Related papers: Picard-Lefschetz theory and alien calculus: a case…
In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series,…
We revisit the path-integral approach to the wave function of the Universe by utilizing Lefschetz thimble analyses and resurgence theory. The traditional Euclidean path-integral of gravity has the notorious ambiguity of the direction of…
We review \'Ecalle's formalism of minors, natural-majors and real-majors, and provide explicit formulas in the Borel plane that show the resurgence of the exponential of the Stirling series. We also discuss its Stokes phenomena in the…
Picard--Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its…
We study three dimensional $\mathcal{N}=2$ supersymmetric abelian gauge theories with various matter contents living on a squashed sphere. In particular we focus on two problems: firstly we perform a Picard-Lefschetz decomposition of the…
We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a…
The present text is an introduction to \'Ecalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for…
Picard-Lefschetz theory is applied to solutions of the Helmholtz equation, formulated in terms of sums of integrals of a proper-time, or `einbein', wave function $\Psi(\Lambda) = \exp(i\mathbb S(\Lambda))$ along complex contours bounded by…
We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach…
We study the resurgence structure of the topological string partition function, with an emphasis on the Borel analysis of the instanton amplitudes. To this end, we introduce a differential operator that implements the pointed alien…
In recent work, we introduced Picard-Lefschetz theory as a tool for defining the Lorentzian path integral for quantum gravity in a systematic semiclassical expansion. This formulation avoids several pitfalls occurring in the Euclidean…
We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the…
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the…
In this work, we first solve complex Morse flow equations for the simplest case of a bosonic harmonic oscillator to discuss localization in the context of Picard-Lefschetz theory. We briefly touch on the exact non-BPS solutions of the…
Quantum cosmology aims at elucidating the beginning of our Universe. Back in early 80's, Vilenkin and Hartle-Hawking put forward the "tunneling from nothing" and "no boundary" proposals. Recently there has been renewed interest in this…
It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of of…
Gompf's end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial ${\bf R^4}$ topology, but for which the exotic differentiable structure is confined to a region which…
We propose a point of view on resurgence theory based on the study of perverse sheaves on the complex line carrying an algebraic structure with respect to additive convolution. In particular, we lift the concept of alien derivatives…
We discuss all-order transseries in one of the simplest quantum mechanical systems: a U(1) symmetric single-degree-of-freedom system with a first-order time derivative term. Following the procedure of the Lefschetz thimble method, we…
Deforming the domain of integration after complexification of the field variables is an intriguing idea to tackle the sign problem. In thimble regularization the domain of integration is deformed into an union of manifolds called Lefschetz…