The Mukai pairing and integral transforms in Hochschild homology

Let $X$ be a separated smooth proper scheme over a field of characteristic 0. Following Shklyarov, we construct a (non-degenerate) pairing on the Hochschild homology of $\operatorname{perf}(X)$, and hence, on the Hochschild homology of $X$. On the other hand the Hochschild homology of $X$ also has the Mukai pairing (see papers by Căldăraru). 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 $\mathrm H_0(\mathcal M_0(2,0))$ 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 Căldăraru's work. This is used to prove that the “more natural” construction of integral transforms in Hochschild homology by Shklyarov coincides with that of Căldăraru. These results give rise to a Hirzebruch–Riemann–Roch theorem for the sheafification of the Dennis trace map.