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UNIVERSITÄT BIELEFELD

Fakultät für Physik

Energy Momentum Tensor Correlators

in Improved Holographic QCD

In Attainment of the Academic Degree

Doctor rerum naturalium

Martin Krššák



July 2013

Energy Momentum Tensor Correlators in Improved Holographic QCD

Martin Krššák

Supervisor: PD Dr. Aleksi Vuorinen

Abstract

In this thesis, we study the physics of the quark gluon plasma (QGP) using holographic methods borrowed from string theory. We start our discussion by motivating the use of such machinery, explaining how recent experimental results from the LHC and RHIC colliders suggests that the created QGP should be described as a strongly coupled liquid with small but nonvanishing bulk and shear viscosities. We argue that holographic dualities are a very efficient framework for studying transport properties in such a medium.

Next, we introduce the underlying physics behind all holographic dualities, the AdS/CFT correspondence, and then motivate the necessity of implementing conformal invariance breaking in them. After this, we present the phenomenologically most successful holographic model of the strong interactions — Improved Holographic QCD (IHQCD).

Working within IHQCD, we next move on to calculate energy momentum tensor correlators in the bulk and shear channels of large- Yang-Mills theory. In the shear channel, we confront our results with those derived in strongly coupled Super Yang-Mills theory as well as weakly interacting ordinary Yang-Mills theory. Close to the critical temperature of the deconfinement transition, we observe significant effects of conformal invariance breaking. In the bulk channel, where the conformal result is trivial, we make comparisons with both perturbative and lattice QCD. We observe that lattice data seem to favor our holographic prediction over the perturbative one over a wide range of temperatures.

The work presented here is based upon the journal papers

  • K. Kajantie, M. Krššák, A. Vuorinen, Energy momentum tensor correlators in hot Yang-Mills theory: holography confronts lattice and perturbation theory, JHEP 1305 (2013) 140, arXiv:1302.1432 [hep-ph],

  • K. Kajantie, M. Krššák, M. Vepsäläinen and A. Vuorinen, Frequency and wave number dependence of the shear correlator in strongly coupled hot Yang-Mills theory, Phys. Rev. D 84, 086004 (2011), arXiv:1104.5352 [hep-ph],

as well as the works published in conference proceedings

  • M. Krššák, Viscosity Correlators in Improved Holographic QCD, Proceedings of the Barcelona Postgrad Encounters on Fundamental Physics, arXiv:1302.3181 [hep-ph],

  • A. Vuorinen, M. Krššák, Y.Zhu, Bulk and shear spectral functions in weakly and strongly coupled Yang-Mills theory, Proceedings of the workshop Xth Quark Confinement and the Hadron Spectrum, PoS ConfinementX (2012) 191, arXiv:1301.3449 [hep-ph].

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Chapter 1 Motivation

Figure 1.1: A schematic cartoon of a heavy ion collision. The figure has been taken from a presentation of S. Bass.

We live in an exciting time for particle physics — and particle physicists. Only recently did the largest particle accelerator ever built, the Large Hadron Collider (LHC) of CERN, start its TeV scale proton-proton and heavy ion runs, with its numerous experiments, such as ATLAS, CMS and ALICE, measuring and analyzing the collision products. Of these collaborations, ATLAS and CMS are primarily dedicated to the study of p-p collisions, aiming to provide answers to two fundamental questions in elementary particle physics. The first of these is to verify the existence of the Higgs boson, which by now seems to indeed have been accomplished [1, 2]. The second question on the other hand has to do with the extensions of the Standard Model, in particular the possible existence of supersymmetry at the TeV scale. If discovered, supersymmetry would not only have the potential to solve many phenomenologically important problems in modern high energy physics, but is also a direct prediction of the most promising candidate for a unified ‘theory of everything’ — string theory. Unfortunately, so far there have been no traces of any beyond the Standard Model physics in the ATLAS and CMS results [3], indicating that if present in the Nature at all, supersymmetry can most likely only be restored at energies above 2 TeV.

In contrast to ATLAS and CMS, the ALICE experiment is mainly dedicated to study the collisions of heavy ions, i.e. the process schematically depicted in fig. fig1?. Here, one starts with two colliding (lead) nuclei that are highly Lorentz contracted as illustrated by their pancake shapes in the figure. Immediately after the collision, the system is thought to be in a complicated initial state, the color glass condensate, that can be described using the so-called McLerran-Venugopalan model (see e.g. [4]). Through their mutual interactions, the constituents of the nuclei then exchange energy and momentum, eventually leading to the expanding system achieving local thermal equilibrium. This state of matter, where the quarks and gluons are liberated and can move freely, is called the quark gluon plasma (QGP). The fireball consisting of the QGP then undergoes a rapid expansion, described most efficiently through relativistic hydrodynamic simulations, and finally leads to recombination and hadronization once the temperature of the system falls below a certain critical value.

Of our interest in this thesis is the description of a near-thermal QGP, which is believed to have existed also in the early universe, shortly after the Big Bang. Its creation in a laboratory environment was first achieved by the experiments at the Relativistic Heavy Ion Collider (RHIC) of the Brookhaven National Laboratory in 2004 [5, 6, 7, 8], and later of course by the ALICE experiment at the LHC [9].

A common conclusion from the analysis of all experiments conducted so far [10, 9] has been that a successful hydrodynamical description of the expanding fireball requires the incorporation of a small but nonvanishing shear viscosity. This poses a direct challenge to the theory community, as one would clearly like to have a first principles prediction for this and other transport coefficients. The smallness of the viscosity — the experiments suggesting its ratio to the entropy being of the order — has in addition been somewhat of a surprise, considering that expectations from perturbative QCD calculations were suggesting parametrically larger values [11, 12].

By now it has been rather widely accepted that observations such as the smallness of the shear viscosity and the rapid apparent thermalization of the heavy ion collision product indicate that the created medium may in fact not be amenable to a description via weakly interacting quasiparticles. Rather, it may be more natural to think of the QGP as a strongly coupled liquid, indicating that fundamentally nonperturbative methods should be used to determine its properties. Due to the restriction of lattice QCD to the Euclidean formulation of the theory, this has unfortunately meant that many interesting dynamical quantities, and even most transport coefficients, are out of the reach of traditional field theory tools (though some progress has lately been achieved in the lattice determination of e.g. the shear viscosity [13, 14, 15].

Whereas lattice simulations are demanding numerical calculations requiring considerable computer power, a complementary and rather elegant approach to strongly coupled field theory was proposed some 15 years ago. This is of course the by now famous AdS/CFT correspondence [16], which relates the physics of strongly interacting field theories to a weakly coupled gravitational theory in one dimension higher. Using this method, many physically interesting quantities have been computed in the infinitely strongly coupled limit of the conformal Super Yang-Mills theory, with some predictions even conjectured to be rather universal. One famous example of this universality is the ratio of the shear viscosity to entropy, which obtains the value in all theories with holographic duals (with two-derivative gravitational actions).

While the prediction is consistent with experimental results, one needs to exercise caution in translating such insights to real world QCD (and QGP). This is primarily due to the conformality of the theory, which is a property of QCD only at very high energies, and obviously calls for the development of non-conformal holographic models. This is exactly what we aim to do in this thesis, namely study the Improved Holographic QCD (IHQCD) model of [17, 18, 19], which not only incorporates a dynamical breaking of conformality, but whose entire structure is designed to systematically mimic the most crucial properties of QCD. In the later chapters of this thesis, we will use this model to study the transport properties of the QGP, and in particular illustrate the effects of the broken conformal invariance on the predictions of holography.

This thesis is organized as follows.

  • In chapter sec2?, we start with an introduction to a selected class of issues in quantum field theory that are required to understand the physics of strong interactions at finite temperature.

  • In chapter sec3?, we present an introduction to holography. We start with a discussion of classical general relativity, and by studying the thermodynamics of black holes motivate the holographic principle. Next, we move on to the best understood realization of the holographic principle obtained from string theory, the AdS/CFT correspondence.

  • In chapter sec4?, we illustrate how the IHQCD model is developed to mimic the crucial properties of QCD. We discuss briefly the thermodynamic properties of IHQCD and our method of numerical integration of the equations of motion within the model.

  • In chapter ch6?, we present our original research, which amounts to the calculation of various energy momentum tensor correlators in IHQCD. We confront our results with those of perturbative and lattice QCD where available.

  • In chapter ch7?, we finally draw conclusions.

Chapter 2 Elements of Quantum Field Theory

To begin, we present a brief review some of the most important elements of Quantum Field Theory (QFT), with emphasis on concepts that we will encounter in the following chapters from a holographic viewpoint. We begin our treatment from scalar field theories, and then move on to gauge fields and their interactions, covering briefly also the large- limit of QCD as well as supersymmetry, before ending up with the basics of thermal field theory. The main references for this chapter are the excellent textbooks [20, 21] on a zero-temperature QFT, as well as the classic introduction to finite-temperature field theory, [22].

2.1 Quantum Fields at Zero Temperature

Consider a real scalar field , with denoting the coordinate of a four-dimensional Minkowskian spacetime with signature . We write the corresponding action in the form

(2.1)

where stands for the (as of yet unspecified) Lagrangian density for the scalar field.

2.1.1 Green’s functions

The most elementary called Green’s function or correlator, standing for the amplitude for a particle to propagate from a spacetime point to , is given by

(2.2)

where denotes the ground (vacuum) state of the field theory. This function can, however, be seen to obtain nonzero values also outside the lightcone, marking a violation of causality. Thus, we often rather work with the vacuum expectation value of the commutator , for which the noncausal parts can be seen to cancel.

Further, we can define two causual correlators, the retarded and advanced Green’s functions, corresponding to particles moving forward (backward) in time,

(2.3)
(2.4)

where is the Heaviside step function. Another important correlator, playing an important role in QFT, is the Feynman Green’s function or time-ordered correlator

(2.5)

where is the time-ordering symbol.

The most convenient way to evaluate the time-ordered correlator in the path integral formulation of QFT is using the generating functional of Green’s functions, . This quantity is defined by

(2.6)

where denotes functional integration over all field configurations. Using this function, we can calculate e.g. the vacuum expectation value of the field operator via

(2.7)

where and denotes a functional derivative. The time-ordered correlation function is further obtained as a second functional derivative

(2.8)

We can see that these results can be easily generalized to higher-point correlator functions by simply taking more functional derivatives, and even for arbitrary (usually local) operators, writing

(2.9)

For us, particlularly important operators , will be components of the energy momentum tensor , as well as the field strength tensor squared, , in the case of gauge theory.

2.1.2 Renormalization and Running Coupling

The simplest QFT one can write down is clearly a theory for one non-interacting scalar field, described by the Lagrangian

(2.10)

This is an exactly solvable model, and, as it turns out, actually the only example of a theory that can be fully analytically solved using standard methods. For example, if we add a quartic interaction to the Lagrangian

(2.11)

the functional integral in (genfunc?) are no longer straightforwardly solvable. What we can do in this case is, however, to use the machinery of perturbation theory, where we expand integrand of the generating functional in a power series in the coupling constant. This perturbative approach, however, works only if the coupling constant can be considered small.

When performing explicit calculations in a given QFT, one very often runs into ultraviolet (UV) divergences in loop diagrams, encountered when perturbative expansions are carried out beyond their leading orders. In order to subvert the UV divergences, the parameters (and sometimes even the fields) of the theory have to be redefined in a process called renormalization. This leads to the emergence of a corrected, or renormalized, action, which is free of divergent quantities. The coupling constant of this new action, called the renormalized coupling , is in general different from the original bare coupling , appearing in (lagphi4?). Upon subtracting the divergences of the latter, the renormalized coupling typically becomes a function of an energy-like parameter, the renormalization scale , giving rise to the term running coupling. This dependence is encoded in the beta function of the theory,

(2.12)

which e.g.  for a scalar theory with a quartic interaction reads

(2.13)

We observe that in this case, the coupling increases with energy, implying that at high-enough energies perturbation theory will inevitably fail. As we will see, this behavior is exactly opposite to that encountered in the theory of the strong nuclear interaction, where the coupling constant in fact decreases with energy.

2.1.3 Conformal Field Theory

As we have seen, quantum field theories are usually defined over Minkowski spacetime, and thus they have to respect the symmetry group of this space. The symmetry group of the Minkowski space is found as the symmetry group under which the metric

(2.14)

is invariant. This is the famous Poincare symmetry, which we know consists of translations as well as special orthogonal transformations, i.e. the Lorentz transformations.

A conformal field theory is a theory that is invariant under more general transformations that leave the metric invariant up to a scale change

(2.15)

It turns out that the conformal symmetry group is the product of translations and the group , representing the Lorentz and scaling symmetries. The presence of the scaling symmetry means that the theory must be free of dimensionfull parameters, such as mass scales. As a consequence, the beta function of conformal field theories has to vanish.

Conformal symmetry also constraints the form of correlation functions much more than Poincare symmetry alone. For example, we find that a general two-point correlation function in conformal field theory is given by

(2.16)

where is the scaling dimension of the operator .

2.1.4 Energy Momentum Tensor

Consider next in somewhat more detail the spacetime translations

(2.17)

under which the scalar field transforms as

(2.18)

Using Noether’s theorem, which states that to each symmetry of a Lagrangian corresponds a conserved current, we find that varying the action (scalaraction?) with respect to provides us with a conservation law

(2.19)

where

(2.20)

is the canonical energy momentum tensor. We note that the canonical energy momentum tensor is not symmetric by definition, which can be seen to result from the freedom to add any divergenceless quantity to . Adding a properly defined such quantity, we can however define a symmetric energy momentum tensor . It can be shown, that this symmetric energy momentum tensor can be alternatively derived through a variation of the action with respect to the spacetime metric,

(2.21)

where is the determinant of . We call the energy momentum tensor derived in this way the metric energy momentum tensor. From now on, unless specified otherwise, when we talk about the energy momentum tensor, we will have the symmetric, or metric, energy momentum tensor in mind.

2.2 Gauge Theory

Nearly all fundamental forces of the nature can be understood through theories with a gauge symmetry. The simplest example of this is electromagnetism, where we have only one gauge field

(2.22)

with the field strength

(2.23)

The electric and magnetic fields are then given by

(2.24)

while the Lagrangian density reads

(2.25)

This theory has a gauge symmetry under the group, meaning that the field strength is invariant under the local gauge transformation

(2.26)

The above relations define a pure Maxwell theory of photons, to which we have to add fermions (electrons) and specify the coupling between these two. Using the minimal coupling prescription, we obtain form here Quantum Electrodynamics, described by the Lagrangian density

(2.27)

where , with the so-called Dirac matrices and the gauge covariant derivative

(2.28)

The gauge symmetry in this case is the requirement that be invariant under the simultaneous transformations (gaugetransf?) of the gauge field and of the following transformation of the fermions

(2.29)

The symmetry group of the transformation (rot4?) is the unitary group , and hence we will call electrodynamics a gauge theory with a symmetry group.

2.2.1 Yang-Mills Theory

If we want to have more gauge fields in our theory, we have to consider their respective commutation relations. If all fields commute with each other, we call such a theory abelian. Electrodynamics is an example of an abelian gauge theory, since we have only one gauge field that trivially commutes with itself. The generalization of this construction to the non-abelian case is in general dubbed Yang-Mills Theory, where we have set of Yang-Mills fields , with an index belonging to some nonabelian gauge group. The gauge group is spanned by the generators belonging some representation and satisfying the Lie algebra of the group,

(2.30)

where are the corresponding structure constants.

The most important examples of non-abelian symmetry groups are the special orthogonal group and the special unitary group . We will be interested primarily in the last one, since is the gauge symmetry group of Quantum Chromodynamics, i.e. the theory of strong interactions. In this case, the index is called a color index and runs from to . The generators of the fundamental representation can be represented by traceless Hermitian matrices, i.e. the Pauli matrices for and the Gell-Mann matrices for .

The adjoint representation of a simple Lie group such as is defined through . This is important because the gluon fields live in the adjoint representation, giving rise to the field strength

(2.31)

Under a gauge transformation , the gluon fields transform as

(2.32)

implying that the field strength is not invariant under gauge transformation, but instead

(2.33)

The trace of the square of the field strength operator, , is however invariant, ensuring that Lagrangian density

(2.34)

is also invariant. This Lagrangian density describes pure Yang-Mills theory or gluodynamics, as so far we do not have any fermionic matter in the theory.

The energy momentum tensor for pure Yang-Mills theory is given by

(2.35)

with

(2.36)
(2.37)

where we have separated the traceless part and the anomalous trace proportional to the -function of the theory. The -function of the pure Yang-Mills theory is in turn given by

(2.38)

We can see that the sign of the beta-function is negative, and as a consequence of this the strength of the gauge interaction decreases with energy. This means that perturbation theory works in the Yang-Mills theory only at high enough energies — the completely opposite of the case of scalar field theory or QED. This property of Yang-Mills theory is known as asymptotic freedom.

2.2.2 Quantum Chromodynamics

Quantum chromodynamics (QCD) is the theory that describes the interactions between quarks and gluons. Formally, it is an Yang-Mills theory coupled to flavors of fundamental quarks, and thus has the Lagrangian density

(2.39)

where and

(2.40)

The beta function for this theory, given for the sake of generality for unspecified and , reads

(2.41)

From (QCDbeta?), we see that the beta function is still negative for the physical case of six quark flavors and three colors, indicating that QCD is, just as pure Yang-Mills theory, asymptotically free. For low energies, the strong coupling constant on is the other hand typically of order (even diverging at some critical scale ), and thus the theory must be treated nonperturbatively. A good example of this is zero-temperature nuclear physics, where the properties of e.g. protons and neutrons cannot be accessed using simple perturbation theory. It is in fact an extraordinary difficult task to calculate even the proton mass using only the structure of the underlying theory and the quark masses as input.

Due to asymptotic freedom, we can, however, use perturbative methods to understand processes well above the QCD energy scale . This is an intriguing consequence of asymptotic freedom, implying that it is e.g. easier to describe at least some features of ultrarelativistic heavy ion collisions than the structure of a proton. This is, of course, a very unsatisfactory situation, and there is a great need to develop nonperturbative methods to access the physics of the strong interaction at low energies. In this context, the holographic principle, introduced in the next chapter, is a prime example of a relatively recent nonperturbative method, which has been used to gain insights into the strongly coupled regime of QCD.

2.2.3 Large- Limit

It was realized by ’t Hooft [23] that gauge theory simplifies in the limit — an observation that will turn out to play a major role in holography. In our previous discussion, we absorbed the gauge coupling into the definition of the field strength . In order to understand, why Yang-Mills theory simplifies in the large- limit, we will now rewrite the action with a different normalization of the gauge field, leading to the Lagrangian

(2.42)

and the partition function

(2.43)

Next, we introduce the ’t Hooft coupling

(2.44)

which allows us to make the important observation that the logarithm of the partition function can be expanded in powers of as

(2.45)

where the are functions of only. What is remarkable about this expansion is that, at a fixed , Feynman diagrams are organized by their topologies. In particular, contains only the simplest diagrams that can be drawn on a planar surface without crossing any lines (so-called planar diagrams). We can see that in the limit , the partition function will be dominated by these diagrams and we can neglect the more complicated non-planar diagrams. This is called the large- limit or planar limit of Yang-Mills theory.

2.2.4 Supersymmetry and SYM theory

Poincare symmetry is defined by the generators of the Lorentz group as well as the generators of the translation group, . In addition, we have seen that in QFTs there exist internal symmetries, represented by the generators , such as the local gauge symmetry of Yang-Mills, satisfying the Lie algebra of (liealg?). It has been shown that it is not possible to combine spacetime and internal symmetries into a larger group, such that , . This result is known as the Coleman-Mandula theorem [24].

Despite the above, it turned out that if we generalize the notion of a Lie algebra to a graded Lie algebra, with some generators satisfying anticommutation laws, the theorem can be evaded. It turns out that these generators , called supercharges, have a natural representation as spinors, and thus produce another spinor field upon acting on a bosonic field. This means that these exotic generators provide a symmetry between bosons and fermions, called supersymmetry. For a more detailed explanation, see e.g. [25].

For our limited purposes, it is sufficient to know that supersymmetric transformations turn fermionic fields into bosonic, and vice versa. A supersymmetric theory with supercharges has a global symmetry, called the -symmetry. It turns out that in four dimensions (without gravity) we can have at most four supercharges, i.e. . A particularly interesting theory is generated when we consider the supersymmetric extension of Yang-Mills theory [26] with this maximal number of four supercharges, called the supersymmetric Yang-Mills (SYM) theory [27].

The bosonic part of the Lagrangian of the SYM theory is given by

(2.46)

It turns out that in this theory, the supersymmetry prevents gauge couplings to obtain corrections at the quantum level, and in fact makes the theory conformal. Thus the symmetry group of SYM theory is the product of the conformal symmetry group and the group corresponding to the -symmetry, whose bosonic part is ,

(2.47)

We will see that this theory will play a highly special role in holography.

Let us finally mention that we have so far discussed how one can add supersymmetry into a gauge theory. However, we can in fact also gauge supersymmetry itself, with the resulting theory being so-called supergravity, described by the action (sugra?). For details of the construction of supergravity theories, see e.g. [28].

2.3 Finite Temperature Field Theory

Consider next the Wick rotation

(2.48)

under which the 3+1-dimensional Minkowski spacetime metric turns into a four-dimensional Euclidean one. Under this transformation, we find that the generating functional (genfunc?) becomes

(2.49)

where is the Euclidean Lagrangian density . With this action, the functional integration converges much better, since the oscillatory exponent is replaced with a damped one, . While this offers us some computational advantages, there is also another advantage of the Euclidean approach; namely, it allows us to make a direct connection to thermal field theory.

Indeed, eq. (partfunc?) is the quantum field theory generalization of the well-known partition function of statistical physics in the path integral formalism. We can namely rewrite the above relation as

(2.50)

where we have exploited the form of the quantum mechanical time evolution operator and limited the temporal integration into a finite interval,

(2.51)

Here is the Hamiltonian, and the parameter is related to the temperature of the system, .

One assumption inherent in the above identification of the Euclidean functional integral as a partition function is that bosonic and fermionic fields satisfy the Kubo-Martin-Schwinger (KMS) relation, i.e. obey (anti)periodic boundary conditions in time,

(2.52)

where the + sign corresponds to bosonic and - sign to fermionic fields. This means that we can express the time dependence of the fields in terms of Fourier sums, writing

(2.53)

where the discrete read

(2.54)
(2.55)

with running over all integers. These frequencies are commonly referred to as Matsubara frequencies.

From the partition function (partfunc?), we can now calculate in principle all equilibrium thermodynamics properties of the system. For example, the pressure and entropy 111We use standard notation where is used for both entropy and action. However, it should be clear from the context which is which. are given simply by the relations

(2.56)
(2.57)

2.3.1 Euclidean Correlators

It is important to stress that the Euclidean approach is best suited for the description of systems in thermal equilibrium. When performing the Wick-rotation (Wick?), we trade the real-valued time coordinate for the Euclidean time , which is not directly related to the dynamics of the system. A consequence of this is that we have formally only one type of correlation function available, namely the Euclidean correlator

(2.58)

where denotes Euclidean time ordering.

Despite the above, also the Minkowski-space correlators , and , defined above in the context, have their counterparts in thermal field theory [29]. It in fact turns out that Euclidean and Minkowskian correlators are closely related, and e.g.the momentum space retarded correlator can be analytically continued to complex values of , where for it reduces to the Euclidean propagator

(2.59)

Similarly, we can write the Euclidean imaginary time correlator in the form

(2.60)

where

(2.61)

is the so-called spectral density.

The relation (euclidean?) has the remarkable property that we can in principle find Minkowskian correlators from the (typically much simpler) Euclidean ones. In order to do so, we however need to invert the integral relation, meaning that the Euclidean correlator must be known for all frequencies, which turns out to be very often a very difficult task. For example, in lattice field theory, where one uses computer simulations to measure the Euclidean correlators, one typically only obtains the results for a finite number of points and with considerable error bars, making the above analytic continuation in practice impossible.

2.4 Hydrodynamic Limit

As the final topic of our field theory chapter, let us briefly comment on the fact that close to thermal equilibrium, many physical systems allow a description using fluid dynamics — a theory that e.g. in the case of guark gluon plasma physics can be viewed as an effective theory for the long wavelength field modes. It is thus suitable for the description of processes involving large distances and time scales [30], understood here as large in comparison with the typical microscopic scales, e.g. the temperature of the system. This means that a hydrodynamic description can be expected to be valid when .

The equations of hydrodynamics are not derived from an action principle, but rather as conservation laws for the energy momentum tensor

(2.62)

Assuming local thermal equilibrium, which implies that the system can be described in terms of a spatially varying ‘temperature field’ and a local fluid velocity , we find that there are only four independent variables, since . This number agrees with the number of equations in (conservation?), and implies that the energy momentum tensor in eq. (conservation?) will be a function of only these four quantities and their derivatives.

At the lowest order in a derivative expansion, we find that the most general form of is that of an ideal fluid, and that the conservation equation (conservation?) is equivalent to the relativistic Euler equation. As it is well-known, ideal hydrodynamics cannot describe dissipation processes, since entropy is conserved. This means that if we desire to take dissipation into account, we have to go to next order, where we write the energy momentum tensor in the form

(2.63)

where is the dissipative part of , proportional to the first derivatives of and . The most general rotationally invariant turns out to be given by

where and are the shear and bulk viscosities, respectively. Using this form of the energy momentum tensor we find that the conservation equations (conservation?) give us a relativistic generalization of the Navier-Stokes equations.

2.4.1 Kubo Formulae for Viscosities

In a field theory, the energy momentum tensor (emtYM?) is expressed as a function of the field strengths , while in hydrodynamics it is given by (emthydro1?)-(emthydro2?). Since hydrodynamics is just an effective theory at large time and space scales, these two must be related. This relation is found using linear response theory, where we couple some sources to bosonic operators , such that action reads

(2.64)

If we assume that the expectation values of the operators vanish when evaluated with the action , and can be considered small, then the expectation values are given by

(2.65)

where is the retarded Green’s function of the operator .

If we now apply linear response theory to the hydrodynamics setting, we find (after some work) that the shear and bulk viscosities are given by the relations

(2.66)
(2.67)

where

(2.68)

and we have defined the shear and bulk channel retarded correlators

(2.69)
(2.70)

These relations are called the Kubo formulae for the viscosities [31].

Chapter 3 Introduction to Holography

Holography [32, 33] is a puzzling idea stating that a theory of quantum gravity in some manifold can be fully described by a non-gravitational field theory living on the boundary . At first sight, it is surprising that such relation between a theory with gravity and one without it could exist. In particular, there are two very curious aspects of this principle.

First, gravity is very different from other field theories. In the modern theory of gravity, general relativity, the motion of a particle in a gravitational field is understood as inertial motion in curved spacetime. This is known as the equivalence principle and, as a consequence, we do not have any appropriate concept of force as in other field theories. Thus it is quite surprising that gravity could be equivalent to some non-gravitational field theory.

The second surprising aspect of this principle is that gravity and its equivalent field theory live on spacetimes of different dimensionality. We will see that this can be understood as a consequence of the very peculiar structure of gravity, where the partition function is given only by a boundary term. As a consequence, the entropy of black holes scales not with volume as it does in any other local field theory, but with the surface of the black hole. Indeed, a study of black hole entropy was one of the primary motivations behind the holographic principle.

In this chapter, we will present a specific realization of the holographic principle, the AdS/CFT correspondence, which is the exact equivalence between a certain string theory in Anti-de Sitter space and the conformal supersymmetric Yang-Mills (SYM) theory in four-dimensional Minkowski space [34, 35]. While we can obtain some insights into the holographic nature of gravity already from general relativity, string theory gives us an exact equivalence of partition functions in both gravity and the corresponding dual field theory, and also provides us with an exact correspondence between observables in these two theories.

In addition to being the most important realization of the holographic principle, the AdS/CFT correspondence has another interesting property. It turns out that it is a strong-weak duality, i.e. it equates a strongly coupled theory on one side with a weakly coupled one on the other side. This is actually the most interesting property of the AdS/CFT correspondence, since it allows us to look into the strongly coupled regimes of both gravitational and gauge theories. In this thesis, we concentrate on using this correspondence to understand strongly coupled field theories in terms of weakly interacting gravity.

At the end of this chapter, we use the AdS/CFT correspondence to calculate some properties of hot SYM theory in the hydrodynamic limit. In particular, we are interested in the viscosities of the theory, and we derive the well-known result that theories with gravitational duals have a universal shear viscosity to entropy ratio.

There are a great number of reviews and introductory texts to this topic. Let us mention here a few of them that we have found helpful [36, 37, 38, 39, 40].

3.1 The Holographic Principle

In general relativity, the gravitational interaction is understood in terms of the curvature of a spacetime manifold. The geometry of a manifold is in turn determined entirely by the metric tensor and the Levi-Civita connection, .

The action that describes the interaction of matter and gravity is given by

(3.1)

where is the action for matter and is the Einstein-Hilbert action describing gravitation. In dimensions it reads

(3.2)

where

(3.3)

are the scalar curvature and Ricci tensor, respectively. Both of these are contractions of the Riemann tensor

(3.4)

The Einstein field equations are then obtained by varying the action (gravaction?) with respect to the metric as

(3.5)

where is the metric energy momentum tensor as defined in the previous chapter.

We can think of the Riemann tensor as playing a similar role as the field strength in Yang-Mills theory, but there is one crucial difference. In Yang-Mills theory, the action contains only first derivatives of gauge fields, and consequently the equations of motion are second order differential equations. This is very different in the gravitational case, where the action (EHaction?) contains second derivatives of the metric but the Einstein equations (eeq?) are also of second order.

The reason for this is that all of the second derivatives in the Einstein-Hilbert action are surface terms, and in the case of a closed manifold, i.e. a manifold which is both compact and without a boundary, they do not contribute to the equations of motion. For a manifold with boundary , it was shown by York [41], and later by Gibbons and Hawking [42], that the appropriate action is

(3.6)

where is the Gibbons-Hawking-York boundary term, is the induced metric on the boundary and is the trace of the exterior curvature of the boundary. The addition of this boundary term to the the action ensures that the variational principle is well-defined.

In the vacuum, the Einstein-Hilbert action has the peculiar property that it vanishes as a consequence of the equations of motion. This means that the on-shell gravitational action is given entirely by the Gibbons-Hawking-York boundary term

(3.7)

The action (gravac?) is important in a construction of the partition function for gravity. The Euclidean partition function can be written [42] as

(3.8)

where represents functional integration over all possible metric configurations. In the following section, we show an interesting example where we evaluate this partition function for a black hole, and we will see how this leads to the Bekenstein-Hawking entropy, which scales with the surface of the black hole.

One way to understand entropy is by the amount of information which is needed in order to fully specify the system, and thus we naturally expect that it will scale with the volume of the space. The scaling of the black hole entropy with the surface lead ’t Hooft [32] to conjecture that gravity could be in its nature holographic, i.e. it could be entirely characterized by its behavior at the boundary of the space. This property has become known as the holographic principle [32, 33].

A theory of quantum gravity in a manifold is equivalent to a non-gravitational field theory living on the corresponding boundary .

This is indeed a very important insight into gravity, and it changes completely our view of the gravitational interaction. Let us note that the holographic principle was originally formulated for theories of quantum gravity, but its consequences even reach the level of classical general relativity.

So far, the way we have stated holographic principle is rather vague, as it only says that gravity is equivalent to some field theory. In order to make the statement more useful, we have to make this relation somewhat more concrete. This was achieved in 1997 by Maldacena [34] who found an explicit realization of the principle in string theory, i.e. the AdS/CFT correspondence, which we will introduce below in section adscft?.

However, before going into string theory let us show two applications of what we have just discussed. First, we calculate the Bekenstein-Hawking entropy of a black hole using the partition function of gravity, and then we show the importance of the boundary term in the definition of the gravitational energy momentum tensor.

3.1.1 Black Hole Entropy

The best-known solution of Einstein equations is the Schwarzchild solution, which is a static and spherically symmetric solution to the vacuum Einstein equations

(3.9)

In four dimensions, the metric of this solution is

(3.10)

where represents the metric of a two-sphere. This solution describes a black hole, which is characterized by the existence of an event horizon, the apparent singularity at . Excellent references for black hole physics are [43] and for more mathematical aspects [44].

Here we would like to discuss only one aspect of black holes, that is their entropy and temperature. We start from the observation that is only an apparent singularity that can be removed by a coordinate transform. In Kruskal coordinates, the metric (schwarz?) becomes regular at the horizon

(3.11)

where the coordinates and are defined as

(3.12)
(3.13)

Let us now perform a Wick transformation, . From (krusk1?)-(krusk2?), it follows then that in Euclidean time the Schwarzchild solution is periodic in with a period . This means that using Euclidean formalism we can assign to the black hole a temperature

(3.14)

This is the famous Hawking temperature [45, 46] of a black hole. From the existence of a temperature, we can calculate directly the entropy of the black hole, as entropy is given by , where is the mass (energy) of the black hole.

However, let us choose a different approach, where we calculate the entropy using the Euclidean partition function of gravity. First, we need to evaluate the gravitational action, which is given only by the boundary term, since by the equations of motion. In order to evaluate the boundary term, we need to specify the boundary , which we choose to be the hypersurface for some constant . Following [42], we find that the Euclidean action is

(3.15)

In the asymptotic limit , only the first term survives. We can then evaluate the partition function, and using (entrop?) we find the entropy of the black hole

(3.16)

where is the area of the surface of the black hole. Alternatively, we can restore the gravitational constant

(3.17)

This is the Bekenstein-Hawking entropy of a black hole. For the original derivation, see [47].

This entropy formula has an interesting consequence, as it says that the maximal amount of information a system can store is proportional to the external surface of the system. In order to see this, consider some region of space with volume and external surface . Now, start adding some matter into this region. Then, at some point we achieve some maximum energy density, and to this configuration corresponds the maximal entropy, . Since entropy is from a microscopic viewpoint interpreted as the amount of information needed to fully specify the system, in a local field theory we expect that it would be proportional to the volume .

However, in gravity the situation is different. It is important to realize that the maximal energy density , which can be achieved in a given region, is the energy density of a black hole. Indeed, adding more matter will only increase the size of the black hole. From (bhent?), it follows that the entropy of a black hole is proportional to the surface . Thus the maximal entropy of the space region is given by the size of the largest black hole it can contain, and thus will be also proportional to the surface area. In the spherical case, this is known as the spherical entropy bound [33]:

(3.18)

For generalizations of this entropy bound, see [48, 49, 50].

The holographic principle is an elegant way to understand the above, since it says that gravity is equivalent to some non-gravitational field theory living on the boundary. The boundary field theory is typically considered to be a local field theory, where entropy scales with the dimensionality of the boundary, i.e. with the surface area. Thus, the holographic principle explains why this holds for gravity in the bulk spacetime.

3.1.2 Holographic Renormalization

An interesting problem in the theory of gravity is the definition of the energy of a gravitational field. Since the Einstein-Hilbert action vanishes on shell, the canonical energy momentum tensor (canten?) associated with this action is zero and the same holds for the metric energy momentum tensor (metem?). In general relativity, there were some attempts to solve this problem by so-called energy momentum pseudotensors, which are supposed to represent the local energy of the gravitational field, but these objects do not transform as tensors [43] and thus the energy of the gravitational field is not observer independent.

It is in fact the boundary term in the action that allows us to define a physical energy momentum tensor for the gravitational field. This is the well-known Brown-York tensor [51]

(3.19)

We can see that the Brown-York tensor is formally similar to the metric energy momentum tensor (metem?), but the variation is performed with respect to the boundary metric and the resulting tensor is defined only on the boundary. We call such a definition quasilocal.

A problem with the Brown-York tensor is that it typically diverges when we take the boundary, where it is defined, to infinity. In [51], a solution for this was proposed, namely that we should subtract the energy momentum tensor defined on the boundary with the same metric , but in some reference spacetime, such as flat space. However, it turns out that it is not possible to do this for a general boundary geometry and reference spacetime.

For asymptotically Anti-de Sitter spacetimes (see Appendix append? for a review of AdS space), an interesting solution to this problem emerged that does not involve any subtractions from reference spacetimes [52]. In this work, it was realized that, in asymptotically AdS spacetimes, we can interpret the divergences of the Brown-York tensor with the divergences of a field theory defined on the boundary. Moreover, they found that the trace of the gravitational energy momentum tensor equals to the trace anomaly of a conformal field theory living on the boundary.

We will see that this is exactly the situation encountered in the AdS/CFT correspondence, where gravity in the AdS space is dual to a conformal field theory on the boundary. The work of [52] was performed after Maldacena discovered the AdS/CFT correspondence, but it does not rely on it. Thus, we can consider it to be an independent motivation leading to the duality between gravity in AdS space and conformal field theories living on its boundary.

As an alternative to the subtraction scheme of Brown and York [51], it was proposed in [52] that divergences can be removed by adding counterterms into the action. This is a procedure very similar to the renormalization procedure in quantum field theory described in the previous chapter, and thus the term holographic renormalization was coined for it.

Holographic renormalization is based on the simple observation that we can freely add to the gravitational action (gravac?) a term that does not affect the equations of motion

(3.20)

where is now the Einstein-Hilbert action including a negative cosmological constant. This term vanishes due to the equations of motion and thus only last two terms give non-zero contribution to the Brown-York tensor. A requirement to cancel divergences determines uniquely form of .

Let us now consider only the case of an space, where is given by

(3.21)

We can then calculate the Brown-York energy momentum tensor

(3.22)

where is the Einstein tensor, , of the boundary metric. We find that the trace of this expression is given by

(3.23)

which can be recognized (up to a constant) to be exactly the trace anomaly of conformal field theory. If we use the relation between and the corresponding coupling constant in SYM from the AdS/CFT correspondence, we see that eq. (trac?) agrees exactly with the prediction of conformal field theory. A truly remarkable result. For futher discussion of the renormalization of the gravitational action, see e.g. [53, 54].

3.2 The AdS/CFT Correspondence

In the previous section, we have motivated the holographic principle using properties of classical general relativity and a study of black hole thermodynamics. However, the holographic principle as we have described it, is somewhat vague, since it does not specify exactly, which field theory should be equivalent to which gravitational theory and under what circumstances this equality holds.

The first explicit realization of the holographic principle was found by Maldacena in 1997 [34], when he found an exact duality in string theory, known today as the AdS/CFT correspondence. Maldacena conjectured that type IIB string theory on is equivalent to four-dimensional conformal Super Yang-Mills (SYM) theory. Later, Witten [55] and Gubser et al. [56], established an exact correspondence between the partition functions and fundamental observables of these theories. The AdS/CFT correspondence has later been generalized to a situation where both supersymmetry and conformal invariance are broken. These models are generally known as gauge/gravity dualities.

We begin our discussion of the AdS/CFT correspondence with a short review of string theory, and then move on to introduce the duality and its generalizations.

3.2.1 String Theory: A Quick Review

Unlike quantum field theory, where elementary particles are considered to be point-like objects, string theory considers them one-dimensional objects, i.e. strings. In string theory, we have two characteristic parameters, the string tension and a dimensionless coupling constant that controls the strength of the interaction. We can write the string tension in the form

(3.24)

where is the string length. The action of a pointlike particle in a relativistic theory is proportional to the length of the particle’s worldline. In string theory, it is correspondingly proportional to the area of the worldsheet, i.e. the two-dimensional generalization of the wordline,

(3.25)

where are coordinates defined on the worldsheet and is the determinant of the worldsheet metric.

Strings are extended objects and thus we have to discuss their endpoints. There are two possibilities for this, as the string can be either closed or open. In the case of open strings, we have to specify boundary conditions for the endpoints, and there are again two possibilities: We impose either the Dirichlet or von Neumann boundary conditions. We will also see that a special role is played in string theory by objects called -branes, which are extended objects at which open strings can end with Dirichlet boundary conditions.

In the case of closed strings, we do not have to specify boundary conditions, since their endpoints are identified and they can freely move in the bulk spacetime. Quantizing the string action (NambuGoto?), we find the spectrum of the excitations of the string. In the case of a closed string, we find also a massless particle of spin two, i.e. a graviton. This is the reason why the theory of closed strings is considered to be a theory of gravity. Moreover, since string theory allows us to calculate quantum corrections and solves certain problems with divergences that plague other approaches to quantize gravity, we consider string theory to be a theory of quantum gravity.

During the quantization of the above action, we find the surprising property that the spacetime Lorentz group is anomalous, unless the string lives in 26 spacetime dimensions. This number of spacetime dimensions is called the critical dimension, and such a theory is called critical string theory. However, string theory with the action (NambuGoto?) describes only bosons, and in order to provide a full description of Nature, we need to introduce fermions into the theory. This is done using supersymmetry, which surprisingly decreases the critical dimension of the theory to only 10.

In order to understand the basics of the AdS/CFT conjecture, it is sufficient for us to consider only type IIB string theory, which in its low-energy limit reduces to supergravity (SUGRA) given by the action

(3.26)

where is the Ricci scalar, is the dilaton field and is the field strength for the four form , and dots represent fermionic terms and other so-called Ramond-Ramond forms that are all irrelevant for our consideration. The gravitational constant, , can be expressed in terms of the string length and coupling constant as

(3.27)

An important point that we would like to stress is that the string coupling is not a free parameter here, but is given by the expectation value of the dilaton field as . When we talk about the string coupling constant, we in fact mean its asymptotic value at infinity, i.e. .

3.2.2 The Maldacena Conjecture

The Maldacena conjecture follows from the observation of the dual role that -brane solutions play in string theory. In the low energy limit of closed string theory, they are solutions of the supergravity action. In open string theory, they play the role of objects on which open strings can end up, and they give the rise to a non-Abelian gauge-theory. Next, we will discuss these two perspectives separately.

Closed String Perspective: -branes as Spacetime Geometry

In 1995, Polchinski [57] discovered that -branes can be identified with the black -branes solutions of SUGRA. The black -branes are higher dimensional generalizations of black hole solutions known from general relativity. We can find the -brane solution considering the metric ansatz

(3.28)

where is an element of the unit 5-sphere, and

(3.29)

Writing down the field equations for the action (sugra?) and imposing self-duality for the field strength, , we can find a solution (see [36] for a detailed derivation).

Instead of writing the solution for a single -brane, we can write it directly for coincident branes,

(3.30)
(3.31)

where the last equation says that due to the existence of -branes, the field strength must be quantized. Observe that the dilaton field is in this example constant and thus the string coupling is also a constant. See also section nonconfmodel? for a generalization where the dilaton field is allowed to vary over space.

The length parameter appearing above is given by

(3.32)

In the limit , we can neglect the in the function and write it as

(3.33)

Then the -term in front of cancels out and we find that the metric for coincident -branes reads

(3.34)
(3.35)

We can see that the first part of (branesol1?) is the metric of a black hole in space in the planar limit (see Appendix A), and the second part is the metric of a 5-dimensional sphere with radius .

In the limit , we find that the metric (branesol1?) simplifies to

(3.36)

Using the coordinate transformation , we rewrite this in the form

(3.37)

where the first part can be recognized as the metric of empty space. Thus, in this special limit, the metric of coincident -branes reduces to the topological product of pure space and an sphere,

Remember that classical SUGRA is a valid approximation of type IIB string theory only if we can neglect both quantum and string effects. String effects are negligible, if the string length can be considered small when compared with the other length scale in the problem, the AdS radius . Quantum effects can on the other hand be considered negligible, if a dimensionless combination of the gravitational constant and the AdS radius is small enough. These two conditions read

(3.38)
Open String Perspective: -branes and Gauge Fields

In open string theory, the -branes are viewed as four-dimensional hypersurfaces, on which open strings can end up with Dirichlet boundary conditions [57]. The action in this case is given by

(3.39)

where is the action of a four-dimensional theory on the brane, is the action describing the interaction in the bulk space, and is the interaction term between these two. It turns out that the bulk theory is SUGRA coupled to the massive string modes.

A crucial observation is that in the low-energy limit , the massive string modes drop out and the bulk theory reduces to free supergravity, while the interaction term, proportional to the , can be neglected. Thus, we obtain two decoupled (non-interacting) theories, free SUGRA in the bulk with the action and a field theory on the brane, .

It turns out that in a low energy limit, if we consider parallel coinciding -branes, the theory on the brane is given by the Lagrangian [58]

(3.40)

which can be recognized as the Lagrangian of SYM theory with the symmetry group , with the coupling constant

(3.41)

The number of colors can be identified with the number of the -branes, i.e. .

The Conjecture and the ’t Hooft Limit

Above we saw that the physics of -branes, particularly when we consider coincident branes, can be described by type IIB string theory in in the closed string perspective or by SYM in the open string theory perspective. The AdS/CFT correspondence [34] is the conjectured equivalence of these two theories

(3.42)

with the parameters of the theories related by

(3.43)

where is the ’t Hooft coupling.

Let us next consider the symmetries of the theories on both sides of the conjecture. In section symch?, we found that the symmetry group of SYM theory is , where represents the conformal symmetry and