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

TL;DR

This paper demonstrates that complexifying integration cycles via Lefschetz thimbles is essential for consistent semi-classical calculations in supersymmetric quantum mechanics, revealing cancellations that preserve supersymmetry.

Contribution

It introduces the necessity of using Lefschetz thimbles and complexified quasi-zero modes for accurate multi-instanton amplitude calculations in supersymmetric theories.

Findings

Instanton-anti-instanton contributions cancel on Lefschetz thimbles due to Hidden Topological Angles.

Complexification of integration cycles ensures consistent semi-classical expansions.

The approach is universal, applicable to both supersymmetric and non-supersymmetric theories.

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.