Seminar on Arithmetic Geometry

Helge Ruddat (University of Stavanger)

Gross-Hacking-Siebert generalized the classical Jacobi theta function from abelian varieties to more general log Calabi-Yau manifolds. Landau-Ginzburg superpotentials in mathematical physics give particular examples of such theta functions. Zaslow, Gräfnitz and I compute the Landau-Ginzburg superpotential of the mirror symmetry dual of P^2 relative a smooth elliptic curve. This infinite power series is tropically defined and can be identified with a generating function for 2-contact point rational Gromov-Witten invariants of (X,E). We found that this series also equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle K_X, so it is closely related to a solution to a Lerche-Mayr system of two differential equations and it is also a generating function of holomorphic disk counts. The fundamental structure used to study theta functions is the wall structure. I am going to explain the background and usefulness of this recent technology.

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