Given a surface with boundary $\Sigma$, its relative mapping class group is the quotient of $\operatorname{Diff}(\Sigma)$ by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If $\Sigma$ has no boundary, then that's the usual mapping class group; if $\Sigma$ is a disc, then that's the group $\operatorname{Diff}(S^1)$ of diffeomorphisms of $S^1$.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.