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

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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.

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Authors and Affiliations

  1. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02138, USA

    Can Kozçaz

  2. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Can Kozçaz

  3. Department of Physics, North Carolina State University, Raleigh, NC, 27695, USA

    Tin Sulejmanpasic & Mithat Ünsal

  4. Philippe Meyer Institute for Theoretical Physics, Physics Department, École Normale Supérieure, PSL Research University, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Tin Sulejmanpasic

  5. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY, 11973, USA

    Yuya Tanizaki

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  1. Can Kozçaz
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Kozçaz, C., Sulejmanpasic, T., Tanizaki, Y. et al. Cheshire Cat Resurgence, Self-Resurgence and Quasi-Exact Solvable Systems. Commun. Math. Phys. 364, 835–878 (2018). https://doi.org/10.1007/s00220-018-3281-y

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