2d TQFTs and baby universes

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.

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.

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 $\widehat{Z}[k]=\sum_{q}\frac{\left|k\right|}{\left|G\right|}\chi_{q}(k)\widehat{Z}_{q}$.

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.

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.