Picard–Lefschetz thimble contributions of real-time saddles

Determine the explicit Picard–Lefschetz thimbles, intersection numbers, and one-loop fluctuation determinants for the singular and periodic real-time saddles of the triple-well-to-double-well homotopy, in order to establish which saddles contribute to the real-time path-integral contour decomposition.

Background

The paper classifies the classical complex solutions and identifies both singular and bounded periodic real-time saddles, but it does not determine their status in the quantum path integral. A complete Picard–Lefschetz analysis would require constructing the relevant thimbles, calculating the intersection numbers that determine contour contributions, and evaluating the one-loop fluctuation determinants around the periodic solutions.

The unresolved issue is therefore whether the classical saddles identified for the triple-well-to-double-well homotopy actually contribute to the real-time path integral and, if so, with what semiclassical weights. The authors explicitly defer this analysis to future work.

References

A full treatment would require constructing these thimbles explicitly and computing the intersection numbers nσ that dictate which saddles actually contribute, together with the one-loop fluctuation determinant around the periodic real-time solutions of Section 4. We leave this analysis, and the resulting statement about which of our singular and periodic saddles survive in the contour decomposition, to future work.

— Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy  (2609.10621 - Loubal et al., 8 Sep 2026) in Section 5, paragraph beginning “Picard-Lefschetz theory”; Section 6, final discussion