Crossing the Boundaries

The following sections are included:

• FOREWORD

• CONTENTS

This is a colloquium style pedagogical introduction to the paradigm of large extra dimensions. This talk was delivered at various universities including UCLA (1999), Technion (2000), DESY (2001), University of Pisa (2001), UBC (2002), NYU (2002), University of Nantes (2003), Bern University (2003), Max-Planck Institute/Munich University (2003), University of Padova (2003), University of Minnesota (2005).

This note was written as an introduction to the volume Vacuum Structure and QCD Sum Rules (North-Holland, 1992). It has never been published.

Two new results on non-Abelian vortices are illustrated. In the first part we discuss the construction of non-Abelian vortices in theories with general gauge groups. The second part is dedicated to the fractional vortices and lumps, which occur in a wide variety of Abelian and non-Abelian generalizations of the Higgs model.

We study magnetic flux tubes in the Higgs vacuum of the mass deformation of SU(N_c), and its large N_c string dual, the Polchinski-Strassler geometry. Choosing equal masses for the three adjoint chiral multiplets, for all N_c we identify a "colour-flavour locked" symmetry, SO(3)_C+F which leaves the Higgs vacuum invariant. At weak coupling, we find explicit non-Abelian k-vortex solutions carrying a ℤ_Nc-valued magnetic flux, with topological winding 0 < k < N_c. These k-strings spontaneously break SO(3)_C+F to U(1)_C+F resulting in an S² moduli space of solutions. The world-sheet sigma model is a nonsupersymmetric ℂℙ¹ model with a theta angle θ₁₊₁ = k(N_c - k)θ₃₊₁ where θ₃₊₁ is the Yang-Mills vacuum angle. We find numerically that k-vortex tensions follow the Casimir scaling law T_k ∝ k(N_c - k) for large N_c. In the large N_c IIB string dual, the SO(3)_C+F symmetry is manifest in the geometry interpolating between AdS₅ × S⁵ and the interior metric due to a single D5-brane carrying D3-brane charge. We identify candidate k-vortices as expanded probe D3-branes formed from a collection of k D-strings. The resulting k-vortex tension exhibits precise Casimir scaling, and the effective world-sheet theta angle matches the semiclassical result. S-duality maps the Higgs to the confining phase so that confining string tensions at strong 't Hooft coupling also exhibit Casimir scaling in theory in the large N_c limit.

We address the problem of non-Abelian super-QCD, with a Fayet-Iliopoulos term, as seen from the vortex worldsheet perspective. Together with the FI term ξ, also a mass µ for the adjoint superfield Φ enters into the game. This mass allows the interpolation between and super-QCD. We distinguish, inside the parameter space spanned by ξ and µ, four different corners where some quantitative statements can be made. We focus on two questions: 1) Is the quantum vortex BPS or non-BPS? 2) What is the phase of the internal non-Abelian moduli? We find that the answer to these questions depends upon the choice of the linear term in the superpotential.

We then address the problem of multiple non-Abelian vortices, in the presence of the deformation. We show that the tension of the "Abelian" vortex is not affected by the heterotic deformation, and in particular it remains BPS-saturated when µ is also switched on.

Instantons are composed of partons.

In the last few years, we have realized the existence of a new class of topological excitations, which are rather distinct from the platonic world of monopoles, monopole-instantons and instantons. All of the latter arise as solutions of the Prasad-Sommerfield type first order differential (self-duality) equations and have been extensively discussed in the context of confinement and chiral symmetry breaking for the last 30 years. However, new calculable deformations of asymptotically free chiral and vector-like gauge theories give us a new picture of these physical phenomena. Most often, the excitations which lead to confinement are not solutions to PS-type equations, they are non-selfdual and they are often bizarre. They are referred to as magnetic bions, triplets, and quintets, due to their composite nature. Bizarre as they are, combined with large-N volume independence, these novel non-self-dual excitations may also provide hope that at least some non-abelian gauge theories may be solvable.

Soliton solutions in a scalar field theory defined on an AdS₁₊₁ background space-time are investigated. An analytic soliton solution is obtained in a polynomial model, and the classical soliton mass is calculated. The fluctuation spectrum around the soliton solution is determined, and the one-loop quantum correction to the soliton mass is computed in the semi-classical approximation.

We review the recent results on a decay rate of metastable topological configurations such as strings and domain walls. The transition from a state with higher tension to a state with lower one proceeds through quantum tunneling or through thermally catalyzed quantum tunneling (at sufficiently small temperatures). It is shown that the effects of the motion in transverse direction lead to the renormalization of a mass (tension) parameter of a particle (string) associated with a boundary of a string (wall) in the semiclassical exponent. For a non-zero temperature we derive the catalysis factor for the decay rates. It is discovered that the catalysis factor is closely related to the probability (effective length) of the collision of the the Goldstone bosons, corresponding to the transverse waves on a string (wall). We find that the destruction of a string only takes place in collisions of even number of the bosons, while the destruction of the wall can occur in a collision of any number of particles.

The study of AdS/CFT (or gauge/gravity) duality has been one of the most active and illuminating areas of research in string theory over the past decade. The scope of its relevance and the insights it is providing seem to be ever expanding. In this talk I briefly describe some of the attempts to explore how the duality works for maximally supersymmetric systems.

We review some recent progress in understanding the spectrum of energies/dimensions of strings/operators in AdS₅ × S⁵ – planar super Yang-Mills correspondence. We consider leading strong coupling corrections to the energy of lightest massive string modes in AdS₅ × S⁵, which should be dual to members of the Konishi operator multiplet in the SYM theory. This determines the general structure of strong-coupling expansion of the anomalous dimension of the Konishi operator. We use 1-loop results for semiclassical string states to extract information about the leading coefficients in this expansion.

We provide a brief introduction to the ABJM theory, the level k U(N) × U(N) superconformal Chern-Simons matter theory which has been conjectured to describe N coincident M2-branes. We discuss its dual formulation in terms of M-theory on AdS₄ × S⁷/ℤ_k and review some of the evidence in favor of the conjecture. We end with a brief discussion of the important role played by the monopole operators.

I review a recent progress in computing scattering amplitudes in strongly coupled gauge theories - a fascinating subject which has been recently boosted by the formulation of the gauge/string duality in maximally supersymmetric Yang-Mills theory. In addition to the conventional symmetry of the underlying Lagrangian, the scattering amplitudes in this theory exhibit a new, dual superconformal symmetry. This symmetry is powerful enough to completely determine the scattering amplitudes for arbitrary coupling in a suitably defined limit.

The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N = 4 Super Yang-Mills in four dimensions.

In honor of Misha Shifman's 60th birthday this note will start with one of Misha's many contributions, the famous NSVZ beta function, and study how it affects the construction of Brane Tilings. The main features of Brane Tilings for 3+1 dimensions SCFT's are reviewed and the generalization to 2+1 dimensional SCFT's is discussed.

We discuss the constraints on SUSY field theories and their IR phases, including bounds on the ratio a/c. Proceeding of talk at Shifmania, in honor of Misha Shifman's 60th birthday.

The high-energy behavior of the of amplitudes in the Regge limit can be calculated order by order in perturbation theory using the high-energy operator expansion in Wilson lines. At large N_c, a typical four-point amplitude is determined by a single BFKL pomeron. The conformal structure of the four-point amplitude is fixed in terms of two functions: pomeron intercept and the coefficient function in front of the pomeron (the product of two residues). The pomeron intercept is universal while the coefficient function depends on the correlator in question. The intercept is known in first two orders in coupling constant : LO BFKL intercept and NLO BFKL calculated in Ref. [1]. As an example of using the Wilson-line OPE, we calculate the coefficient function in front of the pomeron for the correlator of four Z² currents in the leading and next-to-leading order.

AdS/QCD is an extra-dimensional approach to modeling hadronic physics, motivated by the AdS/CFT correspondence in string theory. AdS/QCD models are often more accurate than would have been expected at energies below a few GeV. We address the question of why these models are so successful, and respond to some of the criticisms that have been waged against these models.

We consider zero-temperature transitions from conformal to non-conformal phases in quantum theories. We argue that there are three generic mechanisms for the loss of conformality: (i) fixed point goes to zero coupling, (ii) fixed point runs off to infinite coupling, or (iii) an IR fixed point annihilates with a UV fixed point and they both disappear into the complex plane. We give examples of the last case and show that the critical behavior of the mass gap is similar to that of the inverse correlation length in the finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase transition, ξ ~ exp(c/|T - T_c|^1/2). We speculate that the chiral phase transition in QCD at large number of fermion flavors belongs to this universality class, and attempt to identify the UV fixed point that annihilates with the Banks-Zaks fixed point at the lower end of the conformal window.

We discuss some universal relations that hold for conformal field theories which admit a dual classical gravitational description in anti-de Sitter space. These relations therefore apply in a suitable large N limit, and imply that various hydrodynamic transport coefficients for conserved currents and thermodynamic state functions are entirely fixed by central charges.

We study the approach of the speed of sound towards the value at high temperatures in a general class of strongly interacting theories. This class includes theories holographically dual to a theory of gravity coupled to a single scalar field, representing the operator of the scale anomaly. We show that quite universally the conformal value is approached from below and give a generic argument for why this has to be so.

We incorporate chiral symmetry breaking in a soft-wall version of the AdS/QCD model by using a modified dilaton profile and a quartic term in the bulk scalar potential. This allows one to separate the dependence on spontaneous and explicit chiral symmetry breaking. The resulting mass spectra in the scalar, vector and axial-vector sectors compares favorably with the respective QCD resonances.

A model-independent relation for the long-distance behavior of baryon form factors in the large N_c and chiral limits is introduced and used to probe the consistency of various models of the baryon including recently proposed holographic models. This relation is satisfied by the Skyrme model and all other 4D semiclassical chiral soliton models. The "bottom-up" Pomarol-Wulzer holographic model which treats baryons as 5D Skyrmions also satisfies the relation. However, the "top down" holographic model treating baryons as instantons in the Sakai-Sugimoto model fails to satisfy the relation. This failure can ultimately be traced to the imposition of a scale separation in the model between the curvature scale and the KK scale; such a scale separation has no counterpart in QCD.

We review the connection between QCD and supersymmetric theories. We focus on the non-perturbative large-N (planar) correspondence between one-flavor QCD and pure supersymmetric Yang-Mills theory (). We explain how non-perturbative quantities in QCD, such as the quark condensate, can be evaluated by using the corresponding non-perturbative results in supersymmetric gauge theories. The review consists of three parts. The first part is devoted to a review of pure . In the second part we introduce "orientifold planar equivalence". The third part is devoted to the implications of planar equivalence for QCD.

By superposition of regular gauge instantons or merons, ensembles of gauge fields are constructed which describe the confining phase of SU(2) Yang-Mills theory. Various properties of the Wilson loops, the gluon condensate and the topological susceptibility are found to be in qualitative agreement with phenomenology or results of lattice calculations. Limitations in the application to the glueball spectrum and small size Wilson loops are discussed.

I discuss the subject of powerlike asymptotic behavior of hadronic form factors in pre-QCD analyses of soft (Feynman/Drell-Yan) and hard (West) mechanisms, and also recent derivation of 1/Q² asymptotics of meson form factors in AdS/QCD. At the end, I briefly comment on "light-front holography" ansatz.

I discuss recent applications of QCD light-cone sum rules to various form factors of pseudoscalar mesons. In this approach both soft and hard contributions to the form factors are taken into account. Combining QCD calculation with the analyticity of the form factors, one enlarges the region of accessible momentum transfers.

I review the modern approach to quantum evolution of hadronic observables with energy. Recent developments relating the (eikonal) evolution equation to the QCD Reggeon field theory and the approach to including the Pomeron loops is discussed.

We discuss various manifestations of the "magnetic scenario" for the quark-gluon plasma viewed as a mixture of two plasmas, of electrically (quark and gluons) as well as magnetically charged quasiparticles. Near the deconfinement phase transition, T ≈ T_c very small density of free quarks should lead to negligible screening of electric field while magnetic screening remains strong. The consequence of this should be existence of a "corona" of the QGP, in a way similar to that of the Sun, in which electric fields influence propagation of perturbations and even form metastable flux tubes. The natural tool for its description is (dual) magnetohydrodynamics: among observable consequences is splitting of sound into two modes, with larger and smaller velocity. The latter can be zero, hinting for formation of pressure-stabilized flux tubes. Remarkably, recent experimental discoveries at RHIC show effects similar to expected for "corona structures". In dihadron correlation function with large-pt trigger there are a "cone" and a "hard ridge", while the so called "soft ridge" is a similar structure seen without hard trigger. They seem to be remnants of flux tubes, which – contrary to naive expectations – seem to break less often in near-T_c matter than do confining strings in vacuum.

We conjecture that the confinement–deconfinement phase transition in QCD at large number of colors N and N_f ≪ N at T ≠ 0 and µ ≠ 0 is triggered by the drastic change in θ behavior. The conjecture is motivated by the holographic model of QCD where confinement–deconfinement phase transition indeed happens precisely at T = T_c where θ dependence experiences a sudden change in behavior. The conjecture is also supported by quantum field theory arguments when the instanton calculations (which trigger the θ dependence) are under complete theoretical control for T > T_c, suddenly break down immediately below T < T_c with sharp changes in the θ dependence. Finally, the conjecture is supported by a number of numerical lattice results. We employ this conjecture to study confinement–deconfinement phase transition of hot and dense QCD in large N limit by analyzing the θ dependence. We estimate the critical values for T_c and µ_c where the phase transition happens by approaching the critical values from the hot and/or dense regions where the instanton calculations are under complete theoretical control. We also describe some defects of various codimensions within a holographic model of QCD by focusing on their role around the phase transition point.

The hypothesis of an 'invisible' axion was made by Misha Shifman and others, approximately thirty years ago. It has turned out to be an unusually fruitful idea, crossing boundaries between particle physics, astrophysics and cosmology. An axion with mass of order 10^-5 eV (with large uncertainties) is one of the leading candidates for the dark matter of the universe. It was found recently that dark matter axions thermalize and form a Bose-Einstein condensate (BEC). Because they form a BEC, axions differ from ordinary cold dark matter (CDM) in the non-linear regime of structure formation and upon entering the horizon. Axion BEC provides a mechanism for the production of net overall rotation in dark matter halos, and for the alignment of cosmic microwave anisotropy multipoles. Because there is evidence for these phenomena, unexplained with ordinary CDM, an argument can be made that the dark matter is axions.

I consider models of light super-weakly interacting cold dark matter, with mass, focusing on bosonic candidates such as pseudoscalars and vectors. I analyze the cosmological abundance, the γ-background created by particle decays, the impact on stellar processes due to cooling, and the direct detection capabilities in order to identify classes of models that pass all the constraints. In certain models, variants of photoelectric (or axioelectric) absorption of dark matter in direct-detection experiments can provide a sensitivity to the superweak couplings to the Standard Model which is superior to all existing indirect constraints. In all models studied, the annual modulation of the direct-detection signal is at the currently unobservable level of O(10^-5).

Misha Shifman knows much more about conformal symmetry than I do, but I have been thinking hard about it, and I thought it would be fun to present some of my ideas as a little birthday present.

This article reviews some recent work on a version of the standard model (the Lee-Wick standard model) that contains higher derivative kinetic terms that improve the convergence of loop diagrams removing the quadratic divergence in the Higgs boson mass. Naively higher derivative theories of this type are not acceptable since the higher derivative terms either cause instabilities (from negative energies) or a loss of unitarity (from negative norm states). Lee and Wick provided an interpretation for such theories arguing that theories with higher derivative kinetic terms can be unitary and stable if the states associated with the massive propagator poles, that arise from the higher derivatives, have widths and hence decay and are not in the spectrum of the theory.

We discuss first the flavor mixing of the quarks, using the texture zero mass matrices. Then we study a similar model for the mass matrices of the leptons. We are able to relate the mass eigenvalues of the charged leptons and of the neutrinos to the mixing angles and can predict the masses of the neutrinos. We find a normal hierarchy - the masses are 0.004 eV, 0.01 eV and 0.05 eV. The atmospheric mixing angle is given by the mass ratios of the charged leptons and the neutrinos. We find about 40°, consistent with the experiments. The mixing element, connecting the first neutrino with the electron, is predicted to be 0.05. This prediction can soon be checked by the Daya Bay experiment.

Relying solely on unitarity and the consistency with large-distance black hole physics, we derive model-independent properties of the microscopic black holes and of short-distance gravity in theories with N particle species. In this class of theories black holes can be as light as and be produced in particle collisions above this energy. We show, that the micro black holes must come in the same variety as the species do, although their label is not associated with any conserved charge measurable at large distances. In contrast with big Schwarzschildian ones, the evaporation of the smallest black holes is maximally undemocratic and is biased in favor of particular species. With an increasing mass the democracy characteristic to the usual macro black holes is gradually regained. The lowest possible mass above which black holes become Einsteinian is . This fact uncovers the new fundamental scale (below the quantum gravity scale) above which gravity changes classically, and the properties of black holes and gravity are such as if some extra dimensions open up, although no such input exists in the theory. Our observations indicate that geometry of space may be an emergent concept following from large number of species and the consistency with macro black hole physics. We apply our findings to the phenomenological properties of the micro black holes that can be observed at LHC for large N. Extrapolating our findings to small N, one may ask whether the existence of quark and lepton flavors is already an evidence for emergent extra dimensions at short distances.

The existence of crystalline condensates in the temperature and chemical potential phase diagram of the Gross-Neveu models can be traced to intricate symmetries of the associated inhomogeneous gap equation. The gap equation based on the Ginzburg-Landau expansion is precisely the mKdV or AKNS hierarchy of integrable nonlinear equations for the Gross-Neveu model with discrete or continuous chiral symmetry, respectively. The former model also has a dense-dilute symmetry that is due to the energy-reflection duality of the underlying quasi-exactly soluble spectral operators.

Arguments are summarized, that neutral matter made of helium, carbon, etc., should form a quantum liquid at the above-atomic but below-nuclear densities for which the charged spin-0 nuclei can condense. The resulting substance has distinctive features, such as a mass gap in the bosonic sector and a gap-less spectrum of quasifermions, which determine its thermodynamic properties. I discuss an effective field theory description of this substance, and as an example, consider its application to calculation of a static potential between heavy charged impurities. The potential exhibits a long-range oscillatory behavior in which both the fermionic and bosonic low-energy degree of freedom contribute. Observational consequences of the condensate for cooling of helium-core white dwarf stars are briefly discussed.

We derived the low energy effective action for the collective modes in asymmetric fermionic systems with attractive interaction. We obtained the phase diagram in terms of the chemical potentials. It features a stable gapless superfluidity with one Fermi surface on the BEC side of the resonance. Also we predict a sharp increase in outer core of a vortex, i.e. vortex size, upon entering into the gapless phase. This may serve as a signature of a gapless phase.

A simple approximate solution for the quantum-mechanical quartic oscillator V = m²x² + gx⁴ in the double-well regime m² < 0 at arbitrary g ≥ 0 is presented. It is based on a combining of perturbation theory near true minima of the potential, semi-classical approximation at large distances and a description of tunneling under the barrier. It provides 9-10 significant digits in energies and gives for wavefunctions the relative deviation in real x-space less than ≲ 10^-3.

The following sections are included:

• Pictures

• LIST OF THE PARTICIPANTS