2007

The Mukai pairing, I: a categorical approach

Caldararu, Andrei, Willerton, Simon

Understand

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces.

We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology.
Adjoint functors induce adjoint linear maps with respect to the Mukai pairing.
We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics.

Built on

Mukai, S., Moduli of vector bundles on K3 surfaces, and symplectic manifolds, Sugaku 39 (3) (1987), 216-235 (translated as Sugaku Expositions, Vol. 1, No. 2, (1988), 139-174)
1988
Earlier work this paper cites.
Bondal, A., I., Kapranov, M., M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337; translation in Math. USSR-Izv. 35 (1990), no. 3, 519–541
1990
Earlier work this paper cites.
Bondal, A., I., Kapranov, M., M., Enhanced triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683; translation in Math. USSR-Sb. 70 (1991), no. 1, 93–107
1991
Earlier work this paper cites.
Weibel, C., A., Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 1655-1662
1996
Earlier work this paper cites.
Căldăraru, A., Derived Categories of Twisted Sheaves on Calabi-Yau Manifolds
2000
Earlier work this paper cites.

Similar

May, J., P., The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73
2001
Cited alongside, same era.
Reiten, I., Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366
2002
Cited alongside, same era.
Gelfand, S., I., Manin, Y., I., Methods of homological algebra
2003
Cited alongside, same era.
Orlov, D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, (Russian) Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 240–262; translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 227–248
2004
Cited alongside, same era.
Căldăraru, A., The Mukai pairing. II. The Hochschild-Kostant-Rosenberg isomorphism, Adv. Math. 194 (2005), no. 1, 34–66 (also math.AG/0308080
2005
Cited alongside, same era.
Căldăraru, A., The Mukai pairing, I: the Hochschild structure, preprint math.AG/0308079
Cited in the paper.
Lauda A., D., Pfeiffer, H., Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras, preprint math/0510664
Cited in the paper.
Markarian, N., The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, preprint, math.AG/0610553
Cited in the paper.
Moore, G., Lectures on Branes, K-Theory and RR charges – two variations on the theme of 2d TFT, Clay Mathematical Institute Lectures, available at http://www.physics.rutgers.edu/~gmoore/clay.html
Cited in the paper.
Moore, G., Segal, G., D-branes and K-theory in 2D topological field theory, preprint, hep-th/0609042
Cited in the paper.
Willerton, S., in preparation
Cited in the paper.

Then

Căldăraru, A., Derived categories a skimming, Snowbird lectures in algebraic geometry, 43-75, Contemp. Math., 388, Amer. Math. Soc., Providence, RI, 2005
2005
Later among the works it cites.
May, J., P., Sigurdsson, J., Parametrized Homotopy Theory
2006
Later among the works it cites.
Costello, K., Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165–214 (also math.AG/0412149
2007
Closest in time.
Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667, also preprint, math.AG/0408337
2007
Closest in time.

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