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In these lectures we discuss methods for computing scattering amplitudes in \({\cal N}=4\) super-Yang–Mills theory in both weak and strong coupling expansions.

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I review several old/new approaches to the string/gauge correspondence for the cusped/lightcone Wilson loops. The main attention is payed to SYM perturbation theory calculations at two loops and beyond and to the cusped loop equation.

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We study the semiclassical spectrum of bosonic string theory on AdS\(_{3}\times S^{1}\) in the limit of large AdS angular momentum. At leading semiclassical order, this is a subsector of the IIB superstring on AdS\(_{5}\times S^{5}\). The theory includes strings with \(K\geq 2\) spikes which approach the boundary in this limit. We show that, for all \(K\), the spectrum of these strings exactly matches that of the corresponding operators in the dual gauge theory up to a single universal prefactor which can be identified with the cusp anomalous dimension. We propose a precise map between the dynamics of the spikes and the classical SL\((2,\mathbb {R})\) spin chain which arises in the large-spin limit of \({\cal N}=4\) Super Yang–Mills theory.

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We summarise the status of an intriguing new duality between planar maximally helicity violating scattering amplitudes and light-like Wilson loops in \(\mathcal {N}=4\) super Yang–Mills. In particular, we focus on the role played by (dual) conformal symmetry, which is made predictive by deriving anomalous conformal Ward identities for the Wilson loops. Assuming the duality, the conformal symmetry of the dual Wilson loops becomes an unexpected new symmetry of scattering amplitudes in \(\mathcal {N}=4\) SYM.

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The main features of the cusp anomalous dimension in \({\cal N}=4\) supersymmetric Yang–Mills theory are reviewed. Moreover, the strong coupling expansion of the cusp derived in B. Basso, G.P. Korchemsky, J. Kotanski, Phys. Rev. Lett. 100, 091601 (2008) is presented.

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In this paper we will review the symmetries of the AdS/CFT S-matrix, in particular, the Hopf Algebra and Yangian symmetries.

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Magnon interactions give important contributions to the wrapping interactions of the \(\mathcal {N}=4\) spin-chain. Similar effects are expected for the finite size corrections to the giant magnon energy in \(\mathrm {AdS}_5 \times {\rm S^5}\). In this paper I review the finite gap description of giant magnons and the leading order calculation of the finite size corrections to the giant magnon dispersion relation for multi-magnon states.

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In this talk we study the hard wall AdS/QCD model, the simplest model of bottom-up approach to AdS/QCD. We reveal the relations between fields in the model and operators in QCD, fix parameters of the model and calculate several quantities of interest. We underline the problems of the model and propose the way to solve them.

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The main results concerning the 4-point super-conformal blocks in the \(N=1\) Neveu–Schwarz sector are resumed.

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We review recent applications of gauge/gravity duality to finite-temperature quantum field theory. In particular we describe how the shear viscosity can be computed from gravity duals.

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The contribution presents a summary of the Gauge/Gravity approach to the study of hydrodynamic flow of the quark–gluon plasma formed in heavy-ion collisions. Considering the ideal case of a supersymmetric Yang–Mills theory for which the AdS/CFT correspondence gives a precise form of the Gauge/Gravity duality, the properties of the strongly coupled expanding plasma are put in one-to-one correspondence with the metric of a 5-dimensional black hole moving away in the 5th dimension and its deformations consistent with the relevant Einstein equations. Several recently studied aspects of this framework are recalled and put in perspective. This paper is a written version of the four lectures given by the authors on that subject.

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We briefly discuss our first attempt to describe an anisotropic quark–gluon plasma at strong coupling in the AdS/CFT correspondence framework. We constructed an exact dual gravity solution and found that despite the fact that it is singular it allows for a construction of natural incoming boundary conditions. A study of a small perturbation about this geometry shows that the dispersion relation depends strongly on the relative sign of the wave vector and the sign of the anisotropy. Yet, we did not encounter any instabilities.

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We give a pedagogical review of recent progress towards understanding the response of a strongly coupled plasma at finite temperature to a hard probe. The plasma is that of the \({\mathcal N}=4\) supersymmetric Yang–Mills theory and the hard probe is a virtual photon, or, more precisely, an \({\mathcal R}\)-current. Via the gauge/gravity duality, the problem of the current interacting with the plasma is mapped onto the gravitational interaction between a Maxwell field and a black hole embedded in the AdS\(_5\times {\rm S}^5\) geometry. The physical interpretation of the AdS/CFT results can be then reconstructed with the help of the ultraviolet/infrared correspondence. We thus deduce that, for sufficiently high energy, the photon (or any other hard probe: a quark, a gluon, or a meson) disappears into the plasma via a universal mechanism, which is medium-induced quasi-democratic parton branching: the current develops a parton cascade such that, at any step in the branching process, the energy is almost equally divided among the daughter partons. The branching rate is controlled by the plasma which acts on the coloured partons with a constant force \(\sim T^2\). When reinterpreted in the plasma infinite momentum frame, the same AdS/CFT results suggest a parton picture for the plasma structure functions, in which all the partons have fallen at very small values of Bjorken’s \(x\). For a time-like current in the vacuum, quasi-democratic branching implies that there should be no jets in electron–positron annihilation at strong coupling, but only a spatially isotropic distribution of hadronic matter.

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We study the fate of the energy deposited by a jet in a heavy ion collision assuming that the medium created is opaque (jets quickly lose energy) and its viscosity is so low that the energy lost by the jet is quickly thermalized. The expectation is that under these conditions the energy deposited gives rise to a Mach cone. We argue that, in general, the behavior of the system is different from the naive expectation and it depends strongly on the assumptions made about the energy and momentum deposited by the jet into the medium. We compare our phenomenological hydrodynamic calculations performed in a static medium for a variety of energy-momentum sources (including a pQCD-based calculation) with the exact strong coupling limit obtained within the AdS/CFT correspondence. We also discuss the observability of hydrodynamical features triggered by jets in experimentally measured two-particle correlations at RHIC.

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In this paper we review how a background independent non-pertur- bative regularization of quantum gravity, denoted causal dynamical triangulation (CDT), in the infrared leads to the standard minisuperspace effective action. We show how it is possible to study in detail the quantum fluctuations around the classical solution to the minisuperspace action and outline how one in principle might be able to study the quantum gravity theory in the sub-Planckian regime.

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The method of Causal Dynamical Triangulations is a background independent approach to Quantum Gravity. Imposing causal structure of the universe we observe a classical 4D de Sitter spacetime as a “background” geometry. From the study of the spatial volume fluctuations one can determine the effective action for the scale factor. In this approach one obtains a minisuperspace model which has a maximum symmetry by integrating out all degrees of freedom except the scale factor and not by freezing them.

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We review a recently discovered continuum limit for the one-matrix model which describes “causal” two-dimensional quantum gravity. The behaviour of the quantum geometry in this limit is different from the quantum geometry of Euclidean two-dimensional quantum gravity defined by taking the “standard” continuum limit of the one-matrix model. Geodesic distance and time scale with canonical dimensions in this new limit, contrary to the situation in Euclidean two-dimensional quantum gravity. Remarkably, whenever we compare, the known results of (generalized) causal dynamical triangulations are reproduced exactly by the one-matrix model. We complement previous results by giving a geometrical interpretation of the new model in terms of a generalization of the loop equation of Euclidean dynamical triangulations. In addition, we discuss the time evolution of the quantum geometry.

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We present a semiclassical description of the SU\((N)\) Yang–Mills theory whose partition function at nonzero temperatures is approximated by a saddle point — an ensemble of an infinite number of interacting dyons of \(N\) kinds. The ensemble is governed by an exactly solvable 3D quantum field theory, allowing calculation of correlations functions relevant to confinement. We show that known criteria of confinement are satisfied in this semiclassical approximation: (i) the average Polyakov line is zero below some critical temperature, and nonzero above it, (ii) static quarks in any nonzero \(N\)-ality representation have linear rising potential energy, (iii) the average spatial Wilson loop falls off exponentially with the area, (iv) \(N^{~2}\) gluons are canceled out from the spectrum. The critical temperature is in good agreement with lattice data.

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Renormalization group procedure suggests that the low-energy behavior of effective coupling constant in asymptotically free Hamiltonians is connected with the existence of bound states and depends on how the interactions responsible for the binding are included in the renormalization group equations.

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We review recent solutions of the classical equations of motion corresponding to the effective action of the linearized gravity. The action was derived by Amati, Ciafaloni and Veneziano to describe scattering of gravitating sources at very high energies. The solutions reveal a number of phenomena consistent with existence of a gravitational collapse in such a scattering.

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The partition function of \({1}/16\) BPS states in \(\mathcal {N}=4\) SYM is found using the one loop dilatation operator. The result matches precisely the AdS/CFT prediction, i.e. it coincides with the partition function of the gas of supergravitons in AdS\(_5\times {\rm S}^5\).

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The new mechanism of an interesting phenomenon of the negative radiation pressure is presented. Force exerted by radiation on the kink in a simple toy model is calculated using perturbation scheme. The results are compared with numerical calculations. The interaction of vortices and radiation is discussed and possible explanation of the negative radiation pressure is examined.

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In this paper we investigate the cut-off effects at tree-level of perturbation theory for three different lattice regularizations of fermions — maximally twisted mass Wilson, overlap and Creutz fermions. We show that all three kinds of fermions exhibit the expected \(\mathcal O(a^2)\) scaling behaviour in the lattice spacing. Moreover, the size of these cut-off effects for the considered quantities i.e. the pseudoscalar correlation function \(C_{\rm PS}\), the mass \(m_{\rm PS}\) and the decay constant \(f_{\rm PS}\) is comparable for all of them.