††institutetext: Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy, University of California, Los Angeles, CA
90095, USA

# 2d TQFTs and baby universes

John Gardiner  and Stathis Megas<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we extend the 2d topological gravity model of Marolf:2020xie  to
have as its bulk action any open/closed TQFT obeying Atiyah’s axioms. The
holographic duals of these topological gravity models are ensembles of 1d
topological theories with random dimension. Specifically, we find that the
TQFT Hilbert space splits into sectors, between which correlators of boundary
observables factorize, and that the corresponding sectors of the boundary
theory have dimensions independently chosen from different Poisson
distributions. As a special case, we study in detail the gravity model built
from the bulk action of 2d Dijkgraaf-Witten theory, with or without end-of-
the-world branes, and for arbitrary finite group $G$. The dual of this
Dijkgraaf-Witten gravity model can be interpreted as a 1d topological theory
whose Hilbert space is a random representation of $G$ and whose aforementioned
sectors are labeled by the irreducible representations of $G$.

These holographic interpretations of our gravity models require projecting out
negative-norm states from the baby universe Hilbert space, which in
Marolf:2020xie  was achieved by the (only seemingly) ad hoc solution of adding
a nonlocal boundary term to the bulk action. In order to place their solution
in the completely local framework of a TQFT with defects, we couple the
boundaries of the gravity model to an auxiliary 2d TQFT in a non-gravitational
(i.e. fixed topology) region. In this framework, the difficulty of negative-
norm states can be remedied in a local way by the introduction of a defect
line between the gravitational and non-gravitational regions. The gravity
model is then holographically dual to an ensemble of boundary conditions in an
open/closed TQFT without gravity.

##  1 Introduction

Recent work has re-emphasized the role of the Euclidean path integral in
quantum gravity. Specifically we have seen the revitalization of the old idea
that quantum gravity should be described by a sum over spacetime topologies
and an integration over all metrics within each topology. Making sense of such
a path integral is, of course, difficult or impossible in general, so this
course of action has mostly focused on two dimensions. Despite the notoriously
hard problem of classifying topological manifolds in 3d, some attempts at path
integrating over a subset of 3d manifolds (often Seifert manifolds) have also
been made maloney2020averaging ; cotler2020ads3 ; maxfield2020path .

A helpful point of view into this sum over topologies of spacetimes is
afforded by the old ideas of baby universes Coleman:1988cy ; Giddings:1988wv ;
Polchinski_1994 , where one thinks of non-trivial topologies as the emission
and re-absorption of baby-universes. In Euclidean signature, this emission/re-
absorption process takes the form of a spacetime wormhole.111Spacetime
wormholes differ from the usual Einstein-Rosen wormholes in that the latter
are topological connections between otherwise distant regions of _space_ ,
whereas the former are additional topological connections between regions of
spacetime. Specifically, Coleman:1988cy  considers a formalism in which the
Hamiltonian of the universe contains couplings between the fundamental fields
and “baby universe field” operators $A_{i}$ which describe the creation and
annihilation of different types of baby universes. Because couplings between
the $A_{i}$ and other fields appear in the Hamiltonian, tracing over the
number and types of baby universes in existence (facts that are presumably
unknowable to an observer in the parent universe) produces an effective
evolution that is non-unitary. Crucially however, the baby universe operators
$A_{i}$ commute with each other and with the Hamiltonian. This renders the
non-unitarity relatively benign, as _within_ any eigenspace of $A_{i}$ the
evolution remains unitary. This means ignorance of the number and types of
baby universes does not, in fact, lead to observable decoherence. Rather, such
ignorance is just operationally equivalent to ignorance of some number of
coupling constants in the theory describing parent universe physics. In other
words, the physics of the parent universe is described by a statistical
ensemble of unitary theories. The theories in this ensemble are parametrized
by the simultaneous eigenstates $\ket{\alpha}$ of the $A_{i}$, which are
called alpha-states. The unknown coupling constants of parent universe physics
are the eigenvalues of $A_{i}$, also called alpha-parameters.

The baby universe idea and the appearance of ensembles has a counterpart in
the context of holography. In this context, where partition functions of a
boundary theory are dual to gravity in the bulk, there is a question of
whether the calculation of a boundary partition function with disconnected
spacetime components should include bulk geometries that connect those
components Maldacena:2004rf . However natural a sum including connected
topologies may otherwise be, it destroys the manifest factorization between
spacetime components of the boundary partition function. That is to say, a
rule including contributions from connected geometries cannot a priori be
expected to give boundary partition functions satisfying $Z[M\sqcup
N]=Z[M]Z[N]$ for disconnected spacetimes $M$ and $N$. One way to proceed is to
reinterpret the boundary partition functions appearing in this holographic
context as ensemble averages of partition functions, where bulk gravity is
then interpreted as dual to a statistical ensemble of boundary
theories.222Equally, we could turn around the logic and stipulate
factorization as a _criterion_ for any well-behaved quantum theory of gravity,
thus declaring most generic gravity path integrals to be in the so-called
“swampland” McNamara:2020uza . This leads to what you could call a new entry
in the holographic dictionary:

$\displaystyle\begin{subarray}{c}\text{\large ensemble of non-
gravitational}\\\ \text{\large boundary theories on}\\\ \text{\large
disconnected
boundaries}\end{subarray}~{}~{}\Leftrightarrow\quad\begin{subarray}{c}\text{\large
Euclidean gravity path integral}\\\ \text{\large with contributions
connecting}\\\ \text{\large disconnected boundary
components}\end{subarray}\;,$ (1)

which now has evidence from a number of lines of study, including the JT-
gravity maldacena2016conformal  and SYK model Kitaev_2018  correspondence
saad2019jt ; witten2020matrix , replica wormhole calculations of the Page
curve Almheiri:2019qdq ; penington2020replica , and recent work suggesting a
possible correspondence between 3d gravity and ensembles of 2d conformal field
theories Afkhami-Jeddi:2020ezh ; maloney2020averaging ; cotler2020ads3 ;
Cotler:2020hgz .

All these different ideas merge in Marolf:2020xie , wherein the authors
demonstrate, via logic reminiscent of the older baby universe picture, how
ensembles of boundary theories can naturally emerge from gravity path
integrals, and connect with the replica wormhole discussions of the Page
curve. Important to the present work, the authors of Marolf:2020xie  introduce
and analyze a simple topological model of a gravity path integral as an
explicit example of these ideas. Their gravity model is a sum over 2d surfaces
weighted by a topological action. Subsequently, the authors of
Balasubramanian:2020jhl  extend this model to include surfaces with spin
structures. The present work picks up from where Marolf:2020xie , and in a
certain sense Balasubramanian:2020jhl  left off, by considering more general
topological bulk theories.

The rest of the paper is organized as follows. In section 1.1, we review the
simple topological model described in Marolf:2020xie  which this work can be
seen as generalizing. In section 2, we consider a gravity path integral
constructed from Dijkgraaf-Witten theory in two dimensions. We find (not
unexpectedly Kapec:2019ecr ) that the correlators of boundary insertion
operators factorize between sectors described by irreducible representations
of $G$. What’s more, the boundary dual theory can be interpreted as a 1d
topological theory with global symmetry group $G$ whose Hilbert space is a
random representation of $G$. Specifically, the number of times a copy of an
irreducible representation of $G$ appears in the Hilbert space is given by a
random integer, with integers for different irreducible representations chosen
independently. In section 3, we consider a gravity path integral constructed
from any 2d TQFT as defined by Atiyah’s axioms Atiyah . We show that the
Hilbert space of the boundary theory is the direct sum of sectors labeled by
the eigenstates of the TQFT handle operator with the number of dimensions in
each sector chosen independently from a different Poisson distribution. In
section 4 we generalize further by considering open/closed TQFTs, that is to
say TQFTs with boundary. We describe a gravity path integral constructed from
open/closed TQFTs, again using Dijkgraaf-Witten as a first example. Similar to
the closed case, for open/closed TQFTs we find that correlation functions
factorize between sectors labeled by eigenstates of the handle operator. We
discuss the interpretation of the gravity path integral in terms of a boundary
ensemble theory. We encounter the same difficulty encountered by
Marolf:2020xie  with defining a theory without negative-norm baby universe
states, a difficulty for which they proposed the solution of adding a nonlocal
boundary term to the bulk action. We discuss this difficulty in section 5,
reframing the solution in the language of 2d TQFTs with defects. Along the
way, we introduce generalized boundary observables representing non-
gravitational regions coupled to the gravity region with fluctuating topology.
The gravity region then is dual to an ensemble of boundary conditions for an
open/closed TQFT. Besides motivating a solution to the negative-norm states in
terms of a defect line, this picture also motivates an understanding of the
additional alpha-parameters associated with having end-of-the-world branes (or
more general boundary conditions). In section 6 we discuss directions for
future work, including the aim of studying more realistic models of gravity,
as well as the possibility of exploring aspects of holography such as bulk
reconstruction in the simple setting of topological theories. In appendix A,
we review a state sum formulation of 2d Dijkgraaf-Witten theory with defects,
which is sufficient to calculate all the Dijkgraaf-Witten theory results used
in the paper.

###  1.1 Review of a simple gravity model

The model of Marolf:2020xie  describes a gravity path integral built from a
sum over spacetime topology. The authors consider 2d orientable spacetimes
without metric or other geometric structure and an action that is (nearly)
just the Euler characteristic. It is, in fact the Euler characteristic
together with an additional term depending on the number of boundaries.
Specifically, for a 2d manifold with genus $g$ and $n$ boundaries they assign
the action

$\displaystyle
S[M_{g,n}]=S_{0}\,\chi(M_{g,n})+n\,S_{\partial}=S_{0}(2-2g)+n(S_{\partial}-S_{0})\;,$
(2)

where $\chi$ is the Euler characteristic $2-2g-n$ and $S_{0}$ and
$S_{\partial}$ are parameters of the theory. The gravity path integral for
spacetime with fixed $n$ number of circle boundary components then becomes

$\displaystyle Z^{\text{QG}}[\text{$n$
boundaries}]=\sum_{\begin{subarray}{c}\text{$M$ s.t.\ $\partial M$ has}\\\
\text{$n$ components}\end{subarray}}\mu(M)e^{S_{0}\chi(M)+S_{\partial}n}\;.$
(3)

The measure $\mu(m)$ here is $\mu(M)=\frac{1}{\prod_{g}m_{g}!}$, where $m_{g}$
is the number of components with genus $g$ that are not connected to any
boundary. This accounts for the residual gauge symmetries permuting identical
connected components of $M$.333Permutations that involve components with
boundaries are of course not gauge redundancies. We point out here that for
the particular choice $S_{\partial}=S_{0}$, the number of boundaries $n$
affects the combinatorics both in the sum over topologies and in the measure
$\mu$, but it has no effect on the action.

We should now make two comments. First, by analogy with a correlation function
being computed as a sum over Feynman diagrams with fixed external legs, we
notate the output of the gravity path integral (a sum over spacetimes with
fixed boundaries) as a correlation function, where the fixed boundaries play
the role of operator insertions. So for example the gravity path integral over
manifolds with $n$ circle boundary components will be notated

$\displaystyle\expectationvalue{\widehat{Z}^{n}}\equiv Z^{\text{QG}}[\text{$n$
boundaries}]\;,$ (4)

so that the operator $\widehat{Z}$ simply denotes the insertion of an
additional circle boundary component. Second, we point out that the action
(2), at least with $S_{\partial}$ set to zero, produces the partition function
of a (particularly simple) TQFT, so that the gravity path integral is simply a
sum over TQFT partition functions:

$\displaystyle Z^{\text{QG}}[\text{$n$
boundaries}]=\sum_{\begin{subarray}{c}\text{$M$ s.t.\ $\partial M$ has}\\\
\text{$n$ components}\end{subarray}}\mu(M)Z^{\text{TQFT}}[M]\;.$ (5)

We mention this, as our aim in this paper is to allow $Z^{\text{TQFT}}$ to be
a more general TQFT partition function.

The values of the correlators $\expectationvalue{\widehat{Z}^{n}}$ can be
gathered in the generating function $\expectationvalue{e^{u\widehat{Z}}}$. The
generating function for the connected correlators is simply
$\log\expectationvalue{e^{u\widehat{Z}}}$, which evaluates to

$\displaystyle\log\expectationvalue{e^{u\,\widehat{Z}}}=\sum_{n=0}^{\infty}\frac{u^{n}}{n!}\expectationvalue{\widehat{Z}^{n}}_{\text{conn}}=\lambda
e^{u\,e^{S_{\partial}-S_{0}}}\;,$ (6)

where $\lambda=\sum_{g}e^{S_{0}(2-2g)}=e^{2S_{0}}/(1-e^{-2S_{0}})$ is the
value of the connected vacuum correlator. From the resulting expression for
$\expectationvalue{e^{u\widehat{Z}}}$, the correlators
$\expectationvalue{\widehat{Z}^{n}}$ can be extracted

$\expectationvalue{\widehat{Z}^{n}}=e^{\lambda}B_{n}(\lambda)e^{(S_{\partial}-S_{0})n}\;,$
(7)

where $B_{n}$ here is the $n$-th Touchard (or Bell) polynomial. This can be
written equivalently as

$\expectationvalue{\widehat{Z}^{n}}=\sum_{d=0}^{\infty}\frac{\lambda^{d}}{d!}(d\,e^{S_{\partial}-S_{0}})^{n}\;.$
(8)

The normalized correlators are thus

$\displaystyle\expectationvalue{\widehat{Z}^{n}}/\expectationvalue{{\mathbb{1}}}=e^{-\lambda}\sum_{d=0}^{\infty}\frac{\lambda^{d}}{d!}(d\,e^{S_{\partial}-S_{0}})^{n}=\sum_{d=0}^{\infty}p_{d}(\lambda)(d\,e^{S_{\partial}-S_{0}})^{n}\;,$
(9)

where $p_{d}(\lambda)=e^{-\lambda}\frac{\lambda^{d}}{d!}$ is the Poisson
distribution with mean $\lambda$.

The gravitational path integral also gives a way to define the Hilbert space
of quantum gravity. The construction of the Hilbert space begins by picking
the vacuum, often called the Hartle-Hawking state $\ket{\text{HH}}$, to be the
empty set thought of as a 1d manifold. This is the state of no boundaries. The
rest of the Hilbert space is constructed by application of the $\widehat{Z}$
operators which insert boundaries,
$\widehat{Z}^{n}\ket{\text{HH}}=\ket{\widehat{Z}^{n}}$. The gravitational path
integral then provides the means to calculate inner products of any two such
states, thereby defining the Hilbert space.

The authors of Marolf:2020xie  also consider the above model with the addition
of so-called end-of-the-world (EofW) branes. These are boundaries on which
spacetime ends, but unlike the $\widehat{Z}$ boundaries we have discussed
above, they are taken to be dynamical, in that the gravity path integral
includes a sum over all configurations of such branes. By constrast the
$\widehat{Z}$ boundaries are fixed boundaries and can be considered
observables of the gravity path integral. With the presence of EofW branes, on
which spacetime can end, we now have, in addition to the $\widehat{Z}$ fixed
boundaries, the possibility of another type of fixed boundary. This is an
interval that is bounded on both sides by EofW branes.444In this work, to
match the language of open/closed TQFTs we will sometimes call these interval
boundaries “open sector” boundaries, and we call circle boundary components
like $\widehat{Z}$ “closed sector” boundaries. These intervals bounded by
branes are not dynamical (only the EofW branes are taken to be dynamical). The
gravity path integral now includes manifolds whose boundary components are of
three different types, fixed circles (inserted by operators $\widehat{Z}$),
circles with EofW brane boundary conditions running completely around the
circle, and circles made from alternating fixed and EofW brane intervals.
Given some number of fixed circle and interval boundaries, the gravity path
integral will only include manifolds whose fixed boundaries match those given,
but will include a sum over all the different possible ways of consistently
configuring the EofW branes.

Like the circle boundary components inserted by $\widehat{Z}$, we can
associate an operator with the inclusion of an additional fixed interval
boundary, which we will call $\widehat{S}$. The model of Marolf:2020xie
considers the possibility of having some number $K$ of “flavors” of EofW
branes, index by $a=1,\ldots,K$. These different types of EofW brane differ
only in their label $a=1,\ldots,K$, with the rule that only EofW branes with
the same label can be connected together. The fixed intervals $\widehat{S}$
now have their endpoints labeled by the flavor of EofW brane on which they
end, giving operators $\widehat{S}_{ab}$. Correlation functions of the gravity
model include insertions of both $\widehat{Z}$ and $\widehat{S}_{ab}$
operators. For example:

$\displaystyle\bra{\text{HH}}\widehat{Z}^{n}\widehat{S_{ab}}\widehat{S_{ba}}\widehat{S_{cd}}\widehat{S_{de}}\widehat{S_{ec}}\ket{\text{HH}}\;.$
(10)

The operators $\widehat{Z}$ and $\widehat{S}_{ab}$ all commute within these
Euclidean path integral correlators. Given a configuration of fixed
boundaries, there are many ways we can partition them into “future” and “past”
boundaries, and reinterpret the gravity path integral correlator as an inner
product of states in the baby universe Hilbert space. For example we can write
(10) variously as
$\innerproduct{\widehat{Z}^{n}\widehat{S_{ab}}\widehat{S_{ba}}}{\widehat{S_{cd}}\widehat{S_{de}}\widehat{S_{ec}}}$,
$\innerproduct{\widehat{Z}^{n-m}\widehat{S_{ab}}}{\widehat{Z}^{m}\widehat{S_{ba}}\widehat{S_{cd}}\widehat{S_{de}}\widehat{S_{ec}}}$,
$\innerproduct{\widehat{Z}^{n}\widehat{S_{ab}}\widehat{S_{ba}}\widehat{S_{cd}}\widehat{S_{de}}}{\widehat{S_{ec}}}$,
etc. The baby universe Hilbert space is spanned by states of the form

$\ket{\widehat{Z}^{n}\prod_{a,b}\widehat{S}_{ab}^{n_{ab}}}\equiv\widehat{Z}^{n}\prod_{a,b}\widehat{S}_{ab}^{n_{ab}}\ket{\text{HH}}.$
(11)

From AdS/CFT we recall that the non-normalizable asymptotic modes of fields in
AdS specify sources in the boundary CFT, and a bulk path integral with given
boundaries $\partial M$ is dual to the appropriate CFT partition function on
that boundary. Inspired by this, we might expect that the boundary conditions
for the 2d gravity path integral correspond to partition functions of a 1d
theory. Or, invoking the idea of (1), we might lower our expectations only
slightly to include the possibility that boundary conditions in this case are
dual to an average of partition functions in an ensemble of 1d theories.
Indeed, the authors of Marolf:2020xie  find a dual description of this sort
for the gravity path integral (3). From this point of view, the correlator
$\expectationvalue{\widehat{Z}}$ is no longer a correlator, but the average
value of a partition function $Z$ in a 1d topological theory. Each additional
boundary component $\widehat{Z}$ is another copy of this boundary partition
function, so that the correlators $\expectationvalue{\widehat{Z}^{n}}$ with
multiple fixed boundaries probe higher moments, $\expectationvalue{Z^{n}}$, in
the ensemble distribution.

The only parameter in topological quantum mechanics is the dimension $d$ of
the Hilbert space, so an ensemble of 1d topological theories is a probability
distribution for $d$. We immediately run into a problem, though. For a single
theory within the ensemble

$\displaystyle Z=\tr_{\mathcal{H}}{\mathbb{1}}=d\;,$ (12)

but (9) suggests that $\widehat{Z}$ takes the value $de^{S_{\partial}-S_{0}}$
for nonnegative integer $d$. Expecting a dual boundary interpretation of
$\widehat{Z}$ as a partition function in a 1d topological theory, thus forces
us to set $S_{\partial}=S_{0}$ in (9).

Alternatively (taking a perspective more in line with that taken in the rest
of this paper) we can forgo adding the term $S_{\partial}$ to the boundary and
instead take the holographic map to be rescaled by some factor
$e^{S_{\partial}}$, so that

$e^{S_{\partial}}\widehat{Z}\leftrightarrow\tr({\mathbb{1}})=Z^{\text{boundary}}\;.$
(13)

Then a choice of rescaling given by $e^{S_{\partial}}=e^{S_{0}}$ gives a
sensible boundary dual, whose theories all have integer dimensional Hilbert
spaces. This perspective has the downside, of course, of making the choice of
notation $\widehat{Z}$ for the boundary insertion operators something of a
misnomer.

The boundary interpretation just introduced motivates a conceptually useful
basis for our baby universe Hilbert space: the basis of eigenvectors of
$\widehat{Z}$,

$\displaystyle\widehat{Z}\ket{\alpha}=\alpha\ket{\alpha}\;.$ (14)

The eigenstates $\ket{\alpha}$, called alpha-states, are of course orthogonal:
$\innerproduct{\alpha^{\prime}}{\alpha}\sim\delta_{\alpha^{\prime},\alpha}$.
But they have one very special property. The boundary theories in our ensemble
are characterized by the values they give to the observables $\widehat{Z}$, so
the set of alpha states is precisely the sample space of boundary theories in
our ensemble. The probability distribution over the theories in our ensemble
can be extracted from the overlap between a given alpha-state and the Hartle-
Hawking state:

$p(\alpha)=\frac{\innerproduct{\text{HH}}{\alpha}\innerproduct{\alpha}{\text{HH}}}{\innerproduct{\alpha}{\alpha}\innerproduct{\text{HH}}{\text{HH}}}.$
(15)

Whereas obtaining a sensible boundary interpretation for a theory without end-
of-the-world branes necessitated only a judicious choice of rescaling (13) for
the $\widehat{Z}$ operators, a potentially more serious problem manifests when
we include end-of-the-world branes and their attendant $\widehat{S}_{ab}$
operators. In the 1d dual theory, the interval boundary insertions
$\widehat{S}_{ab}$ have a natural interpretation as inner products of states
induced by boundary conditions $a$ and $b$ at the endpoints of the interval.
So within a particular boundary theory in the ensemble, the operators
$\widehat{S}_{ab}$ should take as values the components of a $K$ by $K$
positive semidefinite Hermitian matrix $M$. A boundary ensemble will be given
by a joint probability distribution over the dimension $d$ and the matrix $M$.
Viewing correlators of $\widehat{Z}$ and $\widehat{S}_{ab}$ operators as
averages in such an ensemble, they are moments of the ensemble probability
distribution. In particular, the generating function for normalized
correlators, which Marolf:2020xie  calculate to be

$\displaystyle\expectationvalue{e^{iu\widehat{Z}+\sum_{a,b}^{K}it_{ab}\widehat{S}_{ab}}}/\expectationvalue{{\mathbb{1}}}=\exp(\lambda\left(\frac{e^{iu}}{\det({\mathbb{1}}-it)}\right)^{e^{S_{\partial}-S_{0}}}-\lambda)\;,$
(16)

should be the inverse Fourier transform of the probability density $p(d,M)$
defining the ensemble. As we have described above, the $\widehat{Z}$ operators
take values $de^{S_{\partial}-S_{0}}$ with probability
$p_{\lambda}(d)=e^{-\lambda}\lambda^{d}/d!$, giving an expansion of (16) as

$\displaystyle\expectationvalue{e^{iu\widehat{Z}+\sum_{a,b}^{K}it_{ab}\widehat{S}_{ab}}}/\expectationvalue{{\mathbb{1}}}=\sum_{d=0}^{\infty}p_{\lambda}(d)e^{iude^{S_{\partial}-S_{0}}}\expectationvalue{e^{\sum_{a,b}^{K}it_{ab}\widehat{S}_{ab}}}_{Z=d\,e^{S_{\partial}-S_{0}}}\;.$
(17)

For a given $d$, the residual probability distribution $p_{d}(M)$ over the
matrices $M$ will then be the Fourier transform of the generating function

$\expectationvalue{e^{\sum_{a,b}^{K}it_{ab}\widehat{S}_{ab}}}_{Z=d\,e^{S_{\partial}-S_{0}}}=\det({\mathbb{1}}-it)^{-de^{S_{\partial}-S_{0}}}\;.$
(18)

Unfortunately, only for certain values does the above generating function have
an inverse Fourier transform that can be interpreted as a valid probability
distribution graczyk2003 .555In which case, it is known as the Wishart
distribution. Specifically, the exponent $de^{S_{\partial}-S_{0}}$ must lie in
the set $\\{0,1,2,\ldots,K-1\\}\cup[K-1,\infty)$, where, as a reminder, $K$ is
the number of flavors of end-of-the-world brane included in the theory. As $d$
runs over all nonnegative integers, the factor $e^{S_{\partial}-S_{0}}$ must
lie in the set $\\{0,1,2,\ldots,K-1\\}\cup[K-1,\infty)$. A natural choice is
to take $S_{\partial}=S_{0}$.

The above argument highlights an important point. It need not be the case that
a theory without factorization has a description as an ensemble. As in the
situation where $S_{\partial}=0$, the correlation functions of a non-
factorizing theory are not necessarily the moments of a probability
distribution. Failure to have an ensemble description is linked with the
presence of negative-norm states in the baby universe Hilbert space. To see
this, consider a gravity theory with boundary insertion operators
$\widehat{Z}_{i}$. The theories in the ensemble will be parametrized by values
these operators $\widehat{Z}_{i}$ take. Assume for simplicity, these values
are continuous, real, and independent. Then we can formally construct the
alpha-states as

$\ket{\alpha}=\int\\!\prod_{i}\left(\frac{du_{i}}{2\pi}\right)e^{i\sum_{i}u_{i}\left(\widehat{Z}_{i}-\alpha_{i}\right)}\ket{\text{HH}}\;,$
(19)

where $\alpha_{i}$ are the values which $Z_{i}$ takes in the theory described
by $\ket{\alpha}$. From this, the inner product of two alpha states is

$\displaystyle\innerproduct{\alpha^{\prime}}{\alpha}$
$\displaystyle=\int\\!\prod_{i}\left(\frac{du^{\prime}_{i}}{2\pi}\frac{du_{i}}{2\pi}\right)e^{-i\sum_{i}u^{\prime}_{i}(\alpha_{i}-\alpha^{\prime}_{i})}\expectationvalue{e^{i\sum_{i}u_{i}(\widehat{Z}_{i}-\alpha_{i})}}$
(20)
$\displaystyle=\delta(\alpha^{\prime}-\alpha)\int\\!\prod_{i}\left(\frac{du_{i}}{2\pi}\right)e^{-i\sum_{i}u_{i}\alpha_{i}}\expectationvalue{e^{i\sum_{i}u_{i}\widehat{Z}_{i}}}$
$\displaystyle=\delta(\alpha^{\prime}-\alpha)\expectationvalue{{\mathbb{1}}}p(\alpha)\;.$

The last equality comes from viewing the correlators of $\widehat{Z}_{i}$ as
moments of a putative probability distribution $p(\alpha)$. Viewed thus, the
generating function $\expectationvalue{e^{i\sum_{i}u_{i}\widehat{Z}_{i}}}$ is
simply the inverse Fourier transform of $p(\alpha)$. This equation (20)
suggests something about theories that fail to be ensembles. If the
distribution $p(\alpha)$ takes negative values, this means both that the
theory does not have an ensemble description and that the baby universe
Hilbert space contains negative-norm states.

Returning again to the gravity model with end of the world branes, one could
complain that in order to cure the boundary interpretation we have ruined the
locality of the bulk TQFT theory. Indeed, the action for $S_{\partial}\neq 0$
no longer depends only on the Euler characteristic of the manifold, so it is
no longer consistent with cuttings and gluings of the spacetime.
Alternatively, if we insist on locality, the $S_{\partial}$ term can be
interpreted as the contribution of local degrees of freedom associated to the
boundaries, both brane and fixed. In that case, however, a question arises of
whether or not we should consider additional $\widehat{S}$ operators
corresponding to these additional degrees of freedom. Doing so would be
equivalent to considering the theory with $S_{\partial}=0$ just with more
flavors of end-of-the-world brane, so the problem would arise again. The
problem seems to require that the degrees of freedom propagate, unprobed,
along the boundaries. We refer the reader to the discussion of this boundary
term and its meaning in Marolf:2020xie . We address the problem as it shows up
in our case in section 5, where we offer a somewhat different description for
these degrees of freedom, and some speculation on their meaning.

The topological action of the model described in this section is, in fact,
that of a 2d TQFT with a one-dimensional (closed sector) Hilbert space. In
some sense it describes the simplest possible 2d TQFT. In what follows, we
will analyze gravity models built from more complicated 2d TQFTs. We will find
that much of this analysis can be reduced to that of the simpler one-
dimensional TQFT.

##  2 2d Dijkgraaf-Witten theory

The simple model of a gravity path integral from Marolf:2020xie  can be
generalized to include any topological action in the bulk. As a first example,
we will examine 2d Dijkgraaf-Witten theory, a topological gauge theory, as our
bulk theory and construct a gravity path integral from that. We will find a
dual interpretation of the gravity path integral in terms of a one-dimensional
ensemble theory whose Hilbert space is a random representation of the gauge
group. We also find that the correlation functions of boundary insertion
operators factorize between the irreducible representations of the gauge
group, similar to the results in Kapec:2019ecr . In section 3 we will see that
analogous features hold in the case of more general 2d TQFTs. Before
describing the gravity path integral, however, we will first briefly review
Dijkgraaf-Witten theory and present the results of Dijkgraaf-Witten theory in
two-dimensions that will be relevant to our construction.

###  2.1 Review of Dijkgraaf-Witten theory

Dijkgraaf-Witten theory is a topological gauge theory with finite symmetry
group $G$ Dijkgraaf:1989pz ; Freed:1991bn . The path integral is given as a
sum over $G$-principal bundles on the spacetime manifold. Given a connected,
manifold without boundary $M$, let ${\mathcal{C}}_{M}$ be the set of
$G$-principal bundles on $M$. A $G$-principal bundle on $M$ can be identified
with a homomorphism from the fundamental group $\pi_{1}(M,x)$ to the group
$G$, where $x$ is some chosen basepoint in $M$. So we will take
${\mathcal{C}}_{M}$ to be the set $\operatorname{Hom}(\pi_{1}(M,x),G)$.
Identified this way, some of the principal bundles can be related to each
other via residual gauge symmetries. Specifically, a gauge transformation
$g\in G$ with support over all of $M$ will act on a bundle
$\phi:\pi_{1}(M,x)\rightarrow G$ via conjugation, like so: $\phi(\cdot)\mapsto
g\,\phi(\cdot)\,g^{-1}$. The gauge invariant path integral must take these
gauge symmetries into account, and is thus over the space of $G$-principal
bundles ${\mathcal{C}}_{M}$, orbifolded by this action of $G$. We’ll denote
this orbifolded space by $\overline{{\mathcal{C}}_{M}}$.

The measure over $\overline{{\mathcal{C}}_{M}}$ induced by this orbifolding
will weight bundles inversely to the size of their stabilizer subgroup under
the action of $G$. Without any further weighting of the bundles beyond this,
the sum over $\overline{{\mathcal{C}}_{M}}$ gives the “untwisted” version of
Dijkgraaf-Witten theory. In that case the partition function for $M$ is
Freed:1991bn

$Z^{\text{DW}}[M]=\sum_{\phi\in\overline{{\mathcal{C}}_{M}}}\frac{1}{\left|\operatorname{Stab}(\phi)\right|}=\frac{\left|{\mathcal{C}}_{M}\right|}{\left|G\right|}\;.$
(21)

The numerator $\left|{\mathcal{C}}_{M}\right|$ is
$\left|\operatorname{Hom}(\pi_{1}(M,x),G)\right|$, the number of homomorphisms
from $\pi_{1}(M,x)$ to $G$. When $M_{g}$ is the connected, closed, oriented
surface of genus $g$, this count is given by a result known as Mednykh’s
formula:

$\left|{\mathcal{C}}_{M_{g}}\right|=\left|G\right|\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g}\;,$
(22)

where $q$ labels the irreducible representations of $G$, and $d_{q}$ are the
dimensions of each irreducible representation. (See jones_1998  and references
therein.) This gives the Dijkgraaf-Witten partition function of $M_{g}$ as

$Z^{\text{DW}}[M_{g}]=\frac{\left|{\mathcal{C}}_{M_{g}}\right|}{\left|G\right|}=\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g}\;.$
(23)

The above partition function is for a closed surface. If our manifold is a
surface with boundaries, the path integral is a sum over $G$-principal bundles
that satisfy given boundary conditions. Specifically each boundary component
is a circle with boundary condition given by a conjugacy class of $G$,
representing the holonomy around that circle. Let
${\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})$ denote the set of $G$-principal
bundles on the genus $g$ surface with $n$ boundary components having holonomy
boundary conditions $k_{1}$, …, $k_{n}$ respectively. The path integral is
again over the orbifolded space
$\overline{{\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})}$, and comes out to
$\operatorname{vol}(\overline{{\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})})=\left|{\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})\right|/\left|G\right|$.

A $G$-principal bundle on a surface $M_{g,n}$ with boundaries is still a
choice of homomorphism from $\pi_{1}(M_{g,n},x)$ to $G$, but now with the
restriction that it is compatible with the boundary conditions on $M_{g,n}$ in
the following sense: if a path $a_{i}\in\pi_{1}(M_{g,n},x)$ is homologous to
the $i$-th boundary, the bundle $\phi:\pi_{1}(M_{g,n},x)\rightarrow G$ must
map $a_{i}$ to an element of $k_{i}$. (Note this notion of compatibility is
well-defined, as all paths in $\pi_{1}$ homologous $a_{i}$ will be conjugates
of each other and hence must map to the same conjugacy class in $G$.) The
partition function will be given by the count
$\left|{\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})\right|$ of such compatible
homomorphisms $\phi$. Mednykh’s formula can be generalized to the case of a
connected surface with boundaries as

$\left|{\mathcal{C}}_{M_{g,n}}(k_{1},\ldots,k_{n})\right|=\left|G\right|\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n}\prod_{i}\frac{\left|k_{i}\right|}{\left|G\right|}\chi_{q}(k_{i}),$
(24)

where $g$ is the genus of the surface, $k_{1}$, …, $k_{n}$ are the respective
boundary conditions of the $n$ boundaries, and $\chi_{q}(k)$ is the
irreducible character $q$ evaluated on an element in $k$. (See Proposition 1
in mednykh_nonequivalent_1984 . This can also be obtained by first obtaining
the result for an $(n+2g)$-holed sphere $M_{0,n+2g}$ with given holonomies on
the boundaries. This is done by counting maps from $\pi_{1}(M_{0,n+2g})$, the
free group on $n+2g$ generators, to $G$ that satisfy the boundary constraints.
Then one can glue $2g$ of the boundaries together in pairs by summing over
boundary conditions.) The path integral on $M_{g,n}$ with boundary conditions
$k_{1}$, …, $k_{n}$ is thus

$Z^{DW}[M_{g,n}\,;\,k_{1},\ldots,k_{n}]=\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n}\prod_{i}\frac{\left|k_{i}\right|}{\left|G\right|}\chi_{q}(k_{i}).$
(25)

We can interpret this path integral on a manifold with boundaries as defining
a multilinear map, from the Hilbert space of $n$ circles to ${\mathbb{C}}$,
that takes the state $\ket{k_{1}}\otimes\cdots\otimes\ket{k_{n}}$ to the
complex number $Z^{{DW}}[M_{g,n}\,;\,k_{1},\ldots,k_{n}]$.

Note that the space of states on a circle is evidently spanned by states
labeled by conjugacy classes. We can take the path integral
$Z^{\text{DW}}[M_{0,2}\,;\,k_{1},k_{2}]$ as defining a bilinear pairing on
this Hilbert space. We get

$\displaystyle\big{(}\ket{k_{1}},\ket{k_{2}}\big{)}=Z^{\text{DW}}[M_{0,2}\,;\,k_{1},k_{2}]=\sum_{q}\frac{\left|k_{1}\right|}{\left|G\right|}\chi_{q}(k_{1})\frac{\left|k_{2}\right|}{\left|G\right|}\chi_{q}(k_{2})=\frac{\left|k_{1}\right|}{\left|G\right|}\delta_{k_{1},k_{2}^{-1}}\;.$
(26)

Under this pairing the states $\ket{k}$ are evidently linearly independent,
but they are not orthogonal. We can switch to a diagonal basis,
$\ket{q}\equiv\sum_{k}\chi_{q}(k^{-1})\ket{k}$, where $\chi_{q}(k^{-1})$ is
the character for irreducible representation $q$ evaluated at an element whose
inverse is in the conjugacy class $k$. In this basis labeled by irreps of $G$,
the pairing above is simply
$\big{(}\ket{q_{1}},\ket{q_{2}}\big{)}=\delta_{q_{1},q_{2}}$, so the $\ket{q}$
are orthogonal.666If we take the $\ket{q}$ basis as a real basis, then complex
conjugation induces the antiunitary map $\ket{k}\mapsto\ket{k^{-1}}$ with the
interpretation of a reflection. This antiunitary composed with the pairing
described above defines an inner product on the Hilbert space, under which the
states $\ket{k}$ are orthogonal and the states $\ket{q}$ are orthonormal. The
partition function for a connected, genus $g$ surface with boundaries labeled
(in this irrep basis) by $q_{1}$, …, $q_{n}$ is simply

$Z^{\text{DW}}[M_{g,n}\,;\,q_{1},\ldots,q_{n}]=\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n}\delta_{qq_{1}\cdots
q_{n}}.$ (27)

Note that this partition function for a connected manifold with boundaries
evaluates to zero, unless all boundaries are labeled by the same irreducible
representation.

In addition to “untwisted” Dijkgraaf-Witten described above, one can also
define a “twisted” generalization of Dijkgraaf-Witten by further weighting
each $G$-principal bundle in the path integral by a $U(1)$-valued
characteristic class $\alpha$ of the bundle. This is equivalent to adding a
term $iS_{\alpha}$ to the action satisfying
$\alpha[\phi]=e^{iS_{\alpha}[\phi]}$. So the twisted partition function for a
closed, connected manifold $M$ is

$Z^{\text{DW},\alpha}[M]=\sum_{\phi\in\overline{{\mathcal{C}}_{M}}}\frac{1}{\left|\operatorname{Stab}(\phi)\right|}\alpha[\phi]=\sum_{\phi\in\overline{{\mathcal{C}}_{M}}}\frac{1}{\left|\operatorname{Stab}(\phi)\right|}e^{iS_{\alpha}[\phi]}.$
(28)

In what follows, we will consider the untwisted case before discussing in
section 2.4 how the results are altered in the twisted case.

###  2.2 A Dijkgraaf-Witten gravity path integral

In preparation for defining a gravity path integral, we will include, in
addition to the Dijkgraaf-Witten action, the topological action term
$S_{0}\chi(M)$ of the simple theory described in Marolf:2020xie . This will
have the effect of suppressing higher genus manifolds in an eventual sum over
topology. With this addition to the action, the bulk theory partition function
is given by

$Z^{\text{bulk}}[M_{g,n}\,;\,q_{1},\ldots,q_{n}]=\sum_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n}\delta_{qq_{1}\cdots
q_{n}}.$ (29)

In the model of Marolf:2020xie  the action includes an additional, nonlocal
term $nS_{\partial}$ proportional to the number of boundaries. This can be
regarded as either the contribution to the action of some additional degrees
of freedom living on the boundary or as a rescaling of boundary insertion
operators in the gravity path integral. We won’t include this term here and
will discuss its meaning and inclusion in section 2.3. For now, we take our
partition function to be that described above, which is the partition function
of a TQFT; in other words, it is local, in the sense of being compatible with
cutting and gluing.

A gravity path integral defined from (29) will take as input a boundary
manifold (so, for a 2d bulk, some number of circles) with specified boundary
conditions, and will output the partition function (29) summed over all
manifolds with the given boundary conditions. Following the notation of
Marolf:2020xie , we denote the inclusion of a circle with boundary condition
$k$ by the operator $\widehat{Z}[k]$. The gravity path integral is

$\expectationvalue{\widehat{Z}[k_{1}]\cdots\widehat{Z}[k_{n}]}=\sum_{\begin{subarray}{c}\text{$M$
s.t. $\partial M$}\\\ \text{is $n$
circles}\end{subarray}}\mu(M)Z^{\text{bulk}}[M\,;\,k_{1},\ldots,k_{n}].$ (30)

Where $\mu(M)$ is the appropriate measure, with a factor of $1/m!$ whenever
$M$ has $m$ identical closed components.

The connected contribution to the vacuum correlator,
$\lambda\equiv\expectationvalue{{\mathbb{1}}}_{\text{conn.}}=\log\expectationvalue{{\mathbb{1}}}$,
is a sum over connected surfaces with no boundary, so in effect a sum over
genus:

$\lambda=\sum_{g}\sum_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{2-2g}=\sum_{q}\frac{\left(\frac{e^{S_{0}}d_{q}}{\left|G\right|}\right)^{2}}{1-\left(\frac{e^{S_{0}}d_{q}}{\left|G\right|}\right)^{-2}}=\sum_{q}\lambda_{q}.$
(31)

Here we’ve denoted $\sum_{g}\left(e^{S_{0}}d_{q}/\left|G\right|\right)^{2-2g}$
by $\lambda_{q}$. The full vacuum correlator is correspondingly
$\expectationvalue{{\mathbb{1}}}=e^{\lambda}=\prod_{q}e^{\lambda_{q}}$.

Calculating the correlators of boundary insertion operators will be easier in
the basis labeled by irreducible representations. To that end, define
operators $\widehat{Z}_{q}\equiv\sum_{k}\chi_{q}(k^{-1})\widehat{Z}[k]$,
corresponding to the TQFT states $\ket{q}$. We can define a generating
function for the general correlator,
$F(u_{q})=\expectationvalue{e^{\sum_{q}u_{q}\widehat{Z}_{q}}}$ with chemical
potentials $u_{q}$ for the insertion of each operator $\widehat{Z}_{q}$. The
logarithm of this generating function will simply be the corresponding
generating function for connected correlators

$\log
F(u_{q})=\sum_{\cdots,n_{q},\cdots}\prod_{q}\frac{1}{n_{q}!}u_{q}^{n_{q}}\expectationvalue{\prod_{q}\widehat{Z}_{q}^{n_{q}}}_{\text{conn.}}.$
(32)

Connected correlators are simple to calculate as the surfaces to be summed
over in the corresponding gravity path integral are parametrized by genus. For
example,

$\displaystyle\expectationvalue{\widehat{Z}_{q_{1}}\cdots\widehat{Z}_{q_{n}}}_{\text{conn.}}$
$\displaystyle=\sum_{g}Z^{\text{bulk}}[M_{g,n}\,;\,q_{1},\ldots,q_{n}]$ (33)
$\displaystyle=\sum_{g}\sum_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n}\delta_{qq_{1}\cdots
q_{n}}$
$\displaystyle=\lambda_{q_{1}}\left(e^{S_{0}}\frac{d_{q_{1}}}{\left|G\right|}\right)^{-n}\delta_{q_{1}\cdots
q_{n}}.$

Note that this connected correlator evaluates to zero unless all boundaries
are labeled by the same irreducible representation. This fact simplifies the
resulting expression for $\log F(u_{q})$. The correlator with $n_{q}$
boundaries for each type $q$ will be zero unless no more than one of the
$n_{q}$ is nonzero. So the sum over all possible numbers of boundary $n_{q}$
for each $q$ reduces to a sum over just one of the $n_{q}$, followed by a sum
over $q$. We obtain

$\log
F(u_{q})=\sum_{q}\sum_{n_{q}}\frac{1}{n_{q}!}u_{q}^{n_{q}}\lambda_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{-n_{q}}=\sum_{q}\lambda_{q}\exp(u_{q}\frac{\left|G\right|}{e^{S_{0}}d_{q}}).$
(34)

The final result of this simplification is that the full generating function
$F(u_{q})$ factorizes between the different labels $q$:

$F(\ldots,u_{q},\ldots)=\prod_{q}e^{\lambda_{q}\exp(u_{q}\left|G\right|/e^{S_{0}}d_{q})}.$
(35)

From (35) the full correlators can be extracted. They are

$\expectationvalue{\prod_{q}\widehat{Z}_{q}^{n_{q}}}=\prod_{q}e^{\lambda_{q}}B_{n_{q}}\\!(\lambda_{q})\left(\frac{\left|G\right|}{e^{S_{0}}d_{q}}\right)^{n_{q}},$
(36)

where $B_{m}$ denotes the $m$-th Touchard, or Bell, polynomial. Note the
normalized correlation functions have the property of factorizing between
boundaries labeled by different irreducible representations:

$\expectationvalue{\prod_{q}\widehat{Z}_{q}^{n_{q}}}/\expectationvalue{{\mathbb{1}}}=\prod_{q}\left(\expectationvalue{\widehat{Z}_{q}^{n_{q}}}/\expectationvalue{{\mathbb{1}}}\right),$
(37)

whereas no such factorization holds between boundaries label by the same
irrep, e.g.
$\expectationvalue{\widehat{Z}_{q}^{n+m}}\nsim\expectationvalue{\widehat{Z}_{q}^{n}}\expectationvalue{\widehat{Z}_{q}^{m}}$.
The correlators for the operators $\widehat{Z}[k]$ can be gotten through the
change of basis back from the $\widehat{Z}_{q}$ operators to the
$\widehat{Z}[k]$ operators, namely
$\widehat{Z}[k]=\sum_{q}\frac{\left|k\right|}{\left|G\right|}\chi_{q}(k)\widehat{Z}_{q}$.

###  2.3 Boundary interpretation

The above result (36) is analogous to having multiple copies of the model
presented in Marolf:2020xie . In fact, each factor labeled by $q$ is
equivalent to one copy of the model of Marolf:2020xie , where
$e^{S_{0}}\rightarrow e^{S_{0}}\frac{d_{q}}{|G|}$. We will see that our
gravity path integral with a Dijkgraaf-Witten bulk likewise has a dual
interpretation as a random theory living on the boundary. This will in fact be
equally true of any 2d TQFT satisfying Atiyah’s axioms, as we will demonstrate
in section 3. We present Dijkgraaf-Witten theory here as a representative
example.

In a standard holographic boundary interpretation the $\widehat{Z}$ operators
would get reinterpreted as the partition functions of a one-dimensional
theory. In our case, however, this is impossible because the correlators of
the $\widehat{Z}$ operators do not factorize. Instead we will look for an
ensemble of one-dimensional theories and interpret the correlator
$\expectationvalue{\widehat{Z}[k_{1}]\cdots\widehat{Z}[k_{n}]}/\expectationvalue{{\mathbb{1}}}$
as the average of the quantity $Z[k_{1}]\cdots Z[k_{n}]$. The boundary
theories in our ensemble are characterized by the value they assign to each
operator $\widehat{Z}[k]$, so the ensemble will be a probability distribution
over the space ${\mathbb{C}}^{r}$, where $r$ is the number of irreducible
representations of $G$ and where each copy of ${\mathbb{C}}$ represents the
values that one of the $\widehat{Z}[k]$ can take. This makes the correlators
$\expectationvalue{\widehat{Z}[k_{1}]\cdots\widehat{Z}[k_{n}]}/\expectationvalue{{\mathbb{1}}}$
interpretable as moments of the ensemble probability distribution. The problem
of finding the ensemble probability distribution from the correlators (36) is
thus an instance of the so-called moment problem, wherein one attempts to find
a probability distribution from its moments.

If we let $\vec{\alpha}\in{\mathbb{C}}^{r}$ index the theories in our
ensemble, the probability distribution $p_{\alpha}$ over the theories should
satisfy
$\expectationvalue{\widehat{Z}_{q_{1}}\cdots\widehat{Z}_{q_{n}}}/\expectationvalue{{\mathbb{1}}}=\int\\!d^{r}\\!\alpha\,p_{\alpha}\alpha_{1}\cdots\alpha_{r}$.
In terms of the generating function $F(u_{1},\ldots,u_{r})$ for the
correlators,

$F(u_{1},\ldots,u_{r})/\expectationvalue{{\mathbb{1}}}=e^{\sum_{q}\lambda_{q}\left(\exp(u_{q}\left|G\right|/e^{S_{0}}d_{q})-1\right)}=\int\\!d^{r}\\!\alpha\,p_{\alpha}e^{\sum_{q}u_{q}\alpha_{q}}.$
(38)

We can extract the function $p_{\alpha}$ by performing a Fourier transform
with respect to the variables $iu_{q}$. The result is

$p_{\alpha}=\prod_{q}\sum_{N_{q}=0}^{\infty}\frac{\lambda_{q}^{N_{q}}}{{N_{q}}!e^{\lambda_{q}}}\delta\left(\alpha_{q}-\frac{\left|G\right|}{e^{S_{0}}d_{q}}{N_{q}}\right)\;.$
(39)

In other words, each $\alpha_{q}$ takes the values
$\frac{\left|G\right|}{e^{S_{0}}d_{q}}N_{q}$ where the $N_{q}$ are random
integers drawn independently from Poisson distributions with respective means
$\lambda_{q}$. Recall that the $\alpha_{q}$ are the values of $Z_{q}$ in the
different theories in our ensemble, so

$Z_{q}=\frac{\left|G\right|}{e^{S_{0}}d_{q}}N_{q}\;.$ (40)

Switching from the $Z_{q}$ to the $Z[k]$ basis gives

$Z[k]=\sum_{q}\frac{\left|k\right|}{e^{S_{0}}d_{q}}\chi_{q}(k)N_{q}\;.$ (41)

Our bulk theory is topological, so it’s boundary dual will likewise be
topological. As the $\widehat{Z}[k]$ operators represent the insertion of a
boundary with holonomy $k$, we are tempted to interpret $Z[k]$ as a partition
function in a one-dimensional topological quantum mechanics theory, with an
insertion of a $G$-symmetry operator with conjugacy class $k$. That is to say,
$Z[k]=\tr(U(g))$, where $U(g)$ is the representation on the Hilbert space of a
group element $g\in k$. In fact, this is only nearly so. Looking at (41) we
see that $Z[k]$ has the form of a trace of an element of $k$ in a
representation that has $\frac{\left|k\right|}{e^{S_{0}}d_{q}}N_{q}$ copies of
the representation $q$, for each $q$. Unfortunately for this interpretation,
$\frac{\left|k\right|}{e^{S_{0}}d_{q}}N_{q}$ is not necessarily an integer,
which it would have to be to avoid the absurdity of a theory with a fractional
number of copies of a representation. One immediate fix would appear to be
picking a specific value for $S_{0}$ such that
$\frac{\left|G\right|}{e^{S_{0}}d_{q}}\in{\mathbb{Z}}$. This is not possible
though, as it would ruin the convergence of eq. (33) and, what’s worse, would
render $\lambda_{q}$ negative, giving negative probabilities in our ensemble
distribution.

On the other hand, going back to the $Z_{q}$ operators, in light of their
definition $Z_{q}=\sum_{k}\chi_{q}(k^{-1})Z[k]$ a natural interpretation would
be for $Z_{q}$ to be the contribution to the partition function of states with
charge $q$. That is to say, $Z_{q}=\tr(P_{q})$, where $P_{q}$ is a projection
onto states living in copies of irreducible representation $q$. Again, this is
nearly so, but unfortunately
$Z_{q}=\frac{\left|G\right|}{e^{S_{0}}d_{q}}N_{q}$ is not an integer for all
$N_{q}$. One possible solution is to identify the size of the $q$-sector,
$\tr(P_{q})$, with a rescaled operator $e^{S_{q}}Z_{q}$ rather than with
$Z_{q}$, where we choose $e^{S_{q}}$ so that $e^{S_{q}}Z_{q}$ is an integer in
every $\alpha$-state. (We will discuss a possible motivation for this
rescaling in section 5.) For now, we can interpret this rescaling as a
modification of the expected holographic map $Z_{q}\leftrightarrow\tr{P_{q}}$
to be $e^{S_{q}}Z_{q}=\tr(P_{q})$. Similarly, we can to identify $\tr(U(g))$,
with an appropriately rescaled operator $e^{S_{k}}Z[k]$, rather than with
$Z[k]$ as is, in order to avoid the situation of having a fractional number of
copies of a representation. A choice of rescalings for the $\widehat{Z}_{q}$
and the $\widehat{Z}[k]$ that avoids noninteger dimensions, that avoids
noninteger copies of irreducible representations, and that respects the
identity $\tr(U(g))=\sum_{q}\tr(P_{q})\chi_{q}(g)/d_{q}$ is

$\displaystyle S_{k}$
$\displaystyle=S_{0}+\log(\left|G\right|/\left|k\right|)$ (42) $\displaystyle
S_{q}$ $\displaystyle=S_{0}+\log d_{q}\;.$ (43)

These result in a consistent interpretation of the gravity model as a boundary
theory with $\frac{|G|}{d_{q}}N_{q}\in{\mathbb{Z}}$ copies of the irreducible
representation $q$.777 This is an integer by the basic result from
representation theory that $|G|/d_{q}\in{\mathbb{Z}}$ for any irreducible
representation.

###  2.4 Twisted Dijkgraaf-Witten

We now turn our attention to twisted Dijkgraaf-Witten theories. After a review
of the essential background, we use twisted Dijkgraaf-Witten in our gravity
models, and describe their boundary interpretation.

####  2.4.1 Background

The purpose of this subsection is to describe a practical way to compute the
partition function of “twisted” 2d Dijkgraaf-Witten. Here we follow the
exposition in Kapec:2019ecr , and refer the reader there for details. See also
Turaev_2007 . We will see that the partition functions of 2d twisted
Dijkgraaf-Witten are given as a sum similar to (25) but over projective
representations of $G$.

As was briefly mentioned above, the different twisted theories are labeled by
characteristic classes of $G$-principal bundles. The different characteristic
classes that describe the possible twisted action terms are classified by
elements of $H^{2}(BG,U(1))$, the second $U(1)$-valued cohomology classes of
the classifying space $BG$ Dijkgraaf:1989pz ; Freed:1991bn . Specifically,
given a class $\alpha$ in $H^{2}(BG,U(1))$, and viewing a $G$-principal bundle
on a manifold $M$ as a map $\phi$ from $M$ to $BG$, consider the pullback of
$\phi^{*}\alpha$ along this map. Evaluating $\phi^{*}\alpha$ on $M$ gives a
phase $\phi^{*}\alpha(M)=\alpha(\phi(M))\in U(1)$ for the bundle $\phi$. The
partition function on $M$ is a sum over inequivalent $G$-principal bundles
$\phi$ weighted by these phases. See equation (28). For simplicity, in what
follows we will only consider the particular case where
$H^{2}(BG,U(1))={\mathbb{Z}}_{N}$.

One simplification afforded by restricting to the case
$H^{2}(BG,U(1))={\mathbb{Z}}_{N}$ is that all the twisted Dijkgraaf-Witten
actions can be described in terms of any characteristic class $\alpha_{0}\in
H^{2}(BG,U(1))$ which generates the others, so
$H^{2}(BG,U(1))={\mathbb{Z}}_{N}=\\{\alpha_{0}^{k}:~{}k=0,...,N-1\\}$. The
first part of this section is devoted to the construction of such a
$\alpha_{0}$. Then, after eq. (46) we consider $\alpha_{0}^{k}$ with general
$k$ and show how the partition functions for general $k$ become sums over
projective representations of $G$.

We will relate $\alpha_{0}$ to the failure of the bundle $\phi$ to lift to a
$\widetilde{G}$-principal bundle when $\widetilde{G}$ is a central extension
of $G$ by $Z_{N}$. Consider a $G$-bundle on a surface $M$ described in terms
of transition functions $g_{ij}$ which satisfy the triple overlap condition

$\displaystyle g_{ij}g_{jk}g_{ki}=1\;.$ (44)

Picking a lift of each transition function $g_{ij}\mapsto\widetilde{g}_{ij}$,
we get

$\displaystyle\widetilde{g}_{ij}\widetilde{g}_{jk}\widetilde{g}_{ki}=\widetilde{c}_{ijk}\;\in\;{\mathbb{Z}}_{N}\;,$
(45)

which for nontrivial $c_{ijk}$ is a violation of the
$\widetilde{G}_{S}$-bundle cocycle condition. Gauge transformations and
changes of lift can change $\widetilde{c}_{ijk}$ locally but in general there
is a global obstruction to removing all such violations. The assignment of an
element of ${\mathbb{Z}}_{N}$ to each triple intersection, modulo gauge
transformations and changes of lift, defines a 2-cocycle

$\displaystyle[\omega_{N}]\in H^{2}(M,{\mathbb{Z}}_{N})\;,$ (46)

which can be paired with the 2-cycle $[M]$ to give an element
$\omega=\expectationvalue{[\omega_{N}],[M]}\in{\mathbb{Z}}_{N}$. Note that the
element $\omega\in{\mathbb{Z}}_{N}$ depends on the bundle $\phi$. The
characteristic class $\alpha_{0}\in H^{2}(BG,U(1))$ is then defined via
$\alpha_{0}(\phi(M))=e^{2\pi i\omega(\phi)/N}$. In fact, these are the phases
by which the Dijkgraaf-Witten theory described by $\alpha_{0}$ weights each
bundle.

The characteristic class $\alpha_{0}$ obtained this way depends on the choice
of the central extension $\widetilde{G}$. We would like to choose
$\widetilde{G}$ so that $\alpha_{0}$ is a generating element of
$H^{2}(BG,U(1))={\mathbb{Z}}_{N}$ so that any Dijkgraaf-Witten action can be
obtained as $\alpha_{0}^{k}$ for some $k$. We are also interested in
$\widetilde{G}$ with the property that all irreducible representations of $G$,
projective or linear, can be lifted to linear representations of
$\widetilde{G}$. There is always a central extension of $G$ that satisfies
these criteria, namely the Schur covering group of $G$. The Schur covering
group $\widetilde{G}_{S}$ is a central extension

$\displaystyle 1\rightarrow H_{2}(G,{\mathbb{Z}})\stackrel{{\scriptstyle
i}}{{\rightarrow}}\widetilde{G}_{S}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}G\rightarrow
1\;.$ (47)

For finite groups $G$,

$\displaystyle H^{2}(BG,U(1))=H^{2}(G,U(1))=H_{2}(G,{\mathbb{Z}})\;.$ (48)

Because we are restricting to the case $H^{2}(BG,U(1))=Z_{N}$, this means the
kernel of the quotient map $\pi:\widetilde{G}_{S}\rightarrow G$ is
$H_{2}(G,{\mathbb{Z}})=Z_{N}$. So $\widetilde{G}_{S}$ is indeed a central
extension of $G$ by ${\mathbb{Z}}_{N}$.

For what follows it is convenient to classify the irreducible representations
of $\widetilde{G}_{S}$ by how they represent the subgroup $\ker\pi=Z_{N}$.
Notate the generator of $\ker\pi=Z_{N}$ by $e^{2\pi i/N}$. For any irreducible
representation $\widetilde{q}$ of $\widetilde{G}_{S}$, the element $e^{2\pi
i/N}$ must get represented as

$\displaystyle U^{(\widetilde{q})}(e^{2\pi i/N})=e^{2\pi
ik^{\prime}(\widetilde{q})/N}{\mathbb{1}}\;,$ (49)

for some $k^{\prime}(\widetilde{q})\in\\{0,1,\ldots,N-1\\}$. Refer to the
value $k^{\prime}(\widetilde{q})$ for a given $\widetilde{q}$ as the $N$-ality
of $\widetilde{q}$. Notice that, of course, a choice of how to represent
$e^{2\pi i/N}$ doesn’t completely specify $\widetilde{q}$, so that there are
many $\widetilde{q}$ of the same $N$-ality. We will see that the partition
function of Dijkgraaf-Witten theory with “twist” $k^{\prime}$ can be written
as a sum over representations with $N$-ality $k^{\prime}$.

Bringing everything together, let’s calculate the partition function of
twisted Dijkgraaf-Witten on a manifold with boundaries. First recall that in
the untwisted case (section 2.2), we were simply counting isomorphism classes
of $G$-bundles (without weighting them with phases, i.e. $k=0$) consistent
with conjugacy classes specified on closed boundaries. A gauge transformation
keeps us in the same isomorphism class and changes the boundary holonomy only
up to conjugacy, so that the counting is well-defined. However, in the twisted
case, although the counting is still a well-defined problem, the weighting is
not well-defined, when there are boundaries of fixed conjugacy class.

On a closed manifold, a gauge transformation will not change the cohomology
class $\omega_{N}$ defined in (46). This is not neccesarily the case, however
on a manifold with boundaries. A gauge transformation that is nonzero at the
boundary can nontrivially transform $\omega_{N}$:

$\displaystyle\omega_{N}\rightarrow\omega_{N}+db$ (50)

where $b$ is a 1-cocycle. Then on a manifold with boundaries

$\displaystyle\delta\omega=\expectationvalue{db,[M]}=\expectationvalue{b,\partial[M]}$
(51)

is potentially nonzero, causing phase ambiguities in the partition function

$\displaystyle e^{2\pi ik\omega(\phi)/N}\rightarrow e^{2\pi
ik\omega(\phi)/N}e^{2\pi ik\delta\omega(\phi)/N}\;.$ (52)

The presence of these phase ambiguities implies that twisted Dijkgraaf-Witten
is not gauge invariant on manifolds with boundary, so on the boundary there
has to be a theory with an ’t Hooft anomaly. To have an unambiguously defined
bulk partition function, we must make an ad hoc choice for those phases. We
can do this by picking a lift $U_{i}\rightarrow\widetilde{U}_{i}$ for the each
specified boundary holonomy $U_{i}$.

We are now equipped to calculate the partition function of twisted Dijkgraaf-
Witten theory on a manifold with boundaries. Remember that for a Riemann
surface with genus $g$ and $n$ boundary components, the holonomies have to be
such that

$\displaystyle V_{1}W_{1}V_{1}^{-1}W_{1}^{-1}\cdots
V_{g}W_{g}V_{g}^{-1}W_{g}^{-1}U_{1}\cdots U_{n}=1_{G}\;,$ (53)

where $g$ is the genus and $U_{i}$ are the holonomies on the $n$ boundaries of
$M$. Trying to uplift in a straightforward way would give

$\displaystyle
u=\widetilde{V}_{1}\widetilde{W_{1}}\widetilde{V_{1}}^{-1}\widetilde{W_{1}}^{-1}\cdots\widetilde{V}_{g}\widetilde{W}_{g}\widetilde{V}_{g}^{-1}\widetilde{W}_{g}^{-1}\widetilde{U}_{1}\cdots\widetilde{U}_{n}\in{\mathbb{Z}}_{N}\;,$
(54)

which means that this $G$-bundle can’t be uplifted to a
$\widetilde{G}_{S}$-bundle. In such a case $u$ is in fact $u=e^{2\pi
i\omega(\phi)/N}$ where $\omega$ is that defined earlier. Although this
$G$-bundle doesn’t uplift, we could excise a disk and impose the holonomy
$u^{-1}$ on this new boundary to create a $\widetilde{G}_{S}$-bundle, albeit
over a manifold with an extra boundary. This gives us a way to count the
number of $G$-bundles that are weighted by the same value $\omega$.
Specifically, we get the number of $G$-bundles on $M_{g;U_{1},\ldots,U_{n}}$
with weight $\omega$ by (correctly) counting the number of
$\widetilde{G}_{S}$-bundles on
$M_{g;\widetilde{U}_{1},\ldots,\widetilde{U}_{n},u^{-1}}$. Counting correctly
here means accounting for the fact that when a $G$-bundle can be uplifted
there are $N^{2g+n}$ possible upliftings. We have

$Z_{G}^{(k)}\big{[}g\,;\,\widetilde{U}_{1},\ldots,\widetilde{U}_{n}\big{]}=\sum_{\phi}\frac{e^{2\pi
ik\omega(\phi)/N}}{|\operatorname{Stab}(\phi)|}=\sum_{\omega=0}^{N-1}e^{2\pi
ik\omega/N}\frac{\big{|}\widetilde{G}\big{|}\,Z^{(0)}_{\widetilde{G}_{S}}\big{[}g\,;\,\widetilde{U}_{1},\ldots,\widetilde{U}_{n},e^{-2\pi
i\omega}\big{]}}{N^{2g+n-1}}$ (55)

where
$Z^{(0)}_{\widetilde{G}_{S}}\big{[}g\,;\,\widetilde{U}_{1},\ldots,\widetilde{U}_{n},e^{-2\pi
i\omega}\big{]}$ is the partition function of untwisted Dijkgraaf-Witten
theory for $\widetilde{G}_{S}$ on
$M_{g;\widetilde{U}_{1},\ldots,\widetilde{U}_{n},u^{-1}}$. Using our known
expression for the partition function of untwisted Dijkgraaf-Witten theory

$Z^{(0)}_{\widetilde{G}_{S}}\big{[}g\,;\,\widetilde{U}_{1},\ldots,\widetilde{U}_{n},u^{-1}\big{]}=\sum_{\widetilde{q}}\left(\frac{d_{\widetilde{q}}}{\big{|}\widetilde{G}\big{|}}\right)^{2-2g-n}\prod_{i=1}^{n}\left(\frac{\big{|}\widetilde{U}_{i}\big{|}}{\big{|}\widetilde{G}\big{|}}\chi_{\widetilde{q}}(\widetilde{U}_{i})\right)\frac{|u^{-1}|}{\big{|}\widetilde{G}\big{|}}\chi_{\widetilde{q}}(u^{-1})$
(56)

and substituting this in to (55) gives the result

$Z_{G}^{(k)}\big{[}g\,;\,\widetilde{U}_{1},\ldots,\widetilde{U}_{n}\big{]}=\sum_{\widetilde{q}}\delta_{k,k^{\prime}({\widetilde{q}})}\left(\frac{d_{\widetilde{q}}}{|G|}\right)^{2-2g-n}\prod_{i=1}^{n}\left(\frac{\big{|}\widetilde{U}_{i}\big{|}}{\big{|}\widetilde{G}\big{|}}\,\chi_{\widetilde{q}}\big{(}\widetilde{U}_{i}\big{)}\right)\;$
(57)

after expanding $\chi_{\widetilde{q}}(u^{-1})=d_{\widetilde{q}}\,e^{-2\pi
ik^{\prime}({\widetilde{q}})\omega/N}$ then simplifying via the identity
$\sum_{\omega}e^{2\pi i\omega(k-k^{\prime})/N}=N\delta_{k,k^{\prime}}$.

####  2.4.2 The gravity path integral

Since the form of eq. (57) is very similar to the untwisted case, the
calculations proceed in the same fashion. We define boundary conditions
labeled by projective representation $\widetilde{q}$ by
$\widehat{Z}^{(k)}_{\widetilde{q}}=\sum_{\widetilde{k}}\chi_{\widetilde{q}}\big{(}\widetilde{k}^{-1}\big{)}\widehat{Z}^{(k)}[\widetilde{k}]$,
where the sum goes over all conjugacy classes $\widetilde{k}$ of
$\widetilde{G}_{S}$. This definition makes sense for any irreducible
representation $\widetilde{q}$ of $\widetilde{G}_{S}$, but the representations
of the wrong $N$-ality will give zero in correlators because

$\displaystyle
Z^{(k)}_{G}\big{[}M_{g,n}\,;\,\widetilde{q}_{1},\ldots,\widetilde{q}_{n}\big{]}=\sum_{\widetilde{q}}\delta_{k,k^{\prime}(\widetilde{q})}\left(e^{S_{0}}\frac{d_{\widetilde{q}}}{\left|G\right|}\right)^{2-2g-n}\delta_{\widetilde{q}\widetilde{q}_{1}\cdots\widetilde{q}_{n}}\;,$
(58)

as can be readily worked out from (57). Thus the most general correlator, one
with $n_{\widetilde{q}}$ insertions of each boundary with conditions
$\widetilde{q}$, is

$\expectationvalue{\prod_{\begin{subarray}{c}\text{$\widetilde{q}$ with}\\\
\text{$N$-ality
$k$}\end{subarray}}\widehat{Z}_{\widetilde{q}}^{n_{\widetilde{q}}}}=\prod_{\begin{subarray}{c}\text{$\widetilde{q}$
with}\\\ \text{$N$-ality
$k$}\end{subarray}}e^{\lambda_{\widetilde{q}}}B_{n_{\widetilde{q}}}\\!(\lambda_{\widetilde{q}})\left(\frac{\left|G\right|}{e^{S_{0}}d_{\widetilde{q}}}\right)^{n_{\widetilde{q}}}\;,$
(59)

with $\lambda_{\widetilde{q}}$ defined analogously to $\lambda_{q}$ in the
untwisted case. From the similarity of these correlators to those of the
untwisted case, we immediately recognize the boundary dual of our twisted
Dijkgraaf-Witten bulk gravity model. Namely, (after making the rescalings
described in section 2.3) it is a 1d topological theory whose Hilbert space
consists of random numbers of _projective_ representations of $G$ that have
the correct $N$-ality. In other words, the boundary dual is an ensemble of 1d
topological theories with anomalous global symmetry $G$.

##  3 General 2d TQFTs

In the previous section we constructed a simple gravity path integral with a
bulk action of Dijkgraaf-Witten theory and found a dual interpretation as an
ensemble of 1d theories on its boundary. We will now show that a general 2d
TQFT bulk theory likewise leads to a model gravity path integral with similar
features. To be precise, we will consider here TQFTs as defined by Atiyah’s
axioms Atiyah  and over the field ${\mathbb{C}}$. Such TQFTs are fairly
simple. They have finite dimensional Hilbert spaces, and, as we will discuss,
can be viewed as a direct sum of theories all with Hilbert space dimension 1.
Though they are simple, we speculate that the important features of our
analysis will extend appropriately to 2d TQFTs more broadly defined, and
perhaps even, in some form, to TQFTs in higher dimensions.

There are many excellent expositions of TQFTs;888See, for example
Carqueville:2017fmn ; or for a more abstract and more general exposition see
lurie . we present here only the most basic sketch for those not familiar. In
the language of category theory, a 2d TQFT can be defined as a functor (with
certain requirements) from the category of 2d cobordisms $\mathbf{Cob}(2)$ to
the category of complex vector spaces $\mathbf{Vect}({\mathbb{C}})$. This
definition, again, for those not familiar with this language, is a concise
codification of the cutting and gluing properties that would naturally be
expected of a path integral. Like a path integral, a 2d TQFT assigns to each
closed 2d manifold a number, the partition function on that manifold. To a
closed 1d manifold it assigns a Hilbert space. To a 2d manifold $M$ with
boundary $\partial M$ it assigns a state in the Hilbert space associated to
the boundary $\partial M$.999Or, alternatively, an object that takes as input
a state and outputs a complex number. These assignments are compatible in the
way expected of the output of a path integral. For example, gluing two
boundary components of a manifold together corresponds to a sum over matching
states on the two components. As another example, if a TQFT assigns to a
circle the Hilbert space ${\mathcal{H}}_{S^{1}}$, then it assigns to the
“handle creation operator” (the manifold as seen in fig. 1) a unitary map
${\mathcal{H}}_{S^{1}}\rightarrow{\mathcal{H}}_{S^{1}}$. From this map and the
state in ${\mathcal{H}}_{S^{1}}$ assigned to a hemisphere, we can construct
any closed manifold by gluing. This handle creation map always has positive,
real eigenvalues Durhuus_1994 . We will denote these eigenvalues by
$\mu_{I}^{-2}$ where $I=1,\ldots,\dim({\mathcal{H}}_{S^{1}})$.

Figure 1: The handle creation operator:
${\mathcal{H}}_{S^{1}}\rightarrow{\mathcal{H}}_{S^{1}}$.

In the eigenbasis, $\ket{I}$, of the handle creation operator, the calculation
of partition functions becomes a simple matter. In particular, the partition
function of a connected manifold $M_{g,n}$ with genus $g$ and $n$ boundaries,
and states $\ket{I_{1}}$,…, $\ket{I_{n}}$ input on the boundaries
respectively, is

$Z^{\text{TQFT}}[M_{g,n}\,;\,I_{1},I_{2},\ldots,I_{n}]=\sum_{I}\mu_{I}^{2-2g-n}\delta_{II_{1}I_{2}\cdots
I_{n}}\;.$ (60)

Note this evaluates to zero if the boundary labels $I_{i}$ are not all the
same.

From the result (60), the generating function for connected correlation
functions of the gravity path integral is

$\displaystyle\log\expectationvalue{e^{\sum_{I}u_{I}\widehat{Z}_{I}}}$
$\displaystyle=\sum_{I}\sum_{n}\frac{u_{I}^{n}}{n!}\sum_{g}\mu_{I}^{2-2g-n}$
(61) $\displaystyle=\sum_{I}\lambda_{I}e^{u_{I}/\mu_{I}}\;,$

where $\lambda_{I}\equiv\mu_{I}^{2}/(1-\mu_{I}^{-2})$. The full generating
function of correlators is then
$\expectationvalue{\exp(\sum_{I}u_{I}\widehat{Z}_{I})}=\prod_{I}\exp(\lambda_{I}e^{u_{I}/\mu_{I}})$,
from which we can extract the result

$\expectationvalue{\prod_{I}\widehat{Z}_{I}^{n_{I}}}=\prod_{I}e^{\lambda_{I}}B_{n_{I}}(\lambda_{I})\mu_{I}^{-n_{I}}\;.$
(62)

This can be written

$\expectationvalue{\prod_{I}\widehat{Z}_{I}^{n_{I}}}/\expectationvalue{{\mathbb{1}}}=\prod_{I}\sum_{N_{I}}p_{\lambda_{I}}(N_{I})\left(\frac{N_{I}}{\mu_{I}}\right)^{n_{I}}$
(63)

where $p_{\lambda_{I}}$ here is the Poisson distribution with mean
$\lambda_{I}$. The normalized correlators then have an interpretation as an
average, where $Z_{I}$ takes the value $N_{I}/\mu_{I}$, and the $N_{I}$ are
independently chosen Poisson random integers. As we saw for the case of
Dijkgraaf-Witten, the $Z_{I}$ don’t quite have the interpretation as partition
functions of a one-dimensional topological theory. If, however, we rescale
each of them by a factor of, in this case, $\mu_{I}$, we do find a nice
interpretation for them. Namely we have as our boundary theory a topological
quantum mechanics with sectors labeled by $I$ and the number of dimensions in
each sector $I$ given by the Poisson distribution with mean $\lambda_{I}$.
Then $\mu_{I}Z_{I}=\tr(P_{I})$ where $P_{I}$ is a projection onto the sector
$I$.

##  4 2d TQFTs with boundaries

In their simple model of a gravity path integral, Marolf:2020xie  consider the
addition of end-of-the-world branes (EofW branes). These are boundaries on
which bulk spacetime ends. Generalizing this construction, we will consider
general boundaries for general 2d TQFTs and the describe the model gravity
path integrals built out of them. As a first example we will again consider
Dijkgraaf-Witten theory, but this time with the addition of EofW branes. As we
will explain, EofW branes are in some sense the simplest boundary conditions
possible in the theory, but we will nonetheless see features that will hold in
the more general case. These include the factorization between different
sectors of the theory, as before, as well as a new difficulty with
interpreting the gravity path integral as a boundary ensemble theory without
an additional modification. We will discuss this difficulty and a solution to
it in section 5.

###  4.1 End-of-the-world branes for Dijkgraaf-Witten

As a first example of an open/closed TQFT we consider Dijkgraaf-Witten theory
with the addition of end-of-the-world (EofW) brane boundaries. Following the
constructions of Marolf:2020xie ; Balasubramanian:2020jhl , we allow for some
number of “flavors” of otherwise identical EofW branes, which we will label by
$a=1,2,\ldots,K$. Though these $K$ differently labeled EofW branes have
identical dynamics, the number $K$ of such branes will play an important role
later. The path integral for Dijkgraaf-Witten theory with EofW branes is still
defined as a sum over gauge backgrounds, but now that our spacetime manifolds
have boundaries, the specification of a gauge background must include two
kinds of data: the holonomies around loops (as before), as well as the
parallel transports along paths that begin and end on EofW branes. Cutting a
manifold with brane boundaries will result in a new, non-brane boundary. These
resulting boundaries, which we will call variously “gluing” boundaries,
“state” boundaries, or “Hilbert space” boundaries, are not EofW brane
boundaries, but rather, correspond to a Hilbert space of states that represent
the input of field configuration data on that boundary. The “gluing”
boundaries, in our case, will have fixed parallel transports along them which
serve as boundary conditions for the possible gauge backgrounds. More
specifically, a circle gluing boundary component will be labeled by the
holonomy about that circle, and an interval gluing boundary component will be
labeled by an element of $G$ representing the parallel transport across that
interval. Note, that the intervals, unlike the circles, are labeled by
_elements_ of $G$, rather than conjugacy classes. This is similar to the
construction of Balasubramanian:2020jhl . Though they consider boundary
conditions for spin structures, these are analogous to our case with the
choice of $G={\mathbb{Z}}_{2}$.101010Where
${\mathbb{Z}}_{2}=\\{\mathrm{id},a\\}$, NS boundary conditions on a circle are
analogous to a holonomy of $\mathrm{id}$ about the circle, and R boundary
conditions are analogous to a holonomy of $a$ around the circle. Likewise, an
interval labeled by the identity (by the non-identity $(-1)^{F}$) is analogous
to an interval with parallel transport $\mathrm{id}$ (parallel transport $a$).
Substituting ${\mathbb{Z}}_{2}$ for $G$ in what follows will largely mirror
much of their discussion, where one keeps in mind that the irreps of an
abelian group like ${\mathbb{Z}}_{2}$ are all one-dimensional.

EofW branes, unlike gluing boundaries, do not get labeled by group elements or
conjugacy classes of $G$. We take the EofW branes to be “decoupled” from the
bulk, in the sense that inserted $G$-symmetry operators are not permitted to
end on EofW branes, which is to say, there is no parallel transport as one
moves along an EofW brane boundary.

When boundary conditions for gauge backgrounds are specified along circle and
interval gluing boundaries, the count of gauge backgrounds will only include
those that match the specified holonomies and parallel transports along those
boundaries. Counting gauge backgrounds can be done using a lattice
description, as explained in appendix A.111111Alternatively, given a connected
manifold $M$ with end-of-the-world-brane boundary $\partial_{\text{brane}}M$,
a choice of gauge background can be identified with a homomorphism from
$\pi_{1}\left(M/\partial_{\text{brane}}M,m\right)$ to $G$, where
$M/\partial_{\text{brane}}M$ is the quotient manifold obtained by identifying
all end-of-the-world brane boundaries to a single point $m$, and where we take
that point $m$ as our basepoint. We list some results, from which all
partition functions can be calculated. First, consider a strip with EofW
branes on both sides and gluing boundaries at the ends, with boundary
conditions $g_{1},g_{2}\in G$ respectively. There is one gauge background
compatible with the boundary conditions if $g_{1}=g_{2}^{-1}$ and zero
otherwise, giving the partition function

$\vbox{\hbox{\includegraphics[width=108.405pt]{DW_pairing_k_basis.pdf}}}=\quad\quad\delta_{g_{1},g_{2}^{-1}}\;.$
(64)

Taking the different boundary conditions $g\in G$ as states in the Hilbert
space ${\mathcal{H}}_{\text{open}}$ associated to the interval, the above
diagram gives a (nondegenerate) pairing on that Hilbert space,
$\left(\ket{g_{1}},\ket{g_{2}}\right)=\delta_{g_{1},g_{2}^{-1}}$. Second,
consider a disk with three gluing boundary intervals labeled by
$g_{1},g_{2},g_{3}\in G$, alternating with three EofW brane intervals. This
path integral gives $\delta(g_{3}g_{2}g_{1},1_{G})$. We can likewise interpret
this as a map from the Hilbert spaces of the intervals to ${\mathbb{C}}$,
namely
$\ket{g_{1}}\ket{g_{2}}\ket{g_{3}}\mapsto\delta(g_{3}g_{2}g_{1},1_{G})$. This
along with the pairing above induces a map
${\mathcal{H}}_{\text{open}}\otimes{\mathcal{H}}_{\text{open}}\rightarrow{\mathcal{H}}_{\text{open}}$,
represented in diagram form by

$\vbox{\hbox{\includegraphics[width=108.405pt]{DW_multiplication_k_basis.pdf}}}=\quad\quad\ket{g_{2}g_{1}}\;.$
(65)

Figure 2: A cylinder with a mixture of end-of-the-world branes and gluing
boundaries with boundary conditions. With the above boundary conditions the
path integral evaluates to $\delta(k,[g^{-1}])$, where $[g^{-1}]$ is the
conjugacy class of the element $g^{-1}\in G$.

Finally, consider a cylinder where one boundary circle is a gluing boundary
with fixed holonomy $k$ and where the other boundary circle is made up of an
EofW brane interval together with a gluing interval with fixed parallel
transport $g$. (See figure 2.) The count of gauge bundles on this manifold is
$\delta(k,[g^{-1}])$, where $[g^{-1}]$ is the conjugacy class of the element
$g^{-1}\in G$. Via the pairing (26), this provides a map from the open sector
Hilbert space ${\mathcal{H}}_{\text{open}}$ to the circle Hilbert space
${\mathcal{H}}_{S^{1}}$ given by the diagram

$\vbox{\hbox{\includegraphics[width=108.405pt]{DW_trace_k_basis.pdf}}}=\quad\quad\frac{\left|G\right|}{\left|[g]\right|}\ket{[g]}\;,$
(66)

where, again, $[g]$ is the conjugacy class of $g\in G$.

These are all the diagrams necessary to compute any partition function. The
computations proceed more easily in bases for the open and closed sectors that
are labeled by irreducible representation. We define bases

$\displaystyle\ket{q}$ $\displaystyle\equiv\sum_{k}\chi_{q}(k^{-1})\ket{k}$
(67) $\displaystyle\ket{q;i,j}$
$\displaystyle\equiv\frac{d_{q}}{\left|G\right|}\sum_{g\in
G}U^{(q)}_{ij}(g^{-1})\ket{g}$ (68)

for the closed and open Hilbert spaces, respectively, where
$i,j=1,\ldots,d_{q}$. In these bases the maps (64), (65), and (66) are

$\displaystyle=\quad\quad\frac{d_{q_{1}}}{\left|G\right|}\delta_{q_{1}q_{2}}\delta_{j_{1}i_{2}}\delta_{j_{2}i_{1}}$
(69)
$\displaystyle=\quad\quad\delta_{q_{1}q_{2}}\delta_{j_{1}i_{2}}\ket{q_{1};i_{1},j_{2}}$
(70) $\displaystyle=\quad\quad\delta_{ij}\ket{q}$ (71)

In particular, these imply the following two diagrams:

$\displaystyle=\quad\quad\sum_{q}d_{q}\ket{q}$ (72)
$\displaystyle=\quad\quad\sum_{q}\delta_{qq_{1}\cdots
q_{n}}\delta_{j_{1}i_{2}}\delta_{j_{2}i_{3}}\cdots\delta_{j_{n}i_{1}}\ket{q}$
(73)

with which, we can calculate any partition function.

A manifold can have boundary components of three different types: closed
sector circles, circles made up of a single EofW brane, and circles made up of
alternating open sector intervals and EofW brane intervals. The implications
of the last two diagrams above, for the partition functions of connected
manifolds, include the following four facts:

  * •

Any connected manifold that has boundaries, of any type, labeled by different
irreps, will evaluate to zero.

  * •

Any connected manifold with a boundary made up of alternating open sector and
EofW interval boundaries, where the i,j indices on the open sector labels do
not match up appropriately, will also evaluate to zero. Matching appropriately
means that the second index of one open sector interval equals the first index
of the next open sector interval, and so on around the circle. In the case
where we have more than one type of EofW brane, we also require that any
indices for the species of EofW brane match similarly.

  * •

A boundary made up of alternating open sector and EofW brane intervals, with
all open sector intervals labeled by q and all the i,j indices matching
appropriately, will simply contribute the same as a closed sector boundary
labeled by q.

  * •

A circular boundary made up of a single EofW brane contributes a factor of
$|G|$ relative to just filling in that boundary with a disk.

Let’s use these facts to obtain the partition function of a general manifold.
We will consider two cases: (1) manifolds with at least one “gluing” boundary
and (2) manifolds with no boundaries other than EofW brane boundaries. For the
first case, let $M$ be a surface of genus $g$, with $n$ closed sector
boundaries, $m$ circle EofW brane boundaries, and $\ell$ boundaries made up of
alternating open sector and EofW brane intervals, where all closed and open
sector boundaries are labeled by $q$ and all $i,j$ indices are matched
appropriately. Note that in this first case at least one of $n$ or $\ell$ must
be greater than $0$. The partition function for such a manifold will be

$Z^{\text{DW}}[M]=\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-n-m-\ell}d_{q}^{m}.$
(74)

The additional $m$ factors of $d_{q}$ come from the last fact above: filling
in a boundary with a disk will contribute a factor of $d_{q}/\left|G\right|$,
and a circular EofW brane boundary will contribute $\left|G\right|$ times
that, so it will contribute an overall factor of $d_{q}$. Now consider the
second case: let $M$ be a manifold with $m$ circular EofW brane boundaries,
but no closed or open sector boundaries. Then the partition function of $M$ is

$Z^{\text{DW}}[M]=\sum_{q}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-m}d_{q}^{m}.$
(75)

###  4.2 The gravity path integral

We are now equipped to build a gravity path integral out of Dijkgraaf-Witten
with EofW branes. The gravity path integral will be a sum over all manifolds
with compatible EofW brane boundaries. Specifically, we take the EofW branes
to be dynamical while we take the open and closed sector “gluing” boundaries
to be fixed. The open sector boundaries are intervals with a fixed label at
each end designating the type of EofW brane found there, and in the gravity
path integral we only allow configurations where an EofW brane of a given type
attaches only to interval endpoints labeled by that type.

Operators inserting fixed circle boundaries will be denoted by
$\widehat{Z}[k]$ or $\widehat{Z}_{q}$, as before. The operators inserting
fixed interval boundaries will be denoted by $\widehat{S}_{gab}$ where $g$
specifies the fixed gauge background boundary conditions (i.e. the parallel
transport $g\in G$ along the interval) and where $a$ and $b$ are the just
mentioned labels designating the type of EofW brane allowed at the endpoints.
Note that the boundary insertion operators are in one-to-one correspondence
with the states of the TQFT: the $\widehat{Z}$ operators correspond to states
in the closed sector, and the $\widehat{S}$ operators labeled with $a$ and $b$
correspond to states in the open sector with boundary conditions $a$ and $b$.
Using the change of basis (68) we can define the more convenient interval
operators $\widehat{S}_{qijab}$ representing the insertion of an interval with
the state $\ket{q;i,j}$ fixed on it.

We will add a term $S_{0}\chi$ proportional to the Euler characteristic to the
Dijkgraaf-Witten action, as we did before. This will have the effect of
suppressing higher genus manifolds, allowing the sum over topologies to
converge. The gravity path integral is the sum of the TQFT partition functions
over every manifold that is compatible with the inserted boundaries with a
measure that takes into account possible residual diffeomorphisms.

We now describe the calculation of correlators of the boundary insertion
operators. Let $m_{a}$ denote the number of circular EofW brane boundaries of
type $a$ on a connected manifold, so that $m=\sum_{a}m_{a}$. Then the
connected vacuum correlator for Dijkgraaf-Witten with EofW branes is

$\displaystyle\lambda^{\prime}$
$\displaystyle\equiv\log\expectationvalue{{\mathbb{1}}}=\expectationvalue{{\mathbb{1}}}_{\text{connected}}$
(76) $\displaystyle=\sum_{g}\sum_{m_{1},\ldots,m_{K}}\frac{1}{m_{1}!\cdots
m_{K}!}\sum_{q}e^{S_{0}(2-2g-\sum_{a}m_{a})}\left(\frac{d_{q}}{\left|G\right|}\right)^{2-2g-\sum_{a}m_{a}}d_{q}^{\sum_{a}m_{a}}$
$\displaystyle=\sum_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{2}\frac{1}{1-\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{-2}}\prod_{a=1}^{K}\exp(\left(\frac{e^{S_{0}}}{\left|G\right|}\right)^{-1})$
$\displaystyle=\sum_{q}\lambda_{q}e^{K\left|G\right|e^{-S_{0}}}$
$\displaystyle=e^{K\left|G\right|e^{-S_{0}}}\lambda,$

where we have used our previous definitions (31) of $\lambda_{q}$ and
$\lambda$. The connected manifolds without fixed boundary are entirely
specified by their genus $g$ and then by the number $m_{a}$ of dynamical EofW
brane circle boundaries they have for each type $a$, hence the sums over $g$
and each $m_{a}$ in the above expression. Also note the factor of $1/m_{a}!$
for each brane type. These are factors in the measure of the path integral and
are due to the $m_{a}!$ large diffeomorphisms that permute the EofW brane
circles (which are dynamical and hence taken to be indistinguishable).
Comparing this expression for the connected vacuum correlator to that of the
theory without EofW branes (31), we see that the effect of the circle EofW
brane boundaries is simply to multiply $\lambda$ by a factor.

Let’s now consider the generating function for all correlators

$F(\vec{u},T)=\expectationvalue{\exp(\sum_{q}u_{q}\widehat{Z}_{q}+\sum_{a,b=1}^{K}\sum_{q}\sum_{i,j=1}^{d_{q}}t_{qijab}\widehat{S}_{qijab})}\;.$
(77)

For convenience, we will collect the chemical potentials $t_{qijab}$ into an
object $T$ and the chemical potentials $u_{q}$ into the object $\vec{u}$.
Also, note that the $i,j$ indices within each $q$ sector play the same role as
the EofW brane labels $a,b$, so to ease notation for the calculation of the
above generating function we will combine $i$ and $a$ into a single index $A$
and similarly $j$ and $b$ into $B$.

Following Marolf:2020xie , and analogously to the procedure in section 2.2 for
the theory without EofW branes, we calculate the logarithm of the generating
function,

$\displaystyle\log
F(\vec{u},T)=\sum_{n}\sum_{k}\frac{1}{n!}\frac{1}{k!}\expectationvalue{\left(\sum_{q}u_{q}\widehat{Z}_{q}\right)^{n}\left(\sum_{qAB}t_{qAB}\widehat{S}_{qAB}\right)^{k}}_{\text{conn}}$
(78)
$\displaystyle=\sum_{q}\sum_{n,k}\frac{1}{n!}u_{q}^{n}\frac{1}{k!}\sum_{A_{1}B_{1}\cdots
A_{k}B_{k}}t_{qA_{1}B_{1}}\cdots
t_{qA_{k}B_{k}}\expectationvalue{\widehat{Z}_{q}^{n}\widehat{S}_{qA_{1}B_{1}}\widehat{S}_{qA_{2}B_{2}}\cdots\widehat{S}_{qA_{k}B_{k}}}_{q,\text{conn}}\;$

where we have used the fact that a connected correlator with boundaries
labeled by different irreps is always zero.

Each correlator in the above will be a sum over genus, a sum over numbers of
circle EofW brane boundaries, and finally a sum over all the ways of
connecting the $\widehat{S}$ operators by EofW branes. We can express this
final sum as a sum over all permutations of $k$ elements, where $k$ is the
number of open sector interval boundaries. Say $\widehat{S}_{qA_{1}B_{1}}$,
$\widehat{S}_{qA_{2}B_{2}}$, …, $\widehat{S}_{qA_{k}B_{k}}$ are the intervals
in our correlator. For a given permutation $\pi\in S(k)$ we attach the
outgoing end of $\widehat{S}_{qA_{i}B_{i}}$ to the ingoing end of
$\widehat{S}_{qA_{\pi(i)}B_{\pi(i)}}$ for every $i$. A particular way of
connecting the branes will only result in a nonzero partition function if all
pairs of connected endpoints have the same index. In other words, for any
$\pi$ the partition function will be proportional to
$\delta_{B_{1},A_{\pi(1)}}\delta_{B_{2},A_{\pi(2)}}\cdots\delta_{B_{k},A_{\pi(k)}}$.
For a permutation $\pi$, denote the number of 1-cycles in $\pi$ by
$a_{1}(\pi)$, the number of 2-cycles by $a_{2}(\pi)$, and so on. The total
number of cycles we will denote by $a(\pi)=\sum_{i}a_{i}(\pi)$, and therefore
the number of alternating-type boundaries will be $\ell=a(\pi)$ for any $\pi$.

We are now ready to calculate the general correlator of $n$ closed sector
boundaries and $k$ open sector interval boundaries. We get

$\displaystyle\langle\widehat{Z}^{n}_{q}\widehat{S}_{qA_{1}B_{1}}$
$\displaystyle\widehat{S}_{qA_{2}B_{2}}\cdots\widehat{S}_{qA_{k}B_{k}}\rangle_{\text{conn}}$
$\displaystyle=\sum_{m_{1},\ldots,m_{K}}\frac{1}{m_{1}!\cdots
m_{K}!}\sum_{g}\sum_{\pi\in
S(k)}\delta_{B_{1},A_{\pi(1)}}\cdots\delta_{B_{k},A_{\pi(k)}}Z^{\text{bulk}}\left[M_{g,n+m+\ell}\,;q,m\right]$
$\displaystyle=\lambda^{\prime}_{q}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{-n}\sum_{\pi\in
S(k)}\delta_{B_{1},A_{\pi(1)}}\cdots\delta_{B_{k},A_{\pi(k)}}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{-a(\pi)}\;,$
(79)

where we have used $Z^{\text{bulk}}\left[M_{g,n+m+\ell}\,;q,m\right]$ to
denote the partition function of a connected manifold of genus $g$ with
$n+m+\ell$ circular boundaries, $n$ of which are closed sector boundaries
labeled by $q$, $m=\sum_{a}m_{a}$ of which are EofW brane boundaries, and
$\ell=a(\pi)$ of which are alternating interval boundaries labeled by $q$.

We can now plug this in to our expansion (78) above. We get

$\log
F(\vec{u},T)=\sum_{q}\lambda^{\prime}_{q}\exp(u_{q}\frac{\left|G\right|}{d_{q}e^{S_{0}}})\sum_{k}\frac{1}{k!}\sum_{\pi\in
S(k)}\left(e^{S_{0}}\frac{d_{q}}{\left|G\right|}\right)^{-\sum_{j}a_{j}(\pi)}\prod_{j}\tr(T_{(q)}^{j})^{a_{j}(\pi)}\;,$
(80)

Note that the functions $a_{j}$ on the permutation group only depend on the
cycle structure of the permutation. So we can replace the sum over
permutations with a sum over cycle structures, using the fact that there are
$\frac{k!}{\prod_{j}a_{j}!j^{a_{j}}}$ permutations with $a_{j}$ cycles of
length $j$ for each $j$. This results in

$\log
F(\vec{u},T)=\sum_{q}\lambda^{\prime}_{q}\exp(u_{q}\frac{\left|G\right|}{d_{q}e^{S_{0}}})\prod_{j=0}^{\infty}\sum_{a_{j}}\frac{1}{a_{j}!}\left(\frac{\left|G\right|}{d_{q}e^{S_{0}}}\frac{1}{j}\tr(T_{(q)}^{j})\right)^{a_{j}}\;.$
(81)

Recognizing the rightmost sum as an exponential and then using the identity
$\exp(\sum_{j}\frac{1}{j}\tr(T^{j}))=\det({\mathbb{1}}-T)^{-1}$, we get a
final expression for the generating function:

$F(\vec{u},T)=\prod_{q}\exp(\lambda^{\prime}_{q}\left(\frac{e^{u}}{\det({\mathbb{1}}-T_{(q)})}\right)^{\frac{\left|G\right|}{d_{q}e^{S_{0}}}})$
(82)

Compare to (16), the analogous result for the simple model of Marolf:2020xie .

###  4.3 Boundary Interpretation

As explained in section 2.3, finding an interpretation of the gravity path
integral in terms of an ensemble of boundary theories reduces to an instance
of the moment problem, the problem of finding a distribution given its
moments. In the case of Dijkgraaf-Witten with EofW branes, the different
boundary theories that could appear in our ensemble are characterized by the
values that the boundary partition functions $Z_{q}$ and $S_{qijab}$ take in
them. These are the alpha-parameters and the (normalized) correlators of the
$\widehat{Z}$ and $\widehat{S}$ operators are moments probing the probability
distribution over the alpha-parameters. From this point of view we obtain the
following relation between the generating function (82) obtained above and the
probability distribution $p(\alpha)$ over the alpha-parameters:

$\prod_{q}\exp(\lambda^{\prime}_{q}\left(e^{u}/\det({\mathbb{1}}-T_{(q)})\right)^{\left|G\right|/d_{q}e^{S_{0}}}-\lambda^{\prime}_{q})=\int\\!\\!d\alpha\,p(\alpha)e^{\sum_{q}u_{q}\alpha_{q}+\sum_{q}\tr(T_{(q)}M_{(q)})}.$
(83)

Here the integral is over all $\alpha\in{\mathbb{C}}^{r+\left|G\right|K^{2}}$
with $\alpha=(\alpha_{1},\ldots,\alpha_{r};M_{(1)},\ldots,M_{(r)})$ where the
$\alpha_{q}$ are the values the $Z_{q}$ take in a particular theory and each
$M_{(q)}$ is a $Kd_{q}$ by $Kd_{q}$ matrix representing the value of the
matrix $S_{(q)}$ in a particular theory. In principle we can extract the
probability distribution $p(\alpha)$ by taking the appropriate Fourier
transform of both sides of the above. In fact, we already know the probability
distribution for the $\alpha_{q}$. These come out nearly the same as in
section 2.3. The result is that $Z_{q}$ takes the value
$Z_{q}=\frac{\left|G\right|}{e^{S_{0}}d_{q}}{N_{q}}$ where $N_{q}$ is an
integer drawn from the Poisson distribution with mean $\lambda^{\prime}_{q}$.

So we are left with the task of determining the probability distribution for
the matrices $M_{(q)}$. To see what that will be, we can consider the
generating function for the $\widehat{S}$ operators but conditioned on chosen
values $\alpha_{q}$ for the $\widehat{Z}$ operators. This can read off from
the result of taking the Fourier transform of both sides of 83 with respect to
the variables $iu_{q}$. We get

$\expectationvalue{\exp(\sum_{a,b=1}^{K}\sum_{q}\sum_{i,j=1}^{d_{q}}t_{qijab}\widehat{S}_{qijab})}_{Z_{q}=\alpha_{q}}=\det({\mathbb{1}}-T_{(q)})^{-\alpha_{q}}\;,$
(84)

so we are left with

$\det({\mathbb{1}}-T_{(q)})^{-\alpha_{q}}=\int\\!\\!dM\,p(\vec{\alpha};M)e^{\sum_{q}\tr(T_{(q)}M_{(q)})},$
(85)

where $dM$ represents an integral over all values for the matrices
$M_{(1)},\ldots,M_{(r)}$. The situation is simply that of (18), encountered
for the simple model of Marolf:2020xie . Namely,
$\det({\mathbb{1}}-T_{(q)})^{-\alpha_{q}}$ is a generating function of moments
of a probability distribution only when the exponent $\alpha_{q}$ takes
certain values graczyk2003 , in this case values in the set
$\\{0,1,2,\ldots,Kd_{q}-1\\}\cup[Kd_{q}-1,\infty)$. Unfortunately,
$\alpha_{q}$ takes values $\frac{\left|G\right|}{e^{S_{0}}d_{q}}N_{q}$ where
$N_{q}$ is a nonnegative integer. For $\alpha_{q}$ to always be in the set
$\\{0,1,2,\ldots,Kd_{q}-1\\}\cup[Kd_{q}-1,\infty)$ would require that
$\frac{\left|G\right|}{e^{S_{0}}d_{q}}$ be an integer. This is not possible as
it would render $\lambda^{\prime}_{q}$ negative, which is nonsensical as
$\lambda^{\prime}_{q}$ is the mean of the Poisson distribution for $N_{q}$. So
we are left with the problem that the gravitational path integral correlators
are not the moments of some probability distribution, meaning it cannot be
dual to an ensemble of boundary theories. This problem is worse than the one
encountered earlier which necessitated identifying the boundary partition
functions with rescaled $\widehat{Z}$ operators. Rescaling the $\widehat{Z}$
or $\widehat{S}$ operators does not change the values that $\alpha_{q}$ takes.

One solution is to modify the theory by adding terms to the TQFT action
proportional to the number of boundary components. This would be for all three
sorts of boundary components: circular EofW branes, closed sector fixed
boundaries, and alternating EofW brane and open sector intervals. Specifically
we can add a term $S_{q}$ in the action for each boundary component labeled by
$q$. Retracing through the calculations of section 4.2 we see how an
additional factor of $e^{S_{q}}$ for each boundary circle modifies the
generating function of correlators. First, considering the boundaries made up
entirely of a single EofW brane component, we see that the connected vacuum
correlator gets modified:
$\lambda^{\prime}_{q}=\lambda_{q}e^{K\left|G\right|e^{-S_{0}}}\rightarrow\lambda_{q}e^{K\left|G\right|e^{S_{q}-S_{0}}}$.
Second, a factor of $e^{S_{q}}$ for circle fixed boundaries gives
$u_{q}\rightarrow e^{S_{q}}u_{q}$. And finally, a factor of $e^{S_{q}}$ for
circles made of alternating brane and fixed boundaries changes the factors
$\tr\left(T_{(q)}^{j}\right)$ like $\tr\left(T_{(q)}^{j}\right)\rightarrow
e^{S_{q}}\tr\left(T_{(q)}^{j}\right)$. Making these substitutions in 81 leads
to a modified generating function of

$F(\vec{u},T)=\prod_{q}\exp(\lambda^{\prime}_{q}\left(\frac{e^{u}}{\det({\mathbb{1}}-T_{(q)})}\right)^{\frac{e^{S_{q}}\left|G\right|}{d_{q}e^{S_{0}}}})\;.$
(86)

Then with a the choice of $S_{q}=S_{0}+\ln d_{q}$ as in (43), for example, the
values of $\alpha_{q}$ are integers, ensuring the generating functional (85)
is indeed the generating function for moments of a probability distribution.

The additional $S_{q}$ terms in the action can be viewed as the contributions
of degrees of freedom that propagate along the boundaries. This picture is
odd, however, in that these degrees of freedom must propagate along both EofW
brane _and_ fixed boundaries. In this solution the “fixed” boundary components
are no longer fixed, but must include these dynamical degrees of freedom that
propagate across them. In some sense the solution requires us to not allow the
full set of $\widehat{S}$ operators, which would otherwise allow us to
completely fix the interval boundary conditions in the gravity path integral.
We discuss the problem in these terms in section 5.

###  4.4 General open/closed TQFTs

In this section we will describe the gravity path integral obtained by
choosing our bulk theory to be a general open/closed TQFT (aka a general 2d
TQFT with boundaries). We’ll see that this has the same features that we found
above for the specific case of Dijkgraaf-Witten theory with end-of-the-world
brane boundaries.

When discussing Dijkgraaf-Witten theory above, we considered only end-of-the-
world brane boundaries. These are, in the context of Dijkgraaf-Witten theory,
the simplest sort of boundary compatible with gauge symmetry. In general, we
could consider more complicated boundaries, and in particular for theories
without gauge symmetry, there is no condition of gauge symmetry that
boundaries must be compatible with. The only conditions that we need to
require of our boundaries are that they be compatible with the possible ways
of cutting and gluing spacetimes with boundaries. For a review of open/closed
TQFTs and the axioms that define them we refer interested readers to
Carqueville_2018 . Here we will simply state some of the results that apply to
our case.

We’ll use the index $a$ to label the different types of boundaries present in
the theory. For any two types $a$ and $b$ we have a corresponding open sector
Hilbert space ${\mathcal{H}}_{ab}$, which is the Hilbert space associated with
an interval with $a$ boundary conditions on one end and $b$ boundary
conditions on the other. In addition to these open sectors we have the closed
sector Hilbert space ${\mathcal{H}}_{S^{1}}$, which is the Hilbert space of
states on a circle. The closed sector makes, in its own right, a 2d TQFT. So,
as described in section 3, the partition function for the closed, connected
manifold of genus $g$ has the form $\sum_{I}\mu_{I}^{2-2g}$, for some
positive, real numbers $\mu_{I}$, and accordingly, the closed sector Hilbert
space ${\mathcal{H}}_{S^{1}}$ has an orthonormal basis $\ket{I}$ labeled by
the index $I$.

(a) multiplication map

(b) pairing

(c) trace map

Figure 3:  Three open/closed TQFT diagrams: (a) A map from
${\mathcal{H}}_{ab}\otimes{\mathcal{H}}_{bc}$ to ${\mathcal{H}}_{ac}$. The
open sector states can be represented as matrices such that this map
implements matrix multiplication. (b) The pairing between Hilbert spaces
${\mathcal{H}}_{ab}$ and ${\mathcal{H}}_{ba}$. (c) The trace map, which maps
${\mathcal{H}}_{aa}$ to ${\mathcal{H}}_{S^{1}}$. With the open sector states
represented as matrices, this map implements the trace of these matrices.

The interval Hilbert space ${\mathcal{H}}_{ab}$ will have the form
${\mathcal{H}}_{ab}\cong\bigoplus_{I}\left({\mathbb{C}}^{d_{aI}}\otimes{\mathbb{C}}^{d_{bI}}\right)$,
where $d_{aI}$ are integers. In fact, we can think of the states $\ket{\psi}$
in ${\mathcal{H}}_{ab}$ as being direct sums of matrices
$\ket{\psi}=\bigoplus_{I}\Psi_{I}$ where the “multiplication” diagrams as in
figure 3(a) are computed via matrix multiplication of the matrices $\Psi_{I}$.
(For example, the multiplication  acting on states
$\ket{\psi}\in{\mathcal{H}}_{ab}$ and $\ket{\phi}\in{\mathcal{H}}_{bc}$
produces a state in ${\mathcal{H}}_{ac}$ described by
$\bigoplus_{I}\Psi_{I}\Phi_{I}$.) We can take as a basis for
${\mathcal{H}}_{ab}$ the matrices with one entry $1$ and the rest zero. Denote
these basis states by $\ket{I;i,j}$ where $I$ labels the block and where
indices $i=1,\ldots,d_{aI}$ and $j=1,\ldots,d_{bI}$ label the position of the
nonzero entry. (To make connection to Dijkgraaf-Witten theory with end-of-the-
world branes, the label $I$ is in that case the label $q$ running over
irreducible representations, the values $\mu_{I}$ are
$e^{S_{0}}d_{q}/\left|G\right|$, and the dimensions $d_{aI}$ are simply
$d_{q}$ for all boundaries. The diagram 70 can be seen to be describing matrix
multiplication.)

There are two additional facts we make use of. First, the partition function
calculated by a strip with boundary type $a$ on one side and boundary type $b$
on the other (see figure 3(b)) induces a pairing between the Hilbert spaces
${\mathcal{H}}_{ab}$ and ${\mathcal{H}}_{ba}$. To be entirely consistent with
the cutting and gluing axioms defining an open/closed TQFT, this pairing must
take the form

$\left(\ket{\psi},\ket{\phi}\right)=\sum_{I}\mu_{I}\tr(\Psi_{I}\Phi_{I})$ (87)

where $\ket{\psi}\in{\mathcal{H}}_{ab}$ and $\ket{\phi}\in{\mathcal{H}}_{ba}$.
Second, we have the map
$\vbox{\hbox{\includegraphics[width=17.34189pt]{open_closed_trace_no_labels_cropped.pdf}}}\\!\\!:{\mathcal{H}}_{aa}\rightarrow{\mathcal{H}}_{S^{1}}$
from any diagonal open sector ${\mathcal{H}}_{aa}$ to the closed sector
${\mathcal{H}}_{S^{1}}$. This map simply implements the trace
$\ket{\psi}\mapsto\sum_{I}\tr(\Psi_{I})\ket{I}$.

With the multiplication, pairing, and trace diagrams in hand we can calculate
the partition function of any manifold with boundaries, and from there define
a gravity path integral as a subsequent sum over all compatible manifolds with
boundaries. The observables in this gravity theory are operators
$\widehat{Z}_{I}$ that insert a circle with state
$\ket{I}\in{\mathcal{H}}_{S^{1}}$, and $\widehat{S}_{abIij}$ which insert the
interval with state $\ket{I;i,j}\in{\mathcal{H}}_{ab}$ on it. The correlators
are given by a sum over manifolds where the boundary configurations are
compatible with the boundary labels of all the insertions
$\widehat{S}_{abIij}$. In our sum over manifolds $M$ we include the
appropriate measure $\mu(M)$. In addition to the factors for having multiple
identical components without fixed boundaries, the measure $\mu(M)$ also
includes a factor of $1/m_{a}!$ whenever a component of $M$ has $m_{a}$ circle
boundaries with boundary conditions $a$ all around the circle. These are due
to residual diffeomorphisms that permute the identical dynamical boundaries.

The calculation of correlators is analogous to the Dijkgraaf-Witten case in
section 4.2. We simply state the results. The connected vacuum correlator is

$\lambda^{\prime}\equiv\log\expectationvalue{{\mathbb{1}}}=\expectationvalue{{\mathbb{1}}}_{\text{connected}}=\sum_{I}\lambda_{I}e^{K_{I}/\mu_{I}}=\sum_{I}\lambda_{I}^{\prime}\;,$
(88)

where $K_{I}=\sum_{a}d_{aI}$ count the boundary degrees of freedom, and the
$\lambda_{I}=\mu_{I}^{2}/(1-\mu_{I}^{-2})$ are as defined in section 3 for the
corresponding closed TQFT. The generating function

$F(u_{I},t_{abIij})=\expectationvalue{e^{\sum_{I}u_{I}\widehat{Z}_{I}+\sum_{ab}\sum_{I,i,j}t_{abIij}\widehat{S}_{abIij}}}$
(89)

is given by

$\displaystyle
F(u,t)=\prod_{I}e^{\lambda_{I}^{\prime}\exp(\mu_{I}^{-1}u_{I}+\mu_{I}^{-1}\sum_{j=1}^{\infty}\frac{1}{j}\tr(T_{I}^{j}))}=\prod_{I}\exp(\lambda_{I}^{\prime}e^{u_{I}/\mu_{I}}\det({\mathbb{1}}-T_{I})^{-1/\mu_{I}})\;,$
(90)

where we have collected the chemical potentials $t_{abIij}$ into $K_{I}$ by
$K_{I}$ matrices $T_{I}$. This is of course the same form as (82). Similar to
the special case of Dijkgraaf-Witten with end-of-the-world branes, correlators
will factorize between the $I$ sectors, and will also fail to correspond to
the moments of an ensemble distribution without negative probabilities. We
discuss this failure and possible solutions in the next section.

##  5 Boundaries and the ensemble problem

In order to get a sensible holographic interpretation from a gravity path
integral built from a closed 2d TQFT, we were forced to introduce in a
seemingly ad hoc fashion a rescaling of the $\widehat{Z}_{I}$ operators.
What’s more, for an open/closed theory, we find that rescalings of the
operators $\widehat{Z}$ and $\widehat{S}$ are no longer enough to land on a
sensible boundary interpretation. The solution discussed in Marolf:2020xie
and reviewed here in section 1.1 is to add a nonlocal boundary term
$S_{\partial}=S_{0}$ to the action. The authors attempt to justify this in
terms of a large number of additional degrees of freedom that are allowed to
propogate on the boundary. We discuss the analogous solution in our case and
attempt to paint a clearer picture of how these degrees of freedom fit into a
local framework.

The observables in our gravity theory are in one-to-one correspondence with
states in a TQFT. By definition, a TQFT is guaranteed to be compatible with
cutting and gluing spacetime manifolds. In particular, upon cutting a
spacetime manifold the newly created boundary is what we will call a “gluing
boundary.” By this we mean this boundary corresponds to a Hilbert space of
states and hence can be glued to a similar gluing boundary by appropriately
summing over states on the two boundaries. So far, the operators describing
observables in the theory (the $\widehat{Z}$ and $\widehat{S}$ operators) are
operators that insert such gluing boundaries in our gravitational spacetime.
This is suggestive. Specifically, it suggests the possibility of gluing the
boundaries of our gravity spacetime to nongravitational TQFT path integrals.
That is to say, we can consider spacetimes with designated nongravitational
regions where topology is fixed, while the rest of the manifold has
unspecified (i.e. summed over) topology.

$\vbox{\hbox{\includegraphics[width=195.12767pt]{GPI_holography_LHS}}}=\vbox{\hbox{\includegraphics[width=195.12767pt]{GPI_holography_RHS}}}$
Figure 4: A gravitational region whose topology is summed over is
holographically dual to an ensemble of boundary conditions. The above figure
is a schematic illustration of a gravity correlator of nongravitational
regions. The correlator is a sum over all manifolds that are compatible with
the boundaries of the nongravitational regions. Equivalently, it is an average
over different boundary conditions $\alpha$ (represented by dashed lines) that
are placed on the boundaries of the nongravitational regions.

Holography on such hybrid gravitational/nongravitational spacetimes will
reduce the gravitational bulk region to a boundary condition on the
nongravitational region. Or, more generally, as in our case, the gravitational
region will be dual to an _ensemble_ of boundary conditions. Specifically,
each alpha-state of the gravitational theory will correspond to a distinct
boundary condition. For each type of boundary $\alpha$, there are
corresponding open sector Hilbert spaces for intervals that end on that
boundary. For example, in a theory that already has boundaries (for example,
these could be end-of-the-world branes) labeled by the index $a$, we will get
additional open sector Hilbert spaces like ${\mathcal{H}}_{a\alpha}$ and
${\mathcal{H}}_{\alpha a}$. Even in a theory without additional boundaries,
such as those considered in sections 2 and 3, upon going to the holographic
dual we will have an open/closed TQFT with boundary $\alpha$ and the open
sector ${\mathcal{H}}_{\alpha\alpha}$.

Taking gravity as a boundary condition provides a new perspective on the
interpretational problems that necessitated rescaling the $\widehat{Z}$
operators and that led to negative probabilities when $\widehat{S}$ operators
are added. We can see how such problems arise in this framework. Consider the
correlator
$\expectationvalue{\widehat{\vbox{\hbox{\includegraphics[width=32.52017pt]{curved_cylinder.pdf}}}}}=\sum_{I}\expectationvalue{\widehat{Z}_{I}\widehat{Z}_{I}}$
, where $I$ labels the closed sector states of the TQFT. This is dual to the
annulus where both boundaries are the gravity boundary. This diagram is a
trace that computes the dimension of the open sector Hilbert space
${\mathcal{H}}_{\alpha\alpha}$. Consequently, it ought be an integer. However,
from the results of section 3,

$\sum_{I}\expectationvalue{\widehat{Z}_{I}\widehat{Z}_{I}}=\sum_{\ldots,N_{I},\ldots}\prod_{I}\frac{\lambda_{I}^{N_{I}}}{N_{I}!e^{N_{I}}}\sum_{I}\frac{N_{I}}{\mu_{I}}\frac{N_{I}}{\mu_{I}}\;,$
(91)

meaning the annulus with gravity boundaries evaluates to
$\sum_{I}\frac{N_{I}^{2}}{\mu_{I}^{2}}$ where $N_{I}$ is a Poisson random
integer. This is not in general an integer. In fact, it cannot be an integer
for all $N_{I}$ if we restrict $\mu_{I}$ to the values that are compatible
with a convergent gravity path integral or with an ensemble without negative
probabilities. Likewise, if we start with a TQFT where boundary conditions are
already present, similar problems arise for the “mixed” open sectors
${\mathcal{H}}_{\alpha a}$ and ${\mathcal{H}}_{a\alpha}$, where $a$ here
labels the original, nongravity, boundary conditions in the theory. An annulus
with one boundary of type $a$ and the other with gravity theory $\alpha$ on
it, will evaluate to $\sum_{I}d_{aI}\frac{N_{I}}{\mu_{I}}$, where $d_{aI}$ are
integers. This, again, is not in general an integer. This implies the
absurbity that the Hilbert spaces ${\mathcal{H}}_{\alpha\alpha}$ and
${\mathcal{H}}_{a\alpha}$ have noninteger dimension. One can equally view this
as a breakdown of locality, in that cutting a gravity boundary $\alpha$ cannot
be done consistently.

The perspective of gravity as dual to a boundary condition on a
nongravitational TQFT also motivates an understanding of what the alpha-states
for a theory with boundaries, like end-of-the-world branes, are. Consider a
state $\ket{I;i,j}$ in an open sector ${\mathcal{H}}_{ab}$ that propogates
from a nongravitational region into the gravitational region.

$\vbox{\hbox{\includegraphics[width=205.97214pt]{GPI_open_sector_LHS.pdf}}}=\vbox{\hbox{\includegraphics[width=205.97214pt]{GPI_open_sector_RHS.pdf}}}$
Figure 5: A strip with $a$ and $b$ boundary conditions that enters a
gravitational region is dual to a strip that ends with gravity boundary
conditions $\alpha$ interpolating between $a$ and $b$. At the junction between
$\alpha$ and $a$ boundary conditions there must be a boundary condition
changing operator, which we call $\psi_{\alpha a}$. Likewise for the junction
between $b$ and $\alpha$ boundary conditions.

This is holographically dual to a half-disk bounded by three segments: a
segment with boundary conditions $a$, a segment with gravity boundary
conditions $\alpha$, and a segment with boundary conditions $b$. (See figure
5.) At the two points on the edge of the half-disk where the boundary
conditions switch, there are so-called boundary condition changing operators
Cardy:2004hm . A boundary condition changing operator between boundaries of
two types, say from $a$ to $b$, is a state in the open sector Hilbert space
${\mathcal{H}}_{ab}$. (This can be seen from considering the slice surrounding
the point where the boundary conditions change. As this slice ends on the
boundaries with conditions $a$ and $b$, the Hilbert space associated to it is
${\mathcal{H}}_{ab}$.) So in this case of a half-disk with three different
boundary types, there must be specified two states, in
${\mathcal{H}}_{b\alpha}$ and ${\mathcal{H}}_{\alpha a}$ respectively, which
represent the boundary condition changing operators. We take the specification
of these states, call them $\ket{\psi_{b\alpha}}\in{\mathcal{H}}_{b\alpha}$
and $\ket{\psi_{\alpha a}}\in{\mathcal{H}}_{\alpha a}$, to be additional alpha
parameters. That is to say, fully specifying the alpha state, and hence the
boundary theory, includes not just specifying the dimensions $d_{\alpha I}$,
but also the boundary condition changing states $\ket{\psi_{a\alpha}}$ that
must appear when switching from the gravity path integral to its holographic
dual.121212Reflection positivity ensures that the state $\ket{\psi_{\alpha
a}}$ switching $\alpha$ boundaries to $a$ boundaries and the state
$\ket{\psi_{a\alpha}}$ in the other direction, are determined by each other.
Represented as matrices they are each other’s Hermitian conjugates. Recalling
from section 4.4 that the states of an open sector ${\mathcal{H}}_{ab}$ can be
regarded as direct sums of $d_{aI}$ by $d_{bI}$ matrices, we can represent the
states $\ket{\psi_{\alpha a}}$ as $\bigoplus_{I}\Psi_{(\alpha aI)}$. We see
that the probability distribution $p(\alpha)$ defining our ensemble should be
over the nonnegative integers $d_{\alpha I}$, for each $I$, and the $d_{\alpha
I}$ by $d_{aI}$ complex matrices $\Psi_{(\alpha aI)}$, for each choice of $a$
and $I$.

We see an immediate problem, however. For the naive gravity path integral the
dimensions $d_{\alpha I}$ are $d_{\alpha I}=N_{I}/\mu_{I}$, and hence, as
explained above, not nonnegative integers for every
$N_{I}\in\\{0,1,2,\ldots\\}$. Defining the boundary theories in our ensemble
then seems to entail choosing the elements of a matrix with noninteger
dimension. We saw that, for a TQFT without additional boundaries, the
holographic dual already has the problem that the Hilbert space
${\mathcal{H}}_{\alpha\alpha}$ does not exist. (It would have noninteger
dimension.) This can be viewed as a violation of locality, insofar as it
forbids us from making cuts in our manifolds that intersect $\alpha$
boundaries. With that restriction on making cuts in place, perhaps it might
otherwise makes sense. In the presence of additional boundaries $a$, however,
the problems with the holographic dual theory become worse. The inability to
make cuts intersecting $\alpha$ boundaries prevents us from relating boundary
condition changing operators to states $\ket{\psi_{\alpha
a}}\in{\mathcal{H}}_{\alpha a}$, as the Hilbert space ${\mathcal{H}}_{\alpha
a}$ does not exist. Instead, while such states $\ket{\psi_{b\alpha}}$ and
$\ket{\psi_{\alpha a}}$ don’t exist, we _do_ have access to the state in
${\mathcal{H}}_{ba}$ that would be their product under open sector
multiplication described in section 4.4 (see figure 3(a)). When represented as
a matrix, the entries of this state in ${\mathcal{H}}_{ba}$ are simply the
values that the operators $\widehat{S}_{abIij}$ take. As we have explained,
the generating function (90), by analogy with the arguments of Marolf:2020xie
, does not lead to a nonnegative probability distribution for the values
$S_{abIij}$, for all values of $d_{\alpha I}=N_{I}/\mu_{I}$. Thus, even with
the locality-violating restriction of disallowing cuts that intersect gravity
boundaries, we are left without a consistent boundary ensemble theory.

So should we give up hope of consistently viewing the gravity path integral as
dual to an ensemble of boundary conditions? In the case of the simple model
with end-of-the-world branes considered in Marolf:2020xie , they discuss, as a
solution to the negative probabilities, adding $e^{S_{0}}$ degrees of freedom
that propagate along the end-of-the-world branes. These degrees of freedom
contribute to the existing bulk action a boundary term $S_{\partial}=S_{0}$
for every boundary in the theory. This effectively rescales the boundary
insertion operators $\widehat{Z}$ and likewise ensures that $d$ in eq. (16) is
an integer. Then the Fourier transform of eq. (18) does, indeed, give a valid
probability distribution, namely the Wishart distribution.

This solution unavoidably has one of two problems, however. On the one hand,
if the degrees of freedom are taken to propagate only along the end-of-the-
world branes, then the open sectors ${\mathcal{H}}_{ab}$ consequently have
$e^{2S_{0}}$ times as many states as they did before, and we have a
corresponding $e^{2S_{0}}$ times as many interval insertion operators
$\widehat{S}$. As the problem of negative probabilities is, roughly speaking,
the problem of having too many boundary flavors, the additional interval
insertion operators $\widehat{S}$ reintroduce the problem that adding the
$e^{S_{0}}$ degrees of freedom might have solved. On the other hand, if we
allow the additional boundary degrees of freedom to propagate along not just
the branes, but also the “gluing” boundaries, these boundaries lose their
interpretation as gluing boundaries. In other words, the bulk action is now
nonlocal.

The perspective of a gravitational region being dual to boundary conditions on
a nongravitational region clarifies what the problem is. Adding, in our case,
$\mu_{I}$ degrees of freedom to each boundary type $a$, implements the change
$d_{aI}\rightarrow d^{\prime}_{aI}=\mu_{I}d_{aI}$. On its face, this seems to
solve the problem of $\dim{\mathcal{H}}_{a\alpha}$ being a noninteger, as
$\sum_{I}d^{\prime}_{aI}\frac{N_{I}}{\mu_{I}}=\sum_{I}d_{aI}N_{I}\in{\mathbb{Z}}$,
but the new size of the open sector Hilbert space ${\mathcal{H}}_{ba}$ is then
inconsistent with a multiplication rule
${\mathcal{H}}_{b\alpha}\otimes{\mathcal{H}}_{\alpha
a}\rightarrow{\mathcal{H}}_{ba}$ that involves matrices with $d_{bI}N_{I}$ and
$d_{aI}N_{I}$ number of entries.131313It also does not solve the problem of
noninteger $\dim{\mathcal{H}}_{\alpha\alpha}$. That is to say, there are no
sizes that two matrices could have such that

  * •

the number of entries in each are $d_{bI}N_{I}$ and $d_{aI}N_{I}$
respectively, and

  * •

their matrix product has $\mu_{I}d_{bI}\mu_{I}d_{aI}$ entries,

other than their inner dimensions being $N_{I}/\mu_{I}$, which is not an
integer for all $N_{I}$. (This makes precise the statement we made above that
the problem is in some sense having too many flavors of boundary.)

The difficulties with the boundary interpretation all ultimately stem from the
fact that the $d_{\alpha I}=N_{I}/\mu_{I}$ are not integers for all $N_{I}$.
This suggests that a solution should be something that adds $\mu_{I}$
additional degrees of freedom to the gravity boundary $\alpha$, rather than
merely adding degrees of freedom to the original boundaries $a$. If we
substitute $d_{\alpha I}\rightarrow d^{\prime}_{\alpha I}=\mu_{I}d_{\alpha
I}$, the Hilbert spaces ${\mathcal{H}}_{\alpha\alpha}$,
${\mathcal{H}}_{a\alpha}$, etc. are all defined as well as the boundary
condition changing operators. With ${\mathcal{H}}_{a\alpha}$ well-defined,
locality would then imply that the $S_{abI}$ matrices factorize as
$S_{abI}=\Psi_{\alpha aI}^{\dagger}\Psi_{\alpha bI}$. Such a solution would
presumably cure the problem of negative probabilities for the values of
$S_{abIij}$.

One solution fitting these requirements involves adding a defect line
separating the gravitational and nongravitational regions in our gravity path
integral. We let this defect line, call it $\varepsilon$, have
$\sum_{I}\mu_{I}$ degrees of freedom, and we take it to be coupled to the 2d
TQFT so that there are $\mu_{I}$ degrees of freedom in each $I$ sector. To
explain what we mean by this, consider that with the addition of such a defect
line our TQFT has additional open sector Hilbert spaces, namely those
corresponding to intervals that cross the defect some number of times. If the
interval crosses the defect $\varepsilon$ once, the corresponding Hilbert
space, call it ${\mathcal{H}}_{a\varepsilon b}$ will be
${\mathcal{H}}_{a\varepsilon
b}\cong\bigoplus_{I}\left({\mathbb{C}}^{d_{aI}}\otimes{\mathbb{C}}^{\mu_{I}}\otimes{\mathbb{C}}^{d_{bI}}\right)$.
In fact, each $I$ sector of the Hilbert space will have an additional tensor
factor of ${\mathbb{C}}^{\mu_{I}}$ for each time the interval crosses a defect
line $\varepsilon$. Likewise, there are additional closed sector Hilbert
spaces, for circles that intersect the defect line some number of times. The
simplest way to define a gravity path integral with the addition of such a
defect line is to consider surfaces where the defect line is placed
surrounding the gravity region, so that it runs along brane boundaries within
the gravity region and along the interface between the gravity and nongravity
regions. To be precise, with the addition of the defect, a boundary $a$ within
the gravity region gets replaced by the combined boundary and defect
$a\otimes\varepsilon$, and at a triple junction of $a$, $\varepsilon$, and
$a\otimes\varepsilon$ we contract the degrees of freedom in the natural way.
(See figure 6.)

Figure 6: The gravitational and non-gravitational regions are separated by the
defect. At the junctions where, for example, $a$, $\varepsilon$, and
$a\otimes\varepsilon$ meet, we have the degrees of freedom of $a$ and the
degrees of freedom of $\varepsilon$ propagate in the obvious way.

Calculating the resulting correlators of $\widehat{Z}_{I}$ and
$\widehat{S}_{abIij}$ is straightforward. For every boundary component,
whether made by the $\widehat{Z}$, or a circle with boundary conditions $a$,
or made from a combination of boundaries $a$ and interval operators
$\widehat{S}$, the TQFT action now has an additional factor of $\mu_{I}$ for
each $I$ sector. The result is that (90) is modified to

$\displaystyle
F(u,t)=\prod_{I}e^{\lambda_{I}e^{K_{I}}\exp(u_{I}+\sum_{j=1}^{\infty}\frac{1}{j}\tr(T_{I}^{j}))}=\prod_{I}\exp(\lambda_{I}e^{K_{I}}e^{u_{I}}\det({\mathbb{1}}-T_{I})^{-1})\;,$
(92)

This is not hard to see: each $\widehat{Z}_{I}$ boundary component comes with
an additional factor $\mu_{I}$, effectively giving
$u_{I}\rightarrow\mu_{I}u_{I}$; circles with boundary conditions $a$ gets an
additional factor $\mu_{I}$ for each $I$ sector, giving
$\lambda_{I}e^{K_{I}/\mu_{I}}\rightarrow\lambda_{I}e^{K_{I}}$; and finally,
each boundary component made of $j$ connected $\widehat{S}_{I}$ intervals gets
a factor of $\mu_{I}$, giving $\tr(T^{j})\rightarrow\mu_{I}\tr(T^{j})$. This
modified generating function for the correlators implies an ensemble of
theories where $Z_{I}=d_{\alpha I}=N_{I}$ are random integers $N_{I}$,
independently chosen from Poisson distributions with respective means
$\lambda_{I}e^{K_{I}}$. Within the space of theories with given $Z_{I}=N_{I}$,
the remaining probability distribution over the values of $S_{abIij}$ has the
generating function $\det({\mathbb{1}}-T_{I})^{-N_{I}}$ for its moments. This
is the generating function for the moments of the Wishart distribution
graczyk2003 . Recognizing this, we can write it suggestively as

$\det({\mathbb{1}}-T_{I})^{-N_{I}}=\pi^{-K_{I}N_{I}}\int\\!\prod_{a}\left(d\Psi_{\alpha
aI}d\overline{\Psi}_{\alpha aI}e^{-\tr(\Psi_{\alpha aI}^{\dagger}\Psi_{\alpha
aI})}\right)\,e^{\sum_{a,b}\tr(T_{abI}\Psi_{\alpha bI}^{\dagger}\Psi_{\alpha
aI})}\;,$ (93)

where by $d\Psi_{\alpha aI}d\overline{\Psi}_{\alpha aI}$ we mean integration
over the $2d_{aI}N_{I}$-dimensional space of complex $N_{I}$ by $d_{aI}$
matrices $\Psi_{\alpha aI}$, and where $T_{abI}$ is the $d_{aI}$ by $d_{bI}$
matrix with entries $t_{abIij}$. We can see immediately that the operator
$\widehat{S}_{abI}$ takes the matrix value $S_{abI}=\Psi_{\alpha
bI}^{\dagger}\Psi_{\alpha aI}$ in a given alpha-state. This is precisely
consistent with the holographic picture of gravity as a boundary condition. In
that case locality implies that the matrix $S_{ab}$ should indeed factorize
into the matrix multiplication of two boundary condition changing states,
namely $\ket{\psi_{b\alpha}}=\bigoplus_{I}\Psi_{\alpha bI}^{\dagger}$ and
$\ket{\psi_{\alpha a}}=\bigoplus_{I}\Psi_{\alpha aI}$. Thus we can replace the
gravity region’s fluctuating topology with an ensemble of boundary conditions
$\alpha$, as illustrated in the example of figure 7. The $\alpha$ boundary
conditions are, as expected completely characterized by the dimensions
$d_{\alpha I}$ and the boundary condition changing states $\ket{\psi_{\alpha
a}}$, where $d_{\alpha I}$ are drawn from Poisson distributions and the
entries in the matrix representation of $\ket{\psi_{\alpha a}}$ are
independent complex Gaussian random variables.

$\vbox{\hbox{\includegraphics[width=195.12767pt]{GPI_holography_defect_LHS.pdf}}}=\vbox{\hbox{\includegraphics[width=216.81pt]{GPI_holography_defect_RHS.pdf}}}$
Figure 7: The gravity path integral where a defect line $\varepsilon$ (green
solid lines) separates the gravitational and nongravitational regions, and the
boundaries within the gravity region now have boundary conditions
$a\otimes\varepsilon$. This gravity region is holographically dual to an
ensemble of boundary conditions $\alpha$ (green lines with red dashed lines)
that are enhanced with the $\mu_{I}$ degrees of freedom from the defect, so
that $d_{\alpha I}=N_{I}$.

There are many other ways we could choose to configure defect lines, and there
may certainly be others that lead to a well-defined ensemble of boundary
conditions. For example, instead of the simple junctions pictured in figure 6,
we could allow the degrees of freedom to mix or to end at the junction. Such
different setups would lead to the inclusion of additional operators in the
traces $\tr(T_{I}^{j})$ in the double exponent of (90). The setup considered
above is simply the most straightforward option, and it does, in fact, cure
the problem of negative ensemble probabilities. We speculate that in a TQFT
that descends from a realistic gravity theory, a defect line separating
gravitational and nongravitational regions may descend from the data that
specify how the geometries of the gravity and nongravity regions are
consistently glued together. For example, in the construction of
Almheiri:2019psf  and subsequent papers, wherein JT gravity with CFT matter
has its boundary glued to a flat region without gravity, the “boundary
graviton” mode of JT gravity appears as a reparametrization of the boundary,
determining how the JT gravity region is glued to the non-gravitational flat
region.

The alpha-states $\ket{\alpha}$, defined similarly to (19), satisfy
$\innerproduct{\alpha^{\prime}}{\alpha}\sim\delta_{\alpha^{\prime},\alpha}p(\alpha)$.
The failure to obtain a well-defined ensemble with nonnegative probabilities
is thus equivalent to the presence of negative norm states in the baby
universe Hilbert space. One possible solution, then, to the problem discussed
in this section would be to simply project out the negative norm states. The
baby universe Hilbert space can be constructed by acting on $\ket{\text{HH}}$
with the single-boundary operators $\widehat{Z}$ and $\widehat{S}$. A
projection on the space of single-boundary operators would thus induce a
projection on the baby universe Hilbert space. It is possible for the states
projected out by such an operation to include the offending negative-norm
states. In fact, the gravity model with defect separating gravity and
nongravity regions is an example of precisely this. The diagram pictured in
figure 6, when viewed from right to left, constitutes a projection.141414The
analogous map on the closed sector Hilbert space ${\mathcal{H}}_{S^{1}}$ does
not project out any states. It is simply a rescaling, though, as explained
above, a crucial one for the interpretation of $\widehat{Z}$ as a partition
function in a 1d topological theory or as boundary conditions in a non-
gravitational TQFT. Namely the large Hilbert space
${\mathcal{H}}_{a\otimes\varepsilon,b\otimes\varepsilon}$ is mapped to the
smaller Hilbert space ${\mathcal{H}}_{ab}$. The observables that non-
gravitational observers have access to only include those built from the
$\dim{\mathcal{H}}_{ab}$ operators $\widehat{S}_{abIij}$, rather than the much
larger number of operators that correspond to states in the
${\mathcal{H}}_{a\otimes\varepsilon,b\otimes\varepsilon}$ Hilbert space.
Specifically, it is the defect degrees of freedom that are inaccessible.

##  6 Future directions

In our work we extended the 2d topological gravity model of Marolf:2020xie  to
a broader class of topological actions. The holographic duals of these gravity
models are ensembles of 1d topological theories with random dimension. This
is, in retrospect, not terribly surprising, as all 2d TQFTs are in a sense
direct sums of the simplest TQFT, whose Hilbert space is one-dimensional and
whose action is proportional to the Euler characteristic, like in
Marolf:2020xie .

Perhaps the most obvious limitation of the present work, then, is our
restriction to TQFTs as defined by Atiyah’s axioms. In particular, TQFTs
satisfying Atiyah’s axioms are always finite dimensional, so this restriction
rules out many TQFTs of physical interest. These include, such TQFTs as the A-
and B-models of topological string theory (both examples of the broader class
of “topological conformal field theories”). Likewise, JT gravity has a
description as a modified BF theory with gauge group
$\operatorname{SL}(2,{\mathbb{R}})$ Blommaert:2018iqz , an infinite
dimensional topological field theory.

Also in the spirit of working towards more realistic theories would be the
extension to higher dimensional spacetimes. Several recent works attempt to
make connections between a 3d bulk gravity and an ensemble of 2d CFTs on the
boundary Afkhami-Jeddi:2020ezh ; maloney2020averaging ; cotler2020ads3 ;
Cotler:2020hgz . Making sense of gravity path integrals in three dimensions,
however, runs up against the difficulty that the equivalent of the genus
expansion, for 3d manifolds, is not so well-behaved. Just as any 2d closed,
connected, oriented manifold is the connected sum of some number of tori, any
3d closed, connected, oriented manifold can be uniquely written as the