We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces functorially induce linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

AC's initial work on this project was supported by an NSF postdoctoral fellowship, and by travel grants and hospitality from the University of Pennsylvania, the University of Salamanca, Spain, and the Newton Institute in Cambridge, England. AC's current work is supported by the National Science Foundation under Grant No. DMS-0556042. SW has been supported by a WUN travel bursary and a Royal Society Conference grant.