Quantum cohomology of twistor spaces and their Lagrangian submanifolds. (English) Zbl 1306.53076

Evans, Jonathan David

J. Differ. Geom. 96, No. 3, 353-397 (2014).

The author studies quantum cohomology of the twistor space of an oriented compact 6-dimensional hyperbolic space. The twistor space \(Z\) of an oriented \(2n\)-dimensional Riemannian manifold \((M,g)\) is the total space of orthogonal almost complex structures \(J:TM\to TM, J^2=-Id, g(J.,J.)=g(.,.)\) compatible with the given orientation. The fibre of the bundle \(\tau:Z\to M\) is the homogeneous space \(F=\mathrm{SO}(2n)/\mathrm{U}(n)\) with a natural Kähler structure \((g_F,j_F,\omega_F)\). The Levi-Civita connection of \(g\) defines a vertical-horizontal spliting \(TZ=\mathcal H\oplus\mathcal V\). This splitting defines a metric \(g_Z\) on \(Z\) given by \(\tau^*g\) on horizontal spaces and \(g_F\) on vertical spaces. On \(Z\) there are two natural orthogonal almost complex structures \(J_{\pm}\) given on \(T_{\psi}Z\) by \(J_{\pm}=(\pm j_F)\oplus\tau^*\psi\). The forms \(\omega_{\pm}=\pm\omega_F\oplus\tau^*\omega_{\psi}\), where \(\omega_{\psi}=g(\psi.,.)\) are nondegenerate and if \((M,g)\) is a hyperbolic space the form \(\omega_-\) is closed so it is a \(J_-\) compatible symplectic form.
The author proves the following theorem: “The small quantum cohomology of the twistor space of a hyperbolic 6-manifold \(M\) with vanishing Stiefel-Whitney classes is \[ QH^*(Z;\Lambda)=H^*(M,\Lambda)/ (\alpha^4=8\alpha\tau^*\chi+8q\alpha^2-16q^2) \] where \(\alpha=c_1(Z)\) and \(\Lambda=\mathbb C[q]\). Moreover \(c_1(\mathcal H)^2=\alpha^2-4q, c_1(\mathcal H)^3=\alpha^3-4\alpha q\) and \(\chi\) is the Euler class of \(M\).”
He considers also 3-dimensional totally geodesic submanifolds \(\Sigma\) of a 6-dimensional hyperbolic space and a Reznikov Lagrangian \(L_{\Sigma^-}\), i.e., a monotone Lagrangian submanifold of \(Z\) associated with \(\Sigma\). He “computes the obstruction term \(\mathfrak m_0\) in the Fukaya-Floer \(A_{\infty}\)-algebra of a Reznikov Lagrangian \(L_{\Sigma}\) and calculates the Lagrangian quantum homology”. The author also computes the eigenvalues of a quantum multiplication by the first Chern class \(c_1(Z)\) on \(QH^*(Z,\Lambda)\).

Reviewer: Włodzimierz Jelonek (Kraków)

MSC:

53D45	Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53C28	Twistor methods in differential geometry

Keywords:

twistor space; quantum cohomology; symplectic manifold; Lagrangian submanifold