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?