Integral transforms and Drinfeld centers in derived algebraic geometry

We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks $X$ and their $\infty$-categories ${QC}(X)$ of quasi-coherent sheaves. (When $X$ is a familiar scheme or stack, ${QC}(X)$ is an enriched version of the usual quasi-coherent derived category $D_{qc}(X)$.) We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on a fiber product with functors between the categories of sheaves on the factors (thus realizing functors as integral transforms, generalizing a theorem of Toën for ordinary schemes). As a first application, for a perfect stack $X$, consider ${QC}(X)$ with its usual monoidal tensor product. Then our main results imply the equivalence of the Drinfeld center (or Hochschild cohomology category) of ${QC}(X)$, the trace (or Hochschild homology category) of ${QC}(X)$ and the category of sheaves on the loop space of $X$. More generally, we show that the $\mathcal{E}_n$-center and the $\mathcal{E}_n$-trace (or $\mathcal{E}_n$-Hochschild cohomology and homology categories, respectively) of ${QC}(X)$ are equivalent to the category of sheaves on the space of maps from the $n$-sphere into $X$. This directly verifies geometric instances of the categorified Deligne and Kontsevich conjectures on the structure of Hochschild cohomology. As a second application, we use our main results to calculate the Drinfeld center of categories of linear endofunctors of categories of sheaves. This provides concrete applications to the structure of Hecke algebras in geometric representation theory. Finally, we explain how the above results can be interpreted in the context of topological field theory.