Chapters 4 and 5 together constitute a primer on gauge/string duality, written for a QCD audience.

Our goal in this section is to state what we mean by gauge/string duality, via a clear statement of the original example of such a duality [594, 392, 803], namely the conjectured equivalence between a certain conformal gauge theory and a certain gravitational theory in anti-de Sitter (AdS) spacetime. We shall do this in Section 4.3. In order to get there, in Section 4.1 we will first motivate from a gauge theory perspective why there must be such a duality. Then, in Section 4.2, we will give the reader a look at all that one needs to know about string theory itself in order to understand Section 4.3, and indeed to read this book.

Since some of the contents of this chapter are by now standard textbook material, in some cases we will not give specific references. The reader interested in a more detailed review of string theory may consult the many textbooks available such as [386, 587, 685, 501, 823, 532, 133, 321]. The reader interested in complementary aspects or extra details about the gauge/string duality may consult some of the many existing reviews, e.g. [29, 319, 596, 749, 672, 601, 340, 398, 730, 687].

Motivating the duality

Although the AdS/CFT correspondence was originally discovered [594, 392, 803] by studying D-branes and black holes in string theory, the fact that such an equivalence may exist can be directly motivated from certain aspects of gauge theories and gravity. In this section we motivate such a direct path from gauge theory to string theory without going into any details about string theory and D-branes.

In Chapter 6 we have described ways in which holographic calculations have yielded insight into various properties of strongly coupled plasma that is at rest and in thermal equilibrium or, in our discussion of transport properties, infinitesimally close to being at rest and in thermal equilibrium. In this chapter, we release these restrictions. By the end of the chapter, we will be analyzing violent dynamical processes that are initially very far from thermal equilibrium and that may provide a caricature of the dynamics in the earliest moments of a heavy ion collision, or at least what that dynamics would be if the physics then was already strongly coupled. Before the development of gauge/gravity duality, there were no reliable quantum field theoretical calculations valid in far-from-equilibrium, highly time-dependent, strongly coupled matter. We shall build up the holographic tools that can now provide such calculations in stages over the course of the chapter. In doing so we shall make the connections to heavy ion physics manifest throughout, but it is important to realize that these tools are of considerable value in any other quantum field theoretical context in which the physics is strongly coupled and the questions of interest include far-from-equilibrium dynamics and thermalization.

We begin by letting the strongly coupled plasma move. In Sections 7.1 and 7.2 we show how to construct the gravitational description of solutions to the hydrodynamic equations for strongly coupled plasma in motion. That is, we continue to assume local thermal equilibrium, but we let the plasma move and flow.

Since the purpose of heavy ion collisions is to study the properties of Quantum Chromodynamics at extreme temperature and energy density, any successful phenomenology must ultimately be based on QCD. However, as discussed in Chapter 2, heavy ion phenomenology requires strong coupling techniques not only for bulk thermodynamic quantities like the QCD equation of state, but also for many dynamical quantities at nonzero temperature, such as transport coefficients, relaxation times and quantities accessed by probes propagating through a plasma. By now, lattice-regularized QCD calculations provide well controlled results for the former, but progress on all the latter quantities is likely to be slow since one needs to overcome both conceptual limitations and limitations in computing power. Alternative strong coupling tools are therefore desirable. The gauge/string duality provides one such tool for performing nonperturbative calculations for a wide class of non-Abelian plasmas.

In this book we have mostly focused on results obtained within one particular example of a gauge/string duality, namely the case in which the gauge theory is N = 4 SYM or a small deformation thereof. One reason for this is pedagogical: N = 4 SYM is arguably the simplest and best understood case of a gauge/string duality. By now many examples are known of more sophisticated string duals of non-supersymmetric, nonconformal QCD-like theories that exhibit confinement, spontaneous chiral symmetry breaking, thermal phase transitions, etc.

As discussed in Section 2.4, heavy quarks and quarkonium mesons, with masses such that M/T ≫ 1, constitute valuable probes of the QGP. Since dynamical questions about these probes are very hard to answer from first principles, here we will study analogous questions in the strongly coupled N = 4 SYM plasma. In this case the gauge/string duality provides the tool that makes a theoretical treatment possible. Although for concreteness we will focus on the N = 4 plasma, many of the results that we will obtain are rather universal in the sense that, at least qualitatively, they hold for any strongly coupled gauge theory with a string dual. Such results may give us insights relevant for the QCD quark–gluon plasma at temperatures at which it is reasonably strongly coupled.

The QGP only exists at temperatures T > Tc, so in QCD the condition M/T ≫ 1 can only be realized by taking M to be large. In contrast, N = 4 SYM is a conformal theory with no confining phase, so all temperatures are equivalent. In the presence of an additional scale, namely the quark or the meson mass, the physics only depends on the ratio M/T. This means that in the N = 4 theory the condition M/T ≫ 1 can be realized by fixing T and sending M to infinity, or by fixing M and sending T → 0; both limits are completely equivalent. In particular, the leading order approximation to the heavy quark or quarkonium meson physics, in an expansion in T/M, may be obtained by setting T = 0. For this reason, this is the limit that we will study first.

As discussed in Sections 2.3 and 2.4, two of the most informative probes of strongly coupled plasma that are available in heavy ion collisions are the rare highly energetic partons and quarkonium mesons produced in these collisions. In this chapter and in Chapter 9, we review results obtained by employing the AdS/CFT correspondence that are shedding light on these classes of phenomena. In Sections 8.1 and 8.2, we describe how a test quark of mass M moving through the strongly coupled N = 4 SYM plasma loses energy and picks up transverse momentum. In Section 8.3 we consider how the strongly coupled plasma responds to the hard parton plowing through it; that is, we describe the excitations of the medium which result. In Section 8.4, we discuss calculations of the stopping distance of a light quark moving through the strongly coupled plasma. Throughout Sections 8.1, 8.2, 8.3 and 8.4 we assume that all aspects of the phenomena associated with an energetic parton moving through the plasma are strongly coupled. In Section 8.5, we present an alternative approach in which we assume that QCD is weakly coupled at the energy and momentum scales that characterize gluons radiated from the energetic parton, while the medium through which the energetic parton and the radiated gluons propagate is strongly coupled. In this case, one uses the AdS/CFT correspondence only in the calculation of those properties of the strongly coupled plasma that arise in the calculation of radiative parton energy loss and transverse momentum broadening.