From VOAs to Short Star Products in SCFT

• Published in: Communications in mathematical physics 2021-05, Vol.384 (1), p.245-277
• Main Author: Dedushenko, Mykola
• Format: Article
• Language: English
• Subjects: Algebra; Bridges; Classical and Quantum Gravitation; Complex Systems; High temperature; Homology; Mathematical and Computational Physics; Mathematical Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum Physics; Quotients; Relativity Theory; Theoretical; Toruses
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-021-04066-2

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 S 3 × S 1 : by sending the radius of S 1 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 S 1 ; (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- 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.