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Quon Language: Surface Algebras and Fourier Duality

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Abstract

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.

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Acknowledgements

The author would like to thank Terry Gannon, Arthur Jaffe, Vaughan F. R. Jones, Shamil Shakirov, Cumrun Vafa, Erik Verlinde, Jinsong Wu, and Feng Xu for helpful discussions. The author was supported by Grants TRT0080 and TRT0159 from the Templeton Religion Trust and an AMS-Simons Travel Grant. The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Operator algebras: subfactors and their applications”.

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  1. Departments of Mathematics and of Physics, Harvard University, Cambridge, USA

    Zhengwei Liu

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Correspondence to Zhengwei Liu.

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Communicated by C. Schweigert

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Liu, Z. Quon Language: Surface Algebras and Fourier Duality. Commun. Math. Phys. 366, 865–894 (2019). https://doi.org/10.1007/s00220-019-03361-3

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