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

Abstract: We propose a SymTFT interpretation of ensemble averaging in low-dimensional holography. The central operation is to keep fixed both the SymTFT and the physical boundary condition, while averaging over topological boundary conditions at the other end of the SymTFT slab. Each such boundary condition gives an absolute completion of the same relative theory, so the ensemble is interpreted as an average over topological completions rather than over arbitrary local dynamics. We formulate this construction in terms of cap functionals and their natural groupoid or Haar-type measures, and illustrate it in two examples. In the closed-string sector of the Marolf--Maxfield model, topological boundary conditions are labelled by finite sets, and the groupoid sum reproduces the Poisson/Bell-polynomial moments. In the Narain case, compact topological boundary conditions of an $\mathbb{R}$-valued BF SymTFT are identified with maximal isotropic subgroups, so that topological-boundary averaging becomes the usual Narain moduli average with Zamolodchikov measure. We also discuss possible extensions to JT gravity, random matrix theory, Virasoro T(Q)FT, and 3D gravity.

• The paper demonstrates that ensemble averaging in holography arises from averaging over topological completions of fixed QFTs, using the SymTFT framework.
• It employs discrete (Marolf–Maxfield) and continuous (Narain moduli) examples to recover Poisson and Siegel–Weil results through specific averaging procedures.
• Its methodology distinguishes averaging over fixed topological caps from boundary dynamics, offering new insights into non-factorizing gravitational path integrals.

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

Overview and Motivation

This paper develops a formal framework interpreting ensemble averaging in low-dimensional holography via topological boundary condition averaging within the formalism of Symmetry Topological Field Theories (SymTFTs). The central thesis is that gravitational path integrals in certain holographic models do not correspond to averages over arbitrary boundary dynamics, but rather to averages over distinct topological completions—labelled by topological boundary conditions on SymTFT slabs—of a fixed relative quantum field theory (QFT). This approach is instantiated in two emblematic cases: the closed-string sector of the Marolf–Maxfield (MM) topological model and the ensemble of Narain compactifications in two-dimensional conformal field theory (CFT), providing both discrete and continuous examples.

Topological Boundary Averaging Prescription

The paper’s formalism centers around placing a fixed SymTFT $\mathfrak{T}_{\mathrm{sym}}$ on a slab $M \times [0,1]$, with a physical boundary condition $B_{\mathrm{phys}}(J)$ determined by metric and source data. The state prepared on a closed $d$-manifold $M$ is $$\left|\Psi_{\mathrm{phys}}(M;J)\right\rangle\in\mathcal{H}_{\mathrm{sym}}(M)$$. The other end of the slab is capped with a topological boundary condition $L$, which defines a cap functional $$\left\langle L;M\right| \in \mathcal{H}_{\mathrm{sym}}(M)^*$$. The absolute partition function is then $$Z_L(M;J) = \left\langle L;M\,|\,\Psi_{\mathrm{phys}}(M;J) \right\rangle$$, and ensemble averaging is formalized as an average over admissible (simple) topological boundary conditions: $$\big\langle Z(M;J) \big\rangle_{\mathrm{top}} = \int_{\mathcal{L}_{\mathrm{top}}} d\mu(L)\, \left\langle L;M\,|\,\Psi_{\mathrm{phys}}(M;J) \right\rangle.$$ Discrete cases employ groupoid sums with automorphism factors, while continuous cases employ Haar-type measures on moduli spaces.

This averaging procedure is sharply distinguished from averaging over arbitrary boundary conditions or local Hamiltonians; only topological caps (carrying zero Hamiltonian) are considered, so physically the ensemble explores different Hilbert spaces and charge lattices but leaves boundary dynamics fixed.

Marolf–Maxfield Model: Discrete Topological Boundary Ensemble

The first example analyzed is the MM topological model's closed-string sector. Here, ensemble moments correspond to those of a Poisson random variable, classically attributed to baby universe contributions in gravitational path integrals, leading to non-factorization. The paper constructs a countably semisimple discrete SymTFT analog $M \times [0,1]$0, whose simple topological boundary conditions are labelled by finite sets up to bijection (objects: finite sets; morphisms: bijections).

Upon capping by a boundary labelled by $M \times [0,1]$1, the absolute completion is a 2D oriented TFT with Frobenius algebra $M \times [0,1]$2. The partition function on $M \times [0,1]$3 is simply $M \times [0,1]$4. Ensemble averaging employs a groupoid sum, weighted by automorphism factors and fugacity $M \times [0,1]$5: $M \times [0,1]$6 where $M \times [0,1]$7 is the Bell polynomial—recovering the MM moments and their non-factorization. Thus, the entire ensemble structure arises from averaging over finite-set caps, not from boundary dynamics.

Narain Moduli Ensemble: Continuous Topological Boundary Integration

The second example is the continuous ensemble of Narain compactifications. The relevant SymTFT is 3D $M \times [0,1]$8-valued abelian BF theory, whose physical boundary prepares a state with fixed oscillator sector and $M \times [0,1]$9-dimensional current algebra, but whose compact topological boundary conditions correspond to choices of even, self-dual, maximal isotropic subgroups of the defect group $B_{\mathrm{phys}}(J)$0. Each subgroup labels a point in the Narain moduli space: $B_{\mathrm{phys}}(J)$1 and the ensemble average becomes an integration over this space with Haar-induced (Zamolodchikov) measure: $B_{\mathrm{phys}}(J)$2 For $B_{\mathrm{phys}}(J)$3, the normalized average converges, and the Siegel–Weil formula yields: $B_{\mathrm{phys}}(J)$4 where $B_{\mathrm{phys}}(J)$5 is the real-analytic Eisenstein series. The $B_{\mathrm{phys}}(J)$6 and $B_{\mathrm{phys}}(J)$7 cases are shown to diverge.

This continuous averaging encapsulates the full Narain moduli ensemble structure, and topological caps precisely specify charge lattices and thus the CFT spectrum, with local boundary dynamics (e.g., Hamiltonian) held fixed.

Extensions and Speculative Directions

The paper discusses speculative extensions of the framework to JT gravity (where a noncompact $B_{\mathrm{phys}}(J)$8 BF-theory structure potentially underlies averaging over one-dimensional quantum mechanics ensembles), as well as pure AdS$B_{\mathrm{phys}}(J)$9 gravity, where Virasoro T(Q)FT provides a SymTFT backbone whose topological boundary condition average could correspond to averaging over CFT data (primary spectra, OPE coefficients) at fixed central charge and Virasoro symmetry. It is conjectured that similar topological boundary averaging mechanisms may explain ensemble holography in these more intricate, non-rational, and non-compact settings.

The classification of permissible topological boundaries for noncompact and non-rational TQFTs—especially Virasoro T(Q)FT—is identified as an open, challenging problem for the field.

Implications and Future Outlook

This paper offers a precise technical elucidation of how ensemble averages in low-dimensional holography are not arbitrary but constrained by topological boundary conditions of fixed SymTFTs. The distinction between averaging over caps (topological completions) and averaging over boundary Hamiltonians provides clarity for interpreting wormhole-connected contributions and non-factorizing path integrals in gravitational settings.

Practically, this formalism enables computation of ensemble averages using groupoid sums or Haar measures, with explicit identification of moduli spaces as spaces of topological caps. The framework illuminates the underlying structure of ensemble holography in both discrete and continuous settings, and recovers strong numerical results such as Poisson/Bell moment generation and the Siegel–Weil formula for Narain averages. Contradicting interpretations that align ensemble averages with arbitrary boundary dynamics, the paper asserts that only topological completions matter for these ensembles.

Theoretically, if extended to more general settings—especially non-rational or noncompact TQFTs—this perspective may organize understanding of ensemble averages in gravity and QFT, potentially linking quantum group, category-theoretic, and functional-analytic structures relevant for the classification of topological boundaries. It suggests concrete future directions: technical development of boundary classification and measure construction for JT gravity and Virasoro T(Q)FT, and explicit computation of ensemble averages beyond the abelian and topological cases.

Conclusion

By rigorously formalizing ensemble averaging as averaging over topological boundary conditions in the SymTFT framework, this paper clarifies the structure of holographic ensembles, highlights mechanisms of non-factorization, and unifies discrete and continuous moduli averaging via cap measures. These results provide robust tools and conceptual advances for low-dimensional holography, ensemble QFTs, and the role of topological completions in quantum field theory and gravity. Future progress in boundary classification and measure theory for irrational and noncompact SymTFTs could further deepen and generalize these insights.