  We show that homologically projectively dual varieties for Grassmannians
Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of
singularities of the corresponding Pfaffian varieties. As an application we
describe the derived categories of linear sections of these Grassmannians and
Pfaffians. In particular, we show that
  (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal
decompositions consisting of 3 exceptional line bundles, and of the derived
category of a K3-surface;
  (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the
corresponding Pfaffian variety are derived equivalent.
  We also conjecture a rationality criterion for cubic 4-folds in terms of
their derived categories.
