1998/04/30 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/9804145
8 pages, AMSLaTeX
arxiv created 1998/04/30 · openalex publication_date 1998/04/30 · 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, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of dh. We also define quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the correspoding quantum Stokes theorem is proved. We also derive quantum hard Lefschetz theorem. By 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. Quantum equivariant de Rham cohomology is defined in the similar fashion.