1404.6407 (with Vasily Golyshev and Hiroshi Iritani).
Next four algebraic properties of quantum cohomology of a Fano manifold are all distinct:
(1) Big quantum cohomology is generically semi-simple (the condition of Dubrovin's conjecture)
(2) Small quantum cohomology is generically semi-simple
(3) "Very small" quantum cohomology (i.e. QH of the monotome symplectic manifold) is semi-simple
(4) A small quantum cohomology algebra (of a monotone symplectic manifold) has a field as a direct summand
In 1404.7388 using mirror symmetry and a simple argument from Ginzburg-Landau theory I show that property (4) should
also hold for all Fano manifolds, and prove it for all toric Fano manifolds,
thus confirming conjecture of Ostrover-Tyomkin.
This is also related to Conjecture O (of 1404.6407) that roughly says that the structure sheaf of a Fano manifold
is mirror dual to the Lagrangian thimble formed by the locus of real positive points.
Ostrover and Tyomkin shown that for some toric Fano fourfolds (3) fails, but (2) is true.
In 1405.3857 we show that for isotropic Grassmannian IG(2,6) (2) fails, but (1) is true;
and also the derived category of coherent sheaves has a full exceptional collection,
so it is the first case where one needs big quantum cohomology to formulate the first part of Dubrovin's conjecture.
Finally, in 1404.3857 we formulate Gamma Conjecture I (related to Conjecture O)
and also Gamma Conjecture II (which is the exact formulation of the third part of Dubrovin's conjecture).
I will provide some confirming examples, and give some ways that could lead to proving these conjectures.