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Twisted Modules for Vertex Operator Algebras

Published online by Cambridge University Press:  06 July 2010

Benjamin Doyon
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK Current address: Department of Mathematical Sciences Durham Unversity, UK
James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
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Summary

Abstract

This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general principles of twisted modules for vertex operator algebras in their powerful formulation using formal series, and derive general relations satisfied by twisted and untwisted vertex operators. Using these, I prove new “equivalence” and “construction” theorems, identifying a set of sufficient conditions in order to have a twisted module for a vertex operator algebra, and a simple way of constructing the twisted vertex operator map. This essentially combines our general relations for twisted modules with ideas of Li (1996), who had obtained similar construction theorems using different relations. Then, I show how to apply these theorems in order to construct twisted modules for the Heisenberg vertex operator algebra. I obtain in a new way the explicit twisted vertex operator map, and in particular give a new derivation and expression for the formal operator Δx constructed some time ago by Frenkel, Lepowsky and Meurman. Finally, I reproduce parts of our joint papers. I use the untwisted relations in the Heisenberg vertex operator algebra in order to understand properties of a certain central extension of a Lie algebra of differential operators on the circle: the connection between the structure of the central term in Lie brackets and the Riemann Zeta function at negative integers.

Type
Chapter
Information
Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 144 - 187
Publisher: Cambridge University Press
Print publication year: 2010

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  • Twisted Modules for Vertex Operator Algebras
    • By Benjamin Doyon, Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK Current address: Department of Mathematical Sciences Durham Unversity, UK
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.008
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  • Twisted Modules for Vertex Operator Algebras
    • By Benjamin Doyon, Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK Current address: Department of Mathematical Sciences Durham Unversity, UK
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.008
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Twisted Modules for Vertex Operator Algebras
    • By Benjamin Doyon, Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK Current address: Department of Mathematical Sciences Durham Unversity, UK
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.008
Available formats
×