String-net models for non-spherical pivotal fusion categories

TL;DR

This paper explores how string-net models behave when the sphericality condition of the underlying fusion categories is relaxed, revealing new topological and algebraic structures.

Contribution

It investigates the effects of dropping the sphericality condition in string-net models, providing explicit examples and connections to topological quantum field theories.

Findings

String-net space counts r-spin structures on surfaces.

String-net space of a sphere with a marked point is non-zero iff certain conditions are met.

Deformation of stress tensor in CFT relates to non-spherical pivotal structures.

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.