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Integrability and Braided Tensor Categories

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Abstract

Many integrable statistical mechanical models possess a fractional-spin conserved current. Such currents have been constructed by utilising quantum-group algebras and ideas from “discrete holomorphicity”. I find them naturally and much more generally using a braided tensor category, a topological structure arising in knot invariants, anyons and conformal field theory. I derive a simple constraint on the Boltzmann weights admitting a conserved current, generalising one found using quantum-group algebras. The resulting trigonometric weights are typically those of a critical integrable lattice model, so the method here gives a linear way of “Baxterising”, i.e. building a solution of the Yang-Baxter equation out of topological data. It also illuminates why many models do not admit a solution. I discuss many examples in geometric and local models, including (perhaps) a new solution.

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Notes

  1. Artin coined the term in 1925, but there are antecedents [46]. Yang was born in 1922, Baxter 1940.

  2. I am grateful to Denis Bernard for this observation, made at my seminar on this work at the MSRI in 2012.

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Acknowledgements

I am very grateful to David Aasen and Roger Mong for collaboration on [14, 44] and for their mentoring in the Way of the Category. I thank Denis Bernard and John Cardy for essential conversations many moons ago, and Niall Mackay, Eric Rowell and Eric Vernier for helpful comments and guidance to the literature. This work was supported by EPSRC grants EP/S020527/1 and EP/N01930X.

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    Paul Fendley

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Fendley, P. Integrability and Braided Tensor Categories. J Stat Phys 182, 43 (2021). https://doi.org/10.1007/s10955-021-02712-6

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