A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli; Giovanni Felder

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 4, page 623-655
  • ISSN: 0012-9593

Abstract

top
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 H H 2 n ( 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.

How to cite

top

Engeli, Markus, and Felder, Giovanni. "A Riemann-Roch-Hirzebruch formula for traces of differential operators." Annales scientifiques de l'École Normale Supérieure 41.4 (2008): 623-655. <http://eudml.org/doc/272132>.

@article{Engeli2008,
abstract = {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 $HH^\{2n\}(\mathcal \{D\}_n,\mathcal \{D\}_n^*)$ 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.},
author = {Engeli, Markus, Felder, Giovanni},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {traces of differential operators; Lefschetz formula; Riemann–Roch–Hirzebruch formula},
language = {eng},
number = {4},
pages = {623-655},
publisher = {Société mathématique de France},
title = {A Riemann-Roch-Hirzebruch formula for traces of differential operators},
url = {http://eudml.org/doc/272132},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Engeli, Markus
AU - Felder, Giovanni
TI - A Riemann-Roch-Hirzebruch formula for traces of differential operators
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 4
SP - 623
EP - 655
AB - 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 $HH^{2n}(\mathcal {D}_n,\mathcal {D}_n^*)$ 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.
LA - eng
KW - traces of differential operators; Lefschetz formula; Riemann–Roch–Hirzebruch formula
UR - http://eudml.org/doc/272132
ER -

References

top
  1. [1] A. A. Beĭlinson & V. V. Schechtman, Determinant bundles and Virasoro algebras, Comm. Math. Phys.118 (1988), 651–701. Zbl0665.17010
  2. [2] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften 298, Springer, 1992. Zbl0744.58001
  3. [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. Zbl0289.57011
  4. [4] J.-L. Brylinski & E. Getzler, The homology of algebras of pseudodifferential symbols and the noncommutative residue, K -Theory 1 (1987), 385–403. Zbl0646.58026
  5. [5] A. Connes, Noncommutative differential geometry, Publ. Math. I.H.É.S. 62 (1985), 257–360. Zbl0592.46056MR823176
  6. [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. Zbl1106.53055
  7. [7] B. Feĭgin, A. Losev & B. Shoikhet, Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics, preprint arXiv:math.QA/0401400. 
  8. [8] B. Feĭgin & B. Tsygan, Riemann-Roch theorem and Lie algebra cohomology. I, Rend. Circ. Mat. Palermo Suppl. 21 (1989), 15–52. Zbl0686.14007
  9. [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. Zbl0239.58004MR440631
  10. [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. Zbl0238.58001
  11. [11] I. M. Gelʼfand, D. A. Každan & D. B. Fuks, Actions of infinite-dimensional Lie algebras, Functional Anal. Appl.6 (1972), 9–13. Zbl0267.18023
  12. [12] A. Jaffe, A. Lesniewski & K. Osterwalder, Quantum K -theory. I. The Chern character, Comm. Math. Phys. 118 (1988), 1–14. Zbl0656.58048
  13. [13] S. Lefschetz, Introduction to topology, Princeton Mathematical Series, vol. 11, Princeton University Press, 1949. Zbl0041.51801MR31708
  14. [14] J.-L. Loday, Cyclic homology, 2 éd., Grund. Math. Wiss. 301, Springer, 1998. Zbl0885.18007MR1600246
  15. [15] V. Lysov, Anticommutativity equations in topological quantum mechanics,, JETP Lett. 76 (2002), 724–727. 
  16. [16] S. MacLane, Homology, 1 éd., Springer, 1967, Die Grundlehren der mathematischen Wissenschaften, Band 114. Zbl0133.26502MR349792
  17. [17] R. Nest & B. Tsygan, Algebraic index theorem, Comm. Math. Phys.172 (1995), 223–262. Zbl0887.58050
  18. [18] A. Ramadoss, Some notes on the Feigin–Losev–Shoikhet integral conjecture, preprint, arXiv:math.QA/0612298. Zbl1194.32010MR2438339
  19. [19] V. V. Schechtman, Riemann-Roch theorem after D. Toledo and Y.-L. Tong, Rend. Circ. Mat. Palermo Suppl.21 (1989), 53–81. Zbl0707.14007MR1009565
  20. [20] F. Trèves, Topological vector spaces, distributions and kernels, Academic Press, 1967. Zbl0171.10402MR225131
  21. [21] M. Wodzicki, Cyclic homology of differential operators, Duke Math. J.54 (1987), 641–647. Zbl0635.18010MR899408

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.