The curious incident of multi-instantons and the necessity of Lefschetz thimbles

TL;DR

This work addresses how supersymmetry shapes multi-instanton amplitudes in semi-classical analyses by complexifying quasi-zero modes and employing Lefschetz thimbles via Picard-Lefschetz theory. Focusing on ${\cal N}=2$ supersymmetric quantum mechanics, the authors decompose the instanton–anti-instanton contribution into fermion-correlated and scalar-correlated channels, showing that their naive leading-order signs and orders in coupling cancel when integrated on thimbles due to a hidden topological angle. The main result is that the leading nonperturbative correction to the ground-state energy vanishes, consistent with unbroken SUSY, and that thimble integration provides a necessary, universal framework for higher-order semi-classics beyond SUSY. The analysis also clarifies the relation to the BZJ prescription and suggests broader implications for instanton physics in QFT, including the absence or cancellation of certain contributions in theories with multiple fermionic zeromodes.

Abstract

We show that compatibility of supersymmetry with exact semi-classics demands that in calculating multi-instanton amplitudes, the "separation" quasi-zeromode must be complexified and the integration cycles must be found by using complex gradient flow (or Picard-Lefschetz equations.) As a non-trivial application, we study $\mathcal N=2$ extended supersymmetric quantum mechanics. Even though in this case supersymmetry is unbroken, the instanton-anti-instanton amplitude (naively calculated) seems to contribute to the ground state energy. We show, however, that the instanton-anti-instanton event consists of two parts: a fermion-correlated and a scalar-correlated event. Although both of these contributions are naively of the same sign and the latter is superficially higher order in the perturbative coupling, we show that the two contributions exactly cancel when they are evaluated on Lefschetz thimbles due to their relative Hidden Topological Angles (HTAs). This gives strong evidence that the semi-classical expansion using Lefschetz thimbles is not only a meaningful prescription for higher order semi-classics, but a necessary one. This deduction seems to be universal and applicable to both supersymmetric and non-supersymmetric theories. In conclusion we speculate that similar conspiracies are responsible for the non-formation of certain molecular contributions in theories where instantons have more than two fermionic zeromodes and do not contribute to the superpotential.

Figures (2)

• Figure 1: Top: a fermion-correlated $I\bar{I}$ event, contributing the first term in Eq. (\ref{['eq:groundstate']}). Bottom: a scalar-correlated $I\bar{I}$ event, contributing the second term in Eq. (\ref{['eq:groundstate']}). The two contributions are proportional to different powers of the perturbative ($g \ll \omega^3$) coupling $g$. In QM, the diagrams are intended to schematically represent the lifting of fermion zero modes by the two mechanisms. In QFT, one can associate the (anti-)instanton vertices with effective 't Hooft interactions and the lines connecting them to free (away from the instanton cores) scalar and fermion propagators.
• Figure 2: The steepest descent cycles for the fermion-correlated channel vs. scalar correlated channels. The blue cycle is the naive cycle in which the separation between the instanton and anti-instanton is interpreted as real. A result compatible with supersymmetry only comes about if we use the critical point cycles.