1998/06/30 by Huai-Dong Cao, Jian Zhou, Cao, Huai-Dong +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.DG #math.QA
paper · pdf · doi:10.48550/arxiv.math/9806157
36 pages, AMS LaTeX
arxiv created 1998/06/30 · openalex publication_date 1998/06/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define quantum exterior product wedgeh and quantum exterior differential dh on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of dh. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. We also define a version of quantum integral, and prove the quantum Stokes theorem. By the trick of replacing d by dh and wedge by wedgeh 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 classical Chern-Weil theory, i.e., they can be represented by expressions of quantum curvature. Quantum equivariant de Rham cohomology is defined in a similar fashion. Calculations are done for some examples, which show that quantum de Rham cohomology is different from the quantum cohomology defined using pseudo-holomorphic curves.