Mirror symmetric Gamma conjecture for del Pezzo surfaces

TL;DR

This paper verifies a version of the Gamma conjecture for del Pezzo surfaces of degree at least 3 by computing oscillatory integrals of their mirror Landau-Ginzburg models and relating them to twisted Chern characters and Gamma classes.

Contribution

It explicitly computes mirror oscillatory integrals for del Pezzo surfaces and demonstrates the Gamma conjecture in this non-toric Fano setting with arbitrary K-group insertions.

Findings

Computed oscillatory integrals for mirror Landau-Ginzburg models of del Pezzo surfaces.

Established the relation between the leading order of integrals, twisted Chern characters, and Gamma classes.

Proved a version of the Gamma conjecture for non-toric Fano surfaces.

Abstract

For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces I". In: Publ. Math. Inst. Hautes Etudes Sci. 122 (2015), pp. 65-168]. We explicitly construct the mirror cycle of a line bundle and show that the leading order of the integral on this cycle involves the twisted Chern character and the Gamma class. This proves a version of the Gamma conjecture for non-toric Fano surfaces with an arbitrary K-group insertion.