---
title: Topological-Boundary Averaging in Field Theories
url: https://www.emergentmind.com/topics/topological-boundary-averaging
type: topic
---

# Topological-Boundary Averaging in Field Theories

Topological-Boundary Averaging denotes a family of constructions in which topological or boundary data are varied, summed, integrated, or compressed while some bulk structure remains fixed. In the cited literature, the expression is used in several distinct ways. In Abelian topological order, it can mean comparing the ground-state degeneracy on manifolds with boundary across admissible gapped boundary conditions, and optionally averaging those values over boundary types [1212.4863]. In SymTFT, it means fixing the SymTFT and the physical boundary condition, while averaging over topological boundary conditions at the other end of a slab, thereby averaging over absolute completions of a fixed relative theory [2605.06653]. In locally topologically ordered quantum spin systems, a canonical quantum channel compresses bulk observables onto a boundary quasi-local algebra, acting as a rigorous boundary-to-bulk averaging map [2307.12552]. In two-dimensional BCFT, boundary averaging is implemented as a Gaussian average over bulk–boundary OPE data, producing wormhole contributions and resolving end-of-the-world brane intersections [2206.03035].

## 1. Terminological scope

The term does not name a single universally standardized construction. Rather, the literature uses it for several boundary-centered operations that share a common pattern: some boundary datum is varied or integrated, while a bulk theory, relative theory, or topological phase is held fixed. This suggests a family resemblance rather than a single invariant definition.

| Setting | Fixed structure | Boundary operation |
|---|---|---|
| Abelian topological order | Bulk phase and geometry | Compare or average boundary GSD over gapped boundary types |
| SymTFT slab | SymTFT and physical boundary condition | Average over topological caps with groupoid/Haar-type measure |
| Local topological order | Net of local ground state projections | Compress bulk observables to boundary algebra via a canonical channel |
| BCFT / AdS\(_3\) | Bulk CFT data and boundary label \(a\) | Gaussian averaging over \(C^a_{p\mathbb{I}}\) |
| Hyperbolic lattices | Bulk Hamiltonian on \(\ell^2(\Gamma)\) | Remove geometric boundaries via converging periodic quotients |

A recurrent distinction is between **topological completion** and **local dynamics**. In the SymTFT formulation, the Hamiltonian comes from the physical boundary condition \(B_{\mathrm{phys}}\), while the topological boundary condition \(L\) supplies the Hilbert space and the spectrum of allowed charge sectors on which that Hamiltonian acts [2605.06653]. In Wang–Wen’s boundary-degeneracy setting, the bulk topological order is fixed, but the open-manifold ground-state degeneracy depends crucially on the boundary gapping conditions [1212.4863].

## 2. Boundary degeneracy and averaging in topological order

Wang and Wen introduced **boundary degeneracy** as the ground-state degeneracy of a topologically ordered system on a compact orientable manifold with boundary, under the assumption that all boundary edge modes are fully gapped [1212.4863]. They emphasized that boundary degeneracy provides richer information than bulk degeneracy, because it depends not only on the bulk and manifold topology but crucially on boundary gapping conditions.

In the Abelian \(K\)-matrix Chern–Simons framework, gapped boundaries arise from condensing a complete set of non-fractionalized particles on the edge through sine-Gordon terms
\[
\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),
\]
with \(\ell_a\in \Gamma_e\), where \(\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}\). The condensed set \(\Gamma^{\partial}\) must satisfy the Boundary Fully Gapping Rules: for all \(\ell_a,\ell_b\in\Gamma^{\partial}\),
\[
\ell_a^T K^{-1}\ell_b=0,
\]
\[
\dim \Gamma^{\partial}=N/2,
\]
\[
\mathrm{signature}(K)=0,
\]
and \(\ell_a\in\Gamma_e\) [1212.4863]. The compatible anyons form a larger lattice \(\Gamma^{\partial}_{qp}\) consisting of quasiparticles with trivial braiding statistics relative to \(\Gamma^{\partial}\).

For a manifold with \(\eta\) boundaries \(\partial_\alpha\), each carrying boundary lattice \(\Gamma^{\partial_\alpha}\), Wang–Wen derived the boundary GSD formula
\[
\mathrm{GSD}=\left|\frac{L_{qp\cap e}}{\bigoplus_{\alpha=1}^{\eta}\Gamma^{\partial_\alpha}}\right|,
\]
where \(L_{qp\cap e}\) is the set of tuples of compatible anyons satisfying the total neutrality constraint \(\sum_\alpha \ell^{\partial_\alpha}_{qp}\in\Gamma_e\) [1212.4863]. On a cylinder or annulus, this reduces to counting deconfined Wilson-line sectors compatible with both boundaries and the neutrality condition.

The standard examples make the diagnostic content explicit. For the \(Z_2\) toric code,
\[
K_{Z_2}=\begin{bmatrix}0&2\\2&0\end{bmatrix},
\]
rough and smooth boundaries correspond to condensates generated by \((2,0)\) and \((0,2)\). On a cylinder, rough–rough or smooth–smooth gives \(\mathrm{GSD}=2\), while rough–smooth gives \(\mathrm{GSD}=1\). For the \(Z_2\) double-semion model,
\[
K=\begin{bmatrix}2&0\\0&-2\end{bmatrix},
\]
Wang–Wen show two boundary gapping lattices \(\{n(2,2)\}\) and \(\{n(2,-2)\}\), but these are identified by \(\Gamma_e\), yielding a single physical boundary condition, and the cylinder GSD remains \(2\) for any pair [1212.4863]. Thus the \(Z_2\) toric code and \(Z_2\) double-semion model have identical bulk GSD on closed surfaces and the same doubled fusion algebra \(\mathbb Z_2\times \mathbb Z_2\), but different boundary GSD on a cylinder.

The paper itself does not define an averaging procedure. A meaningful reinterpretation given in the data is to fix a bulk phase and geometry, let \(B\) be the set of admissible gapped boundary conditions, and define
\[
\langle \mathrm{GSD}\rangle=\frac{1}{|B|^2}\sum_{(L_1,L_2)\in B\times B}\mathrm{GSD}(L_1,L_2).
\]
This “Topological-Boundary Averaging” is not a topological invariant: it depends on the boundary ensemble and weights [1212.4863]. For \(k=2\), the toric code has two distinct boundary types and a uniform average \((2+2+1+1)/4=1.5\), whereas the double-semion theory effectively has one boundary type and \(\langle \mathrm{GSD}\rangle=2\). The averaged value and the full distribution \(\{\mathrm{GSD}(L_1,L_2)\}\) therefore distinguish phases that are indistinguishable by bulk GSD alone.

## 3. SymTFT formulation: averages over topological completions

In the SymTFT formulation, a \(D\)-dimensional QFT with generalized global symmetry is realized as a boundary condition for a \((D+1)\)-dimensional topological theory. The physical boundary condition \(B_{\mathrm{phys}}(J)\) on one end of a slab \(M\times[0,1]\) prepares a partition vector
\[
\big|\Psi_{\mathrm{phys}(M;J)}\big\rangle\in\mathcal H_{\mathrm{sym}(M)},
\]
while a topological boundary condition \(L\) on the other end defines a cap functional
\[
\big\langle L;M\big|\in\mathcal H_{\mathrm{sym}(M)}^*.
\]
The corresponding absolute theory has partition function
\[
Z_L(M;J)=\big\langle L;M\,\big|\,\Psi_{\mathrm{phys}(M;J)}\big\rangle.
\]
Topological-boundary averaging is then
\[
\big\langle Z(M;J)\big\rangle_{\mathrm{top}}
=
\int_{\mathcal L_{\mathrm{top}}} d\mu(L)\,\big\langle L;M\,\big|\,\Psi_{\mathrm{phys}(M;J)}\big\rangle,
\]
with \(d\mu\) the intrinsic groupoid/Haar-type measure on the space of topological boundaries [2605.06653].

A central point is that this is an average over **absolute completions** of a fixed relative theory, not an average over arbitrary local dynamics. Different \(L\) select which bulk topological operators can end on the boundary, thereby fixing global form, charge lattice, and other topological data of the absolute theory. For finite families of caps, the integral is replaced by a groupoid sum
\[
\sum_{[L]}\frac{1}{|\mathrm{Aut}(L)|}\,\big\langle L;M\,\big|\,\Psi_{\mathrm{phys}(M;J)}\big\rangle,
\]
while continuous components use invariant Haar measures descending to quotients of the form \(\Gamma\backslash G/H\) [2605.06653].

Two examples organize the framework. In the Marolf–Maxfield closed-string sector, topological boundary conditions are labeled by finite sets \(S\), the groupoid is \(\mathrm{FinSet}^{\simeq}\), and the groupoid/Haar-type weight is \(1/|\mathrm{Aut}(S)|=1/|S|!\). A cap specified by \(S\) produces a closed 2D oriented TFT with Frobenius algebra
\[
\mathcal A_S=\mathrm{Fun}(S,\mathbb C)=\bigoplus_{s\in S}\mathbb C\,e_s,
\]
and \(Z_S(S^1)=|S|\). With fugacity \(\lambda\), the normalized moments are Bell polynomials,
\[
\big\langle Z^n\big\rangle_{\mathrm{top}}=B_n(\lambda),
\]
with generating function
\[
G(t)=\exp\big(\lambda(e^t-1)\big).
\]
Non-factorization arises only after averaging over finite-set caps with the Poisson weights \(e^{-\lambda}\lambda^d/d!\) [2605.06653].

In the Narain example, the SymTFT is a 3D abelian BF theory with \(\mathbb R\)-valued gauge fields,
\[
S_{\mathrm{BF}}=\frac{1}{2\pi}\int_{X_3}\sum_{i=1}^c a_i\wedge db_i,
\]
and compact topological boundary conditions are maximal isotropic subgroups \(L\subset \mathbb D_c=\mathbb R^c\oplus\mathbb R^c\). They are parametrized by \((G,B)\) through \(O(c,c;\mathbb R)\) and form the Narain moduli space
\[
\mathcal M_{\mathrm{Narain}}
=
O(c,c;\mathbb Z)\backslash O(c,c;\mathbb R)/(O(c)\times O(c)).
\]
The averaging measure is the Haar-induced, equivalently Zamolodchikov, measure
\[
d\mu(G,B)=C_{\mathrm{Nar}}\,
\frac{\prod_{1\le i\le j\le c} dG_{ij}\prod_{1\le i<j\le c} dB_{ij}}{(\det G)^c}.
\]
For \(c>2\), the Siegel–Weil formula gives
\[
\big\langle Z(\tau)\big\rangle_{\mathrm{top}}^{(c)}
=
\frac{E_{c/2}(\tau)}{\tau_2^{c/2}\,|\eta(\tau)|^{2c}},
\]
while for \(c=2\) the theta integral diverges at cusps, and for \(c=1\) the radius average diverges due to noncompact endpoints [2605.06653].

## 4. Operator-algebraic realization in locally topologically ordered systems

An operator-algebraic realization of boundary averaging appears in the theory of local topological order. For a locally topologically ordered quantum spin system on \(\mathbb Z^k\), one has a quasi-local algebra \(\mathcal A\), a translation-invariant net of local algebras \(\Lambda\mapsto\mathcal A(\Lambda)\), and a translation-invariant net of local ground-state projections \(p_\Lambda\). The Local Topological Order axioms include the Hastings condition, boundary factorization, boundary stability/surjectivity, and boundary injectivity [2307.12552].

These axioms produce a net of boundary algebras \(B(I)\) on a codimension-one boundary hyperplane \(\mathcal K\subset\mathbb Z^k\). If \(\Lambda\subset\mathbb H\) and \(\Delta\) surrounds \(\Lambda\) with nonempty \(\partial\Delta\cap\mathcal K\), then the compression \(p_\Delta x p_\Delta\) of a local bulk observable \(x\in\mathcal A(\Lambda)\) lies in a boundary algebra localized on the interval \(I=\partial\Lambda\cap\partial\Delta\). Passing to the inductive limit gives the boundary quasi-local algebra \(B\) [2307.12552].

The key averaging object is the canonical unital completely positive map
\[
\mathfrak E:\mathcal A_{\mathbb H}\to B,
\]
defined by
\[
p_\Delta x p_\Delta=\mathfrak E(x)\,p_\Delta.
\]
It acts as the identity on the boundary subalgebra, but on observables deep in the bulk it reduces to the canonical ground state \(\psi\): if \(\Lambda\ll\Delta\) inside \(\mathbb H\), then \(\mathfrak E|_{\mathcal A(\Lambda)}=\psi\) [2307.12552]. In this precise sense, \(\mathfrak E\) “forgets” microscopic interior details and averages bulk observables to their boundary-compatible component.

This yields a Hamiltonian-free bulk–boundary correspondence. Any boundary state \(\phi_B\) on \(B\) induces a bulk–boundary state \(\phi=\phi_B\circ\mathfrak E\) on \(\mathcal A_{\mathbb H}\), and that state restricts to the canonical ground state \(\psi\) in the interior. The construction is therefore not an average over boundary conditions, but a canonical boundary-to-bulk averaging map.

The framework also identifies the boundary algebra as a complete carrier of bulk topological information. For Levin–Wen models, the boundary net is canonically identified with a fusion categorical net \(F(I)=\mathrm{End}_{\mathcal C}(X^{\# I})\), where \(X=\bigoplus_{c\in\mathrm{Irr}(\mathcal C)} c\). Its DHR category satisfies
\[
\mathrm{DHR}(B)\cong Z(\mathcal C),
\]
recovering the Drinfeld center of the input category, while for the Toric Code one obtains the quantum double of \(\mathbb Z/2\) [2307.12552]. The canonical boundary state is a trace iff \(\mathcal C\) is pointed; otherwise it is a KMS-1 state for a locally representable dynamics, and the associated cone and boundary von Neumann algebras are type III rather than type II [2307.12552].

## 5. Boundary averaging in BCFT and semiclassical gravity

In two-dimensional BCFT, boundary averaging is formulated as a Gaussian ensemble over bulk–boundary OPE coefficients. For a conformal boundary condition \(a\) with boundary state
\[
|B^a\rangle=g^a\sum_p C^a_{p\mathbb I}\,|p\rangle\rangle,
\]
the averaged ensemble treats \(C^a_{p\mathbb I}\) as Gaussian random variables with
\[
\overline{C^a_{p\mathbb I}}=0\quad\text{for }p\neq\mathbb I,
\]
and variance
\[
\overline{C^a_{p\mathbb I}\,C^a_{p\mathbb I}}=(g^a)^{-2}S_{\mathbb I p}^{-1},
\]
where \(S_{ij}\) is the modular \(S\)-matrix [2206.03035].

The annulus partition function in the closed channel is
\[
Z_{ab}(\tau)=g^a g^b \sum_p C^a_{p\mathbb I}\,C^b_{p\mathbb I}\,\chi_p(\tau),
\]
and modular duality gives the open-channel decomposition
\[
Z_{ab}(\tau)=\sum_P n^P_{ab}\,\chi_P\!\left(-\tfrac{1}{\tau}\right),
\]
or, in the holographic/continuous case, a spectral density \(\rho^{ab}(\alpha_P)\). For \(a\neq b\), the average yields
\[
\rho^{ab}(\alpha_P)=
\begin{cases}
g^a g^b\,S_{\mathbb I P}, & \alpha_P\in Q/2+i\mathbb R,\\[2pt]
0, & \text{otherwise},
\end{cases}
\]
so the open spectrum contains only “black-hole” states \(h_P\ge c/24\) and no vacuum [2206.03035].

The disk two-point bootstrap provides the geometric consequence. In Liouville variables \(c=1+6Q^2\) and \(h_i=\alpha_i(Q-\alpha_i)\), the lowest open-channel Liouville momentum is
\[
\alpha_P^{\mathrm{min}}=2\alpha_i.
\]
If \(\mathrm{Re}\,\alpha_i>Q/4\), equivalently \(h_i>c/32\), then \(\mathrm{Re}\,\alpha_P\ge Q/2\) and
\[
h_P\ge \frac{c}{24}.
\]
This matches the threshold at which a heavy bulk operator would otherwise induce an end-of-the-world brane self-intersection, because the conical defect angle satisfies
\[
\delta\theta=2\pi\left(1-\sqrt{1-\frac{24 h_i}{c}}\right),
\]
and \(\delta\theta>\pi\) iff \(h_i>c/32\) [2206.03035]. The averaged boundary bootstrap therefore replaces an unphysical intersecting-brane configuration by a black-hole saddle.

The same Gaussian averaging generates wormhole contributions. Averaged products of disk correlators produce Liouville four-point functions rather than factorized products, and replica wormholes appear in BCFT Rényi computations. This is the basis for the proposal that boundary averaging provides a BCFT dual of the island model [2206.03035]. A persistent misconception is that the average is over arbitrary boundary labels \(a\); the construction described here relies on Gaussian averaging of the bulk–boundary data \(C^a_{p\mathbb I}\), while \(g^a\) is kept fixed.

## 6. Broader boundary-aware averaging in analysis, geometry, and computation

Several neighboring literatures use boundary-aware averaging in ways that are not identical to the preceding topological-field-theoretic constructions, but still fit the pattern of preserving topology while smoothing, summing, or coarse-graining boundary data.

For continuous functions, averaging by a measure \(\mu\) on \([-1,1]\) is defined by
\[
f_\alpha(x)=\int_{-1}^{1} f(x+t\alpha)\,d\mu(t).
\]
Topological stability means that \(f\) and \(f_\alpha\) are topologically equivalent for all sufficiently small \(\alpha>0\). The cited results establish that, for continuous \(f:\mathbb R\to\mathbb R\) with finitely many local extremes, global topological stability of \(f\) under averaging is equivalent to topological stability of the germs at its local extremes, and they provide explicit criteria for measures with locally continuous and locally constant densities, including the \(X_i\)-criterion for piecewise constant \(\rho\) [1601.00151].

For sampled manifolds with boundary, union-of-balls offsets implement a boundary-aware aggregation of data points. If \(M\subset\mathbb R^N\) is a compact differentiable manifold with boundary, \(\bar x\) is \(\varepsilon/2\)-dense in \(M\), and \(\varepsilon<\delta/2\) with \(\delta<\min\{\mathrm{reach}(M),\mathrm{reach}(\partial M)\}\), then
\[
U=\bigcup_{x\in\bar x} B_\varepsilon(x)
\]
deformation retracts to \(M\), with explicit homotopy
\[
F(x,t)=t x+(1-t)\pi(x),
\]
where \(\pi\) is the nearest-point projection [1810.05759]. Here the “averaging” is geometrically realized by thickening the sample while preserving the topology of a manifold with boundary.

In regular hyperbolic tessellations, converging periodic boundary conditions eliminate geometric boundary effects by passing to a nested sequence of finite-index normal subgroups \(N_k\triangleleft \Gamma\) and boundaryless finite quotients \(G_k=\Gamma/N_k\). The normalized traces
\[
\frac{1}{|G_k|}\mathrm{Tr}
\]
converge to the bulk von Neumann trace, so that
\[
\tau(f(\pi_R(h)))=\lim_{k\to\infty}\frac{1}{|G_k|}\mathrm{Tr}\,f(\pi_R(\rho_k(h))).
\]
This is described explicitly as Topological-Boundary Averaging: boundary contributions are absent on the closed quotients, and their finite-size imprints are suppressed by normalized trace averaging [2303.15611].

In computer vision, the TOP+BAC framework combines topology extracted from density-based clustering with Bayesian active contours. The paper states that this combination “suggests a natural pathway to Topological-Boundary Averaging,” formalized as an averaged boundary
\[
\bar C=\arg\min_C \sum_{i=1}^N w_i\,D(C,C_i)+\lambda\,P_{\mathrm{topo}}(C),
\]
or, in level-set form,
\[
\phi_{\mathrm{avg}}=\sum_{i=1}^N w_i\,\phi_i,\qquad C_{\mathrm{avg}}=\{x:\phi_{\mathrm{avg}}(x)=0\},
\]
with topology penalties enforcing desired Betti numbers or Euler characteristic [1910.04778]. This is presented as an extension rather than a definition already present in the original method.

A different computational use appears in the study of boundary criticality in interacting topological insulators. There, “boundary averaging” means averaging boundary observables along the edge,
\[
C_{\mathrm{bdy}}^{SS}(r)=\frac{1}{L}\sum_{i\in\mathrm{bdy}}\langle S_i^y S_{i+r}^y\rangle,\qquad
G_{\mathrm{bdy}}(r)=\frac{1}{L}\sum_{i\in\mathrm{bdy}}\langle c_{i\sigma} c_{i+r,\sigma}^\dagger\rangle,
\]
to improve statistical accuracy and define boundary observables independent of edge position [2504.12600].

## 7. Common structure, distinctions, and diagnostic value

Across these usages, boundary averaging is rarely an invariant in its own right. In Wang–Wen’s setting, the average over boundary GSD values depends on the ensemble of admissible condensates and on the weights assigned to them [1212.4863]. In the SymTFT setting, the measure is intrinsic only after specifying the groupoid or homogeneous-space structure of the topological boundary conditions [2605.06653]. In BCFT, the average depends on the Gaussian statistics assigned to \(C^a_{p\mathbb I}\) and on the universal heavy-asymptotic formula for their variance [2206.03035].

What is common is the diagnostic role of the boundary sector. In Abelian topological order, the pattern of boundary degeneracies distinguishes phases with identical bulk fusion data [1212.4863]. In SymTFT, averaging over caps distinguishes different absolute completions of the same relative theory [2605.06653]. In the operator-algebraic framework, the boundary net \(B\) and its DHR category recover the bulk topological order, while the canonical channel \(\mathfrak E\) makes boundary states parameterize bulk–boundary states [2307.12552]. In AdS/BCFT, boundary averaging generates non-factorization and selects physically acceptable bulk saddles [2206.03035].

A second recurring theme is that “boundary” can refer to very different objects: gapped boundaries of topological phases, topological boundary conditions in a SymTFT slab, boundary states in BCFT, codimension-one boundary algebras in quantum spin systems, open boundaries in hyperbolic approximants, or geometric boundaries in sampled manifolds. A plausible implication is that the most useful encyclopedic understanding of Topological-Boundary Averaging is not as a single theorem, but as a boundary-centric methodology: hold the relevant bulk structure fixed, vary or integrate the admissible boundary data, and extract quantities—degeneracies, partition functions, spectral measures, or operator algebras—that are inaccessible from bulk information alone.

Source: https://www.emergentmind.com/topics/topological-boundary-averaging