Does monodromy act on the derived category of sheaves? Let $\mathcal{X} \to \Delta^* $ be a family of complex projective varieties over a punctured disc.  Then, for any fibre $X$, there is a monodromy action $M: H^* (X) \to H^*(X)$.  


Is there a monodromy action on the K-theory of $X$?  What about on various derived categories of sheaves?


 A: $(1)$ When $X$ is Calabi-Yau, there is a "derived version" of monodromy around a conifold point. It was also investigated in the framework of string theory, see for instance the paper
Paul S. Aspinwall1 and Michael R. Douglas
"D-brane stability and monodromy"
Journal of High Energy Physics (2002).
$(2)$ When $X$ is a $K3$ surface, it is possible to construct a representation of the group of auto-equivalences $\textrm{Aut } D(X)$ on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on $X$, and to relate this representation to the monodromy of the Hilbert schemes $X^{[n]}$ of points on the surface. See
Eyal Markman
"On the monodromy of moduli spaces of sheaves on K3 surfaces", 
J. Algebraic Geom. 17 (2008), 29-99. 
A: When the fibers $X$ are Calabi-Yau threefolds, the monodromy preserves a symplectic form, and $Y$ is their mirror partner, then one expects a corresponding autoequivalence on $Y$.
(The monodromy induces a symplectic auto-equivalence of any fiber, so an auto-equivalence of the Fukaya category, and by Kontsevich's homological mirror symmetry this gives an auto-equivalence of the derived category of coherent sheaves on $Y$.)
In the case of a Dehn twist on the symplectic side, this yields a Seidel-Thomas on the coherent sheaves side, but many other examples have been studied.
