Equivariant gerbes on complex tori (Q1938618)

The aim of this paper, which is a continuation of an earlier article [J. Noncommut. Geom. 6, No. 3, 407--455 (2012; Zbl 1262.53022)], is to 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. The author calculates it explicitly as a central extension of the group of symmetries of the gerbe by the Picard groupoid of the underlying complex torus. He discusses obstructions to equivariance and gives an example of a group of symmetries of a gerbe with respect to which the gerbe cannot be equivariant. Also, he surveys 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.