Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762

 
 

 

On quantum de Rham cohomology theory


Authors: Huai-Dong Cao and Jian Zhou
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 24-34
MSC (1991): Primary 53C15, 58A12, 81R05
DOI: https://doi.org/10.1090/S1079-6762-99-00056-6
Published electronically: April 1, 1999
MathSciNet review: 1679455
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define the quantum exterior product $\wedge _h$ and quantum exterior differential $d_h$ on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of $d_h$. We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing $d$ by $d_h$ and $\wedge$ by $\wedge _h$ in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.


References [Enhancements On Off] (What's this?)

  • [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61-110. MR 58:14737a
  • [2] R. Bott, L. W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. MR 83i:57016
  • [3] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93-114. MR 89m:58006
  • [4] H.-D. Cao, J. Zhou, Quantum de Rham cohomology, preprint, math.DG/9806157, 1998.
  • [5] M. De Wilde, P. B. A. Lecomte, Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), no. 6, 487-496. MR 85j:17021
  • [6] B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), no. 2, 213-238. MR 95h:58062
  • [7] P. Griffiths, J. Harris, Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 80b:14001
  • [8] M. Kontsevich, Deformation quantization of Poisson manifolds, I, preprint, q-alg/9709040.
  • [9] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Élie Cartan (Lyon, 1984). Asterisque 1985, Numero Hors Serie, 257-271. MR 88m:17013
  • [10] H. B. Lawson, Jr., M.-L. Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. MR 91g:53001
  • [11] J. Li, G. Tian, Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint, December 1997, available at alg-geom/9712035.
  • [12] O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995), no. 1, 1-9. MR 96e:58004
  • [13] G. Tian, Quantum cohomology and its associativity, Current developments in mathematics 1995 (Cambridge, MA), pp. 361-401, Internat. Press, Cambridge, MA, 1994. MR 98k:58031
  • [14] C. Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, pp. 96-119, Internat. Press, Hong Kong, 1992. MR 94c:81193
  • [15] I. Vaisman, Lectures on the geometry of Poisson manifolds. Progress in Mathematics, 118, Birkhäuser, Basel, 1994. MR 95h:58057
  • [16] D. Yan, Hodge structure on symplectic manifolds, Adv. Math. 120 (1996), no. 1, 143-154. MR 97e:58004

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 53C15, 58A12, 81R05

Retrieve articles in all journals with MSC (1991): 53C15, 58A12, 81R05


Additional Information

Huai-Dong Cao
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843
Email: cao@math.tamu.edu

Jian Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843
Email: zhou@math.tamu.edu

DOI: https://doi.org/10.1090/S1079-6762-99-00056-6
Received by editor(s): May 7, 1998
Published electronically: April 1, 1999
Additional Notes: Authors’ research was supported in part by NSF grants DMS-96-32028 and DMS-95-04925
Communicated by: Richard Schoen
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society