The basic objects for this talk are motives consisting of a curve together with a K_2 class, and their mixed Hodge-theoretic invariants.  My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves.  (Here "distinguished" turns out not to be "special" in the Tannakian sense discussed in B. Klingler's recent talk, but rather a mixed analogue of rank-1 attractor points.)  By local mirror symmetry, this gives evidence for a conjecture in topological string theory relating enumerative invariants of toric CY 3-folds to spectra of quantum curves.

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