Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
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
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?)


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: http://dx.doi.org/10.1090/S1079-6762-99-00056-6
PII: S 1079-6762(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