## Duke Mathematical Journal

### Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

#### 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 $\widehat{\Gamma}_{F}$. 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 $\widehat{\Gamma}_{F}\operatorname{Ch}(E_{i})$ for an exceptional collection $\{E_{i}\}$ in the derived category of coherent sheaves $\mathcal{D}^{b}_{\mathrm{coh}}(F)$. 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
Revised: 6 September 2015
First available in Project Euclid: 21 April 2016

https://projecteuclid.org/euclid.dmj/1461252850

Digital Object Identifier
doi:10.1215/00127094-3476593

Mathematical Reviews number (MathSciNet)
MR3536989

Zentralblatt MATH identifier
1350.14041

#### 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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