Quantum type cohomologies on contact manifolds (Q2836543)

The author studies quantum type cohomologies on contact manifolds \(M^{2n+1}\). Almost coholomorphic maps on contact manifolds are defined and the moduli space of stable almost coholomorphic maps with marked points which represent a two-dimensional integral homology class of \(M\) are studied. Finally, Gromov-Witten type invariants of the contact manifold \(M\), and the quantum type cohomology of the contact manifold \(M\) of an odd dimension are considered.NEWLINENEWLINE Let \((M,g,\varphi,\eta,\xi)\) be a contact manifold. A smooth map \(u: (\Sigma,j)\to(M,\varphi)\) from a Riemannian surface \((\Sigma,j)\) into \((M,\varphi)\) is said to be \(\varphi\)-coholomorphic if \(du\circ j=\varphi\circ du\).NEWLINENEWLINE A stable map with \(k\) marked points representing a two-dimensional integral homology class \(A\in H_2(M)=H_2(M;\mathbb Z)/\text{Tor}\) consists of a connected reduced curve \((C;z_1,z_2,\dots,z_k)\) with \(k\) marked points and \(u : C\to M\) is a \(\varphi\)-coholomorphic map on each component of \(C\) satisfying certain conditions.NEWLINENEWLINE The author proves that for a \((2n+1)\)-dimensional contact manifold \((M,g,\varphi,\eta,\xi)\) and for a generic almost co-complex structure \(\varphi\) on \(M\), the moduli space \({\mathcal M}(A,\varphi)\) of rational \(\varphi\)-coholomorphic maps which represent the class \(A\), is a smooth manifold with virtual dimension \(2\,c_1({\mathcal D})[A]+2n\), where \({\mathcal D}=\{X;\;\eta(X)=0\}\). Also, it is shown that for the class \(A\in H_2(M;\mathbb Z)\) and a generic almost co-complex structure \(\varphi\) on a compact contact manifold \((M,g,\varphi,\eta,\xi)\), the moduli space \({\mathcal M}_{g,k}(M,A,\varphi)\) of stable \(\varphi\)-coholomorphic maps from a reduced curve of genus \(g\) with \(k\) marked points to \(M\), which represent the class \(A\), is a compact space with virtual dimension \(2\,c_1({\mathcal D})[A]+2(1-g)(n-3)+2k\).NEWLINENEWLINE For a canonical evaluation map \(\text{ev}:{\mathcal M}_{g,k}(M,A,\varphi)\to M^k\), the author defines a Gromov-Witten type invariant by \(\Phi^{M,A,\varphi}_{g,k}:H^*(M^k)\to\mathbb Q\) and a quantum type cohomology by \(QH^*(M)=H^*(M)\otimes\mathbb Q[q]\), where \(\mathbb Q[q]\) is the ring of Laurent polynomials in \(q\) with coefficients in \(\mathbb Q\), to prove that the quantum type cohomology \(QH^*(M)\) of the contact manifold \(M\) is an associative ring under the quantum type product \(*\).