Complexified path integrals, exact saddles and supersymmetry

TL;DR

This paper demonstrates that complexification of the path integral and inclusion of complex saddles are essential for accurate semi-classical analysis in supersymmetric quantum mechanics, revealing new exact saddles and topological insights.

Contribution

It introduces the necessity of complex saddles in semi-classical analysis and uncovers new exact complex solutions in supersymmetric quantum mechanics.

Findings

Complex saddles are essential for consistent semi-classical analysis.

New exact complex saddle solutions are identified.

Multi-valued actions relate to a hidden topological angle.

Abstract

In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semi-classical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of $\pi$ in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.