Resurgence in Deformed Integrable Models

Abstract

Resurgence has been shown to be a powerful and even necessary technique to understand many physical system. The study of perturbative methods in general quantum field theories is hard, but progress is often possible in reduced settings, such as integrable models. In this thesis, we study resurgent effects in integrable deformations of two-dimensional σ-models in two settings.First, we study the integrable bi-Yang-Baxter deformation of the SU (2) principal chiral model (PCM) and find finite action uniton and complex uniton solutions. Under an adiabatic compactification on an S 1 , we obtain a quantum mechanical system with an elliptic Lamé-like potential. We perform a perturbative calculation of the ground state energy of this quantum mechanical system to large orders obtaining an asymptotic series.Using the Borel-Padé technique, we determine that the locations of branch cuts in the Borel plane match the values of the uniton and complex uniton actions. Therefore, we can match the non-perturbative contributions to the energy with the uniton solutions which fractionate upon adiabatic compactification. An off-shoot of the WKB analysis, is to identify the quadratic differential of this deformed PCM with that of an N = 2 Seiberg-Witten theory. This can be done either as an N f = 4 SU (2) theory or as an elliptic SU (2) × SU (2) quiver theory. The mass parameters of the gauge theory are given in terms of the bi-Yang-Baxter deformation parameters.Second, we perform a perturbative expansion of the thermodynamic Bethe ansatz (TBA) equations of the SU (N ) λ-model with WZW level k in the presence of a chemical potential. This is done with its exact S-matrix and the recently developed techniques [1, 2] using a Wiener-Hopf decomposition, which involve a careful matching of bulk and edge ansätze. We determine the asymptotic expansion of this series and compute its renormalon ambiguities in the Borel plane. The analysis is supplemented by a parallel solution of the TBA equations that results in a transseries. The transseries comes with an ambiguity that is shown to precisely match the Borel ambiguity. It is shown that the leading IR renormalon vanishes when k is a divisor of N .ii Contents Foreword viAcknowledgements ix Declaration of Authorship x Resurgence for Laymen xiIn the first place, my gratitude extends to Dan, who gave me not only the chance to embark on this project, but without whose mentor-ship, guidance and encouragement this thesis would have fallen flat on many more occasions.Secondly, I would like to thank S. Prem Kumar, Saskia Demulder and Giacomo Piccinini for useful conversations on this and related topics, as well as my other collaborators Leonardo Santilli, Carlos Nuñez, Andrea Legramandi and Mohammad Akhond. I am supported by The Royal Society through grant RGF\R1 \180087 Generalised Dualities, Resurgence and Integrability. In addition, I wish to thank my other friends in the UK. I thank Mohammad for being an animated speaker and Karol for being a supportive listener. Freya for being a dreamer and Laura fo...