Within condensed matter physics, there are many extremely interesting physical systems which are strongly coupled. Although various approaches have been developed within condensed matter physics to deal with strongly coupled systems, there are many important physically relevant examples where a description in terms of theoretical models has not been successful. It thus appears natural to make use of gauge/gravity duality, which is very effective for describing systems at strong coupling, in this context as well. Of course, the microscopic degrees of freedom in a condensed matter system are very different from those described by a non-Abelian gauge theory at large N. For instance, these systems are non-relativistic in general. Nevertheless, the idea is to make use of universality again and to consider systems at second order phase transitions or renormalisation group fixed points, where the microscopic details may not be important. A prototype example of this scenario is given by quantum phase transitions, i.e. phase transitions at zero temperature which are induced by quantum rather than thermal fluctuations. These transitions appear generically when varying a parameter or coupling, which does not have to be small, such that the use of perturbative methods may not be possible.

In many cases, the study of models relevant to condensed matter physics involves the introduction of a finite charge density in addition to finite temperature. This applies for instance to Fermi surfaces or condensation processes. In the gauge/gravity duality context, this is obtained in a natural way by considering charged black holes of Reissner–Nordström type. Their gravity action involves additional gauge fields. Within this approach, standard thermodynamical quantities such as the free energy and the entropy may be calculated. A further important observable characterising the properties of condensed matter systems is the frequency-dependent conductivity. This is calculated in a straightforward way using gauge/gravity duality techniques.

A very instructive example of a quantum phase transition within gauge/gravity duality is obtained by using a magnetic field as the control parameter.

So far we have studied examples of the AdS/CFT correspondence which are motivated by the near-horizon limit of a stack of D-branes placed either in flat space or in a more involved geometry such as the conifold. In this chapter we will consider examples where additional D-branes are placed in the supergravity solution after the near-horizon limit has been taken. This approach has several motivations. One possible application is to wrap branes on non-trivial cycles in the geometry resulting from the near-horizon limit. Such branes correspond to soliton-like states in the dual conformal field theory. These states are non-perturbative from the point of view of the 1/N expansion. Consequently, they allow information about the stringy nature of the correspondence to be uncovered even in its weakest form, where λ and N are large. The soliton-like field theory states include the pointlike baryon vertex, one-dimensional flux tubes and higher dimensional domain walls.

Here, however, we will focus on the second important application of embedding additional D-branes into the near-horizon geometry, the flavour branes. Adding additional D-branes to the supergravity solution in the near-horizon limit gives rise to a modification of the original AdS/CFT correspondence which involves field theory degrees of freedom that transform in the fundamental representation of the gauge group. This is in contrast to the fields of N = 4 Super Yang–Mills theory which transform in the adjoint representation of the gauge group. This is particularly useful for describing strongly coupled quantum field theories which are similar to QCD, since the quark fields in QCD transform in the fundamental representation. From an anti-fundamental and a fundamental field, a gauge invariant bilinear or meson operator may be formed. The key idea is then to conjecture that the meson operators are dual to the fluctuations of flavour branes embedded in the dual supergravity background.

Scattering experiments are crucial for our understanding of the building blocks of nature. The standard model of particle physics was developed from scattering experiments, including the discovery of the weak force bosons W± and Z0, quarks and gluons, and most recently the Higgs boson.

The key observable measured in particle scattering experiments is the scattering crosssection σ. It encodes the likelihood of a given process to take place as a function of the energy and momentum of the particles involved. A more refined version of this quantity is the differential cross-sectiondσ/ dΩ: it describes the dependence of the cross-section on the angles of the scattered particles.

Interpretation of data from scattering experiments relies heavily on theoretical predictions of scattering cross-sections. These are calculated in relativistic quantum field theory (QFT), which is the mathematical language for describing elementary particles and their interactions. Relativistic QFT combines special relativity with quantum physics and is a hugely successful and experimentally well-tested framework for describing elementary particles and the fundamental forces of nature. In quantum mechanics, the probability distribution |ψ|2 = ψ * ψ for a particle is given by the norm-squared of its complex-valued wavefunction ψ. Analogously, in quantum field theory, the differential cross-section is proportional to the norm-squared of the scattering amplitudeA, dσ/ dΩ ∝ |A|2. The amplitudes A are well-defined physical observables: they are the subject of this book.

Scattering amplitudes have physical relevance through their role in the scattering cross-section. Moreover, it has been realized in recent years that amplitudes themselves have a very interesting mathematical structure. Understanding this structure guides us towards more efficient methods to calculate amplitudes. It also makes it exciting to study scattering amplitudes in their own right and explore (and exploit) their connections to interesting branches of mathematics, including combinatorics and geometry.