What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles

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. For example, in certain cases it has been shown that all information around the instanton saddle is encoded in perturbation theory around the perturbative saddle. In quantum field theory, such as sigma models compactified on a circle, neutral bions provide a semi-classical interpretation of the elusive IR-renormalon, and fractional kink instantons lead to the non-perturbatively induced gap, of order of the strong scale. 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. The multi-valued saddles enter either via resurgent cancellations, or their phase is tied with a hidden topological angle. We emphasize the importance of the destructive/constructive interference effects between equally dominant saddles in the Lefschetz thimble decomposition. This is especially important in the context of the sign problem.