Duke Mathematical Journal

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

Sergey Galkin, Vasily Golyshev, and 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 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.

Article information

Source
Duke Math. J., Volume 165, Number 11 (2016), 2005-2077.

Dates
Received: 18 June 2014
Revised: 6 September 2015
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1461252850

Digital Object Identifier
doi:10.1215/00127094-3476593

Mathematical Reviews number (MathSciNet)
MR3536989

Zentralblatt MATH identifier
1350.14041

Subjects
Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 11G42: Arithmetic mirror symmetry [See also 14J33] 14J45: Fano varieties 14J33: Mirror symmetry [See also 11G42, 53D37]

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

Citation

Galkin, Sergey; Golyshev, Vasily; Iritani, Hiroshi. Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures. Duke Math. J. 165 (2016), no. 11, 2005--2077. doi:10.1215/00127094-3476593. https://projecteuclid.org/euclid.dmj/1461252850


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