Gamma conjecture via mirror symmetry. (English) Zbl 1452.53073

Galkin, Sergey; Iritani, Hiroshi

Hori, Kentaro (ed.) et al., Primitive forms and related subjects – Kavli IPMU 2014. Proceedings of the international conference, University of Tokyo, Tokyo, Japan, February 10–14, 2014. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 83, 55-115 (2019).

Summary: 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. The Gamma conjecture [the first author et al., Duke Math. J. 165, No. 11, 2005–2077 (2016; Zbl 1350.14041)] 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 the 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.
For the entire collection see [Zbl 1446.53004].

MSC:

53D37	Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J45	Fano varieties
14J33	Mirror symmetry (algebro-geometric aspects)
11G42	Arithmetic mirror symmetry

Keywords:

Fano varieties; quantum cohomology; mirror symmetry; Dubrovin’s conjecture; gamma class; apery constant; derived category of coherent sheaves; exceptional collection; Landau-Ginzburg model

Citations:

Zbl 1350.14041

References:

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