Constructing the 0- and 1-Dimensional Part of a Segal Chiral CFT from Conformal Nets

Two-dimensional unitary chiral conformal field theories (CFTs) admit three distinct mathematical formulations: unitary vertex operator algebras (uVOAs), conformal nets, and Segal (functorial) chiral CFTs. With the aim of building fully extended Segal chiral CFTs from the data of a conformal net, we give the assignment to points and 1-dimensional cobordisms. We assign the category of solitonic representations of the net to a point and bimodule categories to intervals, and discuss the functoriality of the assignment under gluing of 0- and 1-dimensional cobordisms.