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I talk about joint work with Helge Ruddat, Eric Zaslow and Benjamin Zhou interpreting
the q-refined theta function of a log Calabi-Yau surface as a natural q-refinement of
the open mirror map, defined by quantum periods of mirror curves for outer
Aganagic-Vafa branes on the local Calabi-Yau threefold. The series coefficients are
all-genus logarithmic two-point invariants, directly extending the relation found by
the first three authors. The main part of the proof is combinatorial in nature, using
a convolution relation for Bell polynomials, and thus works in any dimension. We find
an explicit discrepancy at higher genus in the relation to open Gromov-Witten
invariants of the Aganagic-Vafa brane, expressible in terms of relative invariants of
an elliptic curve.
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