|XXXVIII Workshop on Geometric Methods in Physics||30.06-6.07.2019|
|VIII School on Geometry and Physics||24-28.06.2019|
Algebraic geometric aspects of Isomonodromic Deformations: Helix structures in Quantum Cohomology of Fano Varieties
In occasion of the 1998 ICM in Berlin, B. Dubrovin conjectured an intriguing connection between the enumerative geometry of a Fano manifold $X$ with algebro-geometric properties of exceptional collections in the derived category $\mathcal D^b(X)$. Under the assumption of semisimplicity of the quantum co- homology of $X$, the conjecture prescribes an explicit form for local invariants of $QH^\bullet(X)$, the so-called “monodromy data”, in terms of Gram matrices and characteristic classes of objects of exceptional collections. In this talk, a refinement of this conjecture will be presented, and particular attention will be given to the case of complex Grassmannians. At points of small quantum cohomology, these varieties manifest a coalescence phenomenon, whose occurrence and frequency is surprisingly subordinate to the distribution of prime numbers. A priori, the analytical description of these Frobenius structures cannot be obtained from an immediate application of the theory of isomonodromy deformations. The speaker will show how, under minimal conditions, the classical theory of M. Jimbo, T. Miwa, K. Ueno (1981) can be extended to describe isomonodromy deformations at a coalescing irregular singularity. Furthermore, a property of quasi-periodicity of Stokes matrices associated to the points of small Quantum Cohomology of complex Grassmannians will be discussed.
Based on joint works with B. Dubrovin and D. Guzzetti.
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