Introduction to integrability and one-point functions in N=4 supersymmetric Yang–Mills theory and its defect cousin

This chapter gives a basic introduction to N 7equals; 4 SYM theory and the integrability of its planar spectral problem as seen from the perspective of a recent development, namely the AdS/dCFT correspondence and the application of integrability to the study of one-point functions in a defect CFT version of the theory.

Integrability in 2D fields theory/sigma-models

This is a two-part course about the integrability of two-dimensional non-linear sigma models (2D NLSM). In the first part general aspects of classical integrability are discussed, based on the O(3) and O(4) sigma-models and the field theories related to them. The second part is devoted to the quantum 2D NLSM. Among the topics considered are: basic facts of conformal field theory, zero-curvature representations, integrals of motion, one-loop renormalizability of 2D NLSM, integrable structures in the so-called cigar and sausage models, and their RG flows. The text contains a large number of exercises of varying levels of difficulty.

Integrability in statistical physics and quantum spin chains

This chapter illustrates basic concepts of quantum integrable systems on two important models of statistical physics: the Q-state Potts model and the O(n) model. Both models are transformed into loop and vertex models that provide representations of the dense and dilute Temperley–Lieb algebras. The identification of the corresponding integrable R-matrices leads to the solution of both models by the algebraic Bethe Ansatz technique. Elementary excitations are discussed in the critical case and the link to conformal field theory in the thermodynamic limit is established. The concluding sections outline the solution of a specific model of the theta point of collapsing polymers, leading to a continuum limit with a non-compact target space.

Integrability in sigma-models

The following topics are covered in this chapter: (1) Homogeneous spaces, (2) Classical integrability of sigma-models in two dimensions, (3) Topological terms, (4) Background-field method and beta-function, (5) S-matrix bootstrap in the O(N) model, (6) Supersymmetric coset models and strings on AdS(d) x X.

Localization and N=2 supersymmetric field theory

The lectures give an introduction to supersymmetric gauge theories from a mathematical perspective. Basic notions about Kähler and special Kähler geometry, and electric–magnetic duality are introduced. Supersymmetry and N = 1 and N = 2 supersymmetric gauge theories are defined and described in detail. The last section deals with the Seiberg–Witten integrable system and Hitchin systems

Spectrum of N=4 supersymmetric Yang–Mills theory and the quantum spectral curve

This chapter describes the recently developed quantum spectral curve (QSC) approach for the non-perturbative planar spectrum of integrable N=4 supersymmetric Yang–Mills gauge theory in a pedagogical way, starting from the harmonic oscillator and avoiding a long historical path. Many examples and exercises are provided. At the end a list is given of recent and possible future applications of the QSC.

Applications of integrable models in condensed matter and cold atom physics

This chapter starts with a discussion of approximate physical realizations of 1+1-dimensional integrable models in solids and systems of ultra-cold trapped atoms. It then turns to local properties of energy eigenstates away from the edges of the spectrum. In generic models these states are thermal, while in integrable models non-thermal states with finite entropy densities coexist with thermal states. It discusses how to construct these atypical states by means of the Bethe Ansatz in the Heisenberg model. Finally, it outlines the crucial role these states play in describing the stationary states reached at late times after quantum quenches to integrable theories.

Introduction to scattering amplitudes

This chapter covers the new on-shell methods that have been developed over the past twenty years for computing scattering amplitudes in quantum field theory. These methods break free from the traditional approach of Feynman diagrams. The chapter covers a subset of topics, setting up the basic kinematics, spinor helicities, spinor products, and the calculation of tree amplitudes.

Lectures on the holographic duality of gauge fields and strings

This chapter gives a pedagogical review of the holographic duality between string theory and quantum field theory. The main focus is on the duality of maximally supersymmetric Yang–Mills gauge theory in four dimensions with string theory in asymptotically anti-de Sitter backgrounds. This duality is motivated using the large N expansion in the rank of the gauge group, as well as the D-brane solution for the AdS string theory background. The computation of Wilson loops on both sides of the duality is given as an example.

Three-point functions in N=4 supersymmetric Yang–Mills theory

This is a review of the integrability-based approach to the three-point function in N = 4 supersymmetric Yang–Mills theory. We first discuss the computation of the structure constant at weak coupling and show that the result can be recast as a sum over partitions of the rapidities of the magnons. We then introduce a non-perturbative framework, called the ‘hexagon approach’, and explain how one can use the symmetries and integrability to determine the structure constants.