Mirror enumerative correspondence for Hitchin moduli spaces

The conjecture asks whether holomorphic-disk enumeration on the $PGL(2)$ mirror of an $SU(2)$ Hitchin moduli space matches critical trajectories of quadratic-differential foliations. No source identifies a proposer or gives a definitive formulation of the full singular case.

Known results

• A 2010 paper constructs compatible Gross–Siebert consistent structures for $SU(2)$ Hitchin spaces with prescribed singularities.
• Its instanton corrections from logarithmic morphisms correspond to critical trajectories of quadratic differentials, and the resulting toric degeneration has generic fiber isomorphic to the Hitchin moduli space.
• A 2012 paper establishes the $SL(2)$/$SO(3)$ SYZ-mirror relationship near singular fibers.
• These results concern instanton corrections and mirror geometry, not a rigorous equality of holomorphic-disk counts with trajectory counts.

2010 partial result

The 2010 work presents the construction as a nontrivial mirror-symmetry result, but explicitly notes that the physical-brane and holomorphic-disk interpretation is difficult to justify mathematically in general. No retrieved source reports a later proof, counterexample, or verification of the full enumerative correspondence.

Current status (as of August 2026): The mirror framework and trajectory-side instanton correspondence have substantial partial results, but the full holomorphic-disk enumerative statement, especially with prescribed singularities, remains open.