The differential K-theory is a differential generalized cohomology theory which combines the topological K-theory and the information of the geometric data. It is partly motivated by the study of D-branes in theoretical physics by Witten in 1998, which could be regarded as a smooth version of the arithmetic K-theory in Arakelov geometry. In differential K-theory, the role of the holomorphic torsion in arithmetic K-theory is replaced by the Bismut-Cheeger eta form. In this talk, we introduce two models of the differential K-theory and discuss its equivariant extension and the corresponding differential Grothendieck-Riemann-Roch theorem. Especially, we will mainly focus on the functoriality property of the push-forwards (direct images) by using the functoriality and the embedding property of the equivariant eta forms developed in our recent work.

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