From VOAs to Short Star Products in SCFT

Abstract

We build a bridge between two algebraic structures in superconformal field theories (SCFT): a vertex operator algebra (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 $$\mathcal {A}_H = \mathrm{Zhu}_{s}(V)/N$$ , where $$\mathrm{Zhu}_{s}(V)$$ is the non-commutative Zhu algebra of the VOA V (for $${s}\in \mathrm{Aut}(V)$$ ), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map $$T_{s}: \mathrm{Zhu}_{s}(V) \rightarrow \mathbb {C}$$ determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips $$\mathcal {A}_H$$ with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map $$T_{s}$$ is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non- $$C_2$$ -cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.