Duality structures for representation categories of vertex operator algebras and the Feigin–Fuchs boson

• Robert Allen ^[1] ; Simon Lentner ^[2] ; Christoph Schweigert ^[2] ; Simon Wood ^[1]
  1. [1] Cardiff University

     Cardiff University

     Castle, Reino Unido

     
  2. [2] Algebra and Number Theory, University Hamburg, Hamburg, Germany
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 2, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01032-x
• Enlaces
  • Texto completo
• Resumen

  • Huang, Lepowsky and Zhang developed a representation theory for vertex operator algebras that endows suitably chosen module categories with the structures of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the representation category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fact that a vertex operator algebra and its gradewise dual need not be isomorphic as modules and that the tensor product of modules over vertex operator algebras need not be exact. We develop criteria which allow the detection of ribbon Grothendieck-Verdier equivalences and use them to explore ribbon Grothendieck-Verdier structures in the example of the rankn Heisenberg vertex operator algebra or chiral free boson on a not necessarily full rank even lattice with arbitrary choice of conformal vector. We show that these categories are equivalent, as ribbon Grothendieck-Verdier categories, to certain categories of graded vector spaces and categories of modules over a certain quasi-Hopf algebra.

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