From VOAs to short star products in SCFT

Understand

We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d $\mathcal{N}=2$ theories and an associative algebra in the Higgs sector of 3d $\mathcal{N}=4$.

• The natural setting is a 4d $\mathcal{N}=2$ SCFT placed on $S^3\times S^1$: by sending the radius of $S^1$ to zero, we recover the 3d $\mathcal{N}=4$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle.
• We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the $S^1$; 3) no new cohomology classes are added.
• We show that the algebra in 3d is given by the quotient $\mathcal{A}_H = {\rm Zhu}_{s}(V)/N$, where ${\rm Zhu}_{s}(V)$ is the non-commutative Zhu algebra of the VOA $V$ (for ${s}\in{\rm Aut}(V)$), and $N$ is a certain ideal.