This site is for research level mathematics. Your question is about undergraduate level physics. Maybe try physics stack exchange.

@BenjaminSteinberg Thanks!

There are many wrong and confusing statements here and a long list of questions. I don't think it is reasonable to expect an answer to questions like this from mathematicians. Many of your confusions and questions could be cleared up by reading a graduate level textbook on QFT such as the book by S. Weinberg.

He considers compact simple Lie groups so the first Chern class vanishes.

What do you mean by an application to physics?

I'm downvoting this because I don't see a clear mathematical question. I suggest you read the first few chapters of Weinberg's book on QFT. There is nothing weird about the interaction picture and no, interactions cannot be thought of as a representation.

You asked for a conceptual explanation of the product formula and Carnahan provided one. Now you seem to insist on a conceptual explanation involving the Leech lattice but I see no reason why there should be one. You quoted Carnahan, but it wasn't clear to me you understood his explanation.