Open Gromov-Witten Theory on Calabi-Yau manifolds and symplectic cutting

Details Open Gromov-Witten Theory on Calabi-Yau manifolds and symplectic cutting Open Gromov-Witten Theory on Calabi-Yau manifolds and symplectic cutting

2010-03-23

arxiv.org/abs/1003.4325v1

Mathematics

Symplectic Geometry

Scientific paper

In this paper we will study moduli space of $J$-holomorphic discs in an almost Calabi-Yau $X$ of real dimension $2n$ with boundary on Lagrangian submanifolds which are either diffeomorphic to $S^n$ or $\R P^n$. Our main tool will be symplectic-cut technique. As result will prove rationality of numbers defined in \cite{FO-C} and non-displacability of Lagrangian spheres in dimension bigger than two. In the case that the Lagrangian $L$ is in the fixed point set of some anti-symplectic involution, we show that if $L$ is diffeomorphic to $S^3$ then all open invariants defined using the symmtery are zero and if $L$ is diffeomorphic to $\R P^3$, then there is some relation between open invariants of odd classes in $X$ and closed invariants of another almost Calabi-Yau 3-fold $X_{out}$ constructed from $X$.

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