Tying up instantons with anti-instantons

In quantizing classical mechanical systems, one often sums over the classical trajectories as in localization formulas, but also takes into account the contributions of the “instanton gas”: a set of approximate solutions of the equations of motion. This paper attempts to alleviate some of the frustrations of this 40+ year-old approach by finding the honest solutions of equations of motion of the complexified classical mechanical system. These ideas originated in the Bethe/gauge correspondence. The examples include algebraic integrable systems, from the abstract Hitchin systems to the well-studied anharmonic oscillator. We also speculate on the applications to the black hole radiation. We elucidate the relation between Lefschetz thimbles and the O-deformed B-model. We propose the notion of the topological renormalization group.