$\displaystyle\frac{1}{48\nu}\int_{z_{-}+\epsilon}^{z_{+}-\epsilon}dz\,\frac{V^{\prime\prime}+V^{\prime\prime\prime}z+\frac{1}{2}V^{\prime\prime}z^{2}}{(\frac{1}{2}V^{\prime\prime}(z^{2}-z_{\pm}^{2})+\frac{1}{6}V^{\prime\prime\prime}(z^{3}-z_{\pm}^{3})+\frac{1}{24}V^{\prime\prime\prime\prime}(z^{4}-z_{\pm}^{4}))^{3/2}}$
$\displaystyle\approx\frac{1}{12\sqrt{2}\nu V^{\prime\prime
3/2}}\int_{z_{-}+\epsilon}^{z_{+}-\epsilon}\frac{dz}{(z_{\pm}^{2}-z^{2})^{3/2}}\left(V^{\prime\prime}+\frac{V^{\prime\prime\prime}(z^{2}+z_{\pm}z-z_{\pm}^{2})}{2(z_{\pm}+z)}\right.$
$\displaystyle\hskip 85.35826pt\left.+\frac{V^{\prime\prime\prime
2}(z_{\pm}^{2}+z_{\pm}z+z^{2})(5z_{\pm}^{2}-7z_{\pm}z-7z^{2})}{24V^{\prime\prime}(z_{\pm}+z)^{2}}+\frac{(3z^{2}-z_{\pm}^{2})V^{\prime\prime\prime\prime}}{8V^{\prime\prime}}\right).$
(185)

We split the integral into a piece from $z_{-}+\epsilon$ to zero and a piece
from zero to $z_{+}-\epsilon$. In the first piece we replace $z_{\pm}$ by
$z_{-}$ in the above expression, and in the second piece we replace $z_{\pm}$
by $z_{+}$. The integral evaluates to

$\displaystyle\text{divergent}+\frac{1}{2\sqrt{2}\nu V^{\prime\prime
3/2}}\left(\frac{(21\pi+16)V^{\prime\prime\prime
2}}{432V^{\prime\prime}}-\left(\frac{1}{9z_{-}}+\frac{1}{9z_{+}}\right)V^{\prime\prime\prime}-\frac{\pi
V^{\prime\prime\prime\prime}}{16}\right).$ (186)

The divergent terms are canceled by the circular contours, and the remaining
piece is finite. The final step is to solve for $z_{\pm}$ perturbatively in
$V^{\prime\prime\prime}$. We get

$\displaystyle
z_{\pm}=\pm\sqrt{\frac{2(u^{2}-V)}{V^{\prime\prime}}}+\frac{(V-u^{2})V^{\prime\prime\prime}}{3V^{\prime\prime
2}}+\ldots$ (187)

Terms proportional to $V^{\prime\prime\prime\prime}$ and
$V^{\prime\prime\prime 2}$ in $z_{\pm}$ do not contribute at this order and we
can neglect them. Expanding (186) we get

$\displaystyle\frac{\pi(7V^{\prime\prime\prime
2}-9V^{\prime\prime}V^{\prime\prime\prime\prime})}{288\sqrt{2}V^{\prime\prime
5/2}}.$ (188)

Collecting everything together, (180) becomes

$\displaystyle\frac{1}{\sqrt{2V^{\prime\prime}}}\left(\nu(u^{2}-V)+\frac{\nu(u^{2}-V)^{2}(5V^{\prime\prime\prime
2}-3V^{\prime\prime}V^{\prime\prime\prime\prime\prime})}{48V^{\prime\prime
3}}+\frac{7V^{\prime\prime\prime
2}-9V^{\prime\prime}V^{\prime\prime\prime\prime}}{288\nu V^{\prime\prime
2}}\right)=n+1/2.$ (189)

##  Appendix C Double-twist anomalous dimensions in general $d$

In this appendix we collect the formulas in general dimensions for the
anomalous dimensions to first order in $1/\nu$ and second order in $\mu$. We
find

$\displaystyle\gamma_{1}$
$\displaystyle=-\frac{1}{768k^{d/2}}\left[384k\nu+48(2n+1)(d(d+2)k+(d-2)(d-4))\right.$
(190) $\displaystyle\hskip
14.22636pt\left.+\frac{d(d-2)}{k\nu}\left(6n(n+1)((d+2)(d+4)k^{2}+2(d+2)(d-4)k+(d-4)(d-6))\right.\right.$
$\displaystyle\hskip
71.13188pt\left.\left.+(d+2)(3d+4)k^{2}+6(d+2)(d-4)k+(d-4)(3d-10)\right)\right]+O(1/\nu^{2})\
.$

This formula matches the result of Li:2020dqm  in 4, 6, 8, and 10 dimensions,
see (4.16)-(4.18) in that paper. At the second order we find

$\displaystyle\gamma_{2}$
$\displaystyle=-\frac{1}{18432k^{d}}\left[576(d^{2}k+d^{2}-4d+4)k\nu\right.$
(191) $\displaystyle\left.\hskip
14.22636pt+36((5d^{4}+4d^{3}-12d^{2})k^{2}+(10d^{4}-40d^{3}+40d^{2})k+5d^{4}-44d^{3}+132d^{2}-160d+64)(1+2n)\right.$
$\displaystyle\left.+\frac{d(d-2)}{k\nu}\left(6n(n+1)((11d^{4}+40d^{3}-8d^{2}-88d)k^{3}+(33d^{4}-48d^{3}-108d^{2}+192d-96)k^{2}\right.\right.$
$\displaystyle\left.\left.\hskip
56.9055pt+(33d^{4}-216d^{3}+396d^{2}-240d)k+11d^{4}-128d^{3}+496d^{2}-712d+288)\right.\right.$
$\displaystyle\hskip
56.9055pt\left.\left.+(29d^{4}+64d^{3}-56d^{2}-136d)k^{3}+(87d^{4}-168d^{3}-180d^{2}+384d-96)k^{2}\right.\right.$
$\displaystyle\hskip
56.9055pt\left.\left.+(87d^{4}-528d^{3}+900d^{2}-432d)k+29d^{4}-296d^{3}+1024d^{2}-1336d+480\right)\right]+O(1/\nu^{2}).$

This result agrees with the one in Kulaxizi:2018dxo  in $d=2,4$ and $6$, see
formula (6.39) in that paper.

##  Appendix D Bulk higher spin symmetry and the twist gap

The twist $\tau=\Delta-J$ spectrum of unitary CFTs in $d>2$ has a finite gap
due to the unitarity bound $\tau\geq d-2$. In AdS this corresponds to the fact
that interactions between a pair of objects go to zero at large separation
(this is known as AdS clustering). One way to generate a separation between a
pair of objects is by spinning them around each other. In this way large spin
leads to large separation between a pair of objects, and thanks to the twist
gap, the interaction between them goes to zero Alday:2007mf . The same
phenomenon can be understood by considering the light-cone limit of the
crossing equation Fitzpatrick:2012yx ; Komargodski:2012ek , see also
Alday:2015eya ; Alday:2015ewa ; Alday:2016njk ; Simmons-Duffin:2016wlq ;
Caron-Huot:2017vep . The punchline of this analysis is that the spectrum of
interacting CFTs reproduces the one of the generalized free field (GFF), or,
equivalently, the free field in AdS, at large spin and fixed twist.

It might be instructive to think about this phenomenon in terms of emerging
symmetry. Let us consider the free scalar field in AdS. It admits an infinite
set of conserved higher spin currents in the bulk, which schematically take
the form, see e.g. Bekaert:2014cea ,

$\displaystyle\mathcal{J}_{J}(x,z)=\sum_{k=0}^{J}c_{k}(z\cdot\nabla)^{k}\phi(z\cdot\nabla)^{J-k}\phi+...,$
(192)

where $z^{\mu}$ is a polarization vector and $...$ stand for the trace terms.

The bulk field $\phi$ is dual to the boundary operator $\mathcal{O}$ of
dimension $\Delta$. On the boundary the manifestation of _bulk higher spin
symmetry_ is the statement that correlation functions of the GFF are invariant
under the transformation

$\displaystyle[Q_{J},\mathcal{O}]=\partial^{J-1}\mathcal{O}.$ (193)

Note that bulk higher spin symmetry is different from the CFT higher spin
symmetry considered in Maldacena:2011jn ; Maldacena:2012sf . In the latter
case the boundary theory contains a set of local conserved higher spin
symmetry currents, which is not the case here. Indeed, in the case of the GFF
the spectrum contains the operator $\mathcal{O}$ as well as a set of multi-
twist operators (schematically): $\mathcal{O}\Box^{n}\partial^{J}\mathcal{O}$,
$\mathcal{O}\Box^{n_{1}}\partial^{J_{1}}\mathcal{O}\Box^{n_{2}}\partial^{J}_{2}\mathcal{O}$,
… . The bulk higher spin currents are then some objects made out of double-
twist operators $\mathcal{O}\Box^{n}\partial^{J}\mathcal{O}$ with dimension

$\displaystyle\Delta_{n,J}=2\Delta+2n+J,$ (194)

and spin $J$. The presence of the equidistant operators (194) then can be
understood as a consequence of the bulk higher spin symmetry. The statement of
AdS clustering, or the light-cone bootstrap, can be stated as an emergence of
bulk higher spin symmetry at large spin.222222In fact flat space clustering
can be understood in the same way: in this case we can separately change the
relative distance between two groups of widely separated particles/operators
without changing the amplitude/correlator, see Coleman:1967ad  for a related
discussion.

The presence of the bulk higher spin symmetry makes the theory integrable and
leads to revivals and lack of thermalization. If such states persist in an
interacting theory and are present amidst thermalizing states, they will serve
as an example of many-body scars (a small fraction of non-thermalizing states
in the otherwise thermalizing continuum) in higher-dimensional CFTs. As should
be clear from our analysis the double-twist operators observed in the light-
cone bootstrap analysis of the heavy-heavy-light-light four-point function are
precisely such states, perturbatively in spin. For example, we showed in
Section 6.2 that the one-point function in such states does not obey ETH. In
this context bulk higher spin symmetry is called the spectrum-generating
algebra serbyn ; PhysRevB.102.085140 .

On the other hand, as we have seen, this conclusion would be too fast due to
the presence of important nonperturbative in spin effects (due to the presence
of the black hole horizon) which are not captured by the naive light-cone
bootstrap analysis. These effects remove the scar states from the spectrum by
turning them into long-lived resonances comprised of the superposition of
black hole microstates. A similar phenomenon was considered in the context of
many-body scars in lin2020slow .

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