Family Floer SYZ singularities for the conifold transition

TL;DR

This paper provides a rigorous mathematical realization of the SYZ conjecture for the conifold, explicitly describing singular fibers and confirming mirror symmetry predictions in a non-archimedean setting.

Contribution

It offers a precise, explicit construction of SYZ singularities for the conifold and establishes a mirror relation between smoothing and crepant resolution in a non-archimedean framework.

Findings

Explicit description of singular T-duality fibers.

Confirmation of mirror symmetry conjecture in non-archimedean setting.

New example of affinoid torus fibration with singular extension.

Abstract

We show a mathematically precise version of the SYZ conjecture, proposed in the family Floer context, for the conifold with a conjectural mirror relation between smoothing and crepant resolution. The singular T-duality fibers are explicitly written and exactly correspond to the codimension-2 `missing points' in the mirror cluster variety, which confirms the speculation of Chan, Pomerleano, and Ueda but only in the non-archimedean setting. Concerning purely the area of Berkovich geometry and forgetting all the mirror symmetry background, our B-side analytic fibration is also a new explicit example of affinoid torus fibration with singular extension.