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 nonspherical 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 labeled by a simple object Z of the Drinfeld center. This space is found to be nonzero iff Z
is isomorphic to a nonunit simple object determined by the nonspherical 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.