  We review the Hodge theory of some classic examples from mirror symmetry,
with an emphasis on what is intrinsic to the A-model, and on interesting open
questions and problems. In particular, we illustrate the construction of a
quantum Z-local system on the cohomology of the total space of the canonical
bundle of P^2, and suggest how this should be related to the higher algebraic
cycles studied in our 2011 CNTP article "Algebraic K-theory of toric
hypersurfaces".
