Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems

Abstract

We explore a one parameter $\zeta$-deformation of the quantum-mechanical Sine-Gordon and Double-Well potentials which we call the Double Sine-Gordon (DSG) and the Tilted Double Well (TDW), respectively. In these systems, for positive integer values of $\zeta$, the lowest $\zeta$ states turn out to be exactly solvable for DSG - a feature known as Quasi-Exact-Solvability (QES) - and solvable to all orders in perturbation theory for TDW. For DSG such states do not show any instanton-like dependence on the coupling constant, although the action has real saddles. On the other hand, although it has no real saddles, the TDW admits all-orders perturbative states that are not normalizable, and hence, requires a non-perturbative energy shift. Both of these puzzles are solved by including complex saddles. We show that the convergence is dictated by the quantization of the hidden topological angle. Further, we argue that the QES systems can be linked to the exact cancellation of real and complex non-perturbative saddles to all orders in the semi-classical expansion. We also show that the entire resurgence structure remains encoded in the analytic properties of the $\zeta$-deformation, even though exactly at integer values of $\zeta$ the mechanism of resurgence is obscured by the lack of ambiguity in both the Borel sum of the perturbation theory as well as the non-perturbative contributions. In this way, all of the characteristics of resurgence remains even when its role seems to vanish, much like the lingering grin of the Cheshire Cat. We also show that the perturbative series is Self-resurgent -a feature by which there is a one-to-one relation between the early terms of the perturbative expansion and the late terms of the same expansion -which is intimately connected with the Dunne-Ünsal relation. We explicitly verify that this is indeed the case.

41 pages

References in corpus (16)

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• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
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• Monte Carlo study of real time dynamics
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets
• Global symmetries, volume independence and continuity
• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}{P}^{N-1}$ Models
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Resurgence in $\eta$-deformed Principal Chiral Models
• Real-Time Feynman Path Integral Realization of Instantons

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• A Primer on Resurgent Transseries and Their Asymptotics
• Quasinormal modes of black holes and Borel summation
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• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Multi-flavor massless QED${}_2$ at finite densities via Lefschetz thimbles
• Generating functions and large-charge expansion of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms
• Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories
• Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $\mathcal{N}=2$ field theories
• Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls
• Resurgence in the Bi-Yang-Baxter Model
• Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves
• To the cusp and back: Resurgent analysis for modular graph functions
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Lectures on nonperturbative QCD ( Nonperturbative Topological Phenomena in QCD and Related Theories)
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Resumming perturbative series in the presence of monopole bubbling effects
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• Resurgence of the Effective Action in Inhomogeneous Fields
• Asymptotics in an Asymptotic CFT
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Resurgence of the Tilted Cusp Anomalous Dimension
• Monopole-vortex continuity of $\mathcal{N}=1$ super Yang-Mills theory on ${\mathbb{R}}^2\times S^1\times S^1$ with 't Hooft twist
• Resurgence in 2-dimensional Yang-Mills and a genus altering deformation
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• Resurgence of one-point functions in a matrix model for 2D type IIA superstrings
• Resurgence and $1/N$ Expansion in Integrable Field Theories
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis
• Bions and Instantons in Triple-well and Multi-well Potentials
• Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation
• A duality of the classical action yields a reflection symmetry of the quantum energy spectrum
• Exact-WKB Analysis of Two-level Floquet Systems
• Angular correlation of the two gamma rays produced in the thermal neutron capture on gadolinium-155 and gadolinium-157
• Exact WKB in all sectors II: Potentials with non-degenerate saddles