Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

Abstract

Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about non-perturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NP-saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the $SU(N)$ principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel `fracton' saddle points, which turn out to be the fractionalized constituents of previously observed unstable `uniton' saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of 't Hooft's renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.

80 pages, 13 figures; v2 - various clarifications and minor corrections

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• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• Fractional instantons and bions in the O(N) model with twisted boundary conditions
• Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories
• Interpolating function and Stokes Phenomena
• Tracy-Widom distribution as instanton sum of 2D IIA superstrings
• Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories
• Resurgence and Dynamics of O(N) and Grassmannian Sigma Models