A very successful and important application of gauge/gravity duality has emerged in the context of hydrodynamics. In generalisation of the dynamics of fluids, the term hydrodynamics generically refers to an effective field theory describing long-range, low-energy fluctuations about equilibrium.

Recently, experimental evidence has accumulated that the quark–gluon plasma observed in heavy-ion collision experiments is best described by a strongly coupled relativistic fluid, rather than by a gas of weakly interacting particles. Strongly coupled fluids are intrinsically difficult to describe by standard methods. This explains the success of applying gauge/gravity duality to this area of physics. In particular, gauge/gravity duality has made predictions of universal values of certain transport coefficients in strongly coupled fluids. The most famous example of this is the ratio of shear viscosity over entropy density, which takes a very small value. Beyond these results, gauge/gravity duality has provided a fresh look at relativistic hydrodynamics, for which many new non-trivial properties have been uncovered using the fluid/gravity correspondence.

We will describe these results in some detail. The starting point is to introduce linear response theory and Green's functions which respect the causal structure. Then we move on to an introduction to relativistic hydrodynamics. We consider the energy-momentum tensor and a conserved current and their dissipative contributions in an expansion in derivatives of fluctuations. We define the associated first-order transport coefficients and subsequently relate them to the retarded Green's function by virtue of appropriate Green–Kubo relations. This provides a link between macroscopic hydrodynamic properties and microscopic physics as described by the Green's functions. Using gauge/gravity duality methods to evaluate the relevant Green's functions, we compute the charge diffusion constant and the shear viscosity.

QCD at finite temperature and density

A crucial aspect of the strong interaction as decribed by QCD is the study of its properties at finite temperature and density. In the past decade, significant progress has been achieved towards understanding the phase structure of the strong interaction, both experimentally and theoretically. However, many questions, in particular about the detailed structure of the QCD phase diagram, remain open. To a large extent, this is due to the strong coupling nature of QCD in the relevant energy range.

Phase diagram of QCD

QCD has a very non-trivial phase diagram which is only partially understood both experimentally and theoretically. The picture which is generally believed to emerge is shown schematically in figure 14.1.

As a central feature of this diagram, it is generally expected that there is a deconfinement phase transition from bound states at low temperature and chemical potential to deconfined quarks and gluons at high temperature and chemical potential. The order of this phase transition, which is expected to end in a critical point, denoted by a black dot, is not clear at present. Only at very low μ it is known that there is merely a crossover from confinement to deconfinement when increasing the temperature. Moreover, experiments at the RHIC accelerator in Brookhaven, as well as more recently at the Large Hadron Collider (LHC) at CERN, Geneva, strongly suggest that the quark–gluon plasma observed at high temperatures is still a strongly coupled state of matter for which a hydrodynamical description is appropriate. At low temperatures, when increasing the chemical potential, a phase of dense nuclear matter such as found in neutron stars is reached. At very large chemical potential a colour-flavour locked (CFL) superconducting phase is expected, in which colour and flavour degrees of freedom are coupled to each other.

Quark–gluon plasma

The quark–gluon plasma is a new state of matter which has been studied experimentally in heavy ion collisions at the RHIC accelerator in Brookhaven and more recently at the LHC at CERN.

In theoretical physics, important new results have often been found by realising that two different concepts are related to each other at a deep and fundamental level. Examples of such relations are dualities which relate two seemingly different quantum theories to each other by stating that the theories are in fact equivalent. In particular, the Hilbert spaces and the dynamics of the two theories agree. From a mathematical point of view, this means that the theories are identical. However, from a physical point of view, their descriptions may differ, for instance there may be different Lagrangians for the two theories. The duality examples mentioned in box 5.1 either relate quantum field theories together, or they relate string theories together. The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT), however, is a new type of duality which relates a quantum field theory on flat spacetime to a string theory. This is particularly remarkable since string theory is a very promising candidate for a consistent theory of quantum gravity. Naively, quantum field theory on flat spacetime does not appear to be a theory of quantum gravity. However, the AdS/CFT correspondence, being a duality, implies that the two theories are equivalent. This explains why many scientists think that the AdS/CFT correspondence, discovered by Maldacena in 1997, is one of the most exciting discoveries in modern theoretical physics in the last two decades.

Moreover, the AdS/CFT correspondence is an important realisation of the holographic principle. This principle states that in a gravitational theory, the number of degrees of freedom in a given volume V scales as the surface area ∂V of that volume, as described in box 5.2. The theory of quantum gravity involved in the AdS/CFT correspondence is defined on a manifold of the form AdS × X, where AdS is the Anti-de Sitter space and X is a compact space. The quantum field theory may be thought of as being defined on the conformal boundary of this Anti-de Sitter space.

Gauge/gravity duality is a major new development within theoretical physics. It brings together string theory, quantum field theory and general relativity, and has applications to elementary particle, nuclear and condensed matter physics. Gauge/gravity duality is of fundamental importance since it provides new links between quantum theory and gravity which are based on string theory. It has led to both new insights about the structure of string theory and quantum gravity, and new methods and applications in many areas of physics. In a particular case, the duality maps strongly coupled quantum field theories, which are generically hard to describe, to more tractable classical gravity theories. In this way, it provides a wealth of applications to strongly coupled systems. Examples include theories similar to low-energy quantum chromodynamics (QCD), the theory of strong interactions in elementary particle physics, and models for quantum phase transitions relevant in condensed matter systems.

Gauge/gravity duality realises the holographic principle and is therefore referred to as holography. The holographic principle states that the entire information content of a quantum gravity theory in a given volume can be encoded in an effective theory at the boundary surface of this volume. The theory describing the boundary degrees of freedom thus encodes all information about the bulk degrees of freedom and their dynamics, and vice versa. The holographic principle is of very general nature and is expected to be realised in many examples. In many of these cases, however, the precise form of the boundary theory is unknown, so that it cannot be used to describe the bulk dynamics.

String theory, however, gives rise to a precise realisation of the holographic principle, in which both bulk and boundary theory are known: this is gauge/gravity duality. In this case, a quantum field theory at the boundary, which involves a gauge symmetry, is conjectured to be equivalent to a theory involving gravity in the bulk. Moreover, string theory provides many examples of dualities: a physical theory may generically have different equivalent formulations which are referred to as being dual to each other.

In the preceding chapters we have studied examples of very non-trivial tests of the AdS/CFT correspondence. At the same time we have seen that the AdS/CFT correspondence is a new useful approach for studying strongly coupled field theories by mapping them to a weakly coupled gravity theory. This raises the question whether a similar procedure may be used to study less symmetric strongly coupled field theories, thus generalising the AdS/CFT correspondence to gauge/gravity duality. The prototype example where such a procedure is desirable is Quantum Chromodynamics (QCD), the theory of quarks and gluons, which is strongly coupled at low energies. Although a holographic description of QCD itself is not yet available, decisive progress has been achieved in many respects. We will discuss the achievements and open questions in this direction in chapter 13. Here we begin the discussion of generalisations of the AdS/CFT correspondence in a more modest, though well-controlled and simpler, way by considering the gravity duals of N = 4 Super Yang–Mills theory deformed by relevant and marginal operators. These deformations break part of supersymmetry, and relevant operators also break conformal symmetry.

Renormalisation group flows in quantum field theory

Perturbing UV fixed points

The term interpolating flows refers to renormalisation group flows which connect an unstable UV fixed point to an IR fixed point at which the field theory is conformal again. A flow of this type is obtained for instance by perturbing the theory at a UV fixed point by a relevant or marginal operator. Marginal operators typically lead to a line of fixed points, while relevant operators generate a genuine RG flow, which may end at an IR fixed point. A further issuse is whether the theory flows to a confining theory in the IR, as we will discuss in more detail in chapter 13. A field theory example of an interpolating flow will be given in section 9.1.3 below.

Solving interacting quantum (field) theories exactly for all values of the coupling constant, and not just for very small coupling constant where perturbation theory is applicable, is a long-standing open problem of theoretical physics. By exactly solving we mean diagonalising the corresponding Hamiltonian, such that both eigenstates and eigenvalues are known explicitly. This is an extraordinarily difficult task: for instance, for QCD this would mean finding the complete mass spectrum from first principles from the QCD Lagrangian using analytical methods, as well as the associated eigenstates. This is certainly impossible at present for such a complicated theory.

The same problem occurs also for quantum mechanics. For most quantum systems, the complete set of eigenstates is not known. However, within quantum mechanics there are some cases where an exact solution is possible: the harmonic oscillator and the hydrogen atom, for instance. These systems should be viewed as toy models since they are ideal approximations to systems realised in nature. For instance, to describe real oscillators in solids, higher order interaction terms have to be added.

In quantum field theory there are also exactly solvable toy models: however, they are defined in low spacetime dimensions, for instance in d = 1 + 1. An example is the Thirring model which is integrable in the sense that it has an infinite number of conserved quantities. Are there any interacting exactly solvable quantum field theories also in 3+1 dimensions? The surprising answer is yes: there is a large amount of evidence that N = 4 Super Yang– Mills theory has an integrable structure, at least in the planar (large N) limit. The evidence for such an integrable structure at strong coupling has been found using the AdS/CFT correspondence. Since N = 4 Super Yang–Mills is a conformal field theory, in the planar limit the theory is characterised completely by the scaling dimensions Δ of the composite local operators built from products of elementary fields.