On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfacesXwith isolated cyclic quotient singularities such thatXadmits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomialsfin two variables, and that the quantum period of such a surfaceX, which is a generating function for Gromov–Witten invariants ofX, coincides with the classical period of its mirror partnerf. In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with 13 $\begin{array}{} \frac{1}{3} \end{array} $ (1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.