Family Floer and Kontsevich–Soibelman Wall Crossing

Definitions and mathematical statement

A wall-crossing formula describes how specified mathematical data change when a parameter crosses a specified wall in a parameter space. In this problem, the formulas to be compared are those arising from family Floer homology and the Kontsevich--Soibelman wall-crossing formulas. Refined tropical counts are counts of tropical objects with additional weighting or refinement data; the source does not specify the objects, weights, or refinement parameter intended here. These descriptions identify the subject matter but do not supply the missing definitions for a particular comparison.

The source asks: How can wall-crossing formulas from family Floer homology be related to the Kontsevich--Soibelman wall-crossing formulas, and thereby to refined tropical counts?

The intended task is to formulate and establish such a relation in an explicitly specified mathematical setting. The supplied record does not identify the underlying symplectic or geometric space, a family of Lagrangian objects, the parameter space and its walls, the relevant Floer constructions, the version of the Kontsevich--Soibelman formula, or the refined tropical enumerative problem. It also does not specify whether the desired relation is an equality, a specialization, a correspondence, or another construction. Consequently, the record determines a research question but does not determine a single quantified conjecture or comparison identity. Any precise theorem statement requires these choices and their hypotheses to be recovered from the source or supplied by mathematical review.

The supplied background further describes the target as a general comparison between refined tropical counts and relevant Fukaya or mirror-symmetry counts. Neither these latter counts nor their relationship to the family Floer formulation is specified in the record.

Short English statement

Explain how wall crossing in family Floer homology relates to Kontsevich--Soibelman wall crossing and how this relation connects to refined tropical counts. The supplied question leaves the geometric setting, counting theories, and form of the comparison unspecified, so a precise quantified statement remains unresolved.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002872$, title "Family Floer and Kontsevich--Soibelman Wall Crossing," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied source citation: Tropical Aspects in Geometry, Topology and Physics (2016), Oberwolfach Reports, pp. 1255--1316, DOI 10.4171/owr/2015/23. Supplied source URL and literature reference "Oberwolfach Report 23": https://doi.org/10.4171/owr/2015/23. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No individual contribution or problem-specific page is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the target as a general relation between refined tropical counts and relevant Fukaya or mirror-symmetry counts. It reports that known results cover special one-disc or restricted cases rather than the general comparison. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification. The sole supplied literature source is Oberwolfach Report 23 at the DOI link above.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is supplied editorial metadata and does not establish mathematical correctness or independent source verification of this draft.