in blue. This is because in that case the $\text{Long}_{\Delta,0}^{[0000]}$
multiplet that gives the ${\cal N}=8$ bound is replaced by an ${\cal N}=8$
conserved current multiplet $(A,\text{cons.})_{1,0}^{[0000]}$ which no longer
contributes to the $\text{Long}_{\Delta,0}^{[000],2}$. The gray dot in Figure
6 instead comes from a $\text{Long}_{2,0}^{[0000]}$ multiplet in ${\cal N}=8$.

Figure 6: Upper bounds on the scaling dimension of the lowest dimension
$\ell=0$ long multiplet in terms of $c_{T}$ for the second superconformal
structure $\text{Long}_{\Delta,0}^{[000],2}$, which for parity preserving
theories has the opposite parity as the superprimary. The orange shaded region
is allowed, and the plot ranges from the GFFT limit $c_{T}\to\infty$ to the
free theory $c_{T}=16$. The black line denotes the $\mathcal{N}=6$ upper bound
computed in this work with $\Lambda=39$, the blue lines denote the
$\mathcal{N}=8$ upper bounds computed in [10] with $\Lambda=43$. At $c_{T}=16$
the $\mathcal{N}=8$ upper bound does not apply, as the $\mathcal{N}=8$
superblock becomes a conserved current that does not decompose to
$\text{Long}_{\Delta,0}^{[000],2}$. The red dashed line denotes the spectrum
read off from the functional saturating the lower bound in Figure 1, which we
identify with the $U(1)_{2M}\times U(1+M)_{-2M}$ theory. The gray dots denote
the GFFT and free theory values from Table 6.

For the $\Delta_{(2,2)}$ and $\Delta_{(2,3)}$ bounds, we should recall that,
at unitarity, the $(A,0)_{\Delta,\ell+1}^{[000],2}$ and
$(A,0)_{\Delta,\ell+1}^{[000],3}$ superblocks become
$(A,1)_{\ell+5/2,\ell+1/2}^{[100],1}$ and
$(A,1)_{\ell+5/2,\ell+1/2}^{[100],2}$, as per (2.56). Thus, the bound
$\Delta_{(2,n)}$, with $n=2,3$ depends on what we assume about
$(A,1)_{\ell+5/2,\ell+1/2}^{[100],n-1}$. Our first result is that if we assume
that $(A,1)_{7/2,3/2}^{[100],n-1}$ does not appear in the $S\times S$ OPE, we
find that the long multiplet bound is at the unitarity value
$\Delta_{(2,n)}=3$, which implies that the assumption that
$(A,1)_{7/2,3/2}^{[100],n-1}$ does not appear in the $S\times S$ OPE was
incorrect. Thus, all ${\cal N}=6$ SCFTs must contain $(A,1)_{7/2,3/2}^{[100]}$
multiplets! This is consistent with the resuult in [9] that all
$\mathcal{N}=8$ SCFTs must contain an $\mathcal{N}=8$ $(A,2)_{3,1}^{[0020]}$
multiplet, which reduces to the $(A,1)_{7/2,3/2}^{[100]}$ $\mathcal{N}=6$
multiplet as per (D.4).

Next, we can derive revised bounds $\Delta_{(2,n)}$, with $n=2,3$, under the
assumption that the $S\times S$ OPE contains the $(A,1)_{7/2,3/2}^{[100],n-1}$
superblocks. As we can see from Figure 7, we found that the bounds
$\Delta_{(2,n)}$ are slightly above $5$ for all $c_{T}$, with little
dependence on $c_{T}$. This is consistent with the value at both GFFT and free
theory. For comparison, we also showed the second lowest operator for
$\mathcal{N}=8$ theories, which corresponds to the lowest long spin 2
$\mathcal{N}=8$ operator.212121It may again seem curious that at $c_{T}=16$,
where the free theory has ${\cal N}=8$ SUSY, the gray dot does not obey the
${\cal N}=8$ bound. At exactly the free theory point, this $\mathcal{N}=8$
operator becomes a conserved current which no longer decomposes to a parity
odd $\mathcal{N}=6$ long multiplet, which is why the $\mathcal{N}=6$ free
theory value of the second lowest operator, denoted by the second lowest gray
dot, does not coincide with the $\frac{16}{c_{T}}\to 1$ limit of the
$\mathcal{N}=8$ upper bound. We do not show any extremal functional results
for these plots, because we do not yet have sufficient numerical precision.

Figure 7: Upper bounds on the scaling dimension of the lowest dimension
$\ell=2$ long multiplet in terms of $c_{T}$ for the
$\text{Long}_{\Delta,2}^{[000],2}$ (left) and
$\text{Long}_{\Delta,2}^{[000],3}$ (right) superconformal structures, which
for parity preserving theories have the opposite parity as the superprimary,
and for $\mathcal{Z}$ preserving theories has the same charge for
$\text{Long}_{\Delta,2}^{[000],2}$ and the opposite charge for
$\text{Long}_{\Delta,2}^{[000],3}$. The orange shaded region is allowed, and
the plot ranges from the GFFT limit $c_{T}\to\infty$ to the free theory
$c_{T}=16$. The black lines denotes the $\mathcal{N}=6$ upper bounds computed
in this work with $\Lambda=39$, the blue lines denote the $\mathcal{N}=8$
upper bounds computed in [10] with $\Lambda=43$. At $c_{T}=16$ the
$\mathcal{N}=8$ upper bound does not apply, as the $\mathcal{N}=8$ superblock
becomes a conserved current that does not decompose to
$\text{Long}_{\Delta,2}^{[000],2}$ or $\text{Long}_{\Delta,2}^{[000],3}$. The
gray dots denote the GFFT and free theory values from Table 6.

###  4.5 Islands for semishort OPE coefficients

In the previous subsection we discussed numerical bounds that apply to all 3d
$\mathcal{N}=6$ SCFTs. In particular, we noticed that the upper/lower bounds
on $(A,+)_{\ell+2,\ell}^{[002]}$ for $\ell=1/2,5/2$ were extremely
constraining. This implies that for a given value of $c_{T}$, we could find a
small island in the space of
$(\lambda^{2}_{(A,+)_{5/2,1/2}^{[002]}},\lambda^{2}_{(A,+)_{9/2,5/2}^{[002]}})$
using the OPE island algorithm described in the previous subsection. We can
make this island even smaller, and correlate it to a physical theory, by
imposing values of $c_{T}$ and $\lambda^{2}_{(B,2)_{{2,0}}^{[022]}}$ computed
using localization in Section 3. We show the results of these plots for
$U(N)_{k}\times U(N)_{-k}$ for a variety of $N,k$ in the Figure 8. Note that
the islands are small enough that we can distinguish each value of $N$ and
$k$, which allows us to non-perturbatively interpolate between M-theory at
small $k$ and Type IIA at large $k$.

Figure 8: Islands in the space of the semi-short OPE coefficients
$\lambda_{(A,+)_{5/2,1/2}^{[002]}}^{2}$,
$\lambda_{(A,+)_{9/2,5/2}^{[002]}}^{2}$ (to be defined precisely later) for
$U(N)_{k}\times U(N)_{-k}$ ABJM theory for various $N,k$. These bounds are
derived from the $\mathcal{N}=6$ bootstrap with $\Lambda=39$ derivatives, and
with the short OPE coefficients (i.e. $c_{T}$ and
$\lambda^{2}_{(B,2)_{2,0}^{[022]}}$) fixed to their values in each theory
using the exact localization results of [9] for $N=2,3,4$ as shown in Table 9
and the all orders in $1/N$ formulae in [10] for $N=10$ as shown in Table 10.

One difficulty with trying to fix a physical theory by imposing two exactly
computed quantities, $c_{T}$ and $\lambda^{2}_{(B,2)_{{2,0}}^{[022]}}$, is
that the most general $\mathcal{N}=6$ ABJ(M) theory has gauge group
$U(N)_{k}\times U(N+M)_{-k}$ and so is described by 3 parameters $N,k,M$.
While for physical theories these parameters should be integers, we expect
that the numerical bootstrap should find theories with any real value of these
parameters, so we are effectively trying to parameterize a 3-dimensional space
of theories. Since we are only imposing two quantities, these islands are
expected to have a finite area even at high numerical precision corresponding
to the third direction in “theory space”. Thankfully, this third direction
appears to be very small. We can quantify this by fixing $N=k=10$ and
computing islands for several different values of $M\leq k/2=5$. As shown in
Figure 9, the island is not very sensitive to the value of $M<N$, which
explains why we were able to get such small islands in a 3-dimensional
conformal manifold by just imposing two values of the parameters.

Figure 9: Islands in the space of the semi-short OPE coefficients
$\lambda_{(A,+)_{5/2,1/2}^{[002]}}^{2}$,
$\lambda_{(A,+)_{9/2,5/2}^{[002]}}^{2}$ for $U(10)_{10}\times U(10+M)_{-10}$
ABJM theory for $M<k/2=5$. These bounds are derived from the $\mathcal{N}=6$
bootstrap with $\Lambda=39$ derivatives, and with $c_{T}$ and
$\lambda^{2}_{(B,2)_{2,0}^{[022]}}$ fixed to their values in each theory using
the all orders in $1/N$ localization formulae in [10] for $N=10$. Note that
the axes describe a very narrow range in parameter space.

##  5 Conclusion

In this paper, we have three main results. Firstly, we setup up the conformal
bootstrap for the stress tensor multiplet scalar bottom component four-point
function $\langle SSSS\rangle$, by computing the superblock decomposition and
deriving the four independent crossing equations. Secondly, we computed both
$c_{T}$ and the short OPE coefficient squared
$\lambda^{2}_{(B,2)_{2,0}^{[022]}}$ using the supersymmetric localization [52]
formula for the mass deformed free energy in both the $U(N)_{k}\times
U(N+M)_{-k}$ and $SO(2)_{2k}\times USp(2+2M)_{-k}$ ABJM theories for a wide
range of $N,k,M$, including the exact results for the vector-like limit $N=1$
and arbitrary $k,M$, which extends the all orders in $1/N$ results for the
$M\leq k\leq N$ case [10]. Finally, we combined all these ingredients to non-
perturbatively study the ABJM theories. In particular, by inputting the exact
values of $c_{T}$ and $\lambda^{2}_{(B,2)_{2,0}^{[022]}}$ for a given $k,N$
and $M=0$, we found precise rigorous islands in the space of semishort OPE
coefficients that interpolate between M-theory at small $k$ and type IIA
string theory at $k\sim N$. We also conjectured that in the infinite precision
limit, the numerical lower bound on $\lambda^{2}_{(B,2)_{2,0}^{[022]}}$ is
saturated by the family of $U(1)_{2M}\times U(1+M)_{-2M}$ theories, which
allowed us to non-rigorously read off all CFT data in $\langle SSSS\rangle$
using the extremal functional method. Interestingly, in the regime of large
$c_{T}$, we found a spin zero long multiplet whose scaling dimension
approaches a zero spin conserved current multiplet at large $c_{T}$, as
expected from weakly broken higher spin symmetry.

There are several ways we can improve upon our 3d $\mathcal{N}=6$ bootstrap
study. From the numerical perspective, it will be useful to improve the
precision of our study. This is parameterized by the parameter $\Lambda$
defined in the main text. While we used $\Lambda=39$ in this work, which is
close to the $\Lambda=43$ values used in the analogous $\mathcal{N}=8$ studies
[10, 12], for $\mathcal{N}=6$ this value has not led to complete convergence.
For instance, we found the lower bound $c_{T}\geq 15.5$, compared to the
$\mathcal{N}=8$ result $c_{T}\geq 15.9$; both are expected to converge to the
free theory $c_{T}=16$. More physically, we expect that approximately
conserved currents should appear in the extremal functional that conjecturally
describes the $U(1)_{2M}\times U(1+M)_{-2M}$ theory. We found such an operator
in the zero spin sector as shown in Figure 6, but for higher spin these
operators are hard to see from the numerics at large $c_{T}$, since their OPE
coefficients start at $O(c_{T}^{-1})$. The main obstacle to increasing
$\Lambda$ at the moment is not SDPB, which due to the recent upgrade [80] can
easily handle four crossing equations at very high $\Lambda$, but simply the
difficulty in computing numerical approximations to the superblocks at large
$\Lambda$. In particular it would be extremely useful to have an efficient
code for approximations of linear combinations of conformal blocks with
$\Delta$ dependent coefficients around the crossing symmetric point. Currently
the code scalar_blocks code, found on the bootstrap collaboration
website,222222This code can be found at
https://gitlab.com/bootstrapcollaboration/scalar_blocks/blob/
master/Install.md. is only able to efficiently compute single conformal
blocks.

From the analytical perspective, it would be interesting to derive the
$1/c_{T}$ corrections to $\langle SSSS\rangle$ for ABJ theories in the vector-
like limit, such as for $U(N)_{k}\times U(N+M)_{-k}$ at finite $N$ and large
$M,k$ with fixed $M/k$. This would complement the order $1/c_{T}$ correlator
that was computed in the supergravity limit in [81] and was successfully
matched to numerical bootstrap results in [82]. We expect that the vector-like
correlator can be computed by generalizing [83, 84] to $\mathcal{N}=6$, and
hope to report on these results soon in a future publication.

We could also make further use of localization to improve our results. In this
study we were only able to impose two analytic constraints, $c_{T}$ and
$\lambda^{2}_{(B,2)_{2,0}^{[022]}}$, while ABJM is parameterized by three
parameters $N,M,k$. For this reason there are not enough constraints to
uniquely pick out a single ABJM theory and so we should not expect our islands
to shrink indefinitely as we increase $\Lambda$. We think this is the reason
why the islands shown in Figure 8, while small, are still much bigger than the
$\mathcal{N}=8$ islands computed in [12]. Localization conveniently provides
us a third quantity given by four mixed mass derivatives of the mass deformed
free energy, which as shown [60] is related to an integral of $\langle
SSSS\rangle$. While this integrated constraint can be imposed analytically in
a large $N$ expansion as in [74, 60], it is not yet known how to impose it on
the numerical bootstrap in our case. Perhaps the method used in [85], where a
similar similar integrated constraint was successfully imposed on the
numerical bootstrap of a certain supersymmetric 2d theory, could be applied to
our case. Another option would be to look at a larger system of correlators
involving fermions. This would allow us to impose parity, which would restrict
the set of known ${\cal N}=6$ SCFTs to a few families such as $U(N)_{k}\times
U(N)_{-k}$ parameterized by only two parameters each.

Once we can fully fix the three-parameters ABJM theory, it would be
interesting to see if we can match integrability results computed for the
lowest dimension singlet scaling dimension in the leading large $N$ ’t Hooft
limit at fixed $\lambda_{\text{'t Hooft}}=N/k$ and $M=0$. On the integrability
side some results are available, for instance, in [86, 87]. On the
localization side, we would need to compute the derivatives of the mass
deformed free energy in the $1/N$ expansion at finite $\lambda_{\text{'t
Hooft}}$. In fact, the zero mass free energy has already been computed in this
limit in [88] by applying topological recursion to the Lens space $L(2,1)$
matrix model, so computing $c_{T}$ and $\lambda^{2}_{(B,2)_{2,0}^{[022]}}$
should correspond to just computing two- and four-body operators in this
matrix model. This could potentially lead to the first precise comparison
between integrability and the numerical conformal bootstrap.

Finally, it would be interesting to consider the superconformal block
decomposition of other correlators involving operators which are less than
half-BPS but still have scalar superprimaries, which makes them feasible to
numerical bootstrap. For instance, in ${\mathcal{N}}=4$ theories the stress
tensor multiplet is only $1/4$-BPS but still has a scalar superprimary.
Similarly, while conserved current multiplets are half-BPS for
${\mathcal{N}}=4$ theories (and were studied using the numerical bootstrap in
[18]), for ${\mathcal{N}}=3$ theories they are $1/3$-BPS and for
${\mathcal{N}}=2$ theories are only $1/4$-BPS. Many localization results exist
which could be applied to these cases, including ${\mathcal{N}}=4$ results
computed in [89].

## Acknowledgments

We thank Ofer Aharony, Oren Bergman, Simone Giombi, and Yuji Tachikawa for
useful discussions and correspondence. DJB, MJ, and SSP are supported in part
by the Simons Foundation Grant No. 488653, and by the US NSF under Grant No.
1820651. DJB is also supported in part by the General Sir John Monash
Foundation. SMC is supported by the Zuckerman STEM Leadership Fellowship.

##  Appendix A Multiplets with only $\bf 15$s and $\bf 1$s in the $S\times S$
OPE

In the main text we were able to restrict the possible superconformal blocks
where operators in the $\bf 84$, $\bf 45$, $\overline{\bf 45}$ or $\bf
20^{\prime}$ are exchanged. This leaves us to consider superblocks where the
only exchanged operators are in the $\bf 1$ or $\bf 15$. We will analyze this
possibility using the superconformal Ward identities.

Let us fix some supermultiplet ${\mathcal{M}}$ and define

${\mathcal{S}}_{\bf r}^{({\mathcal{M}})}(U,V)=\sum_{(\Delta,\ell,{\bf
r})\in{\mathcal{M}}}a_{\Delta,\ell,\bf r}g_{\Delta,\ell}(U,V)$ (A.1)

to be the contribution from $s$-channel ${\mathcal{M}}$ exchange to
${\mathcal{S}}_{\bf r}(U,V)$. The superconformal Ward identities apply to each
superblock independently, and so ${\mathcal{S}}_{\bf
r}^{({\mathcal{M}})}(U,V)$ must satisfy the Ward identities (2.10). If we
demand that no operators in the $\bf 84$, $\bf 45$, $\overline{\bf 45}$ or
$\bf 20^{\prime}$ are exchanged, we then find that

$\begin{split}U^{2}\partial_{U}\left({\mathcal{S}}_{\bf
1}^{({\mathcal{M}})}+{\mathcal{S}}_{{\bf
15}_{s}}^{({\mathcal{M}})}\right)+\left[2U(U+2V-2)\partial_{U}+4UV\partial_{V}-2(V-1)\right]{\mathcal{S}}_{{\bf
15}_{a}}^{({\mathcal{M}})}&=0\\\ U\partial_{V}({\mathcal{S}}_{\bf
1}^{({\mathcal{M}})}+{\mathcal{S}}_{{\bf
15}_{s}}^{({\mathcal{M}})})+\left[-4U\partial_{U}-2U\partial_{V}+2\right]{\mathcal{S}}_{{\bf
15}_{a}}^{({\mathcal{M}})}&=0\,.\end{split}$ (A.2)

To make further progress we can consider the correlator

$\displaystyle\langle S(\vec{x}_{1},X_{1})S(\vec{x}_{2},X_{2})$ $\displaystyle
P(\vec{x}_{3},X_{3})P(\vec{x}_{4},X_{4})\rangle=\frac{1}{x_{12}^{2}x_{34}^{4}}\sum_{i=1}^{6}{\mathcal{P}}^{i}(U,V){\cal
B}_{i}\,,$ (A.3)

where ${\cal B}_{i}$ are defined as in (2.4). The Ward identities relating
${\mathcal{P}}^{i}(U,V)$ to ${\mathcal{S}}^{i}(U,V)$ have been computed in
[60]. Note that because both $S$ and $P$ transform in the $\bf 15$ of
$\mathfrak{so}(6)$, the correlators $\langle SSSS\rangle$ and $\langle
SSPP\rangle$ have the same R-symmetry structures. Since we are interested in
$s$-channel conformal block expansion, we are led to define
${\mathcal{P}}_{\bf r}(U,V)$ in an analogous fashion to ${\mathcal{S}}_{\bf
r}(U,V)$ in (2.8).

If $s$-channel exchange of ${\mathcal{M}}$ only contributes to the $\bf 1$ and
$\bf 15$ channels in the $\langle SSSS\rangle$ correlator, this must also be
true of $\langle SSPP\rangle$ and so

${\mathcal{P}}_{\bf 20^{\prime}}^{({\mathcal{M}})}(U,V)={\mathcal{P}}_{{\bf
45}\oplus\overline{\bf 45}}^{({\mathcal{M}})}(U,V)={\mathcal{P}}_{\bf
84}^{({\mathcal{M}})}(U,V)=0\,.$ (A.4)

Combining this with (A.2) and the Ward identities derived in [60], we find
that

${\mathcal{D}}{\mathcal{S}}^{({\mathcal{M}})}_{\bf
1}(U,V)={\mathcal{D}}{\mathcal{S}}^{({\mathcal{M}})}_{{\bf
15}_{s}}(U,V)={\mathcal{D}}{\mathcal{S}}^{({\mathcal{M}})}_{{\bf
15}_{a}}(U,V)=0\,,$ (A.5)

where ${\mathcal{D}}$ is the differential operator

${\mathcal{D}}=2U^{2}\partial_{U}^{2}+2(U+V-1)\partial_{U}\partial_{V}+2UV\partial_{V}^{2}+U\partial_{U}+(1+2U-V)\partial_{V}\,.$
(A.6)

Our next step is to rewrite the cross-ratios $(U,V)$ using radial coordinates
$(r,\eta)$

$U=\frac{16r^{2}}{(1+r^{2}+2r\eta)^{2}}\,,\qquad
V=\frac{(1+r^{2}-2r\eta)^{2}}{(1+r^{2}+2r\eta)^{2}}\,.$ (A.7)

Conformal blocks have a relatively simple form in radial coordinates:

$g_{\Delta,\ell}(r,\eta)=r^{\Delta}\sum_{k=0}^{\infty}r^{2k}p_{\Delta,\ell,k}(\eta)\,,$
(A.8)

where each $p_{\Delta,\ell,k}(\eta)$ is polynomial in $\eta$ [90]. In
particular, the leading term is given by

$p_{\Delta,\ell,0}(\eta)=P_{\ell}(\eta)\,,$ (A.9)

where $P_{n}(x)$ is the $n^{\text{th}}$ Legendre polynomial. Since
${\mathcal{S}}_{\bf r}^{({\mathcal{M}})}(U,V)$ is the sum of a finite number
of conformal blocks, we expect that

${\mathcal{S}}_{\bf r}^{({\mathcal{M}})}(U,V)=r^{\Delta}\left(q_{\bf
r}(\eta)+O(r^{2})\right)$ (A.10)

for some polynomial $q_{\bf r}(\eta)$.

Let us translate (A.5) into radial coordinates:

$\left[r^{2}(r^{2}-1)\partial_{r}^{2}+2r^{3}\partial_{r}-(r^{2}-1)(\eta^{2}-1)\partial_{\eta}^{2}-2(r^{2}-1)\eta\partial_{\eta}\right]{\mathcal{S}}_{\bf
r}^{({\mathcal{M}})}(r,\eta)=0\,.$ (A.11)

Substituting (A.10) into this equation we find that $q_{\bf r}(\eta)$
satisfies Legendre’s equation

$(1-\eta^{2})q_{\bf r}^{\prime\prime}(\eta)-2\eta q_{\bf
r}^{\prime}(\eta)+\Delta(\Delta-1)q_{\bf r}(\eta)=0\,.$ (A.12)

Hence $q_{\bf r}(\eta)$ is a polynomial if and only if $\Delta\in\mathbb{Z}$,
in which case ${q_{\bf r}(\eta)=aP_{\Delta+1}(\eta)}$ for some arbitrary
constant $a$. Since unitarity implies that $\Delta\geq 0$, we conclude that
${\mathcal{S}}_{\bf r}^{({\mathcal{M}})}(r,\eta)$ includes a contribution from
either an operator with twist $\Delta-\ell=1$, or else from the identity
operator $\Delta=\ell=0$.

Any operator in a superconformal multiplet has twist greater than or equal to
the twist of the superconformal primary. Thus if ${\mathcal{M}}$ is not the
trivial supermultiplet then its superconformal primary must have twist one.
Examining Table 2, we see that aside from the stress-tensor multiplet the only
other such multiplets are conserved currents: A-type multiplets whose
superprimary is an R-symmetry singlet with conformal dimension
$\Delta=\ell+1$. We conclude that any superblock in which the only exchanged
operators transform in the $\bf 1$ or $\bf 15$ must correspond to the exchange
of the trivial, stress-tensor, or a conserved current multiplet.

##  Appendix B Superconformal algebra

We follow the conventions of Appendix B of [9] for both the conformal and
superconformal generators. In particular, the commutation relations for the
conformal generators are

$\begin{split}[M_{\alpha}^{\,\,\beta},P_{\gamma\delta}]&=\delta_{\gamma}^{\,\,\beta}P_{\alpha\delta}+\delta_{\delta}^{\,\,\beta}P_{\alpha\gamma}-\delta_{\alpha}^{\,\,\beta}P_{\gamma\delta}\,,\\\
[M_{\alpha}^{\,\,\beta},K^{\gamma\delta}]&=-\delta_{\alpha}^{\,\,\gamma}K^{\beta\delta}-\delta_{\alpha}^{\,\,\delta}K^{\beta\gamma}+\delta_{\alpha}^{\,\,\beta}K^{\gamma\delta}\,,\\\
[M_{\alpha}^{\,\,\beta},M_{\gamma}^{\,\,\delta}]&=-\delta_{\alpha}^{\,\,\delta}M_{\gamma}^{\,\,\beta}+\delta_{\gamma}^{\,\,\beta}M_{\alpha}^{\,\,\delta}\,,\\\
[D,P_{\alpha\beta}]&=P_{\alpha\beta}\,,\\\
[D,K^{\alpha\beta}]&=-K^{\alpha\beta}\,,\\\
[K^{\alpha\beta},P_{\gamma\delta}]&=4\delta_{(\gamma}^{\,\,(\alpha}M_{\delta)}^{\,\,\beta)}+4\delta_{(\gamma}^{\,\,\alpha}\delta_{\delta)}^{\,\,\beta}D\,.\end{split}$
(B.1)

In radial quantization, the conjugation properties of the conformal generators
are

$\begin{split}(P_{\alpha\beta})^{\dagger}&=K^{\alpha\beta}\,,\qquad(K_{\alpha\beta})^{\dagger}=P^{\alpha\beta}\,,\\\
(M_{\alpha}{}^{\beta})^{\dagger}&=M_{\beta}{}^{\alpha}\,,\qquad
D^{\dagger}=D\,.\end{split}$ (B.2)

The extension of the conformal algebra to the
$\mathfrak{osp}({\mathcal{N}}|4)$ superconformal algebra is given by

$\displaystyle\\{Q_{\alpha r},Q_{\beta s}\\}$
$\displaystyle=2\delta_{rs}P_{\alpha\beta}\,,\qquad$
$\displaystyle\\{S^{\alpha}_{\,\,r},S^{\beta}_{\,\,s}\\}$
$\displaystyle=-2\delta_{rs}K^{\alpha\beta}\,,$ (B.3)
$\displaystyle[K^{\alpha\beta},Q_{\gamma r}]$
$\displaystyle=-i\left(\delta_{\gamma}^{\,\,\alpha}S^{\beta}_{\,\,r}+\delta_{\gamma}^{\,\,\beta}S^{\alpha}_{\,\,r}\right)\,,\qquad$
$\displaystyle[P_{\alpha\beta},S^{\gamma}_{\,\,r}]$
$\displaystyle=-i\left(\delta_{\alpha}^{\,\,\gamma}Q_{\beta
r}+\delta_{\beta}^{\,\,\gamma}Q_{\alpha r}\right)\,,$ (B.4)
$\displaystyle[M_{\alpha}^{\,\,\beta},Q_{\gamma r}]$
$\displaystyle=\delta_{\gamma}^{\,\,\beta}Q_{\alpha
r}-\frac{1}{2}\delta_{\alpha}^{\,\,\beta}Q_{\gamma r}\,,$
$\displaystyle[M_{\alpha}^{\,\,\beta},S^{\gamma}_{\,\,r}]$
$\displaystyle=-\delta_{\alpha}^{\,\,\gamma}S^{\beta}_{\,\,r}+\frac{1}{2}\delta_{\alpha}^{\,\,\beta}S^{\gamma}_{\,\,r}\,,$
(B.5) $\displaystyle[D,Q_{\alpha r}]$ $\displaystyle=\frac{1}{2}Q_{\alpha
r}\,,$ $\displaystyle[D,S^{\alpha}_{\,\,r}]$
$\displaystyle=-\frac{1}{2}S^{\alpha}_{\,\,r}\,,$ (B.6)
$\displaystyle[R_{rs},Q_{\alpha t}]$
$\displaystyle=i\left(\delta_{rt}Q_{\alpha s}-\delta_{st}Q_{\alpha
r}\right)\,,$ $\displaystyle[R_{rs},S^{\alpha}_{\,\,t}]$
$\displaystyle=i\left(\delta_{rt}S^{\alpha}_{\,\,s}-\delta_{st}S^{\alpha}_{\,\,r}\right)\,,$
(B.7) $\displaystyle[R_{rs},R_{tu}]$
$\displaystyle=i\left(\delta_{rt}R_{su}+\cdots\right)\,,\qquad$
$\displaystyle\\{Q_{\alpha r},S^{\beta}_{\,\,s}\\}$
$\displaystyle=2i\left(\delta_{rs}\left(M_{\alpha}^{\,\,\beta}+\delta_{\alpha}^{\,\,\beta}D\right)-i\delta_{\alpha}^{\,\,\beta}R_{rs}\right)\,,$
(B.8)

where $R_{rs}$ are the anti-symmetric generators of the
$\mathfrak{so}({\mathcal{N}})$ R-symmetry. In addition to (B.2), we also have
the following conjugation properties:

$\begin{split}(Q_{\alpha
r})^{\dagger}&=-iS^{\alpha}_{r}\,,\qquad(S^{\alpha}_{r})^{\dagger}=-iQ_{\alpha
r}\,,\\\ (R_{rs})^{\dagger}&=R_{rs}\,.\end{split}$ (B.9)

##  Appendix C Characters of $\mathfrak{osp}(6|4)$

In this section we will review the character formulas of
$\mathfrak{osp}(6|4)$, which were computed in [27], as well as their
decomposition under
$\mathfrak{osp}(6|4)\rightarrow\mathfrak{so}(3,2)\oplus\mathfrak{so}(6)$. This
decomposition was used in Section 2 to determine which conformal primaries
reside in each supermultiplet appearing in the $S\times S$ OPE.

The $\mathfrak{osp}(6|4)$ characters are defined in terms of the quantum
numbers and generators given in Section 2.2 as

$\displaystyle\chi_{(\Delta;j;\mathrm{r})}(s,x,y)\equiv\mathrm{Tr}_{{\mathcal{R}}_{(\Delta;j;\mathrm{r})}}\left(s^{2D}x^{2J_{3}}y_{1}^{H_{1}}y_{2}^{H_{2}}y_{3}^{H_{3}}\right)\,.$
(C.1)

Their explicit form for the multiplets we consider are

$\displaystyle\chi^{(A,\pm)}_{(\Delta;j;r,r,r)}(s,x,y)$
$\displaystyle=s^{2\Delta}P(s,x)\sum_{a_{1}\,,a_{2}\,,a_{3}=0}^{2}\,\sum_{\bar{a}_{1}\,,\bar{a}_{2}\,,\bar{a}_{3}=0}^{1}s^{a_{1}+a_{2}+a_{3}+\bar{a}_{1}+\bar{a}_{2}+\bar{a}_{3}}\chi_{2j+\bar{a}_{1}+\bar{a}_{2}+\bar{a}_{3}}(x)$
$\displaystyle\times\left(\prod_{i=1}^{3}\chi_{j_{a_{i}}}(x)\right)\,\chi_{(r+\bar{a}_{1}-a_{1}\,,r+\bar{a}_{2}-a_{2}\,,\pm
r\pm\bar{a}_{3}\mp a_{3})}(y)\,,$ (C.2)
$\displaystyle\chi^{(B,+)}_{(\Delta;0;r,r,r)}(s,x,y)$
$\displaystyle=s^{2\Delta}P(s,x)\sum_{a_{1}\,,a_{2}\,,a_{3}=0}^{2}s^{a_{1}+a_{2}+a_{3}}\left(\prod_{i=1}^{3}\chi_{j_{a_{i}}}(x)\right)\,\chi_{(r-a_{1}\,,r-a_{2}\,,r-a_{3})}(y)\,,$
(C.3)
$\displaystyle\chi^{(A,n)}_{(\Delta;j;r_{1},\ldots,r_{1},r_{n+1},\ldots,r_{3})}(s,x,y)$
$\displaystyle=s^{2\Delta}P(s,x)\sum_{a_{1}\,,a_{2}\,,a_{3}=0}^{2}\sum_{\bar{a}_{n+1},\ldots,\bar{a}_{3}=0}^{2}\sum_{\bar{a}_{1},\ldots,\bar{a}_{n}=0}^{1}s^{a_{1}+a_{2}+a_{3}+\bar{a}_{1}+\bar{a}_{2}+\bar{a}_{3}}\chi_{2j+\bar{a}_{1}+\cdots+\bar{a}_{n}}(x)$
$\displaystyle\times\left(\prod_{i=n+1}^{3}\chi_{j_{\bar{a}_{i}}}(x)\right)\left(\prod_{i=1}^{3}\chi_{j_{a_{i}}}(x)\right)\,\chi_{(r_{1}+\bar{a}_{1}-a_{1}\,,r_{2}+\bar{a}_{2}-a_{2}\,,r_{3}+\bar{a}_{3}-a_{3})}(y)\,,$
(C.4)
$\displaystyle\chi^{(B,n)}_{(\Delta;0;r_{1},\ldots,r_{1},r_{n+1},\ldots,r_{3})}(s,x,y)$
$\displaystyle=s^{2\Delta}P(s,x)\sum_{a_{1}\,,a_{2}\,,a_{3}\,,\bar{a}_{n+1},\ldots,\bar{a}_{3}=0}^{2}s^{a_{1}+a_{2}+a_{3}+\bar{a}_{n+1}+\cdots+\bar{a}_{3}}\left(\prod_{i=n+1}^{3}\chi_{j_{\bar{a}_{i}}}(x)\right)$
$\displaystyle\times\left(\prod_{i=1}^{3}\chi_{j_{a_{i}}}(x)\right)\,\chi_{(r_{1}-a_{1},\ldots,r_{1}-a_{n},r_{n+1}+\bar{a}_{n+1}-a_{n+1},\ldots,r_{3}+\bar{a}_{3}-a_{3})}(y)\,,$
(C.5)

where the long multiplet corresponds to $(A,0)$, we define $j_{a}\equiv
a\,\,(\\!\\!\\!\\!\mod{2})$, the $\mathfrak{su}(2)$ and $\mathfrak{so}(6)$
characters are

$\displaystyle\chi_{j}(x)$
$\displaystyle=\frac{x^{j+1}-x^{-j-1}}{x-x^{-1}}\,,$ (C.6)
$\displaystyle\chi_{\mathrm{r}}(y)$
$\displaystyle=\frac{\det\left[y_{i}^{r_{j}+3-j}+y_{i}^{-r_{j}-3+j}\right]+\det\left[y_{i}^{r_{j}+3-j}-y_{i}^{-r_{j}-3+j}\right]}{2\prod_{1\leq
i<j\leq 3}(y_{i}+y_{i}^{-1}-y_{j}-y_{j}^{-1})}\,,$ (C.7)

and the function $P(s,x)$ is related to the $\mathfrak{so}(3,2)$ character and
takes the form

$\begin{split}P(s,x)&=\frac{1}{1-s^{4}}\sum_{n=0}^{\infty}s^{2n}\chi_{2n}(x)\,.\end{split}$
(C.8)

The products of the $\mathfrak{su}(2)$ characters in (C.2)–(C.5) are easily
transformed into sums of such characters by decomposing $\mathfrak{su}(2)$
tensor products. After doing so, we see that (C.2)–(C.5) become sums over
$\mathfrak{so}(3,2)\oplus\mathfrak{so}(6)$ characters, as
desired.232323Sometimes the $\mathfrak{so}(6)$ characters in (C.2)–(C.5)
appear with negative Dynkin labels. One can then try to use the identity
$\displaystyle\chi_{\mathrm{r}^{\omega}}(y)=(-)^{\ell(\omega)}\chi_{\mathrm{r}}(y)\,,$
to obtain a character with non-negative Dynkin labels. In this identity
$\omega$ is an element of the $\mathfrak{so}(6)$ Weyl group $S_{4}$,
$\mathrm{r}^{\omega}=\omega(\mathrm{r}+\rho)-\rho$ is a Weyl reflection,
$\rho=(2,1,0)$ is the Weyl vector, and $(-)^{\ell(\omega)}$ is the signature
of the Weyl transformation. If there is no Weyl transformation such that
$\mathrm{r}^{\omega}$ correspond to non-negative integer Dynkin labels, then
$\chi_{\mathrm{r}}=0$.

##  Appendix D Decomposing $\mathcal{N}=8$ superblocks to $\mathcal{N}=6$

In this appendix we discuss how the superblocks that appeared in the four-
point function of the $\mathcal{N}=8$ stress tensor superprimary
$\overline{S}$ decompose into the $\mathcal{N}=6$ superblocks discussed for
$\langle SSSS\rangle$ in the main text. This serves as both a consistency
check of our $\mathcal{N}=6$ superblocks, and also allows us to translate the
$\mathcal{N}=8$ numerical bootstrap results of [11, 10, 9, 12] into
$\mathcal{N}=6$ language, which we will use to compare to the $\mathcal{N}=6$
results in Section 4. $\overline{S}$ transforms in the $\bf 35_{c}$ of the
$\mathcal{N}=8$ R-symmetry group $SO(8)$, which decomposes to $SO(6)\times
U(1)$ as

$\begin{split}{\bf 35}\to{\bf 15}_{0}\oplus{\bf 10}_{2}\oplus\overline{\bf
10}_{-2}\,,\end{split}$ (D.1)

so $\overline{S}$ decomposes to $S$ as well as the superprimaries of the
multiplets $(B,+)_{1,0}^{020}$ and $(B,+)_{1,0}^{002}$ that are charged under
$U(1)$. Since we are only interested in correlators of $S$, we will always
restrict to $U(1)$ singlets when decomposing from $\mathcal{N}=8$ to
$\mathcal{N}=6$ in this appendix, which we will denote using an arrow instead
of an equality.

Type | $(\Delta,\ell)$ |  $\mathfrak{so}(8)_{R}$ irrep | spin $\ell$ | BPS
---|---|---|---|---
$(B,+)$ | $(2,0)$ | ${\bf 294}_{c}=[0040]$ | $0$ | $1/2$
$(B,2)$ | $(2,0)$ | ${\bf 300}=[0200]$ | $0$ | $1/4$
$(B,+)$ | $(1,0)$ | ${\bf 35}_{c}=[0020]$ | $0$ | 1/2
$(A,+)$ | $(\ell+2,\ell)$ | ${\bf 35}_{c}=[0020]$ | even | $1/4$
$(A,2)$ | $(\ell+2,\ell)$ | ${\bf 28}=[0100]$ | odd | $1/8$
Long | $\Delta\geq\ell+1$ | ${\bf 1}=[0000]$ | even | $0$
$(A,\text{cons.})$ | $(\ell+1,\ell)$ | ${\bf 1}=[0000]$ | even | $5/16$
Table 13: The possible superconformal multiplets in the
$\overline{S}\times\overline{S}$ OPE. The
$\mathfrak{so}(3,2)\oplus\mathfrak{so}(8)_{R}$ quantum numbers are those of
the superconformal primary in each multiplet.

The $\mathcal{N}=8$ multiplets that appear in $\overline{S}\times\overline{S}$
are listed in Table 13.242424In [11], the long multiplet was denoted as
$(A,0)$. We can decompose the characters for these superblocks, as computed in
[11], into the characters of the $\mathcal{N}=6$ superblocks as computed from
the previous Appendix to get the following decomposition of multiplets:

$\begin{split}(B,+)^{[0020]}_{1,0}\qquad&\to\qquad(B,2)^{[011]}_{1,0}\,,\\\
(B,+)^{[0040]}_{2,0}\qquad&\to\qquad(B,2)^{[022]}_{2,0}\,,\\\
(B,2)^{[0200]}_{2,0}\qquad&\to\qquad(B,2)^{[022]}_{2,0}\oplus(B,1)^{[200]}_{2,0}\oplus(A,2)^{[011]}_{2,0}\oplus(A,0)^{[000]}_{2,0}\,,\\\
(A,+)^{[0020]}_{\ell+2,\ell}\qquad&\to\qquad(A,+)^{[020]}_{\ell+5/2,\ell+1/2}\oplus(A,-)^{[002]}_{\ell+5/2,\ell+1/2}\oplus(A,2)^{[011]}_{\ell+2,\ell}\oplus(A,2)^{[011]}_{\ell+3,\ell+1}\,,\\\
(A,2)^{[0100]}_{\ell+2,\ell}\qquad&\to\qquad(A,2)^{[011]}_{\ell+2,\ell}\oplus(A,2)^{[011]}_{\ell+3,\ell+1}\oplus
2\times(A,1)^{[100]}_{\ell+5/2,\ell+1/2}\oplus(A,0)^{[000]}_{\ell+3,\ell+1}\oplus(A,0)^{[000]}_{\ell+2,\ell}\,,\\\
\text{Long}^{[0000]}_{\Delta,\ell}\qquad&\to\qquad\text{Long}^{[000]}_{\Delta,\ell}\oplus\text{Long}^{[000]}_{\Delta+1,\ell-1}\oplus
2\times\text{Long}^{[000]}_{\Delta+1,\ell}\oplus\text{Long}^{[000]}_{\Delta+1,\ell+1}\oplus\text{Long}^{[000]}_{\Delta+2,\ell}\,,\\\
(A,\text{cons.})^{[0000]}_{\ell+1,\ell}\qquad&\to\qquad(A,\text{cons.})^{[000]}_{\ell+1,\ell}\oplus(A,\text{cons.})^{[000]}_{\ell+2,\ell+1}\,,\\\
\end{split}$ (D.2)

where $2\times$ denotes that the multiplet appears twice.

The $\mathcal{N}=8$ stress tensor correlator was written in [11] in the basis

$\begin{split}\langle\overline{S}(\vec{x}_{1},Y_{1})\overline{S}(\vec{x}_{2},Y_{2})\overline{S}(\vec{x}_{3},Y_{3})\overline{S}(\vec{x}_{4},Y_{4})\rangle&=\frac{1}{x_{12}^{2}x_{34}^{2}}\biggl{[}\overline{{\mathcal{S}}^{1}}(U,V)Y_{12}^{2}Y_{34}^{2}+\overline{{\mathcal{S}}^{2}}(U,V)Y_{13}^{2}Y_{24}^{2}+\overline{{\mathcal{S}}^{3}}(U,V)Y_{14}^{2}Y_{23}^{2}\\\
&{}+\overline{{\mathcal{S}}^{4}}(U,V)Y_{13}Y_{14}Y_{23}Y_{24}+\overline{{\mathcal{S}}^{5}}(U,V)Y_{12}Y_{14}Y_{23}Y_{34}+\overline{{\mathcal{S}}^{6}}(U,V)Y_{12}Y_{13}Y_{24}Y_{34}\biggr{]}\,,\\\
\end{split}$ (D.3)

where $Y$ are $\mathfrak{so}(8)$ null vectors. As shown in Section 2.7.1 of
[60],252525This was for a basis of $SO(8)$ matrices $\overline{X}$, but as
noted there its the exact same decomposition for the basis of $Y$’s. this
decomposes to the $\mathcal{N}=6$ basis in (2.3) as

$\begin{split}\\{\overline{\mathcal{S}^{1}}\,,\overline{\mathcal{S}^{2}}\,,\overline{\mathcal{S}^{3}}\,,\overline{\mathcal{S}^{4}}\,,\overline{\mathcal{S}^{5}}\,,\overline{\mathcal{S}^{6}}\\}\to\\{\mathcal{S}^{1}\,,\mathcal{S}^{2}\,,\mathcal{S}^{3}\,,4\mathcal{S}^{4}\,,4\mathcal{S}^{5}\,,4\mathcal{S}^{6}\\}\,.\end{split}$
(D.4)

Finally, we can decompose the explicit $\mathcal{N}=8$ superblocks
$\mathfrak{G}_{\mathcal{M}_{\Delta,\ell}^{[d_{1}d_{2}d_{3}d_{4}]}}(U,V)$ given
in [11] into the $\mathcal{N}=6$ superblocks
$\mathfrak{G}_{\mathcal{M}_{\Delta,\ell}^{[d_{1}d_{2}d_{3}]}}(U,V)$ given in
the attached Mathematica file to get

$\begin{split}\mathfrak{G}_{(B,+)_{1,0}^{[0020]}}\qquad&\to\qquad\frac{1}{4}\mathfrak{G}_{(B,2)_{1,0}^{[011]}}\,,\\\
\qquad\mathfrak{G}_{(B,+)_{2,0}^{[0040]}}\qquad&\to\qquad\frac{1}{4}\mathfrak{G}_{(B,2)_{2,0}^{[022]}}\,,\\\
\mathfrak{G}_{(B,2)_{2,0}^{[0200]}}\qquad&\to\qquad\frac{1}{16}\mathfrak{G}_{(B,2)_{2,0}^{[022]}}+\frac{1}{3}\mathfrak{G}_{(A,2)_{2,0}^{[011]}}+\frac{4}{35}\mathfrak{G}_{\text{Long}_{2,0}^{[000],1}}+\frac{1}{8}\mathfrak{G}_{(B,1)_{2,0}^{[200]}}\,,\\\
\mathfrak{G}_{(A,+)_{\ell+2,\ell}^{[0020]}}\qquad&\to\qquad\frac{1}{4}\mathfrak{G}_{(A,2)_{\ell+2,\ell}^{[011]}}+\frac{(4+\ell)^{2}}{(5+2\ell)(7+2\ell)}\mathfrak{G}_{(A,2)_{\ell+3,\ell+1}^{[011]}}+\frac{1+\ell}{4+2\ell}\mathfrak{G}_{(A,+)_{\ell+5/2,\ell+1/2}^{[020]}}\,,\\\
\mathfrak{G}_{(A,2)_{\ell+2,\ell}^{[0020]}}\qquad&\to\qquad-\frac{1}{4}\mathfrak{G}_{(A,2)_{\ell+2,\ell}^{[011]}}-\frac{(3+\ell)^{2}}{(3+2\ell)(5+2\ell)}\mathfrak{G}_{(A,2)_{\ell+3,\ell+1}^{[011]}}-\frac{\ell^{2}}{3+4\ell(2+\ell)}\mathfrak{G}_{\text{Long}_{\ell+2,\ell}^{[011]}}\\\
&-\frac{(5+\ell)^{2}}{(7+2\ell)(9+2\ell)}\mathfrak{G}_{\text{Long}_{\ell+3,\ell+1}^{[000]}}-\frac{1+\ell}{9+3\ell}\mathfrak{G}_{(A,1)_{\ell+5/2,\ell+1/2}^{[000],1}}-\mathfrak{G}_{(A,1)_{\ell+5/2,\ell+1/2}^{[000],2}}\,,\\\
\mathfrak{G}_{\text{Long}_{\Delta,\ell}^{[0000]}}\qquad&\to\qquad\mathfrak{G}_{\text{Long}_{\Delta,\ell}^{[000],1}}+\frac{4(\ell-1)^{2}(-\Delta+\ell+1)(\Delta+\ell)}{(2\ell-1)(2\ell+1)(\ell-\Delta)(\Delta+\ell+1)}\mathfrak{G}_{\text{Long}_{\Delta+1,\ell-1}^{[000]}}\\\
&+\frac{4\Delta(-\Delta+\ell+1)(\Delta+\ell)}{3(\Delta+2)(\ell-\Delta)(\Delta+\ell+1)}\mathfrak{G}_{\text{Long}_{\Delta+1,\ell}^{[000],2}}+\frac{4(-\Delta+\ell+1)(\Delta+\ell)}{(\ell-\Delta)(\Delta+\ell+1)}\mathfrak{G}_{\text{Long}_{\Delta+1,\ell}^{[000],3}}\\\
&+\frac{4(\ell+1)(\ell+2)(\Delta+\ell)(\Delta+\ell+2)}{(2\ell+1)(2\ell+3)(\Delta+\ell+1)(\Delta+\ell+3)}\mathfrak{G}_{\text{Long}_{\Delta+1,\ell+1}^{[000]}}\\\
&+\frac{4(\Delta+4)^{2}(-\Delta+\ell+1)(\Delta+\ell)}{(2\Delta+5)(2\Delta+7)(\ell-\Delta)(\Delta+\ell+1)}\mathfrak{G}_{\text{Long}_{\Delta+2,\ell}^{[000],1}}\,,\\\
\mathfrak{G}_{(A,\text{cons}.)_{\ell+1,\ell}^{[0000]}}\qquad&\to\qquad\mathfrak{G}_{(A,\text{cons}.)_{\ell+1,\ell}^{[000]}}+\frac{4(\ell+1)(\ell+2)}{(2\ell+2)(2\ell+4)}\mathfrak{G}_{(A,\text{cons}.)_{\ell+2,\ell+1}^{[000]}}\,,\end{split}$
(D.5)

where for $\mathfrak{G}_{\text{Long}_{\Delta,0}^{[0000]}}$ we should ignore
the $\mathfrak{G}_{\text{Long}_{\Delta+1,-1}^{[000]}}$ and
$\mathfrak{G}_{\text{Long}_{\Delta+1,0}^{[000],3}}$ terms on the RHS.

##  Appendix E Simplifying the $S^{3}$ partition function

In this appendix we describe how to simplify the $S^{3}$ partition function
for various ${\mathcal{N}}=6$ theories. In Section E.1 we consider the
$U(N)_{k}\times U(N+M)_{-k}$, and in Section E.2 we consider the
$SO(2)_{2k}\times USp(2+2M)_{-2k}$ theory. We close by demonstrating that
additional $U(1)$ factors do not change the $S^{3}$ partition function

###  E.1 ABJ theory

We will begin with the generalization of (3.11) to the case of two mass
deformations $m_{\pm}$:

$\begin{split}Z_{M,N,k}&(m_{+},m_{-})\\\ &=\int d^{M+N}\mu\,d^{N}\nu\
\frac{e^{-i\pi
k(\sum_{i}\mu_{i}^{2}-\sum_{a}\nu_{a}^{2})}\prod_{i<j}4\sinh^{2}\left[\pi(\mu_{i}-\mu_{j})\right]\prod_{a<b}4\sinh^{2}\left[\pi(\nu_{a}-\nu_{b})\right]}{\prod_{i,a}4\cosh\left[\pi(\mu_{i}-\nu_{a})+\frac{\pi
m_{+}}{2}\right]\cosh[\pi(\mu_{i}-\nu_{a})+\frac{\pi
m_{-}}{2}]}\,,\end{split}$ (E.1)

which we aim to simplify from $2N+M$ integrals to an expression with merely
$N$ integrals. To achieve this we will follow the methods of [76] which
considered the special case $m_{+}=m_{-}=0$. We are ultimately only interested
in computing $Z$ up to an overall normalization constant $Z_{0}$ which is
independent of $m_{\pm}$.

Our first step is to use the determinant formula:

$\begin{split}&\frac{\prod_{i<j}2\sinh\frac{x_{i}-x_{j}}{2}\prod_{a<b}2\sinh\frac{y_{a}-y_{b}}{2}}{\prod_{i,a}2\cosh\frac{x_{i}-y_{a}}{2}}=\prod_{i=1}^{N+M}e^{-\frac{1}{2}Mx_{i}}\prod_{a=1}^{N}e^{\frac{1}{2}My_{a}}\det\left(A(x,y)\right)\end{split}$
(E.2)

where $A(x,y)$ is the matrix

$A_{ij}(x,y)=\frac{\theta_{N,i}}{2\cosh\frac{x_{i}-y_{j}}{2}}+e^{(N+M+1/2-j)y_{i}}\theta_{j,N+1}\,,\quad\text{where}\quad\theta_{i,j}=\begin{cases}1&i\geq
j\\\ 0&\text{otherwise}\end{cases}\,,$ (E.3)

which is proven in [76] by using a generalization of the Cauchy determinant
formula. Applying this formula with $x_{i}=2\pi\mu_{i}$ and
$y_{i}=2\pi\nu_{i}+\pi m_{-}$, we can rewrite (E.1) as

$\begin{split}&Z_{M,N,k}(m_{+},m_{-})=(N+M)!\,e^{-\frac{\pi}{2}MN(m_{+}+m_{-})}\int
d^{M+N}\mu\,d^{N}\nu\ \prod_{j=1}^{N+M}e^{-\pi(ik\mu_{j}^{2}+2M\mu_{j})}\\\
\qquad&\times\prod_{a=1}^{N}\frac{e^{\pi(ik\nu_{a}^{2}+2M\nu_{a})}}{2\cosh\left[\pi(\mu_{a}-\nu_{a})+\frac{\pi
m_{+}}{2}\right]}\prod_{j=N+1}^{N+M}e^{(2(N+M-j)+1)\pi\mu_{j}}\\\
\qquad&\times\sum_{\text{perms
}\sigma}(-1)^{\text{sgn}(\sigma)}\prod_{i=1}^{N+M}\left(\frac{\theta_{N,\sigma(i)}}{2\cosh\left[\pi(\mu_{i}-\nu_{\sigma(i)})+\frac{\pi
m_{-}}{2}\right]}+e^{(2(N+M-{\sigma(i)})+1)\pi\mu_{i}}\theta_{{\sigma(i)},N+1}\right)\,,\\\
\end{split}$ (E.4)

where the sum over $\sigma$ is a sum over all permutations $\sigma(i)$ of
$N+M$ elements.

We next take the Fourier transform of the coshines

$\frac{1}{2\cosh(\pi
p)}=\frac{1}{\pi}\int_{-\infty}^{\infty}dx\,\frac{e^{2ipx}}{2\cosh(x)}\,.$
(E.5)

The $\mu$ and $\nu$ integrals then become Gaussian and can be easily
performed. We thus find that

$\begin{split}Z_{M,N,k}&(m_{+},m_{-})\propto
e^{-\frac{\pi}{2}MN(m_{+}+m_{-})}\\\
&\times\sum_{\sigma}(-1)^{\sigma}\Bigg{(}\int
d^{N}x\,d^{N+M}y\prod_{a=1}^{N}\frac{e^{-\frac{2i}{k\pi}x_{a}(y_{a}-y_{\sigma(a)})+\frac{2}{k}M(y_{a}-y_{\sigma(a)})+i(x_{a}m_{+}+y_{a}m_{-})}}{4\cosh(x_{a})\cosh(y_{a})}\\\
&\times\prod_{l=N+1}^{N+M}\left[\pi
e^{-\frac{i\pi}{k}(N+\frac{1}{2}-l)^{2}}\delta\left(y_{l}+i\pi(N+M+1/2-l)\right)e^{\frac{i}{k\pi}y_{l}^{2}+\frac{2}{k}(N+1/2-l)y_{\sigma(l)}}\right]\Bigg{)}\,.\end{split}$
(E.6)

So long as $2M<|k|+1$, we can integrate over $x_{i}$, leaving

$\begin{split}Z_{M,N,k}(m_{+},m_{-})&\propto
e^{-\frac{\pi}{2}MN(m_{+}+m_{-})}\sum_{\sigma}(-1)^{\sigma}\int\,d^{N+M}y\prod_{a=1}^{N}\frac{e^{\frac{2}{k}M(y_{a}-y_{\sigma(a)})+iy_{a}m_{-}}}{4\cosh\left[\frac{y_{a}-y_{\sigma(a)}}{k}-\frac{\pi
m_{+}}{2}\right]\cosh(y_{a})}\\\ &\times\prod_{l=N+1}^{N+M}\left[\pi
e^{-\frac{i\pi}{k}(N+\frac{1}{2}-l)^{2}}\delta\left(y_{l}+i\pi(N+M+1/2-l)\right)e^{\frac{i}{k\pi}y_{l}^{2}+\frac{2}{k}(N+1/2-l)y_{\sigma(l)}}\right]\,.\end{split}$
(E.7)

After a change of variable $y_{a}\rightarrow y_{a}/2$ and judicious use of the
equation $\sum_{a}y_{a}=\sum_{a}y_{\sigma(a)}$ we find that

$\begin{split}Z_{M,N,k}(m_{+},m_{-})&\propto
e^{-\frac{\pi}{2}MN(m_{+}+m_{-})}\sum_{\sigma}(-1)^{\sigma}\int\,d^{N+M}y\prod_{a=1}^{N}\frac{e^{\frac{i}{2}y_{a}m_{-}}}{2\cosh\left[\frac{y_{a}}{2}\right]}\\\
&\times\prod_{l=N+1}^{N+M}\left[\pi
e^{-\frac{i\pi}{k}(N+\frac{1}{2}-l)^{2}}\delta\left(y_{l}+i\pi(N+M+1/2-l)\right)e^{\frac{i}{4k\pi}y_{l}^{2}-\frac{M}{k}y_{l}}\right]\\\
&\times\det\left(\frac{\theta_{N,l}}{2\cosh\frac{y_{j}-y_{l}+k\pi
m_{+}}{2k}}+e^{\frac{1}{k}(N+M+1/2-l)y_{j}}\theta_{l,N+1}\right)\,.\end{split}$
(E.8)

Applying (LABEL:cauchDet) again, integrating over $y_{N+1}\,,\dots\,,y_{N+M}$,
and then performing a final change of variables $y_{a}\rightarrow 2\pi y_{a}$,
we arrive at our final expression

$\begin{split}Z_{M,N,k}(m_{+},m_{-})\qquad\qquad&\\\
=\frac{e^{-\frac{\pi}{2}MNm_{-}}Z_{0}}{\cosh^{N}\frac{\pi m_{+}}{2}}&\int
d^{N}y\prod_{a<b}\frac{\sinh^{2}\frac{\pi(y_{a}-y_{b})}{k}}{\cosh\left[\frac{\pi(y_{a}-y_{b})}{k}+\frac{\pi
m_{+}}{2}\right]\cosh\left[\frac{\pi(y_{a}-y_{b})}{k}-\frac{\pi
m_{+}}{2}\right]}\\\ &\times\prod_{a=1}^{N}\left(\frac{e^{i\pi
y_{a}m_{-}}}{2\cosh\left(\pi
y_{a}\right)}\prod_{l=0}^{M-1}\frac{\sinh\left[\frac{\pi\big{(}y_{a}+i(l+1/2)\big{)}}{k}\right]}{\cosh\left[\frac{\pi\big{(}y_{a}+i(l+1/2)\big{)}}{k}-\frac{\pi
m_{+}}{2}\right]}\right)\,,\end{split}$ (E.9)

Equation (3.12) in the main text is simply the special case of this where we
set $m_{-}=m$ and $m_{+}=0$.

###  E.2 $SO(2)\times USp(2+2M)$ theory

In this section we shall reduce the $SO(2)_{2k}\times USp(2+2M)_{-k}$ sphere
partition function (3.29) down to a single integral. As in the previous
section, we will work with a slightly more general partition function

$\begin{split}Z_{M,k}&(m_{+},m_{-})\propto\int d\mu\,d^{M}\nu\,e^{2\pi
ik(\mu^{2}-\sum_{a}\nu_{a}^{2})}\\\
&\times\frac{\prod_{a}\sinh^{2}\left(2\pi\nu_{a}\right)\prod_{a<b}\sinh^{2}\left[\pi(\nu_{a}+\nu_{b})\right]\sinh^{2}\left[\pi(\nu_{a}-\nu_{b})\right]}{\prod_{b}\cosh\frac{2\pi(\mu-\nu_{b})+\pi
m_{+}}{2}\cosh\frac{2\pi(\mu+\nu_{b})+\pi
m_{+}}{2}\cosh\frac{2\pi(\mu-\nu_{b})+\pi
m_{-}}{2}\cosh\frac{2\pi(\mu+\nu_{b})+\pi m_{-}}{2}}\,,\end{split}$ (E.10)

where we have two mass deformations $m_{\pm}$. We follow the derivation in
[79], which considered the special case $m_{+}=m_{-}=0$. They however consider
the general $SO(2N)\times USp(2N+2M)$ theory, while here we only focus on the
$N=1$ case, for which the manifest ${\mathcal{N}}=5$ SUSY is enhanced to
${\mathcal{N}}=6$.

We begin by rewriting (E.10):

$\begin{split}Z_{M,k}&(m_{+},m_{-})\propto\int
d\mu\,d^{M+1}\nu\,e^{\frac{i}{4\tilde{k}\pi}(\mu^{2}-\sum_{a}\nu_{a}^{2})}\\\
&\times\frac{\prod_{a}\sinh^{2}\frac{\nu_{a}}{\tilde{k}}\prod_{a<b}\sinh^{2}\frac{\nu_{a}+\nu_{b}}{2\tilde{k}}\sinh^{2}\frac{\nu_{a}-\nu_{b}}{2\tilde{k}}}{\prod_{b}\cosh\left(\frac{\mu-\nu_{b}}{2\tilde{k}}+\frac{m_{+}}{2}\right)\cosh\left(\frac{\mu+\nu_{b}}{2\tilde{k}}+\frac{m_{+}}{2}\right)\cosh\left(\frac{\mu-\nu_{b}}{2\tilde{k}}+\frac{m_{-}}{2}\right)\cosh\left(\frac{\mu+\nu_{b}}{2\tilde{k}}+\frac{m_{-}}{2}\right)}\\\
\end{split}$ (E.11)

with $\tilde{k}=2k$, in order to match the conventions of [79]. Our first
major step is to use the Cauchy-Vandermonde determinant, given in (2.6) of
[79]:

$\begin{split}&\frac{\prod_{a}\sinh\nu_{a}\prod_{a<b}\sinh\frac{\nu_{a}+\nu_{b}}{2{\tilde{k}}}\sinh\frac{\nu_{a}-\nu_{b}}{2{\tilde{k}}}}{\prod_{b}\cosh\frac{\mu-\nu_{b}+{\tilde{k}}m_{+}}{2{\tilde{k}}}\cosh\frac{\mu+\nu_{b}+{\tilde{k}}m_{+}}{2{\tilde{k}}}}\\\
&=\det\begin{pmatrix}\left[\frac{\sinh\frac{\nu_{a}}{2{\tilde{k}}}}{\cosh\frac{\mu+{\tilde{k}}m_{+}-\nu_{a}}{2{\tilde{k}}}\cosh\frac{\mu+{\tilde{k}}m_{+}+\nu_{a}}{2{\tilde{k}}}}\right]\\\
\left[\sinh\frac{(2b-1)\nu_{a}}{2{\tilde{k}}}\right]_{b=1,\dots,M}\end{pmatrix}\,.\end{split}$
(E.12)

We can simplify this expression by introducing canonical position and momentum
operators $\hat{q}$ and $\hat{p}$ which satisfy $[\hat{q},\hat{p}]=2\pi
i\tilde{k}$. We denote the $\hat{q}$ eigenstates by $\left|\nu\right\rangle$,
and introduce states $|m]]$ such that

$2\sinh\frac{m\nu_{\tilde{k}}}{\tilde{k}}=[[m\left|\nu_{\tilde{k}}\right\rangle\,,$
(E.13)

allowing us to simplify the lower block of (E.12). The upper block can be
simplified using the Fourier transform, giving us (2.9) of [79]:

$\frac{\sinh\frac{\nu_{a}}{2{\tilde{k}}}}{\cosh\frac{\mu+{\tilde{k}}m_{+}-\nu_{a}}{2{\tilde{k}}}\cosh\frac{\mu+{\tilde{k}}m_{+}+\nu_{a}}{2{\tilde{k}}}}\propto\langle\nu_{a}|\frac{1}{2\sinh\frac{\hat{p}}{2}}\hat{\Pi}_{+}|\mu+{\tilde{k}}m_{+}\rangle\,.$
(E.14)

We then find that our version of equation (2.11) is [79]

$\begin{split}Z(m_{+},m_{-})&\propto\int
d\mu\,d^{M+1}\nu\frac{\prod_{a}\sinh\frac{\nu_{a}}{\tilde{k}}\prod_{a<b}\sinh\frac{\nu_{a}+\nu_{b}}{2{\tilde{k}}}\sinh\frac{\nu_{a}-\nu_{b}}{2{\tilde{k}}}}{\prod_{b}\cosh\frac{\mu-\nu_{b}+{\tilde{k}}m_{-}}{2{\tilde{k}}}\cosh\frac{\mu+\nu_{b}+{\tilde{k}}m_{-}}{2{\tilde{k}}}}\\\
&\times\langle\mu+{\tilde{k}}m_{+}|e^{\frac{i}{2\hbar}\hat{p}^{2}}e^{\frac{i}{2\hbar}\hat{q}^{2}}\frac{1}{2\sinh\frac{\hat{p}}{2}}\hat{\Pi}_{-}e^{-\frac{i}{2\hbar}\hat{q}^{2}}e^{-\frac{i}{2\hbar}\hat{p}^{2}}|\nu_{1}\rangle\prod_{m=1}^{M}[[m|e^{-\frac{i}{2\hbar}\hat{q}^{2}}e^{-\frac{i}{2\hbar}}|\nu_{m+1}\rangle\,.\end{split}$
(E.15)

Applying (2.12-5) of [79], we find that

$\begin{split}Z(m_{+},m_{-})&\propto\int d\mu\,d^{M+1}\nu\
\frac{\prod_{a}\sinh\frac{\nu_{a}}{\tilde{k}}\prod_{a<b}\sinh\frac{\nu_{a}+\nu_{b}}{2{\tilde{k}}}\sinh\frac{\nu_{a}-\nu_{b}}{2{\tilde{k}}}}{\prod_{b}\cosh\frac{\mu-\nu_{b}+{\tilde{k}}m_{-}}{2{\tilde{k}}}\cosh\frac{\mu+\nu_{b}+{\tilde{k}}m_{-}}{2{\tilde{k}}}}\\\
&\times\frac{\delta(\mu+\tilde{k}m_{+}-\nu_{1})}{\sinh\frac{\mu+\tilde{k}m_{+}}{2}}\prod_{m=1}^{M}\delta(\nu_{m+1}-2\pi
im)\\\ &\propto\frac{1}{\cosh\frac{m_{+}-m_{-}}{2}}\int
d\mu\frac{\sinh\left[\frac{\mu}{2k}+m_{+}\right]}{\sinh\left[\frac{\mu+2km_{+}}{2}\right]\cosh\left[\frac{\mu}{2k}+\frac{m_{+}+m_{-}}{2}\right]}\\\
&\times\frac{\prod_{l=1}^{M}\sinh\left[\frac{\mu+2\pi
il}{4k}+\frac{m_{+}}{2}\right]\sinh\left[\frac{\mu-2\pi
il}{4k}+\frac{m_{+}}{2}\right]}{\prod_{l=1}^{M}\cosh\left[\frac{\mu+2\pi
il}{4k}+\frac{m_{-}}{2}\right]\cosh\left[\frac{\mu-2\pi
il}{4k}+\frac{m_{-}}{2}\right]}\\\
&\propto\frac{1}{\cosh\frac{m_{+}-m_{-}}{2}}\int\frac{d\mu\
\sinh\left[\frac{\mu}{2k}+m_{+}\right]\cosh\left[\frac{\mu}{4k}+\frac{m_{-}}{2}\right]}{\sinh\left[\frac{\mu+2km_{+}}{2}\right]\cosh\left[\frac{\mu}{2k}+\frac{m_{+}+m_{-}}{2}\right]\sinh\left[\frac{\mu}{4k}+\frac{m_{+}}{2}\right]}\\\
&\times\prod_{l=-M}^{M}\frac{\sinh\left[\frac{\mu+2\pi
il}{4k}+\frac{m_{+}}{2}\right]}{\cosh\left[\frac{\mu+2\pi
il}{4k}+\frac{m_{-}}{2}\right]}\,,\end{split}$ (E.16)

and this completes the derivation.

###  E.3 Additional $U(1)$ factors

In this section we shall show that given an ${\mathcal{N}}=6$ Chern-Simon
gauge theory with gauge group $G=SU(N)\times U(N+M)$ or $G=USp(2+2M)$, the
$S^{3}$ partition function for the theory $G\times U(1)^{L}$ is equivalent to
that of the theory $G\times U(1)$, up to an overall constant. To this end we
note that the mass-deformed $S^{3}$ partition function for the $G\times
U(1)^{L}$ can be generically written as

$Z_{G\times U(1)^{L}}(m_{+},m_{-})=\int d\chi_{1}\dots
d\chi_{N}\,e^{i\pi\sum_{ab}K_{ab}\chi_{a}\chi_{b}}Z_{G}(m_{+}+2q\cdot\chi,m_{-}+2q\cdot\chi)\,,$
(E.17)

where $K_{ab}$ is the matrix of Chern-Simons levels for the $U(1)$s
$q=(q_{1},\dots,q_{L})$ are the charges of the (bi)fundamentals under each
$U(1)$, and $Z_{G}(m_{+},m_{-})$ is the $S^{3}$ partition function for the
theory without any $U(1)$ factors. In order for $G\times U(1)^{N}$ to have
${\mathcal{N}}=6$ supersymmetry, $K_{ab}$ and $q_{a}$ must satisfy the
condition

$\sum_{a,b}K^{ab}q_{a}q_{b}=\frac{1}{k}_{G}$ (E.18)

for some $G$ dependent constant $k_{G}$, where $K^{ab}$ is the inverse of
$K_{ab}$ [38].

To simplify (E.17), we first perform a change of basis of $\chi_{a}$ such that
$q_{a}=(1,0,\dots,0)$. Because $K_{ab}$ is symmetric, we can then perform a
second change of basis to $\chi_{2}\,,\dots\,,\chi_{L}$, so that $K_{ab}$ take
the form

$K_{ab}=\begin{pmatrix}K_{11}&K_{12}&0&\ldots\\\ K_{12}&1&0&\ldots\\\
0&0&1&\ldots\\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}\,.$ (E.19)

We can now integrate over $\chi_{2}\,,\chi_{3}\,,\dots$ leaving us with

$Z_{G\times U(1)^{L}}(m_{+},m_{-})\propto\int
d\chi_{1}\,e^{i\pi(K_{11}-K_{12}^{2})\chi^{2}}Z_{G}(m_{+}+2\chi_{1},m_{-}+2\chi_{1})\,,$
(E.20)

We then note that, in this basis, the condition (E.18) becomes:

$K_{11}-K_{12}^{2}=k_{G}\,,$ (E.21)

and so

$Z_{G\times U(1)^{L}}(m_{+},m_{-})\propto\int d\chi_{1}\,e^{i\pi
k_{G}\chi^{2}}Z_{G}(m_{+}+2\chi_{1},m_{-}+2\chi_{1})\,.$ (E.22)

We now simply recognize the right-hand side of this equation is the partition
function for the $G\times U(1)$ theory, and have hence shown what we set out
to prove.

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# Constraining logarithmic $f(R,T)$ model using Dark Energy density parameter
$\Omega_{\Lambda}$ and Hubble parameter $H_{0}$

Biswajit Deb 111Electronic<EMAIL_ADDRESS>, Atri Deshamukhya
222Electronic Address<EMAIL_ADDRESS>
Department of Physics, Assam University, Silchar, India

###### Abstract

Of many extended theories of gravity, $f(R,T)$ gravity has gained reasonable
interest in recent times as it provides interesting results in cosmology.
Logarithmic corrections in modified theories of gravity has been studied
extensively. In this work, we considered logarithmic correction to the trace
term T and take the functional form as $f(R,T)=R+16\pi G\alpha\ln T$ where
$\alpha$ is a free parameter. The free parameter is constrained using dark
energy density parameter $\Omega_{\Lambda}$ and Hubble parameter $H_{0}$. The
lower bound is found to be $\alpha\geq-9.93\times 10^{-29}$. The cosmological
implications are also studied.

##  1 Introduction

The remarkable discovery in 1998 by Riess et. al and Perlmutter et. al paused
a fundamental question on the classical General Relativity (GR) [1, 2]. The
standard GR predicts that the expansion of the Universe should be decelerating
with time. Instead, it is observed that current Universe has entered a second
phase of accelerated expansion which started around $z=1$ [1, 2]. Besides
this, it is observed that nearly 70% of the total energy density of the
Universe is in some mysterious form called Dark Energy [1, 2]. Whether this
Dark Energy is driving the present expansion of the Universe is a matter of
investigation.

Einstein field equations, in their original form can’t explain this late time
acceleration because it gives an equation of state (EoS) parameter $w=0$ [3].
Whereas, the present late time acceleration demands EoS parameter to be
$w<-\frac{1}{3}$. Now, this can be explained if one considers the Universe
filled with some exotic form of fluid with EoS parameter $w<-\frac{1}{3}$ [3].
That means the dominant Dark Energy constituent is driving the expansion.
Apart from this, cosmological constant with EoS parameter $w=-1$ is a viable
alternative explanation for Dark Energy as well as late expansion of the
Universe [3]. In fact the $\Lambda$CDM model gives the best fit results to the
present observed Universe [4]. But the Cosmological constant $\Lambda$ is into
fine tuning problem [5].

$f(R)$ theory, a class of modified gravity, emerged in 1980’s and gained
popularity as it can explain the early inflationary Universe without
considering the scalar fields in the theory [6]. Since then different
modifications of gravity viz. $f(G)$, $f(R,G)$, $f(G,T)$, $f(R,T)$, $f(Q)$,
$f(Q,T)$ etc. have been proposed and studied in literature where R, Q, G and T
are the Ricci scalar, non-metricity, Gauss-Bonnet scalar and Trace of the
Energy momentum tensor respectively [7, 8, 9, 10, 11, 12]. After the discovery
of late time acceleration, studies of modified theories of gravity gained even
more interest as it has the potential to explain late time acceleration of the
Universe without invoking Dark Energy in the theory [13].

$f(R,T)$ theory of gravity, proposed by Harko et. al is found to be extremely
sound in explaining cosmological phenomena. It has been studied with reference
to inflation [14, 15, 16, 17], dark energy [18, 19, 20, 21, 22], dark matter
[23], wormhole [24, 25, 26, 27, 28, 29, 30], pulsar [31, 32], white dwarfs
[33], gravitational waves [34, 35], scalar field models [36, 37], anisotropic
models [38, 39], bouncing cosmology [40, 41], big-bang neucleosysnthesis [42],
baryogenesis [43], brane world [44, 45] etc. Further, the energy conditions
and junction conditions in $f(R,T)$ gravity have also been studied [46, 47,
48, 49]. Snehasish et. al developed a novel way to impose lower bound on the
model parameter $\lambda$ of the simplest $f(R,T)=R+2\lambda T$ model through
the equation relating the cosmological constant and critical density of the
universe [18]. This method can be applied to other complex forms of $f(R,T)$
to constrain the model parameter(s). This might reveal interesting results.

Logarithmic correction in modified gravity theories has been studied
extensively. The first Logarithmic correction to trace term T in $f(R,T)$
theory has been proposed by Elizalde et. al where they have taken the function
as $f(R,T)=R+\alpha R^{2}+2\beta\ln(T)$ and studied the stability conditions &
energy conditions of the model [50]. In this work, we have considered the
simplest $f(R,T)$ model with logarithmic correction to trace term T as
$f(R,T)=R+16\pi G\alpha\ln T$, where $\alpha$ is the model parameter and will
use the methodology developed by Snehasish et. al to impose a bound on the
model parameter.

The paper has been organised as follows: In section 2, we present mathematical
framework to impose bound on the model parameter. In section 3, we present our
conclusion. Here, we will use (+,-,-,-) sign convention for the metric tensor.

##  2 Constraining model parameter

Considering $f(R,T)=R+16\pi G\alpha\ln T$, the Einstein-Hilbert action
becomes,

$S=\int\sqrt{-g}\left[\frac{R}{16\pi G}+\alpha\ln T+L_{m}\right]d^{4}x$ (1)

where $L_{m}$ is the matter Lagrangian. This leads to field equations,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}^{(eff)}$ (2)

where $T_{\mu\nu}^{(eff)}$ is the effective stress-energy tensor given by,

$T_{\mu\nu}^{(eff)}=T_{\mu\nu}-\frac{2\alpha}{T}\left(T_{\mu\nu}-\frac{T}{2}g_{\mu\nu}\ln
T+\Theta_{\mu\nu}\right)$ (3)

where the term $\Theta_{\mu\nu}$, which plays a crucial role in f(R,T) gravity
as it contains matter Lagrangian $L_{m}$, is given by

$\Theta_{\mu\nu}=g^{\beta\gamma}\frac{\delta T_{\beta\gamma}}{\delta
g^{\mu\nu}}=-2T_{\mu\nu}+g_{\mu\nu}L_{m}-2\frac{\delta^{2}L_{m}}{\delta
g^{\mu\nu}\delta g^{\beta\gamma}}$ (4)

Clearly, depending on the nature of the matter field, the field equation for
$f(R,T)$ gravity will be different. Now, assuming the Universe is dominated by
perfect fluid, the energy-momentum tensor is

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (5)

and hence the matter Lagrangian density can be assumed as $L_{m}=-p$. Now, let
us assume the Friedmann-Lemaitre-Robertson-Walkar (FLRW) metric in spherical
coordinate for flat Universe,

$ds^{2}=dt^{2}-a(t)^{2}\left[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]$ (6)

where a(t) denotes scale factor of the Universe. Now, considering the FLRW
metric, the 00 component of field Eq. 2 yields first modified Friedmann
equation as,

$3H^{2}=8\pi
G\left[\rho+\frac{2\alpha}{\rho-3p}\left(\rho+p+\frac{\rho-3p}{2}\ln(\rho-3p)\right)\right]$
(7)

where H is Hubble parameter, $\rho$ is energy density and p is pressure of the
Universe. Further, $T=\rho-3p$ is the trace of the EM tensor. Now using
equation of state parameter $w=p/\rho$, the above Friedmann equation can be
rearranged as,

$3H^{2}=8\pi
G\left[\rho+\frac{2\alpha}{(1-3w)\rho}\left((1+w)\rho+\frac{(1-3w)\rho}{2}\ln(1-3w)\rho\right)\right]$
(8)

Current observation [4] suggest that $w=-1$ and hence substituting this value
in Friedmann Eq. yields,

$3H^{2}=8\pi G(\rho+\alpha\ln 4\rho)$ (9)

Now, Friedmann equation in GR with Cosmological Constant $\Lambda$ is given by
[51],

$3H^{2}=8\pi G\rho+\Lambda c^{2}$ (10)

Equating Eq.9 and Eq.10 yields,

$\Lambda c^{2}=8\pi G\alpha\ln 4\rho$ (11)

Now, Cosmological Constant $\Lambda$ can be defined in terms of present value
of Hubble parameter $H_{0}$ and dark energy density parameter
$\Omega_{\Lambda}$ as [52]

$\Lambda=3\left(\frac{H_{0}}{c}\right)^{2}\Omega_{\Lambda}$ (12)

Substituting Eq.12 in Eq.11 we get,

$\Omega_{\Lambda}=\frac{8\pi G\alpha}{3H_{0}^{2}}\ln 4\rho$ (13)

The present Universe is spatially flat. As a result the total density
parameter is $\Omega_{0}=\rho/\rho_{cr}=1$ [4]. Hence we can replace $\rho$ by
$\rho_{cr}$ in the previous equation.

$\Omega_{\Lambda}=\frac{8\pi G\alpha}{3H_{0}^{2}}\ln 4\rho_{cr}$ (14)

Now the critical density of the Universe is defined as [53]

$\rho_{cr}=\frac{3H_{0}^{2}}{8\pi G}$ (15)

From Eq.14 and Eq.15 we get the final expression of the model parameter
$\alpha$ in terms of Hubble parameter and dark energy density parameter as

$\Omega_{\Lambda}=\frac{8\pi G\alpha}{3H_{0}^{2}}\ln\frac{3H_{0}^{2}}{2\pi G}$
(16)

From Planck 2018 data [4], we have $H_{0}=67.4\pm 0.5Kms^{-1}MPc^{-1}$ and
$\Omega_{\Lambda}=0.6889\pm 0.0056$. Substituting these values in Eq.16 we
found the lower bound on the model parameter is

$\alpha\geq-9.93\times 10^{-29}$ (17)

##  3 Conclusion

Free parameters in modified gravity theories are trivial and hold significant
role. It allows a particular gravity model to be consistent with observational
results. In this work we tried to constrain the simplest $f(R,T)$ model with
logarithmic correction, using Hubble parameter and dark energy density
parameter. We report the lower bound on model parameter as $-9.93\times
10^{-29}$.

This method of imposing lower bound on the model parameter by relating
equation to the cosmological constant and critical density of the Universe
developed by Snehasish et. al is model dependent. For more complex forms of
$f(R,T)$, more input parameters are required to constraint the model. Further,
this method can be applied to constrain other modified gravity models but it
may require more constraining parameters beside dark energy density parameter.

Snehasish et. al obtained the lower bound for $f(R,T)=R+2\lambda T$ model
which is of the order of $10^{-8}$. In case $f(R,T)=R+16\pi G\alpha\ln T$, the
lower bound obtained is of the order of $10^{-29}$. In both the cases, the
lower bound on the model parameter is quite small. As a result, these bounds
need to be validated from other sources like spectral indices, tensor-to-
scalar ratio etc which is beyond the scope of this work.

As a possible extension of this work, one can apply this methodology to other
$f(R,T)$ models, specially non-minimally coupled ones. Further, this method
can also be applied to other modified gravity theories like $f(R,G),f(G,T)$
etc. which may generate interesting results.

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