Topological-Boundary Averaging in Field Theories

• Topological-boundary averaging is a framework that varies and integrates boundary data while keeping the bulk structure fixed, offering clear diagnostics beyond standard bulk observables.
• It applies to multiple settings, from Abelian topological order and SymTFT formulations to BCFT and quantum spin systems, using methods like Gaussian and Haar-type measures.
• The technique allows computing physical quantities such as ground-state degeneracies, partition functions, and operator algebras, thereby distinguishing phases not discernible by bulk properties alone.

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 (Wang et al., 2012). 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 (Yu, 7 May 2026). 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 (Jones et al., 2023). 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 (Kusuki, 2022).

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 (Yu, 7 May 2026). 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 (Wang et al., 2012).

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 (Wang et al., 2012). 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 $a$0 must satisfy the Boundary Fully Gapping Rules: for all $a$1,

$a$2

$a$3

$a$4

and $a$5 (Wang et al., 2012). The compatible anyons form a larger lattice $a$6 consisting of quasiparticles with trivial braiding statistics relative to $a$7.

For a manifold with $a$8 boundaries $a$9, each carrying boundary lattice $C^a_{p\mathbb{I}}$0, Wang–Wen derived the boundary GSD formula

$C^a_{p\mathbb{I}}$1

where $C^a_{p\mathbb{I}}$2 is the set of tuples of compatible anyons satisfying the total neutrality constraint $C^a_{p\mathbb{I}}$3 (Wang et al., 2012). 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 $C^a_{p\mathbb{I}}$4 toric code,

$C^a_{p\mathbb{I}}$5

rough and smooth boundaries correspond to condensates generated by $C^a_{p\mathbb{I}}$6 and $C^a_{p\mathbb{I}}$7. On a cylinder, rough–rough or smooth–smooth gives $C^a_{p\mathbb{I}}$8, while rough–smooth gives $C^a_{p\mathbb{I}}$9. For the $\ell^2(\Gamma)$0 double-semion model,

$\ell^2(\Gamma)$1

Wang–Wen show two boundary gapping lattices $\ell^2(\Gamma)$2 and $\ell^2(\Gamma)$3, but these are identified by $\ell^2(\Gamma)$4, yielding a single physical boundary condition, and the cylinder GSD remains $\ell^2(\Gamma)$5 for any pair (Wang et al., 2012). Thus the $\ell^2(\Gamma)$6 toric code and $\ell^2(\Gamma)$7 double-semion model have identical bulk GSD on closed surfaces and the same doubled fusion algebra $\ell^2(\Gamma)$8, 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 $\ell^2(\Gamma)$9 be the set of admissible gapped boundary conditions, and define

$B_{\mathrm{phys}}$0

This “Topological-Boundary Averaging” is not a topological invariant: it depends on the boundary ensemble and weights (Wang et al., 2012). For $B_{\mathrm{phys}}$1, the toric code has two distinct boundary types and a uniform average $B_{\mathrm{phys}}$2, whereas the double-semion theory effectively has one boundary type and $B_{\mathrm{phys}}$3. The averaged value and the full distribution $B_{\mathrm{phys}}$4 therefore distinguish phases that are indistinguishable by bulk GSD alone.

3. SymTFT formulation: averages over topological completions

In the SymTFT formulation, a $B_{\mathrm{phys}}$5-dimensional QFT with generalized global symmetry is realized as a boundary condition for a $B_{\mathrm{phys}}$6-dimensional topological theory. The physical boundary condition $B_{\mathrm{phys}}$7 on one end of a slab $B_{\mathrm{phys}}$8 prepares a partition vector

$B_{\mathrm{phys}}$9

while a topological boundary condition $L$0 on the other end defines a cap functional

$L$1

The corresponding absolute theory has partition function

$L$2

Topological-boundary averaging is then

$L$3

with $L$4 the intrinsic groupoid/Haar-type measure on the space of topological boundaries (Yu, 7 May 2026).

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$5 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

$L$6

while continuous components use invariant Haar measures descending to quotients of the form $L$7 (Yu, 7 May 2026).

Two examples organize the framework. In the Marolf–Maxfield closed-string sector, topological boundary conditions are labeled by finite sets $L$8, the groupoid is $L$9, and the groupoid/Haar-type weight is $K$0. A cap specified by $K$1 produces a closed 2D oriented TFT with Frobenius algebra

$K$2

and $K$3. With fugacity $K$4, the normalized moments are Bell polynomials,

$K$5

with generating function

$K$6

Non-factorization arises only after averaging over finite-set caps with the Poisson weights $K$7 (Yu, 7 May 2026).

In the Narain example, the SymTFT is a 3D abelian BF theory with $K$8-valued gauge fields,

$K$9

and compact topological boundary conditions are maximal isotropic subgroups $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$0. They are parametrized by $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$1 through $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$2 and form the Narain moduli space

$\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$3

The averaging measure is the Haar-induced, equivalently Zamolodchikov, measure

$\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$4

For $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$5, the Siegel–Weil formula gives

$\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$6

while for $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$7 the theta integral diverges at cusps, and for $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$8 the radius average diverges due to noncompact endpoints (Yu, 7 May 2026).

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 $\int dt\,dx \sum_a g_a \cos(\ell_{a,I}\Phi_I),$9, one has a quasi-local algebra $\ell_a\in \Gamma_e$0, a translation-invariant net of local algebras $\ell_a\in \Gamma_e$1, and a translation-invariant net of local ground-state projections $\ell_a\in \Gamma_e$2. The Local Topological Order axioms include the Hastings condition, boundary factorization, boundary stability/surjectivity, and boundary injectivity (Jones et al., 2023).

These axioms produce a net of boundary algebras $\ell_a\in \Gamma_e$3 on a codimension-one boundary hyperplane $\ell_a\in \Gamma_e$4. If $\ell_a\in \Gamma_e$5 and $\ell_a\in \Gamma_e$6 surrounds $\ell_a\in \Gamma_e$7 with nonempty $\ell_a\in \Gamma_e$8, then the compression $\ell_a\in \Gamma_e$9 of a local bulk observable $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$0 lies in a boundary algebra localized on the interval $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$1. Passing to the inductive limit gives the boundary quasi-local algebra $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$2 (Jones et al., 2023).

The key averaging object is the canonical unital completely positive map

$\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$3

defined by

$\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$4

It acts as the identity on the boundary subalgebra, but on observables deep in the bulk it reduces to the canonical ground state $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$5: if $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$6 inside $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$7, then $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$8 (Jones et al., 2023). In this precise sense, $\Gamma_e=\{\sum_J c_J K_{IJ}\mid c_J\in\mathbb Z\}$9 “forgets” microscopic interior details and averages bulk observables to their boundary-compatible component.

This yields a Hamiltonian-free bulk–boundary correspondence. Any boundary state $a$00 on $a$01 induces a bulk–boundary state $a$02 on $a$03, and that state restricts to the canonical ground state $a$04 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 $a$05, where $a$06. Its DHR category satisfies

$a$07

recovering the Drinfeld center of the input category, while for the Toric Code one obtains the quantum double of $a$08 (Jones et al., 2023). The canonical boundary state is a trace iff $a$09 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 (Jones et al., 2023).

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$10 with boundary state

$a$11

the averaged ensemble treats $a$12 as Gaussian random variables with

$a$13

and variance

$a$14

where $a$15 is the modular $a$16-matrix (Kusuki, 2022).

The annulus partition function in the closed channel is

$a$17

and modular duality gives the open-channel decomposition

$a$18

or, in the holographic/continuous case, a spectral density $a$19. For $a$20, the average yields

$a$21

so the open spectrum contains only “black-hole” states $a$22 and no vacuum (Kusuki, 2022).

The disk two-point bootstrap provides the geometric consequence. In Liouville variables $a$23 and $a$24, the lowest open-channel Liouville momentum is

$a$25

If $a$26, equivalently $a$27, then $a$28 and

$a$29

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

$a$30

and $a$31 iff $a$32 (Kusuki, 2022). 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 (Kusuki, 2022). A persistent misconception is that the average is over arbitrary boundary labels $a$33; the construction described here relies on Gaussian averaging of the bulk–boundary data $a$34, while $a$35 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 $a$36 on $a$37 is defined by

$a$38

Topological stability means that $a$39 and $a$40 are topologically equivalent for all sufficiently small $a$41. The cited results establish that, for continuous $a$42 with finitely many local extremes, global topological stability of $a$43 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 $a$44-criterion for piecewise constant $a$45 (Maksymenko et al., 2016).

For sampled manifolds with boundary, union-of-balls offsets implement a boundary-aware aggregation of data points. If $a$46 is a compact differentiable manifold with boundary, $a$47 is $a$48-dense in $a$49, and $a$50 with $a$51, then

$a$52

deformation retracts to $a$53, with explicit homotopy

$a$54

where $a$55 is the nearest-point projection (Wang et al., 2018). 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 $a$56 and boundaryless finite quotients $a$57. The normalized traces

$a$58

converge to the bulk von Neumann trace, so that

$a$59

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 (Lux et al., 2023).

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

$a$60

or, in level-set form,

$a$61

with topology penalties enforcing desired Betti numbers or Euler characteristic (Luo et al., 2019). 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,

$a$62

to improve statistical accuracy and define boundary observables independent of edge position (Ge et al., 17 Apr 2025).

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 (Wang et al., 2012). In the SymTFT setting, the measure is intrinsic only after specifying the groupoid or homogeneous-space structure of the topological boundary conditions (Yu, 7 May 2026). In BCFT, the average depends on the Gaussian statistics assigned to $a$63 and on the universal heavy-asymptotic formula for their variance (Kusuki, 2022).

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 (Wang et al., 2012). In SymTFT, averaging over caps distinguishes different absolute completions of the same relative theory (Yu, 7 May 2026). In the operator-algebraic framework, the boundary net $a$64 and its DHR category recover the bulk topological order, while the canonical channel $a$65 makes boundary states parameterize bulk–boundary states (Jones et al., 2023). In AdS/BCFT, boundary averaging generates non-factorization and selects physically acceptable bulk saddles (Kusuki, 2022).

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.