Equivariant gerbes on complex tori
Abstract
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).
- Publication:
-
Journal of Geometry and Physics
- Pub Date:
- February 2013
- DOI:
- 10.1016/j.geomphys.2012.10.012
- arXiv:
- arXiv:1102.2312
- Bibcode:
- 2013JGP....64..209B
- Keywords:
-
- Mathematics - Algebraic Geometry;
- High Energy Physics - Theory;
- Mathematics - Representation Theory;
- 53C08;
- 20C25;
- 20J06
- E-Print:
- J. Geom. Phys. 64 (2013) 209-221