  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.
