Let $(X,omega)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(Sigma,partialSigma)$ to the pair
$(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $Lcap V=emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $phi$ on $X$ such that $L$ is the fixed point set of $phi$ and $V$ is $phi$-anti-invariant, then we define the so-called relatively open invariants for the tuple $(X,omega,V,phi)$ if $L$ is orientable and dim$Xle 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$Xle4$ and some additional restrictions on the number of marked points on each boundary component of the domain.
Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-
Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.
