On Quantum de Rham Cohomology

TL;DR

This paper introduces quantum analogues of classical differential geometric objects on Poisson and symplectic manifolds, establishing quantum de Rham and Dolbeault cohomologies, and proving fundamental theorems like quantum Lefschetz and Stokes.

Contribution

It defines quantum exterior product, differential, and cohomology theories, extending classical geometry into the quantum setting with new theorems and examples.

Findings

Quantum de Rham cohomology differs from quantum cohomology based on pseudo-holomorphic curves.

Quantum Lefschetz theorem is proved.

Calculations show differences between quantum de Rham and quantum cohomology.

Abstract

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. 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 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 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.