Hamilton dynamics for Lefschetz-thimble integration akin to the complex Langevin method

The Lefschetz-thimble method, i.e., integration along the steepest descent cycles, is a way to avoid the sign problem by complexifying the theory. We discuss that such steepest descent cycles can be identified as ground-state wave functions of a supersymmetric Hamilton dynamics, which is described with a framework akin to the complex Langevin method. We numerically construct the wave functions on a grid using a toy model and confirm their well-localized behavior.