ | 5 | 9 | 1 | 1 | 0 | 0 | 0
1 | 3 | 6 | 91 | 8 | 8 | 0 | 0 | 0
$\frac{3}{2}$ | $\frac{5}{2}$ | 7 | 511 | 85 | 83 | 2 | 2 | 0
2 | 2 | 8 | 1369 | 445 | 426 | 19 | 19 | 0
$\frac{5}{2}$ | $\frac{3}{2}$ | 9 | 1898 | 953 | 924 | 29 | 28 | 1
3 | 1 | 10 | 1456 | 961 | 945 | 16 | 16 | 0
$\frac{7}{2}$ | $\frac{1}{2}$ | 11 | 633 | 505 | 495 | 10 | 10 | 0
4 | 0 | 12 | 136 | 136 | 128 | 8 | 8 | 0
Table 2:  For each charge sector $R=R_{1}=R_{2}=R_{3}$ and $J$ at level
$j=24$, we present the numbers of letters, of independent basis operators with
the permutation property, of $Q$-closed operators, of $Q$-exact operators, and
therefore of $Q$-cohomologies. Then we compare with the number of gravitons
with the same permutation property to determine the number of non-graviton
(black hole) cohomologies in each charge sector.

From the fact that we have constructed and counted all operators and their
$Q$-actions in the $R_{1}=R_{2}=R_{3}$ charge sectors at $j=24$ order with the
permutation property, we can also prove the non-existence of any other $SU(3)$
singlet non-graviton cohomology at $j=24$ order. Recall that the result of
step 2 is a complete basis of all operators, in a given charge sector $(R,J)$
and with permutation property that is designed to include all $SU(3)$
singlets. There are further linear relations between $Q$-actions of these
basis operators, reducing the number of independent $Q$-exact operators at
charge sector $(R+\frac{1}{2},J-\frac{1}{2})$ in step 3. The reduced operators
correspond to the $Q$-closed operators at charge sector $(R,J)$:

$\text{(\\#closed)}_{(R,J)}=\text{(\\#basis)}_{(R,J)}-\text{(\\#exact)}_{(R+\frac{1}{2},J-\frac{1}{2})}~{}.$

Then the number of $Q$-cohomologies is given by

$\text{(\\#coh.)}_{(R,J)}=\text{(\\#closed)}_{(R,J)}-\text{(\\#exact)}_{(R,J)}~{}.$

Meanwhile, we can also count the number of independent graviton cohomologies
in these charge sectors and with the same permutation property, like we
counted the full set of gravitons in subsection 3.1. The number of non-
graviton cohomologies is given by

$\text{(\\#BH
coh.)}_{(R,J)}=\text{(\\#coh.)}_{(R,J)}-\text{(\\#gravitons)}_{(R,J)}~{}.$

We present all the numbers mentioned in this paragraph in Table 2. We find
only one non-graviton cohomology in the $(R,J)=(\frac{5}{2},\frac{3}{2})$
sector, which is the fermionic cohomology presented in (4.1). Since the
operators with the permutation property in the $R_{1}=R_{2}=R_{3}$ charge
sectors include all $SU(3)$ singlets, we conclude that (4.1) is the only non-
graviton cohomology that is an $SU(3)$ singlet at order $j=24$.

The computation presented in this subsection, of constructing the basis
operators and counting independent ones between them and their $Q$-actions, is
essentially the sort of computation that was performed in [15], although we
find our numerics-assisted approach to be more efficient. Moreover, we have
only performed this computation in the $R_{1}=R_{2}=R_{3}$ charge sectors at
$j=24$ order in the BMN sector, and further restricted to operators with
certain permutation property. This is because we focused on the $SU(3)$
singlet sector at order $j=24$, where the non-graviton index indicated the
existence of a non-graviton cohomology.

##  5 Future directions

In this paper we studied the $Q$-cohomologies of 4d maximal super-Yang-Mills
theory for the local BPS operators at 1-loop level. We detected new non-
graviton cohomologies from the index for the $SU(3)$ and $SU(4)$ gauge
theories, and constructed the apparent $SU(3)$ threshold cohomology. A goal of
this program is to identify and characterize the microstates of BPS black
holes in the $SU(N)$ theory with parametrically large $N$. Although we are
currently very far from this goal, several novel structures are observed for
$SU(3)$ and $SU(4)$ theories which we hopefully think will shed light on the
large $N$ black hole physics.

Constructing new cohomologies (non-graviton type, or black hole type) requires
us to find operators which become $Q$-closed only after using the trace
relations of finite-sized matrices. Although the basic principle for the trace
relations should be simple (e.g. repeated uses of the Cayley-Hamilton
identities), it is hard in general to analytically generate useful trace
relations. In this paper, we have developed a semi-systematic way of
constructing $Q$-closed operators using certain trace relations. Within this
framework, the main technical bottleneck is proving that the constructed
$Q$-closed operator is not $Q$-exact. This demands us to check if the target
$Q$-closed operators are $Q$-exact or not after using all possible trace
relations. We developed numerics-assisted computational strategies to check
this on a computer.

In [17], $Q$-exactness was easy to show for a particular class of operators in
the BMN sector. Namely, if a BMN operator contains a term without any scalars,
this operator cannot be $Q$-exact. This is because $Q$ acting on the BMN
fields always generate one of more scalars, so that $Q$ cannot generate a term
without scalars. An infinite number of cohomologies with this property was
found in [17], whose checks of $Q$-nonexactness were trivial. These
cohomologies are beyond our ansatz in this paper. This implies that our ansatz
is far from sufficient to generate all the non-graviton cohomologies, even
within the BMN sector. It would be highly desirable, if possible, to combine
the analytic insights learned in [17] and in this paper.

Merely knowing the index $Z-Z_{\rm grav}$ over non-graviton cohomologies is
very useful to learn their novel spectral structures. Section 3.1 has
extensively discussed the $SU(3)$ non-graviton index up to charge $j=54$,
finding the hints of novel partial no-hair behaviors and the tower structures.
Technically, the full BMN index $Z$ is relatively easy to compute at not too
large $N$, by computing the residue sum for the integral formula (2.11). The
harder part is to compute the finite $N$ graviton index $Z_{\rm grav}$ by
taking into account the trace relations. This counting problem reduces to
counting independent polynomials of $4(N-1)$ bosonic and $3(N-1)$ fermionic
variables subject to constraints. In principle this counting problem can be
solved completely by knowing the so-called Gröbner basis of constraints. In
practice, constructing the Gröbner bases is very cumbersome, even on computer.
We have partially obtained the Gröbner bases for the $SU(3)$ BMN gravitons in
the subsector not containing the letter $f$. To expand these results to the
full BMN sector including all letters, we did a rather brutal computations on
computer order by order in the charges. For $SU(4)$, the uses of the Gröbner
bases were more limited to obtain results of section 4.2. It will be very
desirable to compute the full $Z_{\rm grav}$, and thus $Z-Z_{\rm grav}$,
exactly. There are several features that we would like to check with these
exact results. For instance, many towers of states were observed in $Z-Z_{\rm
grav}$ in the $SU(3)$ theory till $j=54$, and to us it is unclear whether
these towers continue indefinitely or not.

One may wonder if the analysis of spectrum will be simpler at large $N$,
ignoring $\frac{1}{N}$ corrections in the computations. However, whether a
state is BPS or not is an exact property so that $\frac{1}{N}$ corrections may
affect the answer. Nevertheless, there should be large $N$ simplifications for
studying the near-BPS operators with small anomalous dimensions, say, at order
$\frac{1}{S}\sim\frac{1}{N^{2}}$ [46]. Some studies in this direction were
made in [18, 19].

Acknowledgements

We thank Minkyoo Kim, Eunwoo Lee, Masaki Shigemori and especially Shiraz
Minwalla for helpful discussions and comments. We also thank Goojin Kwon for
independently deriving some results in section 2.1. This work is supported in
part by the NRF grant 2021R1A2C2012350 (JC, SK, JL), a KIAS Individual Grant
PG081602 at Korea Institute for Advanced Study (SC), the DoE grant DE-
SC0007859 (SL) and Rackham Predoctoral Fellowship (SL). This research was
supported in part through computational resources and services provided by
Advanced Research Computing (ARC), a division of Information and Technology
Services (ITS) at the University of Michigan, Ann Arbor.

##  Appendix A Graviton trace relations

In this appendix, we first list the trace relations between the graviton
cohomologies in the BMN sector of the $SU(3)$ theory. Then we construct the
relations of relations at $j=24$ and $j=30$ which are singlets under the
$SU(3)\subset SU(4)_{R}$ global symmetry. These are the two sectors in which
the index predicted fermionic cohomologies in the $SU(3)$ singlet. The results
at $j=24$ are used in section 4.1 to construct the threshold cohomology. The
results at $j=30$ provide ansätze for the $Q$-closed operators, whose
$Q$-exactness are not checked in this paper.

The trace relations are the linear dependence between the multi-trace
operators, up to $Q$-exact operators, due to the finite size of the matrices.
In this appendix, we shall only consider the trace relations between
gravitons. Let us first arrange the trace relations by their level $j$ and
distinguish them into two types; fundamental ones and the others. The
fundamental trace relations at level $j$ cannot be written as linear
combinations of the trace relations at lower levels $j^{\prime}(<j)$,
multiplied by the gravitons at level $j-j^{\prime}$. All trace relations of
gravitons can be expressed as linear combinations of the fundamental trace
relations with the coefficients being graviton cohomologies. We explicitly
constructed the fundamental trace relations till certain levels, which will be
presented below.

The single-trace generators of the $SU(3)$ BMN gravitons are given by

$\displaystyle u^{ij}\equiv$
$\displaystyle\;\textrm{tr}\left(\phi^{(i}\phi^{j)}\right)\ ,\ \
u^{ijk}\equiv\;\textrm{tr}\left(\phi^{(i}\phi^{j}\phi^{k)}\right)\ ,$ (A.1)
$\displaystyle{v^{i}}_{j}\equiv$
$\displaystyle\;\textrm{tr}\left(\phi^{i}\psi_{j}\right)-{\textstyle\frac{1}{3}}\delta^{i}_{j}\textrm{tr}\left(\phi^{a}\psi_{a}\right)\
,\ \
{v^{ij}}_{k}\equiv\;\textrm{tr}\left(\phi^{(i}\phi^{j)}\psi_{k}\right)-{\textstyle\frac{1}{4}}\delta^{i}_{k}\textrm{tr}\left(\phi^{(j}\phi^{a)}\psi_{a}\right)-{\textstyle\frac{1}{4}}\delta^{j}_{k}\textrm{tr}\left(\phi^{(i}\phi^{a)}\psi_{a}\right)\
,$ $\displaystyle w^{i}\equiv$
$\displaystyle\;\textrm{tr}\left(f\phi^{i}+{\textstyle\frac{1}{2}}\epsilon^{ia_{1}a_{2}}\psi_{a_{1}}\psi_{a_{2}}\right)\
,\ \
w^{ij}\equiv\;\textrm{tr}\left(f\phi^{(i}\phi^{j)}+\epsilon^{a_{1}a_{2}(i}\phi^{j)}\psi_{a_{1}}\psi_{a_{2}}\right)\
,$

where we suppressed the subscript of $u_{n},v_{n},w_{n}$ since it can be
easily read off from the number of the indices. Note that the $Q$-actions on
$\phi,\psi,f$ are given by

$Q\phi^{m}=0\ ,\quad Q\psi_{m}=-\frac{i}{2}\epsilon_{mnp}[\phi^{n},\phi^{p}]\
,\quad Qf=-i[\phi^{m},\psi_{m}]\ .$ (A.2)

We would like to find the fundamental trace relations of (A.1).

It is helpful to start from the Gröbner basis for the trace relations. The
Gröbner basis contains all fundamental trace relations. In general, the
Gröbner basis also contains some non-fundamental trace relations. We shall
obtain the fundamental trace relations from the Gröbner basis by induction.

At the lowest level of the trace relations, all of them are fundamental.
Namely, every generator of the Gröbner basis at such level are the fundamental
relations. For the $SU(3)$ theory, the lowest level is $j=10$. In order to
organize them into covariant forms in the $SU(3)$ global symmetry, we use the
following computational strategy (which also proves useful at higher orders).
We list the polynomials of (A.1) which have the same representations as the
lowest fundamental trace relations at $j=10$. Among them, we should find
particular linear combinations which vanish when all off-diagonal elements of
$\phi^{m},\psi_{m},f$ are turned off, since the graviton trace relations
vanish with diagonal fields. Once such combinations are identified, keeping
$\phi,\psi,f$ general in this combination will yield the $Q$-exact operators
for the lowest fundamental trace relations. This way, we can find the
fundamental trace relations at the lowest level.777There can be linear
combinations which vanish even when the off-diagonal elements are turned on.
In principle, they can also be the trace relations but most of them are just
the identities that hold at arbitrary $N$. In practice, we only find them as
mesonic identities between (A.1) rather than the trace relations.

Now, suppose that we found all fundamental trace relations until the level
$j$. We can construct the fundamental trace relations at $j+2$ as follows. We
first construct all non-fundamental trace relations at level $j+2$ by
multiplying suitable graviton cohomologies to the fundamental ones below the
level $j$. Not all of them are linearly independent so we should extract a
linearly independent set among them. This lets us to compute the $SU(3)$
character of the non-fundamental trace relations at level $j+2$. Next, we
consider a union of the non-fundamental trace relations and the Gröbner bases
at level $j+2$. Note that the Gröbner basis will contain all fundamental trace
relations and some non-fundamental ones. We extract a linearly independent set
among such union, which contains all fundamental and non-fundamental
relations. We also compute the $SU(3)$ character over them. Finally, we
subtract the former character from the latter, which yields the $SU(3)$
character of the fundamental trace relations at level $j+2$. Then we list the
multi-trace operators using (A.1) which can account for it as before. Among
them, we find particular linear combinations which vanish when all off-
diagonal elements of $\phi,\psi,f$ are turned off, and which are linearly
independent from the non-fundamental trace relations we constructed above. The
final results are the fundamental trace relations at level $j+2$. In this way,
one can construct the fundamental trace relations inductively.

In principle, one can obtain all fundamental trace relations of gravitons from
the above induction. For the $SU(2)$ theory, it can be easily done. We found a
66-dimensional Gröbner basis, and there exist 48 fundamental trace relations
among them. However, for the $SU(3)$ theory, we could not do a similar
calculation since the construction of the Gröbner basis is time-consuming. We
constructed it only in two subsectors: (1) all trace relations between
$u_{2},u_{3},v_{2},v_{3}$, and (2) trace relations between
$u_{2},u_{3},v_{2},v_{3},w_{2},w_{3}$ until $j\leq 20$. From the subsector
(1), which has 1170 generators, we obtained all fundamental trace relations
between $u_{2},u_{3},v_{2},v_{3}$, i.e. the relations which do not involve
$f$’s. There are in total 287 relations whose lowest level is $j=10$ and the
highest level is $j=30$. On the other hand, from the subsector (2), we could
generate the fundamental trace relations involving $f$’s until $j\leq 20$.
There are in total 130 relations involving $f$’s between $14\leq j\leq 20$.
These are enough to construct relations of relations at $j=24$.

Before presenting their explicit forms, we first explain our notation. When we
write down certain operator in the irreducible representation $\mathbf{R}$
under $SU(3)\subset SU(4)_{R}$ as
$O^{i_{1}i_{2}i_{3}...}_{j_{1}j_{2}j_{3}...}$, the actual form of such an
operator should be understood as $O^{i_{1}i_{2}i_{3}...}_{j_{1}j_{2}j_{3}...}$
subtracted by its trace part to make it traceless, like

$\displaystyle[n,0]:O^{i_{1}i_{2}i_{3}\cdots i_{n}}\to
O^{i_{1}i_{2}i_{3}\cdots i_{n}}\ ,\qquad[0,n]:O_{i_{1}i_{2}i_{3}\cdots
i_{n}}\to O_{i_{1}i_{2}i_{3}\cdots i_{n}}\ ,$ (A.3)
$\displaystyle[1,1]:O^{i}_{j}\to
O^{i}_{j}-{\textstyle\frac{1}{3}}\delta^{i}_{j}O^{a}_{a}\ ,$
$\displaystyle[2,1]:O^{ij}_{k}\to
O^{ij}_{k}-{\textstyle\frac{1}{2}}\delta^{(i}_{k}O^{j)a}_{a}\
,\qquad[1,2]:O^{i}_{jk}\to
O^{i}_{jk}-{\textstyle\frac{1}{2}}\delta^{i}_{(j}O^{a}_{k)a}\ ,$
$\displaystyle[3,1]:O^{ijk}_{l}\to
O^{ijk}_{l}-{\textstyle\frac{3}{5}}\delta^{(i}_{l}O^{jk)a}_{a}\
,\qquad[1,3]:O^{i}_{jkl}\to
O^{i}_{jkl}-{\textstyle\frac{3}{5}}\delta^{i}_{(j}O^{a}_{kl)a}\ ,$
$\displaystyle[2,2]:O^{ij}_{kl}\to
O^{ij}_{kl}-{\textstyle\frac{4}{5}}\delta^{(i}_{(k}O^{j)a}_{l)a}+{\textstyle\frac{1}{10}}\delta^{(i}_{(k}\delta^{j)}_{l)}O^{a_{1}a_{2}}_{a_{1}a_{2}}\
,$

and so on. Here, $[\cdot,\cdot]$ are the Dynkin labels for $SU(3)$.

Below, we list the explicit forms of the fundamental trace relations according
to their level $j$ and representation under $SU(3)\subset SU(4)_{R}$ as
$t^{j}[R_{1}^{\prime},R_{2}^{\prime}]$. The relations which do not involve
$f$’s are given as follows:

$\displaystyle
t^{10}[1,2](u_{2}u_{3}):(R_{10}^{(0,0)})^{i}_{jk}=\epsilon_{a_{1}a_{2}(j}\epsilon_{k)b_{1}b_{2}}u^{a_{1}b_{1}}u^{ia_{2}b_{2}}$
$\displaystyle
t^{12}[0,0](u_{2}u_{2}u_{2}):R_{12}^{(0,0)}=\epsilon_{a_{1}a_{2}a_{3}}\epsilon_{b_{1}b_{2}b_{3}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{a_{3}b_{3}}$
$\displaystyle
t^{12}[2,2](u_{2}u_{2}u_{2},u_{3}u_{3}):(R_{12}^{(0,0)})^{ij}_{kl}=\epsilon_{a_{1}a_{2}(k}\epsilon_{l)b_{1}b_{2}}\left(u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{ij}+6u^{a_{1}b_{1}(i}u^{j)a_{2}b_{2}}\right)$
$\displaystyle
t^{12}[0,3](u_{2}v_{3}):(R_{12}^{(0,1)})_{ijk}=\epsilon_{(i|a_{1}a_{2}}\epsilon_{|j|b_{1}b_{2}}u^{a_{1}b_{1}}{v^{a_{2}b_{2}}}_{|k)}$
$\displaystyle
t^{12}[1,1](u_{2}v_{3},u_{3}v_{2}):(R_{12}^{(0,1)})^{i}_{j}=\epsilon_{ja_{1}a_{2}}\left(4u^{a_{1}b}{v^{ia_{2}}}_{b}+3u^{ia_{1}b}{v^{a_{2}}}_{b}\right)$
$\displaystyle
t^{12}[2,2](u_{2}v_{3},u_{3}v_{2}):(R_{12}^{(0,1)})^{ij}_{kl}=\epsilon_{a_{1}a_{2}(k}\left(u^{a_{1}(i}{v^{j)a_{2}}}_{l)}+u^{ija_{1}}{v^{a_{2}}}_{l)}\right)$
$\displaystyle
t^{14}[1,0](u_{2}u_{2}v_{2}):(R_{14}^{(0,1)})^{i}=\epsilon_{a_{1}a_{2}a_{3}}u^{ia_{1}}u^{ba_{2}}{v^{a_{3}}}_{b}$
$\displaystyle
t^{14}[0,2](u_{2}u_{2}v_{2},u_{3}v_{3}):(R_{14}^{(0,1)})_{ij}=\epsilon_{a_{1}a_{2}(i|}\left(\epsilon_{b_{1}b_{2}b_{3}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}{v^{b_{3}}}_{|j)}-2\epsilon_{|j)b_{1}b_{2}}u^{a_{1}b_{1}c}{v^{a_{2}b_{2}}}_{c}\right)$
$\displaystyle
t^{14}[2,1](u_{2}u_{2}v_{2},u_{3}v_{3}):(R_{14}^{(0,1)})^{ij}_{k}=\epsilon_{ka_{1}a_{2}}\left(3u^{(a_{1}b}u^{ij)}{v^{a_{2}}}_{b}+4u^{a_{1}b}u^{a_{2}(i}{v^{j)}}_{b}+24u^{a_{1}b(i}{v^{j)a_{2}}}_{b}\right)$
$\displaystyle
t^{14}[1,3](u_{2}u_{2}v_{2},u_{3}v_{3}):(R_{14}^{(0,1)})^{i}_{jkl}=\epsilon_{(j|a_{1}a_{2}}\epsilon_{|k|b_{1}b_{2}}\left(u^{a_{1}b_{1}}u^{a_{2}b_{2}}{v^{i}}_{|l)}+6u^{ia_{1}b_{1}}{v^{a_{2}b_{2}}}_{|l)}\right)$
$\displaystyle
t^{14}[3,2](u_{2}u_{2}v_{2},u_{3}v_{3}):(R_{14}^{(0,1)})^{ijk}_{lm}=\epsilon_{a_{1}a_{2}(l}\left(u^{(a_{1}i}u^{jk)}{v^{a_{2}}}_{m)}+6u^{a_{1}(ij}{v^{k)a_{2}}}_{m)}\right)$
$\displaystyle
t^{14}[1,3](v_{2}v_{3}):(R_{14}^{(0,2)})^{i}_{jkl}=\epsilon_{a_{1}a_{2}(j}{v^{a_{1}}}_{k}{v^{ia_{2}}}_{l)}$
$\displaystyle
t^{16}[0,1](u_{2}v_{2}v_{2},v_{3}v_{3}):(R_{16}^{(0,2)})_{i}=\epsilon_{ia_{1}a_{2}}\left(12u^{bc}{v^{a_{1}}}_{b}{v^{a_{2}}}_{c}+13u^{a_{1}b}{v^{a_{2}}}_{c}{v^{c}}_{b}+12{{v}^{a_{1}b}}_{c}{v^{a_{2}c}}_{b}\right)$
$\displaystyle
t^{16}[1,2](u_{2}v_{2}v_{2},v_{3}v_{3}):(R_{16}^{(0,2)})^{i}_{jk}=\epsilon_{a_{1}a_{2}(j}\\!\left(3u^{ib}{v^{a_{1}}}_{k)}{v^{a_{2}}}_{b}\\!-\\!7u^{ia_{1}}{v^{b}}_{k)}{v^{a_{2}}}_{b}\\!+\\!6u^{a_{1}b}{v^{i}}_{k)}{v^{a_{2}}}_{b}\\!+\\!24{v^{a_{1}b}}_{k)}{v^{ia_{2}}}_{b}\right)$
$\displaystyle
t^{16}[2,3](u_{2}v_{2}v_{2},v_{3}v_{3}):(R_{16}^{(0,2)})^{ij}_{klm}=\epsilon_{a_{1}a_{2}(k}\left({u^{a_{1}(i}{v^{j)}}_{l}v^{a_{2}}}_{m)}+3{v^{a_{1}(i}}_{l}{v^{j)a_{2}}}_{m)}\right)$
$\displaystyle
t^{18}[0,0](u_{3}v_{2}v_{2}):R_{18}^{(0,2)}=\epsilon_{a_{1}a_{2}a_{3}}u^{a_{1}bc}{v^{a_{2}}}_{b}{v^{a_{3}}}_{c}$
$\displaystyle
t^{20}[1,0](v_{2}v_{2}v_{3}):(R_{20}^{(0,3)})^{i}=2{v^{a}}_{c}{v^{b}}_{a}{v^{ic}}_{b}-3{v^{i}}_{a}{v^{c}}_{b}{v^{ab}}_{c}$
$\displaystyle
t^{22}[2,0](u_{2}v_{2}v_{2}v_{2}):(R_{22}^{(0,3)})^{ij}=u^{ij}{v^{a}}_{b}{v^{b}}_{c}{v^{c}}_{a}-3u^{a(i}{v^{j)}}_{b}{v^{b}}_{c}{v^{c}}_{a}+3u^{ab}{v^{(i}}_{a}{v^{j)}}_{c}{v^{c}}_{b}$
$\displaystyle
t^{24}[0,0](u_{2}v_{2}v_{2}v_{3}):R_{24}^{(0,3)}=\epsilon_{a_{1}a_{2}a_{3}}u^{a_{1}b}{v^{a_{2}}}_{b}{v^{a_{3}c}}_{d}{v^{d}}_{c}$
$\displaystyle
t^{26}[1,0](v_{2}v_{2}v_{2}v_{3}):(R_{26}^{(0,4)})^{i}={v^{i}}_{a}{v^{a}}_{b}{v^{d}}_{c}{v^{bc}}_{d}$
$\displaystyle
t^{30}[0,0](v_{2}v_{2}v_{2}v_{2}v_{2}):R_{30}^{(0,5)}={v^{a}}_{b}{v^{b}}_{c}{v^{c}}_{d}{v^{d}}_{e}{v^{e}}_{a}$
$\displaystyle
t^{30}[3,0](v_{2}v_{2}v_{2}v_{2}v_{2}):(R_{30}^{(0,5)})^{ijk}=\epsilon^{a_{1}a_{2}(i}{v^{j}}_{a_{1}}{v^{k)}}_{a_{2}}{v^{b}}_{c}{v^{c}}_{d}{v^{d}}_{b}\
.$ (A.4)

Here, the superscripts of $R$ denote $(n_{f},n_{\psi})$ of the terms with
maximal $n_{f}$ in the trace relations and the subscripts denote their $j$.
Their $SU(3)$ representations can be read off from the number of upper and
lower indices. The listed trace relations vanish up to $Q$-exact operators
whose explicit form will be discussed below. As explained before, this is the
exhaustive set of the fundamental trace relations of gravitons which do not
involve $f$’s. The fundamental trace relations involving $f$’s until $j\leq
20$ are given by

$\displaystyle
t^{14}[0,2](v_{2}v_{3},u_{2}w_{3}):(R_{14}^{(1,0)})_{ij}=\epsilon_{a_{1}a_{2}(i}\left(8{v^{a_{1}b}}_{j)}{v^{a_{2}}}_{b}+5\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}w^{a_{2}b_{2}}\right)$
$\displaystyle
t^{14}[2,1](v_{2}v_{3},u_{2}w_{3},u_{3}w_{2}):(R_{14}^{(1,0)})^{ij}_{k}=2{v^{(i}}_{a}{v^{j)a}}_{k}-5{v^{a}}_{k}{v^{ij}}_{a}+3\epsilon_{ka_{1}a_{2}}u^{a_{1}(i}w^{j)a_{2}}+3\epsilon_{ka_{1}a_{2}}u^{ija_{1}}w^{a_{2}}$
$\displaystyle t^{16}[0,1](v_{3}v_{3},u_{2}v_{2}v_{2},u_{2}u_{2}w_{2}):$
$\displaystyle\hskip
14.22636pt(R_{16}^{(1,0)})_{i}=\epsilon_{ia_{1}a_{2}}\left(48{v^{a_{1}b_{1}}}_{b_{2}}{v^{a_{2}b_{2}}}_{b_{1}}+9u^{b_{1}b_{2}}{v^{a_{1}}}_{b_{1}}{v^{a_{2}}}_{b_{2}}-13\epsilon_{b_{1}b_{2}b_{3}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}w^{b_{3}}\right)$
$\displaystyle
t^{16}[1,2](v_{3}v_{3},u_{2}v_{2}v_{2},u_{3}w_{3},u_{2}u_{2}w_{2}):$
$\displaystyle\hskip
14.22636pt(R_{16}^{(1,0)})^{i}_{jk}=\epsilon_{a_{1}a_{2}(j|}\left(24{v^{ia_{1}}}_{b}{v^{ba_{2}}}_{|k)}+2u^{ia_{1}}{v^{a_{2}}}_{b}{v^{b}}_{|k)}\\!-\\!6u^{a_{1}b}{v^{a_{2}}}_{b}{v^{i}}_{|k)}\right.$
$\displaystyle\hskip
71.13188pt\left.+6\epsilon_{|k)b_{1}b_{2}}u^{ia_{1}b_{1}}w^{a_{2}b_{2}}+\epsilon_{|k)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}w^{i}\right)$
$\displaystyle
t^{16}[3,1](v_{3}v_{3},u_{2}v_{2}v_{2},u_{3}w_{3},u_{2}u_{2}w_{2}):$
$\displaystyle\hskip
14.22636pt(R_{16}^{(1,0)})^{ijk}_{l}=24{v^{(ij}}_{a}{v^{k)a}}_{l}+7u^{(ij}{v^{k)}}_{a}{v^{a}}_{l}-6u^{a(i}{v^{j}}_{a}{v^{k)}}_{l}+18\epsilon_{la_{1}a_{2}}u^{a_{1}(ij}w^{k)a_{2}}+3\epsilon_{la_{1}a_{2}}u^{(ij}u^{k)a_{1}}w^{a_{2}}$
$\displaystyle
t^{16}[1,2](v_{2}w_{3},v_{3}w_{2}):(R_{16}^{(1,1)})^{i}_{jk}=\epsilon_{a_{1}a_{2}(j}\left({v^{a_{1}}}_{k)}w^{a_{2}i}+{v^{a_{1}i}}_{k)}w^{a_{2}}\right)$
$\displaystyle
t^{18}[0,0](v_{2}v_{2}v_{2},u_{2}v_{2}w_{2}):R_{18}^{(1,1)}={v^{a_{1}}}_{a_{2}}{v^{a_{2}}}_{a_{3}}{v^{a_{3}}}_{a_{1}}-3\epsilon_{a_{1}a_{2}a_{3}}u^{a_{1}b}{v^{a_{2}}}_{b}w^{a_{3}}$
$\displaystyle t^{18}[1,1](v_{2}v_{2}v_{2},v_{3}w_{3},u_{2}v_{2}w_{2}):$
$\displaystyle\hskip
14.22636pt(R_{18}^{(1,1)})^{i}_{j}=9{v^{i}}_{a_{1}}{v^{a_{1}}}_{a_{2}}{v^{a_{2}}}_{j}-24\epsilon_{ja_{1}a_{2}}{v^{ia_{1}}}_{b}w^{ba_{2}}$
$\displaystyle\hskip
71.13188pt-13\epsilon_{ja_{1}a_{2}}u^{ia_{1}}{v^{a_{2}}}_{b}w^{b}-16\epsilon_{ja_{1}a_{2}}u^{ib}{v^{a_{1}}}_{b}w^{a_{2}}+5\epsilon_{ja_{1}a_{2}}u^{a_{1}b}{v^{i}}_{b}w^{a_{2}}$
$\displaystyle t^{18}[0,3](v_{2}v_{2}v_{2},v_{3}w_{3},u_{2}v_{2}w_{2}):$
$\displaystyle\hskip
14.22636pt(R_{18}^{(1,1)})_{ijk}=\epsilon_{a_{1}a_{2}(i|}\\!\left(3{v^{a_{1}}}_{|j|}{v^{a_{2}}}_{b}{v^{b}}_{|k)}\\!-\\!3\epsilon_{b_{1}b_{2}|j}{v^{a_{1}b_{1}}}_{k)}w^{a_{2}b_{2}}\\!-\\!\epsilon_{b_{1}b_{2}|j}u^{a_{1}b_{1}}{v^{a_{2}}}_{k)}w^{b_{2}}\right)$
$\displaystyle
t^{18}[2,2](v_{2}v_{2}v_{2},v_{3}w_{3},u_{2}v_{2}w_{2}):(R_{18}^{(1,1)})_{ij}^{kl}=2{v^{(i}}_{a}{v^{j)}}_{(k}{v^{a}}_{l)}-6\epsilon_{a_{1}a_{2}(k}{v^{a_{1}(i}}_{l)}w^{j)a_{2}}-\epsilon_{a_{1}a_{2}(k}u^{ij}{v^{a_{1}}}_{l)}w^{a_{2}}$
$\displaystyle
t^{20}[0,2](v_{2}v_{2}w_{2},u_{2}w_{2}w_{2},w_{3}w_{3}):(R_{20}^{(2,0)})_{ij}=2\epsilon_{a_{1}a_{2}(i|}{v^{a_{1}}}_{b}{v^{b}}_{|j)}w^{a_{2}}-3\epsilon_{a_{1}a_{2}a_{3}}{v^{a_{1}}}_{i}{v^{a_{2}}}_{j}w^{a_{3}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\epsilon_{ia_{1}a_{2}}\epsilon_{jb_{1}b_{2}}u^{a_{1}b_{1}}w^{a_{2}}w^{b_{2}}+3\epsilon_{ia_{1}a_{2}}\epsilon_{jb_{1}b_{2}}w^{a_{1}b_{1}}w^{a_{2}b_{2}}\
.$ (A.5)

The relations involving one $f$ appear from $j=14$ and those involving two
$f$’s appear from $j=20$. We do not find any relations involving three $f$’s
until $j\leq 20$.

As explained before, the trace relations (A), (A) vanish up to $Q$-exact
operators, which we now construct explicitly. In principle, one should first