Topological Quantum Field Theory in Two Dimensions BV algebra,Quantum Master Equation,etc

In my book, I give the analogy of the main results in one of Kevin Costello s paper for open-closed topological conformal field theory. In other words, I show that there is a Batalin-Vilkovisky algebraic structure on the open-closed moduli space (moduli space of Riemann surface with boundary and marked points) , which is defined by Harrelson, Voronov and Zuniga, and the most important, there is a solution up to homotopy to the quantum master equation of that BV algebra if the initial condition is given, under the assumption that a new geometric chain theory gives rise to ordinary homology. This solution is expected to encode the fundamental chain of compactified open-closed moduli space, which is studied thoroughly by C.-C.Liu, as exactly in the closed case. We hope this result can give new insights into the mysterious two dimensional open-closed field theory.