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From VOAs to Short Star Products in SCFT

Dedushenko, Mykola (2021) From VOAs to Short Star Products in SCFT. Communications in Mathematical Physics, 384 (1). pp. 245-277. ISSN 0010-3616. doi:10.1007/s00220-021-04066-2.

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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.

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Dedushenko, Mykola0000-0002-9273-7602
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 25 January 2020; Accepted 10 March 2021; Published 12 April 2021. I thank M. Fluder and Y. Wang for collaborations on related projects [22] and [82]. I also acknowledge useful conversations, comments, and/or correspondence with: T. Arakawa, T. Creutzig, P. Etingof (whom I also thank for sending the draft of [15]), B. Feigin, L. Rastelli, D. Simons-Duffin. This work was supported in part by the Walter Burke Institute for Theoretical Physics, with the additional support from the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No de-sc0011632, as well as the Sherman Fairchild Foundation.
Group:Walter Burke Institute for Theoretical Physics
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Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Sherman Fairchild FoundationUNSPECIFIED
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Issue or Number:1
Record Number:CaltechAUTHORS:20200205-144527344
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Official Citation:Dedushenko, M. From VOAs to Short Star Products in SCFT. Commun. Math. Phys. 384, 245–277 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101145
Deposited By: Joy Painter
Deposited On:06 Feb 2020 15:40
Last Modified:11 Aug 2021 16:53

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