  The paper continues the discussion of symplectic aspects of Picard-Lefschetz
theory begun in "Vanishing cycles and mutation" (this archive). There we
explained how to associate to a suitable fibration over a two-dimensional disc
a triangulated category, the "derived directed Fukaya category" which describes
the structure of the vanishing cycles. The present second part serves two
purposes. Firstly, it contains various kinds of algebro-geometric examples,
including the "mirror manifold" of the projective plane. Secondly there is a
(largely conjectural) discussion of more advanced topics, such as (i)
Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse
theory, (iii) a proposed "dimensional reduction" algorithm for doing certain
Floer cohomology computations.
