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Classification of screening systems for lattice vertex operator algebras

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Abstract

We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra with an associated vertex operator (super)algebra. Screening pairs arising from weight one primary vectors acting commutatively on a lattice vertex operator algebra (the vacuum module) are classified into four general types; one type of which has been shown to play an important role in the construction and study of certain important families of \({\mathcal {W}}\)-vertex algebras. These types of screening pairs which we go on to study in detail through the notion of a system of screeners are lattice elements or “screening momenta” which give rise to screening pairs. We classify screening systems for all positive definite integral lattices of rank two, and for all positive definite even lattices of arbitrary rank when these lattices are generated by a screening system.

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Acknowledgements

This paper consists, essentially, of the first three chapters of the second author’s Ph.D. Thesis [56] under the supervision of the first author. The authors would like to thank Antun Milas, Drazan Adamovic, and Jinwei Yang for useful comments and guidance in the writing of the thesis from which this paper is derived. The authors thank the referee for pointing out significant points of clarification in the expository parts of the paper as well as some mathematical improvements. The first author is the recipient of Simons Foundation Collaboration Grant 282095, and greatly appreciates their support.

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  1. Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, USA

    Katrina Barron

  2. Department of Mathematics, University of Nebraska at Kearney, Kearney, Nebraska, USA

    Nathan Vander Werf

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  1. Katrina Barron
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  2. Nathan Vander Werf
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Correspondence to Katrina Barron.

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K. Barron was supported by Simons Foundation Collaboration Grant 282095.

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Barron, K., Vander Werf, N. Classification of screening systems for lattice vertex operator algebras. Lett Math Phys 109, 1573–1610 (2019). https://doi.org/10.1007/s11005-019-01161-3

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