# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1907.12532