An analogue of Dubrovin's conjecture

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.

arXiv comment42 pages. To appear at Annales de l'Institut Fourier

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• The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods
• Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetry
• Categorification of Legendrian knots