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Gauge theories, TQFTs and integrable systems joint seminar
Gauge theories, TQFTs and integrable systems joint seminar
Zamolodchikov’s tetrahedron equation: solutions from 2-shifted Poisson structures
Zamolodchikov’s tetrahedron equation: solutions from 2-shifted Poisson structures
Organizers
Speaker
Cameron Kemp
Time
Thursday, November 26, 2026 10:00 AM - 11:30 AM
Venue
A3-3-301
Online
Zoom 559 700 6085
(BIMSA)
Abstract
Just as the Yang-Baxter equation (YBE) guarantees the integrability of certain 2D models, so too the Zamolodchikov tetrahedron equation (ZTE) guarantees the integrability of certain 3D models. Likewise, just as Joyal and Street showed that braided monoidal categories satisfy the YBE, Kapranov and Voevodsky showed that braided monoidal 2-categories satisfy the ZTE. This talk will begin with an overview of the relevant 2-categorical notions needed for the construction of concrete braided monoidal 2-categories. We will start by defining cochain 2-categories/functors as categories/functors enriched over the category of cochain complexes concentrated in degrees {-1,0}. We will then specify the notion of pseudonatural transformations and modifications to this context so that we may define braided monoidal cochain 2-categories.
Cartier introduced the notion of an infinitesimal braiding and showed how to produce braided monoidal categories from them via Drinfeld's deformation quantisation results. The notion of an infinitesimal 2-braiding and the Knizhnik-Zamolodchikov (KZ) 2-connection is due to Cirio and Faria Martins, they showed that pseudonaturality of the infinitesimal 2-braiding leads to the KZ 2-connection being a fake flat 2-connection thus one can study its 2-holonomy. We thus finish by showing how to construct infinitesimal 2-braidings from 2-shifted Poisson structures and how to construct hexagonators and the pentagonator from the 2-holonomy of the KZ 2-connection.
Cartier introduced the notion of an infinitesimal braiding and showed how to produce braided monoidal categories from them via Drinfeld's deformation quantisation results. The notion of an infinitesimal 2-braiding and the Knizhnik-Zamolodchikov (KZ) 2-connection is due to Cirio and Faria Martins, they showed that pseudonaturality of the infinitesimal 2-braiding leads to the KZ 2-connection being a fake flat 2-connection thus one can study its 2-holonomy. We thus finish by showing how to construct infinitesimal 2-braidings from 2-shifted Poisson structures and how to construct hexagonators and the pentagonator from the 2-holonomy of the KZ 2-connection.