Higher-genus homotopies and the interplay of genus, singularities, and resurgence

Investigate higher-genus homotopies with n ≥ 5 that connect non-consecutive well-counts, and determine how the resulting hyperelliptic structure interacts with singularity formation and resurgent behavior.

Background

The studied triple-well-to-double-well interpolation is an elliptic, genus-one system with an elementary closed-form instanton solution. The paper notes that homotopies with n ≥ 5 lead to higher-genus hyperelliptic systems, for which no elementary closed-form instanton solution is available and which necessarily skip intermediate well-counts.

Beyond the technical challenge of solving these higher-genus systems—potentially using uniform WKB and Bender–Wu methods—the authors identify an unresolved conceptual issue: how the genus of the underlying system affects the singularity structure of analytically continued instantons and their resurgent behavior.

References

Because such homotopies necessarily skip intermediate well-counts (Section 2), they would connect potentials of non-consecutive well-number directly which is a qualitatively different construction from the one studied here, and one where the interplay between genus, singularity structure, and resurgent behaviour remains open.

— Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy  (2609.10621 - Loubal et al., 8 Sep 2026) in Section 6, paragraph beginning “The genus classification of Section 2”