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

TL;DR

This paper proves that in the non-archimedean SYZ mirror construction, the critical values of the superpotential correspond to eigenvalues of quantum multiplication by the first Chern class, confirming a longstanding conjecture.

Contribution

It establishes the folklore conjecture relating superpotential critical values to quantum eigenvalues within the non-archimedean SYZ framework, under mild unobstructedness assumptions.

Findings

Critical values match eigenvalues of quantum product by c_1.

Results rely on weak unobstructedness, often guaranteed by Solomon's work.

Explicit examples are provided in recent literature.

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).