Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry
David Ben-Zvi, John Francis, David Nadler
TL;DR
This work develops a derived-categorical framework linking geometry of derived stacks with algebraic operations on their quasi-coherent sheaves. Central results identify the derived fiber product QC(X×_Y X') with both QC(X)⊗_{QC(Y)}QC(X') and with the category of QC(Y)-linear functors, enabling integral-transform descriptions. Consequently, the Drinfeld center and Hochschild (co)homology of QC(X) are realized as QC( LX ) on the loop space LX, with higher ℰ_n-structures corresponding to mapping spaces X^{S^n}, validating geometric instances of Deligne and Kontsevich conjectures in this setting. Applications to affine Hecke categories and topological field theory illustrate the framework's power, connecting categorical centers to local systems and TFTs. The results extend to relative bases, convolution categories, and higher centers, offering a robust bridge between derived algebraic geometry and quantum/topological field theories.
Abstract
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 oo-categories QC(X) of quasicoherent sheaves. 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 Toen 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 E_n-center and the E_n-trace (or 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 all of the above results can be interpreted in the context of topological field theory.