  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.
