abstract

The eigenvalues of Wilson loop matrices in SU\((N)\) gauge theories in dimensions 2, 3, 4 at infinite \(N\) are supported on a small arc on the unit circle centered at \(z=1\) for small loops, but expand to the entire unit circle for large loops. These two regimes are separated by a large \(N\) phase transition whose universal properties are the same in \(d=2,3\) and 4. Hopefully, this large \(N\) universality could be exploited to bridge traditional perturbation theory calculations, valid for small loops, with effective string calculations for large loops. A concrete case of such a calculation would obtain analytically an estimate of the large \(N\) string tension in terms of the perturbative scale \({\mit \Lambda }_{{\rm SU}(N)}\).

abstract

These are notes associated with three lectures given at the XLIX Cracow School of Theoretical physics where a pedagogical explanation of the Gross–Witten transition, Eguchi–Kawai reduction and continuum reduction were given, followed by a description of the numerical computation of fermionic observables in the ’t Hooft limit of large \(N\) gauge theory.

abstract

I begin this paper by describing some of the useful things that we have learned about large-\(N\) gauge theories using lattice simulations. For example that the theory is confining in that limit, that for many quantities SU\((3) \simeq {\rm SU}(\infty )\), and that this includes the strongly coupled gluon plasma just above \(T_{\rm c}\), thus providing some justification for the use of gauge-gravity duality in analysing QCD at RHIC/LHC temperatures. I then turn, in a more detailed discussion, to recent progress on the problem of what effective string theory describes confining flux tubes. I describe lattice calculations of the energy spectrum of closed loops of confining flux, and some dramatic analytic progress in extending the ‘universal Luscher correction’ to terms that are of higher order in \(1\) / \(l^2\), where \(l\) is the length of the string. Both approaches point increasingly to the Nambu–Goto free string theory as being the appropriate starting point for describing string-like degrees of freedom in SU(\(N\)) gauge theories.

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We study the fluid-like dynamics of eigenvalues of the Wilson operator in the context of the order–disorder (Durhuus–Olesen) transition in large \(N_{c}\) Yang–Mills theory. We link the universal behavior at the closure of the gap found by Narayanan and Neuberger to the phenomenon of spectral shock waves in the complex Burgers equation, where the role of viscosity is played by \(1\) / \(N_{c}\). Next, we explain the relation between the universal behavior of eigenvalues and certain class of random matrix models. Finally, we conclude the discussion of universality by recalling exact analogies between Yang–Mills theories at large \(N_{c}\) and the so-called diffraction catastrophes.

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I summarize a number of ideas about the confinement mechanism which are currently under active investigation. These include confinement via center vortices, monopoles, and calorons, Coulomb confinement and the Gribov horizon, Dyson–Schwinger equations, and vacuum wavefunctionals.

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Using as inspiration the well known chiral effective Lagrangian describing the interactions of pions at low energies, in these lectures we review the quantization procedure of Einstein gravity in the spirit of effective field theories. As has been emphasized by several authors, quantum corrections to observables in gravity are, by naive power counting, very small. While some quantities are not predictable (they require local counterterms of higher dimensionality) others, non local, are. A notable example is the calculation of quantum corrections to Newton’s law. Albeit tiny these corrections are of considerable theoretical importance, perhaps providing information on the ultraviolet properties of gravity. We then try to search for a situation where these non local corrections may be observable in a cosmological context in the early universe. Having seen that gravity admits an effective treatment similar to the one of pions, we pursue this analogy and propose a two-dimensional toy model where a dynamical zwei-bein is generated from a theory without any metric at all.

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We discuss the arbitrariness in the choice of cutoff scheme in calculations of beta functions. We define a class of “pure” cutoff schemes, in which the cutoff is completely independent of the parameters that appear in the action. In a sense they are at the opposite extreme of the “spectrally adjusted” cutoffs, which depend on all the parameters that appear in the action. We compare the results for the beta functions of Newton’s constant and of the cosmological constant obtained with a typical cutoff and with a pure cutoff, keeping all else fixed. We find that the dependence of the fixed point on an arbitrary parameter in the pure cutoff is rather mild. We then show in general that if a spectrally adjusted cutoff produces a fixed point, there is a corresponding pure cutoff that will give a fixed point in the same position.

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In this article we review the blackfold approach, a recently developed effective worldvolume description of higher-dimensional black holes, that captures the long wavelength dynamics of black holes with horizons exhibiting two characteristic lengths of very different size. In this regime the black hole is regarded as a black brane curved into a submanifold of a background spacetime, and can be formulated in terms of an effective fluid that lives on a dynamical worldvolume. We discuss the resulting blackfold equations, which separate into a set of intrinsic and extrinsic equations. The general solution of the intrinsic fluid equations for stationary configurations is presented along with a class of novel stationary black hole solutions. We also comment on how the formalism can be used to study time evolution and stability of blackfolds.

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Causal dynamical triangulations (CDT) can be used as a regularization of quantum gravity. In two dimensions the theory can be solved anlytically, even before the cut-off is removed and one can study in detail the how to take the continuum limit. We show how the CDT theory is related to Euclidean 2d quantum gravity (Liouville quantum gravity), how it can be generalized and how this generalized CDT model has a string field theory representation as well as a matrix model representation of a new kind, and finally how it examplifies the possibility that time in quantum gravity might be the stochastic time related to the branching of space into baby universes.

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We give an elementary introduction to the construction of probability distributions on sets of infinite graphs, called random graphs, as limits of ensembles of finite graphs motivated by a brief discussion of the incipient infinite cluster in bond percolation on an infinite graph. The Hausdorff and spectral dimension of random graphs are introduced. For illustrational purposes some concrete examples of random combs are considered, for which it is shown that their Hausdorff and spectral dimensions equal \(1\) except on a critical curve on which they exhibit non-trivial behaviour. Models of so-called generic planar random trees are defined and a proof that their Hausdorff and spectral dimensions equal \(2\) and 4/3, respectively, is outlined. In a concluding section we describe some open problems.

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We introduce the topic of dynamical breaking of the electroweak symmetry and its link to unparticle physics and cosmology. The knowledge of the phase diagram of strongly coupled theories plays a fundamental role when trying to construct viable extensions of the standard model (SM). Therefore, we present the state-of-the-art of the phase diagram for SU, Sp and SO gauge theories with fermionic matter transforming according to arbitrary representations of the underlying gauge group. We summarize several analytic methods used recently to acquire information about these gauge theories. We also provide new results for the phase diagram of the generalized Bars–Yankielowicz and Georgi–Glashow chiral gauge theories. These theories have been used for constructing grand unified models and have been the template for models of extended technicolor interactions. To gain information on the phase diagram of chiral gauge theories we will introduce a novel all orders beta function for chiral gauge theories. This permits the first unified study of all non-supersymmetric gauge theories with fermionic matter representation both chiral and non-chiral. To the best of our knowledge the phase diagram of these complex models appears here for the first time. We will introduce recent extensions of the SM featuring minimal conformal gauge theories known as minimal walking models. Finally, we will discuss the electroweak phase transition at nonzero temperature for models of dynamical electroweak symmetry breaking.

abstract

We discuss the holographic duals of non-relativistic conformal field theories and their realisation in string theories. Based on lectures given at the XLIX Cracow School of Theoretical Physics.

abstract

This is a sequel to the paper published under the same title [Acta Phys. Pol. B 35, 2249 (2004)] in which the integral representation of the matrix element \(\langle u|\exp (-\sigma C_1)|u\rangle \), where \(|u\rangle =\exp (-iS(u))|0\rangle \) is the quantum Coulomb field and \(C_1=-({1}\ /\ {2})M_{\,\mu \nu }\,M^{\,\mu \nu }\) is the first Casimir operator of the proper, orthochronous Lorentz group, was given. In this paper another integral representation of the same matrix element is given. In this new representation contributions from the bound state, which belongs to the supplementary series, and from the continuous spectrum, which belongs to the main series, are separated. This allows to calculate the asymptotic behaviour of the matrix element for \(\sigma \to \infty \). The matrix element \(\langle u|\exp (-\sigma C_1)|u\rangle \) is a non-analytic function of \(\sigma \) at \(\sigma =0\). The nature of this non-analyticity is clarified by means of a representation of the relevant integrals with the help of the function \(g(x)=\sum _{n=-\infty }^{+\infty }\exp (-\pi n^2x)\) which satisfies the well known functional equation \(g(x)=x^{-1\,/\,2}g(1\,/\,x),\, x\gt 0\).