Abstract
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".
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