  This paper generalizes Bismut's equivariant Chern character to the setting of
abelian gerbes. In particular, associated to an abelian gerbe with connection,
an equivariantly closed differential form is constructed on the space of maps
of a torus into the manifold. These constructions are made explicit using a new
local version of the higher Hochschild complex, resulting in differential forms
given by iterated integrals. Connections to two dimensional topological field
theories are indicated. Similarly, this local higher Hochschild complex is used
to calculate the 2-holonomy of an abelian gerbe along any closed oriented
surface, as well as the derivative of 2-holonomy, which in the case of a torus
fits into a sequence of higher holonomies and their differentials.
