@MartinSleziak Yes, thanks!

@მამუკაჯიბლაძე Yes, but in the vertex algebra world, we typically add finiteness conditions to our definition of module.

@ReimundoHeluani I also don't have the first reference, but "holomorphic conformal field theory" was used in a manner consistent with "vertex algebra with trivial representation theory" by Dixon-Ginsparg-Harvey (section 4 of "Beauty and the Beast") in 1988. I think this supports Theo's explanation.

@მამუკაჯიბლაძე There is no canonical reference at this time. I've been thinking of starting up something like the stacks project but for vertex algebras for some time now, but there are so many other things to do! Anyway, for (1) and (2), chapter 5 of Frenkel-Ben-Zvi can be reworked into a proof. For (3) and (4) I don't remember, but some papers by Arakawa (maybe annals?) may be helpful.

@erz In contrast to მამუკა ჯიბლაძე's comment, I don't think it is necessary to leave mathematics if you have career-related doubts. Some people see mathematics as a "this is absolutely my life's path", others see it as "just a job", and I think both types, and many others, should be welcomed. If circumstances make it too difficult to continue for now, we as a community should try our best to leave doors open for your possible return.

The short article you linked is not a memoir. It is the text of a public ("culture") lecture at the Japanese Electrophoresis Society's 50th meeting. However, it does have lots of interesting anecdotes about his contemporaries together with reflections on people who inspired him.