(Colloquium) What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles
Abstract: This is an introductory level review of recent applications of resurgent trans-series and Picard-Lefschetz theory to quantum mechanics and quantum field theory. Resurgence connects local perturbative data with global topological structure. In quantum mechanical systems, this program provides a constructive relation between different saddles. In quantum field theory, such as sigma models compactified on a circle, neutral bions provide a semi-classical interpretation of the elusive IR-renormalon of 't Hooft, and fractional kink instantons lead to the non-perturbatively induced mass gap, of order of the strong scale. In the recent few years, we learned that in the path integral formulation of quantum mechanics, saddles must be found by solving the holomorphic Newton's equation in the inverted (holomorphized) potential. Some saddles are complex, multi-valued, and even singular, but of finite action, and their inclusion is strictly necessary to prevent inconsistencies.