Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory
math-ph, cond-mat.str-el, hep-th, math.MP, math.QA
Submitted: 2026-09-25
Updated: 2026-09-25
Comments: 53 pages; 24 figures
License: http://creativecommons.org/licenses/by/4.0/
The gist: In this work, we define (3+1) -dimensional topological quantum field theories (TQFTs) and their lattice realizations with input data given by a pivotal fusion 2-category; more precisely, our
Terminology
Abstract
In this work, we define (3+1) -dimensional topological quantum field theories (TQFTs) and their lattice realizations with input data given by a pivotal fusion 2-category; more precisely, our framework applies in wider generality to presemisimple locally fusion pivotal tensor 2-categories, which are equipped with a so-called shadow trace. We introduce a decoration of triangulated manifolds by the input data, and develop a 20j-symbol formalism for computing 4-simplex scattering amplitudes. We verify the topological invariance of 20 j-symbol summations under the 4-dimensional Pachner moves. Furthermore, we also construct explicitly a 3-dimensional Levin-Wen-type lattice Hamiltonian model realizing the state-sum invariant. Our construction can be understood as a higher-dimensional analogue of the spin-networks construction for the Turaev-Viro invariant. Examples arising from 2-groups are discussed from the perspective of higher gauge theory.
Sources
- A hierarchy of topological tensor network states
- Kitaev models based on unitary quantum groupoids
- Kitaev lattice models as a Hopf algebra gauge theory
- Boundary and domain wall theories of 2d generalized quantum double model
- On weak Hopf symmetry and weak Hopf quantum double model
- 2-Drinfel'd double symmetry of the 4d Kitaev model
- String-net condensation: A physical mechanism for topological phases
- String-net model of Turaev-Viro invariants
- Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
- Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
- A lattice realization of general three-dimensional topological order
- Invariants of Piecewise-Linear 3-Manifolds
- Hopf 2-algebras and Braided Monoidal 2-Categories
- Tannaka-Krein reconstruction for fusion 2-categories
- Topological phases from higher gauge symmetry in 3+1D
- On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups
- Dynamical generalization of Yetter's model based on a crossed module of discrete groups
- Excitations in strict 2-group higher gauge models of topological phases
- Lie crossed modules and gauge-invariant actions for 2-BF theories
- Hamiltonian analysis of the BFCG theory for a strict Lie 2-group
Related papers
- Pure gapped ground states of spin chains are short-range entangled
- Multipartite entanglement spreads
- Central Limit Theorems for Outcome Records in Disordered Quantum Trajectories
- The Classification of 3+1d Symmetry Enriched Topological Order
- Large Deviations for the Bose--Einstein Condensate of the Ideal Gas in the Canonical Ensemble
- A Time-Frequency Framework for GKP Codes