We find that multiple vertex algebras can arise from a single 4d $\mathcal{ N}=2$ superconformal field theory (SCFT). The connection is given by the BPS monodromy operator $M$, which is a wall-crossing invariant quantity that captures the BPS spectrum on the Coulomb branch. For a class of low-rank Argyres-Douglas theories, we find that the trace of the multiple powers of the monodromy operator $\mathrm{ Tr} M^N$ yield modular functions that can be identified with the vacuum characters of certain vertex algebra for each $N$. In particular, we realize unitary VOAs of the Deligne-Cvitanovi\'c exceptional series type $(A_2)_1$, $(G_2)_1$, $(D_4)_1$, $(F_4)_1$, $(E_6)_1$ from Argyres-Douglas theories. We also find the modular invariant characters of the `intermediate vertex algebras' $(E_{ 7\frac{ 1}{ 2}})_1$ and $(X_1)_1$. Our analysis allows us to construct 3d $\mathcal{ N}=2$ gauge theories that flow to $\mathcal{ N}=4$ SCFTs in the IR, whose specialized half-index can be identified with these modular invariant characters.
Categories: hep-th math.QA