A Riemann-Roch-Hirzebruch formula for traces of differential operators
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 4, p. 623-655

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology HH 2n (𝒟 n ,𝒟 n * ) of the algebra of differential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula.

Soit D un opérateur différentiel holomorphe opérant sur les sections d’un fibré vectoriel holomorphe sur une variété complexe de dimension n. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme intégrale d’une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle du générateur canonique de la cohomologie de Hochschild HH 2n (𝒟 n ,𝒟 n * ) de l’algèbre des opérateurs différentiels sur un entourage formel d’un point. Si D est l’identité, la formule se réduit à la formule de Riemann-Roch-Hirzebruch.

@article{ASENS_2008_4_41_4_623_0,
     author = {Engeli, Markus and Felder, Giovanni},
     title = {A Riemann-Roch-Hirzebruch formula for traces of differential operators},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {4},
     year = {2008},
     pages = {623-655},
     doi = {10.24033/asens.2077},
     zbl = {1163.32009},
     mrnumber = {2489635},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2008_4_41_4_623_0}
}
Engeli, Markus; Felder, Giovanni. A Riemann-Roch-Hirzebruch formula for traces of differential operators. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 4, pp. 623-655. doi : 10.24033/asens.2077. http://www.numdam.org/item/ASENS_2008_4_41_4_623_0/

[1] A. A. Beĭlinson & V. V. Schechtman, Determinant bundles and Virasoro algebras, Comm. Math. Phys. 118 (1988), 651-701. | Zbl 0665.17010

[2] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften 298, Springer, 1992. | Zbl 0744.58001

[3] I. N. Bernšteĭn & B. I. RosenfelʼD, Homogeneous spaces of infinite-dimensional Lie algebras and the characteristic classes of foliations,, Russian Math. Surveys 28 (1973), 107-142. | Zbl 0289.57011

[4] J.-L. Brylinski & E. Getzler, The homology of algebras of pseudodifferential symbols and the noncommutative residue, K-Theory 1 (1987), 385-403. | Zbl 0646.58026

[5] A. Connes, Noncommutative differential geometry, Publ. Math. I.H.É.S. 62 (1985), 257-360. | Numdam | MR 823176 | Zbl 0592.46056

[6] B. Feĭgin, G. Felder & B. Shoikhet, Hochschild cohomology of the Weyl algebra and traces in deformation quantization, Duke Math. J. 127 (2005), 487-517. | Zbl 1106.53055

[7] B. Feĭgin, A. Losev & B. Shoikhet, Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics, preprint arXiv:math.QA/0401400.

[8] B. Feĭgin & B. Tsygan, Riemann-Roch theorem and Lie algebra cohomology. I, Rend. Circ. Mat. Palermo Suppl. 21 (1989), 15-52. | Zbl 0686.14007

[9] I. M. GelʼFand, The cohomology of infinite dimensional Lie algebras: some questions of integral geometry, in Actes du Congrès International des Mathématiciens, Nice, 1970, Gauthier-Villars, 1971, 95-111. | MR 440631 | Zbl 0239.58004

[10] I. M. GelʼFand & D. A. Každan, Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields, Soviet Math. Dokl. 12 (1971), 1367-1370. | Zbl 0238.58001

[11] I. M. GelʼFand, D. A. Každan & D. B. Fuks, Actions of infinite-dimensional Lie algebras, Functional Anal. Appl. 6 (1972), 9-13. | Zbl 0267.18023

[12] A. Jaffe, A. Lesniewski & K. Osterwalder, Quantum K-theory. I. The Chern character, Comm. Math. Phys. 118 (1988), 1-14. | Zbl 0656.58048

[13] S. Lefschetz, Introduction to topology, Princeton Mathematical Series, vol. 11, Princeton University Press, 1949. | MR 31708 | Zbl 0041.51801

[14] J.-L. Loday, Cyclic homology, 2 éd., Grund. Math. Wiss. 301, Springer, 1998. | MR 1600246 | Zbl 0885.18007

[15] V. Lysov, Anticommutativity equations in topological quantum mechanics,, JETP Lett. 76 (2002), 724-727.

[16] S. Maclane, Homology, 1 éd., Springer, 1967, Die Grundlehren der mathematischen Wissenschaften, Band 114. | MR 349792 | Zbl 0133.26502

[17] R. Nest & B. Tsygan, Algebraic index theorem, Comm. Math. Phys. 172 (1995), 223-262. | Zbl 0887.58050

[18] A. Ramadoss, Some notes on the Feigin-Losev-Shoikhet integral conjecture, preprint, arXiv:math.QA/0612298. | MR 2438339 | Zbl 1194.32010

[19] V. V. Schechtman, Riemann-Roch theorem after D. Toledo and Y.-L. Tong, Rend. Circ. Mat. Palermo Suppl. 21 (1989), 53-81. | MR 1009565 | Zbl 0707.14007

[20] F. Trèves, Topological vector spaces, distributions and kernels, Academic Press, 1967. | MR 225131 | Zbl 0171.10402

[21] M. Wodzicki, Cyclic homology of differential operators, Duke Math. J. 54 (1987), 641-647. | MR 899408 | Zbl 0635.18010