  We prove that finite-index conformal nets are fully dualizable objects in the
3-category of conformal nets. Therefore, assuming the cobordism hypothesis
applies, there exists a local framed topological field theory whose value on
the point is any finite-index conformal net. Along the way, we prove a
Peter-Weyl theorem for defects between conformal nets, namely that the annular
sector of a finite defect is the sum of every sector tensor its dual.
