The Mukai pairing and integral transforms in Hochschild homology

Let $X$ be a smooth proper scheme over a field of characteristic 0. Following D. Shklyarov [10], we construct a (non-degenerate) pairing on the Hochschild homology of $\per{X}$, and hence, on the Hochschild homology of $X$. On the other hand the Hochschild homology of $X$ also has the Mukai pairing (see [1]). If $X$ is Calabi-Yau, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in $\text{H}_{0}({\mathcal M}_0(2,0))$ on the algebra of closed states of a version of the B-Model on $X$. We show that these pairings "almost" coincide. This is done via a different view of the construction of integral transforms in Hochschild homology that originally appeared in Caldararu's work [1]. This is used to prove that the more "natural" construction of integral transforms in Hochschild homology by Shklyarov [10] coincides with that of Caldararu [1]. These results give rise to a Hirzebruch Riemann-Roch theorem for the sheafification of the Dennis trace map.