Valley Views: Instantons, Large Order Behaviors, and Supersymmetry

TL;DR

This paper unifies perturbative and non-perturbative analyses of instantons using a geometric valley approach, clarifies Borel ambiguities, confirms large order predictions, and discovers a new supersymmetry consistent with known theorems.

Contribution

It introduces a geometric valley framework to unify instanton properties, resolves Borel ambiguities, and reveals a novel supersymmetry in quantum systems.

Findings

Unified perturbation and non-perturbation calculations

Confirmed large order behavior up to 478th order

Discovered a new supersymmetry consistent with non-renormalization

Abstract

The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A `proof' of Bogomolny's ``trick'' is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schr\"{o}dinger equation.