Equivariant gerbes on complex tori

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We explore a new direction in representation theory which comes from holomorphic gerbes on complex tori. The analogue of the theta group of a holomorphic line bundle on a (compact) complex torus is developed for gerbes in place of line bundles. The theta group of symmetries of the gerbe has the structure of a Picard groupoid. We calculate it explicitly as a central extension of the group of symmetries of the gerbe by the Picard groupoid of the underlying complex torus. We discuss obstruction to equivariance and give an example of a group of symmetries of a gerbe with respect to which the gerbe cannot be equivariant. We calculate the obstructions to invariant gerbes for some group of translations of a torus to be equivariant. We survey various types of representations of the group of symmetries of a gerbe on the stack of sheaves of modules on the gerbe and the associated abelian category of sheaves on the gerbe (twisted sheaves).

Original languageEnglish
Pages (from-to)209-221
Number of pages13
JournalJournal of Geometry and Physics
Issue number1
StatePublished - Feb 2013
Externally publishedYes


  • Brauer group
  • Complex tori
  • Gerbes
  • Heisenberg group

ASJC Scopus subject areas

  • Mathematical Physics
  • General Physics and Astronomy
  • Geometry and Topology


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