In solvable models the quantum stability path enters the semiorthogonal selection region at finite time and never leaves; the cubic-fourfold chamber theorem and the full determinant dictionary remain conjectural.
An analogue of Dubrovin's conjecture
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abstract
We propose an analogue of Dubrovin's conjecture for the case where Fano manifolds have quantum connections of exponential type. It includes the case where the quantum cohomology rings are not necessarily semisimple. The conjecture is described as an isomorphism of two linear algebraic structures, which we call "mutation systems". Given such a Fano manifold $X$, one of the structures is given by the Stokes structure of the quantum connection of $X$, and the other is given by a semiorthogonal decomposition of the derived category of coherent sheaves on $X$. We also prove the conjecture for a class of smooth Fano complete intersections in a projective space.
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K3 atoms of the cubic fourfold and the BPS structure of the Painlev\'e I determinant line
In solvable models the quantum stability path enters the semiorthogonal selection region at finite time and never leaves; the cubic-fourfold chamber theorem and the full determinant dictionary remain conjectural.