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. 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.

20 pages, 5 figures, The 33rd International Symposium on Lattice Field Theory

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Cited by in corpus (15)

• New Methods in QFT and QCD: From Large-N Orbifold Equivalence to Bions and Resurgence
• TBA equations and resurgent Quantum Mechanics
• Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}{P}^{N-1}$ Models
• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Thimble regularization at work: from toy models to chiral random matrix theories
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Non-BPS exact solutions and their relation to bions in $\mathbb{C}{P}^{N-1}$ models
• Resurgence in the O(4) sigma model
• Approaching the Self-Dual Point of the Sinh-Gordon model
• The two-dimensional O(3) model at nonzero density: from dual lattice simulations to repulsive bosons
• Infrared renormalon in the supersymmetric $\mathbb{C}{P}^{N-1}$ model on $\mathbb{R}\times S^1$
• Resurgent transseries $\&$ Dyson-Schwinger equations
• Quantum Yang-Mills Theory in Two Dimensions: Exact versus Perturbative
• The Perturbative Approach to Path Integrals: A Succinct Mathematical Treatment
• The Lefschetz thimble and the sign problem