Family Floer superpotential’s critical values are eigenvalues of quantum product by \(c_1\)

In the setting of the non-archimedean SYZ mirror construction (Yuan in Family Floer program and non-archimedean SYZ mirror construction, Diss. State University of New York at Stony Brook, 2021), 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 (Solomon in Adv Math 367:107107, 2020) on anti-symmetric Lagrangians. Lastly, we note that some explicit examples are presented in the recent work (Yuan in Family Floer mirror space for local SYZ singularities, Forum Mathe Sigma 12:e119, 2024).