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Abelian 3D TQFT gravity, ensemble holography and stabilizer states

Nikolaos Angelinos

TL;DR

This work defines a 3D abelian TQFT-based model of gravity by summing over all 3D topologies with genus-$g$ boundary circles, where each topology’s TQFT path-integral prepares a stabilizer state. It develops a dual description as a weighted ensemble of boundary 2D CFTs through the $\lambda$-matrix, which relates bulk topology weights to boundary CFT weights, and demonstrates $Sp(2g,\mathbb Z)$-invariance of the gravity state. By classifying $\eta$- and $\mathfrak q$-Lagrangian submodules and constructing topological boundary conditions via surface operators and interfaces, the paper provides a concrete, computable framework for ensemble holography in 3D. The results are illustrated with explicit examples (cyclic and higher-rank groups), showing how handlebody and non-handlebody topologies contribute to a holographic boundary spectrum, and offering a path toward a universal, genus-compatible bulk weighting scheme compatible with holography.

Abstract

We construct a model of 3D quantum gravity based on abelian topological quantum field theory (TQFT), by defining the gravitational path-integral as a sum over all 3D topologies with genus-$g$ boundary $Σ_g$. The path-integral of an abelian TQFT $\mathcal T$ on any single topology with boundary $Σ_g$ prepares a stabilizer state. This way, $\mathcal T$ partitions all these topologies into finitely many equivalence classes, where each topology within a class is associated with the same stabilizer state. The gravitational path-integral can thus be rephrased as a weighted sum over representative topologies, which are further organized into orbits under the mapping class group of $Σ_g$. One orbit is represented by handlebodies, whose average reproduces the ``Poincaré series of the vacuum", while additional orbits describe non-handlebody topologies. The resulting quantum gravity state is $Sp(2g,\mathbb Z)$-invariant and can be expressed as a weighted average of 2D CFT partition functions on $Σ_g$. This establishes a duality between a weighted sum over bulk topologies and a weighted sum over boundary CFTs. We introduce the ``$λ$-matrix", which relates bulk and boundary weights. The $λ$-matrix can be fully determined by the set of topological boundary conditions that the TQFT admits, and we present a systematic procedure to construct this set. Using this framework, we evaluate the $λ$-matrix and the TQFT gravity state in several tractable examples.

Abelian 3D TQFT gravity, ensemble holography and stabilizer states

TL;DR

This work defines a 3D abelian TQFT-based model of gravity by summing over all 3D topologies with genus- boundary circles, where each topology’s TQFT path-integral prepares a stabilizer state. It develops a dual description as a weighted ensemble of boundary 2D CFTs through the -matrix, which relates bulk topology weights to boundary CFT weights, and demonstrates -invariance of the gravity state. By classifying - and -Lagrangian submodules and constructing topological boundary conditions via surface operators and interfaces, the paper provides a concrete, computable framework for ensemble holography in 3D. The results are illustrated with explicit examples (cyclic and higher-rank groups), showing how handlebody and non-handlebody topologies contribute to a holographic boundary spectrum, and offering a path toward a universal, genus-compatible bulk weighting scheme compatible with holography.

Abstract

We construct a model of 3D quantum gravity based on abelian topological quantum field theory (TQFT), by defining the gravitational path-integral as a sum over all 3D topologies with genus- boundary . The path-integral of an abelian TQFT on any single topology with boundary prepares a stabilizer state. This way, partitions all these topologies into finitely many equivalence classes, where each topology within a class is associated with the same stabilizer state. The gravitational path-integral can thus be rephrased as a weighted sum over representative topologies, which are further organized into orbits under the mapping class group of . One orbit is represented by handlebodies, whose average reproduces the ``Poincaré series of the vacuum", while additional orbits describe non-handlebody topologies. The resulting quantum gravity state is -invariant and can be expressed as a weighted average of 2D CFT partition functions on . This establishes a duality between a weighted sum over bulk topologies and a weighted sum over boundary CFTs. We introduce the ``-matrix", which relates bulk and boundary weights. The -matrix can be fully determined by the set of topological boundary conditions that the TQFT admits, and we present a systematic procedure to construct this set. Using this framework, we evaluate the -matrix and the TQFT gravity state in several tractable examples.

Paper Structure

This paper contains 48 sections, 5 theorems, 182 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Abelian TQFT and Stabilizer Formalism
  3. Anyon basis and Weyl operators
  4. Clifford group and its $\mathsf O(\mathsf G)\times \operatorname{Sp}(2g,\mathbb Z_N)$ subgroup
  5. Stabilizer groups and stabilizer states
  6. Gauging 1-form symmetries in 3D
  7. Lagrangian Submodules and their Stabilizer States
  8. Classification of $\eta$-Lagrangian submodules and representative topologies
  9. Classification of $\mathfrak q$-Lagrangian submodules and topological boundary conditions
  10. CFT partition functions from $\mathfrak q$-Lagrangian submodules
  11. The Quantum Gravity Path-Integral
  12. Average over topologies
  13. A prescription for the weights
  14. Constructing Topological Boundary Conditions
  15. Construction and classification of surface operators
  16. ...and 33 more sections

Key Result

Theorem 1

Let $C$ be a submodule of $\mathbb Z_N^{2g}$ and let $N=\prod_{i=1}^k p_i^{s_i}$ be the prime factorization of $N$. Define the $g$-tuple $\boldsymbol{d}=(d_1,\dots,d_g)$ with Then $C$ is a Lagrangian submodule with respect to $\eta$ if and only if there exists $\gamma\in \operatorname{Sp}(2g,\mathbb Z)$ such that is a generator matrix for $C$.

Figures (3)

  • Figure 1: The network that performs the gauging of the symmetry $\mathsf H\subset\mathsf G$ in the handlebody $\mathcal{V}_g$ reduces to $g$ loops around the non-trivial cycles. It is necessary for the quadratic form $\mathfrak q$ to vanish on $\mathsf H$.
  • Figure 2: Gauging of $\mathsf H\subseteq\mathsf G$ on a surface $\Sigma_g$, for some choice of discrete torsion $\nu\in H^2(\mathsf H,\mathbb C^\times)$.
  • Figure 3: A representation of the topology $\mathcal{V}_g^{\boldsymbol{d}}$, corresponding to (\ref{['gen']}) with $\gamma=\mathds{1}$, as a singular handlebody. The dashed line labeled $d_i$ is a condensation defect that can absorb and emit anyons with charges in $(N/d_i)\mathsf G$, modifying the first homology of the $i$-th factor to $\mathbb Z_{N/d_i}\times\mathbb Z_{d_i}$. The stabilizer state $|\Omega_{\boldsymbol{d}};\gamma\rangle$ can be obtained by acting with the mapping class group element $\gamma\in \operatorname{Sp}(2g,\mathbb Z)$ on the above configuration.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1: Classification of symplectic Lagrangian submodules
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2: Classification of orthogonal Lagrangian submodules
  • Definition 3
  • Lemma 1
  • ...and 1 more