Gauge Field Theory

The onset of dynamical chaos is studied numerically in (2+1)-dimensional non-Abelian field theory with the Chern-Simons topological term. In the limit of strong fields, slowly varying in space (spatially homogeneous fields), this theory is an analog to a system of three charged particles moving in a plane in an orthogonal magnetic field and under the influence of a quartic potential. The ‘‘phase transition’’ (order chaos) is observed within a narrow energy range. The threshold of the transition depends on the sign of the angular momentum of the field reflecting parity violation in the underlying field theory. The transition region is investigated in some detail and the hyperfine structure of order-chaos-order-... transitions is observed suggesting the necessity of probabilistic description.

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic offdiagonal metrics and generalized symmetries in modern gravity define nonholonomic spacetime manifolds with uncompactified extra dimensions.

This paper is devoted to the calculation by Mellin-Barnes transform of a especial class of integrals. It contains double integrals in the position space in d = 4 − 2ǫ dimensions, where ǫ is parameter of dimensional regularization. These integrals contribute to the effective action of the N = 4 supersymmetric Yang-Mills theory. The integrand is a fraction in which the numerator is a logarithm of ratio of spacetime intervals, and the denominator is the product of powers of spacetime intervals. According to the method developed in the previous papers, in order to make use of the uniqueness technique for one of two integrations, we shift exponents in powers in the denominator of integrands by some multiples of ǫ. As the next step, the second integration in the position space is done by Mellin-Barnes transform. For normalizing procedure, we reproduce first the known result obtained earlier by Gegenbauer polynomial technique. Then, we make another shift of exponents in powers in the denominator to create the logarithm in the numerator as the derivative with respect to the shift parameter δ. We show that the technique of work with the contour of the integral modified in this way by using Mellin-Barnes transform repeats the technique of work with the contour of the integral without such a modification. In particular, all the operations with a shift of contour of integration over complex variables of twofold Mellin-Barnes transform are the same as before the δ modification of indices, and even the poles of residues coincide. This confirms the observation made in the previous papers that in the position space all the Green function of N = 4 supersymmetric Yang-Mills theory can be expressed in terms of UD functions.

Exact solutions to the low-energy effective action (LEEA) of the four-dimensional N = 2 supersymmetric gauge theories are known to be obtained either by quantum field theory methods from S-duality in the Seiberg-Witten approach, or by the Type-IIA superstring/M-Theory methods of brane technology. After a brief review of the standard field-theoretical results for the N = 2 gauge (Seiberg-Witten) LEEA, we consider a field-theoretical derivation of the exact hypermultiplet LEEA by using the N = 2 harmonic superspace methods. We illustrate our techniques on a number of explicit examples. Our main purpose, however, is to discuss the existing analytical (calculational) support for the alternative methods of brane technology. We summarize known exact solutions to the eleven-dimensional and ten-dimensional type-IIA supergravities, which describe classical configurations of intersecting BPS branes with eight supercharges relevant to the non-perturbative N = 2 gauge theory with fundamental hypermultiplet matter. The crucial role of the M-Theory in providing a classical resolution of singularities in the ten-dimensional (Type-IIA superstring) brane picture, as well as the N = 2 extended supersymmetry in four dimensions, are

A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space-time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized zero-curvature formulation and infinitely many conserved quantities. A close relation between the Lagrange density of the integrable models and the pullback of the pertinent volume form on target space is established. Moreover, we show that the conserved currents are Noether currents generated by the volume preserving diffeomorphisms. Further, we show how such models may emerge via abelian projection of some gauge theories.

The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.

The purpose of this paper is to present a generalized hole argument for gauge field theories and their geometrical setting in terms of fiber bundles. The generalized hole argument is motivated and extended from the spacetime hole arguments which appear in spacetime theories based on differentiable manifolds such as general relativity. Analogously, the generalized hole argument rules out fiber bundle substantivalism and, thus, a relationalistic interpretation of the geometry of fiber bundle spaces is favoured. Along the way, the concept of gauge field theories will be analyzed via considering the gauge principle and thereby hopefully clarifying certain terminological ambiguities.

We propose a distinction between the physical and the mathematical parts of gauge field theories. The main problem we face is to uphold a strong and meaningful criterion of what is physical. We like to call it "Field's dilemma", referring to Hartry Field's nominalist proposal which we consider to be inadaequate. The resolution to the dilemma, we believe, is implicitly

With the two most profound conceptual revolutions of XX th century physics, quantum mechanics and relativity, which have culminated into relativistic spacetime geometry and quantum gauge field theory as the principles for gravity and the three other known fundamental interactions, the physicist of the XXI st century has inherited an unfinished symphony: the unification of the quantum and the continuum. As an invitation to tomorrow's quantum geometers who must design the new rulers by which to size up the Universe at those scales where the smallest meets the largest, these lectures review the basic principles of today's conceptual framework, and highlight by way of simple examples the interplay that presently exists between the quantum world of particle interactions and the classical world of geometry and topology.

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold M , i.e. the critical points of the functional PYM(D) = 1 2 M π H R D 2 θ ∧ (dθ) n , where D is a connection in a Hermitian CR-holomorphic vector bundle (E, h) → M . Let Ω = {ϕ < 0} ⊂ C n be a smoothly bounded strictly pseuodoconvex domain and g the Bergman metric on Ω. We show that boundary values D b of Yang-Mills fields D on (Ω, g) are pseudo Yang-Mills fields on ∂Ω, provided that i T R D b = 0 and i N R D = 0 on H(∂Ω). If S 1 → C(M ) π → M is the canonical circle bundle and π * D is a Yang-Mills field with respect to the Fefferman metric F θ of (M, θ) then D is a pseudo Yang-Mills field on M . The Yang-Mills equations δ π * D R π * D = 0 project on the Euler-Lagrange equations δ D b R D = 0 of the variational principle δ PYM(D) = 0, provided that i T R D = 0. When M has vanishing pseudohermitian Ricci curvature the pullback π * D of the (CR invariant) Tanaka connection D of (E, h) is a Yang-Mills field on C(M ). We derive the second variation formula {d 2 PYM(D t )/dt 2 } t=0 = M S D b (ϕ), ϕ θ ∧ (dθ) n , D t = D + A t (provided that D is a pseudo Yang-Mills field and ϕ ≡ {dA t /dt} t=0 ∈ Ker(δ D )), and show that S D b (ϕ) ≡ ∆ D b ϕ + R D b (ϕ), ϕ ∈ Ω 0,1 (Ad(E)), is a subelliptic operator.

Abelian duality on the closed three-dimensional Riemannian manifold M is discussed. Partition functions for the ordinary U(1) gauge theory and a circle-valued scalar field theory on M are explicitly calculated and compared. It is shown that the both theories are mutually dual.

For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.

The phenomenon of jet supression observed in highly energetic heavy ion collisions is discussed. The focus is devoted to the stunning applications of the AdS/CFT correspondence [1] to describe these real time processes, hard to be illuminated by other means. In particular, the introduction of as many flavors as colors into the quark-gluon plasma is scrutinized.

In this paper, the analog of Maxwell electromagnetism for hydrodynamic turbulence, the metafluid dynamics, is extended in order to reformulate the metafluid dynamics as a gauge field theory. That analogy opens up the possibility to investigate this theory as a constrained system. Having this possibility in mind, we propose a Lagrangian to describe this new theory of turbulence and, subsequently, analyze it from the symplectic point of view. From this analysis, a hidden gauge symmetry is revealed, providing a clear interpretation and meaning of the physics behind the metafluid theory. Also, the geometrical interpretation to the gauge symmetries is discussed.

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A Γ ⊂ A of local observables invariant under Γ. A set of positive energy A Γ modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of SL(2, Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules.

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables “up to n loops ” where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green’s functions.

In previous work [Rosenbaum M. et al., J. Phys. A: Math. Theor. 40 (2007), 10367-10382] we have shown how for canonical parametrized field theories, where spacetime is placed on the same footing as the other fields in the theory, the representation of space-time diffeomorphisms provides a very convenient scheme for analyzing the induced twisted deformation of these diffeomorphisms, as a result of the space-time noncommutativity. However, for gauge field theories (and of course also for canonical geometrodynamics) where the Poisson brackets of the constraints explicitely depend on the embedding variables, this Poisson algebra cannot be connected directly with a representation of the complete Lie algebra of space-time diffeomorphisms, because not all the field variables turn out to have a dynamical character [Isham C.J., Kuchař K.V., Ann. Physics 164 , [316][317][318][319][320][321][322][323][324][325][326][327][328][329][330][331][332][333]. Nonetheless, such an homomorphic mapping can be recuperated by first modifying the original action and then adding additional constraints in the formalism in order to retrieve the original theory, as shown by Kuchař and Stone for the case of the parametrized Maxwell field in [Kuchař K.V., Stone S.L., Classical Quantum Gravity 4 (1987), 319-328]. Making use of a combination of all of these ideas, we are therefore able to apply our canonical reparametrization approach in order to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms as well as to consider how gauge transformations act on the twisted algebras of gauge and particle fields. Thus, hopefully, adding clarification on some outstanding issues in the literature concerning the symmetries for gauge theories in noncommutative space-times.

The aim of this paper is twofold: First, to present an examination of the principles underlying gauge field theories. I shall argue that there are two principles directly connected to the two well-known theorems of Emmy Noether concerning global and local symmetries of the free matter-field Lagrangian, in the following referred to as "conservation principle" and "gauge principle". Since both these express nothing but certain symmetry features of the free field theory, they are not sufficient to derive a true interaction coupling to a new gauge field. For this purpose it is necessary to advocate a third, truly empirical principle which may be understood as a generalization of the equivalence principle. The second task of the paper is to deal with the ontological question concerning the reality status of gauge potentials in the light of the proposed logical structure of gauge theories. A nonlocal interpretation of topological effects in gauge theories and, thus, the non-reality of gauge potentials in accordance with the generalized equivalence principle will be favoured.

A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, di, F ijk lmn , δ ijk ). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the "topological entropy" which directly measures the quantum dimension D = P i d 2 i .

Polyvector-valued gauge field theories in noncommutative Clifford spaces are presented. The noncommutative star products are associative and require the use of the Baker-Campbell-Hausdorff formula. Actions for pbranes in noncommutative (Clifford) spaces and noncommutative phase spaces are provided. An important relationship among the n-ary commutators of noncommuting spacetime coordinates [X 1 , X 2 , ......, X n ] with the poly-vector valued coordinates X 123...n in noncommutative Clifford spaces is explicitly derived [X 1 , X 2 , ......, X n ] = n! X 123...n . The large N limit of n-ary commutators of n hyper-matrices Xi 1 i 2 ....in leads to Eguchi-Schild p-brane actions for p + 1 = n. Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Cliffordspaces would require quantum Hopf algebraic deformations of Clifford algebras. 1 Polyvector Gauge Field Theories in Noncommutative Clifford Spaces Clifford algebras are deeply related and essential tools in many aspects in Physics. The Extended Relativity theory in Clifford-spaces ( C-spaces ) is a natural extension of the ordinary Relativity theory [3] whose generalized polyvectorvalued coordinates are Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in D-dimensional target spacetime backgrounds.

We consider QCD-like theories with one massless fermion in various representations of the gauge group SU$(N)$. The theories are formulated on $R_3\times S_1$. In the decompactification limit of large $r(S_1)$ all these theories are characterized by confinement, mass gap and spontaneous breaking of a (discrete) chiral symmetry ($\chi$SB). At small $r(S_1)$, in order to stabilize the vacua of these theories at a center-symmetric point, we suggest to perform a double trace deformation. With these deformation, the theories at hand are at weak coupling at small $r(S_1)$ and yet exhibit basic features of the large-$r(S_1)$ limit: confinement and $\chi$SB. We calculate the string tension, mass gap, bifermion condensates and $\theta$ dependence. The double-trace deformation becomes dynamically irrelevant at large $r(S_1)$. Despite the fact that at small $r(S_1)$ confinement is Abelian, while it is expected to be non-Abelian at large $r(S_1)$, we argue that small and large-$r(S_1)$ physics are continuously connected. If so, one can use small-$r(S_1)$ laboratory to extract lessons about QCD and QCD-like theories on $R_4$.

We give an example of a purely bosonic model -a rotor model on the 3D cubic lattice -whose low energy excitations behave like massless U (1) gauge bosons and massless Dirac fermions. This model can be viewed as a "quantum ether": a medium that gives rise to both photons and electrons. It illustrates a general mechanism for the emergence of gauge bosons and fermions known as "stringnet condensation." Other, more complex, string-net condensed models can have excitations that behave like gluons, quarks and other particles in the standard model. This suggests that photons, electrons and other elementary particles may have a unified origin: string-net condensation in our vacuum.

We analyse D-branes on orbifolds with discrete torsion, extending earlier results. We analyze certain abelian orbifolds of the type Bbb C3/Gamma, where Gamma is given by Bbb Zm × Bbb Zn, for the most general choice of discrete torsion parameter. By comparing with the AdS/CFT correspondence, we can consider different geometries which give rise to the same physics. This identifies

The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end.

We present a 1-loop toroidal membrane winding sum reproducing the conjectured $M$-theory, four-graviton, eight derivative, $R^4$ amplitude. The $U$-duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for $M$-theory compactified on a 3-torus, where the target-space $Sl(3,\Zint)\times Sl(2,\Zint)$ $U$-duality and $Sl(3,\Zint)$ world-volume modular groups are embedded in $E_{6(6)}(\Zint)$. Unlike previous semi-classical expansions, $U$-duality is built in manifestly and realized at the quantum level thanks to Fourier invariance of cubic characters. In addition to winding modes, a pair of new discrete, flux-like, quantum numbers are necessary to ensure invariance under the larger group. The action for these modes is of Born-Infeld type, interpolating between standard Polyakov and Nambu-Goto membrane actions. After integration over the membrane moduli, we recover the known $R^4$ amplitude, including membrane instantons. Divergences are disposed of by trading the non-compact volume integration for a compact integral over the two variables conjugate to the fluxes -- a constant term computation in mathematical parlance. As byproducts, we suggest that, in line with membrane/fivebrane duality, the $E_6$ theta series also describes five-branes wrapped on $T^6$ in a manifestly U-duality invariant way. In addition we uncover a new action of $E_6$ on ten dimensional pure spinors, which may have implications for ten dimensional super Yang--Mills theory. An extensive review of $Sl(3)$ automorphic forms is included in an Appendix.

The purpose of this paper is to present a generalized hole argument for gauge field theories and their geometrical setting in terms of fiber bundles. The generalized hole argument is motivated and extended from the spacetime hole arguments which appear in spacetime theories based on differentiable manifolds such as general relativity. Analogously, the generalized hole argument rules out fiber bundle

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables "up to n loops" where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions.

We obtain an analytic solution for a pseudo-scalar massless perturbation of a non-supersymmetric deformation of the warped deformed conifold. This allows us to study Dstrings in the infrared limit of non-supersymmetric deformations of the Klebanov-Strassler background. They are interpreted as axionic strings in the dual field theory. Following the arguments of hep-th/0405282, the axion is a massless pseudo-scalar glueball which is present in the supergravity fluctuation spectrum and it is interpreted as the Goldstone boson of the spontaneously broken U(1) B baryon number symmetry, being the gauge theory on the baryonic branch.

We briefly review one of the current applications of the AdS/CFT correspondence known as AdS/QCD and discuss about the calculation of four-point quark-flavour current correlation functions and their applications to the calculation of observables related to neutral kaon decays and neutral kaon mixing processes.

We present a new version of the CompHEP program (version 4.4). We describe shortly new issues implemented in this version, namely, simplification of quark flavor combinatorics for the evaluation of hadronic processes, Les Houches Accord-based CompHEP-PYTHIA interface, processing the color configurations of events, implementation of MSSM, symbolical and numerical batch modes, etc. We discuss how the CompHEP program is used for preparing event generators for various physical processes. We mention a few concrete physics examples for CompHEP-based generators prepared for the LHC and Tevatron. r

We study a new class of infinite-dimensional Lie algebras W ∞ (N + , N − ) generalizing the standard W ∞ algebra, viewed as a tensor operator algebra of SU (1, 1) in a grouptheoretic framework. Here we interpret W ∞ (N + , N − ) either as an infinite continuation of the pseudo-unitary symmetry U (N + , N − ), or as a "higher-U (N + , N − )-spin extension" of the diffeomorphism algebra diff(N + , N − ) of the N = N + + N − torus U (1) N . We highlight this higher-spin structure of W ∞ (N + , N − ) by developing the representation theory of U (N + , N − ) (discrete series), calculating higher-spin representations, coherent states and deriving Kähler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W ∞ symmetries and algebras of symbols of U (N + , N − )-tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds.

Restricting the functional integral to the Gribov region Ω leads to a deep modication of the behavior of Euclidean Yang-Mills theories in the infrared region. For example, a gluon propagator of the Gribov type, k 2 k 4 +γ 4 , can be viewed as a propagating pair of unphysical modes, called here i-particles, with imaginary masses ±iγ 2 . From this viewpoint, gluons are unphysical and one can see them as being conned. We introduce a simple toy model describing how a suitable set of composite operators can be constructed out of i-particles whose correlation functions exhibit only real branch cuts, with associated positive spectral density. These composite operators can thus be called physical and are the toy analogy of glueballs in the Gribov-Zwanziger theory.

A physical interpretation of the two-sheeted space, the most fundamental ingredient of noncommutative spectral geometry proposed by Connes as an approach to unification, is presented. It is shown that the doubling of the algebra is related to dissipation and to the gauge structure of the theory, the gauge field acting as a reservoir for the matter field. In a regime of completely deterministic dynamics, dissipation appears to play a key rôle in the quantization of the theory, according to 't Hooft's conjecture. It is thus argued that the noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.

Chiral orbifold models are defined as gauge field theories with a finite gauge group $\Gamma$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group $\Gamma$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^{\Gamma}\subset A$ of local observables invariant under $\Gamma$. A set of positive energy $A^{\Gamma}$ modules is constructed whose characters span, under some assumptions on $\Gamma$, a finite dimensional unitary representation of $SL(2,Z)$. We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of $W_{1+\infty}$ that appear to provide a bridge between two approaches to the quantum Hall effect.

We use two renormalization techniques, Effective Field Theory and the Similarity Renormalization Group, to solve simple Schr{\&quot;o}dinger equations with delta-function potentials in one and two dimensions. The familiar one-dimensional delta-function does not require renormalization, but it provides the simplest example of a local interaction that can be replaced by a sequence of effective cutoff interactions that produce controllable power-law errors.

We construct a generalization of the quantum Hall effect where particles move in an eight dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configurations spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20 dimensional manifold of with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14 dimensional manifold with an internal component of $G_2$ holonomy.

We study a new class of infinite-dimensional Lie algebras W_\infty(p,q) generalizing the standard W_\infty algebra, viewed as a tensor operator algebra of SU(1,1) in a group-theoretic framework. Here we interpret W_\infty(p,q) either as an infinite continuation of the pseudo-unitary symmetry U(p,q), or as a "higher-U(p,q)-spin extension" of the diffeomorphism algebra diff(p,q) of the N=p+q torus U(1)^N. We highlight this higher-spin structure of W_\infty(p,q) by developing the representation theory of U(p,q) (discrete series), calculating higher-spin representations, coherent states and deriving K\"ahler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W_\infty symmetries and algebras of symbols of U(p,q)-tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds.

Various features of domain walls in supersymmetric gluodynamics are discussed. We give a simple field-theoretic interpretation of the phenomenon of strings ending on the walls recently conjectured by Witten. An explanation of this phenomenon in the framework of gauge field theory is outlined. The phenomenon is argued to be particularly natural in supersymmetric theories which support degenerate vacuum states with distinct physical properties. The issue of existence (or non-existence) of the BPS saturated walls in the theories with glued (super)potentials is addressed. The amended Veneziano-Yankielowicz effective Lagrangian belongs to this class. The physical origin of the cusp structure of the effective Lagrangian is revealed, and the limitation it imposes on the calculability of the wall tension is explained. Related problems are considered. In particular, it is shown that the so called discrete anomaly matching, when properly implemented, does not rule out the chirally symmetric phase of supersymmetric gluodynamics, contrary to recent claims.

The quarks of quark models cannot be identified with the quarks of the QCD Lagrangian. We review the restrictions that gauge field theories place on any description of physical (colour) charges. A method to construct charged particles is presented. The solutions are applied to a variety of applications. Their Green's functions are shown to be free of infra-red divergences to all orders in perturbation theory. The interquark potential is analysed and it is shown that the interaction responsible for anti-screening results from the force between two separately gauge invariant constituent quarks. A fundamental limit on the applicability of quark models is identified.

We report on a numerical simulation of the classical evolution of the plane-wave matrix model with semiclassical initial conditions. Some of these initial conditions thermalize and are dual to a black hole forming from the collision of D-branes in the plane-wave geometry. In particular, we consider a large fuzzy sphere (a D2-brane) plus a single eigenvalue (a D0-particle) going exactly through the center of the fuzzy sphere and aimed to intersect it. Including quantum fluctuations of the off-diagonal modes in the initial conditions, with sufficient kinetic energy the configuration collapses to a small size. We also find evidence for fast thermalization: rapidly decaying autocorrelation functions at late times with respect to the natural time scale of the system.

In this paper it is stressed that there is no physical reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the theory finite W -algebras, which is an important class of non-linear symmetries. In particular, we discuss both the classical and quantum theory and elaborate on several aspects of their representation theory. Some new results are presented. These include finite W coadjoint orbits, real forms and unitary representation of finite W -algebras and Poincare-Birkhoff-Witt theorems for finite W -algebras. Also we present some new finite W -algebras that are not related to sl(2) embeddings. At the end of the paper we investigate how one could construct physical theories, for example gauge field theories, that are based on non-linear algebras. §

We study the predictions of holographic QCD for various observable four-point quark flavour current-current correlators. The dual 5-dimensional bulk theory we consider is a SU(3)_L × SU(3)_R Yang Mills theory in a slice of AdS_5 spacetime with boundaries. Particular UV and IR boundary conditions encode the spontaneous breaking of the dual 4D global chiral symmetry down to the SU(3)_V subgroup. We explain in detail how to calculate the 4D four-point quark flavour current-current correlators using the 5D holographic theory, including interactions. We use these results to investigate predictions of holographic QCD for the Δ I = 1/2 rule for kaon decays and the B_K parameter. The results agree well in comparison with experimental data, with an accuracy of 25 contributions of the meson resonances to the four-point correlators. The correlators agree well in the low-momentum and high-momentum limit, in comparison with chiral perturbation theory and perturbative QCD results, respectively.

The complete tables of Clebsch-Gordan (CG) coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given. Explicit expressions of states of all corresponding multiplets under standard model gauge group G 321 = SU (3) C × SU (2) L × U (1) Y , necessary for evaluation of the CG coefficients are presented. The SUSY SO(10) GUT model considered here includes most of the Higgs irreducible representations usually used in the literature: 10, 45, 54, 120, 126, 126 and 210. Mass matrices of all G 321 multiplets are found for the most general superpotential. These results are indispensable for the precision calculations of the gauge couplings unification and proton decay, etc.

We describe a new paradox for ideal fluids. It arises in the accretion of an ideal fluid onto a black hole, where, under suitable boundary conditions, the flow can violate the generalized second law of thermodynamics. The paradox indicates that there is in fact a lower bound to the correlation length of any real fluid, the value of which is determined by the thermodynamic properties of that fluid. We observe that the universal bound on entropy, itself suggested by the generalized second law, puts a lower bound on the correlation length of any fluid in terms of its specific entropy. With the help of a new, efficient estimate for the viscosity of liquids, we argue that this also means that viscosity is bounded from below in a way reminiscent of the conjectured Kovtun-Son-Starinets lower bound on the ratio of viscosity to entropy density. We conclude that much light may be shed on the Kovtun-Son-Starinets bound by suitable arguments based on the generalized second law.

This paper is devoted to the calculation by Mellin-Barnes transform of a especial class of integrals. It contains double integrals in the position space in d = 4 − 2ǫ dimensions, where ǫ is parameter of dimensional regularization. These integrals contribute to the effective action of the N = 4 supersymmetric Yang-Mills theory. The integrand is a fraction in which the numerator is a logarithm of ratio of spacetime intervals, and the denominator is the product of powers of spacetime intervals. According to the method developed in the previous papers, in order to make use of the uniqueness technique for one of two integrations, we shift exponents in powers in the denominator of integrands by some multiples of ǫ. As the next step, the second integration in the position space is done by Mellin-Barnes transform. For normalizing procedure, we reproduce first the known result obtained earlier by Gegenbauer polynomial technique. Then, we make another shift of exponents in powers in the denominator to create the logarithm in the numerator as the derivative with respect to the shift parameter δ. We show that the technique of work with the contour of the integral modified in this way by using Mellin-Barnes transform repeats the technique of work with the contour of the integral without such a modification. In particular, all the operations with a shift of contour of integration over complex variables of two-fold Mellin-Barnes transform are the same as before the δ modification of indices, and even the poles of residues coincide. This confirms the observation made in the previous papers that in the position space all the Green function of N = 4 supersymmetric Yang-Mills theory can be expressed in terms of UD functions.