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Gray categories with duals and their diagrams (2024)
Journal Article
Barrett, J. W., Meusburger, C., & Schaumann, G. (2024). Gray categories with duals and their diagrams. Advances in Mathematics, 450, Article 109740. https://doi.org/10.1016/j.aim.2024.109740

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray categ... Read More about Gray categories with duals and their diagrams.

Spectral estimators for finite non-commutative geometries (2019)
Journal Article
Barrett, J. W., Druce, P., & Glaser, L. (2019). Spectral estimators for finite non-commutative geometries. Journal of Physics A: Mathematical and Theoretical, 52(27), https://doi.org/10.1088/1751-8121/ab22f8

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract information about these geometries from the spectrum of the... Read More about Spectral estimators for finite non-commutative geometries.

Dichromatic state sum models for four-manifolds from pivotal functors (2017)
Journal Article
Bärenz, M., & Barrett, J. W. (2018). Dichromatic state sum models for four-manifolds from pivotal functors. Communications in Mathematical Physics, 360(2), https://doi.org/10.1007/s00220-017-3012-9

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ri... Read More about Dichromatic state sum models for four-manifolds from pivotal functors.

Monte Carlo simulations of random non-commutative geometries (2016)
Journal Article
Barrett, J. W., & Glaser, L. (2016). Monte Carlo simulations of random non-commutative geometries. Journal of Physics A: Mathematical and Theoretical, 49(24), Article 245001. https://doi.org/10.1088/1751-8113/49/24/245001

© 2016 IOP Publishing Ltd. Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. The cases with the simplest types of Clifford algebra are... Read More about Monte Carlo simulations of random non-commutative geometries.

Two-dimensional state sum models and spin structures (2014)
Journal Article
Barrett, J. W., & Tavares, S. O. G. (2014). Two-dimensional state sum models and spin structures. Communications in Mathematical Physics, 336(1), 63-100. https://doi.org/10.1007/s00220-014-2246-z

The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai are generalised by allowing algebraic data from a non-symmetric Frobenius algebra. Without any further data, this leads to a state sum model on the sphere. When the data is... Read More about Two-dimensional state sum models and spin structures.

Integrability for Relativistic Spin Networks (2001)
Journal Article
Barrett, J. W., & Baez, J. C. (2001). Integrability for Relativistic Spin Networks. Classical and Quantum Gravity, 18(4683-4),

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lo... Read More about Integrability for Relativistic Spin Networks.

A Lorentzian Signature Model for Quantum General Relativity (2000)
Journal Article
Barrett, J. W., & Crane, L. (2000). A Lorentzian Signature Model for Quantum General Relativity

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral over suitable representations of... Read More about A Lorentzian Signature Model for Quantum General Relativity.