1

The holographic correspondence

A detected gravitational wave signal from the merger of a binary black hole. The late-time ringdown after the black hole merger is described by damped oscillations, called quasinormal modes. Via holographic duality, this key behavior of black holes describes dissipation into a strongly interacting medium. This chapter will introduce holographic duality. Image adapted from [4] with permission.

1.1 Historical context I: Quantum matter without quasiparticles

A ubiquitous property of the higher temperature superconductors (found in the cuprates, the pnictides, and related compounds) is an anomalous ‘strange metal’ state found at temperatures T above the superconducting critical temperature, Tc [679]. Its transport properties deviate strongly from those of conventional metals described by Fermi liquid theory, and spectral probes display the absence of long-lived quasiparticle excitations. The value of Tc is determined by a balance between the free energies of the superconductor and the strange metal, and so a theory for the strange metal is a pre-requisite for any theory of Tc. This is one of the reasons for the great interest in developing a better understanding of strange metals.

Speaking more broadly, we may consider the strange metal as an important realization of a state of quantum matter without quasiparticle excitations. We can roughly define a quasiparticle as a long-lived, low energy elementary excitation of a many-body state with an ‘additive’ property: we can combine quasiparticles to create an exponentially large number of composite excitations. In Landau theory [628], the energy of a multi-quasiparticle excitation is given by sum of the energies of the quasiparticles and an additional mean-field interaction energy, dependent upon the quasiparticle density. It is important to note that the quasiparticles need not have the same quantum numbers as a single electron (or whatever happens to be the lattice degree of freedom in the model). In quantum states with long-range quantum entanglement, such as the fractional quantum Hall states, the quasiparticles are highly non-trivial combinations of the underlying electrons, and carry exotic quantum numbers such as fractional charge, and obey fractional statistics. Nevertheless, Landau’s quasiparticle framework can be used to build multi-particle excitations even in such cases.

In the states of quantum matter of interest to us here, no quasiparticle description exists. However, such a definition is unsatisfactory because it makes it essentially impossible to definitively establish whether a particular state is in a non-quasiparticle category. Given a state and a Hamiltonian, we can propose some set of possible quasiparticles, and rule them out as valid excitations. However, it is difficult to rule out all possible quasiparticles: short of an exact solution, we cannot be sure that there does not exist a ‘dual’ description of the model in which an unanticipated quasiparticle description can emerge.

A more satisfactory picture emerges when we view the issue from a dynamic perspective. We focus on one of the defining properties of a quasiparticle, that it is ‘long-lived’. Imagine perturbing the system of interest a little bit from a state at thermal equilibrium at a temperature T. If quasiparticles exist, they will be created by the perturbation; eventually they will collide with each other and establish local thermal equilibrium. We denote the time required to establish local thermal equilibrium by τφ; alternatively, we can also consider τφ to be the time over which the local quantum phase coherence is lost after the external perturbation. The upshot of this discussion is that states with quasiparticles are expected to have a long τφ, while those without quasiparticles have a short τφ. Can we make this more precise? In a Fermi liquid, a standard Fermi’s golden rule computation shows that τφ diverges as T → 0 as 1/T2. In systems with an energy gap, this time is even longer, given the diluteness of the thermal quasiparticle excitations, and we have τφ eΔ/T [166], where Δ is the energy gap. But how short can we make τφ in a system without quasiparticles? By examining a variety of models with and without quasiparticles, and also by arguments based upon analytic continuation of equivalent statistical mechanics models in imaginary time, it was proposed [673] that there is a lower bound

which is obeyed by all infinite many-body quantum systems as T → 0. Here C is a dimensionless number of order unity which is independent of T. Note that even without specifying C, (1.1) is a strong constraint e.g. it rules out systems in which 1.1) as T → 0 i.e. τφ 1/T.

In recent work, using inspiration from gravitational analogies, Maldacena, Shenker, and Stanford [559] have pointed out the fundamental role of many-body quantum chaos. They focused attention on a Lyapunov time, τL, defined as a measure of the time for a many-body system to lose memory of its initial state [510]. This can be measured by considering the magnitude-squared of the commutator of two observables a time t apart: the growth of the commutator with t is then a measure of how the quantum state at the later time has been perturbed the initial observation. With a suitable choice of observables, the growth is initially exponential, exp(t/τL), and this defines τL. Modulo some reasonable physical assumptions, they established a lower bound for τL

1.2): (i) a mean-field model of a strange metal called the Sachdev-Ye-Kitaev (SYK) model [671; 484] (see §5.11), and (ii) the holographic duals of black holes in most theories of gravity. From a gravitational perspective, the key feature of quantum systems which are holographically dual to gravity is the presence of many-body quantum chaos with τL /(kBT). The complementary, and closely related [364; 773], condensed matter perspective is that systems without quasiparticles locally thermalize or dephase in the shortest possible ‘Planckian’ time [772] of τφ /(kBT).

1.2 Historical context II: Horizons are dissipative

The most interesting results in holographic condensed matter physics are made possible by the fact that the classical dynamics of black hole horizons is thermodynamic and dissipative. This is the single most powerful and unique aspect of the holographic approach. In this subsection we briefly outline these basic intrinsic properties of event horizons, independently of any holographic interpretation. Later, via the holographic correspondence, the classical dynamics of event horizons will be reinterpreted as statements about the deconfined phase of gauge theories in the ’t Hooft matrix large N limit. The ’t Hooft limit will be the subject of the following §1.3.

To emphasize the central points, we discuss stationary, neutral and non-rotating black holes. The important quantities characterizing the black hole are then its area A, mass M and surface gravity κ. The surface gravity is the force required at infinity to hold a unit mass in place just above the horizon. The horizon is the surface of no return – this irreversibility, inherent in classical gravity, underpins the connection with thermodynamics. Bardeen, Carter and Hawking showed that the classical Einstein equations imply [76]:

1. The surface gravity κ is constant over the horizon.

2. Adiabatic changes to the mass and size of the black hole are related by

Here GN is Newton’s constant.

3. The horizon area always increases: δA ≥ 0.

These are clearly analogous to the zeroth through second laws of thermodynamics. Hawking subsequently showed that black holes semiclassically radiate thermal quanta, and thereby obtained the temperature and entropy [378]

ℏ. This shows that while Hawking radiation is a quantum mechanical effect, the entropy of the black hole is so large (the area is measured in Planck units Lp) that the thermodynamic nature of black holes is imprinted on the classical equation (1.3) – the factors of in the temperature and entropy have cancelled. This is an important lesson for our purposes: black holes are able to geometrize thermodynamics because they describe systems with a huge number of degrees of freedom. Granted that black holes in equilibrium behave as a thermodynamic system, the next question is whether perturbations of stationary black holes are described by dissipative linear response theory. Early work in the 1970s by Hawking, Hartle, Damour and Znajek showed that perturbed black holes exhibited dissipative processes such as Joule heating. These observations were formalized as the ‘membrane paradigm’ of black holes [642; 732; 616]. For example, near an event horizon electromagnetic radiation must be ‘infalling’ (that is, directed into rather than out of the black hole). Maxwell’s equations then relate the electric and magnetic fields parallel to the horizon as

is a spacelike outward pointing unit normal to the horizon. Because the horizon acts as a boundary of the spacetime, properly satisfying the variational principle leading to the Maxwell equations requires the association of charge and current densities to the horizon. The charge density is found to be proportional to the electric flux through the horizon while the current density j along the horizon is related to the magnetic field at the horizon via

is the electromagnetic coupling evaluated at the horizon (allowing in general for a space-dependent ‘dilaton’ coupling). Putting the previous two equations together gives Ohm’s law on the horizon

, much like a quantum critical medium in two spatial dimensions. Solving the Einstein equations at quadratic order in the electromagnetic fields one finds that the area of the horizon increases in precisely the way required to describe the entropy generated by this current and resistivity.

The membrane description of black hole dissipation in asymptotically flat spacetime was complicated by the fact that such black holes have an unstable underlying thermodynamics (in particular, a negative specific heat) as well as finite size horizons. As we shall see, the holographic correspondence gives a clean framework for making sense of – and putting to use – the dissipative dynamics of horizons. The membrane paradigm was directly incorporated into holography in [498; 431; 226], to be discussed below.

The membrane paradigm describes perturbations of the event horizon by long-lived modes of currents and conserved charges; holographically, these are hydrodynamic modes. More general perturbations of a black hole relax quickly to equilibrium [746]. For the black holes appearing in holographic scenarios, the thermalization timescale characterizing this decay is found to be τφ /(kBT), as first emphasized in [403]. More recently, it was recognized [691] that black holes are maximally chaotic, and their dynamics implies a saturation of the Lyapunov time τL in (1.2). The saturation of the timescale (1.1) for local equilibration indicates that the degrees of freedom underlying black hole thermodynamics must have strongly interacting non-quasiparticle dynamics. This leads us to the following §1.3.

1.3 Historical context III: The ’t Hooft matrix large N limit

The essential fact about useful large N limits of quantum fields theories is that there exists a set of operators {𝒪i} with no fluctuations to leading order at large N [763; 151]:

Operators that obey this large N factorization are classical in the large N limit. One might generally hope that, for a theory with many degrees of freedom ‘per site’, each configuration in the path integral has a large action, and so the overall partition function is well approximated by a saddle point computation. However, the many local fields appearing in the path integral will each be fluctuating wildly. To exhibit the classical nature of the large N limit one must therefore identify a set of ‘collective’ operators {𝒪i} that do not fluctuate, but instead behave classically according to (1.8). Even if one can identify the classical collective operators, it will typically be challenging to find the effective action for these operators whose saddle point determines their expectation values. The complexity and richness of different large N limits depends on the complexity of this effective action for the collective operators.

A simple set of examples are given by vector large N limits. One has, for instance, a bosonic field with N components and, crucially, interactions are restricted to be O(N) symmetric. This means the interaction terms must be functions of

). This is the desired effective action for the collective field σ that can now be treated in a saddle point approximation in the large N limit, see e.g. [779].

For our purposes the vector large N limits are too simple. The fact that one can obtain the effective action for σ from a few functional determinants translates into the fact that the theory is essentially a weakly interacting theory in the large N limit. As we will emphasize repeatedly throughout this review, weak interactions means long lived quasiparticles, which in turn mean essentially conventional phases of matter and conventional Boltzmann descriptions of transport. The point of holographic condensed matter is precisely to realize inherently strongly interacting phases of matter and transport that are not built around quasiparticles. Thus, while vector large N limits do admit interesting holographic duals [299], they will not be further discussed here. As we will explain below, strongly interacting large N theories are necessary to connect directly with the classical dissipative dynamics of event horizons discussed in the previous §1.2.

The ’t Hooft matrix large N limit [725], in contrast to the vector large N limit, has the virtue of admitting a strongly interacting saddle point description. The fields in the theory now transform in the adjoint rather than the vector representation of some large group such as U(N). Thus the fields are large N × N matrices ΦI and interactions are functions of traces of these fields

These ‘single trace’ operators are straightforwardly seen to factorize and hence become classical in the large N limit [763; 151]. There are vastly more classical operators of the form (1.10) than there were in the vector large N limit case (1.9). Furthermore there is no obvious prescription for obtaining the effective action whose classical equations of motion will determine the values of these operators. For certain special theories in the ’t Hooft limit, this is precisely what the holographic correspondence achieves. The fact that was unanticipated in the 70s is that the effective classical description involves fields propagating on a higher dimensional curved spacetime.

In the simplest cases, the strength of interactions in the matrix large N limit is controlled by the ’t Hooft coupling λ [725]. When the Lagrangian is itself a single trace operator then this coupling appears in the overall normalization as, schematically,

When λ is small, the large N limit remains weakly coupled and can be treated perturbatively. In this limit the theory is similar to the vector large N limit. When λ is large, however, solving the theory would require summing a very large class of so-called planar diagrams. Through the holographic examples to be considered below, we will see that the large λ limit is indeed strongly coupled when the theory is gapless: there are no quasiparticles and the single trace operators, while classical, acquire significant anomalous dimensions. This book is a study of the phenomenology of these strongly interacting large N theories.

A separate large N limit has been the focus of much recent attention. This is intermediate between the vector and matrix large N limits described above [710], and is realized in the Sachdev-Ye-Kitaev models. This limit does not have quasiparticles, and will be discussed in §5.11.

1.4 Maldacena’s argument and the canonical examples

We will now outline Maldacena’s argument that connects the physics of event horizons described in §1.2 with the ’t Hooft matrix large N limit of §1.3 [562; 557]. This argument involves concepts from string theory. Experimentalists have our permission to skip this section on a first reading. However, we will only attempt to get across the essential logic of the argument, which we hope will be broadly accessible. This argument is the source of the best understood, canonical examples of holographic duality.

The central idea is to describe a certain physical system in two different limits. Specifically, we consider a large number N of ‘Dp branes’ stacked on top of each other, in an otherwise empty spacetime. Dp branes are objects in string theory with p spatial dimensions. Thus a D0 brane is a point particle, a D1 brane is an extended string, a D2 brane is a membrane, and so on. As a physical system, the coincident Dp branes have certain low energy excitations. We proceed to describe these excitations in two limits. These limits are illustrated in Figure 1.1 and explained in more detail in the following.

Figure 1.1
Low energy excitations of a stack of D3 branes at weak and strong ’t Hooft coupling. Weak coupling: N2 light string states connecting the N D3 branes. Strong coupling: classical gravitational excitations that are strongly red-shifted by an event horizon.

The simplest case of all is for D3 branes, so we will start with these. D3 branes are objects with three spatial dimensions, extended within the nine spatial dimensions of (type IIB) string theory. Like all objects, the D3 branes gravitate. How strongly they gravitate depends on their tension and the strength of the gravitational force. The gravitational backreaction turns out to be determined by the number N of superimposed D3 branes and the dimensionless string coupling constant gs. It is found that when

then the gravitational effects of the D3 branes are negligible. This is because the tension of the N D3 branes goes like N/gs, whereas the strength of gravity goes like 𝒩 = 4 super Yang-Mills theory. This theory describes excitations that propagate on the three spatial dimensions of the D3 branes and, schematically, has Lagrangian density

This expression describes a theory with an SU(N) field strength F coupled to bosonic fields Φ and fermionic fields Ψ, all in the adjoint representation. In 𝒩 = 4 super Yang-Mills there are six bosonic fields and four fermionic fields. All of these fields are N × N matrices (matrix indices give the branes on which strings begin/end). The above Lagrangian therefore takes the form we discussed previously in (1.11), with λ in (1.12) now revealed as the ’t Hooft coupling! Therefore, we have found that the low energy excitations of the D3 branes in the regime (1.12) are described by a weakly interacting matrix large N theory.

In the opposite regime

the D3 branes gravitate very strongly. Under the strong effects of gravity the branes collapse upon themselves and form what is known as a black brane. These black branes are described by an appropriate analogue of well-known black hole solutions of general relativity. The horizon of the black brane is, by definition, a surface of infinite gravitational redshift. Therefore, for a far away observer, the low energy excitations of the system are any excitations that occur very close to the horizon. This region is called the near horizon geometry. This (ten dimensional) geometry is known to be a solution called AdS5 × S5, with spacetime metric

surrounds the D3 branes. The horizon itself is at r → ∞ in these coordinates. The scale L is called the AdS radius and can be related (see e.g. [557]) to the two fundamental scales in string theory, the string length Ls and the Planck length Lp

1.14) and at large N 1, the radius of curvature L of the AdS5 × S5 geometry is very large compared to both the string and the Planck lengths. It follows that for many purposes we can neglect effects due to highly excited string states (controlled by Ls) as well as quantum gravity effects (controlled by Lp). The excitations of the near horizon region will simply be described by classical gravitational perturbations of the background (1.15).

The upshot of the previous two paragraphs is that the low energy excitations of N D3 branes look very different in two different limits. At weak ’t Hooft coupling they are described by a weakly interacting matrix large N field theory. At strong ’t Hooft coupling they are described by gravitational perturbations about AdS5 × S5 spacetime (1.15). The original version of the holographic correspondence [562] was the conjecture that there existed a decoupled set of degrees of freedom that interpolated between these weak and strong coupling descriptions. In particular, this implies that the classical gravitational dynamics of AdS5 × S5 is precisely the strong coupling description of large N 𝒩 = 4 super Yang-Mills theory. This is the long-sought effective classical description of a matrix large N limit. Remarkably, the effective classical fields are gravitating and live in a higher number of spacetime dimensions! We now gain our first glimpse of a microscopic understanding of the dissipative dynamics of horizons that we discussed in §1.2: gravity is in fact the dynamics of a strongly interacting matrix large N quantum field theory in disguise.

The logic of the above argument can be applied to a very large number of configurations of branes in string theory. Many ‘dual pairs’ of quantum field theories and classical gravitational theories are obtained in this way. In Table 1.1, we list what might be considered the canonical, best understood examples of holographic duality for field theories in 3+1, 2+1 and 1+1 spacetime dimensions. We will not explain the terminology appearing in this table. Entry points to the literature include the original papers [562; 15] and the review [16]. Our main objective here is to make clear that explicit examples of the duality are known in various dimensions and that they are found by using string theory as a bridge between quantum field theory and gravity.

Table 1.1
The canonical examples of holographic duality for field theories in 3+1, 2+1 and 1+1 spacetime dimensions. The first column shows the string theory setup of which one should consider the low energy excitations. In the limit of no gravitational backreaction the low energy degrees of freedom are described by a quantum field theory, given in the second column. In the limit of strong gravitational back reaction, the low energy degrees of freedom are described by classical gravitational dynamics about the backgrounds shown in the third column.

1.5 The essential dictionary

1.5.1 The GKPW formula

While string theory is useful in furnishing explicit examples of holographic duality, much of the machinery of the duality is quite general and can be described using only concepts from quantum field theory (QFT) and gravity.

Basic observables that characterize the QFT are the multi-point functions of operators in the QFT. In particular, at large N, the basic observables are multi-point functions of the single trace operators 𝒪i, described above in (1.10). As examples of such operators, we can keep in mind charge densities ρ = Jt and current densities Ji, associated with symmetries of the theory. The multi-point functions can be obtained if we know the generating functional

Observables in gravitating systems can be difficult to characterize, because the spacetime itself is dynamical. In the case where the spacetime has a boundary, however, observables can be defined on the boundary of the spacetime. We can, for instance, consider a Dirichlet problem in which the values of all the ‘bulk’ fields (i.e. the dynamical fields in the theory of gravity) are fixed on the boundary. The boundary itself is not dynamical, giving the observer a ‘place to stand’. We can then construct the partition function of the theory as a function of the boundary values {hi(x)} of all the bulk fields {ϕi}

It is a nontrivial mathematical problem to show that the Dirichlet problem for gravity is in fact well-posed, even in the classical limit (see e.g. [566] for a recent discussion). However, we will see below how in practice the boundary data determines the bulk spacetime.

Suppose that a given QFT and theory of gravity are holographically dual. The essential fact relating observables in the two dual descriptions (of the same theory) is that there must be a one-to-one correspondence between single trace operators 𝒪i in the QFT, and dynamical fields ϕi of the bulk theory. For example, a given scalar operator such as tr(Φ2) in the QFT with schematic Lagrangian (1.13) will be dual to a particular scalar field ϕ in the bulk theory. The scalar ϕ will have its own bulk dynamics given by the action S in (1.18). We will be more explicit about the bulk action shortly. Having matched up bulk operators and boundary fields in this way, we can write the essential entry in the holographic dictionary, as first formulated by Gubser-Klebanov-Polyakov and Witten (GKPW) [764; 318]:

𝒪i taking boundary value hi. This relationship is illustrated in Figure 1.2.

Figure 1.2
Essential dictionary: The boundary value h of a bulk field ϕ is a source for an operator 𝒪 in the dual QFT.

The reader may have many immediate questions: How do we know which bulk field corresponds to which operator? There are a very large number of single trace operators, and so won’t the bulk description involve a very large number of fields and hence be unwieldy? Before addressing these questions, we will illustrate the dictionary (1.19) at work.

1.5.2 Fields in AdS spacetime

The large N limit is supposed to be a classical limit, so let us evaluate the bulk partition function semiclassically

.

We need to specify the bulk action. The most important bulk field is the metric gab. The QFT operator corresponding to this bulk field will be the energy momentum tensor Tμν. This means that the boundary value of the bulk metric (more precisely, the induced metric on the boundary) is a source for the energy momentum tensor in the dual QFT, which seems natural. In writing down an action for the bulk metric, we will consider an expansion in derivatives, as one usually does in effective field theory. Later in this section we describe the circumstances in which this is a correct approach. The general bulk action for the metric involving terms that are at most quadratic in derivatives of the metric is

1). The second term is a negative cosmological constant characterized by a lengthscale L (holography with a positive cosmological constant remains poorly understood [44]). The equations of motion following from this action are

This spacetime has multiple symmetries. The easiest to see is the dilatation symmetry: 1.23) dually describes a ‘boundary’ conformal field theory (CFT) in d + 1 spacetime dimensions. Let us see this explicitly.

Perturbing the metric itself is a little complicated, so consider instead an additional field ϕ in the bulk with an illustrative simple action

As noted above, ϕ will correspond to some particular scalar operator in the dual quantum field theory. For concreteness we can have in the back of our mind an operator like tr(Φ2). To see the effects of adding a source h for this operator, we must solve the classical bulk equations of motion

1.20). We have already noted that the horizon of AdSd+2 is the surface r → ∞ in (1.23) – this is properly called the Poincaré horizon. This is the surface of infinite redshift where gtt → 0. The ‘conformal boundary’ at which we impose ϕh is the opposite limit, r → 0. In the metric (1.23) this corresponds to an asymptotic region where the volume of constant r slices of the spacetime are becoming arbitrarily large. This should be thought of as imposing boundary conditions ‘at infinity’.

The equation of motion (1.25) is a wave equation in the AdSd+2 background (1.23). We can solve this equation by decomposing the field into plane waves in the t and x directions

To understand the asymptotic boundary conditions, we should solve this equation in a series expansion as r → 0. This is easily seen to take the form

There are two constants of integration ϕ(0) and ϕ(1). The first of these is what we previously called the boundary value h of ϕ. We shall understand the meaning of the remaining constant ϕ(1) shortly. In (1.28) we see that in order to extract the boundary value h of the field, one must first strip away some powers of the radial direction r. These powers of r will now be seen to have an immediate physical content. We noted below (1.23) that rescaling 1.17), we see the dual operator must rescale as

𝒪. Equation (1.29) is a very important result that relates the mass of a bulk scalar field to the scaling dimension of the dual operator in a CFT. We can see that for relevant perturbations (Δ < d + 1), the ϕ(0) term in (1.28) goes to zero at the boundary r → 0, while for irrelevant perturbations (Δ > d + 1), this term grows towards the boundary. This will fit perfectly with our interpretation in §1.6 below of the radial direction as capturing the renormalization group of the dual quantum field theory, with r → 0 describing the UV.

For a given bulk mass, equation (1.29) has two solutions for Δ. For most masses we must take the larger solution Δ+ in order for ϕ(0) to be interpreted as the boundary value of the field in (1.28). Exceptions to this statement will be discussed later. Note that d + 1 − Δ± = Δ.

1.5.3 Simplification in the limit of strong QFT coupling

We can now address the concern above that a generic matrix large N quantum field theory has a very large number of single trace operators. For instance, with only two matrix valued fields one can construct operators of the schematic form 2 The heavy fields can be neglected for many purposes. This leads to a tractable bulk theory.

In the concrete string theory realizations of holographic duality discussed above in §1.4, the hierarchy in the bulk can be easily understood. The excitations of the theory are string states. From the relation quoted in (1.16) we see that large ’t Hooft coupling λ 1 implies that L/Ls 1. The mass of a typical excited string state is m ∼ 1/Ls, and therefore Lm 1 for these states. Only a handful of low energy string states survive. This perspective also justifies a derivative expansion of the bulk action (and hence the neglect of higher derivative terms in (1.21) and (1.24)). Higher derivative terms are suppressed by powers of Ls/L, as they arise due to integrating out heavy string states.3

Beyond specific string theory realizations, the lesson of the previous paragraph is that a holographic approach is useful for QFTs in which two simplifications occur. Firstly,

While the second condition may appear restrictive, string theory constructions show that these simplifications do indeed arise in concrete models. The most important point is that this is a way (a large N limit with a gap in the operator spectrum) to simplify the description of QFTs without going to a weakly interacting quasiparticle regime. We will repeatedly see below how this fact allows holographic theories to capture generic behavior of, for instance, transport in strongly interacting systems that is not accessible otherwise. In contrast, weakly interacting large N limits, such as the vector large N limit, cannot have such a gap in the operator spectrum. This is because to leading order in large N the dimensions of operators add, and in particular operators such as give an evenly spaced spectrum of conserved currents with no gap. Beyond the holographic approach discussed here, the development of purely quantum field theoretic methods exploiting the existence of an operator gap has recently been initiated [274; 247; 272; 347].

1.5.4 Expectation values and Green’s functions

A second result that can be obtained from the asymptotic expansion (1.28) is a formula for the expectation value of an operator. From (1.17), (1.19) and (1.20) above

1.24) becomes a boundary term when evaluated on a solution. Because the volume of the boundary is diverging as r → 0, we take a cutoff boundary at r = 𝜖. Then

In the first line γ is the induced metric on the boundary (i.e. put r = 𝜖 in (1.23)) and n is an outward pointing unit normal (i.e. nr = −𝜖/L). In the second line we have used the boundary expansion (1.28). Upon taking 𝜖 → 0, the first term in (1.34) diverges while the second is finite. The remaining terms in general include subleading divergences and terms that vanish as 𝜖 → 0. Divergent terms are uninteresting contact terms, to be dealt with more carefully in §1.6.1. We ignore it here. In evaluating the second term, note that because the bulk field ϕ satisfies a linear equation of motion, then ϕ(1)x = (δϕ(1)/δϕ(0))xyϕ(0)y holds as a matrix equation. Thus we obtain

We have not given the constant of proportionality here, as obtaining the correct answer requires a more careful computation, which will appear in §1.6. The important point is that we have discovered the following basic relations:

This connection will turn out to be completely general. For instance, the source could be the chemical potential and the expectation value the charge density. Or, the source could be an electric field and the expectation value could be the electric current.

For a given set of sources {hi} at r → 0, regularity of the fields in the interior of the spacetime (e.g. at the horizon r → ∞) will fix the bulk solution completely. In particular, the subleading behavior near the boundary will be fixed. Therefore, by solving the bulk equations of motion subject to specified sources and regularity in the interior, we will be able to solve for the expectation values 𝒪i. We can illustrate this with the simple case of the scalar field. The radial equation of motion (1.27) is solved by Bessel functions. The boundary condition at the horizon is that the modes of the bulk wave equation must be ‘infalling’, that is, energy flux must fall into rather than come out of the horizon. We will discuss infalling boundary conditions in detail below. For now, we quote the fact that the boundary conditions at the horizon and near the asymptotic boundary pick out the solution (let us emphasize that t here is real time, we will also be discussing imaginary time later)

The overall normalization is easily computed but unnecessary and not illuminating. Here K is a modified Bessel function and Δ is the larger of the two solutions to (1.29). By expanding this solution near the boundary r → 0, we can extract the expectation value (1.35). In momentum space, the retarded Green’s function of the boundary theory is given by the ratio of the expectation value by the source:

𝒪 with dimension Δ in a CFT. This confirms our expectation that perturbations about the AdSd+2 background (1.23) will dually describe the excitations of a CFT.

1.5.5 Bulk gauge symmetries are global symmetries of the dual QFT

The computation of the Green’s function (1.38) illustrates how knowing the bulk action and the bulk background allows us to obtain correlators of operators in the dual strongly interacting theory. For this to be useful, we would like to know which bulk fields correspond to which QFT operators. This can be complicated, even when the dual pair of theories are known explicitly. Symmetry is an important guide. In particular, gauge symmetries in the bulk are described by bulk gauge fields. These include Maxwell fields Aa, the metric gab and also nonabelian gauge bosons. The theory must be invariant under gauge transformations of these fields, including ‘large’ gauge transformations, where the generator of the transformation remains constant on the asymptotic boundary. We can illustrate this point easily with a bulk gauge field Aa, which transforms to Aa → Aa + ∇aχ for a scalar function χ. If χ is nonzero on the boundary, then the boundary coupling in the action transforms to

where we integrated by parts in the last step. Invariance under the bulk gauge transformation requires that ∇μJμ = 0, so that the current is conserved (in all correlation functions). Therefore:

Similarly, fields that are charged under a bulk gauge field will be dual to operators in the QFT that carry the corresponding global charge. Thus quantities such as electric charge and spin must directly match up in the two descriptions.

Matching up operators beyond their symmetries is often not possible in practice, and indeed is not really the right question to ask. The bulk is a self-contained description of the strongly coupled theory. The spectrum and interactions of bulk fields define the correlators and all other properties of a set of dual operators. Reference to a weakly interacting QFT Lagrangian description is not necessary and potentially misleading. Nonetheless, it can be comforting to have in the back of our minds some familiar operators. Thus a low mass scalar field ϕ in the bulk might be dual to QFT operators such as trF2, tr Φ2 or tr 4

The general condensed matter systems we wish to study are not CFTs (although CFTs do comprise an interesting subset). Instead there will be multiple scales of interest: temperature, chemical potential, and scales generated by renormalization group flow of relevant operators. These are all understood within the framework of holographic renormalization, that we turn to next.

1.6 The emergent dimension I: Wilsonian holographic renormalization

An essential aspect of the duality map (1.19) is that the gravitating ‘bulk’ spacetime has an extra spatial dimension relative to the dual ‘boundary’ CFT. In this section and the following we will gain some intuition for the meaning of this extra ‘radial’ dimension. The short answer is that the radial dimension geometrizes the renormalization group: processes close to the boundary of the bulk correspond to high energy physics in the dual QFT while dynamics deep in the interior of the bulk describes low energy physics in the QFT. This will be one of the conceptual pillars of holographic condensed matter physics, another being the already mentioned fact that horizons geometrize dissipation. The relation is illustrated in Figure 1.3.

Figure 1.3
The radial direction: Events in the interior of the bulk capture long distance, low energy dynamics of the dual field theory. Events near the boundary of the bulk describe short distance, UV dynamics in the field theory dual. The simplest way to think about this is that the interior events are increasingly redshifted relative to the boundary energy scales (a similar logic was used in the decoupling argument of §1.4 to derive holographic duality, here we are discussing redshifts within the near horizon AdS region itself).

The radial or ‘holographic’ dimension captures all of the standard renormalization physics: (i) isolating universal low energy dynamics while parameterizing our ignorance about short distance physics, (ii) computing beta functions for running couplings and (iii) determining the structure of short distance divergences as a UV regulator is removed. In this section we work through the illustrative case of a scalar field in a fixed background geometry, as in the previous section. We will develop the concepts further as we need them throughout the review.

1.6.1 Bulk volume divergences and boundary counterterms

We have already encountered a divergence when we tried to evaluate the on-shell action in (1.34). This divergence is due to the infinite volume of the bulk spacetime, that is integrated over in evaluating the action. In (1.34) we regulated the divergence by cutting off the spacetime close to the boundary at r = 𝜖. We will see momentarily that this cutoff appears in the dual field theory as a short distance regulator of UV divergences. This connection between infinitely large scales in the bulk and infinitesimally short scales in the field theory is sometimes called the UV/IR correspondence [717]. Taking our cue from perturbative renormalization in field theory, we can regulate the bulk action by adding a ‘counterterm’ boundary action. For the case of the scalar, let

1.34). Because it is a boundary term defined in terms of purely intrinsic boundary data, it neither changes the bulk equations of motion nor the boundary conditions of the bulk field. The counterterm action furthermore contributes to the finite term in the expectation value of the dual operator: We can now write (1.35) more precisely as

𝜖 → 0. A bulk action that is finite as 𝜖 → 0 allows us to think of the dual field theory as being defined by starting from a UV fixed point. This is the first of several instances we will encounter in holography where boundary terms in the action must be considered in more detail than one might be accustomed to.

The counterterm action (1.41) is a local functional of boundary data. It is a nontrivial fact that all divergences that arise in evaluating on shell actions in holography can be regularized with counterterms that are local functionals of the boundary data. The terms that arise precisely mimic the structure of UV divergences in quantum field theory in one lower dimension, even though the bulk divergences are entirely classical. This identification of the counterterm action is called holographic renormalization. Entry points into the vast literature include [187; 694]. An important conceptual fact is that in order for the bulk volume divergences to be local in boundary data, the growth of the volume towards the boundary must be sufficiently fast. Spacetimes that asymptote to AdS spacetime are the best understood case where such UV locality holds.

1.6.2 Wilsonian renormalization as the Hamilton-Jacobi equation

In this review we will frequently be interested in universal emergent low energy dynamics. For such questions, as in conventional quantum field theory, the precise short distance regularization or completion of the theory is unimportant. To understand the running of couplings and emergent dynamics we require a Wilsonian perspective on holographic theories. We outline how this works following [383; 266], which built on earlier work [185].

Recall that a QFT is defined below some UV cutoff Λ by the path integral

Here I0[Φ] is the microscopic action and IΛ[Φ] are the terms obtained by integrating out all degrees of freedom at energy scales above the cutoff scale Λ. By requiring that the partition function Z is independent of Λ we obtain the renormalization group equations for IΛ[Φ]. By taking the cutoff to low energies we can obtain (in principle) the emergent low energy dynamics. By seeing how this structure emerges in holography we can gain some intuition for the meaning of the radial direction.

In the presence of a UV cutoff, there is a generalization of the fundamental holographic dictionary (1.19). Consider a radial cutoff in the bulk at r = 𝜖. The cutoff gravitational partition function is now expressed in terms of the values of fields 1.18) becomes

This quantity is naturally related to the partition function of the dual QFT integrated only over modes below some UV cutoff Λ. In particular, it is natural to generalize (1.19) to

1.43). Thus, the values of the bulk fields at the cutoff boundary are now interpreted as sources in an effective QFT valid at scales below a field theoretic cutoff Λ. While we schematically write Λ 1/𝜖, the precise nature of the field theory cutoff Λ corresponding to the bulk cutoff 𝜖 is not known. For instance, the regulator preserves any gauge invariance of the boundary field theory and is therefore not a hard momentum cutoff. Determining precisely the field theory renormalization group scheme defined by a cutoff in the bulk would probably amount to a proof of holography. The perspective we will take is that the radial cutoff is a natural UV regulator in theories with holographic gravity duals, and we wish to check now that it leads to sensible consequences. In particular, we proceed to derive the renormalization group equations from the bulk. The following few paragraphs are a little technical, but allow us to show how the second order bulk equations of motion are related to first order renormalization group flow equations.

The full bulk partition function (1.18) with source h at the boundary is related to the truncated partition function (1.44) by gluing path integrals together in the usual way (see Figure 1.4):

Figure 1.4
The Wilsonian cutoff: The cutoff is at r = 𝜖, while the UV fixed point theory is at r = 0. The partition function Z𝜖[ϕ𝜖] is over all modes at r > 𝜖, subject to the boundary condition ϕ(𝜖) = ϕ𝜖. The partition function 𝜖 and r = 0, with boundary conditions at both ends.

Here 1.43) we obtain

𝒪 (a single trace operator built from the field theory degrees of freedom Φ) obtained by integrating out high energy modes above the cutoff Λ. It is given by an integral transform of the bulk partition function where we only integrate over modes living on the bulk spacetime between the boundary at r = 0 and r = 𝜖, subject to boundary conditions at both ends.

The semiclassical limit of (1.47) is the Legendre transform relation

𝒪] can now be obtained using manipulations familiar from Hamilton-Jacobi theory.

To be concrete, consider the previous bulk action for the scalar (1.24) generalized to arbitrary potential

1.48) one can obtain the (Legendre transformed) Hamilton-Jacobi functional equation

In the above we assumed for simplicity that the metric takes the form ds2 = grrdr2 + γμν(r)dxμdxν, with no cross terms between the radial and boundary directions. ∇(d+1) denotes the derivative along the boundary directions only. The second equation in (1.50) follows from (1.48).

The functional equation (1.50) is to be solved for IΛ[𝒪], and gives the evolution of the effective field theory action as a function of the cutoff 𝜖. If we expand the action in powers of 𝒪 then we can write

1.50), we obtain the beta functions for all of the couplings λn. Therefore, we have shown that varying the bulk cutoff 𝜖 leads to explicit first order flow equations of exactly the kind we anticipate for couplings in quantum field theories as function of some Wilsonian cutoff Λ. As we noted below (1.45) above, the exact nature of the QFT cutoff Λ is not known.

1.6.3 Multi-trace operators

It is useful to note that the Wilsonian effective action (1.52) includes multi-trace operators 𝒪n. Even – as is often the case – we start with a single trace action in the UV, these terms are generated under renormalization. Explicit holographic computations typically only make use of the bulk field ϕ, which is dual to the single trace operator 𝒪. However, ϕ satisfies a second order equation in the bulk. What has happened is that the first order beta function equations for all the couplings (including the multi-trace couplings) have been repacked into a second order equation for a single bulk field ϕ. This approach, which is useful at large N, has been called the quantum renormalization group in [518].

Multi-trace operators can also be inserted directly into the UV action. It can be shown that these correspond to changing the boundary conditions of the field as r → 0 [766; 567]. To deform the field theory action by

1.28). This boundary condition is

1.17). That case corresponds to the simple linear relation .

An important special case is the double trace deformation 1.41), the alternate quantization requires instead

Here Δ = Δ is now the smaller of the two exponents. Unlike the previous counterterm (1.41), this expression includes a derivative normal to the boundary. This is the term responsible for changing the boundary conditions so that the Δ+ rather than Δ term becomes the source. The expression (1.42) for the expectation value still holds.

1.6.4 Geometrized versus non-geometrized low energy degrees of freedom

From the Wilsonian holographic renormalization picture outlined above it is clear, in principle, how to zoom in on the low energy universal physics. We need to consider the dynamics of the fields in the far interior of the bulk spacetime. We will also need to understand how to match this interior dynamics to the sources that appear as boundary conditions on the asymptotic spacetime. We shall do this explicitly several times later in this review.

There is an interesting caveat to the previous paragraph. Holography geometrizes the renormalization of the order N2 matrix degrees of freedom in the dual large N field theory. Sometimes, however, there can be order 1 N2 light degrees of freedom that are missed by this procedure. These degrees of freedom are extended throughout the bulk spacetime and not localized in the far interior. Effectively, they are not fully geometrized. An example are Goldstone modes of spontaneously broken symmetries. A framework, known as semi-holography, has been developed to characterize these cases, e.g. [267; 602; 266]. We shall describe these methods later, as the various modes manifest themselves as poles in Green’s functions that we will be computing. The contribution of these (low energy, non geometrized) modes running in loops in the bulk can lead to non-analyticities in low energy observables at subleading order in the 1/N expansion, causing subtle breakdowns of the large N expansion. Examples are quantum oscillations due to bulk Fermi surfaces (§4.6.2), destruction of long-range order in low dimensions by fluctuating Goldstone modes (§6.4.3) and long time tails in hydrodynamics (§3.5).

1.7 The emergent dimension II: Entanglement entropy

The previous section showed how the radial direction of the bulk spacetime geometrized the energy scale of the dual quantum field theory. The energy scale was understood within a functional integral, Wilsonian perspective on the QFT. An alternative way to conceive the bulk radial direction is as a certain geometric re-organization of the QFT Hilbert space.

Shortly after the discovery of the holographic correspondence, it was realized that the bulk can be thought of as being made up of order N2 QFT degrees of freedom per AdS radius [717]. The recent discovery [664; 663; 604] and proof [520] of the Ryu-Takayangi formula for entanglement entropy in theories with holographic duals suggests that a more refined understanding of how the bulk reorganizes the QFT degrees of freedom is within reach. In the simplest case, in which the bulk is described by classical Einstein gravity coupled to matter, the Ryu-Takayanagi formula states that the entanglement entropy of a region A in the QFT is given by

where GN is the bulk Newton’s constant and AΓ is the area of a minimal surface (i.e. a soap bubble) Γ in the bulk that ends on the boundary of the region A. The region A itself is at the boundary of the bulk spacetime. We illustrate this formula in Figure 1.5. The formula (1.56) generalizes the Bekenstein-Hawking entropy formula (1.4) for black holes. This last statement is especially clear for black holes describing thermal equilibrium states, which correspond to entangling the system with a thermofield double [563].

Figure 1.5
Minimal surface Γ in the bulk, whose area gives the entanglement entropy of the region A on the boundary.

Let us now illustrate the Ryu-Takayanagi formula, and in particular how it reveals the physics of the bulk radial direction, by computing the entanglement entropy of a spherical region of radius R in a CFT, following [663]. By spherical symmetry, we know that the minimal surface will be of the form r(ρ), where ρ is the radial coordinate on the boundary (so that 1.23) to obtain:

Ωd−1 denotes the volume of the unit sphere Sd−1. It is simple to check that

1.58). This is therefore the Ryu-Takayanagi surface. Note that dr/dρ = −ρ/r.

The entanglement entropy (1.56) is given by evaluating (1.58) on the solution (1.59). For dimensions d > 1,

This integral is dominated near r = 0. Switching to the variable y = Rρ, we obtain that

where δ is a cutoff on the bulk radial dimension near the boundary that, as in the previous Wilsonian section, we will want to interpret as a short distance cutoff in the dual field theory. This result becomes more transparent in terms of the equivalent cutoff on the bulk coordinate r = 𝜖; using (1.59) one has that . Hence, the entanglement entropy is

1.62) is called the area law of entanglement [709]. It can be intuitively understood as follows – entanglement in the vacuum is due to excitations which reside partially inside the sphere, and partially outside. In a local QFT, we expect that the number of these degrees of freedom is proportional to the area of the sphere in units of the short distance cutoff, (R/𝜖)d−1. This is true regardless of whether or not the state is gapped. Equation (1.62) therefore gives another perspective on the identification of a near-boundary cutoff on the bulk geometry with a short-distance cutoff in the dual QFT. In the holographic result (1.62), the proportionality coefficient Ld/GN (L/Lp)d 1. This is consistent with the general expectation that there must be a large number of degrees of freedom “per lattice site” in microscopic QFTs with classical gravitational duals. While the coefficient of the area law is sensitive to the UV cutoff 𝜖 (since we have to count the number of degrees of freedom residing near the surface), some subleading coefficients are universal [663].

In d = 1, the story is a bit different. Now,

The prefactor of 2 is related to the fact that the semicircular minimal surface has two sides. Using the result that the central charge c of the dual CFT2, which counts the effective degrees of freedom, is given by [113]

Note that the area law, which would have predicted S = constant, has logarithmic violations in a CFT2. We will see an intuitive argument for this shortly, as shown in Figure 1.6.

Figure 1.6
MERA network and a network geodesic that cuts a minimal number of links allowing it to enclose a region of the physical lattice at the top of the network. Each line between two points can be thought of as a maximally entangled pair in the auxiliary Hilbert space, while the points themselves correspond to projections that glue the pairs together. At the top of the network, the projectors map the auxiliary Hilbert space into the physical Hilbert space. The entanglement entropy of the physical region is bounded by the number of links cut by the geodesic: SElogD.

1.7.1 Analogy with tensor networks

It was pointed out in [722] that the Ryu-Takayangi formula resembles the computation of the entanglement entropy in quantum states described by tensor networks. We will now describe this connection, following the exposition of [348]. While, at the time of writing, these ideas remain to be fleshed out in any technical detail, they offer a useful way to think about the emergent spatial dimension in holography. For a succinct introduction to the importance of entanglement in real space renormalization, see [744].

Consider a system with degrees of freedom living on lattice sites labelled by i, taking possible values si. For instance each si could be the value of a spin. Tensor network states (see e.g. [246]) are certain wavefunctions ψ({si}). The simplest example of a tensor network state is a Matrix Product State (MPS) for a one dimensional system with L sites. For every value that the degrees of freedom si can take, construct a D × D matrix Tsi. Here D is called the bond dimension. The physical wavefunction is now given by

For the case that each si describes a spin half, these Matrix Product States parametrize a 2LD2 dimensional subspace of the full 2L dimensional Hilbert space. The importance of these states is that they capture the entanglement structure of the ground state of gapped systems, as we now recall.

To every site i in the lattice, associate two auxiliary vector spaces of dimension D. Let these have basis vectors |ni1 and |ni2 respectively, with n = 1, , D. Now form a state, in this auxiliary space, made of maximally entangled pairs between neighboring sites

11. It is clear that if we trace over some number of adjacent sites, the entanglement entropy of the reduced density matrix will be 2logD, corresponding to maximally entangled pairs at each end of the interval we have traced over. To obtain a state in the physical Hilbert space, we now need to project at each site. That is, let Pi: D × D2 be a projection, then the physical state

.5 In gapped systems, the entanglement entropy of the ground state grows with the correlation length. At a fixed correlation length, the entanglement structure of the state can therefore be well approximated by a MPS state with some fixed bond dimension D (if the inequality SE ≤ 2logD is approximately saturated).

For gapless systems such as CFTs, a more complicated structure of tensor contractions is necessary to capture the large amount of entanglement. A structure that realizes a discrete subgroup of the conformal group is the so-called MERA network [744], illustrated in Figure 1.6. Viewing the network as obtained by projecting maximally entangled pairs, the entanglement bound discussed above for MPS states generalizes to the statement that the entanglement entropy of a sequence of adjacent sites is bounded above logD, where is the number of links in the network that must be cut to separate the entangled sites from the rest of the network. The strongest bound is found from the minimal number of such links that can be cut. This effectively defines a ‘minimal surface’ within the network. As with the minimal surface appearing in the Ryu-Takayanagi formula, the entanglement entropy is proportional (if the bound is saturated) to the length of the surface. Such a surface in a MERA network is illustrated in Figure 1.6.

The similarity between the tensor network computation of the entanglement entropy and the Ryu-Takayangi formula suggests that the extended tensor network needed to capture the entanglement of gapless states can be thought of as an emergent geometry, analogous to that arising in holography [722]. From this perspective, the emergent radial direction is a consequence of the large amount of entanglement in the QFT ground state, and the tensor network describing the highly entangled state is the skeleton of the bulk geometry. Further discussion may be found in [722; 721; 618]. A key challenge for future work is to show how the large N limit can flesh out this skeleton to provide a local geometry at scales below the AdS radius.

1.8 Microscopics: Kaluza-Klein modes and consistent truncations

This section may be skipped by readers of a more pragmatic bent. The following discussion is however conceptually and practically important for understanding the simplest explicit holographic dual pairs, such as those in Table 1.1 above.

In our discussion of the gap in the operator spectrum in §1.5 we glossed over an important issue. In many explicit examples of holographic duality, e.g. those in Table 1.1, the bulk spacetime has additional spatial dimensions relative to the dual QFT, beyond the ‘energy scale’ dimension discussed in previous sections. These dimensions are referred to as the ‘internal space’. For instance in the AdS5 × S5 spacetime (1.15), the five sphere is the internal space. We can see that near the boundary r → 0, the five sphere remains of constant size while the AdS5 part of the metric becomes large. This means that the sphere is not part of the boundary dimensions. Here is the complication: while the statement in (1.32) that there is a small number of bulk fields as λ → ∞ remains true, these fields can have an arbitrary dependence on the internal dimensions. Thus, for every field ϕ in the full spacetime we obtain an infinite tower of fields in AdS5, one for each ‘spherical harmonic’ on the internal space. These are called Kaluza-Klein modes and their dimensions do not show a gap (because LMKK ∼ L/L ∼ 1). That is, while the large λ limit does get rid of many operators (those corresponding to excited string states in the bulk), it does not remove the infinitely many Kaluza-Klein modes. These modes are mostly a nuisance from the point of view of condensed matter physics. While emergent locality in the radial direction – explored in the previous two sections – usefully geometrized the renormalization group, emergent locality in the internal space leads to the QFT growing undesired extra spatial dimensions at low energy scales. In this section we discuss these modes and how to avoid them.

The most compelling strategy would be to get rid of the internal space completely. This is difficult because the best understood consistent bulk theories of quantum gravity require ten or eleven spacetime dimensions. However, ‘non-critical’ string theories – defined directly in a lower number of spacetime dimensions – do exist. In fact, non-critical string theory lead to early intuition about the existence of holographic duality [640]. Holographic backgrounds of non-critical string theory have been considered in, for instance [503; 485; 94]. The technical difficulty with these solutions is that while Kaluza-Klein modes can be eliminated or reduced, the string scale is typically comparable to the AdS radius (i.e. the ’t Hooft coupling λ ∼ 1) and therefore the infinite tower of excited string states must be considered – undoing our motivation for considering these models in the first place.

The second best way to get rid of the Kaluza-Klein modes would be to find backgrounds in which the internal space is parametrically smaller than the AdS radius (unlike, e.g. the AdS5 × S5 spacetime (1.15), in which the AdS5 and the internal space have the same radius of curvature). This way, all the Kaluza-Klein modes acquire a large bulk mass and can be mostly ignored (i.e. LMKK ∼ L/Linternal 1). The necessary tools to find backgrounds with a hierarchy of lengthscales have been developed in the context of the ‘string landscape’ [191]. A first application of these methods to holography can be found in [635]. In our opinion this direction of research is underdeveloped relative to its importance.

The most developed method for getting rid of the Kaluza-Klein modes is called consistent truncation. Here, one shows that it is possible to set all but a finite number of harmonics on the internal space to zero and that these harmonics are not sourced by the handful of modes that are kept. Because the equations of motion in the bulk are very nonlinear, and typically all modes are coupled, it can be technically difficult to identify a consistent set of modes that may be retained. While symmetries can certainly help, a general method does not exist – finding consistent truncations is something of an art form. While less satisfactory than the other approaches mentioned above, consistent truncation gives a manageable bulk action with only one extra spatial dimension relative to the dual QFT. Furthermore, the dynamics of this action consistently captures the dynamics of a subset of operators in the full theory. An important drawback of consistent truncations is that they can miss instabilities in modes that have been thrown out in the truncation. They may, therefore, mis-identify the ground state.

Classical instabilities arising from Kaluza-Klein modes can potentially be avoided in constructions built using ‘baryonic’ branes [390]. More generally, string theoretic realizations of holographic backgrounds may be subject to additional quantum instabilities. It has been suggested that all non-supersymmetric holographic flux vacua are unstable [609].

We proceed to give a list of simple examples of bulk actions that can be obtained by consistent truncation, referring to the literature for details. These examples have been chosen (from a large literature) to illustrate and motivate the kind of bulk dynamics we will encounter later in this review. These constructions lead to so-called “top-down holography” and are in contrast to “bottom-up” holography, wherein one postulates a reasonable bulk action with convenient properties. The objective of bottom-up holography is to explore the parameter space of possible strong coupling dynamics. It is useful to have a flavor for what descends from string theory and supergravity to build a sensible bottom-up model.

1. Einstein-Maxwell theory: The four dimensional bulk action

is a consistent truncation of M theory compactified on any Sasaki-Einstein seven-manifold, see e.g. [391; 290; 192]. Therefore this action captures the dynamics of a sector of a large class of dual 2 + 1 dimensional QFTs. The simplest Sasaki-Einstein seven-manifold is S7. The bulk Maxwell field, F = dA, is dual to a particular U(1) symmetry current operator Jμ, called the R-symmetry, in the dual superconformal field theory. Similarly, the five dimensional action

is a consistent truncation of IIB supergravity compactified on any Sasaki-Einstein five-manifold, e.g. [290]. This theory describes a sector of a large number of dual 3 + 1 dimensional QFTs. The simplest Sasaki-Einstein five-manifold is S5, and so these theories include 𝒩 = 4 SYM theory. The additional Chern-Simons term in the above action is common in five bulk spacetime dimensions and dually describes an anomaly in the global U(1) R-symmetry of the QFT [764].

Einstein-Maxwell theory is the workhorse for much of holographic condensed matter physics.

2. Einstein-Maxwell theory with charged scalars: The above consistent truncations to Einstein-Maxwell theory by compactification on Sasaki-Einstein manifolds may be extended to include additional bulk fields. In particular the five dimensional action (1.70) can be extended to [322; 327]

Here η and θ are the magnitude and phase of a complex scalar field that is charged under the Maxwell field A. The two functions appearing in the action are

1.70). The complex scalar field is dual to a particular charged scalar operator 𝒪 in the dual QFT [322]. Similarly the four dimensional action (1.69) can be extended to [192; 288; 289; 233]

1.69) – the Sasaki-Einstein compactification needs to be ‘skew-whiffed’, as described in the above references. This theory contains a neutral pseudoscalar field σ in addition to the charged scalar {η, θ}. The functions in the action are now

Note that in a background with both electric and magnetic charges, the pseudoscalar is sourced by the final term in the action and hence cannot be set to zero in general. On the other hand we can set η = 0 and remove the complex scalar.

Charged scalars in the bulk will play a central role in the theory of holographic superconductivity.

3. Einstein-Maxwell theory with neutral scalars: Both of the next two examples are taken from [162; 330]. The five dimensional bulk action

𝒩 = 4 SYM theory. Here α is a neutral scalar field. Similarly the four dimensional action

𝜖abcdFabFcd = 0. If this last condition does not hold, an additional pseudoscalar must be included – similarly to in (1.73) above – and the action becomes more complicated [162].

Exponential functions of scalars multiplying the Maxwell action are ubiquitous in consistent truncations and will underpin our discussions of Lifshitz geometries and hyperscaling violation.

The interested reader will find many more instances of consistent truncations by looking through the references in and citations to the papers mentioned above. It is also possible to find consistent truncations in which the four and five bulk dimensional actions above (and others) are coupled to fermionic fields. The resulting actions are a little complicated and we will not give them here, see [287; 82; 286; 198; 199; 201; 152]. Bulk fermion fields will be important in our discussion below of holographic Fermi surfaces.

Exercises

1.1.  A stack of D3 branes. A consistent truncation of the action of type IIB supergravity in 10 spacetime dimensions is to an Einstein-Maxwell-dilaton sector:

The 5-form F is required to be self-dual, F = *F. Now consider a stack of N 1 D3 branes. These are objects extended in 4 spacetime dimensions. So let us divide up flat 10 dimensional spacetime into (xμ, yA), μ = 1, …, 4, A = 1, …, 6, and consider placing this stack of D3 branes at yA = 0. We take one of the x-directions to be timelike. As we noted in §1.4, when N is large enough this stack is very heavy and will backreact on the asymptotically flat geometry.

a) Let us begin by looking for the most symmetric solution possible to the classical supergravity equations, with ϕ = 0. Denote ρ2 = yAyA. Argue that a symmetric solution must take the form

b) Impose the equations of motion following from the action (1.78) and show that a solution is:

This solution describes the gravitational backreaction of the D3 branes. The original derivation of these types of ‘black brane’ spacetimes, including generalization to nonzero temperature, is [414].

c) Show that as ρ → 0, (1.81) reduces to (1.15) after a coordinate change. This is the near-horizon geometry.

d) The 5-form flux F is associated with D3 brane charge, and string theory tells us that for a stack of N branes, and 5-dimensional surface C enclosing yA = 0:

1.16).

e) Finally, consider fluctuations of the dilatonic field ϕ within linear response (neglect the gravitational backreaction), and in the AdS (ρ → 0) limit. By writing , with YS5 a spherical harmonic, show that there is a tower of Kaluza-Klein modes with masses m2L2 = n(n + 4). for n = 0, 1, 2, …. What are the operator dimensions of the holographically dual operators? The n ≠ 0 modes are the problematic Kaluza-Klein modes discussed in §1.8.

1.2.  AdS and the conformal group. Euclidean AdSd+2 can be embedded in a flat space in d + 3 dimensions. Global AdS is a connected half (Y > 0) of the hyperboloid

where d+2, representing the unit sphere Sd+1. What is the resulting metric?

b) Show that the coordinates

1.23) with AdS radius L = 1, but that these coordinates only cover half of the spacetime.

c) The Euclidean time conformal group is SO(d + 2, 1). This group has an obvious linear action on the coordinates (Y, Z, Xi) on the hyperboloid (1.83). Find the (nonlinear) group action on (z, xi) – which is manifestly the isometry group of AdSd+2.

d) By thinking about the action of these isometries on points with z → 0, argue that an isometry transformation in the bulk should correspond to a global conformal transformation in the boundary field theory.

1.3.  Three-point functions. In this exercise you will use the holographic dictionary to compute a three point function of a scalar operator 𝒪. Consider the (Euclidean time) holographic theory

with Λ = −d(d + 1)/2 and m2 = Δ(Δ − d − 1).

In the following y = (r, x) refers to a point in the bulk (1.23), where x is a boundary spacetime coordinate and r a radial position.

a) Suppose that we source the operator dual to the bulk scalar field ϕ with boundary profile h(x) = δ(xx0). Show that in the bulk, neglecting interactions (u = 0), the scalar field

Fix the constant C by requiring that ϕ(x, r → 0) ≈ rd+1−Δδ(xx0), in the sense of distributions. Hint: There are at least two ways to proceed. One is to directly Fourier transform (1.37). A more elegant approach begins by showing that if f(r) solves the scalar equation in the bulk, so does f(r/(r2 + (xx0)2) [764].

b) Using real space Feynman rules in the bulk, show that

1.87) and (1.88), show that

𝒪. See [280] for the original computation of higher point functions from AdS.

1.4.  Mutual information. The mutual information of two regions A and B in quantum field theory is defined as

with S the entanglement entropy. It tells us how much the regions A and B are entangled with each other. Use holography to compute the mutual information of the following two intervals of a CFT in 1 + 1 dimensions: A = [−La,a], B = [a, a + L], with a, L > 0. What forms can the Ryu-Takayangi surface take? Sketch the result for the mutual information as a function of L/a and comment on what happens. Hint: (1.63) and (1.65) describe the most general geodesics in AdS3; further computations should be minimal. See [381] for further discussion.

Notes

1 In this book, a d-dimensional system always refers to a boundary theory in d spatial dimensions and one time dimension. This is the convention of condensed matter physics. We warn readers that much of the (early) holographic literature uses d for the number of spacetime dimensions.

2 We are ignoring the presence of so-called bulk Kaluza-Klein modes in this discussion. These modes can lead to many light bulk fields even in the strong ’t Hooft coupling limit λ → ∞. See § 1.8.

3 A handful of higher derivative terms with unsuppressed couplings are in principle allowed, and can be traded for additional fields. This often leads to ghosts (fields with wrong-sign kinetic terms) or other pathologies [123]. A finite number of unsuppressed and well-behaved higher derivative terms would correspond to fine tuning the bulk theory. We have already encountered an instance of fine tuning: to obtain the basic AdSd+2 solution (1.23), the cosmological constant term in the action balances the Ricci scalar term, which has more derivatives. This is only possible because the cosmological constant has been tuned to be very small in Planck units (L/Lp 1). In the dual QFT, this tuning is just the fact that we have taken N to be large! The minimal statement for a bulk theory dual to a QFT with an operator gap is that there cannot be an infinite number of unsuppressed higher derivative terms in the bulk, which would lead to a non-local bulk action [382].

4 Single trace operators in QFT are dual to fields in the bulk. What about multi-trace operators? The insertion of multi-trace operators in the action in the sense of (1.17) will be discussed in the following section on holographic renormalization. A simpler aspect of multi-trace operators relates to the operator-state correspondence for CFTs. Here the single trace operator 𝒪 corresponds to the state created by a quantum of the dual field ϕ. The double-trace operator 𝒪2 corresponds to creating two quanta of the dual field. In the bulk, quantum effects are suppressed at large N and so these different quanta are simply superimposed solutions of free bulk wave equations. For this reason, single trace operators in large N theories (whose dimension does not scale with N) are sometimes referred to as generalized free fields, e.g. [247].

5 This follows from the fact that, focusing on a single cut between two sites for simplicity, the state (1.67) in the auxiliary space can be written in the physical Hilbert space on either side of the cut. Hence the entanglement entropy is bounded above by logD.