Two questions regarding tensor product of modules over vertex / chiral algebras:
First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a structure of a vertex tensor category on a certain category of modules, see e.g. this paper by Haung and Lepowsky.
Is the structure of a vertex tensor category equivalent / comparable to that of a module category for the modular / cyclic operad?
Second question: For a chiral algebra (in the sense of Beilinson and Drinfeld), we can define the category of chiral modules supported at a given finite set of points. Moving the points, we get a family of categories that factorizes in a certain sense, see e.g. here.
As far as I know there's no known general theory of fusion product of chiral modules - is there anything known about that, such as a description in special cases? And if so is there a comparison to the theory of fusion product of vertex modules mentioned above?
Thanks!
