The Mukai pairing, I: the Hochschild structure
By Andrei Căldăraru. 2003
81 citations recorded by the source corpus.
We study the Hochschild structure of a smooth space or orbifold, emphasizing the importance of a pairing defined on Hochschild homology which generalizes a similar pairing introduced by Mukai on the cohomology of a K3 surface. We discuss those properties of the structure which can be derived without appealing to the Hochschild-Kostant-Rosenberg isomorphism and Kontsevich formality, namely: -- functoriality of homology, commutation of push-forward with the Chern character, and adjointness with respect to the generalized pairing; -- formal Hirzebruch-Riemann-Roch and the Cardy condition from physics; -- invariance of the full Hochschild structure under Fourier-Mukai transforms. Connections with homotopy theory and TQFT's are discussed in an appendix. A separate paper treats consequences of the HKR isomorphism. Applications of these results to the study of a mirror symmetric analogue of Chen-Ruan's orbifold product will be presented in a future paper.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/391a605df86101583494fa2eb8a973c40e2c3983
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In Geometry and Topology
- Richard Thomas
- Simon Willerton
- Andrei Căldăraru
- Mikhail Khovanov
- Aaron D. Lauda
- David E. V. Rose
- Sabin Cautis
- Marko Stosic
In Mathematical Physics
- Richard Thomas
- Simon Willerton
- Andrei Căldăraru
- Mikhail Khovanov
- Aaron D. Lauda
- David E. V. Rose
- Sabin Cautis
- Marko Stosic
Citations in the corpus
- The Mukai pairing, I: a categorical approach (2007, methodology): “SW has been supported by a WUN travel bursary and a Royal Society Conference grant.”
- The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism (2003, background): “Here Tr is the Serre duality trace ([7, 2.3]), SX = ωX [dimX] is the dualizing object of Dbcoh(X) (also to be thought of as the functor − ⊗X SX), S∆ = ∆∗SX is the object whose associated integral transform is SX , and Φ µ X→X is the natural transformation 1X ⇒ SX associated to µ (2.2).”
- Braid cobordisms, triangulated categories, and flag varieties (2006, background): “rojective, but all the functors we use have obvious Fourier-Mukai kernels). However, this natural transformation Φ from Db(X 1 ×X2) to Fun(Db(X1),Db(X2)) is neither full nor faithful (see for example [Ca]), though it can be made to be so by working with dg-categories instead [To]. Again, this will n”