The 2-Hilbert Space of a Prequantum Bundle Gerbe

Severin Bunk, Christian Saemann, Richard J. Szabo

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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 prequantisation, 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 prequantisation. 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 prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.
Original languageEnglish
Number of pages101
JournalReviews in Mathematical Physics
Volume30
Publication statusPublished - 30 Aug 2016

Keywords

  • math-ph
  • hep-th
  • math.CT
  • math.DG
  • math.MP
  • math.SG

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