Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory

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 \textit{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 20j-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.

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BibTeX RIS

BibTeX

@misc{chen2026categorical,
  title = {{Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory}},
  author = {Hank Chen and Zhian Jia and Ran Luo and Wei Cui},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31365},
  eprint = {2609.31365},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31365},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory
AU  - Hank Chen
AU  - Zhian Jia
AU  - Ran Luo
AU  - Wei Cui
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31365
UR  - https://arxiv.org/abs/2609.31365
AB  - 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 \textit{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 20j-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.
ER  -

Abstract

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