Q (rational number) is used as a ground field to study virtual  fundamental chain of the moduli space of pseudo-holomorphic  curve, because it is basically fundamental class of an orbifold. In our work with Oh-Ohta-Ono and etc. we use real number R  in many cases since we use de Rham cohomology. Using de Rham cohomology makes it easier to keep more symmetry. In this talk I will explain some example where working over Q  (or its finite extension) is interesting and explain an idea how we can  work over Q to study open closed Gromov Witten invariant.
(The latter part is mostly work in progress.)