Alpha states demystified
Towards microscopic models of AdS2 holography

Andreas Blommaert1, Luca V. Iliesiu2 and Jorrit Kruthoff$\,{}^{2}$

1SISSA, Via Bonomea 265, 34127 Trieste, Italy

2Department of Physics, Stanford University, Stanford, CA 94305, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We continue our study of factorizing theories of dilaton gravity,
characterized by a universal bilocal interaction. All such factorizing
theories can be shown to have discrete spectra, distinguished only by their
local dilaton potentials. We show how such theories can be used to construct
all alpha-states in the Hilbert space of baby universes of ordinary JT
gravity. Large classes of these theories with different local potentials are
found to be non-perturbatively equivalent and have identical discrete spectra.
This is a concrete example of how different bulk descriptions can give rise to
the same boundary theory. Such equivalences manifest themselves as null
states, which have to be quotiented out in order to construct a proper baby
universe Hilbert space. Our results also allow us to revisit the mechanism
discussed by Coleman, Giddings and Strominger and concretely link ensemble
averaging to the appearance or disappearance of spacetime wormholes.

We then investigate JT gravity deformed only by the universal bilocal
interaction. In this theory, the only terms that do not cancel in a
topological expansion are disks, which capture perturbative fluctuations
around a two-dimensional black hole saddle. We find that this theory of black
holes has an evenly spaced spectrum, instead of a quantum chaotic one. We
present a dual quantum mechanical system with exactly the same discrete
spectrum, and propose that this is an example of a new holographic duality
between a two-dimensional theory of quantum gravity and a conventional quantum
mechanics.

###### Contents

  1. 1 Introduction and summary
  2. 2 Review of gravity factorized
    1. 2.1 Geometric argument
    2. 2.2 Matrix integral argument
  3. 3 Alpha states in JT gravity
    1. 3.1 Constructing alpha-states
    2. 3.2 Checking orthogonality
    3. 3.3 The Coleman-Giddings-Strominger mechanism revisited
    4. 3.4 The baby universe Hilbert space of each factorizing model is one dimensional
  4. 4 Null states in JT gravity
    1. 4.1 Discreteness implies redundancies
    2. 4.2 Null states
    3. 4.3 Examples of null states and their spacetime interpretation
  5. 5 Black holes without quantum chaos
    1. 5.1 Orthogonal polynomials
    2. 5.2 No random matrix statistics
    3. 5.3 Quantum mechanical dual of canonical JT gravity
    4. 5.4 Bulk interpretation of null states
  6. 6 Conclusion and outlook

##  1 Introduction and summary

There is a widely held belief that UV complete theories of quantum gravity in
AdS are dual to individual non-gravitational quantum mechanical systems on the
boundary of AdS [1]. On the other hand, if we consider two copies of such a
boundary quantum system, in gravity, we are instructed to sum over all
geometries that end on two copies of such asymptotically AdS boundary. This
includes summing over wormhole geometries connecting the two boundaries. Those
have desirable properties, explaining many non-trivial predictions of the
boundary quantum mechanics from the bulk point of view, like the Page curve
[2, 3, 4], the non-decaying behavior of correlation functions [5, 6, 7],
complexity [8] and the lack of global symmetries [9, 10]. To explore such
features of gravity we seemingly need wormholes. However wormholes also
introduce a tension with the dual quantum mechanics, known as the
factorization puzzle - naively, wormholes imply a nontrivial variance for say
the two-boundary partition function $Z(\beta_{1},\beta_{2})\neq
Z(\beta_{1})Z(\beta_{2})$, in contradiction with the dual quantum mechanics
where partition functions are just ordinary numbers. This means that gravity
systems are naively dual to ensembles [11, 6, 2, 3, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35],
unless something cancels the variance in this ensemble. Since we believe that
UV complete theories are dual to individual quantum mechanical systems and
therefore do factorize, we should identify the effect that cancels the
variance due to wormholes.

In [36] we investigated, in a bottom-up approach, how factorization of UV
complete models trickles down to the low energy EFT. In models that reduce to
JT gravity in the IR, we found that factorization demands the existence of a
tiny but universal bilocal correction (in spacetime), which can counter the
variance due to wormholes. Schematically

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.65}{0.0}{0.0}{0.65}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.55}]{Factorization_leading_cancellation.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,=\,\,0\,,$ (1.1)

where the wiggly blue line represent the effect of the bilocal correction
which makes seemingly disconnected parts of the spacetime interact. The
bilocal is universal because the same interaction results in factorization for
all the dilaton gravities obtained by adding any possible local dilaton
potential to JT [37, 38]

$I[U_{\text{local}}]=-\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\,\Phi(R+2)-\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\,U_{\text{local}}(\Phi)+\text{universal
bilocal.}$ (1.2)

We find that this bilocal interaction always results in a discrete spectrum in
the gravitational path integral of (1.2): in other words, at least in our
models, factorization always implies spectral discreteness. This discrete
spectrum of black hole states is determined entirely by the local dilaton
potential $U_{\text{local}}(\Phi)$.

The construction of such theories gives us the opportunity to make a few
widely discussed concepts in quantum gravity rather concrete, which will be
the purpose of this paper. Our main results are the following:

  1. 1.

Explicit construction of alpha-states in JT gravity. The baby universe Hilbert
space of any theory can be constructed by acting on a state with no-
boundaries, typically called the Hartle-Hawking state, with boundary creation
operators. Alpha-states are the special states in this space that are
eigenfunctions of all such operators, leading to inner products in the baby
universe Hilbert space that factorize. Marolf and Maxfield [12] constructed
such states in a 2d topological model that does not exhibit any of the rich
physics of black holes. One of the achievement in this paper is to give a
concrete geometric construction for all the alpha-states of JT gravity, a
model which has served as a great exploration ground for black hole physics.
We find that each factorizing model of dilation gravity (1.2) can be mapped to
an _explicit_ operator $e^{-\widehat{I}_{\rm deform}[U_{\text{local}}]}$
(3.9), which is formed from boundary creation operators and can thus act on
the Hartle-Hawking state. When acting on this state, such operators create the
alpha-states of JT gravity

$e^{-I[U_{\text{local}}]}=e^{-I_{\rm JT}-I_{\rm
local}-I_{\text{nonlocal}}}\quad\Leftrightarrow\quad\ket{\alpha}=\mathcal{N}_{\alpha}\,e^{-\widehat{I}_{\rm
deform}[U_{\text{local}}]}\ket{\text{HH}}\,.$ (1.3)

Each alpha-state $\ket{\alpha}$ is associated to an energy spectrum that is
the same as that of the factorizing dilaton gravity with potential
$U_{\text{local}}$. All spectra can be obtained by choosing suitable
potentials $U_{\text{local}}$ and, therefore, we obtain a basis of alpha-
states that spans the entire baby universe Hilbert space. Formula (1.3) shows
two perspectives on alpha-states in JT. Either an alpha-state provides a
complicated prescription for the path integral where we sum over all
geometries that end on the many boundaries, present in $e^{-\widehat{I}_{\rm
deform}[U_{\text{local}}]}$. Alternatively, a more physical picture is that an
alpha-state can be viewed as not including additional boundaries, but as
specifying the spacetime action to $I[U_{\text{local}}]$ in (1.2) of a
factorizing dilaton gravity theory. The fact that these two perspectives are
the same is a type of open-closed duality that we shall explain in section 3.
Having the exact form of all alpha-states (1.3) allows to explicitly revisit
and shed light on the original proposals of Coleman, Giddings and Strominger
[39, 40]: wormholes can be “integrated-out” by taking an ensemble average over
couplings in bulk theories that preserve locality,111Not to be confused with
the ensemble average typically discussed in the boundary theory. or they can
be ”integrated-in” by considering an average in bulk theories where the
universal non-local term is present.222This latter ensemble average turns out
to be equivalent to the ensemble typically discussed in the boundary theory.

  2. 2.

Coexistence of different bulk descriptions yields null states. As described
above, theories with different $U_{\text{local}}$ can have identical spectra
and can therefore be thought of as quantum gravities with different actions
that have identical non-perturbative UV completions. In section 4 we present
and investigate large classes of such examples. Denoting the local potentials
of two theories with an identical energy spectrum by $U_{\text{local}}^{(1)}$
and $U_{\text{local}}^{(2)}$, the associated alpha-states will be
indistinguishable by any measurement, which means they are physically
equivalent. This causes the difference of these states to be null (meaning it
has zero overlap with all states, including itself)

$\left(\mathcal{N}_{\alpha^{(1)}}e^{-\widehat{I}_{\rm
deform}[U_{\text{local}}^{(1)}]}-\mathcal{N}_{\alpha^{(2)}}e^{-\widehat{I}_{\rm
deform}[U_{\text{local}}^{(2)}]}\right)\ket{\text{HH}}\sim 0\,.$ (1.4)

The physical Hilbert space is obtained after quotienting out such null states
[12]. This highlights a physical interpretation of null states, which was not
obvious in the multi-boundary description of Marolf and Maxfield. They
describe redundancies in the spacetime action of the gravitational path
integral: theories with different actions can be non-perturbatively
equivalent, even if they have different semiclassical descriptions. For
instance, they can even have different black hole solutions:

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.45}]{deformeddisk.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hskip 8.5359pt+\hskip
8.5359pt\text{subleading corrections}\quad=\quad\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.4}]{disk.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hskip 8.5359pt+\hskip
8.5359pt\text{ subleading corrections, }$ (1.5)

where on the left, we have schematically represented in the semi-classical
contribution to the partition function in theory 1, while on the right we
represented it in theory 2. The meaning of the subleading corrections will be
concretely addressed. This is an explicit example showing that _multiple bulk
descriptions can coexist_. This is a concrete realization in gravity of this
idea put forward in [41] in the context of the SYK model.

  3. 3.

A quantum mechanical dual for black holes without a chaotic spectrum. In
section 5 we study _canonical_ JT gravity, the factorized version of old-
school JT gravity: (1.2) with $U_{\text{local}}=0$. In the geometric
expansion, it only receives contributions from the disk geometry

$Z(\beta)=\quad\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.4}]{disk.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hskip 8.5359pt+\hskip
8.5359pt\text{ non-perturbative effects}\,,$ (1.6)

where this geometry captures fluctuations around JT gravity’s black hole
saddle.333This makes it similar to what [42, 43] call canonical JT gravity.
There the disk was defined to be the sole contribution. The important
distinction between our approach and that of [42, 43] is that we have a matrix
integral description which allows us to explore the discreteness of the
spectrum and the precise quantum mechanical dual of the bulk theory.
Nevertheless, we show via the deformed matrix integral that our version of
canonical JT gravity has a _discrete_ spectrum, instead of a continuous one
[42]. This spectrum is obtained from the following QM problem444This
Hamiltonian was also recently discussed in [44], however that paper concerns
only the ensemble averaged version of JT gravity, which does not factorize,
nor have a discrete spectrum. The point of [44] is that in some sense, even
within the ensemble, there is a “preferred” spectrum, whereas we are studying
the microscopic theory with that spectrum which can be explicitly determined
from the path integral in the gravitational theory.

${\textsf{H}_{\text{JT}}}\,\psi(x,\lambda)=\lambda\,\psi(x,\lambda)\,,\quad{\textsf{H}_{\text{JT}}}=-e^{-2\textsf{S}_{0}}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+u(x)\,,\quad\psi(0,\lambda)=0\,,$
(1.7)

where $u(x)$ is obtained by solving the JT string equation, which to leading
order is $\sqrt{u}I_{1}(2\pi\sqrt{u})=2\pi x$. Apart from the crucial
Dirichlet boundary condition at $x=0$ this is the same QM problem that one
considers in the orthogonal polynomials approach to matrix models [45, 46]
(see [47, 48, 49, 44] for more recent discussions of this approach in the
context of JT gravity). Surprisingly, since this should be a theory of black
holes [50], this spectrum does not have random matrix level statistics, but
rather, adjacent eigenvalues behave as,555Here, $\rho_{0,{\rm JT}}(\lambda)$
is the leading density of stats in JT gravity.

$\lambda_{i+1}-\lambda_{i}=\frac{1}{\rho_{0,{\rm JT}}(\lambda)}\,,$ (1.8)

and are thus (locally) evenly spaced. We also study the null deformations of
this theory in detail in the semiclassical limit and see that they give rise
to different semiclassical bulk descriptions.

##  2 Review of gravity factorized

The idea of [36] was to imagine starting from a factorizing UV complete bulk
theory. After integrating out all degrees of freedom in the decoupling limit
of a large near extremal black hole in such a theory, except for the 2d metric
$g$ and the dilaton $\Phi$, one should also obtain a factorizing theory in 2d.
Such a theory is given by JT gravity at leading order in $e^{\textsf{S}_{0}}$,
with $\textsf{S}_{0}$ the entropy of the black hole

$\displaystyle-I_{\rm
JT}=\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\,\Phi\,(R+2)+\int_{\partial\Sigma}\mathrm{d}u\sqrt{h}\,\Phi\,(K-1)+\textsf{S}_{0}\chi(\Sigma)\,,$
(2.1)

however it should also have both local and non-local corrections in its
effective action. The goal of [36] was to find what kind of non-local terms
are required in order for this 2d theory to factorize.

Surprisingly, the only non-local terms that result in a theory that factorizes
to all orders in a genus expansion was universal (it was independent of the
detailed spectrum of black hole microstates that the resulting factorizing
theory could have) and unique (other models do not factorize). Explicitly, the
theory factorizes exactly if the non-local interaction is purely bilocal

$\displaystyle-I_{\text{nonlocal}}$
$\displaystyle=-\frac{1}{2}\,e^{-2\textsf{S}_{0}}\,\int_{0}^{\infty}\mathrm{d}b\,b\int_{\Sigma_{1}}\mathrm{d}^{2}x_{1}\sqrt{g_{1}}\int_{\Sigma_{2}}\mathrm{d}^{2}x_{2}\sqrt{g_{2}}\,e^{-2\pi\left(\Phi_{1}+\Phi_{2}\right)}\,\cos(b\Phi_{1})\,\cos(b\Phi_{2})$
$\displaystyle=-\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,,$
(2.2)

where we introduced a basis of functions ${{\mathcal{O}}}_{\text{G}}(b,\Phi)$,
insertions of which correspond with inserting geodesic boundaries in the
gravitational path integral [36]

${{\mathcal{O}}}_{\text{G}}(b,\Phi)=e^{-\textsf{S}_{0}}\int\mathrm{d}^{2}x\sqrt{g}\,e^{-2\pi\Phi(x)}\cos\left(b\Phi(x)\right)\quad\Leftrightarrow\quad\raisebox{-15.649pt}{\includegraphics[scale={0.125}]{Geobdy.pdf}}\,\,\,b$
(2.3)

The most general factorizing dilaton gravity has besides this bilocal also a
local dilaton potential, which can be expanded in the same basis

$-I_{\rm local}-I_{\text{nonlocal}}=\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\rm
brane}(b)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)-\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,.$
(2.4)

As we shall see, the expansion coefficient $Z_{\rm brane}(b)$ does not
influence factorization, or the fact that the theory becomes discrete. Its
role will be to determine the specific discrete spectrum of the theory and
thus encode its “microstructure”.

###  2.1 Geometric argument

Suppose we compute the partition function in this deformed theory

$Z(\beta)=\frac{1}{\mathcal{Z}^{\rm
factorized}}\sum_{\begin{subarray}{c}\text{geometries with}\\\ \text{one
boundary}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{JT}-I_{\rm
local}-I_{\rm nonlocal}}\,,$ (2.5)

where $\mathcal{Z}^{\rm factorized}$ is a normalization factor given by the
gravitational path integral performed on geometries with no asymptotic
boundary,

$\mathcal{Z}^{\rm factorized}=\sum_{\begin{subarray}{c}\text{geometries
with}\\\ \text{no
boundaries}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{JT}-I_{\rm
local}-I_{\rm nonlocal}}\,.$ (2.6)

We now expand out $I_{\rm local}+I_{\rm nonlocal}$, just like one does with
interaction vertices when computing Feynman diagrams in an interacting QFT.
The Feynman rules are that we insert geodesic boundaries with _smeared_
boundary conditions, from expanding out $I_{\text{local}}$

$\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\rm
brane}(b)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\quad\Leftrightarrow\quad\raisebox{-17.07164pt}{\includegraphics[scale={0.12}]{onebrane.pdf}}$
(2.7)

Expanding out $I_{\rm nonlocal}$ gives _correlated_ geodesic boundaries, or
correlated branes

$\displaystyle-\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\quad\Leftrightarrow\quad\raisebox{-19.91692pt}{\includegraphics[scale={0.35}]{2Branebdy.pdf}}$
(2.8)

Expanding local interactions, one sums over all possible brane insertions
(2.7), schematically

$Z(\beta)=\raisebox{-28.45274pt}{\includegraphics[width=312.20692pt]{expression_local_only.pdf}}+\text{bilocal
contrib.}\,$ (2.9)

It is important to emphasize that one must in principle include disconnected
closed spacetimes, with no asymptotic boundaries. When only local interactions
are present, the partition function is normalized by dividing by such closed
spacetimes. However, the bilocal interactions (2.8) can connect closed
universes to universes that have an asymptotic boundary, and we end up with an
expansion of the type

$Z(\beta)=\raisebox{-25.60747pt}{\includegraphics[width=325.215pt]{expression_local_and_nonlocal.pdf}}\,.$
(2.10)

Crucially, the bilocal can also connect disconnected components of spacetimes
with different asymptotic boundaries, as in (1.1), this is the key mechanism
for factorization, as we now review.

Consider the sum of all connected contributions to the $n$-boundary
gravitational partition function. We can organize this sum by grouping
together geometries that share the part $\Sigma$ below

$\sum_{k=0}^{n}\sum_{\sigma_{k}^{n}}\;\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.9}{0.0}{0.0}{0.9}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{nBdyConnectedRelation.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad\;\;=\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.9}{0.0}{0.0}{0.9}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{nBdyConnectedRelation_Copy.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\sum_{k=0}^{n}(-1)^{k}{n\choose
k}=0\,.$ (2.11)

Here, $\Sigma$ could contain any number of handles and any numbers of branes
that could either be connected or disconnected. Using the explicit form of the
bilocal interaction (2.2), one finds that one can replace each connected brane
homotopic to an asymptotic boundary by minus a wormhole

= - (2.12)

From this, the rewriting of the sum as in (2.11) follows and therefore, the
connected contribution to the gravitational path integral vanishes, meaning
that the theory factorizes

$Z(\beta_{1},\,\dots,\,\beta_{n})=Z(\beta_{1})\,\dots\,Z(\beta_{n})\,.$ (2.13)

Applying this cancellation mechanism between wormhole geometries and bilocal
interactions, one finds that the computation of the gravitational path
integral with a single asymptotic boundary reduces to only the disk and the
spacetime with a single brane (called half-wormhole in [41, 35])

$Z(\beta)=\quad\raisebox{-28.45274pt}{\includegraphics[scale={0.35}]{Halfwormhole_contribution.pdf}}+\quad\text{non-
perturbative corrections.}$ (2.14)

The local dilaton potential $U_{\rm local}(\Phi)$ encoded in
$Z_{\text{brane}}(b)$ (2.4) enters only through this half-wormhole.

One might want to argue at this point that adding the bilocal term to the
action is equivalent to an ad-hoc removal of all wormhole geometries.666We
thank Juan Maldacena for bringing up this point. We emphasize that this is not
the case. The mechanism is a numerical cancellation between wormhole
amplitudes, and Feynman diagrams for bilocal interactions. Before expanding
out the interaction vertices in (2.5) we just have a non-local interacting
field theory and a sum over geometries. The fact that, in the end, the
calculation of expectation values in this theory simplifies to only two
integrals (2.14) is a happy and surprising conclusion. We hope that this
effective field theory which solves factorization provides an exploration
ground to understand the origin of the degrees of freedom that were integrated
out, in particular, those responsible for the bi-local interaction.

Perhaps more importantly, there is an important difference in formulas between
our theory and the ad-hoc theory with no wormholes. In our case one finds non-
perturbative corrections to (2.14). Indeed, as we now review, the deformation
(2.5) has an equivalent description in the matrix integral [11]. One finds
that the spectrum of (2.14) is _discrete_ because of the bilocal deformation,
regardless of the specific form of $Z_{\text{brane}}(b)$. This non-
perturbative effect can, as far as we can tell, not be derived in the ad-hoc
theory. We need wormholes, albeit cancelling ones, and their associated matrix
integral description for that.

###  2.2 Matrix integral argument

Observables in (undeformed) JT gravity can be computed using a matrix integral
[11]

$\mathcal{Z}_{\rm JT}=\int\mathrm{d}H\,e^{-L\Tr
V_{\text{JT}}(H)}=\prod_{i=1}^{L}\int_{\mathcal{{{\mathcal{C}}}}}\mathrm{d}\lambda_{i}\exp\bigg(-L\sum_{i=1}^{L}V_{\text{JT}}(\lambda_{i})+\sum_{i\neq
j}^{L}\log(\lambda_{i}-\lambda_{j})\bigg{missing})\,,$ (2.15)

where $\lambda_{i}$ are the random eigenvalues of $H$. JT gravity is a large
$L$ double scaling limit of this [11], but that does not play an important
role in this paper.

The dictionary between gravity and matrix integrals is that inserting
boundaries in the gravitational path integral corresponds with computing
expectation values of single-trace operators in the matrix integral - for
instance

$Z(\beta_{1}\dots\beta_{n})=\frac{1}{\mathcal{Z}_{\text{JT}}}\int\mathrm{d}H\,\Tr\left(e^{-\beta_{1}H}\right)\dots\Tr\left(e^{-\beta_{n}H}\right)e^{-L\Tr
V_{\text{JT}}(H)}=\left\langle\Tr(e^{-\beta_{1}H})\,\dots\,\Tr(e^{-\beta_{n}H})\right\rangle\,.$
(2.16)

In particular, inserting a geodesic boundary in the gravity path integral is
dual to inserting [36, 51]

${{\mathcal{O}}}_{\rm
G}(b)=\frac{2}{b}\Tr\cos(bH^{1/2})-\int_{0}^{\infty}\mathrm{d}E\,\rho_{0,{\rm
JT}}(E)\,\frac{2}{b}\cos(bE^{1/2})\quad\Leftrightarrow\quad\raisebox{-15.649pt}{\includegraphics[scale={0.12}]{Geobdy.pdf}}\,\,\,b$
(2.17)

So the expansion of the deformations in (2.5) corresponds in the matrix
integral with expanding out

$\displaystyle e^{-I_{\rm local}-I_{\rm nonlocal}}$
$\displaystyle\Leftrightarrow\quad\exp\bigg(\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,z_{\text{brane}}(b)+\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b_{1}\,b_{1}\int_{0}^{\infty}\mathrm{d}b_{2}\,b_{2}\,{{\mathcal{O}}}_{\rm
G}(b_{1}){{\mathcal{O}}}_{\rm
G}(b_{2})\,z_{\text{brane}}(b_{1},b_{2})\bigg{missing})\,,$ (2.18)

where the correspondence between the smearing functions $z_{\rm brane}(b)$ and
$z_{\rm brane}(b_{1},b_{2})$ and the local and nonlocal dilaton potentials,
written in terms of ${{\mathcal{O}}}_{\text{G}}(b,\Phi)$ as in (2.2) and
(2.4), is to be determined.

Naively this looks trivial, however there is one key subtlety in the
dictionary between gravity and random matrix theory. In the gravitational path
integral there are no degenerate cylinders (i.e. cylinders of zero surface
that end on two geodesics of equal length). However, in the matrix integral,
degenerate cylinders contribute contact terms in the correlators of the
operator ${{\mathcal{O}}}_{\rm G}(b)$.777The matrix integral assigns the first
term in $\left\langle{{\mathcal{O}}}_{\rm G}(b_{1}){{\mathcal{O}}}_{\rm
G}(b_{2})\right\rangle_{\text{conn}}=\delta(b_{1}-b_{2})/b_{1}+O(e^{-2\textsf{S}_{0}})$
to a degenerate cylinder. Consequently, they give a non-trivial contribution
in the expansion of (2.18) in the matrix integral (2.15). One should therefore
view $z_{\text{brane}}(b_{1},b_{2})$ as a _bare_ brane propagator, while the
propagator between branes seen through (2.2) should be understood as the
_dressed_ propagator. The dressed propagator (2.2) is obtained from the bare
propagator $z_{\text{brane}}(b_{1},b_{2})$ by resumming the Dyson series of
bare propagators connected via degenerate cylinders

(2.19)

In formulas this becomes

$\displaystyle-\frac{1}{b_{1}}\delta(b_{1}-b_{2})=z_{\text{brane}}(b_{1},b_{2})+\int_{0}^{\infty}\mathrm{d}b_{3}b_{3}\,z_{\text{brane}}(b_{1},b_{3})z_{\text{brane}}(b_{3},b_{2})+\dots$
(2.20)

whose solution one finds to be
$z_{\text{brane}}(b_{1},b_{2})=-q\delta(b_{1}-b_{2})/b_{1}$ with $q\to\infty$.
The degenerate cylinders similarly affect the relation between $z_{\rm
brane}(b)$ (red) and $Z_{\rm brane}(b)$ (blue), one obtains the Dyson equation

(2.21)

Plugging in the the solution for $z_{\text{brane}}(b_{1},b_{2})$ this equation
becomes

$Z_{\text{brane}}(b)=\lim_{q\to\infty}\frac{1}{1+q}\,z_{\text{brane}}(b)\quad\Rightarrow\quad
z_{\text{brane}}(b)=(q+1)\,Z_{\text{brane}}(b)\,.$ (2.22)

Thus according to (2.18) we should compute observables in the deformed matrix
integral

$\displaystyle\mathcal{Z}^{\text{factorized}}$
$\displaystyle=\int\mathrm{d}He^{-L\Tr
V_{\text{JT}}(H)+(q+1)\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,Z_{\text{brane}}(b)-\frac{q}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b){{\mathcal{O}}}_{\rm G}(b)\,}$ (2.23)

Looking at this one might think that the perturbative expansion in $q$ is
horrible, but it is not, precisely because we can always resum the Dyson
series (2.19).888We thank Steve Shenker for emphasizing this point. One then
just recovers the gravitational genus expansion of the previous subsection, as
in (2.10).

More importantly, instead of expanding out this matrix integral deformation
perturbatively, we can enforce the limit $q\to\infty$ directly in the matrix
integral action. Because we are being forced to take the $q\to\infty$ limit in
order to match the geometric expansion in the gravitational path integral, the
matrix integral in (2.23) _localizes_ at solutions to its saddle-point
equations. Doing the integrals over $b$ in (2.23), we find that the large $q$
saddle-point equations can be explicitly written as

$\forall\,i=1\dots L:\quad
LV^{\prime}[Z_{\text{brane}}](\lambda_{i})=\frac{1}{2\lambda_{i}}+2\sum_{j\neq
i}^{L}\frac{1}{\lambda_{i}-\lambda_{j}}\,,\quad
V[Z_{\text{brane}}]=V_{\text{JT}}+V_{\text{local}}[Z_{\text{brane}}]\,.$
(2.24)

The local deformation $V_{\text{local}}$ of the dilaton potential for generic
$Z_{\text{brane}}$ is worked out in section 4.1, where we need it explicitly.
This set of $L$ equations has (typically) a unique dominant saddle for the $L$
eigenvalues $\lambda_{i}$. The matrix integral localizes to this saddle, so
when we compute the partition function $Z(\beta)=\left\langle\Tr e^{-\beta
H}\right\rangle$ we find the discrete answer expected of a conventional QM
system

$Z(\beta)=\sum_{i=1}^{L}e^{-\beta\lambda_{i}}\,,$ (2.25)

where $\lambda_{i}$ are the saddle point of (2.24), these depend on the brane
one-point function through $V[Z_{\text{brane}}]$. It is important that when
computing observables in matrix integrals, we normalize by
$\mathcal{Z}_{\text{factorized}}$ from (2.23) before taking $q\to\infty$, such
that the $q$-and $\lambda_{i}$ dependent one-loop determinants cancel in the
expectation value of all observables.

##  3 Alpha states in JT gravity

In this section we explain how the discrete and factorizing models of dilaton
gravity with deformation $I_{\text{local}}+I_{\text{nonlocal}}$ (2.5) give
rise to the alpha-states of pure JT gravity (the theory with no deformations).
We give explicit expressions for these states in dilaton gravity variables,
but use the non-perturbative matrix integral setup to show that they indeed
form a basis of the baby universe Hilbert space.

Before getting there it will be useful to quickly review the construction of
the baby universe Hilbert space applied in AdS/CFT [12]. The idea is that one
can interpret multi-boundary gravitational path integrals as expectation
values of boundary creating operators in a no-boundary state
$\ket{\text{HH}}$. Explicitly in JT multi-boundary partition functions can be
viewed as

$Z(\beta_{1}\dots\beta_{n})=\bra{\text{HH}}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\text{HH}}\,\,\sim\sum_{\begin{subarray}{c}\text{geometries
with}\\\ \text{$n$
boundaries}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{\text{JT}}}\,,$
(3.1)

with $\widehat{Z}(\beta)$ an operator that creates a boundary with usual
asymptotic AdS2 boundary conditions [52, 53, 54]. Other boundary conditions
[55] have their own operators, they are linear combinations of
$\widehat{Z}(\beta)$. This is similar to how we compute correlators in QFT in
the path integral by summing over Feynman diagrams (left), or by computing
expectation values of $\widehat{\phi}$ in a vacuum by expanding out in
creation and annihilation operators (right) [21]. One key difference is that
here all creation operators $\widehat{Z}(\beta)$ are assumed to commute [12].

The states

$\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{k})\ket{\text{HH}}\,,$ (3.2)

span the Hilbert space of baby universes [12] and all gravitational path
integrals (3.1) can be viewed as inner products between two states in this
space. Since the $\widehat{Z}(\beta)$ commute, they can be simultaneously
diagonalized by so-called alpha-states

$\widehat{Z}(\beta)\ket{\alpha}=Z_{\alpha}(\beta)\ket{\alpha}\,.$ (3.3)

Gravity path integrals (3.1) then indeed naturally acquire the structure of an
ensemble average

$Z(\beta_{1}\dots\beta_{n})=\bra{\text{HH}}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\text{HH}}=\sum_{\alpha}P_{\alpha}\,Z(\beta_{1}\dots\beta_{n})_{\alpha}\,,\quad
P_{\alpha}=|\bra{\alpha}\ket{\text{HH}}|^{2}\,,$ (3.4)

and gravitational amplitudes within one alpha-state trivially factorize

$Z(\beta_{1}\dots\beta_{n})_{\alpha}=\bra{\alpha}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\alpha}=Z_{\alpha}(\beta_{1})\dots
Z_{\alpha}(\beta_{n})\,.$ (3.5)

Therefore, formally, the factorization puzzle can be resolved by stating that
factorizing gravity theories should correspond to alpha-states [12]. The
difficulty has simply been shifted to concretely construct the alpha-states in
a given model, and to understanding their gravitational interpretation. We
will do so now.999Eigenbranes are another way to construct alpha-states in JT
[14, 15], but their semiclassical interpretation is unclear, and null states
are not obvious. The current approach is more intuitive and should generalize
better to higher dimensions.

A second important point in constructing a baby universe Hilbert space is to
ensure that the inner product on it (provided by the gravitational path
integral as in (3.1)) is non-degenerate. This means that if there are null
states, they should be quotiented out. This will be the subject of section 4.

###  3.1 Constructing alpha-states

Following the results of [36] as reviewed above, it is straightforward to see
how we can construct explicit alpha-states. The only thing one has to do is to
start with the state without any boundaries, the Hartle-Hawking state, and act
on it such that we create the correlated geodesics boundaries discussed in
section 2. Specifically, we define an operator $\widehat{Z}_{\text{G}}(b)$ in
the baby universe Hilbert space, whose action is to create a geodesic boundary
of the type (2.3)

$\widehat{Z}_{\text{G}}(b)\quad\text{creates}\quad\raisebox{-15.649pt}{\includegraphics[scale={0.125}]{Geobdy.pdf}}\,\,\,b$
(3.6)

These operators allow us to construct a more convenient span for the baby
universe Hilbert space than (3.2)

$\widehat{Z}_{\text{G}}(b_{1})\dots\widehat{Z}_{\text{G}}(b_{k})\ket{\text{HH}}\,.$
(3.7)

We can now construct normalized alpha-states in gravity as follows,

$\ket{\alpha_{{\textsf{H}_{0}}}}=\mathcal{N}_{\textsf{H}_{0}}e^{-\widehat{I}_{\text{deform}}({\textsf{H}_{0}})}\ket{\text{HH}}\,,$
(3.8)

where the normalization $\mathcal{N}_{{\textsf{H}_{0}}}$ is determined in
section 3.2 and $\widehat{I}_{\text{deform}}({\textsf{H}_{0}})$ is just the
deformation in the gravity action
$I_{\text{local}}({\textsf{H}_{0}})+I_{\text{nonlocal}}$ (2.4) translated to
the baby universe operator formalism

$-\widehat{I}_{\text{deform}}({\textsf{H}_{0}})=-\frac{1}{2}\int\mathrm{d}b\,b\,\widehat{Z}_{G}(b)\,\widehat{Z}_{G}(b)+\int\mathrm{d}b\,b\,Z_{\text{brane}}(b,{\textsf{H}_{0}})\,\widehat{Z}_{G}(b)\quad\Leftrightarrow\quad-
I_{\text{local}}({\textsf{H}_{0}})-I_{\text{nonlocal}}\,.$ (3.9)

This is the explicit realization of the mapping between alpha-states and
factorizing theories of dilaton gravity that we announced in (1.3). The
operator $\widehat{Z}_{\text{G}}(b)$ in the second term creates a single
geodesic boundary of length $b$ which is smeared by the function
$Z_{\text{brane}}(b,{\textsf{H}_{0}})$. This function is chosen such that the
corresponding factorizing theory has a spectrum given by the eigen-energies of
a given Hamiltonian ${\textsf{H}_{0}}$.101010One concrete choice that
reproduces the spectrum of a theory with Hamiltonian ${\textsf{H}_{0}}$ is
[36]
$Z_{\text{brane}}(b,{\textsf{H}_{0}})=\frac{2}{b}\Tr\cos(b{\textsf{H}_{0}}^{1/2})-\int_{0}^{\infty}\mathrm{d}E\,\rho_{0,\rm{JT}}(E)\,\frac{2}{b}\cos(bE^{1/2})\,.$
(3.10) In section 4 we point out that this choice of
$Z_{\text{brane}}(b,{\textsf{H}_{0}})$ is not unique, large classes of other
functions $Z_{\text{brane}}(b,{\textsf{H}_{0}})$ result in the same spectrum.
The operator $\widehat{Z}_{G}(b)\widehat{Z}_{G}(b)$ in the first term creates
a correlated set (or an entangled set in the baby universe Hilbert space) of
two geodesic boundary conditions each with length $b$. Schematically, we can
thus represent $\ket{\alpha_{{\textsf{H}_{0}}}}$ as

$\ket{\alpha_{{\textsf{H}_{0}}}}=\mathcal{N}_{{\textsf{H}_{0}}}\exp\bigg(\hskip
5.69046pt\leavevmode\hbox to0pt{\vbox
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2.84544pt+\,\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
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1.42271pt\bigg{missing})\ket{{\rm HH}}\,,$ (3.11)

where the blue line connecting the two branes is given by the integral in
(3.9).

To prove that (3.8) are alpha-states it suffices to prove that

$\bra{\text{HH}}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\alpha_{\textsf{H}_{0}}}=\Tr\left(e^{-\beta_{1}{\textsf{H}_{0}}}\right)\dots\Tr\left(e^{-\beta_{n}{\textsf{H}_{0}}}\right)\bra{\text{HH}}\ket{\alpha_{\textsf{H}_{0}}}\,.$
(3.12)

Indeed, because
$\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\text{HH}}$ for all
values of $n$ and $\beta_{1}$,…,$\beta_{n}$, span the baby universe Hilbert
space, proving (3.12) for all $n$ is equivalent to proving that
$\ket{\alpha_{\textsf{H}_{0}}}$ are eigenstates of $\widehat{Z}(\beta)$, which
is indeed the defining property of alpha-states,

$\widehat{Z}(\beta)\ket{\alpha_{\textsf{H}_{0}}}=\Tr(e^{-\beta{\textsf{H}_{0}}})\ket{\alpha_{\textsf{H}_{0}}}\,.$
(3.13)

As discussed in the introduction, inner products in the baby universe Hilbert
space are obtained from the gravitational path integral with boundary
conditions associated to the boundary creation operators evaluated inside the
inner-product. For the inner product (3.12), modulo the
$\mathcal{N}_{{\textsf{H}_{0}}}$ in the alpha-state, the left-hand side is
given by the Euclidean path integral with asymptotic boundaries
$\beta_{1},\dots,\beta_{n}$, closing off smoothly or ending on any of the
boundaries in the alpha-state (3.11). As we saw in section 2, this path
integral is that of the deformed theory (2.5). Thus, we find that

$\displaystyle\bra{\text{HH}}\widehat{Z}(\beta_{1})\cdots\widehat{Z}(\beta_{n})\ket{\alpha_{{\textsf{H}_{0}}}}$
$\displaystyle=\frac{\mathcal{N}_{{\textsf{H}_{0}}}}{\mathcal{Z}_{\rm
JT}}\sum_{\begin{subarray}{c}\text{geometries}\\\ \text{with $n$
asymptotic}\\\
\text{boundaries}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{\text{JT}}-I_{\rm
local}-I_{\rm nonlocal}}\,.$ (3.14)

where we normalized $\ket{\text{HH}}$ such that
$\bra{\text{HH}}\ket{\text{HH}}=1$ and

$\mathcal{Z}_{\rm JT}=\sum_{\begin{subarray}{c}\text{geometries}\\\
\text{without asymptotic}\\\
\text{boundaries}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi
e^{-I_{JT}}\,.$ (3.15)

Using the fact that this gravitational path integral (3.14) factorizes with
the only remaining contributions being given by disks and half-wormholes as
well as the fact that the matrix integral yields a theory whose partition
function is $Z(\beta)=\Tr(e^{-\beta{\textsf{H}_{0}}})$, we obtain the desired
answer

$\displaystyle\bra{\text{HH}}\widehat{Z}(\beta_{1})\cdots\widehat{Z}(\beta_{n})\ket{\alpha_{{\textsf{H}_{0}}}}$
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$\displaystyle=\bra{\text{HH}}\ket{\alpha_{{\textsf{H}_{0}}}}\Tr(e^{-\beta_{1}{\textsf{H}_{0}}})\cdots\Tr(e^{-\beta_{n}{\textsf{H}_{0}}})\,.$
(3.16)

Here the factor $\left\langle\text{HH}|\alpha_{{\textsf{H}_{0}}}\right\rangle$
(modulo a factor $\mathcal{N}_{\textsf{H}_{0}}$ which is identical on the left
and right in this equation) corresponds to the contribution of closed
universes without asymptotic boundaries, which is (3.14) with $n=0$. The
factors of $1/\mathcal{Z}_{\rm JT}$ are identical left and right too.

This proves that (3.8) are indeed the alpha-states of JT gravity. Since all
possible resulting spectra can be obtained by appropriately choosing $Z_{\rm
brane}(b,\,{\textsf{H}_{0}})$ through the explicit construction in (3.10),
these alpha-states span the baby universe Hilbert space.111111One can check
that the alpha-states are also eigenstates of $\widehat{Z}_{\text{G}}(b)$.
This is a bit subtle due to the ambiguity related to the presence of
degenerate cylinder. The unambiguous way to check this is to reintroduce the
degenerate cylinders that feature in the matrix integral and carefully take
$q\to\infty$.

To summarize, the operators
$e^{-\widehat{I}_{\text{deform}}({\textsf{H}_{0}})}$ used to construct
$\ket{\alpha_{{\textsf{H}_{0}}}}$ can each be mapped to one of the factorizing
theories discussed in section 2. In turn, each such theory can be mapped to a
matrix integral deformation that localizes the integral over all matrices to
${\textsf{H}_{0}}$. Thus, we obtain the triality121212The matrix integral will
be defined further on as $\int\mathrm{d}He^{-L\Tr
V_{\text{JT}}(H)+qI(H,{\textsf{H}_{0}})}$.

BU Hilbert space $\displaystyle\widehat{I}_{\rm
deform}({\textsf{H}_{0}})=\frac{1}{2}\int\mathrm{d}b\,b\,\widehat{Z}_{\text{G}}(b)\,\widehat{Z}_{\text{G}}(b)-\int\mathrm{d}b\,b\,\widehat{Z}_{\text{G}}(b)\,Z_{\text{brane}}(b,{\textsf{H}_{0}})$
$\displaystyle\Leftrightarrow\quad$ dilaton gravity $\displaystyle
I_{\text{deform}}({\textsf{H}_{0}})=\frac{1}{2}\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)-\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,Z_{\text{brane}}(b,{\textsf{H}_{0}})$
$\displaystyle\Leftrightarrow\quad$ matrix integral $\displaystyle
qI(H,{\textsf{H}_{0}})=-\frac{q}{2}\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b)\,{{\mathcal{O}}}_{\text{G}}(b)+(q+1)\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b)\,Z_{\text{brane}}(b,{\textsf{H}_{0}})$
(3.17)

where the objects $\widehat{Z}_{\text{G}}(b)$,
${{\mathcal{O}}}_{\text{G}}(b,\Phi)$ and ${{\mathcal{O}}}_{\text{G}}(b)$
create geodesic boundaries in the different languages

BU Hilbert space $\displaystyle\widehat{Z}_{G}(b)\,\text{creates geodesic
boundary of length $b$}$ $\displaystyle\Leftrightarrow\quad$ dilaton gravity
$\displaystyle{{\mathcal{O}}}_{\text{G}}(b,\Phi)=e^{-\textsf{S}_{0}}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g(x)}\,e^{-2\pi\Phi(x)}\,\cos(b\Phi(x))$
$\displaystyle\Leftrightarrow\quad$ matrix integral
$\displaystyle{{\mathcal{O}}}_{\text{G}}(b)=\frac{2}{b}\Tr\cos(bH^{1/2})-\int_{0}^{\infty}\mathrm{d}E\,\rho_{0,{\rm
JT}}(E)\,\frac{2}{b}\cos(bE^{1/2})\,.$ (3.18)

###  3.2 Checking orthogonality

The alpha-states we have just constructed are eigenstates of
$\widehat{Z}(\beta)$ and should thus form a complete set of states for the
baby universe Hilbert space. This means the overlap between two alpha-states
should be a delta function with unit coefficient. Geometrically this is subtle
to confirm. However, orthogonality can easily be seen directly at the level of
the matrix integral, where the inner product is131313Taking $q$, $q^{\prime}$
to not be independent but rather be the same parameter in fact yields similar
results.

$\bra{\alpha_{\textsf{H}_{0}}}\ket{\alpha_{\textsf{H}_{0}}^{\prime}}=\mathcal{N}_{{\textsf{H}_{0}}}^{*}\mathcal{N}_{{\textsf{H}_{0}}^{\prime}}\lim_{q,q^{\prime}\to\infty}\int\mathrm{d}\Lambda\,e^{-L\tr
V_{\rm
JT}(\Lambda)}e^{qI(\Lambda,{\textsf{H}_{0}})+q^{\prime}I(\Lambda,{\textsf{H}_{0}}^{\prime})}\,,$
(3.19)

where the deformation of the action $I(\Lambda,{\textsf{H}_{0}})$ is

$qI(\Lambda,{\textsf{H}_{0}})=\frac{q}{2}\sum_{i\neq
j}^{L}\log(\lambda_{i}-\lambda_{j})+\frac{q}{2}\sum_{i=1}^{L}P(\lambda_{i},\lambda_{i})-(q+1)\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})+L\sum_{i=1}^{L}V_{\rm
JT}(\lambda_{i})\,,$ (3.20)

and $P(E,\lambda)$ given in (4.3) and $E_{i}$ are the eigenvalues of
${\textsf{H}_{0}}$. Let us now take $q$ to infinity first and then take
$q^{\prime}$ to infinity. Taking $q$ large localizes the eigenvalue integral
$\Lambda$ to the eigenvalues of ${\textsf{H}_{0}}$ (denoted by $\Lambda_{0}$),
so we get

$\displaystyle\lim_{q,q^{\prime}\to\infty}\int\mathrm{d}\Lambda e^{-L\tr
V_{\rm
JT}(\Lambda)}e^{qI(\Lambda,{\textsf{H}_{0}})+q^{\prime}I(\Lambda,{\textsf{H}_{0}}^{\prime})}=\lim_{q\to\infty}(\mathcal{Z}_{{\textsf{H}_{0}}}^{\text{factorized}})\lim_{q^{\prime}\to\infty}\left(\int\mathrm{d}\Lambda
e^{q^{\prime}I(\Lambda,{\textsf{H}_{0}}^{\prime})}\delta(\Lambda-\Lambda_{0})\right)$
(3.21)

where $Z_{{\textsf{H}_{0}}}^{\text{factorized}}$ is defined in (2.23) for a
single-trace deformation that results in a spectrum associated to the
Hamiltonian ${\textsf{H}_{0}}$. Now taking $q^{\prime}$ large as well, we
obtain

$\left\langle\alpha_{\textsf{H}_{0}}|\alpha_{{\textsf{H}_{0}}^{\prime}}\right\rangle=\mathcal{N}_{{\textsf{H}_{0}}}^{*}\mathcal{N}_{{\textsf{H}_{0}}^{\prime}}\frac{\lim_{q\to\infty}(\mathcal{Z}_{{\textsf{H}_{0}}}^{\text{factorized}})\lim_{q^{\prime}\to\infty}(\mathcal{Z}_{{\textsf{H}_{0}}^{\prime}}^{\text{factorized}})}{\mathcal{Z}_{\rm
JT}^{2}}\frac{\delta({\textsf{H}_{0}}-{\textsf{H}_{0}}^{\prime})}{P_{\rm
JT}({\textsf{H}_{0}})},$ (3.22)

where

$P_{\rm JT}({\textsf{H}_{0}})=\frac{1}{\mathcal{Z}_{\text{JT}}}e^{-L\Tr
V({\textsf{H}_{0}})}\,,\quad\mathcal{Z}_{\rm JT}=\int\mathrm{d}\Lambda
e^{-L\Tr V_{\rm JT}(\Lambda)}\,.$ (3.23)

is the (normalized) probability distribution of the original JT matrix
integral. To get a unit coefficient of the delta function we fix the
normalisation factor $\mathcal{N}_{{\textsf{H}_{0}}}$ to be (up to an
irrelevant phase)

$\mathcal{N}_{{\textsf{H}_{0}}}=\frac{\sqrt{P_{\rm
JT}({\textsf{H}_{0}})}\mathcal{Z}_{\rm
JT}}{\lim_{q\to\infty}(\mathcal{Z}_{{\textsf{H}_{0}}}^{\text{factorized}})}\,,$
(3.24)

which thus implies that

$\left\langle\alpha_{\textsf{H}_{0}}|\alpha_{{\textsf{H}_{0}}^{\prime}}\right\rangle=\delta({\textsf{H}_{0}}-{\textsf{H}_{0}}^{\prime})\,.$
(3.25)

This answer makes a lot of sense, since it means that finding a particular
alpha-state in the Hartle-Hawking state is given by $P_{\rm
JT}({\textsf{H}_{0}})$ [12]

$|\left\langle\alpha_{{\textsf{H}_{0}}}|\text{HH}\right\rangle|^{2}=|\mathcal{N}_{{\textsf{H}_{0}}}|^{2}\left(\frac{\lim_{q\to\infty}(\mathcal{Z}_{{\textsf{H}_{0}}}^{\text{factorized}})}{\mathcal{Z}_{JT}}\right)^{2}=P_{\rm
JT}({\textsf{H}_{0}})\,,$ (3.26)

which is indeed how one should interpret the Hartle-Hawking state. It is the
state in the baby universe Hilbert space that gives the ensemble, i.e.
correlators evaluated in that state are given by the pure JT gravity path
integral. Written differently, we have

$\ket{\text{HH}}=\int\mathrm{d}{\textsf{H}_{0}}\sqrt{P_{JT}({\textsf{H}_{0}})}\ket{\alpha_{{\textsf{H}_{0}}}}.$
(3.27)

###  3.3 The Coleman-Giddings-Strominger mechanism revisited

In this context, we want to revisit the mechanisms discussed by Coleman,
Giddings and Strominger to account for the effects of wormholes. Our results
makes precise how ensemble averaging can either be used to “integrate out”
wormholes, through the mechanism discussed in our previous paper [36], which
we called “fighting ensemble with ensemble”. Alternatively, we can “integrate
in” wormholes, through the mechanism similar to that discussed by Coleman,
Giddings and Strominger [39, 40, 12, 56], though in our example the
distribution of the required ensemble can be precisely determined and shown to
be non-Gaussian.

### Integrating out wormholes: fighting ensemble with ensemble

Any factorizing theory with the universal bilocal can alternatively be written
as a bulk ensemble average [36] over a Hubbard-Stratanovich field
$Q(b)$141414Here the $Q$ integral is along the imaginary axis. This is similar
to what was obtained in [57] in the context of SYK.:

$\displaystyle\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{\rm JT}-I_{\rm
local}-I_{\rm nonlocal}}$
$\displaystyle=\int\mathcal{D}g\,\mathcal{D}\Phi\,\mathcal{D}Q(b)\,e^{-I_{\rm
JT}+\int_{0}^{\infty}\mathrm{d}b\,b\,Q(b)\,{{\mathcal{O}}}_{\rm
G}(b,\Phi)}\,e^{\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b\,(Q(b)-Z_{\text{brane}}(b))^{2}}$
$\displaystyle=\left\langle\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{\rm
JT}+\frac{1}{2}\int\mathrm{d}^{2}x\sqrt{g}\,U_{\rm
local}(\Phi,Q(b))}\right\rangle_{\text{couplings}}\,,$ (3.28)

where the local dilaton potential is now given $U_{\rm
local}(\Phi,Q(b))=e^{-2\pi\Phi}\int db\,bQ(b)\cos(b\Phi(x))$ and we have to
ensemble average over the coupling $Q(b)$ with a Gaussian weight centered
around $Z_{\text{brane}}(b)$.

Thus, all the factorizing theories described in this paper can alternatively
be viewed as Gaussian ensembles of different local bulk theories of dilaton
gravity. The integral over $Q(b)$ then has the role of integrating-out
wormholes: while each theory in the ensemble has wormholes in its geometric
expansion, the resulting theory has only disks and half-wormhole remain in its
geometric expansion. Once again, all wormhole contributions cancel. This
implies that each alpha-states can also be viewed, not only to correspond to a
single gravitational theory that is non-local and factorizable, but also as an
ensemble average over bulk theories:

$\left\langle\int\mathcal{D}g\mathcal{D}\Phi e^{-I_{\rm
JT}+\frac{1}{2}\int\mathrm{d}^{2}x\sqrt{g}\,U_{\rm
local}(\Phi,Q(b))}\right\rangle_{\text{couplings}}\,\Leftrightarrow\qquad\ket{\alpha_{\textsf{H}_{0}}}=\left\langle
e^{\int\mathrm{d}b\,b\,Q(b)\,\widehat{Z}_{\text{G}}(b)}\right\rangle_{\text{couplings}}\ket{\text{HH}}\,.$
(3.29)

### Integrating in wormholes: an ensemble average over alpha-states

Alternatively, we can instead start with the alpha-states, which are described
by the factorizing theory with all wormhole contributions cancelling, and try
to integrate-in the wormholes. This is again given by an ensemble average that
is no longer Gaussian, but instead is given by the JT gravity matrix integral
itself. Explicitly, starting with the partition function
$Z_{\text{JT}}(\beta_{1},\dots,\beta_{n})=\bra{\text{HH}}\widehat{Z}(\beta_{1})\cdots\widehat{Z}(\beta_{n})\ket{\text{HH}}$
and introducing the resolution of the identity
$\mathbb{1}=\int\mathrm{d}{\textsf{H}_{0}}\ket{\alpha_{{\textsf{H}_{0}}}}\bra{\alpha_{{\textsf{H}_{0}}}}$
we can write,

$\displaystyle Z_{\rm
JT}(\beta_{1},\dots,\beta_{n})=\bra{\text{HH}}\widehat{Z}(\beta_{1})\cdots\widehat{Z}(\beta_{n})\ket{\text{HH}}$
$\displaystyle\,\,\,\,=\int\mathrm{d}{\textsf{H}_{0}}\,P_{\rm
JT}({\textsf{H}_{0}})\,Z(\beta_{1},{\textsf{H}_{0}})\cdots
Z(\beta_{n},{\textsf{H}_{0}}),$
$\displaystyle\,\,\,\,=\int\underbrace{\mathrm{d}{\textsf{H}_{0}}P_{\rm
JT}({\textsf{H}_{0}})}_{\begin{subarray}{c}\text{Average over}\\\ \text{bulk
couplings}\end{subarray}}\,\,\overbrace{\bigg{(}\leavevmode\hbox
to128.84pt{\vbox to44.28pt{\pgfpicture\makeatletter\hbox{\hskip
31.88799pt\lower-35.94543pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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theories with couplings fixed by }{\textsf{H}_{0}}}\,.$ (3.30)

Each term in the above expansion is a factorized theory in which all wormholes
can be cancelled, these theories depend via the brane coupling $Z_{\rm
brane}(b,{\textsf{H}_{0}})$ on ${\textsf{H}_{0}}$. The integral over
${\textsf{H}_{0}}$ can be viewed as an ensemble over couplings in the
factorized theories that is needed in order to “integrate in” the wormholes
present in the geometric expansion of $Z_{\rm JT}(\beta_{1},\dots,\beta_{n})$.
This is the mechanism discussed by Coleman, Giddings and Strominger [39, 40]
made precise in the context in JT gravity.

The results obtained above are not very surprising, because we start with a
factorizing theory from the beginning, so averaging over ${\textsf{H}_{0}}$
again should give the usual JT answers back. From this, one can already infer
(3.30). Nevertheless it is amusing to see that using the matrix integral
(together with the degenerate cylinders that cause the appearance of $q$) we
can derive these results in a very concrete way.

Let us emphasize the difference between the ensemble averages that integrate
out (3.29) and integrate in the wormholes (3.30): while the former is a simple
universal Gaussian path integral (whose origin is the universal form of the
bilocal interaction), the latter is a complicated non-Gaussian matrix integral
where we have to consider $e^{\textsf{S}_{0}}$ couplings. Integrating-in the
wormholes is much more complicated.151515On a technical level, this is due to
the fact that the contribution of geometries of a certain genus in JT gravity
depend on the complicated quotient by the mapping class group that has to be
performed in the gravitational path integral. In contrast, when integrating
out the wormholes the form of the mapping class group for a manifold with
wormholes of a given topology is unimportant, as discussed in [36].

###  3.4 The baby universe Hilbert space of each factorizing model is one
dimensional

We should contrast the description found above for the baby universe Hilbert
space of JT gravity, $\mathcal{H}_{\text{BU}}^{\text{JT}}$, with that of the
baby universe Hilbert space, $\mathcal{H}_{\text{BU}}^{\rm factorizing}$, of
the factorizing theories found in [36]. In the latter, for a factorizing
theory with some local deformation $I_{\rm local}$ and the universal bi-local
deformation $I_{\rm nonlocal}$, inner products with the Hartle-Hawking state
$\ket{{\text{HH}_{\rm fact}}}$ are computed as

$\displaystyle\frac{\bra{{\text{HH}_{\rm
fact}}}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{{\text{HH}_{\rm
fact}}}}{\bra{{\text{HH}_{\rm fact}}}\ket{{\text{HH}_{\rm fact}}}}$
$\displaystyle=\frac{1}{\mathcal{Z}^{\rm
factorizing}}\sum_{\begin{subarray}{c}\text{geometries}\\\ \text{with $n$
asymptotic}\\\
\text{boundaries}\end{subarray}}\int\mathcal{D}g\,\mathcal{D}\Phi\,e^{-I_{\rm
JT}-I_{\rm local}-I_{\rm nonlocal}}$
$\displaystyle=\Tr(e^{-\beta_{1}{\textsf{H}_{0}}})\dots\Tr(e^{-\beta_{n}{\textsf{H}_{0}}})\,,$
(3.31)

where we suppressed the fact that the definition of $\ket{{\text{HH}_{\rm
fact}}}$ depends on ${\textsf{H}_{0}}$. Consequently, in contrast to the case
of undeformed JT gravity, (3.31) shows that $\ket{{\text{HH}_{\rm fact}}}$ is
an alpha-state in $\mathcal{H}_{\text{BU}}^{\rm factorizing}$.

Are there other alpha-states or does $\ket{{\text{HH}_{\rm fact}}}$ span all
of $\mathcal{H}_{\text{BU}}^{\rm factorizing}$? To find the dimension of
$\mathcal{H}_{\text{BU}}^{\rm factorizing}$, we can compute the inner-product
between two arbitrary states in this Hilbert space161616We thank D. Stanford
and Z. Yang for useful comments in this direction.

$\langle\Psi_{1}|\Psi_{2}\rangle=\alpha^{\dagger}_{1}\cdot\mathcal{Z}\cdot\alpha_{2}\,,\qquad\text{
where
}\qquad|\Psi_{a}\rangle=\sum_{i=0}^{n_{\text{bdies}}}\alpha^{i}_{a}\,\widehat{Z}(\beta)^{i}|{\text{HH}_{\rm
fact}}\rangle\,,$ (3.32)

where $n_{\text{bdies}}$ is the maximum number of boundaries involved in the
state $|\Psi_{a}\rangle$ (which can be taken $\infty$) and where, for
simplicity, we restrict to creating asymptotic boundaries with the same
$\beta$. (3.31) implies that the inner-product matrix (whose dimension is
$n_{\text{bdies}}\times n_{\text{bdies}}$) is

$\mathcal{Z}_{ij}=\Tr(e^{-\beta{\textsf{H}_{0}}})^{i+j}$ (3.33)

The rank of $\mathcal{Z}$ gives the dimension of $\mathcal{H}_{\text{BU}}^{\rm
factorizing}$, and the above matrix has rank one, because it can be written as
the square of a vector. Following a similar construction for inner products of
states created by inserting asymptotic boundaries with different $\beta_{i}$,
one finds that $\mathcal{H}_{\text{BU}}^{\rm factorizing}$ is one-dimensional.
So $|{\text{HH}_{\rm fact}}\rangle$ is the unique state in
$\mathcal{H}_{\text{BU}}^{\rm factorizing}$, which is radically different from
the infinite dimensional $\mathcal{H}_{\text{BU}}^{\rm JT}$. The factorizing
theory with BU Hilbert space $\mathcal{H}_{\text{BU}}^{\rm factorizing}$ is
associated with one specific alpha-state $\ket{\alpha_{\textsf{H}_{0}}}$ in
$\mathcal{H}_{\text{BU}}^{\text{JT}}$, therefore $\mathcal{H}_{\text{BU}}^{\rm
factorizing}$ is the subspace of $\mathcal{H}_{\text{BU}}^{\text{JT}}$ spanned
by $\ket{\alpha_{\textsf{H}_{0}}}=\ket{{\text{HH}_{\rm fact}}}$.

To emphasize, the non-trivial point result is that we find that the dimension
of the BU Hilbert space in the factorizing models of [36] is one dimensional,
a property that had been predicted for UV complete theories in [12, 58].

##  4 Null states in JT gravity

In the previous section we constructed explicit alpha-states and showed that
they form a complete basis for the baby universe Hilbert space, because they
cover the whole ensemble of random Hamiltonians ${\textsf{H}_{0}}$. When
constructing any Hilbert space one needs to quotient by potential null states.
However, since we used the matrix integral to show that the alpha-states (3.8)
form a complete basis, there are simply no null states to account for in the
matrix integral. In a sense, they have already been eliminated since each
different matrix in the ensemble has a different spectrum. This leads to the
conclusion that the matrix integral only knows about the Hilbert space _after_
the quotienting has been performed, also called GNS Hilbert space.

Nevertheless, in situations where we do not have the luxury of considering the
matrix integral or something equivalent as for instance would be the case in
higher dimensions, the question remains whether there is a geometric or
perhaps semiclassical understanding of the null states. In general this is a
hard question to answer and hints at the mechanism emphasized in [41], where
it was argued in the context of the $G\Sigma$ action of SYK that there could
be multiple bulk descriptions which ultimately give the same physics. In our
model we can make this mechanism very precise by studying two alpha-states
that within the matrix model give the same boundary spectrum, but when going
to a geometric picture or in a naive semiclassical approximation, there is a
clear difference between the two. In fact, we will show that there is an
infinite number of geometric theories that give rise to the same boundary
spectrum. Subtracting (or taking the appropriate linear combinations of)
alpha-states associated to theories that have the same spectrum yields null
states that are not part of the baby universe Hilbert space.

###  4.1 Discreteness implies redundancies

The key to understanding why two superficially different alpha-states give the
same boundary spectrum is the discreteness of the spectrum. Namely, this
implies that as long as we localize on the chosen spectrum at large $q$ we can
change the deforming operator in the matrix integral to whatever way we wish.
A bit more concretely, when we alter $Z_{\text{brane}}$ in such a way that the
localization equations (2.24) remain the same and we do not spoil the
stability of the saddle (i.e. the Hessian around the saddle has the correct
sign for all its eigenvalues), the effect in the matrix integral is absolutely
nothing. No observable will change whatsoever, we call such changes in
$Z_{\text{brane}}$ _null deformations_. The name of the game is then to see
whether we can translate this change back to an ordinary geometric
understanding.

To do so, let us first map out a subset of all the null states and translate
those back to deformations of the Euclidean path integral. To simplify the
discussion we consider as our initial theory JT gravity with just the bilocal
deformation turned on, so $I_{\rm local}=0$ in (2.4). Generalizations to cases
with non-zero $Z_{\rm brane}$ as the initial theory are straightforward. We
then consider general deformations around this theory, and seek for
deformations that do not affect the spectrum.

General deformations around the theory with $I_{\rm local}=0$ (which we call
canonical JT gravity) correspond with the following insertion in the JT matrix
integral

$\displaystyle\exp\bigg((q+1)\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,\delta
Z_{\text{brane}}(b)-\frac{q}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b){{\mathcal{O}}}_{\rm G}(b)\bigg{missing})\,,$ (4.1)

with $q\to\infty$ and $\delta Z_{\text{brane}}(b)$ parameterized as171717The
dilaton potential and the genus zero spectral density are linearly related
when the latter vanishes at $E=0$ [59, 60]. In those cases $\delta\rho_{0}(E)$
is the change in the genus zero spectral density if one would not turn on the
bilocal. In other cases, the would-be genus zero spectral density
$\rho_{0,\text{JT}}(E)$ would receive subleading corrections in the
deformation of the action. None of this is relevant in our factorizing models,
where localization determines the discrete spectrum.

$\delta
Z_{\text{brane}}(b)=\int_{0}^{\infty}\mathrm{d}E\,\delta\rho_{0}(E)\,\frac{2}{b}\cos(bE^{1/2})\,.$
(4.2)

Here $\delta Z_{\rm brane}$ is actually of the most general form, since
$b\,Z_{\rm brane}(b)$ needs to be an even function of $b$. For this particular
deformation to not change anything about the spectrum $\delta Z_{\rm brane}$
needs to satisfy some conditions. These conditions can be derived by
considering the large $q$ saddle-point equations for the discrete set of
eigenvalues $\lambda_{i}$181818Here $P(\lambda,E)$ is a regularized version of
$2\log\absolutevalue{\lambda-E}$ with the regulator only relevant when
$\delta\rho_{0}(E)$ has delta spikes [36] $\displaystyle
P(\lambda,E)=\log\Big(\left(E_{1}^{1/2}-E_{2}^{1/2}\right)^{2}+\varepsilon^{2}\Big{missing})+\log\Big(\left(E_{1}^{1/2}+E_{2}^{1/2}\right)^{2}+\varepsilon^{2}\Big{missing})\,,$
(4.3)

$LV_{\text{JT}}^{\prime}(\lambda_{i})+\int_{0}^{\infty}\mathrm{d}E\,\delta\rho_{0}(E)\,\partial_{\lambda}P(\lambda,E)\rvert_{\lambda=\lambda_{i}}=\frac{1}{2\lambda_{i}}+2\sum_{j\neq
i}\frac{1}{\lambda_{i}-\lambda_{j}}\,.$ (4.4)

One can view the second term as introducing a change in the electrostatic
potential for the eigenvalues

$L(V_{\text{JT}}^{\prime}(\lambda_{i})+\delta
V^{\prime}(\lambda_{i}))=\frac{1}{2\lambda_{i}}+2\sum_{j\neq
i}\frac{1}{\lambda_{i}-\lambda_{j}}\,.$ (4.5)

For smooth functions $\delta\rho_{0}(E)$ you recover the standard relation
[11, 61, 36] between changes in the matrix integral potential and changes in
the disk spectral density

$L\delta
V^{\prime}(\lambda)=2\fint_{0}^{\infty}\mathrm{d}E\,\delta\rho_{0}(E)\,\frac{1}{\lambda-E}\,.$
(4.6)

where $\fint$ is the principal value integral.

Let us define the eigenvalues $\lambda_{\text{JT}\,i}$ of a matrix
${\textsf{H}_{\text{JT}}}$ to be the solutions of the electrostatics problem
for canonical JT gravity

$LV_{\text{JT}}^{\prime}(\lambda_{\text{JT}\,i})=\frac{1}{2\lambda_{\text{JT}\,i}}+2\sum_{j\neq
i}\frac{1}{\lambda_{\text{JT}\,i}-\lambda_{\text{JT}\,j}}\,.$ (4.7)

Suppose now that we consider deforming this theory by adding fine-tuned
changes in the potential that depend specifically on the
$\lambda_{\text{JT}\,i}$ in such a way that the deformation satisfies

$L\delta V^{\prime}(\lambda_{\text{JT}\,i})=0\,,$ (4.8)

whilst not spoiling the signs of the eigenvalues of the Hessian, then the
electrostatics problem (4.5) has exactly the same solutions
$\lambda_{\text{JT}\,i}$. This means that the large $q$ matrix integrals with
and without these _null deformation_ are equivalent, as they localize onto the
same eigenvalues. In particular all observables calculated in the matrix model
at large $q$ will agree, for instance in both theories

$Z(\beta_{1}\dots\beta_{n})=\Tr(e^{-\beta_{1}{\textsf{H}_{\text{JT}}}})\dots\Tr(e^{-\beta_{n}{\textsf{H}_{\text{JT}}}})\text{
with or without null deformation}\,.$ (4.9)

This means that in the large $q$ matrix integral with the bilocal interaction,
there is a vast redundancy in description when specifying the theory by its
potential $V(\lambda)$

$\displaystyle V_{\text{JT}}(\lambda)\quad\text{ indistinguishable from }\quad
V_{\text{JT}}(\lambda)+\delta V_{\text{null}}(\lambda)\,$ (4.10)

The fundamental reason for this redundancy is that in the large $q$ matrix
integral we are only imposing a _discrete_ set of equations of motion (4.5).
For instance, a non-trivial solution can be obtained by solving (4.6)

$\fint_{0}^{\infty}\mathrm{d}E\,\delta\rho_{0\,\text{null}}(E)\,\frac{1}{\lambda-E}=G(\lambda)\,,\quad
G(\lambda_{\text{JT}\,i})=0\,.$ (4.11)

If we find a function $G(\lambda)$ with zeros on the spectrum
$\lambda_{\text{JT}\,i}$ we can invert this relation using standard methods
for singular integral equations [62] provided $G$ decays sufficiently fast to
zero at large $\lambda$, from that we can determine a null deformation
$Z_{\text{brane\,null}}$ using (4.2), which in turn determines a
$I_{\text{local\,null}}$ in the dilaton gravity action via (2.4).

We show in section 4.3 that there is one deformation
$\delta\rho_{0\,\text{null}}(E)$ corresponding to the trivial solution
$G(\lambda)=0$. All other null deformations will be immediate consequences of
discreteness, the fact that we can have many nontrivial functions $G(\lambda)$
that vanish on a _discrete_ set of points $\lambda_{\text{JT}\,i}$.

###  4.2 Null states

In section 3, we have described how $\mathcal{H}_{\rm BU}^{\rm JT}$ can be
constructed by acting on $\ket{\rm HH}$ with boundary creation operators and
then taking linear combinations of the resulting states,

$\widehat{Z}_{G}(b_{1})\dots\widehat{Z}_{G}(b_{n})\ket{\rm HH}\,.$ (4.12)

As pointed out by Marolf and Maxfield [12], in this construction of
$\mathcal{H}_{\rm BU}^{\rm JT}$, we need to be careful to eliminate the linear
combinations that have zero norm, i.e. the possible null states in this
construction.

The deformations of the matrix integral that we have described above, that
leave the localization locus invariant, have a natural interpretation in terms
of null states. As mentioned earlier, to appreciate this relation with null
states it is essential to distinguish the large $q$ matrix integral
perspective, where degenerate cylinders are included, from the spacetime path
integral, where degenerate cylinders do not exist. In the large $q$ matrix
integral, we are interested in the deformation

$qI(H,{\textsf{H}_{\text{JT}}},\mathsf{s})=-\frac{q}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,{{\mathcal{O}}}_{\rm
G}(b)+(q+1)\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,Z_{\text{brane}\,\text{null}}(b,{\textsf{H}_{\text{JT}}},\mathbf{s})\,,$
(4.13)

with

$Z_{\text{brane}\,\text{null}}(b,{\textsf{H}_{\text{JT}}},\mathbf{s})=\sum_{a}s_{a}\,Z^{(a)}_{\text{brane}\,\text{null}}(b,{\textsf{H}_{\text{JT}}})$
(4.14)

with $a$ labelling all solutions $\delta\rho^{(a)}_{0\,\text{null}}$ of (4.11)
that gives the null-state deformation
$Z^{(a)}_{\text{brane}\,\text{null}}(b,{\textsf{H}_{\text{JT}}})$, and $s_{a}$
some complex numbers whose range is constrained so as to ensure stability of
the large $q$ saddle point. In the baby universe Hilbert space language these
correspond via (3.17) with the states

$\ket{\alpha_{{\textsf{H}_{\text{JT}}}},\mathbf{s}}=\mathcal{N}_{{\textsf{H}_{\text{JT}}}\,\mathsf{s}}e^{-\widehat{I}_{\rm
deform}({\textsf{H}_{\text{JT}}},\mathsf{s})}\ket{\text{HH}}\,,$ (4.15)

with

$\widehat{I}_{\rm
deform}({\textsf{H}_{\text{JT}}},\mathsf{s})=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b\,b\,\widehat{Z}_{\text{G}}(b)\,\widehat{Z}_{\text{G}}(b)-\int_{0}^{\infty}\mathrm{d}b\,b\,\widehat{Z}_{\text{G}}(b)\,Z_{\text{brane}\,\text{null}}(b,{\textsf{H}_{\text{JT}}},\mathbf{s})$
(4.16)

The null states $\ket{\psi}$ are hence given by differences of alpha-states
with different values of parameters $s_{a}$

$\ket{\alpha_{\textsf{H}_{\text{JT}}},\mathbf{s}}-\ket{\alpha_{\textsf{H}_{\text{JT}}},\mathbf{s}^{\prime}}\,\text{
is a null-state if $\mathbf{s}\neq\mathbf{s}^{\prime}$}\,,$ (4.17)

or more generally if there are $k$ parameters $s_{a}$ and with $\Gamma$
denoting the stable deformations

$\int_{\Gamma}\mathrm{d}^{k}\mathbf{s}\,f(\mathbf{s})\ket{\alpha_{\textsf{H}_{\text{JT}}},\mathbf{s}}\,\text{
is a null-state if }\int_{\Gamma}\mathrm{d}^{k}\mathbf{s}\,f(\mathbf{s})=0\,.$
(4.18)

This is simply because

$\int_{\Gamma}\mathrm{d}^{k}\mathbf{s}\,f(\mathbf{s})\bra{{\rm
HH}}\widehat{Z}(\beta_{1})\dots\widehat{Z}(\beta_{n})\ket{\alpha_{\textsf{H}_{\text{JT}}},\mathbf{s}}=\Tr\left(e^{-\beta_{1}{\textsf{H}_{\text{JT}}}}\right)\dots\Tr\left(e^{-\beta_{n}{\textsf{H}_{\text{JT}}}}\right)\int_{\Gamma}\mathrm{d}^{k}\mathbf{s}\,f(\mathbf{s})=0\,,$
(4.19)

for all $n$. Once again, using the fact that
$\widehat{Z}(\beta_{1}),\,\dots,\,\widehat{Z}(\beta_{n})\ket{{\rm HH}}$ span
the Hilbert space, it then follows that (4.17) and (4.18) are null.

We remark here that even though from the matrix model perspective (as
$q\to\infty$) the $\alpha$-states
$\ket{\alpha_{{\textsf{H}_{\text{JT}}}},\mathbf{s}}$ do not depend on $s_{a}$,
they will actually lead to _inequivalent_ gravitational descriptions of the
bulk. Said differently, since in gravity we do not consider the degenerate
cylinders, the parameter $q$ does not appear and the operators with or without
null deformations turned on are different, as in section 4.3.

These null states described here are quite similar to the examples constructed
by Marolf and Maxfield [12] in pure $2$d Einstein-Hilbert gravity. In that
model one can construct null deformations analogous to our (4.16) by inserting
in the gravity path integral for an alpha-state the operator

$\exp\bigg(\mathrm{i}\sum_{n=1}^{\infty}s_{n}\,2\pi
n\widehat{Z}\bigg{missing})\,.$ (4.20)

These are null deformations because

$\widehat{Z}\ket{m}=m\ket{m}\,.$ (4.21)

whose analog we will describe shortly. In both our and their models we can
have null deformations because there is a fundamental discreteness in the
spectrum of the theory. Nevertheless, in our case we will see a rich set of
null states that can depend on the exact details (not only discreteness) of
the alpha-states whose null deformations we study.

###  4.3 Examples of null states and their spacetime interpretation

One benefit that we have, as compared to the simple model of [12], is that we
can map null deformations to changes in the dilaton gravity potential. Indeed,
using (3.17) we see that there are inequivalent dilaton gravity actions
associated with our class of matrix integrals (4.13)

$I_{\text{deform}}({\textsf{H}_{\text{JT}}},\mathsf{s})=\frac{1}{2}\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)-\int\mathrm{d}b\,b\,{{\mathcal{O}}}_{\text{G}}(b,\Phi)\,Z_{\text{brane}}(b,{\textsf{H}_{\text{JT}}},\mathsf{s})\,,$
(4.22)

which can be written explicitly in dilaton and metric variables as in (2.2)
and (2.4) using

$\displaystyle{{\mathcal{O}}}_{\text{G}}(b,\Phi)=e^{-\textsf{S}_{0}}\int\mathrm{d}^{2}x\sqrt{g}\,e^{-2\pi\Phi(x)}\cos\left(b\Phi(x)\right)\,.$
(4.23)

The change in the dilaton action always involves the bilocal term to ensure
factorization and is universal, but changes to the local dilaton potential
itself can be null. This means that gravitational path integrals with two
potentials related by a null deformation are inequivalent perturbatively, but
once the matrix model non-perturbative effects are included they are
equivalent.

For instance, if one wants to calculate the semiclassical entropy of the black
hole in two null-related dilaton gravities, one would obtain identical answer
to leading order, but they would start deviating at subleading orders in
$e^{\textsf{S}_{0}}$. We expand more on this in section 5.4.

### A first example: intuitive null operators and null states

A null operator can be constructed by using the discreteness of the spectrum
and inserting,191919We thanks Douglas Stanford and Henry Maxfield for bringing
this up.

$e^{2\pi\mathrm{i}k\widehat{N}(E_{1},E_{2})}=\exp\bigg(2\pi\mathrm{i}k\int_{E_{1}}^{E_{2}}\mathrm{d}E\,\widehat{\rho}(E)\bigg{missing})\,\qquad\text{
for }k\in\mathbb{Z}\,,$ (4.24)

When acting on any alpha-state we expect this operator gives one because the
spectrum is discretized and consequently the eigenvalues of
$\widehat{N}(E_{1},E_{2})$ are given by $N(E_{1},E_{2})\in\mathbb{Z}^{+}$.
Consequently,

$\left(e^{2\pi\mathrm{i}k\widehat{N}(E_{1},E_{2})}-\mathbb{1}\right)\ket{\alpha_{\textsf{H}_{0}}}\sim
0\,,$ (4.25)

is null for all ${\textsf{H}_{0}}$ for which $E_{1}$ and $E_{2}$ do not belong
to the spectrum.202020We should be careful when $E_{1}$ and $E_{2}$ are part
of the spectrum of ${\textsf{H}_{0}}$. In such corner cases, the operator in
(4.25) might not be null depending on the definition of the integral in
(4.24). Thus, perhaps it would be better to call this operator almost null
since there is a set of measure zero set of states for which its action might
not result in a null state.

We can now see where this operator comes from in our formalism and whether the
null states that it generates can indeed be though of as those described in
section 4.1 and 4.2. First, to define $\widehat{\rho}$ we take an inverse
Laplace transform of $\widehat{Z}(\beta)$,

$\widehat{\rho}(E)=\rho_{0,{\rm
JT}}(E)+\int_{0}^{\infty}b\mathrm{d}b\frac{\cos(b\sqrt{E})}{2\pi\sqrt{E}}\widehat{Z}_{G}(b)$
(4.26)

where we remind the reader that $\rho_{0,{\rm JT}}(E)$ is the leading density
of states in JT gravity given by the disk contribution. If we now insert this
in (4.24) we can read off what $\delta Z_{\rm brane}(b)$ needs to be by
brining the integral over $E$ inside. The constant coming from the smooth disk
density of states will give a constant that is the same for all our alpha-
states and can be factored out. The result for $\delta Z_{\rm brane}(b)$ is

$\delta Z_{\rm
brane}(b)=2\mathrm{i}\,k\,\frac{\sin(b\sqrt{E_{1}})-\sin(b\sqrt{E_{2}})}{b}.$
(4.27)

Inverting the relation (4.2) we get a $\delta\rho_{0}(E)$ given by

$\delta\rho_{0}(E)=\frac{4\mathrm{i}\,k\,\sqrt{E}}{\pi}\left[\frac{\sqrt{E_{2}}}{E_{2}-E}-\frac{\sqrt{E_{1}}}{E_{1}-E}\right].$
(4.28)

To check whether this is indeed a null deformation as we have defined it in
(4.8), we simply need to calculate (4.6) and show that it gives a constant
shift in the potential. This is indeed the case as long as $\lambda\neq
E_{i}$, which is always the case since we want to integrate the density in
(4.24) always in between the delta peaks not on top of them and so we always
have $\delta V^{\prime}(\lambda_{\text{JT}\,i})=0$ as required. When
$k\in\mathbb{Z}$, the state
$e^{2\pi\mathrm{i}k\widehat{N}(E_{1},E_{2})}\ket{\alpha_{\textsf{H}_{0}}}$ is
unit normalized and thus the general framework described in section section
4.1 and 4.2 indeed describes the null states in (4.25). More generally, we can
construct null operators by considering functions of
$\widehat{N}(E_{1},E_{2})$ that vanish on the integers - since such functions
can be constructed by taking linear combinations of the operators in (4.24)
the null states described by such operators are simply linear combinations of
those found in (4.25). However, not all null states can be obtained from null
operators and we give two examples of such cases below.212121This is the case
even when taking $k\,\not{\in}\,\mathbb{Z}$. In such a case, (4.25) needs to
be modified by multiplying $e^{2\pi\mathrm{i}k\widehat{N}(E_{1},E_{2})}$ by a
normalization factor that is dependent on the specific alpha-state upon which
we act.

### A second trivial solution

To get some feeling for the null deformations, we first consider the case
$G(\lambda)=0$ which has solutions [62]

$\delta\rho_{0\,\text{null}}(E)=s\,\frac{1}{E^{1/2}}.$ (4.29)

Plugging this in (4.11), this gives zero, so that this particular $\delta\rho$
induces a constant shift of the potential

$\delta V(\lambda)=\text{ constant }\quad\Rightarrow\quad\delta
V^{\prime}(\lambda)=0\,,$ (4.30)

so there is no change to the saddle point equations and the spectrum on which
we localize is not affected. The brane one-point function associated to this
deformation of the matrix model potential is

$Z_{\rm
brane\,null}(b)=s\,\int_{0}^{\infty}\mathrm{d}E\,\frac{2}{b}\frac{1}{E^{1/2}}\cos(bE^{1/2})=s\,\frac{\delta(b)}{b}.$
(4.31)

In the dilaton gravity action this deformation gives rise to a change of the
dilaton potential of the form

$\delta
U(\Phi)=s\,e^{-\textsf{S}_{0}}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\,e^{-2\pi\Phi}\,.$
(4.32)

Geometrically, this corresponds to the insertion of a gas of cusps. The point
is that we have shown here that non-perturbatively, when we include the
bilocal interaction, this gas of cusps no effects whatsoever on the resulting
discrete spectrum.

Following the description in section 4.2, from the deformation (4.32) we find
the following null states for example

$\left(\mathcal{N}_{{\textsf{H}_{\text{JT}}}\,0},\,e^{-\frac{1}{2}\int\mathrm{d}b\,b\,\widehat{Z}_{G}(b)\,\widehat{Z}_{G}(b)}-\mathcal{N}_{{\textsf{H}_{\text{JT}}},\,s}\,e^{-\frac{1}{2}\int\mathrm{d}b\,b\,\widehat{Z}_{G}(b)\,\widehat{Z}_{G}(b)+s\,\widehat{Z}_{\text{G}}(0)}\right)\ket{\text{HH}}\sim
0\,.$ (4.33)

While the operator acting on $\ket{\text{HH}}$ is not null, the resulting
state is. Such states again have to be eliminated in order to construct
$\mathcal{H}^{\text{JT}}_{\text{BU}}$, which practically means we should
include only one of the states (4.15) because all resulting theories have the
same spectrum.

### Non-trivial null states: the determinant solution

Another example that comes to mind is by taking $G_{\rm JT}(\lambda)$ in
(4.11) to be the characteristic polynomial of ${\textsf{H}_{\text{JT}}}$

$\fint_{0}^{+\infty}\mathrm{d}E\,\delta\rho_{0\,\text{null}}(E)\,\frac{1}{\lambda-E}=\det(\lambda-{\textsf{H}_{\text{JT}}})\,e^{-\frac{L}{2}V_{\text{JT}}(\lambda)}\,.$
(4.34)

To get something useful out of this we can use the following fact, proven in
section 5 below. To leading order in large $L$, the characteristic polynomial
of ${\textsf{H}_{\text{JT}}}$ equals the ensemble average of the
characteristic polynomial of $H$ in the original undeformed JT gravity
ensemble

$\det(\lambda-{\textsf{H}_{\text{JT}}})=\left\langle\det(\lambda-H)\right\rangle_{\text{JT}}\,.$
(4.35)

The potential term in (4.34) makes for a finite double scaling limit, upon
which the determinant reduces to the Baker-Akhiezer function
$\psi_{\text{JT}}(\lambda)$ of JT gravity ($x=0$ in (5.30) below). The leading
order behavior of the Baker-Akhiezer functions can be obtained using a WKB or
disks and cylinders approximation [63, 11] and one finds up to a
proportionality factor (that can be absorbed in the parameter $s$ multiplying
the null deformation)

$\psi_{\text{JT}}(\lambda)\sim\frac{1}{\lambda^{1/4}}\cos\bigg(\pi\int_{0}^{\lambda}\mathrm{d}E\,\rho_{0,\text{JT}}(E)-\frac{\pi}{4}\bigg{missing})\,,$
(4.36)

and so

$\fint_{0}^{+\infty}\mathrm{d}E\,\delta\rho_{0\,\text{null}}(E)\,\frac{1}{\lambda-E}=\frac{1}{\lambda^{1/4}}\cos\bigg(\pi\int_{0}^{\lambda}\mathrm{d}E\,\rho_{0,\text{JT}}(E)-\frac{\pi}{4}\bigg{missing})\,.$
(4.37)

Using standard techniques for singular integral equations one can solve for
$\delta\rho_{0,\rm null}$ and deduce from that the explicit null deformation
of the dilaton potential. The most practical way to do so is to write the
Baker-Akhiezer function in the Fourier domain, using a generalization of the
Kontsevich [64] integral representation of the Airy function built to
reproduce the WKB approximation above. Inverting (4.37), one finds that at
large $\int_{0}^{E}\mathrm{d}M\,\rho_{0,{\rm JT}}(M)$ the approximate null
deformation is given by

$\delta\rho_{0\,\text{null}}(E)=s\,\frac{1}{E^{1/4}}\sin\bigg(\pi\int_{0}^{E}\mathrm{d}M\,\rho_{0,{\rm
JT}}(M)-\frac{\pi}{4}\bigg{missing})\,.$ (4.38)

This can be generalized by inserting polynomial functions of the determinant
as left-hand side in (4.34). One ultimately finds the following null
deformations of the dilaton potential (to leading order)

$\delta
U(\Phi)=e^{-\textsf{S}_{0}}\,\Phi\sum_{a=1}^{\infty}s_{a}\,\sin\bigg(a\pi\int_{0}^{\Phi^{2}}\mathrm{d}M\,\rho_{0,{\rm
JT}}(M)-\frac{\pi}{4}\bigg{missing})\,e^{-2\pi\Phi}\,.$ (4.39)

Turning on these deformations would semiclassically appear to have non-trivial
effects (see section 5.4), but non-perturbatively they do not change the
theory at all (as long as we have the bilocal interaction).

##  5 Black holes without quantum chaos

We saw in section 4 that there is a spectrum $\textsf{H}_{\rm JT}$ that is
intrinsic to JT gravity: it is uniquely determined by the matrix integral
potential $V_{\text{JT}}(\lambda)$ since it is the spectrum of the theory of
dilaton gravity whose action is

$I=I_{\rm JT}+I_{\rm non-local}\,.$ (5.1)

As mentioned in the introduction, due to the lack of a correction to the local
dilaton potential, the geometric expansion in this model, seen in (1.6), only
contains the disk contribution as the half-wormhole contribution vanishes due
to $Z_{\rm brane}=0$. Notably this system does not require fine-tuning $\sim
e^{\textsf{S}_{0}}$ parameters in the dilaton-gravity action, unlike the
models studied in [36].

We now study the properties of this spectrum ${\textsf{H}_{\text{JT}}}$, and
through this analysis, present a dual QM description to the bulk theory (5.1).

Let us summarize the main steps that result in this QM description.

  1. 1.

The spectrum $\lambda_{\text{JT}}$ of ${\textsf{H}_{\text{JT}}}$ are the
solutions to the electrostatic problem (4.7). We will prove that (to leading
order in large $L$) the solutions are the zeros of the orthogonal polynomial
$P_{\text{JT}\,L}(\lambda)$

$LV_{\text{JT}}^{\prime}(\lambda_{\text{JT}\,i})=\frac{1}{2\lambda_{\text{JT}\,i}}+2\sum_{j\neq
i}^{L}\frac{1}{\lambda_{\text{JT}\,i}-\lambda_{\text{JT}\,j}}\quad\Leftrightarrow\quad
P_{\text{JT}\,L}(\lambda_{\text{JT}\,i})=0\,.$ (5.2)

  2. 2.

These zeros are approximately evenly spaced with coarse grained spectral
density $\rho_{0,\text{JT}}(\lambda)$, this is called _clock behavior_ , and
should be contrasted with the fluctuations in level spacing in quantum chaotic
systems (which follows the Wigner surmise, or has random matrix statistics
[65, 66])

$\lambda_{\text{JT}\,i+1}-\lambda_{\text{JT}\,i}=\frac{1}{\rho_{0,\text{JT}}(\lambda)}\,.$
(5.3)

  3. 3.

The orthogonal polynomials with respect to the JT potential
$P_{\text{JT}\,n}(\lambda)$ satisfy a recursion relation, which can be written
as a matrix equation featuring an $\infty$ dimensional matrix $Q$

$\lambda\,P_{\text{JT}}(\lambda)=Q\cdot P_{\text{JT}}(\lambda)\,.$ (5.4)

Imposing the quantization condition (5.2) $P_{\text{JT}\,L}(\lambda)=0$
reduces this to an eigenvalue equation for the reduction $Q_{L}$ of $Q$ to its
first $L$ rows and columns, thus the eigenvalues of $Q_{L}$ are the zeros of
$P_{\text{JT}\,L}(\lambda)$, which in turn (as we saw above) are the spectrum
of ${\textsf{H}_{\text{JT}}}$. So $Q_{L}$ and ${\textsf{H}_{\text{JT}}}$ have
identical eigenvalues, and can be identified to leading order in large $L$

$P_{\text{JT}\,L}(\lambda)=\det(\lambda-
Q_{L})=\det(\lambda-{\textsf{H}_{\text{JT}}})\quad\Leftrightarrow\quad
Q_{L}={\textsf{H}_{\text{JT}}}\,.$ (5.5)

  4. 4.

Upon double scaling [11], orthogonal polynomials become Baker-Akhiezer
functions and the matrix $Q$ becomes the operator $\widehat{Q}$ in the Lax
formalism [67, 63, 28, 68, 69]. The recursion relation (5.4) becomes a
Schrodinger equation, and crucially (5.2) becomes a Dirichlet boundary
condition which _discretizes_ the spectrum

$\lambda\,\psi_{\text{JT}}(x,\lambda)=\widehat{Q}\,\psi_{\text{JT}}(x,\lambda)\,,\quad\widehat{Q}=\widehat{H}_{\text{JT}}=-\frac{1}{e^{2\textsf{S}_{0}}}\partial_{x}^{2}+u_{\text{JT}}(x)\,,\quad\psi_{\text{JT}}(0,\lambda)=0\,.$
(5.6)

The partition function $Z(\beta)=\Tr\big(e^{-\beta\widehat{Q}}\big{missing})$
involves no projection operators, therefore this QM is a dual description of
gravity - unlike when this $\widehat{Q}$ appears in the Lax description of
matrix integrals.

In the remainder of this section we provide more details and interpretation
for these steps, and we drop the subscripts JT for notational comfort.

###  5.1 Orthogonal polynomials

Let us quickly review some facts about orthogonal polynomials on the real
line, but see [68, 67] for more details. Consider polynomials $P_{n}(\lambda)$
orthogonal with respect to the measure
$e^{-LV(\lambda)}\mathrm{d}\lambda$222222We take the measure to be normalized
such that $P_{0}(\lambda)=1$.

$\int_{-\infty}^{+\infty}\mathrm{d}\lambda\;e^{-LV(\lambda)}P_{n}(\lambda)P_{m}(\lambda)=\delta_{nm}\,,\quad
P_{n}(\lambda)=\frac{1}{\sqrt{h_{n}}}\lambda^{n}+\text{lower degree}\,.$ (5.7)

We are interested in large $L$ large and in even polynomials $V(\lambda)$. We
can expand $\lambda P_{n}(\lambda)$ in $P_{m}(\lambda)$ with $m\leq n+1$,
because of the maximal degree of each polynomial

$\lambda\,P_{n}(\lambda)=\sum_{m=0}^{n+1}Q_{nm}\,P_{m}(\lambda)\,.$ (5.8)

Because the inner product of $\lambda\,P_{n}(\lambda)$ and $P_{m}(\lambda)$ is
symmetric under exchanging $n$ and $m$, $Q$ is symmetric and we have only
three terms in the expansion. Moreover the diagonal is zero, as the integral
of $\lambda\,P_{n}(\lambda)^{2}$ vanishes for even potentials, leaving only
$Q_{nn+1}=Q_{n+1n}$ as nonzero matrix elements. Comparing the terms of highest
degree gives finally $Q_{nn+1}=\sqrt{h_{n+1}}/\sqrt{h_{n}}$, such that the
recursion relation is

$\lambda P_{n}(\lambda)=a_{n+1}P_{n+1}(\lambda)+a_{n}P_{n-1}(\lambda)\,,\quad
a_{n}=\frac{\sqrt{h_{n}}}{\sqrt{h_{n-1}}}\,.$ (5.9)

Very explicitly the so-called Jacobi matrix $Q$ reads

$Q=\begin{pmatrix}0&a_{1}&0&0&\dots\\\ a_{1}&0&a_{2}&0&\dots\\\
0&a_{2}&0&a_{3}&\dots\\\ 0&0&a_{3}&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\end{pmatrix}\,.$ (5.10)

The polynomials can thus be found recursively from knowledge of $a_{n}$, which
in term can be found recursively from the potential $V(\lambda)$. Indeed, we
obtain the following equation, sometimes referred to as the discrete string
equation [67]

$\displaystyle 0$
$\displaystyle=\int\mathrm{d}\lambda\,\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(P_{n}(\lambda)P_{n-1}(\lambda)e^{-LV(\lambda)}\right)$
$\displaystyle=\int\mathrm{d}\lambda\left(P_{n}^{\prime}(\lambda)P_{n-1}(\lambda)e^{-LV(\lambda)}+P_{n}(\lambda)P_{n-1}^{\prime}(\lambda)e^{-LV(\lambda)}-LV^{\prime}(\lambda)P_{n}(\lambda)P_{n-1}(\lambda)e^{-LV(\lambda)}\right)$
$\displaystyle=\frac{n}{a_{n}}-LV^{\prime}(Q)_{nn-1}\,.$ (5.11)

In the first and second term we used (5.7) and the definition of $a_{n}$ in
(5.9) to expand the derivative of an orthogonal polynomial in terms of
orthogonal polynomials of lower degree as
$P^{\prime}_{n}(\lambda)=n/a_{n}P_{n-1}(\lambda)+\text{lower degree}$, then
most terms except one cancel using orthogonality. For the third term we can
Taylor expand $LV^{\prime}(\lambda)$ in $\lambda$ and then use that powers of
$Q$ act as powers of $\lambda$ on the polynomials (5.8). Since $Q$ depends
only on $a_{n}$, the above is an equation that can be used to solve for the
$a_{n}$. For instance when $V(x)=x^{4}/4-t\,x^{2}/2$ this becomes

$\frac{n}{La_{n}}=-t\,a_{n}+a_{n}(a_{n+1}^{2}+a_{n}^{2}+a_{n-1}^{2})\,,$
(5.12)

which can be solved recursively for $a_{n}$. This is then enough to find the
orthogonal polynomials recursively from knowledge of the potential
$V(\lambda)$.

### Electrostatics and zeros of orthogonal polynomials

To prove (5.2), we need to take a different route. The proof follows
essentially by deriving a second order differential equation for orthogonal
polynomials, which was achieved by [70, 71]. Though elementary, the derivation
would distract a bit too much from our point, so we simply state the result

$P_{n}^{\prime\prime}(\lambda)-\bigg{(}LV^{\prime}(\lambda)+\frac{A_{n}^{\prime}(\lambda)}{A_{n}(\lambda)}\bigg{)}P_{n}^{\prime}(\lambda)+S_{n}(\lambda)P_{n}(\lambda)=0\,.$
(5.13)

The form of $S_{n}(\lambda)$ will prove to be irrelevant, and $A_{n}(\lambda)$
is explicitly [70, 71]

$A_{n}(\lambda)=\int_{-\infty}^{+\infty}\mathrm{d}\mu\frac{V^{\prime}(\lambda)-V^{\prime}(\mu)}{\lambda-\mu}P_{n}(\mu)^{2}\,e^{-LV(\mu)}\,.$
(5.14)

This differential equation can be used to obtain an equation for the zeros of
the orthogonal polynomials [70]. Suppose that $\lambda_{i}$ are the zeros of
$P_{L}(\lambda)$, then the derivatives at those zeros become

$\sqrt{h_{L}}\,P_{L}(\lambda)=\prod_{i=1}^{L}(\lambda-\lambda_{i})\,,\quad\sqrt{h_{L}}\,P_{L}^{\prime}(\lambda_{i})=\prod_{j\neq
i}^{L}(\lambda_{j}-\lambda_{i})\,,\quad\sqrt{h_{L}}\,P_{L}^{\prime\prime}(\lambda_{i})=2\sum_{k=1}^{L}\prod_{i\neq
j\neq k}^{L}(\lambda_{j}-\lambda_{i})$ (5.15)

Evaluating (5.13) at these zeros (for which the last term vanishes) results in
the equation

$2\sum_{i\neq
j}\frac{1}{\lambda_{i}-\lambda_{j}}=LV^{\prime}(\lambda_{j})+\frac{A_{L}^{\prime}(\lambda_{j})}{A_{L}(\lambda_{j})}$
(5.16)

To leading order in large $L$ we can ignore the second term, which is order
one, and the same is true for the $1/2\lambda_{i}$ term in (5.2). Therefore to
leading order in large $L$ we indeed find that the solutions to our
electrostatics problem (the eigenvalues to which the matrix integral localizes
when we only turn on the bilocal deformation) are the zeros or the orthogonal
polynomial $P_{L}(\lambda)$ of the original matrix ensemble

$2\sum_{i\neq
j}\frac{1}{\lambda_{i}-\lambda_{j}}=LV^{\prime}(\lambda_{j})\quad\Leftrightarrow\quad
P_{L}(\lambda_{i})=0\,.$ (5.17)

This means in particular that the zeros of this polynomial match the
eigenvalues of ${\textsf{H}_{\text{JT}}}$ (everything from hereon is
implicitly to leading order in large $L$)

$\sqrt{h_{L}}\,P_{L}(\lambda)=\det(\lambda-{\textsf{H}_{\text{JT}}})\,.$
(5.18)

### Jacobi matrix as the Hamiltonian

Now we want to prove that the spectrum of ${\textsf{H}_{\text{JT}}}$ matches
that of $Q_{L}$. For this we reconsider the recursion relation (5.8) but with
the constraint that $P_{L}(\lambda)=0$. Now we see that the recursion relation
closes on the polynomials $P_{0}(\lambda)\dots P_{L-1}(\lambda)$, in
particular equation (5.9) fixes $P_{L-1}(\lambda)$ in terms of just
$P_{L-2}(\lambda)$, resulting in the matrix equation

$\lambda\,P(\lambda)=Q_{L}\cdot P(\lambda)\,,\quad P_{L}(\lambda)=0\,,$ (5.19)

where $P(\lambda)$ is a column with entries $P_{0}(\lambda)\dots
P_{L-1}(\lambda)$. This is now just a standard eigenvalue problem, searching
for the eigenvalues of $Q_{L}$. The $L$ solutions are by construction the $L$
zeros of $P_{L}(\lambda)$, so that one can expand this polynomial as

$\sqrt{h_{L}}\,P_{L}(\lambda)=\det(\lambda-Q_{L})\,.$ (5.20)

Up to unitary transformations232323We care only about eigenvalues of
${\textsf{H}_{\text{JT}}}$ throughout this paper, for some discussion on the
gravitational interpretation of the eigenvectors (or unitary transformations)
see [15, 72, 6]. combined with (5.18) this means that the restriction of the
Jacobi matrix to the first $L$ rows and columns is actually the Hamiltonian of
the QM that we are localizing to in the matrix integral, so the Jacobi matrix
has an important physical meaning

${\textsf{H}_{\text{JT}}}=Q_{L}\,.$ (5.21)

Using the same argument one proves that

$\sqrt{h_{n}}\,P_{n}(\lambda)=\det(\lambda-Q_{n})\,.$ (5.22)

We also note that the orthogonal polynomials can can alternatively be computed
as the expectation values of determinants in $n$ dimensional random matrix
theory, using the same potential $V(\lambda)$

$\sqrt{h_{n}}\,P_{n}(\lambda)=\left\langle\det(\lambda-H)\right\rangle_{n}\,,$
(5.23)

which means that the eigenvalues of ${\textsf{H}_{\text{JT}}}$ are the typical
zeros of the determinant of $H$ in random matrix theory, which is not the same
as the typical eigenvalues of $H$. Indeed, as we now discuss, the eigenvalue
statistics of ${\textsf{H}_{\text{JT}}}$ is quite _atypical_ for the random
matrix ensemble.

###  5.2 No random matrix statistics

Now that we have a realization of ${\textsf{H}_{\text{JT}}}$ we want to know
the properties of its spectrum. The level density follows to leading order in
large $L$ the averaged level density $\rho_{0,{\rm JT}}(\lambda)$ in the
original ensemble, we choose to present the proof of this only in the double
scaling limit in section 5.3. Here instead we focus on the correlation between
nearby energy levels, which has more surprising properties. The discussion of
this section carries over unmodified to gravity in the double scaling limit,
the only thing that changes really upon double scaling is the averaged level
density.

In GUE random matrix theory there is quadratic repulsion between neighboring
energy levels, which is most clearly visible as a quadratic zero in the so-
called Wigner surmise [65, 66] as shown in Figure 1.

Figure 1: The spacing distribution $P(s)$ of adjecent energy levels obtained
from a Gaussian Hermitian random matrix at $L=10^{4}$. The histogram is a
function of the difference between adjecent eigenvalues, denoted by $s$ and we
focussed on $1272$ eigenvalues $E$ within the bin $[-1/10,1/10]$. The solid
blue curve is the infinite $L$ result given by
$\frac{32s^{2}}{\pi^{2}}e^{-4s^{2}/\pi}$.

This plot indicates that eigenvalues repel each other, it is unlikely to find
eigenvalues very close together in quantum chaotic systems. This should be
contrasted with Poisson behavior, where eigenvalues have a certain density but
are otherwise uncorrelated, and can lump together.

Quadratic level repulsion is actually the bell cow of GUE random matrix
statistics, in the literature it is often used as _the_ defining property of
quantum chaotic systems [66, 65]. Black holes on the other hand are certainly
classically chaotic systems, which are characterized by Lyapunov growth -
exponential sensitivity to changes in initial conditions, and observations in
black hole backgrounds have this property, because early perturbations get
exponentially blueshifted as they fall in black holes [73, 74, 75, 76, 77, 78,
79, 80].

It then seems logical to assume that systems which exhibit classical chaos,
such as black holes, also exhibit quantum chaos. This has led to the idea that
black holes should have GUE random matrix level statistics [50]. One way to
visualize random matrix level statistics, is to plot the spectral form factor
in some microcanonical energy window $\delta$ around $\lambda$

$\text{mSFF}(t)=\sum_{i,j\in\delta}e^{\mathrm{i}t(\lambda_{i}-\lambda_{j})}\,.$
(5.24)

We can compare this quantity for a typical draw of random matrix theory with
the spectrum obtained by solving our electrostatics problem (5.2), see Figure
2

Figure 2: Left: Microcanonical spectral form factor for GUE as a function of
time for a bin of size $1/10$ centered around zero energy. Right: Same
microcanonical spectral form factor but with a spectrum obtained from the
electrostatics problem (5.2) with a Gaussian potential $V=1/2x^{2}$. We
clearly see a huge difference between the plot on the right, not only are
there the Poincare recurrences, but there is no ramp region as well. In both
plots we used $L=10^{4}$ and the bin contained $1272$ eigenvalues. We verified
that the recurrence time is indeed $2\pi\rho(0)=4\times 10^{4}$ even if
appearing non-uniform in the figure above due to the log-log scale.

At early times these are identical, but at exponentially late times they are
very different, systems with random matrix statistics have a linear ramp
followed by a plateau starting at $t=2\pi\rho(\lambda)$, whereas our system
has almost perfect Poincare recurrences at times $t=2\pi n\rho(\lambda)$ and
no visible ramp.

The reason for this is that the spectrum of our theory
${\textsf{H}_{\text{JT}}}$ is approximately equally spaced, on scales where
$\rho(\lambda)$ is approximately constant, and this periodicity is responsible
for the Poincare recurrences. This is already obvious when we consider the
electrostatic problem (5.2). This model has an attractive force coming from
the potential and a repulsive force coming from the bilocal deformation. The
potential is a large distance force, saying how much $\lambda_{i}$ should be
packed in some smaller interval. Within those smaller intervals electrostatic
repulsion wins, and the eigenvalues are _maximally_ separated. So locally the
eigenvalues are equally spaced, and globally there is an envelope determined
by the potential. Let us now show this in more detail by studying the
distribution of zeros of $P_{L}(\lambda)$.

Consider the orthogonal polynomials $P_{L}(\lambda)$ associated with some
potential $V(\lambda)$. The zeros of $P_{L}(\lambda)$ are always in some
finite region (or set of regions) along the real axis242424At least for the
potentials that have certain growth properties at large $|\lambda|$ [67] and
in particular when $L$ becomes larger, the zeros of $P_{L}(\lambda)$ become
more and more tightly packed. Let us consider some $\lambda_{0}$ and consider
a set of zeros $\lambda_{0}^{(j)}$ around this value,

$\dots<\lambda_{0}^{(-1)}<\lambda_{0}=\lambda_{0}^{(0)}<\lambda_{0}^{(1)}<\dots$
(5.25)

To study the spacing of these zeros, we follow Lubinsky [81] (see also [82])
and consider the Christoffel-Darboux kernel associated to the orthogonal
polynomials

$K_{L}(\lambda_{1},\lambda_{2})=a_{L}\frac{P_{L}(\lambda_{1})P_{L-1}(\lambda_{2})-P_{L-1}(\lambda_{1})P_{L}(\lambda_{2})}{\lambda_{1}-\lambda_{2}}.$
(5.26)

This object has two interesting properties. First, if $P(\lambda_{1})=0$ then
$K_{L}(\lambda_{1},\lambda_{2})=0$ if $P(\lambda_{2})=0$ and
$\lambda_{1}\neq\lambda_{2}$. Second, if $\lambda_{1}-\lambda_{2}\sim
1/\rho(\lambda)\sim 1/L\to 0$ the Christoffel-Darboux kernel becomes the sine
kernel

$\lim_{L\to\infty}\frac{1}{\rho(\lambda_{0})}K_{L}\left(\lambda_{0}+\frac{a}{\rho(\lambda_{0})},\lambda_{0}+\frac{b}{\rho(\lambda_{0})}\right)=\frac{\sin\pi(a-b)}{\pi(a-b)}$
(5.27)

The relation (5.27) holds for any potential and is the manifestation of random
matrix theory universality. The sine kernel vanished for
$a-b\in\mathbb{Z}_{0}$ and so combining this with the first property, we see
that when we have a zero at $\lambda_{0}^{(j)}$, then the at large $L$ the
next zero $\lambda_{0}^{(j+1)}$ lies as

$\lambda_{0}^{(j+1)}-\lambda_{0}^{(j)}=\frac{1}{\rho(\lambda_{0})}$ (5.28)

Thus the spectrum is locally equally spaced and in the orthogonal polynomial
literature it is called _clock behaviour_. This rigidity is also clearly
visible when we plot the average level separation, the analogue to the Wigner
surmise, see Figure 1, where indeed we see this is highly peeked around
$1/\rho(\lambda_{0})$ as shown in Figure 3

Figure 3: Spacing distribution typical for _clock behaviour_. The right is a
zoom-in of the left plot that emphasizes the small spread of the distribution.
Here we used the electrostatics problem for a quartic potential
$V=-1/2x^{2}+x^{4}$, again we used the binsize $1/10$ centered around zero and
took $L=10^{4}$.

This leads us to a rather surprising result when we consider the double
scaling limit to gravity. By construction we are studying a theory of black
holes, after all we are investigating pure JT gravity with a bilocal
interaction turned on. The spectrum ${\textsf{H}_{\text{JT}}}$ described the
exact quantum spectrum of these black holes, and we found that this spectrum
has clock statistics, _not_ random matrix statistics. So there are theories of
black holes without random matrix statistics, which seems to contradict at
least superficially the claim of [50].

Some comments are in order.

  1. 1.

Clock behaviour of the zeros of orthogonal polynomials follows from the sine-
kernel and is thus not only a very universal property, but also a non-
perturbative (in $L$) one. To see this, notice that the energies $a$ and $b$
are proportional to $\rho(\lambda)\sim L$ so we have an oscillating
exponential of $L$. Furthermore, universality also tells us that when we go to
the gravitational description, i.e. when we double scale, clock behaviour will
persist, the sine kernel is always there [65, 66, 83].

  2. 2.

The ramp has an interpretation in gravity as due to a wormhole stretching
between two asymptotic boundaries in ordinary JT gravity (without local or
bilocal deformations) [11, 6, 5]. This computation is a good approximation for
a typical member of the random matrix ensemble. Clearly it is _not_ a good
approximation to our canonical version of JT gravity with only the bilocal
turned on, since we see no ramp even under time averaging. That is fine,
because this is an atypical draw from the point of random matrix theory (even
though, in some sense, it is the most likely draw). But this atypical draw is
very natural from the bulk point of view.

  3. 3.

When adding matter in the probe limit, our theory still gives the Lyapunov
growth of the OTOC, simply because the coarse grained spectrum is the same as
in ordinary JT gravity. The scrambling time scale is too short to distinguish
our discretized spectrum from the continuous one.

  4. 4.

The argument that black holes are quantum chaotic seems more robust in higher
dimensions, simply because there are more degrees of freedom, so this could be
a pathology of our lower dimensional setup. It is fathomable that the clock
behavior is not robust under such deformations.

###  5.3 Quantum mechanical dual of canonical JT gravity

We now consider the double scaling limit where one takes $L$ to infinity and
scales towards the spectral edge of the spectrum [11]. The precise double
scaling procedure is not very important, here we will limit ourselves to
sketching how different quantities map to one another and state the
corresponding double scaled structures, which we then check gives the desired
answers.

### The auxiliary quantum mechanics reloaded

In the double scaling limit orthogonal polynomials become the Baker-Akhiezer
functions of the relevant double scaled matrix model [84]

$P_{n}(\lambda)\to\psi(x,\lambda)\,,$ (5.29)

where $L-n$ becomes proportional to a rescaled version of the (now continuous)
coordinate $x$, and $\lambda$ has been scaled towards the edge, furthermore we
left out prefactors for comfort. For large $e^{\textsf{S}_{0}}$, which
replaces $L$ upon double scaling, we have the following approximation for the
Baker-Akhiezer functions (up to normalization that should be fixed later) for
$\lambda>u(x)$ [63, 44]

$\psi(x,\lambda)=\frac{1}{(\lambda-u(x))^{1/4}}\cos\bigg(e^{\textsf{S}_{0}}\int_{u(x)}^{\lambda}dv\,\mathcal{F}^{\prime}(v)(\lambda-v)^{1/2}+\frac{\pi}{4}\bigg{missing})\,,\quad
x=\frac{\sqrt{u}}{2\pi}I_{1}(2\pi\sqrt{u})=\mathcal{F}(u)\,,$ (5.30)

where the last expression is the leading approximation to the JT string
equation [28]. For $\lambda>u(x)$ this function is exponentially decaying.
Perhaps the easiest way to understand the mapping (5.29) is that Baker-
Akhiezer functions are by definition expectation values of determinants in
double scaled matrix integrals, just like the orthogonal polynomials (5.23).
This definition leads straight to (5.30) [63, 85].

Baker-Akhiezer functions satisfy two differential equations (this is the Lax
formalism), one of which is [63, 28]

$\lambda\,\psi(x,\lambda)=\widehat{Q}\,\psi(x,\lambda)\,,\quad\widehat{Q}=-\frac{1}{e^{2\textsf{S}_{0}}}\partial_{x}^{2}+u(x)\,.$
(5.31)

One recognizes (5.30) indeed as the WKB approximation for the solutions to
this differential equation. To leading order $u(x)$ satisfies
$x=\mathcal{F}(u)$, in general is is a more complicated function, see section
5. The parameter $x$ is the coupling constant $t_{0}$ of the KdV hierarchy,
which usually is not thought about as physically relevant, all dilaton gravity
models sit at $t_{0}=0$.

The point is that the differential operator $\widehat{Q}$ can be obtained
directly by double scaling the Jacobi matrix $Q$, in the usual way that
difference equations such as (5.9) become differential equations when $n$
becomes continuous. The constraint $P_{L}(\lambda)=0$, which according to
(5.2) is equivalent to finding the eigenvalues of ${\textsf{H}_{\text{JT}}}$,
gets replaced with a Dirichlet boundary condition upon double scaling

$\psi(0,\lambda)=0\,.$ (5.32)

The spectrum of $\widehat{Q}$ with these boundary conditions will then, by
construction, reproduce the spectrum of ${\textsf{H}_{\text{JT}}}$. This means
that we can interpret this QM system as a _dual_ description of JT gravity
with only the bilocal deformation, which we refer to as canonical JT gravity.
See figure 4 for the spectrum, wavefunctions and the potential $u(x)$.
252525The spectrum being given by the zero’s of the Baker-Akhiezer functions
also appeared in a recent work by Clifford Johnson [44] (based on his earlier
work [47, 48, 49]). There it was noted that this spectrum is naturally
embedded in the matrix integral by studying the probability distribution of
the first, second, etc. eigenvalues of the matrix in the double scaling limit.
The peaks of these probability distributions then coincide (approximately)
with the zeros of the Baker-Akhiezer function. It was then argued that on the
disk this discrete spectrum should be present, which at large energy is just
the continuous Schwarzian density of states. Higher genus corrections would
then fill in the gaps and produce a continuous spectrum. This is not what we
are doing here. We have the spectrum given by the zeros of the Baker-Akhiezer
function to be the non-perturbative density of states in the sense that when
we compute any correlator of $\rho(E)$ in the matrix integral we get, first of
all, a factorizing answer, and second the density is a sum of delta functions
on this spectrum. To get these two features it is essential to deform the
matrix integral (and gravity theory) with the bilocal deformation as shown in
[36].

Figure 4: Left: First seven normalized eigenstates $\psi_{i}(x)$ and energies
of $\widehat{Q}$ for $\textsf{S}_{0}=5$. We also plotted the potential $u(x)$
in black. The more red the coloring the higher the energy. The left vertical
axis is for $E_{i}$ and $u(x)$, whereas the right one is for $\psi_{i}$.
Right: Integrated spectral density
$n(E)=\int^{E}_{0}\rho(E^{\prime})\mathrm{d}E^{\prime}$ for the first 1500
eigenvalues of ${\textsf{H}_{\text{JT}}}$ (orange dots) and the JT gravity
result (blue).

The partition function of canonical JT can be computed literally as

$Z(\beta)=\Tr\big(e^{-\beta\widehat{Q}}\big{missing})\,,$ (5.33)

the spectral form factor is
$\Tr\big(e^{-\beta_{1}\widehat{Q}}\big{missing})\Tr\big(e^{-\beta_{2}\widehat{Q}}\big{missing})$
etcetera. One can check that wavefunctions for different solutions
$\lambda_{i}$ and $\lambda_{j}$ are orthogonal and can be normalized to one on
$x>0$ using formula (3.37) in [86], which is a continuous version of the
Christoffel-Darboux formula (5.26)

$\int_{0}^{\infty}\mathrm{d}x\,\psi(x,\lambda_{i})\psi(x,\lambda_{j})=\frac{\psi(0,\lambda_{i})\partial_{x}\psi(0,\lambda_{i})-\psi(0,\lambda_{j})\partial_{x}\psi(0,\lambda_{i})}{\lambda_{i}-\lambda_{j}}=0\text{
when }\lambda_{i}\neq\lambda_{j}\,,$ (5.34)

where one uses that $\psi(0,\lambda_{i})=\psi(0,\lambda_{j})=0$ according to
the Dirichlet boundary condition (5.32).

There are several ways to confirm that the clock behavior (5.28) is respected
in this double scaled theory. One is to use that the Christoffel-Darboux
kernel reduces to the sine kernel for nearby energies. Another is to use that
$\psi(0,\lambda)$ according to (5.30) becomes (4.36), the spacing of zeros
follows then from the quasi-periodicity of the cos. The simplest though, it to
simply use Bohr-Sommerfeld quantization for the Schrodinger equation (5.31)
with Dirichlet boundary conditions

$\pi(n+1/2)=e^{\textsf{S}_{0}}\int_{0}^{\mathcal{F}(E)}\mathrm{d}x\,\sqrt{E-u(x)}=e^{\textsf{S}_{0}}\int_{0}^{E}\mathrm{d}u\,\mathcal{F}^{\prime}(u)\,\sqrt{E-u}\,,$
(5.35)

where the $x=0$ turning point comes from the hard Dirichlet wall. The coarse
grained density of states is $\mathrm{d}n/\mathrm{d}E$, which results in

$\rho(E)=\frac{\mathrm{d}n}{\mathrm{d}E}=\frac{e^{\textsf{S}_{0}}}{2\pi}\int_{0}^{E}\mathrm{d}u\frac{\mathcal{F}^{\prime}(u)}{\sqrt{E-u}}=e^{\textsf{S}_{0}}\frac{\sinh{2\pi\sqrt{E}}}{4\pi^{2}},$
(5.36)

which reproduces indeed the coarse grained spectral density of our canonical
JT gravity, which should be indeed that of the original ensemble.

We stress that the true spectrum of our QM is _discrete_ , this sets it aside
from, say, Schwarzian quantum mechanics [87], but also from the auxiliary
quantum mechanical description of matrix integrals in the Lax formalism, where
the same Hamiltonian $\widehat{Q}$ features, but without Dirichlet boundary
conditions. To drive home the point that discreteness is conserved upon double
scaling, consider for instance the fact that double scaling the Hermite
polynomials results in the Airy function, which does not have a continuum of
zeros (and is the simplest Baker-Akhiezer function).

### Path integral formulation

It is straightforward, at least to leading order in $e^{\textsf{S}_{0}}$ to
write down an action for our quantum mechanical problem, which is describing a
particle moving in a potential $u(x)$. The partition function is

$Z(\beta)=\int\mathcal{D}x\,\exp\bigg(-\int_{0}^{\beta}\mathrm{d}\tau\bigg{(}\frac{e^{2\textsf{S}_{0}}}{4}\,\dot{x}^{2}+u(x)\bigg{)}\bigg{missing})\,,$
(5.37)

where the sum is over periodic paths $x(\tau+\beta)=x(\tau)$ and we have the
constraint $x(\tau)>0$ for all $\tau$. This is the way to enforce Dirichlet
boundary conditions in path integrals, the wavefunction must vanish at the
boundary, so the particles can never travel beyond $x=0$. One can think of the
potential $u(x)$ as being infinite for $x<0$.

To leading order in $e^{\textsf{S}_{0}}$ we can solve this path integral by
saddle-point. As there is no $e^{\textsf{S}_{0}}$ multiplying the potential,
the leading order equation of motion are simply $\ddot{x}=0$, so the unique
periodic solutions are constants

$x(\tau)=x_{0}\,,\quad 0<x_{0}<\infty$ (5.38)

Notice that $x_{0}$ is a parameter of the on-shell solution and we need to
integrate over it in order to sum over all saddles. To leading order we can
ignore the effects of the constraint $x(\tau)>0$ on the Gaussian fluctuations
around the saddle, and we can evaluate $u(x)$ on shell, so we just need to
compute standard Gaussian integrals262626The $\sqrt{\beta}$ comes from
choosing an orthonormal basis for $x$ on the circle, the path integral measure
is then $dx_{0}\sqrt{\beta}$.

$Z(\beta)=\sqrt{\beta}\int_{0}^{\infty}\mathrm{d}x_{0}\,e^{-\beta
u(x_{0})}\int_{-\infty}^{+\infty}\prod_{n>0}\mathrm{d}a_{n}\mathrm{d}b_{n}\exp\bigg(-\frac{e^{2\textsf{S}_{0}}\pi^{2}n^{2}}{\beta^{2}}(a_{n}^{2}+b_{n}^{2})\bigg{missing})$
(5.39)

After zeta regularization this results indeed in the Laplace transform of
(5.36)

$Z(\beta)=\sqrt{\beta}\int_{0}^{\infty}\mathrm{d}x_{0}\,e^{-\beta
u(x_{0})}\frac{e^{\textsf{S}_{0}}}{\beta\sqrt{4\pi}}=\frac{e^{\textsf{S}_{0}}}{4\sqrt{\pi\beta}}\int_{0}^{\infty}\mathrm{d}u\,\mathcal{F}^{\prime}(u)\,e^{-\beta
u}=\frac{e^{\textsf{S}_{0}}}{4\pi^{1/2}\beta^{3/2}}e^{\frac{\pi^{2}}{\beta}}\,.$
(5.40)

An alternative way to obtain this is by writing (5.37) as a phase space path
integral (introducing a field $p(\tau)$), and rescaling $\tau$ by
$e^{\textsf{S}_{0}}$ such that $e^{\textsf{S}_{0}}$ stands everywhere where
one usually has $1/\hbar$ in QM. Then taking $\hbar\to 0$ localizes this phase
space path integral to a classical phase space integral over constants $p_{0}$
and $x_{0}$ as explained in [88]. With the phase space measure
$dp_{0}\,dx_{0}/2\pi\hbar$ one recovers the above answer

$Z(\beta)=\frac{e^{\textsf{S}_{0}}}{2\pi}\int_{0}^{\infty}\mathrm{d}x_{0}\int_{-\infty}^{+\infty}\mathrm{d}p_{0}\,e^{-\beta(p_{0}^{2}+u(x_{0}))}\,.$
(5.41)

So where is the discreteness in this calculation? Even for extremely large
$e^{\textsf{S}_{0}}$, where we can certainly trust this saddle-point, our
quantum mechanics still has a discrete spectrum, which we remind the reader is
a direct consequence of the Dirichlet boundary conditions (5.32). This sits in
the fact that we have ignored the effects of the constraint $x(\tau)>0$ on the
fluctuations, instead we have essentially solved the path integral with
$x_{0}>0$ constraints. This path integral will never result in a discrete
spectrum, not even if we include the expansion of $u(x)$ around the saddle-
point to arbitrary order in the fluctuations.

So discreteness must trickle in through the constraints that $x(\tau)>0$ puts
on the integration range in the mode expansion, the allowed integration range