Building Vertex Algebras from Parts

Communications in Mathematical Physics volume 373, pages1–43(2020)Cite this article

Abstract

Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg–Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an “evenness” problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order two currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.

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Acknowledgements

The author would like to thank Tomoyuki Arakawa, Bruce Bartlett, Richard Borcherds, Thomas Creutzig, Yoshitake Hashimoto, André Henriques, Gerald Höhn, Satoshi Kondo, Ching-Hung Lam, Jacob Lurie, Chris Schommer-Pries, Hiroki Shimakura, Hiromichi Yamada, and the anonymous referee for helpful conversations and advice. This research was partly supported by NSF Grant DMS-0354321, JSPS Kakenhi Grant-in-Aid for Young Scientists (B) 24740005, the Program to Disseminate Tenure Tracking System, MEXT, Japan, and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

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    Scott Carnahan

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Carnahan, S. Building Vertex Algebras from Parts. Commun. Math. Phys. 373, 1–43 (2020). https://doi.org/10.1007/s00220-019-03607-0

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