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

Understand

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).