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Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields $$ a(z) = \sum_{n\in \mathbb Z} a_nz^{-n-1} $$ are replaced by $$ \sum_{n\in \log \mathbb Q} a_nz^{-n} $$ with some assumption on the support of $a_n$, say that it is discrete as a subset of $\mathbb R$?

Obviously, something has got to give... it is not clear to me whether there is any replacement for Virasoro algebra, maybe some kind of Hecke algebra? What would be the analog of operator product expansion?

Any references or suggestions?

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  • $\begingroup$ There's arXiv:math/0602072 of Bakalov and Kac where they study some generalized vertex algebras where you allow the grading by complex numbers. Most generalizations along these lines end up studying the Heisenberg vertex algebra plus a bunch of its modules and then forming the intertwining algebra, as a generalization of the lattice vertex algebra (when you only consider integral heighest weights). $\endgroup$
    – Reimundo Heluani
    Commented Oct 24, 2014 at 15:41

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