  Consistent boundary conditions for Alexandrov-Kontsevich-Schwartz-Zaboronsky
(AKSZ) sigma models and the corresponding boundary theories are analyzed. As
their mathematical structures, we introduce a generalization of differential
graded symplectic manifolds, called twisted QP manifolds, in terms of graded
symplectic geometry, canonical functions, and QP pairs. We generalize the AKSZ
construction of topological sigma models to sigma models with Wess-Zumino terms
and show that all the twisted Poisson-like structures known in the literature
can actually be naturally realized as boundary conditions for AKSZ sigma
models.
