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Journal of physics / A 55(44), 443001 (2022). doi:10.1088/1751-8121/ac8380, This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments., Published by IOP Publ., Bristol

This review describes a link between Lax operators, embedded surfaces and Thermodynamic Bethe Ansatz equations for integrable quantum field theories. This surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off-critical generalisation of the ODE/IM correspondence, which initially involved only conformal invariant quantum field theories. We will mainly focus of the KdV and sinh-Gordon models. However, various aspects of other interesting systems, such as affine Toda field theories and non-linear sigma models, will be mentioned. We also discuss the implications of these ideas in the AdS/CFT context, involving minimal surfaces and Wilson loops. This work is a follow-up of the ODE/IM review published more than ten years ago by JPA, before the discovery of its off-critical generalisation and the corresponding geometrical interpretation. (Partially based on lectures given at the Young Researchers Integrability School 2017, in Dublin.), Comment: Partially based on lectures given at the Young Researchers Integrability School 2017,in Dublin. 62 pages, 8 figures. Minor amendments. Version published in J. Phys A "Special Issue on Recent Advances in AdS/CFT Integrability"

Journal of physics / A 53(45), 453002 (2020). doi:10.1088/1751-8121/abb0fe, A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7���8 Sept. 2017. We attempt to provide a broad, bird���s-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof., Published by IOP Publ., Bristol

One of the simplest examples of a \( \mathcal{P}\mathcal{T} \)-symmetric quantum system is the scaling Yang-Lee model, a quantum field theory with cubic interaction and purely imaginary coupling. We give a historical review of some facts about this model in d ≤ 2 dimensions, from its original definition in connection with phase transitions in the Ising model and its relevance to polymer physics, to the role it has played in studies of integrable quantum field theory and \( \mathcal{P}\mathcal{T} \)-symmetric quantum mechanics. We also discuss some more general results on \( \mathcal{P}\mathcal{T} \)-symmetric quantum mechanics and the ODE/IM correspondence, and mention applications to magnetic systems and cold atom physics.

We show that SU(n) Bethe Ansatz equations with arbitrary `twist' parameters are hidden inside certain nth order ordinary differential equations, and discuss various consequences of this fact., 19 pages and no figures. LaTex2e, uses amssymb and cite.V2: Note added, minor typos corrected

We explore boundary scattering in the sine-Gordon model with a non-integrable family of Robin boundary conditions. The soliton content of the field after collision is analysed using a numerical implementation of the direct scattering problem associated with the inverse scattering method. We find that an antikink may be reflected into various combinations of an antikink, a kink, and one or more breathers, depending on the values of the initial antikink velocity and a parameter associated with the boundary condition. In addition we observe regions with an intricate resonance structure arising from the creation of an intermediate breather whose recollision with the boundary is highly dependent on the breather phase., v3: typos fixed in Journal of Physics A, Volume 49, Number 16, 2016

We study kink-antikink collisions in the one-dimensional non-integrable scalar phi^6 model. Although the single-kink solutions for this model do not possess an internal vibrational mode, our simulations reveal a resonant scattering structure, thereby providing a counterexample to the standard belief that existence of such a mode is a necessary condition for multi-bounce resonances in general kink-antikink collisions. We investigate the two-bounce windows in detail, and present evidence that this structure is caused by existence of bound states in the spectrum of small oscillations about a combined kink-antikink configuration., Comment: 4 pages, 5 ps figures. v2: 3 references added

The mergings of energy levels associated with the breaking of PT symmetry in the model of Bender and Boettcher, and in its generalisation to incorporate a centrifugal term, are analysed in detail. Even though conventional WKB techniques fail, it is shown how the ODE/IM correspondence can be used to obtain a systematic approximation scheme which captures all previously-observed features. Nonperturbative effects turn out to play a crucial role, governing the behaviour of almost all levels once the symmetry-breaking transition has been passed. In addition, a novel treatment of the radial Schrodinger equation is used to recover the values of local and non-local conserved charges in the related integrable quantum field theories, without any need for resummation even when the angular momentum is nonzero., 29 pages, Latex, many figures. v2: sign typos fixed, and a reference added

We study the P"oschl-Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is a interesting interpretation in terms of finite lattice spin (L-2)/2 XXZ quantum chain with Delta= cos(pi/L) (for free-free boundary conditions), or Delta=-cos(pi/L) (for periodic boundary conditions). This result generalises the findings of Fridkin, Stroganov and Zagier. We also discuss the continuous (field theory) limit of these systems in view of the so-called ODE/IM correspondence., 18+1 pages, 4 figures, typo corrected, references added

The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems., Comment: 32 pages, 12 figures, Latex2e, amssymb, cite and graphicx. v2: References, and a new section with two further dualities, added; typos corrected. v3: Ref [8] corrected. v4: 5 more references,and more typos corrected v5: New title, note added at end and 3 new refs

The appearances of complex eigenvalues in the spectra of PT-symmetric quantum-mechanical systems are usually associated with a spontaneous breaking of PT. In this letter we discuss a family of models for which this phenomenon is also linked with an explicit breaking of supersymmetry. Exact level-crossings are located, and connections with N-fold supersymmetry and quasi-exact solvability in certain special cases are pointed out., 11 pages, 12 figures, Latex2e, amssymb, cite and graphicx. v2: a comment and two references added. v3: minor changes, and two references updated

We find the spectrum of boundary bound states for the sine-Gordon model with Dirichlet boundary conditions, closing the bootstrap and providing a complete description of all the poles in the boundary reflection factors. The boundary Coleman-Thun mechanism plays an important role in the analysis. Two basic lemmas are introduced which should hold for any 1+1-dimensional boundary field theory, allowing the general method to be applied to other models., Comment: 26 pages, LaTeX2e, uses amsfonts.sty. V2: typos corrected, and equation numbering changed to agree with published version

The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for $D(E)$. Generalising this result, we conjecture a relationship between the $x^{2M}$ anharmonic oscillators and the $A_{2M-1}$ TBA systems. Finally, spectral determinants for general $|x|^{\alpha}$ potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models., Comment: 11 pages, Latex 2e, uses amssymb, cite

We point out some problems with the previously-proposed phase diagram of the $Z_6$ spin models. Consideration of the diagram near to the decoupling surface using both exact and approximate arguments suggests a modification which remedies these deficiencies. With the aid of a new parametrisation of the phase space, we study the models numerically, with results which support our conjectures., Comment: 10 pages, Latex 2e, 4 figures, uses epsf, amssymb, cite

The theory of classically integrable nonlinear wave equations, and the Bethe Ansatz systems describing massive quantum field theories defined on an infinite cylinder, are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper we shall describe this link for the case of the classical and quantum versions of the (Tzitz\'eica-)Bullough-Dodd model., Comment: 20 pages

A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than two. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain second-order cases, and extended to the higher-order problems., 14 pages, Latex

Within the framework of the ODE/IM correspondence, we show that the minimal conformal field theories with c, 12 pages, 3 figures; typos corrected and reference added

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained., 102 pages, 27 figures. v2: added references