On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

TL;DR

This paper advances the understanding of quantum periods of certain orbifold del Pezzo surfaces with specific singularities, confirming a conjecture relating their quantum invariants to mirror symmetry via Laurent polynomials.

Contribution

It proves a significant part of a conjecture linking quantum periods of del Pezzo surfaces with $rac{1}{3}(1,1)$ singularities to mirror symmetry, using advanced computational techniques.

Findings

Confirmed the conjectural correspondence for surfaces with $rac{1}{3}(1,1)$ singularities.

Computed quantum periods using Quantum Lefschetz theorem and Abelian/non-Abelian Correspondence.

Provided evidence supporting conjectural generalizations to orbifold cases.

Abstract

In earlier joint work with our collaborators Akhtar, Coates, Corti, Heuberger, Kasprzyk, Prince and Tveiten, we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces $X$ with isolated cyclic quotient singularities such that $X$ admits a $\mathbb{Q}$-Gorenstein toric degeneration correspond under Mirror Symmetry to maximally mutable Laurent polynomials $f$ in two variables, and that the quantum period of such a surface $X$, which is a generating function for Gromov-Witten invariants of $X$, coincides with the classical period of its mirror partner $f$. In this paper, we prove a large part of this conjecture for del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities, by computing many of the quantum periods involved. Our tools are the Quantum Lefschetz theorem and the Abelian/non-Abelian Correspondence; our main results are contingent on, and give strong evidence for, conjectural generalizations of these results to the orbifold setting.