  In this paper we consider a conjecture formulated by the second author in
occasion of the 1998 ICM in Berlin (arXiv:math/9807034v2). This conjecture
states the equivalence, for a Fano variety $X$, of the semisimplicity condition
for the quantum cohomology $QH^\bullet(X)$ with the existence condition of full
exceptional collections in the derived category of coherent sheaves $\mathcal
D^b(X)$. Furthermore, in its quantitative formulation, the conjecture also
prescribes an explicit relationship between the monodromy data of
$QH^\bullet(X)$ and characteristic classes of both $X$ and objects of the
exceptional collections. In this paper we reformulate a refinement of
(arXiv:math/9807034v2), which corrects a previous ansatz (lecture of the second
author at Strasbourg) for what concerns the conjectural expression of the
central connection matrix. We clarify the precise relationship between the
refined conjecture presented in this paper and $\Gamma$-conjecture II of S.
Galkin, V. Golyshev and H. Iritani (arXiv:1404.6407v4, arXiv:1508.00719v3).
Through an explicit computation of the monodromy data and a detailed analysis
of the action of the braid group on both the monodromy data and the set of
exceptional collections, we prove the validity of our refined conjecture for
all complex Grassmannians $\mathbb G(r,k)$. From these results, it is outlined
an explicit description of the "geography" of the exceptional collections
realizable at points of the small quantum cohomology of Grassmannians, i.e.
corresponding to the monodromy data at these points. In particular, it is
proved that Kapranov's exceptional collection appears at points of the small
quantum cohomology only for Grassmannians of small dimension (namely, less or
equal than 2). Finally, a property of quasi-periodicity of the Stokes matrices
of complex Grassmannians, along the locus of the small quantum cohomology, is
described.
