From VOAs to Short Star Products in SCFT

We build a bridge between two algebraic structures in superconformal field theories (SCFT): vertex operator algebra (VOA) the Schur sector of 4d $$\mathcal {N}=2$$ and an associative Higgs 3d {N}=4$$ . The natural setting is SCFT placed on $$S^3\times S^1$$ : by sending radius $$S^1$$ to zero, we recover theory, corresponding VOA torus degenerates circle. prove that: (1) branch operators remain cohomology; (2) all non-Higgs type are lifted line wrapped ; (3) no new cohomology classes added. show that given quotient {A}_H = \mathrm{Zhu}_{s}(V)/N$$ , where $$\mathrm{Zhu}_{s}(V)$$ non-commutative Zhu V (for $${s}\in \mathrm{Aut}(V)$$ ), N certain ideal. This ideal null space (s-twisted) trace map $$T_{s}: \mathrm{Zhu}_{s}(V) \rightarrow \mathbb {C}$$ determined 1-point function high temperature (or small complex structure) limit. It therefore equips {A}_H$$ with non-degenerate (twisted) trace, leading short star-product according recent results Etingof Stryker. $$T_{s}$$ easy determine for unitary VOAs, but has much subtler structure non-unitary non- $$C_2$$ -cofinite VOAs our interest. comment relation Beem-Rastelli conjecture associated variety. A companion paper will explore further details, examples, some applications these ideas.