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From VOAs to short star products in SCFT

Dedushenko, Mykola (2019) From VOAs to short star products in SCFT. .

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We build a bridge between two algebraic structures in SCFT: a 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 AH = Zhus(V )/N, where Zhus(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 Ts : Zhus(V ) → C determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips AH with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map Ts 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.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Dedushenko, Mykola0000-0002-9273-7602
Additional Information: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. Significant part of this work was completed when I was a member at (and was supported 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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Department of Energy (DOE)DE-SC0011632
Sherman Fairchild FoundationUNSPECIFIED
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Record Number:CaltechAUTHORS:20200205-144527344
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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:06 Feb 2020 15:40

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