Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry

By David Ben-Zvi, J. Francis, D. Nadler. 2008

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6cce5aecdd2e64ea631ed46db743fd61b60aa7d7

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Citations in the corpus

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A universal characterization of higher algebraic K-theory (2010, methodology): “In addition to Lurie’s work, the paper of Ben-Zvi, Francis, and Nadler [2] provides an excellent exposè of this theory in the context of the study of geometric function theory from a perspective with its origin in Thomason-Trobaugh; we refer the interested reader to sections 2 and 4.1 of that paper”
Ind-coherent sheaves (2011, background): “ding Coh(X) →QCoh(X). Lemma 1.1.6. Assume that Sis a regular classical scheme. Then ΨS is an equivalence. Remark 1.1.7. As we shall see in Sect. 1.6, the converse is also true. Proof. Recall that by [BFN], Prop. 3.19, extending the arguments of [Ne], the category QCoh(S) is compactly generated by it”
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Loop spaces and connections (2010, background): “Such a theory is developed in [22, 2,4] including appropriate notions of algebra objects, module categories over algebra objects, and so on.”