## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Primitive Forms and Related Subjects — Kavli IPMU 2014, K. Hori, C. Li, S. Li and K. Saito, eds. (Tokyo: Mathematical Society of Japan, 2019), 55 - 115

### Gamma conjecture via mirror symmetry

Sergey Galkin and Hiroshi Iritani

#### Abstract

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold $F$ defines a characteristic class $A_F$ of $F$, called the principal asymptotic class. Gamma conjecture [29] of Vasily Golyshev and the present authors claims that the principal asymptotic class $A_F$ equals the Gamma class $\widehat{\Gamma}_F$ associated to Euler's $\Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

#### Article information

**Dates**

Received: 14 June 2015

Revised: 23 June 2017

First available in Project Euclid:
26 December 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1577379883

**Digital Object Identifier**

doi:10.2969/aspm/08310055

**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] 14J45: Fano varieties 14J33: Mirror symmetry [See also 11G42, 53D37] 11G42: Arithmetic mirror symmetry [See also 14J33]

**Keywords**

Fano varieties quantum cohomology mirror symmetry Dubrovin's conjecture Gamma class Apery constant derived category of coherent sheaves exceptional collection Landau–Ginzburg model

#### Citation

Galkin, Sergey; Iritani, Hiroshi. Gamma conjecture via mirror symmetry. Primitive Forms and Related Subjects — Kavli IPMU 2014, 55--115, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08310055. https://projecteuclid.org/euclid.aspm/1577379883