Critical Points at Infinity, Non-Gaussian Saddles, and Bions

TL;DR

This paper explores how complex field configurations, especially non-BPS multi-instantons and critical points at infinity, influence non-perturbative phenomena in quantum theories, revealing new insights into their contributions via Picard-Lefschetz theory.

Contribution

It introduces the interpretation of non-BPS multi-instanton configurations as critical points at infinity and analyzes their Lefschetz thimbles with non-Gaussian quasi-zero modes, advancing the understanding of semi-classical contributions.

Findings

Critical points at infinity contribute to non-perturbative effects.

Non-Gaussian quasi-zero mode thimbles play a key role in these contributions.

In supersymmetric theories, the semi-classical contribution from critical points at infinity vanishes, but non-trivial effects remain.

Abstract

It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of of Picard-Lefschetz theory. In this work we compute the contribution from non-BPS multi-instanton configurations, such as instanton-anti-instanton $[{I}\bar{I}]$ pairs, and argue that these contributions should be interpreted as exact critical points at infinity. The Lefschetz thimbles associated with such critical points have a specific structure arising from the presence of non-Gaussian, quasi-zero mode (QZM), directions. When fermion degrees of freedom are present, as in supersymmetric theories, the effective bosonic potential can be written as the sum of a classical and a quantum potential. We show that in this case the semi-classical contribution of the critical point at infinity vanishes, but there is a non-trivial contribution that arises from its associated non-Gaussian QZM-thimble. This approach resolves several puzzles in the literature concerning the semi-classical contribution of correlated $[{I}\bar{I}]$ pairs. It has the surprising consequence that the configurations that dominate the expansion of observables, and the critical points that define the Lefschetz thimble decomposition need not be the same, a feature not present in the traditional Picard-Lefschetz approach.