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

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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.

- F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,
*Deformation theory and quantization. I. Deformations of symplectic structures*, Ann. Physics**111**(1978), no. 1, 61–110. MR**496157**, DOI https://doi.org/10.1016/0003-4916%2878%2990224-5 - Raoul Bott and Loring W. Tu,
*Differential forms in algebraic topology*, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR**658304** - Jean-Luc Brylinski,
*A differential complex for Poisson manifolds*, J. Differential Geom.**28**(1988), no. 1, 93–114. MR**950556** - H.-D. Cao, J. Zhou,
*Quantum de Rham cohomology*, preprint, math.DG/9806157, 1998. - Marc De Wilde and Pierre 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**728644**, DOI https://doi.org/10.1007/BF00402248 - Boris V. Fedosov,
*A simple geometrical construction of deformation quantization*, J. Differential Geom.**40**(1994), no. 2, 213–238. MR**1293654** - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725** - M. Kontsevich,
*Deformation quantization of Poisson manifolds, I*, preprint, q-alg/9709040. - Jean-Louis Koszul,
*Crochet de Schouten-Nijenhuis et cohomologie*, Astérisque**Numéro Hors Série**(1985), 257–271 (French). The mathematical heritage of Élie Cartan (Lyon, 1984). MR**837203** - H. Blaine Lawson Jr. and Marie-Louise Michelsohn,
*Spin geometry*, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR**1031992** - J. Li, G. Tian,
*Comparison of the algebraic and the symplectic Gromov-Witten invariants*, preprint, December 1997, available at alg-geom/9712035. - Olivier Mathieu,
*Harmonic cohomology classes of symplectic manifolds*, Comment. Math. Helv.**70**(1995), no. 1, 1–9. MR**1314938**, DOI https://doi.org/10.1007/BF02565997 - Gang Tian,
*Quantum cohomology and its associativity*, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 361–401. MR**1474981** - Cumrun Vafa,
*Topological mirrors and quantum rings*, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR**1191421** - Izu Vaisman,
*Lectures on the geometry of Poisson manifolds*, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. MR**1269545** - Dong Yan,
*Hodge structure on symplectic manifolds*, Adv. Math.**120**(1996), no. 1, 143–154. MR**1392276**, DOI https://doi.org/10.1006/aima.1996.0034

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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