2D topological quantum field theory (2D TFT) was invented by E. Witten and has huge influence on mathematics. In closed string theory, one realization comes from the supersymmetric Calabi-Yau sigma-model which corresponds to the Gromov-Witten theory in symplectic topology and the other realization comes from the Landau-Ginzburg model which corresponds to the quantum singularity theory invented by Fan-Jarvis-Ruan. In either side, there is analogous quantum structure and the mirror symmetry phenomena. In open string case, things will become more complicated. 2D TFT will be related to the category of D-branes and its algebraic structures. One central problem is the Homological Mirror symmetry conjecture proposed by M.Konstevich. I will talk about the problems and recent progress in this field.
