Tying up instantons with anti-instantons

TL;DR

This paper explores exact solutions of complexified classical systems to improve the understanding of instanton contributions in quantum mechanics, with implications for integrable systems and black hole physics.

Contribution

It introduces a method to find honest solutions of complexified equations of motion, connecting instanton analysis with Bethe/gauge correspondence and Lefschetz thimbles.

Findings

Derived exact solutions for complexified classical systems.

Linked instanton contributions to algebraic integrable systems.

Suggested applications to black hole radiation phenomena.

Abstract

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+ years old approach by finding the honest solutions of equations of motion of the complexified classical mechanical system. These ideas originate 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 $\Omega$-deformed $B$-model. We propose the notion of the topological renormalization group.