Introduction to classical and quantum integrability

Retore, Ana L.

doi:10.1088/1751-8121/ac5a8e

Cited by 13 publications

(18 citation statements)

References 97 publications

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“…Indeed, a systematic approach to construct Lax pairs in full generality is lacking. 10 Proving that a system is not classically integrable is therefore not straightforward as well, and it is usually only done by showing that the dynamics exhibits chaotic behaviour.…”

Section: Lax Pair Formulationmentioning

confidence: 99%

“…See e.g. [31,32] and the recent notes [10], as well as a few more comments in the next section. For us the Sklyanin relations are already sufficient in the sense that they ensure strong classical integrability.…”

Section: Charges In Involutionmentioning

confidence: 99%

“…Imagine that one wants to reverse the logic, and not check integrability for a particular Lagrangian or Hamiltonian theory, but rather wants to find (or even classify) consistent integrable structures, in particular r-matrices, and construct from them the charges and the S-matrix (see e.g. [10,31]). We have seen here that properties of large classes of classical r-matrices, namely those that solve the CYBE (3.34), will strongly restrict the quantum integrable structure through the QYBE.…”

Section: Comments On Quantum Integrability and Properties Of R-matricesmentioning

confidence: 99%

“…to quantum integrability (symmetries, S-matrices and Bethe Ansatz techniques) is also given by [9]. Recent notes are [10] on classical and quantum integrability and [11] on integrable deformations of sigma models.. Finally, we recommend also the notes [12] which shows the close connection between integrable deformations of sigma models and generalised worldsheet dualities (i.e.…”

Section: Introductionmentioning

confidence: 97%

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Modave Lectures on Classical Integrability in $2d$ Field Theories

Driezen¹

2021

Preprint

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These lecture notes are based on a blackboard course given at the XVII Modave Summer School in Mathematical Physics held from 13-17 September 2021 in Brussels (Belgium), and aimed at Ph.D. students in High Energy Theoretical Physics. We start with introducing classical integrability in finite-dimensional systems to set the stage for our main purpose: introducing two-dimensional classical field theories which are integrable. We focus on their zero-curvature formulation through the so-called Lax connection, which ensures the existence of an infinite tower of conserved charges. We then move on to their Poisson bracket structure, known as the Sklyanin or Maillet structure, which ensures complete classical integrability. All the concepts that we encounter will be illustrated with the integrable Principal Chiral Model, which is the canonical sigma-model that appears (or its generalisations) on the worldsheet of many string backgrounds. Along the way we briefly comment on the properties of integrability at the quantum level.

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Section: Lax Pair Formulationmentioning

confidence: 99%

Section: Charges In Involutionmentioning

confidence: 99%

Section: Comments On Quantum Integrability and Properties Of R-matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Modave Lectures on Classical Integrability in $2d$ Field Theories

Driezen¹

2021

Preprint

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“…For an extensive review of classical Liouville integrability in Hamiltonian systems and 2d integrable models we refer the reader to the lecture notes [84][85][86][87][88]. However, we will highlight some aspects here relevant to the remainder of the thesis.…”

Section: Integrabilitymentioning

confidence: 99%

Resurgence in Deformed Integrable Models

Schepers¹

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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...

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Integrable coupled bosonic massive Thirring model and its nonlocal reductions

Basu-Mallick,

Sinha

2024

J. High Energ. Phys.

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A coupled bosonic massive Thirring model (BMTM), involving an interaction between the two independent spinors, is introduced and shown to be integrable. By incorporating suitable reductions between the field components of the coupled BMTM, five novel integrable models with various type of nonlocal interactions are constructed. Lax pairs satisfying the zero curvature condition are obtained for the coupled BMTM and for each of the related nonlocal models. An infinite number of conserved quantities are derived for each of these models which confirms the integrability of the systems. It is shown that the coupled BMTM respects important symmetries of the original BMTM such as parity, time reversal, global U(1)-gauge and the proper Lorentz transformations. Similarly, all the nonlocal models obtained from the coupled BMTM remain invariant under combined operation of parity and time reversal transformations. However, it is found that only one of the nonlocal models is invariant under proper Lorentz transformation and two other models are invariant under global U(1)-gauge transformation. By using ultralocal Poisson bracket relations among the elements of the Lax operator, it is shown that the coupled BMTM and one of the nonlocal models are completely integrable in the Liouville sense.

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Introduction to classical and quantum integrability

Cited by 13 publications

References 97 publications

Modave Lectures on Classical Integrability in $2d$ Field Theories

Modave Lectures on Classical Integrability in $2d$ Field Theories

Resurgence in Deformed Integrable Models

Integrable coupled bosonic massive Thirring model and its nonlocal reductions

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