From Baby Universes to Narain Moduli:
Topological Boundary Averaging in SymTFTs

We start with a simple observation. In a $(d+1)$-dimensional topological field theory, a filling of a $d$-manifold $M$ is not merely seen as a geometry by itself, but as a linear functional on the state space associated to $M$ [Atiyah:1989vu]. Let $\mathfrak{T}$ be a $(d+1)$-dimensional TFT. It assigns to a closed $d$-manifold $M$ a vector space

$$\mathcal{H}_{\mathfrak{T}}(M).$$

(2.1)

A bordism

$$Y:M\longrightarrow\emptyset$$

(2.2)

defines a map

$$Z_{\mathfrak{T}}(Y):\mathcal{H}_{\mathfrak{T}}(M)\longrightarrow\mathcal{H}_{\mathfrak{T}}(\emptyset)\cong\mathbb{C}.$$

(2.3)

Thus $Y$ defines an element of the dual space,

$$Z_{\mathfrak{T}}(Y)\in\mathcal{H}_{\mathfrak{T}}(M)^{*}.$$

(2.4)

In this sense, a bordism from $M$ to the empty set is a cap.

A sum over bordisms with fixed boundary $M$ therefore gives a distinguished element of $\mathcal{H}_{\mathfrak{T}}(M)^{*}$:

$$\mathcal{C}_{\mathrm{bord}}(M)=\sum_{[Y:M\to\emptyset]}w(Y)\,Z_{\mathfrak{T}}(Y)\in\mathcal{H}_{\mathfrak{T}}(M)^{*}.$$

(2.5)

The coefficient $w(Y)$ denotes whatever weight the theory assigns to the bordism $Y$. At this stage we will not specify it. In an ordinary gravitational path integral it would come from the action and the path-integral measure. In a purely topological or finite groupoid version it includes the usual symmetry factors. The point of (2.5) is not the precise choice of weight, but the fact that the topology sum has become a sum of cap functionals.

This is a useful way to state the problem because a topological boundary condition also naturally defines a cap. If $L$ is a topological boundary condition of $\mathfrak{T}$, then putting $\mathfrak{T}$ on a collar ending on $L$ gives, for each $M$, a linear functional

$$\langle L;M|\in\mathcal{H}_{\mathfrak{T}}(M)^{*}.$$

(2.6)

This functional is not an arbitrary element of the dual vector space. It is one that can be realized by a boundary condition of the same topological theory.

The construction that we will use is that, in some cases, the bordism sum (2.5) can be reorganized in terms of these topological caps:

$$\mathcal{C}_{\mathrm{bord}}(M)=\sum_{[L]}W_{M}(L)\,\langle L;M|.$$

(2.7)

Here $W_{M}(L)$ is the total weight of all bordisms that give the same cap, or more generally the same cap data, in the TFT. Equivalently, the statement is that the image of the bordism sum in $\mathcal{H}_{\mathfrak{T}}(M)^{*}$ factors through the set of topological boundary conditions.

This should not be interpreted as saying that a geometric filling $Y$ is literally the same thing as a topological boundary condition $L$. The statement is weaker, and more natural from the TFT point of view. After applying the functor $Z_{\mathfrak{T}}$, the filling $Y$ is remembered only through the functional, or physically speaking the “wave function” $Z_{\mathfrak{T}}(Y)$. Distinct fillings may therefore define the same cap. If the caps that occur in the sum are naturally labelled by topological boundary conditions, then the topology sum descends to a sum over such boundary conditions.

This is the sense in which we will relate topology sums to boundary condition averaging. The geometric objects may be handlebodies or more exotic fillings. The TFT sees only the corresponding elements of the dual Hilbert space. In favorable examples, these elements are classified by Lagrangian data. This is what happens in the Narain example [2006.04855]: for connected $\Sigma$, a handlebody determines a Lagrangian sublattice of $H_{1}(\Sigma,\mathbb{Z})$; after rewriting the formula in these terms, the same expression continues to make sense also when $\Sigma$ is disconnected, even though there is no longer a preferred ordinary handlebody associated to each Lagrangian sublattice. This suggests that the more intrinsic object is the Lagrangian cap data, not the ordinary smooth manifold itself.

We now specialize this observation to SymTFTs. Let $\mathfrak{T}_{\mathrm{sym}}$ be the $(d+1)$-dimensional SymTFT associated with a $d$-dimensional QFT. The SymTFT is placed on a slab

$$M\times[0,1].$$

(2.8)

One end of the slab is coupled to the physical $d$-dimensional relative theory [1212.1692, 2212.00195]. This boundary in general is not topological. It depends on the metric and possible source data on $M$, which we denote collectively by $J$. We then write the physical boundary condition as

$$B_{\mathrm{phys}}(J).$$

(2.9)

This physical boundary provides a state in the Hilbert space on $M$:

$$|\Psi_{\mathrm{phys}}(M;J)\rangle\in\mathcal{H}_{\mathrm{sym}}(M),$$

(2.10)

which is the partition vector, instead of a number, of the relative physical boundary theory.

To obtain an absolute theory, one chooses a topological boundary condition $L$ at the other end of the slab. $L$ specifies which bulk topological operators can end on the topological boundary. The remaining bulk operators, modulo those trivialized by $L$, become the symmetry operators of the absolute theory. In finite semisimple SymTFTs, $L$ is often described as a Lagrangian algebra of the category of bulk topological operators [hep-th/0204148, 1008.0654, 1012.0911]. In the continuous abelian examples considered in the following sections, the analogous object will be a maximal isotropic subset in the continuous family of topological operators in the SymTFT [hep-th/9812012, 2010.15890, 2203.09537, 2306.11783].

For each $M$, $L$ defines a cap functional

$$\langle L;M|\in\mathcal{H}_{\mathrm{sym}}(M)^{*}.$$

(2.11)

The partition function of the absolute theory specified by $L$ is

$$Z_{L}(M;J)=\langle L;M|\Psi_{\mathrm{phys}}(M;J)\rangle.$$

(2.12)

We here emphasize an important restriction in this construction. We do not average over arbitrary boundary conditions at the second end of the slab. A general boundary condition $B$ would also define a linear functional

$$\langle B;M,J_{B}|:\mathcal{H}_{\mathrm{sym}}(M)\longrightarrow\mathbb{C},$$

(2.13)

which can be regarded as another physical boundary with reversed orientation of $M$. However, such a boundary condition would usually carry its own local degrees of freedom and its own dependence on metric and coupling data. In Hamiltonian language, such a boundary would come with its own nontrivial boundary Hamiltonian. Summing over general boundary conditions would therefore be a sum over local dynamics. This is too large an operation for our purposes. In the ensemble examples we have in mind, the average is over a specified family of theories, such as a fixed theory with random couplings drawn from a prescribed ensemble with measure. An arbitrary sum over boundary conditions would not single out such an ensemble, and would not be determined by the SymTFT alone.

The SymTFT construction uses a smaller class of caps. The boundary $L$ is topological: it carries zero Hamiltonian at the end of the slab. In this sense, the Hamiltonian of the boundary absolute QFTs solely come from $B_{\mathrm{phys}}(J)$, while $L$ supplies which Hilbert space the Hamiltonian acting upon.

We may now construct the object whose measure will be discussed in the next subsection. Let $\mathcal{L}_{\mathrm{top}}$ denote the collection of admissible topological boundary conditions³³3More precisely, simple topological boundary conditions are those cannot be reduced to simpler topological boundaries. Since here we are only interested in semisimple TFTs, simple topological boundaries are equivalently those which are indecomposible into sums of other topological boundaries. of the SymTFT. Before specifying a measure, the possible averaged cap functionals lie in the span

$$\operatorname{Span}\left\{\langle L;M|:L\in\mathcal{L}_{\mathrm{top}}\right\}\subset\mathcal{H}_{\mathrm{sym}}(M)^{*}.$$

(2.14)

An averaged partition function will be obtained by choosing a linear combination, or in continuous examples an integral, of these cap functionals and applying it to $|\Psi_{\mathrm{phys}}(M;J)\rangle$.

The analogy with summing over topologies is now direct. A bordism $Y:M\to\emptyset$ defines a cap in $\mathcal{H}_{\mathrm{sym}}(M)^{*}$. A topological boundary condition $L$ also defines a cap in $\mathcal{H}_{\mathrm{sym}}(M)^{*}$. If, after applying the SymTFT functor, the bordism sum only depends on the filling through its induced topological cap, then the sum over topologies can be reorganized as a sum over topological boundary conditions. Schematically,

$$\text{bordisms }Y:M\to\emptyset\xrightarrow{\;Z_{\mathrm{sym}}\;}\text{cap functionals in }\mathcal{H}_{\mathrm{sym}}(M)^{*}\leadsto\text{topological boundary conditions }L.$$

(2.15)

Note that we by no means claim that every ensemble in holography arises this way. We also do not claim that every geometric filling is literally a topological boundary condition. The claim is that, for a fixed SymTFT, the natural topological caps are supplied by its topological boundary conditions, and in examples where the bulk topology sum is only sensitive to this cap data, the topology sum can be rewritten as an average over such boundary conditions.

It remains to specify how the different caps should be weighted. For a finite SymTFT, the natural object is not a set but a groupoid: topological boundary conditions can have automorphisms, and equivalent boundary conditions should not be counted independently. The corresponding average is therefore a groupoid sum over isomorphism classes of topological boundary conditions, in the same spirit as the groupoid-cardinality factors that appear in sums over fields or bordisms. The same principle applies to discrete, possibly infinite, families whenever the sum is well-defined. For a continuous SymTFT, this counting problem is replaced by a measure problem. The space of Lagrangian boundary data should be equipped with a measure invariant under the natural duality action of the SymTFT. We now turn to this question.

Let us specify weights in the sum over topological caps. If the weights depended on the spacetime $M$, or on the sources $J$, then the resulting object would not be an ensemble average in the usual sense. It would instead be a separate prescription for each observable. An ensemble measure should be chosen once and for all on the space of caps, and the same measure should then be used for all boundary manifolds and all insertions.

Let

$$F_{M,J}(L)=Z_{L}(M;J)=\langle L;M|\Psi_{\mathrm{phys}}(M;J)\rangle$$

(2.16)

be the function on the space of topological boundary conditions obtained by evaluating the physical boundary state against the cap, i.e. topological boundary $L$. The problem is to define an integral of this function over the allowed topological boundary conditions.

We first consider the case where the collection of topological boundary conditions is finite. The correct notion is generally not a merely set but a groupoid⁴⁴4For the physics reader comfortable with categories, a groupoid can be regarded as a special category with all objects and morphisms invertible.. The associated physical facts include that two topological boundary conditions may be equivalent, and/or a given topological boundary condition may have automorphisms. Therefore one should not simply sum over all topological boundaries with the same weight. For a finite groupoid $\mathcal{G}$, the natural integral of a function $F$ on its objects is the groupoid cardinality integral

$$\int_{\mathcal{G}}F:=\sum_{[x]\in\pi_{0}(\mathcal{G})}\frac{1}{|\operatorname{Aut}_{\mathcal{G}}(x)|}\,F(x).$$

(2.17)

Here the sum is over isomorphism classes of objects, and $\operatorname{Aut}_{\mathcal{G}}(x)$ is the automorphism group of the object $x$. This is the same counting rule that appears in finite group gauge theory (see, e.g., [1511.00295]) and in summed-bordism constructions (see, e.g., [2201.00903]): configurations with nontrivial automorphisms are weighted by the inverse order of their automorphism group.

Applying this rule to the groupoid $\mathcal{L}_{\mathrm{top}}$ of topological boundary conditions gives

$$\boxed{\left\langle Z(M;J)\right\rangle_{\mathrm{top}}=\sum_{[L]}\frac{1}{|\operatorname{Aut}(L)|}\,\langle L;M|\Psi_{\mathrm{phys}}(M;J)\rangle.}$$

(2.18)

This formula is the finite version of the topological-boundary ensemble. The automorphism group in (2.18) is the automorphism group of the cap as an object of the groupoid of topological boundary conditions. It is not an automorphism group of the spacetime $M$. The spacetime data have already entered through the function $F_{M,J}(L)=\langle L;M|\Psi_{\mathrm{phys}}(M;J)\rangle$.

The same principle applies to discrete but possibly infinite families of caps. If the set of isomorphism classes is countable and the automorphism groups are finite, the ensemble averaging is again (2.18). Now, however, convergence is part of the problem. The counting prescription tells us how each cap should be weighted, but it does not by itself guarantee that the total sum defines a finite partition function.

For continuous SymTFTs, the groupoid sum is replaced by the integral with a measure. The space of topological boundary conditions can have continuous components, and the sum over $[L]$ is promoted to an integral

$$\boxed{\left\langle Z(M;J)\right\rangle_{\mathrm{top}}=\int_{\mathcal{L}_{\mathrm{top}}}d\mu(L)\,\langle L;M|\Psi_{\mathrm{phys}}(M;J)\rangle.}$$

(2.19)

The measure $d\mu(L)$ should be intrinsic to the SymTFT. In the examples of interest, there is a natural duality group acting on the space of topological boundary, and the measure is fixed by invariance under this action.

More explicitly, suppose that a generic component of the space of topological boundary conditions reads

$$\mathcal{L}_{\mathrm{gen}}\simeq\Gamma\backslash G/H.$$

(2.20)

Here $G$ is the continuous group of automorphisms of the topological defect data preserving the bulk TFT structure, e.g., braiding or quadratic form, $H$ is the stabilizer of a reference topological boundary condition, and $\Gamma$ is the discrete duality group by which physically equivalent data are identified. The measure on $G/H$ is induced from Haar measure [9006cc9e-2dcc-3fd8-aada-e4af19b6e225] on $G$, and it descends to a measure on $\Gamma\backslash G/H$. Equivalently, it is characterized by

$$d\mu(gL)=d\mu(L),\qquad g\in G,$$

(2.21)

together with the quotient by $\Gamma$.

If the quotient has finite volume, one can either use the unnormalized measure or normalize it to unit total volume. Thus one may define

$$\left\langle Z(M;J)\right\rangle_{\mathrm{top}}^{\mathrm{norm}}=\frac{1}{\operatorname{Vol}_{\mu}(\mathcal{L}_{\mathrm{top}})}\int_{\mathcal{L}_{\mathrm{top}}}d\mu(L)\,Z_{L}(M;J),$$

(2.22)

provided

$$\operatorname{Vol}_{\mu}(\mathcal{L}_{\mathrm{top}})=\int_{\mathcal{L}_{\mathrm{top}}}d\mu(L)<\infty.$$

(2.23)

In many cases of interest, the overall normalization is conventional, so we will not specify it.

The finite and continuous prescriptions are compatible. If the quotient $\Gamma\backslash G/H$ has orbifold points, then the invariant measure should be understood as an orbifold measure. Locally, if a point has a finite stabilizer group $G_{L}$, integration over a small neighborhood $U/G_{L}$ is defined by

$$\int_{U/G_{L}}f=\frac{1}{|G_{L}|}\int_{U}f.$$

(2.24)

Thus the familiar factor $1/|\operatorname{Aut}(L)|$ is the discrete version of the same principle.

We begin by recalling the general baby-universe interpretation of ensemble averaging, and then specialize it to the closed-string sector of the Marolf–Maxfield topological model [2002.08950]. This review is slightly longer than what is strictly needed for the computation, but it will be useful for explaining what the SymTFT construction in the next subsection is supposed to reproduce.

Consider a gravitational path integral with asymptotic boundary conditions $J_{i}$ imposed on $n$ disconnected boundary components. Schematically one writes

$$\big\langle Z[J_{1}]\cdots Z[J_{n}]\big\rangle_{\mathrm{grav}}=\int_{\Phi\sim\{J_{i}\}}\mathcal{D}\Phi\,e^{-S[\Phi]}.$$

(3.1)

If the bulk path integral includes connected geometries whose conformal boundary has several connected components, then the answer need not factorize. For example,

$$\big\langle Z[J_{1}]Z[J_{2}]\big\rangle_{\mathrm{grav}}\neq\big\langle Z[J_{1}]\big\rangle_{\mathrm{grav}}\big\langle Z[J_{2}]\big\rangle_{\mathrm{grav}}.$$

(3.2)

The connected contribution is interpreted as a spacetime wormhole between the two asymptotic boundaries. This is the basic origin of the ensemble interpretation.

A convenient way to organize this non-factorization is to cut open the gravitational path integral along an intermediate slice. The slice may contain components that do not reach any asymptotic boundary. These closed spatial components are called baby universes. The states obtained in this way span the baby-universe Hilbert space [Coleman:1988cy, Giddings:1988cx, Giddings:1988wv], denoted $\mathcal{H}_{\mathrm{BU}}$. Given $n$ boundary insertions, there is a corresponding state

$$\big|Z[J_{1}]\cdots Z[J_{n}]\big\rangle\in\mathcal{H}_{\mathrm{BU}}.$$

(3.3)

The state with no asymptotic boundary is the Hartle–Hawking state,

$$|\mathrm{HH}\rangle\in\mathcal{H}_{\mathrm{BU}},\qquad\mathcal{N}:=\langle\mathrm{HH}|\mathrm{HH}\rangle=\langle 1\rangle_{\mathrm{grav}}.$$

(3.4)

Here $\mathcal{N}$ is the no-boundary amplitude, or cosmological partition function. In a completely precise construction one should also quotient by null states in the inner product defined by the gravitational path integral. We will not need this refinement explicitly in the closed-string sector example below.

For each asymptotic boundary condition $J$, one defines an operator $\widehat{Z}[J]$ on $\mathcal{H}_{\mathrm{BU}}$ by adding one more boundary component:

$$\widehat{Z}[J]\,\big|Z[J_{1}]\cdots Z[J_{n}]\big\rangle=\big|Z[J]Z[J_{1}]\cdots Z[J_{n}]\big\rangle.$$

(3.5)

Since the order of the boundary components in the gravitational path integral is irrelevant, these operators commute:

$$[\widehat{Z}[J_{1}],\widehat{Z}[J_{2}]]=0.$$

(3.6)

Thus they can be diagonalized simultaneously. We denote their common eigenstates by $\alpha$ :

$$\widehat{Z}[J]|\alpha\rangle=Z_{\alpha}[J]\,|\alpha\rangle,\qquad\forall\,J.$$

(3.7)

Inserting a resolution of the identity in the $\alpha$-basis gives

$$\frac{\big\langle Z[J_{1}]\cdots Z[J_{n}]\big\rangle_{\mathrm{grav}}}{\langle 1\rangle_{\mathrm{grav}}}=\sum_{\alpha}p_{\alpha}\,Z_{\alpha}[J_{1}]\cdots Z_{\alpha}[J_{n}],\qquad p_{\alpha}=\frac{|\langle\mathrm{HH}|\alpha\rangle|^{2}}{\langle\mathrm{HH}|\mathrm{HH}\rangle}.$$

(3.8)

This is the ensemble interpretation. A fixed $\alpha$-sector is a factorizing theory, while the Hartle–Hawking state prepares a probability distribution over such sectors. If the $\alpha$-spectrum is continuous, the sum in (3.8) is replaced by an integral with the corresponding probability measure.

We now specialize this discussion to the closed-string sector of the Marolf–Maxfield topological model. By “closed-string sector” we mean the sector with closed circular asymptotic boundaries and no end-of-the-world branes. There is only one type of closed asymptotic boundary $S^{1}$, so there is a single boundary observable, which we denote by $Z$. Equivalently, the general operator $\widehat{Z}[J]$ above reduces to one baby-universe operator, which we denote by a reduced symbol $\widehat{Z}$. The gravitational path integral with $n$ labelled circular boundaries computes the moment $\big\langle Z^{n}\big\rangle_{\mathrm{MM}}$. The boundaries are held fixed and labelled. Thus automorphisms of a bulk surface are required to act trivially on the boundary components.

The combinatorics of the closed sector is simple. A connected component of the bulk can end on any nonempty subset of the $n$ boundary circles. After summing over the genus of that connected component, and after absorbing the boundary weight into the normalization of $Z$, every connected component with at least one boundary contributes the same effective factor. We denote this factor by $\lambda$. Equivalently, this is to say that all connected closed-boundary amplitudes are normalized to the same $\lambda$:

$$\frac{\big\langle Z^{m}\big\rangle_{\mathrm{conn}}}{\langle 1\rangle_{\mathrm{MM}}}=\lambda,\qquad m\geq 1.$$

(3.9)

The detailed dependence of $\lambda$ on the topological action is not important for us. It can be regarded as the effective fugacity for one connected bulk component that touches the asymptotic boundary.

For $n$ labelled boundaries, a bulk configuration is therefore specified by a partition $\pi$ of the set $[n]=\{1,\ldots,n\}$. Each block of $\pi$ records the set of boundary circles that lie on one connected component of the bulk. Since each block contributes a factor $\lambda$, we obtain

$$\frac{\big\langle Z^{n}\big\rangle_{\mathrm{MM}}}{\langle 1\rangle_{\mathrm{MM}}}=\sum_{\pi\in\operatorname{Part}([n])}\lambda^{|\pi|}=B_{n}(\lambda)\equiv e^{-\lambda}\sum_{d=0}^{\infty}\frac{\lambda^{d}}{d!}\,d^{n}.$$

(3.10)

where $B_{n}(\lambda)$ is the Bell (or Touchard) polynomial. The normalized generating function thus is

$$\frac{\big\langle e^{uZ}\big\rangle_{\mathrm{MM}}}{\langle 1\rangle_{\mathrm{MM}}}=\exp\!\left[\lambda(e^{u}-1)\right].$$

(3.11)

Equations (3.11) show that the closed-string sector Marolf–Maxfield moments are precisely the moments of a Poisson random variable of mean $\lambda$. Thus the $\alpha$-sectors can be labelled by a non-negative integer $d$, and the baby-universe operator has spectrum

$$\widehat{Z}|d\rangle=d\,|d\rangle,\qquad d\in\mathbb{Z}_{\geq 0}.$$

(3.12)

The Hartle–Hawking state prepares the probability distribution

$$p_{d}=\frac{|\langle\mathrm{HH}|d\rangle|^{2}}{\langle\mathrm{HH}|\mathrm{HH}\rangle}=e^{-\lambda}\frac{\lambda^{d}}{d!}.$$

(3.13)

In a fixed $d$-sector the theory factorizes:

$$\big\langle Z^{n}\big\rangle_{d}=d^{n}.$$

(3.14)

The non-factorizing Marolf–Maxfield answer (3.10) is obtained only after averaging over the $d$-sectors with the Poisson weights (3.13).

We now give a SymTFT realization of the closed-string sector ensemble averaging problem reviewed above. The model does not enjoy a nice Lagrangian description such as BF or Chern–Simons theory. The fixed parent object we use is instead a discrete, stacky, groupoid-completed 2D SymTFT analogue, which we denote by $\mathfrak{S}_{\mathrm{Fin}}$. It is not the ordinary TFT associated with one chosen finite set. Rather, it is a single parent object whose admissible topological boundary conditions, to be summed over below, are labelled by finite sets. Since the closed-string sector of Marolf–Maxfield has only one asymptotic boundary observable $Z$, the topological data needed to reproduce their result is very small and explicit.

A mathematical fact we will use is that oriented 2D TFTs can be constructed via Frobenius algebras [Abrams:1996ty, Kock_2003, 2206.12448]⁵⁵5We refer the reader to [2311.16230] for a physics friendly introduction to Frobenius algebra.. We will use this fact only after a topological cap has been chosen. Let $S$ be a finite set. The cap labelled by $S$ produces an ordinary absolute closed 2D TFT whose Frobenius algebra is

$$\mathcal{A}_{S}=\operatorname{Fun}(S,\mathbb{C})=\bigoplus_{s\in S}\mathbb{C}e_{s},$$

(3.15)

where $e_{s}$ is the delta-function supported at $s$. The categorical data of this algebra is labeled by the Frobenius object $(\mathcal{A}_{S},\mu,\eta,\delta,\varepsilon)$:

$$\begin{split}\text{multiplication }\mu&:\mathcal{A}_{S}\otimes\mathcal{A}_{S}\rightarrow\mathcal{A}_{S},\\ \text{unit }\eta&:I\rightarrow\mathcal{A}_{S},\\ \text{comultiplication }\delta&:\mathcal{A}_{S}\rightarrow\mathcal{A}_{S}\otimes\mathcal{A}_{S},\\ \text{counit }\varepsilon&:\mathcal{A}_{S}\rightarrow I.\end{split}$$

(3.16)

which respectively read

$$e_{s}e_{s^{\prime}}=\delta_{s,s^{\prime}}e_{s},\qquad 1_{S}=\sum_{s\in S}e_{s},\qquad\Delta_{S}(e_{s})=e_{s}\otimes e_{s},\qquad\varepsilon_{S}(e_{s})=1.$$

(3.17)

The Frobenius pairing is

$$\langle e_{s},e_{s^{\prime}}\rangle_{S}=\varepsilon_{S}(e_{s}e_{s^{\prime}})=\delta_{s,s^{\prime}}.$$

(3.18)

By the mathematical construction of oriented 2D TFTs via Frobenius algebras, our algebra $\mathcal{A}_{S}$ defines an oriented closed worldsheet TFT, which we denote by $\mathcal{T}_{S}$. Its state space on a circle is

$$\mathcal{H}_{\mathcal{T}_{S}}(S^{1})=\mathcal{A}_{S}.$$

(3.19)

The case $S=\varnothing$ will also be included. We regard it as the formal zero-dimensional sector

$$\mathcal{A}_{\varnothing}=0.$$

(3.20)

For $d>0$, we choose the representative

$$[d]=\{1,\ldots,d\},\qquad\mathcal{A}_{[d]}\cong\mathbb{C}^{d}.$$

(3.21)

We should not regard the theories $\mathcal{T}_{S}$ as different parent SymTFTs that are subsequently summed over. Rather, the fixed parent object is a discrete analogue

$$\mathfrak{S}_{\mathrm{Fin}},$$

(3.22)

whose simple topological boundary conditions form the groupoid

$$\operatorname{TopBdy}(\mathfrak{S}_{\mathrm{Fin}})\simeq\mathsf{FinSet}^{\simeq}.$$

(3.23)

Since finite sets can have arbitrary cardinality, $\mathfrak{S}_{\mathrm{Fin}}$ is not a finite SymTFT in the usual finite-semisimple sense; it is a countably semisimple, discrete non-compact SymTFT. A choice of topological boundary condition $\mathsf{B}_{S}$ labelled by a finite set $S$ produces the absolute closed TFT $\mathcal{T}_{S}$ described above. Thus

$$\mathcal{T}_{S}(S^{1})=\mathcal{A}_{S}=\operatorname{Fun}(S,\mathbb{C})$$

(3.24)

is the closed algebra of the fixed-cap absolute theory, not the parent SymTFT itself.

This is analogous to a finite discrete gauge-theory SymTFT. For example, in 2D $\mathbb{Z}_{N}$ BF theory, with schematic action

$$\frac{iN}{2\pi}\int a_{0}\,db_{1},$$

(3.25)

one fixed SymTFT controls a family of absolute theories labelled by a discrete parameter in $\mathbb{Z}_{N}$ . A topological boundary condition selects one member of that family. In the present finite-set model, the discrete label $r\in\mathbb{Z}_{N}$ is replaced by a finite set $S$, and the selected absolute theory is $\mathcal{T}_{S}$.

For $c=1$, the SymTFT action (4.1) reduces to the $\mathbb{R}$-valued BF theory [2401.10165]

$$S_{\mathrm{BF}}={1\over 2\pi}\int_{X_{3}}a\wedge db,\qquad a,b\in\Omega^{1}(X_{3};\mathbb{R}).$$

(4.12)

The topological line operators are

$$U_{\alpha}[\gamma]=\exp\left(i\alpha\oint_{\gamma}a\right),\qquad V_{\beta}[\gamma]=\exp\left(i\beta\oint_{\gamma}b\right),\qquad\alpha,\beta\in\mathbb{R}.$$

(4.13)

Their braiding is

$$\left\langle U_{\alpha}[\gamma_{1}]V_{\beta}[\gamma_{2}]\right\rangle=\exp\left(2\pi i\,\alpha\beta\,\mathrm{Link}(\gamma_{1},\gamma_{2})\right).$$

(4.14)

Equivalently, the defect group is

$$\mathbb{D}_{1}=\mathbb{R}\oplus\mathbb{R}$$

(4.15)

with Dirac pairing

$$\big\langle(\alpha,\beta),(\alpha^{\prime},\beta^{\prime})\big\rangle=\alpha\beta^{\prime}+\alpha^{\prime}\beta\quad\mathrm{mod}\ \mathbb{Z}.$$

(4.16)