Unitary TQFTs, unitary disk-like n-categories, and higher Hilbert spaces
Abstract
We introduce the notion of a unitary disk-like n-category, which is a disk-like n-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like n-category is defined to be the local field data of a fully extended (n+1)D unitary TQFT, we propose a complete finite unitary disk-like n-category as the definition of a finite (n+1)-Hilbert space for all n. For n=1 and n=2 we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For n=1 we recover a unitary refinement of Schommer-Pries' classification of oriented (1+1)D TQFTs in terms of H*-Morita equivalence classes of H*-algebras, and for n=2 we categorify this to classify oriented (2+1)D unitary TQFTs by H*-Morita equivalence classes of H*-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like n-categories.
Related
- Involutions on (∞,n)-categories of bordisms and reflection TQFTs
- Anomalies of TQFTs coupled to parameter spaces and constraints on ground-state degeneracies
- Models for dagger (∞,1)-categories I: Dagger simplicial sets and unitary cores
- Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture
- Rhaly and generalized Hilbert operators between spaces of analytic functions
BibTeX
@misc{wesley2026unitary,
title = {{Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces}},
author = {Greyson Wesley},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19713},
eprint = {2609.19713},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19713},
}
RIS
TY - UNPB
TI - Unitary TQFTs, unitary disk-like n-categories, and higher Hilbert spaces
AU - Greyson Wesley
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19713
UR - https://arxiv.org/abs/2609.19713
AB - We introduce the notion of a unitary disk-like n-category, which is a disk-like n-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like n-category is defined to be the local field data of a fully extended (n+1)D unitary TQFT, we propose a complete finite unitary disk-like n-category as the definition of a finite (n+1)-Hilbert space for all n. For n=1 and n=2 we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For n=1 we recover a unitary refinement of Schommer-Pries' classification of oriented (1+1)D TQFTs in terms of H^*-Morita equivalence classes of H^*-algebras, and for n=2 we categorify this to classify oriented (2+1)D unitary TQFTs by H^*-Morita equivalence classes of H^*-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like n-categories.
ER -