15 August 2016 Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures
Sergey Galkin, Vasily Golyshev, Hiroshi Iritani
Duke Math. J. 165(11): 2005-2077 (15 August 2016). DOI: 10.1215/00127094-3476593


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 AF to a Fano manifold F. We say that F satisfies Gamma conjecture I if AF equals the Gamma class ΓˆF. 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 ΓˆFCh(Ei) for an exceptional collection {Ei} in the derived category of coherent sheaves Dcohb(F). Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.


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Sergey Galkin. Vasily Golyshev. Hiroshi Iritani. "Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures." Duke Math. J. 165 (11) 2005 - 2077, 15 August 2016. https://doi.org/10.1215/00127094-3476593


Received: 18 June 2014; Revised: 6 September 2015; Published: 15 August 2016
First available in Project Euclid: 21 April 2016

zbMATH: 1350.14041
MathSciNet: MR3536989
Digital Object Identifier: 10.1215/00127094-3476593

Primary: 53D37
Secondary: 11G42 , 14J33 , 14J45 , 14N35

Keywords: abelian/nonabelian correspondence , Apery limit , derived category of coherent sheaves , Dubrovin’s conjecture , exceptional collection , Fano varieties , Frobenius manifolds , Gamma class , Grassmannians , Landau–Ginzburg model , mirror symmetry , quantum cohomology , quantum Satake principle

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 11 • 15 August 2016
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