Stability conditions, wall-crossing and weighted Gromov–Witten invariants

We extend B. Hassett's theory of weighted stable pointed curves to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories (cf. works by A. Gorodentsev, S. Kuleshov, and A. Rudakov, T. Bridgeland, D. Joyce).
We introduce virtual fundamental classes and thus obtain weighted Gromov–Witten invariants. We show that by including gravitational descendants, one obtains an $L$-algebra as introduced by A. Losev and Yu. Manin as a generalization of a cohomological eld theory.