Duke Mathematical Journal
- Duke Math. J.
- Volume 165, Number 11 (2016), 2005-2077.
Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures
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 to a Fano manifold . We say that satisfies Gamma conjecture I if equals the Gamma class . When the quantum cohomology of is semisimple, we say that satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by for an exceptional collection in the derived category of coherent sheaves . Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.
Duke Math. J., Volume 165, Number 11 (2016), 2005-2077.
Received: 18 June 2014
Revised: 6 September 2015
First available in Project Euclid: 21 April 2016
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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
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