Family Floer superpotential's critical values are eigenvalues of quantum product by $c_1$

Abstract

In the setting of the non-archimedean SYZ mirror construction (arXiv:2003.06106), we prove the folklore conjecture that the critical values of the mirror superpotential are the eigenvalues of the quantum multiplication by the first Chern class. Our result relies on a weak unobstructed assumption, but it is usually ensured in practice by Solomon's results on anti-symmetric Lagrangians. Lastly, we note that some explicit examples are presented in the recent work (arXiv:2206.04652).