The 2-Hilbert space of a prequantum bundle gerbe

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Abstract

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier–Douady class is torsion. Analogously to usual prequantization, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf’s version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantization. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant–Souriau prequantization in this setting, including its dimensional reduction to ordinary prequantization.

Original languageEnglish
Article number1850001
JournalReviews in Mathematical Physics
Volume30
Issue number1
Early online date30 Oct 2017
DOIs
Publication statusPublished - Feb 2018

Keywords

  • 2-Hilbert spaces
  • Bundle gerbes
  • higher geometric quantization
  • transgression

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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