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arXiv · 1404.6407

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

Sergey Galkin·Vasily Golyshev·Hiroshi Iritani
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Abstract

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class G_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by G_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_coh(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces and Grassmannians.

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Sergey Galkin, Vasily Golyshev, Hiroshi Iritani. 2014-04-25. Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures. https://doi.org/10.1215/00127094-3476593

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