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 N=2 theories and an associative algebra in the Higgs sector of 3d N=4. The natural setting is a 4d N=2 SCFT placed on S³×S¹: by sending the radius of S¹ to zero, we recover the 3d 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 S¹; (3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient A_H=Zhu_s(V)/N, where Zhu_s(V) is the non-commutative Zhu algebra of the VOA V (for s ∈ Aut(V)), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map T_s:Zhu_s(V)→C determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips 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-C₂-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.