String-net models for non-spherical pivotal fusion categories

Ingo Runkel

doi:10.1142/s0218216520500352

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arXiv · 1907.12532

String-net models for non-spherical pivotal fusion categories

Ingo Runkel

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Abstract

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the corresponding Turaev-Viro topological quantum field theory. In the present work some effects of dropping the sphericality condition are investigated. In one example of non-spherical pivotal fusion categories, the string-net space counts the number of r-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labelled by a simple object Z of the Drinfeld centre. This space is found to be non-zero iff Z is isomorphic to a non-unit simple object determined by the non-spherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.

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Ingo Runkel. 2019-07-29. String-net models for non-spherical pivotal fusion categories. https://doi.org/10.1142/s0218216520500352

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