Mirror counts of spectral curves

Speaker​ Minghuan Hu, ​SDU-QM

​Given a toric surface, we count rational nodal curves passing through fixed points on the toric boundary, which can be realized as a log Gromov-Witten invariant. Following the idea of Yau-Zaslow, this count coincides with the Euler number of the moduli space of a family of compactified Jacobians. Using homological mirror symmetry, we identify this space with the moduli space of constructible sheaves microsupported in a Legendrian link. We then show that this moduli space admits a stratification called ``ruling decomposition'', and thereby computes its Euler number using combinatorial data. We conjecture that the ruling decomposition also recovers higher-genus invariants. This is a joint work with Tom Graber and Eric Zaslow.