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

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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.

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*Deformation theory and quantization. I. Deformations of symplectic structures*, Ann. Physics **111** (1978), no. 1, 61–110.
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*Lectures on the geometry of Poisson manifolds*. Progress in Mathematics, 118, Birkhäuser, Basel, 1994.
- D. Yan,
*Hodge structure on symplectic manifolds*, Adv. Math. **120** (1996), no. 1, 143–154.

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Additional Information

**Huai-Dong Cao**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843

MR Author ID:
224609

ORCID:
0000-0002-4956-4849

Email:
cao@math.tamu.edu

**Jian Zhou**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843

Email:
zhou@math.tamu.edu

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