# AdS3 Pure Gravity and Stringy Unitarity

Gabriele Di Ubaldo, Eric Perlmutter  Université Paris-Saclay, CNRS, CEA,
Institut de Physique Théorique, 91191, Gif-sur-Yvette, France

###### Abstract

We construct a unitary, modular-invariant torus partition function of a two-
dimensional conformal field theory with a Virasoro primary spectral gap of
$\Delta_{*}=\frac{c-1}{12}$ above the vacuum. The twist gap is identical,
apart from two states $\mathcal{O}_{*}$ with spin scaling linearly in the
central charge $c$. These states admit an AdS3 interpretation as strongly
coupled strings. All other states are black hole microstates.

The quest for AdS3 pure gravity still beckons.

It is not fully known whether, or in what precise sense, a consistent such
theory exists, either quantum mechanically or in the semiclassical limit. The
latter is of particular physical interest, due to the existence of black holes
and the emergence of spacetime.

Holographically speaking, the outstanding spectral problem is to find a torus
partition function of a two-dimensional conformal field theory (CFT) that is
mutually compatible with unitarity (a non-negative Virasoro primary spectral
density) and modularity (exact $SL(2,\mathbb{Z})$-invariance of the partition
function), while preserving the spectral gaps of a dual bulk theory with only
black holes above a normalizable AdS3 ground state. No known partition
function satisfies these basic requirements.

There exists a diverse set of approaches to this problem which, famous as it
is, we describe in condensed fashion. Summing over all smooth on-shell
3-manifolds $\mathcal{M}$ with $\partial\mathcal{M}=T^{2}$ [1], namely the
$SL(2,\mathbb{Z})$ family of BTZ black holes, generates a negative density of
states in two regimes [1, 2, 3]: at large spin $j\rightarrow\infty$ near
extremality,

$\int_{0}^{t_{0}}dt\,\rho_{{\rm MWK},\,j}(t)\sim(-1)^{j}e^{\pi\sqrt{\xi
j}},\quad t_{0}\sim e^{-2\pi\sqrt{\xi j}}$ (1)

where

$t:=\text{min}(h,\overline{h})-\xi\,,\quad
j=h-\overline{h}\,,\quad\xi:=\frac{c-1}{24}\,,$ (2)

and at the scalar black hole threshold,

$\rho_{{\rm MWK},0}(t)=-6\delta(t)+(t>0~{}\text{continuum})\,.$ (3)

The property (1) is especially severe: an exponentially large negative density
despite an exponentially small window. From the bulk perspective, seeking a
consistent pure gravity path integral requires reckoning with the sum over
topologies; for related work, see [4, 5, 6, 7, 8, 9, 10, 11]. (We note here
some recent work in AdS3/CFT2 that studies fixed bulk topologies [12, 13, 14,
15, 16, 17].)

Some valuable progress has been made. Explicit restoration of unitarity may be
achieved by retreating from pure gravity [18, 19], adding heavy point-particle
matter which admits a geometric bulk interpretation. The construction of [20],
which preserves the pure gravity spectrum, uses dimensional reduction to JT
gravity to fix (1) with an infinite sum over off-shell Seifert manifolds,
though it remains a mostly [21] implicit construction away from extremality
and leaves (3) intact. Other approaches that forego a subset of the above
conditions include [22, 23, 24].

##  I Partition function

The Virasoro primary partition function is defined as

$Z_{p}(\tau)=\sqrt{y}|\eta(\tau)|^{2}Z(\tau)$ (4)

where $Z(\tau)={\rm Tr}_{\mathcal{H}}(q^{L_{0}-{c\over
24}}\overline{q}^{\bar{L}_{0}-{c\over 24}})$ is the torus partition function
(non-holomorphic) and $\tau:=x+iy$. The following modular-invariant
$Z_{p}(\tau)$ is unitary at sufficiently large $\xi$:

$\mathcal{Z}(\tau)=Z_{\rm MWK}(\tau)+Z_{\rm string}(\tau)$ (5)

where

$\begin{split}Z_{\rm MWK}(\tau)&:=\sum_{\gamma\in
SL(2,\mathbb{Z})/\Gamma_{\infty}}\sqrt{\text{Im}(\gamma\tau)}\,|q_{\gamma}^{-\xi}(1-q_{\gamma})|^{2}\\\
Z_{\rm string}(\tau)&:=\sum_{\gamma\in
SL(2,\mathbb{Z})/\Gamma_{\infty}}\sqrt{\text{Im}(\gamma\tau)}\,\left(2q_{\gamma}^{\xi/4}\overline{q}_{\gamma}^{-\xi/4}+\text{c.c.}\right)\end{split}$
(6)

with $q_{\gamma}:=e^{2\pi i\gamma\tau}$. These are Poincaré sums over
$SL(2,\mathbb{Z})$ modulo $\Gamma_{\infty}$, the set of modular
$T$-transformations 111We thus formally demand that $\xi\in 2\mathbb{Z}_{+}$,
though we note that $\mathcal{Z}(\tau)$ remains unitary if we shift the seed
quantum numbers while preserving $T$-invariance. In particular,
$\mathcal{Z}(\tau)$ remains unitary if we slightly shift $\Delta_{*}>2\xi$
while keeping $t_{*}=-{\xi\over 4}$ fixed; for simplicity, we focus on the
case $\Delta_{*}=2\xi$ in the main text.. As we substantiate below, the
unitary range of $\xi$ includes $\xi\gg 1$, and provisionally appears to hold
for all $\xi\in 2\mathbb{Z}_{+}$. The reason for the “string” moniker will be
explained momentarily.

From a CFT point of view, $Z_{\rm string}(\tau)$ is a Poincaré sum over two
copies of a Virasoro primary seed state ${\cal O}_{*}$ with quantum numbers

$(\Delta_{*},j_{*})=\left(2\xi,{\xi\over
2}\right)\quad\Leftrightarrow\quad(t_{*},\overline{t}_{*})=\left(-{\xi\over
4},{\xi\over 4}\right)$ (7)

and its parity image with $h_{*}\leftrightarrow\overline{h}_{*}$. We have
employed the “reduced twist” variable $t$ along with its partner
$\overline{t}:=\text{max}(h,\overline{h})-\xi$. We have chosen the state in
(6) to be doubly-degenerate, a natural choice that preserves integrality, but
$\mathcal{Z}(\tau)$ is unitary for a finite range of degeneracies $d_{*}>1$
(see Appendix B, e.g. Fig. 3).

Let us state the spectral properties of the partition function
$\mathcal{Z}(\tau)$, deferring its unitarity to the next subsection. The
spectrum is shown in Fig. 1. The gap in conformal dimension above the vacuum
is exactly

$\Delta_{*}=\frac{c-1}{12}$ (8)

with no corrections. This is the value anticipated by the Virasoro modular
bootstrap program (e.g. [26, 27, 28, 29, 30]) as the optimal gap at large $c$,
on the basis of black hole universality: the conformal dimension (8)
corresponds to the massless limit of the semiclassical BTZ black hole. The
state-of-the-art bootstrap upper bound on the spectral gap at large $c$ is the
numerical result [29]

$\Delta_{*}\lesssim\frac{c}{9.08}\qquad(c\gg 1)$ (9)

with a slightly weaker analytical bound [30]. (See [31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51] for further
bootstrap work on Virasoro spectra at large $c$.) The explicit realization by
$\mathcal{Z}(\tau)$ of the gap (8) while preserving unitarity at $\xi\gg 1$
(the first such example, to our knowledge) is also noteworthy because of the
paucity of pure CFT arguments that a gap this large is possible. Conversely,
$\mathcal{Z}(\tau)$ shows constructively that without incorporating
discreteness into the modular bootstrap [49], the optimal bound on the gap
cannot be lower than $\Delta_{*}=2\xi$. This statement applies for all values
of $\xi$ for which $\mathcal{Z}(\tau)$ is unitary.

Figure 1: The Virasoro primary spectrum of $\mathcal{Z}(\tau)$. Green dots
denote the vacuum state and two (parity-invariant) states with
$(h_{*},\overline{h}_{*})=(\frac{5}{4}\xi,\frac{3}{4}\xi)$, interpretable in
AdS3 as strongly coupled strings. All other states exceed the semiclassical
black hole threshold: min$(h,\overline{h})\geq\xi$. The density of states is
positive.

As for the twist spectrum, all Virasoro primaries besides $\mathbbold{1}$ and
${\cal O}_{*}$ have $t\geq 0$. There is a positive integer number of scalar
states at $t=0$ (see (13)). The spectrum of $t>0$ states is continuous. This
can be understood rather generally in terms of coarse-graining. At large
$\xi$, this can be thought of as a consequence of ignorance of exponentially
small effects in $c$ – for example, smearing over the mean level spacing $\sim
e^{-S_{{\rm Cardy},j}(t)}$. We explain these points of interpretation in Sec.
IV.

###  I.1 Density of states

The corresponding Virasoro primary density of states, related to our partition
function as

${\mathcal{Z}(\tau)\over\sqrt{y}}=\sum_{j=0}^{\infty}(2-\delta_{j,0})\cos(2\pi
jx)\int_{\mathbb{R}}d\Delta\,e^{-2\pi y(\Delta-2\xi)}\rho_{j}(\Delta)$ (10)

can be derived straightforwardly using existing methods for Poincaré sums. We
have, in terms of reduced twist $t$,

$\rho_{j}(t)=\rho_{{\rm MWK},\,j}(t)+\rho_{{\rm string},\,j}(t)$ (11)

for every spin $j$. The MWK density $\rho_{{\rm MWK},\,j}(t)$ is recalled in
(26). The new term is, for $j\neq 0$,

$\begin{split}\rho_{{\rm
string},\,j}(t)&=\frac{4}{\sqrt{t\,\overline{t}}}\sum_{s=1}^{\infty}f_{j,j_{*};s}\cos\Big(\frac{2\pi}{s}\sqrt{\xi\overline{t}}\Big{missing})\cosh\Big(\frac{2\pi}{s}\sqrt{\xi
t}\Big{missing})\\\ &+(j_{*}\rightarrow-
j_{*},t\leftrightarrow\overline{t})\end{split}$ (12)

where $f_{j,j_{*};s}:={S(j,j_{*};s)/s}$ with $S(j,j_{*};s)$ a Kloosterman sum
(27). For $j=0$, such sums must be regularized; using standard methods nicely
summarized in [19], the result is the $j=0$ specialization of the $j\neq 0$
densities, augmented by a constant subtraction; see (28) and (30).

There are two hurdles to establishing positivity: one must cancel the
negativity of the MWK partition function in the $j\rightarrow\infty$ regime,
and at the scalar threshold $t=0$, both without introducing new negativity.

At $j\rightarrow\infty$, the negativity (1) is resolved by construction: we
have added states with reduced twist $t_{*}=-\xi/4$, designed precisely to
avoid the large-spin negativity in accordance with the arguments of [3] and
the subsequent approach of [18, 19]. (We added two such states, but any number
$d_{*}>1$ would do; we review this in Appendix B.) The states ${\cal O}_{*}$
have asymptotically large spin as $\xi\rightarrow\infty$. It is exactly this
property which admits the novelty of a spectral gap $\Delta_{*}=2\xi$ without
introducing further negativity elsewhere in the spectrum – and indeed, as we
now show, curing the scalar negativity (3) in the process.

The scalar density of states is

$\rho_{0}(t)=\delta(t+\xi)+(-6+8\sigma_{0}(j_{*}))\delta(t)+\tilde{\rho}_{0}(t)\,.$
(13)

The first term is the vacuum state. The second, formerly problematic, term has
been rendered strictly positive, for any $j_{*}=\xi/2$. Happily, it is also an
integer, a welcome surprise. Unlike previous approaches to this negativity,
its resolution does not require the addition of an “extra” ad hoc
$+6\delta(t)$ [2], instead coming for free in $\rho_{{\rm string},0}(t)$. We
note a number-theoretic feature of this degeneracy: if $j_{*}$ is prime, then
$\sigma_{0}(j_{*})=2$.

The last term, $\tilde{\rho}_{0}(t)$, is the continuum with support on $t>0$,
given explicitly in (33). Its positivity requires a more careful analysis
because various large-$\xi$ suppression factors are absent when $j=0$, i.e.
$t=\overline{t}$, as can be seen in (12); however, $\tilde{\rho}_{0}(t)$ is
indeed positive for all $t\geq 0$. We provide details in Appendix B, but can
sketch the essential point here. In the regime $\xi t\gg 1$, the scalar MWK
density is positive and exponentially larger in magnitude than the string
density. As $\xi t\sim{\cal O}(1)$, positivity is non-trivial as both
densities are of the same order and the string density is term-wise
oscillatory in $t$. With an eye toward semiclassical gravity, we focus on
$\xi\gg 1$, taking the regime of $x:=2\pi\sqrt{\xi t}$ fixed. The proof of
positivity proceeds in two steps: first, we show that the sum of the $s=1$ and
$s=2$ terms in (33) is positive; and second, we show that the $s\geq 2$ terms
are individually positive.

Numerical evaluation of the scalar density at large but finite $\xi$ confirms
these analytic results, as shown in Fig. 2. Indeed, we see that positivity
appears to hold all the way down to $j_{*}=1$, i.e. $\xi=2$, formally the
smallest central charge in our construction 222One can generalize our
construction to the case of $j_{*}\in\mathbb{Z}_{+}-{1\over 2}$, whereupon the
stabilizer group $\Gamma_{\infty}$ in (6) would instead be the set of $T^{2}$
transformations. Analogous extensions to supersymmetric partition functions
would also be straightforward. It would be interesting to ask whether the
requisite positivity is satisfied in each of these cases. See [1, 32, 82, 38,
47] for related material..

Figure 2: Plot of the regularized scalar primary density of states
$\tilde{\rho}_{0}(x)$ of the partition function $\mathcal{Z}(\tau)$, as a
function of $x=2\pi\sqrt{\xi t}$, with $\xi$ ranging from $\xi=2$ (red) to
$\xi=102$ (blue) in steps of 10. The curves are positive for all $x\geq 0$.
(Obtained by summing over $s\leq 200$ in (33).)

##  II A bulk string interpretation

The above construction is purely on the CFT side. Is there an AdS3 gravity
interpretation of the highly-spinning operator ${\cal O}_{*}$ and its modular
images?

One appealing answer is that ${\cal O}_{*}$ is a strongly coupled string, and
its modular images, stringy contributions to the black hole spectrum. While an
operator like ${\cal O}_{*}$ with $t<0$ and $\overline{t}>0$ cannot be dual to
a smooth BTZ black hole nor to a conical defect (such solutions with real mass
and angular momenta do not exist in semiclassical AdS3 gravity coupled to
point particles), spinning strings in AdS3 can, and indeed do, satisfy this
condition.

The spectrum of folded, spinning Nambu-Goto strings coupled to gravity in AdS3
was studied in [53] in the classical limit. The Virasoro primary string
spectrum is parameterized by a string tension $\lambda$ and an angular
velocity $\omega$. The string tension is given in terms of AdS, string and
Planck scales as

$\lambda={1\over 2\pi}{L_{\rm AdS}\over\ell_{s}}{\ell_{p}\over\ell_{s}}$ (14)

where $\ell_{p}=8\pi G_{N}$.

For a given $\lambda$, the string spectrum starts at the origin
$t=\overline{t}=-\xi$ and ends at the extremality bound $t=0$ or
$\overline{t}=0$. Matching the string spectrum to the quantum numbers (7) of
the operator ${\cal O}_{*}$ yields the unique result

$\lambda_{*}=1\,,\quad\omega_{*}=2\,.$ (15)

This string is strongly coupled: from (14), an AdS-sized string with
$\lambda_{*}=1$ requires $\ell_{p}/\ell_{s}\sim{\cal O}(1)$, which is the
ratio that defines an effective string coupling
$g_{s}=(\ell_{p}/\ell_{s})^{>0}$ (where the exponent depends on the details of
the putative string background 333For example, in the D1-D5 system
$\lambda\sim g_{6}/\sqrt{c}$ where $g_{6}$ is the six-dimensional effective
string coupling, so $\lambda=1$ is highly quantum. In general,
$\lambda\sim{\cal O}(1)$ captures a “very strongly coupled” limit of fixed
coupling $g_{s}$, rather than a ’t Hooft-type double-scaling limit. One can
see this by rewriting $\lambda=6(L/\ell_{s})^{2}/c$ using the Brown-Henneaux
relation [83].).

The specific value $\lambda_{*}=1$ happens to enjoy a certain synergy with the
equations of [53]. For generic $\lambda$ and $\omega$, the solutions of [53]
are given in terms of elliptic integrals. However, at $\lambda=1$ – and only
at $\lambda=1$ – the solution simplifies dramatically, as the string embedding
equation becomes algebraic. It is simple enough to recall explicitly in a few
lines. The Lorentzian spacetime metric outside the string is locally AdS3 with
the corresponding mass and angular momentum,

$ds^{2}={1\over 16}(-du^{2}+dv^{2})-\left(z-{1\over
256z}\right)dudv+{dz^{2}\over 4z^{2}}$ (16)

where, in the conventions of [53], the conformal boundary is at
$z\rightarrow\infty$. The string embedding is determined by functions
$u(\sigma,\tau),v(\sigma,\tau)$ and $z(\sigma)$, where $(\sigma,\tau)$ are
worldsheet coordinates with induced metric

$\begin{split}h&=\Omega^{2}(\sigma)(-d\tau^{2}+d\sigma^{2})\\\
\Omega^{2}(\sigma)&=3{(z_{L}-z(\sigma))(z(\sigma)-z_{R})\over
z(\sigma)}\end{split}$ (17)

where (15) implies $z_{L}={3\over 16},z_{R}=-{1\over 48}$, and

$z(\sigma)=\frac{\left(32-25\cos^{2}\sigma+5\sqrt{25\cos^{4}\sigma-64\cos^{2}\sigma+64}\right)}{384}$
(18)

Opposite points on the string are identified, “sewing up” the spacetime
444This identification does not happen at fixed $t$, so the spacetime does not
“pinch off” on a constant time slice.. The outermost radius of the string
(where it folds) is at $z=z_{L}$, while the center of the string is at
$z=z(0)={1/12}$. The spacetime ends at the string, avoiding a naked
singularity.

So we see that the state ${\cal O}_{*}$ admits an interpretation as a highly-
spinning string coupled to gravity in AdS3. That it is strongly coupled
dovetails nicely with how AdS3 pure gravity could possibly arise in string
theory: strong coupling is necessary to gap the light string modes to the
Planck scale.

###  II.1 Black hole microstates

Our construction adds not only the states ${\cal O}_{*}$, but their
$SL(2,\mathbb{Z})$ images too. These states are heavy, but are not BTZ black
holes (fully captured by the MWK sum over smooth Euclidean saddles) nor their
orbifolds (which are modular images of conical defect geometries).

Instead, these are new black hole microstates made of strongly coupled
strings. The Euclideanized, modular-transformed solutions of [53] are small
black strings, in the following specific sense: whereas a BTZ/conical defect
solution with the same quantum numbers would be nakedly singular, the strings
shroud this region by terminating the spacetime. These geometries may be
thought of as quantum AdS3 versions of the stringy cloak of [56] and other
small black strings (e.g. [57, 58]). That they are “small” – the modular
transforms of a single string, rather than a parametrically large number of
them – is also visible thermodynamically in the different functional forms of
the stringy and BTZ microcanonical entropies: $\rho_{{\rm string},j}(t)$ is
oscillatory as a function of $t$, unlike the BTZ density $\rho_{{\rm
MWK},j}(t)$, and is exponentially subleading to $\rho_{{\rm MWK},j}(t)$, term-
by-term in the modular sum, away from the near-extremal regime $\xi
t\lesssim{\cal O}(1)$ where the BTZ black hole becomes highly quantum 555This
is within the “enigmatic” regime [34] where such corrections may be large,
consistent with modularity and sparseness.. This fluctuating behavior signals
that the stringy degrees of freedom give genuinely new contributions to the
black hole Hilbert space, distinct from the semiclassical BTZ geometries or
quotients thereof.

##  III $SL(2,\mathbb{Z})$ spectral representation

As a slight detour, it is enlightening to give an alternative representation
of $Z_{\rm string}(\tau)$. The spectral gap condition $\Delta_{*}=2\xi$
implies that $Z_{\rm string}(\tau)\in L^{2}(\mathcal{F})$, and hence admits a
harmonic decomposition in the $SL(2,\mathbb{Z})$ spectral eigenbasis,
comprised of the completed Eisenstein series $E_{{1\over
2}+i\omega}^{*}(\tau)$ with $\omega\in\mathbb{R}$ and Maass cusp forms
$\phi_{n}(\tau)$ (e.g. [60, 61, 62]). Denoting their spin-$j$ Fourier
coefficients as $\mathsf{a}^{(s)}_{j}$ and $\mathsf{b}^{(n)}_{j}$,
respectively, and using the conventions of [21], we have

$\begin{split}Z_{\rm string}(\tau)&=\int_{\mathcal{C}_{\rm
crit}}\,\mathsf{a}^{(s)}_{j_{*}}\,{\Gamma\left(\frac{{1\over
2}-s}{2}\right)\Gamma\left(\frac{s-{1\over
2}}{2}\right)\over\Lambda(s)\Lambda(1-s)}E^{*}_{s}(\tau)\\\
&~{}~{}\,+\sum_{n=1}^{\infty}\mathsf{b}^{(n)}_{j_{*}}\Gamma\Big{(}-{i\omega_{n}\over
2}\Big{)}\Gamma\Big{(}\frac{i\omega_{n}}{2}\Big{)}\phi_{n}(\tau)\end{split}$
(19)

where $\mathcal{C}_{\rm crit}$ denotes ($(4\pi i)^{-1}$ times) contour
integration along $s={1\over 2}+i\omega$, and
$\Lambda(s):=\pi^{-s}\Gamma(s)\zeta(2s)$ is the completed Riemann zeta
function. (See Appendix C.)

Presenting $Z_{\rm string}(\tau)$ in spectral form reveals some interesting
features and curiosities.

First, the modular average of $Z_{\rm string}(\tau)$ vanishes:

$\langle Z_{\rm string}\rangle:=\int_{\mathcal{F}}{dxdy\over y^{2}}Z_{\rm
string}(\tau)=0\,.$ (20)

This follows from the vanishing of the Eisenstein spectral overlap in (19) at
$s=0$, which defines the modular average in general. We note that this
property is shared by Narain CFTs [61].

Next, $Z_{\rm string}(\tau)$ may be written as the action of an
$SL(2,\mathbb{Z})$ Hecke operator $T_{\xi/2}$ 666Acting on non-holomorphic
$SL(2,\mathbb{Z})$-invariant functions,
$T_{j}f(\tau)={1\over\sqrt{j}}\sum_{a,b,d}f\quantity({a\tau+b\over d})$ where
$ad=j,\,d>0$ and $0\leq b\leq d-1$. on a “primitive” partition function,
$\mathcal{Z}_{\rm string}(\tau)$, defined as $Z_{\rm string}(\tau)$ but with
the Fourier coefficients evaluated at $j_{*}=1$:

$\mathcal{Z}(\tau)=Z_{\rm MWK}(\tau)+T_{\xi/2}\,\mathcal{Z}_{\rm
string}(\tau)\,.$ (21)

In this way, the entire family of unitary partition functions indexed by $\xi$
may be generated by a Hecke action, implementing shifts in central charge.
This shares a superficial likeness with Witten’s construction of holomorphic
extremal CFT partition functions [64], with obvious differences.

Finally, there is a profound conjecture in number theory, the “horizontal”
Sato-Tate conjecture for Maass cusp forms of $SL(2,\mathbb{Z})$, which has
interesting consequences for the spectral decomposition [65, 66, 67, 68]. The
conjecture states that for prime $j\rightarrow\infty$ and any fixed $n$, the
normalized Fourier coefficients $\mathsf{b}^{(n)}_{j}/\mathsf{b}^{(n)}_{1}$
are equidistributed with respect to Wigner’s semicircle distribution. This
(and $\mathsf{b}^{(n)}_{1}\neq 0$, which follows from Hecke relations applied
to Hecke-Maass cusp forms) implies that

$\lim_{j\rightarrow\infty}\langle\\!\langle{\mathsf{b}^{(n)}_{j}}\rangle\\!\rangle=0\qquad(\text{fixed
{\hbox{n}}})$ (22)

where $\langle\\!\langle\cdot\rangle\\!\rangle$ indicates a statistical
average. Therefore, even though $(Z_{\rm
string},\phi_{n})\propto\mathsf{b}^{(n)}_{j_{*}}\neq 0$, they vanish on
average in the large central charge limit $j_{*}\rightarrow\infty$ 777The
conjecture is widely held to be true on the basis of stringent numerical
checks, limited proofs, and relation to Sato-Tate conjectures in other
contexts. Note that at finite $j_{*}$, the “vertical” Sato-Tate conjecture
[84], would only imply vanishing coefficients as $n\rightarrow\infty$.. In
this sense, the Eisenstein term seems to more directly underlie the unitarity
of $\mathcal{Z}(\tau)$. It would be nice to understand this from a physical,
quantum chaos point of view.

##  IV Summary and Random (matrix) comments

Our main result is the construction of the unitary partition function
$\mathcal{Z}(\tau)$ given in (5), with the spectral gaps depicted in Fig. 1.

From the AdS3 gravity point of view, despite the dimension gap above the
vacuum state to the black hole threshold $\Delta_{*}={c-1\over 12}$, this is
not a semiclassical pure gravity path integral in the strict sense, due to the
spinning states ${\cal O}_{*}$ with sub-threshold twist. At any finite spin,
these states are not visible, and the theory contains only black hole states.
The degeneracies of all discrete states are integers.

We have advanced a bulk interpretation of ${\cal O}_{*}$ as a strongly coupled
spinning string, though other interpretations may well be possible (or
preferred). We view this as an indicative toy model for a genuine string
theory compactification to AdS3 pure gravity. A complete approach would
include higher Regge trajectories; corrections to the spectrum from
excitations around the spin-$j$ ground states of [53]; and the other
ingredients, such as fluxes and their brane sources, required to solve the
strongly coupled string field equations (whatever they may be).

###  IV.1 Randomness

Our construction cures the negativity from the sum over smooth bulk saddles
semiclassically, rather than quantum mechanically. Quantum effects are not
just present in a consistent theory, but are expected to be crucial in the
engineering of a bona fide theory of AdS3 pure gravity: there are strong
indications that if such a theory exists, off-shell geometries encoding random
matrix behavior of the chaotic spectrum play a central role in unitarizing the
spectrum [70, 8, 20]. An explicit determination of the leading-order random
matrix contribution to the semiclassical path integral of pure gravity with
torus boundary, denoted as $Z_{\rm RMT}(\tau)$, was made in [21].

In any theory of semiclassical AdS3 gravity (pure or otherwise), the black
hole spectrum is chaotic, and its path integral should encode random
fluctuations for quantum consistency. Such random matrix contributions are
absent in $\mathcal{Z}(\tau)$. We may explain this fact, as well as the
continuous spectrum in the chaotic regime $t>0$, by interpreting
$\mathcal{Z}(\tau)$ as the partition function of a microscopic compact CFT
that has been subject to coarse-graining.

As shown in [21] using a formalism built on a CFT trace formula, the random
matrix contribution to the density of states, properly understood, vanishes
upon coarse-graining the spectrum over a suitable microcanonical window in
twist, $\delta t$ 888This coarse-graining may be performed using convolution
as
$\overline{f(t)}:=\int_{0}^{\infty}dt^{\prime}\,W(t-t^{\prime})f(t^{\prime})$
where $W(t)$ has characteristic width $\delta t$ and obeys
$\int_{0}^{\infty}dt\,W(t)=1$.. Because this window is necessarily larger than
the exponentially small mean level spacing of the chaotic spectrum, the
coarse-graining simultaneously explains both the absence of random matrix
contributions to (5) and its continuous spectrum while remaining compatible
with a microscopic CFT interpretation. Given our explicit construction, we can
determine $\delta t$: it is the characteristic wavelength of the oscillations
of $\rho_{{\rm string},j}(t)$ in (12), namely, $\delta t\sim 1/\xi$. We
emphasize that this coarse-graining interpretation does not rely on a $\xi\gg
1$ limit, and is compatible with compactness of a putative underlying CFT;
there could, of course, be as-yet-unknown bootstrap constraints that rule this
possibility out.

Note that in a $\xi\gg 1$ limit, $\mathcal{Z}(\tau)$ is also compatible with
other interpretations, in particular with a hypothetical ensemble average over
(possibly near-)CFTs, or with other, perhaps independent, constructions of
“approximate CFT” [72]. While we have presented a microscopic CFT
interpretation in part to emphasize that a departure from standard AdS/CFT
physics is not required at this level, semiclassical gravity seems unable to
distinguish among these [73, 72, 74], at least perturbatively in $G_{N}$.

A complementary view on this coarse-grained interpretation comes from the
formalism of [21]. Since $Z_{\rm string}(\tau)$ is the modular completion of a
non-black hole state, we do not expect it to encode random matrix behavior per
se [75, 72, 21, 76]. Applying the results of [21] to $Z_{\rm string}(\tau)$
helps to ratify this perspective. In (19) we provided the $SL(2,\mathbb{Z})$
spectral decomposition of $Z_{\rm string}(\tau)$. A canonical diagnostic of
random matrix universality is the presence of a linear ramp in the coarse-
grained spectral form factor, with a specific coefficient prescribed by the
random matrix ensemble. We can ask whether $Z_{\rm string}(\tau)$ generates
this ramp after squaring and taking the diagonal approximation. A necessary
and sufficient condition for the ramp was derived in [21], as an exponential
decay condition on the spectral overlaps at $\omega\rightarrow\infty$. One
readily checks that $Z_{\rm string}(\tau)$ does not satisfy this criterion,
instead decaying as a power law 999Similarly, the Hecke projection [21] of
$Z_{{\rm string}}(\tau_{1})Z_{{\rm string}}(\tau_{2})$ is not a wormhole
amplitude..

###  IV.2 Stringiness

On the other hand, $Z_{\rm string}(\tau)$ exhibits some behavior that lies
somewhere “in between” chaotic and non-chaotic. Define a microcanonical
coarse-graining over mean twist,

$\overline{f(t_{1})f(t_{2})}:=\int_{0}^{\infty}dt^{\prime}\,f(t^{\prime}+\epsilon)f(t^{\prime}-\epsilon)W(t-t^{\prime})$
(23)

where $t={t_{1}+t_{2}\over 2}$ and $\epsilon={t_{1}-t_{2}\over 2}$. Applying
this to $f(t)=\rho_{{\rm string},j}(t)$ at fixed $j$ using (12) produces a
non-zero variance upon coarse-graining over windows $\delta
t\gtrsim\frac{1}{\xi}$. However, its oscillatory behavior leads to suppression
relative to the disconnected average. In particular, at $\xi\gg 1$,

$\frac{\text{Var}(\rho_{j}(t))}{\bar{\rho}_{j}(t)^{2}}\approx
e^{-4\pi\sqrt{\xi(t+j)}}\qquad(\xi\gg 1)$ (24)

where $\bar{\rho}_{j}(t)=\rho_{{\rm MWK},j}(t)$. In the extremal limit
$t\rightarrow 0$, the suppression factor is $e^{-S_{0,j}}$, where
$S_{0,j}=4\pi\sqrt{\xi j}$ is the extremal spin-$j$ BTZ black hole entropy. In
contrast, wormholes encoding chaotic behavior are suppressed as
$e^{-2S_{0,j}}$ in the extremal limit [78, 79, 70, 8, 21]. It would be
worthwhile to understand this intermediate behavior as a non-perturbative
effect, possibly associated to strongly coupled strings, in a UV complete AdS3
gravity path integral.

## Acknowledgments

We thank Jacob Abajian, Veronica Collazuol, Scott Collier, Henry Maxfield,
Dalimil Mazac, Sridip Pal, Yiannis Tsiares, and Pierfrancesco Urbani for
helpful discussions. EP and GD thank the Kavli Institute for Theoretical
Physics, Santa Barbara for support during the course of this work. EP also
thanks the ICTP Trieste and Kavli IPMU for hospitality. GD also thanks the
ICISE in Quy Nhon, Vietnam for hospitality. This research was supported by ERC
Starting Grant 853507, and in part by the National Science Foundation under
Grant No. NSF PHY-1748958.

##  Appendix A Density of states

We write the total spin-$j$ density of states as

$\rho_{j}(t)=\rho_{{\rm MWK},\,j}(t)+\rho_{{\rm string},\,j}(t)$ (25)

The MWK density of states may be written as [3, 19]

$\begin{split}&\rho_{{\rm MWK},j}(t)=\\\
&\frac{2}{\sqrt{t\overline{t}}}\sum_{s=1}^{\infty}\bigg{[}f_{j,0;s}\cosh(\frac{4\pi}{s}\sqrt{\xi\overline{t}})\cosh(\frac{4\pi}{s}\sqrt{\xi
t})\\\
&-f_{j,-1;s}\cosh(\frac{4\pi}{s}\sqrt{\xi\overline{t}})\cosh(\frac{4\pi}{s}\sqrt{(\xi-1)t})\\\
&-f_{j,1;s}\cosh(\frac{4\pi}{s}\sqrt{(\xi-1)\overline{t}})\cosh(\frac{4\pi}{s}\sqrt{\xi
t})\\\
&+f_{j,0;s}\cosh(\frac{4\pi}{s}\sqrt{(\xi-1)\overline{t}})\cosh(\frac{4\pi}{s}\sqrt{(\xi-1)t})\bigg{]}\end{split}$
(26)

where $f_{j,k;s}:={S(j,k;s)/s}$ and $S(j,k;s)$ is a Kloosterman sum,

$S(j,k;s)=\sum_{0\leq d\leq s-1,\,(d,s)=1}e^{2\pi i{d\over s}j+{d^{-1}\over
s}k}$ (27)

where $d^{-1}\in\mathbb{Z}$ is the multiplicative inverse of $d$ mod $s$. The
scalar density requires regularization [1, 2]. It is comprised of a delta
function piece given by (3), and a continuous piece which we denote by
$\tilde{\rho}_{{\rm MWK},0}(t)$:

$\begin{split}&\tilde{\rho}_{{\rm MWK},0}(t)=\\\
&\frac{2}{t}\sum_{s=1}^{\infty}\bigg{\\{}{\phi(s)\over
s}\bigg{[}\sinh^{2}\left(\frac{4\pi}{s}\sqrt{\xi
t}\right)+\sinh^{2}\left(\frac{4\pi}{s}\sqrt{(\xi-1)t}\right)\bigg{]}\\\
&-2\,{\mu(s)\over s}\bigg{[}\cosh(\frac{4\pi}{s}\sqrt{\xi
t})\cosh(\frac{4\pi}{s}\sqrt{(\xi-1)t})-1\bigg{]}\bigg{\\}}\end{split}$ (28)

where $\phi(s)=S(0,0;s)$ is the Euler totient function and $\mu(s)=S(0,1;s)$
is the Mobius function.

The string density for $j\neq 0$ was given in (12), which we repeat here for
convenience:

$\begin{split}\rho_{{\rm
string},\,j}(t)&=\frac{4}{\sqrt{t\,\overline{t}}}\sum_{s=1}^{\infty}f_{j,j_{*};s}\cos\Big(\frac{2\pi}{s}\sqrt{\xi\overline{t}}\Big{missing})\cosh\Big(\frac{2\pi}{s}\sqrt{\xi
t}\Big{missing})\\\ &+(j_{*}\rightarrow-
j_{*},t\leftrightarrow\overline{t})\end{split}$ (29)

Similarly to the MWK case, the scalar density requires regularization. The
regularized density contains a delta function piece given in (13), and a
continuous piece which we denote by $\tilde{\rho}_{{\rm string},0}(t)$:

$\begin{split}\tilde{\rho}_{{\rm
string},0}(t)=&\frac{8}{t}\sum_{s=1}^{\infty}\frac{c_{s}(j_{*})}{s}\times\\\
&\bigg{[}\cos(\frac{2\pi}{s}\sqrt{\xi t})\cosh(\frac{2\pi}{s}\sqrt{\xi
t})-1\bigg{]}\end{split}$ (30)

where $c_{s}(j_{*})$ is a Ramanujan sum,

$c_{s}(j_{*})=\sum_{1\leq d\leq s,\,(d,s)=1}e^{2\pi i{d\over s}j_{*}}$ (31)

The total regularized scalar density is given by the sum of these two
contributions, together with the delta functions in (13):

$\rho_{0}(t)=\delta(t+\xi)+(-6+8\sigma_{0}(j_{*}))\delta(t)+\tilde{\rho}_{0}(t)$
(32)

where

$\begin{split}\tilde{\rho}_{0}(t)=\tilde{\rho}_{{\rm
MWK},0}(t)+\tilde{\rho}_{{\rm string},0}(t).\end{split}$ (33)

##  Appendix B Positivity

We treat in turn the positivity of the large spin $j\rightarrow\infty$, finite
spin $j\geq 1$, and scalar $j=0$ sectors of the density (25), focusing mostly
on the regime $\xi\gg 1$. Actually, we consider a more general case in which
we have $d_{*}$ string states: namely, $Z_{p}(\tau)=Z_{\rm
MWK}(\tau)+{d_{*}\over 2}\,Z_{\rm string}(\tau)$, and correspondingly for the
densities. For the partition function $\mathcal{Z}(\tau)$ defined in the main
text, $d_{*}=2$.

###  B.1 Positivity at large spin

The MWK density (26) is known to be negative in the extremal limit
$t\rightarrow 0$ of large spin $|j|\rightarrow\infty$ (more precisely, for
$t\lesssim t_{0}\sim e^{-2\pi\sqrt{\xi j}}$, dropping a numerical prefactor).
We cure this negativity with the states ${\cal O}_{*}$ by design, having
chosen their twist to be $t_{*}=-\frac{\xi}{4}$. The mechanism is the same as
described in Sec. 2.1 of [19], building on [18, 3]. In the regime $t<t_{0}$
with $j\rightarrow\infty$, the MWK density is approximately equal to [19]

$\begin{split}\rho_{{\rm
MWK},\,j}(t)\approx\frac{(-1)^{j}}{\sqrt{jt}}\quantity(e^{2\pi\sqrt{\xi
j}}+e^{2\pi\sqrt{(\xi-1)j}})\end{split}$ (34)

In the same regime, the string density (29) has the same exponential behavior
but with a positive coefficient, coming from the state of spin $-j_{*}$:

$\begin{split}\rho_{{\rm
string},\,j}(t)\approx\frac{d_{*}}{\sqrt{jt}}e^{2\pi\sqrt{\xi j}}.\end{split}$
(35)

The string density cancels the odd-spin negativity for $d_{*}>1$. Whereas at
$d_{*}=1$ there are subleading negativities to take care of, requiring the
addition of higher-twist states [19], choosing $d_{*}=1+\delta$ for any finite
$\delta$ gives a positive extremal density. In the construction of
$\mathcal{Z}(\tau)$ in (5), we chose $d_{*}=2$, the smallest integer
degeneracy which guarantees positivity, as a matter of naturalness. The above
discussion applies equally to the regime of large negative spin
$j\rightarrow-\infty$, whereupon the negativity is cured by the state ${\cal
O}_{*}$ with spin $+j_{*}$.

###  B.2 Positivity at $j\geq 1$

We now consider finite spin $j\geq 1$. We also take $\xi\gg 1$. There are two
regimes of twist $t$ to consider: the extremal limit $t\rightarrow 0$, and
fixed $t$.

In the extremal limit, the arguments given just above are again sufficient to
guarantee positivity. In particular, we have $\xi j\gg 1$ in the present
regime of interest; one may confirm upon inspection that the $\xi$\- and
$j$-dependence of $\rho_{{\rm MWK},\,j}(t)$ and $\rho_{{\rm string},\,j}(t)$
are such that at $\xi j\gg 1$, even for finite $j$, the result of the previous
subsection carries through.

Now we consider the regime of fixed $t$. Since $\xi j\gg 1$, terms of the form
$\cosh(\frac{4\pi}{s}\sqrt{\xi(t+j)})$ for $s\geq 2$ are exponentially
suppressed with respect to the $s=1$ term. At fixed $t$, the MWK density is
therefore well-approximated by the $s=1$ term:

$\begin{split}\frac{\sqrt{t\overline{t}}}{2}&\rho_{{\rm
MWK},\,j}(t)\approx\quantity[\cosh(4\pi\sqrt{\xi
t})-\cosh(4\pi\sqrt{(\xi-1)t})]\\\
&\times\quantity[\cosh(4\pi\sqrt{\xi(t+j)})-\cosh(4\pi\sqrt{(\xi-1)(t+j)})]\end{split}$
(36)

This is manifestly positive, and scales as $\sim e^{4\pi\sqrt{\xi j}}$ times
an ${\cal O}(1)$ coefficient. The string density at leading order in $\xi j\gg
1$ is

$\begin{split}\frac{\sqrt{t\overline{t}}}{2d_{*}}&\rho_{{\rm
string},\,j}(t)=\cos(2\pi\sqrt{\xi t})\cosh(2\pi\sqrt{\xi(t+j)})\\\
&+\sum_{s=1}^{\infty}f_{j,j_{*};s}\cos(\frac{2\pi}{s}\sqrt{\xi(t+j)})\cosh(\frac{2\pi}{s}\sqrt{\xi
t})\end{split}$ (37)

In the first line we dropped the exponentially-suppressed $s>1$ terms (this is
allowed because the sum over $s$ cannot lead to exponential enhancement),
whereas no such suppression is present in the second line. Noting that (37)
scales as $\sim e^{2\pi\sqrt{\xi j}}$, we see that the sum of (36) and (37) is
positive, as the latter is exponentially suppressed in $\xi j\gg 1$. As an
aside, note that this hierarchy can be overcome if $d_{*}$ is exponentially
large in $\xi$, a possibility that we discard (in the next subsection we bound
$d_{*}$ by an ${\cal O}(1)$ number).

Summarizing so far, we have shown that $\rho_{j}(t)>0$ for $j\geq 1$ and all
$t$ at $\xi\gg 1$.

###  B.3 Positivity at $j=0$

The scalar sector requires slightly more attention since there is no longer a
parametrically large scale that suppresses $s>1$ terms in the density. Since
the MWK density is exponentially large and positive as $\xi t\gg 1$, any
possible negativity will arise only for $\xi t\lesssim{\cal O}(1)$. We can
then study the density at fixed $x:=2\pi\sqrt{\xi t}$, where we also take
$\xi\gg 1$.

We divide the proof into two parts: showing that the sum of $(s=1)$ and
$(s=2)$ terms is positive for $d_{*}$ below a critical value; and showing that
$s>2$ terms are individually positive.

Figure 3: Plot of the regularized scalar density of states
$\tilde{\rho}_{0}(x)$ with $\xi=1000$, as a function of $x=2\pi\sqrt{\xi t}$,
with degeneracy ranging from $d_{*}=3.3$ (red) to $d_{*}=7.3$ (blue) in half-
integer steps. For $d_{*}\gtrsim 7.3$, the density develops a negative region.
(Obtained by summing over $s\leq 200$ in (33).)

####  B.3.1 $(s=1)+(s=2)$

The sum of the $s=1,2$ terms of (28) and (30) (times ${d_{*}\over 2}$) is, at
leading order in large $\xi$,

$\begin{split}\frac{t}{2}\tilde{\rho}_{0}(t)\big{|}_{s\leq
2}&=2\sinh^{2}x+2d_{*}(\cos x\cosh x-1)\\\
&+d_{*}(-1)^{j_{*}}\quantity(\cos(\frac{x}{2})\cosh(\frac{x}{2})-1).\end{split}$
(38)

One can easily see numerically that upon increasing $d_{*}$, this function
develops a minimum $x_{\rm min}$ which eventually becomes negative,
approximately given by

$\begin{split}d_{*}&\lesssim 4.910,\quad x_{\rm min}\approx
1.851\qquad(j_{*}~{}\text{even})\\\ d_{*}&\lesssim 5.236,\quad x_{\rm
min}\approx 1.847\qquad(j_{*}~{}\text{odd})\end{split}$ (39)

If $d_{*}$ obeys these bounds, then (38) is positive. We can check how these
bounds compare to the full sum over $s$ at finite but large central charge:
see Fig. 3. Summing up to $s=200$ for $\xi=1000$, which easily ensures
convergence, we observe numerically that the density becomes negative for
$d_{*}\gtrsim 7.3$, not far from the limited analytic bound obtained above.
The growth of the upper bound as we include more terms in the sum is due to
the positivity of the $s>2$ terms, as we will show next. Note that at smaller
$\xi$, the upper bound actually grows, as seen in Fig. 4: for the smallest
value $\xi=2$ allowed within our construction, we observe positivity for
$d_{*}\lesssim 19.2$.

Figure 4: Plot of the regularized scalar density of states
$\tilde{\rho}_{0}(x)$ with $\xi=2$, as a function of $x=2\pi\sqrt{\xi t}$,
with degeneracy ranging from $d_{*}=3.2$ (red) to $d_{*}=19.2$ (blue) in steps
of two. For $d_{*}\gtrsim 19.2$, the density develops a negative region – a
larger critical value than for $\xi=1000$. (Obtained by summing over $s\leq
200$ in (33).)

####  B.3.2 $s\geq 3$ terms

At leading order in $\xi\gg 1$, the density of states for $s\geq 3$ is:

$\begin{split}{ts\over 2}\tilde{\rho}_{0}(t)\big{|}_{s\geq
3}&=2(\phi(s)-\mu(s))\sinh^{2}(2x_{s})\\\ &+2d_{*}c_{s}(j_{*})\quantity(\cos
x_{s}\cosh x_{s}-1),\end{split}$ (40)

where

$x_{s}:=\frac{x}{s}\,.$ (41)

Denoting the right-hand side of (40) as $f(x_{s})$, we observe that there is a
minimum at $x_{s}=0$ for which $f(0)=f^{\prime}(0)=0$. As a consequence, one
way to ensure positivity is to demand convexity, $f^{\prime\prime}(x_{s})>0$,
for all $x_{s}>0$; this is of course not a necessary condition, but it is
sufficient to achieve our goal of demonstrating existence of a range of
$d_{*}$ in which these terms are positive. Imposing convexity gives the
inequality

$\begin{split}16(\phi(s)-\mu(s))\cosh(4x_{s})-4d_{*}c_{s}(j_{*})\sin
x_{s}\sinh x_{s}>0\,.\end{split}$ (42)

Using the bounds

$|c_{s}(j_{*})|<\phi(s)\,,\quad|\mu(s)|\leq 1\,,\quad\phi(s\geq 3)\geq 2$ (43)

gives rise to the strongest inequality,

$d_{*}<2{\cosh(4x_{s})\over\sinh(x_{s})}\,.$ (44)

If this is satisfied for all $x_{s}$, then so is (42). Minimizing the right-
hand side gives

$d_{*}\lesssim 11.888\,.$ (45)

This ensures positivity of each individual $s\geq 3$ term in the density. This
is compatible with the previously derived bounds for the $s=1,2$ terms.

Altogether, we conclude that for a finite range of $d_{*}>1$, the density of
states is positive, $\rho_{j}(t)>0$, for all spins $j$ and twists $t$ at
$\xi\gg 1$.

##  Appendix C Spectral decomposition of $Z_{\rm string}(\tau)$

In this appendix we derive (19). We directly present the relevant
calculations, directing the reader to [60, 61, 62] for details on the
$SL(2,\mathbb{Z})$ spectral formalism, and [80, 76, 21] for its further use in
the 2d CFT context.

We wish to compute the Petersson inner product

$(Z_{\rm string},\psi_{\omega}):=\int_{\mathcal{F}}{dxdy\over y^{2}}Z_{\rm
string}(\tau)\overline{\psi}_{\omega}(\tau)$ (46)

where $\psi_{\omega}(\tau)=\\{E_{{1\over 2}+i\omega}(\tau),\phi_{n}(\tau)\\}$
are the $SL(2,\mathbb{Z})$ eigenbasis elements. Since $Z_{\rm string}(\tau)$
is a Poincaré sum, the overlaps with the Eisenstein series and Maass cusp
forms can be easily computed using the “unfolding trick.” This results in the
following integral for the Eisenstein series:

$\begin{split}(Z_{\rm string},E_{{1\over
2}+i\omega})=\frac{4\mathsf{a}_{j_{*}}^{({1\over
2}+i\omega)}}{\Lambda\quantity({1\over
2}-i\omega)}\int_{0}^{\infty}\frac{dy}{y}K_{i\omega}(2\pi j_{*}y)\end{split}$
(47)

where $\mathsf{a}_{j}^{({1\over 2}+i\omega)}$ are the Eisenstein Fourier
coefficients

$\mathsf{a}_{j}^{({1\over 2}+i\omega)}={2\sigma_{2i\omega}(j)\over
j^{i\omega}}\,,$ (48)

which obey reflection symmetry, $\mathsf{a}_{j}^{({1\over
2}+i\omega)}=\mathsf{a}_{j}^{({1\over 2}-i\omega)}$. The cusp form overlap is
obtained similarly:

$\begin{split}(Z_{\rm
string},\phi_{n})=4\mathsf{b}_{j}^{(n)}\int_{0}^{\infty}\frac{dy}{y}K_{i\omega}(2\pi
j_{*}y),\end{split}$ (49)

where $\mathsf{b}_{j}^{(n)}$ are the cusp form Fourier coefficients, known
only approximately via numerics [81]. The integral is divergent at the origin.
Regularizing the divergence is straightforward: introducing

$\begin{split}(Z_{\rm string},E_{{1\over
2}+i\omega})&{}_{\epsilon}:=\frac{4\mathsf{a}_{j_{*}}^{\quantity({1\over
2}+i\omega)}}{\Lambda\quantity({1\over
2}-i\omega)}\int_{0}^{\infty}\frac{dy}{y^{1-\epsilon}}K_{i\omega}(2\pi
j_{*}y)\\\ &=\frac{(\pi
j_{*})^{-\epsilon}\mathsf{a}_{j_{*}}^{\quantity({1\over
2}+i\omega)}}{\Lambda\quantity({1\over
2}-i\omega)}\Gamma\quantity(\frac{\epsilon-i\omega}{2})\Gamma\quantity(\frac{\epsilon+i\omega}{2})\end{split}$
(50)

the overlaps may be defined by removing the regulator,

$\begin{split}(Z_{\rm string}E_{{1\over
2}+i\omega})&=\lim_{\epsilon\rightarrow 0}(Z_{\rm string},E_{{1\over
2}+i\omega})_{\epsilon}\\\ &=\frac{\mathsf{a}_{j_{*}}^{\quantity({1\over
2}+i\omega)}}{\Lambda\quantity({1\over
2}-i\omega)}\Gamma\quantity(-\frac{i\omega}{2})\Gamma\quantity(\frac{i\omega}{2}).\end{split}$
(51)

and likewise for the cusp form overlap (49). This yields the spectral
decomposition (19). We note that the regularization used here is equivalent to
the following standard regularization of Poincaré sums over seed primaries of
fixed dimensions,

$\begin{split}Z_{h,\overline{h}}^{\epsilon}(\tau):=\sum_{\gamma\in
SL(2,\mathbb{Z})/\Gamma_{\infty}}\text{Im}(\gamma\tau)^{{1\over
2}+\epsilon}q_{\gamma}^{h-\xi}\overline{q}_{\gamma}^{\overline{h}-\xi}\,,\end{split}$
(52)

where $(h,\overline{h})=({5\over 4}\xi,{3\over 4}\xi)$ for our state ${\cal
O}_{*}$.

###  C.1 Re-deriving the scalar density

As a consistency check, we can re-derive the scalar density $\rho_{{\rm
string},0}(t)$ from the spectral decomposition. The scalar piece of the
regularized partition function is

$\frac{Z_{\rm string,0}^{\epsilon}(y)}{2\sqrt{y}(\pi
j_{*})^{-\epsilon}}=\int_{\mathcal{C}_{\rm
crit}}\frac{y^{i\omega}\mathsf{a}_{j_{*}}^{({1\over
2}+i\omega)}}{\Lambda\quantity({1\over
2}-i\omega)}\Gamma\quantity(\frac{\epsilon+i\omega}{2})\Gamma\quantity(\frac{\epsilon-i\omega}{2})$
(53)

where $\int_{\mathcal{C}_{\rm crit}}={1\over
4\pi}\int_{-\infty}^{\infty}d\omega$ is the integration along the critical
line. In writing (53) we have used the scalar Fourier modes
$E^{*}_{s,0}(y)=\Lambda(s)y^{s}+\Lambda(1-s)y^{1-s}$ and $\phi_{n,0}(y)=0$. We
now perform contour integration for complex $z:=i\omega$ by deforming to a new
contour, $\mathcal{C}$, a semicircle in the left half plane $\text{Re}(z)<0$
such that $y^{z}$ decays at infinity. The integrand vanishes factorially on
the arc at infinity due to the $\Lambda\quantity({1\over 2}-i\omega)$ in the
denominator. The poles inside the contour come from
$\Gamma\quantity(\frac{\epsilon+z}{2})$ at $z=-2k-\epsilon$ with
$k=0,1,\dots$. The integral (53) is then given as a sum over residues,

$\begin{split}&\frac{Z_{\rm string,0}^{\epsilon}(y)}{\sqrt{\pi}(\pi
j_{*})^{-\epsilon}}=\text{($k=0$ term) }+\\\
&\sqrt{y}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k!}\frac{4\sigma_{4k+2\epsilon}(j_{*})\Gamma(k+\epsilon)}{\Gamma({1\over
2}+\epsilon+2k)\zeta(1+2\epsilon+4k)}\quantity(\frac{\pi}{j_{*}y})^{2k+\epsilon}\end{split}$
(54)

where we used $\Lambda(s)=\pi^{-s}\Gamma(s)\zeta(2s)$ and the explicit Fourier
coefficients (48). We have separated the $k=0$ term because as we remove the
regulator $\epsilon\rightarrow 0$, two simple poles at $z=\pm\epsilon$
coalesce into a double pole at $z=0$, on the contour. We will thus treat
separately the $k>0$ terms, for which the regulator can be trivially removed,
and the $z=\pm\epsilon$ poles.

Let us now Laplace transform to the density of states,

$Z_{{\rm string},0}(y)=\sqrt{y}\int_{0}^{\infty}dt\,e^{-4\pi yt}\rho_{{\rm
string},0}(t).$ (55)

We have written the density in terms of the reduced twist, $t=\Delta/2-\xi$
for scalars, which is related to the density as a function of dimension
$\Delta$ through $\rho(\Delta)d\Delta=\rho(t)dt$. The regularized partition
function (54) gives a regularized density

$\begin{split}&\rho_{\rm string,0}(t)=\text{($k=0$ term) }+\\\
&{8\sqrt{\pi}\over
t}\sum_{k=1}^{\infty}\frac{(-1)^{k}\sigma_{4k}(j_{*})}{\Gamma(1+2k)\Gamma({1\over
2}+2k)\zeta(1+4k)}\quantity(\frac{4\pi^{2}t}{j_{*}})^{2k}\end{split}$ (56)

where we have removed the regulator in the second line. Using the identity

$\sigma_{z}(j)=\zeta(z+1)j^{z}\sum_{s=1}^{\infty}\frac{c_{s}(j)}{s^{z+1}}\,,$
(57)

swapping the order of the sums and performing some simplifications,

$\begin{split}\rho_{\rm string,0}(t)&=\text{($k=0$ term) }+\\\
&\frac{8}{t}\sum_{s=1}^{\infty}\frac{c_{s}(j_{*})}{s}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{(4k)!}\quantity(\frac{2\pi\sqrt{2\xi
t}}{s})^{4k}\,.\end{split}$ (58)

The second line can be resummed to reproduce (30), the continuous part of the
scalar density.

Finally, we return to the $k=0$ term, still with the regulator, which is
explicitly given by

$\begin{split}\rho_{\rm string,0}(t)\big{|}_{k=0}=\frac{4\sqrt{\pi}}{t(\pi
j_{*})^{\epsilon}}\frac{\sigma_{2\epsilon}(j_{*})}{\Gamma({1\over
2}+\epsilon)\zeta(1+2\epsilon)}\quantity(\frac{4\pi^{2}t}{j_{*}})^{\epsilon}\end{split}$
(59)

To regulate the divergence as $\epsilon\rightarrow 0$ and $t\rightarrow 0$,
similarly to [19] we integrate from $t=0$ up to some $t_{*}>0$ and use
$\epsilon\zeta(1+2\epsilon)\rightarrow{1\over 2}$ as $\epsilon\rightarrow 0$
to arrive at

$\lim_{\epsilon\rightarrow 0}\left[\int_{0}^{t_{*}}dt\rho_{\rm
string,0}(t)\big{|}_{k=0}\right]=8\sigma_{0}(j_{*}).$ (60)

Together with the continuous part of the density derived above, this
reproduces the full result (32).

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