On quantum de Rham cohomology theory
Authors:
HuaiDong Cao and Jian Zhou
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 2434
MSC (1991):
Primary 53C15, 58A12, 81R05
Published electronically:
April 1, 1999
MathSciNet review:
1679455
Fulltext PDF Free Access
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Abstract: We define the quantum exterior product and quantum exterior differential on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of . 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 by and by 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 ChernWeil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.
 [1]
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 0496157
(58 #14737a)
 [2]
Raoul
Bott and Loring
W. Tu, Differential forms in algebraic topology, Graduate
Texts in Mathematics, vol. 82, SpringerVerlag, New YorkBerlin, 1982.
MR 658304
(83i:57016)
 [3]
JeanLuc
Brylinski, A differential complex for Poisson manifolds, J.
Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
(89m:58006)
 [4]
H.D. Cao, J. Zhou, Quantum de Rham cohomology, preprint, math.DG/9806157, 1998.
 [5]
Marc
De Wilde and Pierre
B. A. Lecomte, Existence of starproducts and of formal
deformations of the Poisson Lie algebra of arbitrary symplectic
manifolds, Lett. Math. Phys. 7 (1983), no. 6,
487–496. MR
728644 (85j:17021), http://dx.doi.org/10.1007/BF00402248
 [6]
Boris
V. Fedosov, A simple geometrical construction of deformation
quantization, J. Differential Geom. 40 (1994),
no. 2, 213–238. MR 1293654
(95h:58062)
 [7]
Phillip
Griffiths and Joseph
Harris, Principles of algebraic geometry, WileyInterscience
[John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
(80b:14001)
 [8]
M. Kontsevich, Deformation quantization of Poisson manifolds, I, preprint, qalg/9709040.
 [9]
JeanLouis
Koszul, Crochet de SchoutenNijenhuis et cohomologie,
Astérisque Numero Hors Serie (1985), 257–271
(French). The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837203
(88m:17013)
 [10]
H.
Blaine Lawson Jr. and MarieLouise
Michelsohn, Spin geometry, Princeton Mathematical Series,
vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
(91g:53001)
 [11]
J. Li, G. Tian, Comparison of the algebraic and the symplectic GromovWitten invariants, preprint, December 1997, available at alggeom/9712035.
 [12]
Olivier
Mathieu, Harmonic cohomology classes of symplectic manifolds,
Comment. Math. Helv. 70 (1995), no. 1, 1–9. MR 1314938
(96e:58004), http://dx.doi.org/10.1007/BF02565997
 [13]
Gang
Tian, Quantum cohomology and its associativity, Current
developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge,
MA, 1994, pp. 361–401. MR 1474981
(98k:58031)
 [14]
Cumrun
Vafa, Topological mirrors and quantum rings, Essays on mirror
manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR 1191421
(94c:81193)
 [15]
Izu
Vaisman, Lectures on the geometry of Poisson manifolds,
Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel,
1994. MR
1269545 (95h:58057)
 [16]
Dong
Yan, Hodge structure on symplectic manifolds, Adv. Math.
120 (1996), no. 1, 143–154. MR 1392276
(97e:58004), http://dx.doi.org/10.1006/aima.1996.0034
 [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, 61110. MR 58:14737a
 [2]
 R. Bott, L. W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. SpringerVerlag, New YorkBerlin, 1982. MR 83i:57016
 [3]
 J.L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93114. 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 starproducts and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), no. 6, 487496. MR 85j:17021
 [6]
 B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), no. 2, 213238. MR 95h:58062
 [7]
 P. Griffiths, J. Harris, Principles of algebraic geometry. Pure and Applied Mathematics. WileyInterscience [John Wiley & Sons], New York, 1978. MR 80b:14001
 [8]
 M. Kontsevich, Deformation quantization of Poisson manifolds, I, preprint, qalg/9709040.
 [9]
 J.L. Koszul, Crochet de SchoutenNijenhuis et cohomologie, The mathematical heritage of Élie Cartan (Lyon, 1984). Asterisque 1985, Numero Hors Serie, 257271. 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 GromovWitten invariants, preprint, December 1997, available at alggeom/9712035.
 [12]
 O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995), no. 1, 19. MR 96e:58004
 [13]
 G. Tian, Quantum cohomology and its associativity, Current developments in mathematics 1995 (Cambridge, MA), pp. 361401, Internat. Press, Cambridge, MA, 1994. MR 98k:58031
 [14]
 C. Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, pp. 96119, 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, 143154. MR 97e:58004
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Additional Information
HuaiDong 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/S1079676299000566
PII:
S 10796762(99)000566
Received by editor(s):
May 7, 1998
Published electronically:
April 1, 1999
Additional Notes:
Authors’ research was supported in part by NSF grants DMS9632028 and DMS9504925
Communicated by:
Richard Schoen
Article copyright:
© Copyright 1999
American Mathematical Society
