In this talk we introduce differential cohomology with compact support. There are several different models for differential cohomology. We use the model of differential characters which is originally due to J. Cheeger and J. Simons.
We start by recalling the construction of cohomology with compact support both in the setting of de Rham cohomology and smooth singular cohomology. Then we discuss the appropriate way to adapt these constructions to differential cohomology.
In the end, differential cohomology with compact support has all nice properties it should have: it is functorial with respect to open embeddings, it has exact sequences comparing it to compactly supported differential forms and ordinary cohomology with compact support, and it has a module structure over the differential cohomology ring.
Applications will appear in the quantization of higher abelian gauge theories, especially for self-dual theories (although this is beyond the scope of the present talk).
All this is joint work in progress with Marco Benini (Potsdam), Alexander Schenkel (Edinburgh) and Richard Szabo (Edinburgh).