On Quantum De Rham Cohomology

We define quantum exterior product ∧h 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 ∧ by ∧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. Recently, the quantum cohomology rings have generated a lot of researches. Many mathematicians have contributed to this rapidly progressing field of mathematics. We will not described the history here, but refer the interested reader to the orignal papers and surveys (e.g., [2], [3], [4], [13], [14], [18], [19], [22]–[30], [32]–[40], [42], [43] and the references therein). The purpose of this paper is to give the construction of another deformation of the de Rham cohomology ring. The existence of a different deformation should not be a surprise, since there is no reason to expect the deformation of the cohomology to be unique. A remarkable feature of our construction is that it follows the traditional construction of the de Rham cohomology. More precisely, we construct a quantum wedge product ∧h on exterior forms, and a quantum exterior differential dh, which satisfy the usual property of the calculus of differential forms. This quantum calculus allows us to “deformation quantize” many differential geometric objects, i.e. our quantum objects is a polynomial in an indeterminate h, whose zeroth order terms are the classical objects. (In this sense, h should be regarded as the Planck constant.) For example, we will define quantum curvature of an ordinary connection, and define quantum characteristic classes in the same fashion as the classical Chern-Weil theory. Our construction has the following features which are not shared by the quantum cohomology: 1. Quantum de Rham cohomology can be defined for Poisson manifolds, not necessarily compact, or closed. 2. The proof of associative is of elementary nature. 3. It is routine to define quantum Dolbeault cohomology. 4. It is routine to define quantum characteristic class. 5. It is routine to define quantum equivariant de Rham cohomology. 6. The computations for homogeneous examples are elementary. Both authors are supported in part by NSF 1 2 HUAI-DONG CAO & JIAN ZHOU Our construction is motivated by Moyal-Weyl multiplication and Clifford multiplication. For any finite dimensional vector space V with a basis {e1, · · · , em}, let {e1, · · · , em} be the dual basis. Assume that w = wei ⊗ ej ∈ V ⊗ V , then w defines a multiplication ∧w on Λ(V ∗), and a multiplication ∗w on S(V ∗), such that e ∧w e = e ∧ e + w , e ∗w e = e ⊙ e + w . If w ∈ S(V ), then ∧w is the Clifford multiplication. If w ∈ Λ(V ) is nondegenerate, ∗w is the Moyal-Weyl multiplication. If w ∈ Λ(V ), then ∧w is what we call a quantum exterior product (or a quantum Clifford multiplication). It is elementary to show that this mutiplication is associative. We will use it to obtain a quantum calculus on any Poisson manifold. The main results we obtained in this paper have been announced in [10]. The layout of this paper is clear from the following