Topological field theory and quantum master equation in two dimensions.

Abstract

In our thesis, I give the analogy of the main results in Kevin Costello's paper "The Gromov-Witten potential associated to a TCFT" 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 in "The open-closed moduli space and related algebraic structure", 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 hoped to encode the fundamental chain of compactified open-closed moduli space, studied thoroughly by C.-C.Liu, as exactly in the closed case (Deligne-Mumford space in this case). We hope this result can give new insights to the mysterious two dimensional open-closed field theory.