The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

The Generalization of Chern-Simons Current and the Topological Tensor Current of <Emphasis Type="Italic">p</Emphasis>-Branes

To make the gauge field theory foundation of the topological current of p-branes introduced in our previous work, we present a novel topological tensor current in SO(N) gauge field theory. This non-Abelian gauge field tensor current is the straightforward generalization of the Chern-Simons topological current of strings. By making use of the SO(N) gauge potential decomposition theory and the φ-mapping topological current theory, it is proved that the p-brane is created at every isolated zero of the Clifford vector field \(\overrightarrow{\phi }(x)\) and the charges carried by p-branes are topologically quantized and labelled by the winding number of the φ-mapping.     

Noncommutative SU(N) and gauge invariant baryon operator

We propose a constraint on the noncommutative gauge theory with U(N) gauge group which gives rise to a noncommutative version of the SU(N) gauge group. The baryon operator is also constructed.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Phase portrait ofSU(5) model. II intermediate phasesSU(3)×[U(1)]2 and [SU(2)]2×[U(1)]2

The effective potential of the scalar field in theSU(5) model has extrema with symmetry:SU(5),SU(4)×U(1),SU(3)×SU(2)×U(1),SU(3)×[U(1)]², [SU(2)]²×[U(1)]². In our recent paper it was shown that theSU(4)×U(1) phase as well asSU(3)×SU(2)×U(1) phase were stable at the nonzero temperature in a vast region of parameters. In the present paper it is found that the [SU(2)]²×[U(1)]² symmetric vacuum is unstable and theSU(3)×[U(1)]² symmetric vacuum can be metastable in the certain interval of the temperature. Domains of the three phases:SU(4)×U(1),SU(3)×SU(2)×U(1),SU(3)×[U(1)]²-could co-exist in the early. Universe.     

Non-abelian D = 11 supermembrane

We obtain a U(M) action for super membranes with central charges in the Light Cone Gauge (LCG). The theory realizes all of the symmetries and constraints of the supermembrane together with the invariance under a U(M) gauge group with M arbitrary. The worldvolume action has (LCG) N = 8 supersymmetry arid it corresponds to M parallel supermembranes minimally immersed on the target M g × T ² (MIM2). In order to ensure the invariance under the symmetries and to close the corresponding algebra, a star-product determined by the central charge condition is introduced. It is constructed with a nonconstant symplectic two-form where curvature terms are also present. The theory is in the strongly coupled gauge-gravity regime. At low energies, the theory enters in a decoupling limit and it is described by an ordinary N = 8 SYM in the IR phase for any number of M2-branes.     

N = ∞ phase transition in a class of exactly soluble model lattice gauge theories

An exactly soluble class of model U(N) lattice gauge theories is considered. The ground state is discussed as a separable N-fermion problem solved by mathieu functions. Some exact correlation functions are presented. The N = ∞ limit exhibits a third order phase transition demarcating the strong and weak phases at (g²N)^?1 ≈ 0.55.     

A scaling limit with many noncommutativity parameters

We derive the worldsheet propagator for an open string with different magnetic fields at the two ends, and use it to compute two distinct noncommutativity parameters, one at each end of the string. The usual scaling limit that leads to noncommutative Yang–Mills can be generalized to a scaling limit in which both noncommutativity parameters enter. This corresponds to expanding a theory with U(N) Chan–Paton factors around a background U(1)^N gauge field with different magnetic fields in each U(1).     

A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.     

Boundaries forSU(3) c xU(1)el lattice gauge theory with a chemical potential

It is shown forSU(N) andU(1) gauge groups that periodic spatial boundary conditions, as commonly used in lattice simulations, are not possible in the charged sectors of a local gauge theory. For charge-conjugate (C-)periodic boundary conditions the effective gauge action of fermions is derived. For nonzero chemical potential, the breakdown of translational invariance induced by the breakdown ofC symmetry is discussed. If translational invariance is abandoned, (anti)periodic spatial b.c. for fermions and for theSU(3) gauge field andC-periodic b.c. for theU(1) gauge field can be used.     

Noncommutative two-dimensional gauge theories

We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-NSU(N) gauge theory with adjoint matter. The noncommutative version of the ?t Hooft model receives a non-trivial contribution to the vacuum polarization starting from three-loops order. As a result the mass spectrum of the noncommutative theory is expected to be different from that of the commutative theory.     

Phase portrait ofSU(5) grand unified model

The one-loop effective potential of theSU(5) model is investigated both in high and low temperature approximation. We find the regions of values of coupling constants and temperatures where theSU(5), theSU(4)×U(1) and theSU(3)×SU(2)×U(1) symmetric states are metastable. A general method of such an investigation is proposed. We observe that the domain structure of the Universe with the simultaneous existence of the gauge symmetriesSU(4)×U(1) andSU(3)×SU(2)×U(1) in different domains could take place.     

Phase structure of U(N→∞) gauge theory on a two-dimensional lattice for a broad class of variant actions

It is pointed out that finding the partition function for U(N) gauge theory on a two-dimensional lattice in the limit N→∞ reduces, for a broad class of single-plaquette actions, to a well-known and solved mathematical problem. The case where in the single plaquette action the matrix U + U⁺ occuring in Wilson's formula is replaced by an arbitrary polynomial in this matrix, is discussed in detail and explicit results for the second-order polynomial are presented. A rich phase structure with second- and third-order phase transitions is found. The results are shown to have at the qualitative level a simple thermodynamical interpretation. They support the view that the phase structure of a lattice gauge theory is an artifact of the lattice action used rather than some reflection of the underlying group structure.     

TheY(Y L ), a compositeness-inspired extra gauge boson

We show that the (isoscalar)Y(Y _L ) boson, originally introduced on the basis of compositeness andW, Y(Y _L ) dominance can be understood as an extra gauge boson within aU(1) _B ×SU(2) _L ×U(1) _Y gauge theory broken by an appropriate choice of the Higgs sector. Radiative-correction effects can thus be calculated as reliably as in theSU(2) _L ×U(1) _Y , theory when comparing the predictions with forthcoming precision data in theSLC/LEP energy range.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.     

Light quarks and instantons

Instantons and anti-instantons can profoundly influence the structure of a non-Abelian gauge theory involving N flavors of massless quarks. Interactions of the quarks with these pseudoparticles can spontaneously generate a quark mass, break the theory's SU(N) × SU(N) chiral symmetry and bind quark-antiquark pairs to form N² ? 1 Goldstone bosons. If the spontaneously generated quark mass is small, multipseudoparticle configurations can be treated in a dilute gas approximation.     

Finite size scaling for O(N) φ4-theory at the upper critical dimension

A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N) φ⁴-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ⁴₄-theories for all N.     

A Monte Carlo analysis of an SU(5) scalar model

Spontaneous symmetry breaking is studied in an SU(5) model of adjoint scalars with quartic couplings λ₁ (tr φ²)² + λ₂ trφ⁴ and with a negative mass term. The breaking for λ₂ > 0 is to SU(3)?SU(2)?U(1), and for λ₂ > 0 to SU(3)?SU(2)?U(1), in agreement with the standard tree-level analysis of the scalar potential.     

Computing parallel/coincident phase D-brane superpotentials and Type-Ⅱ/F-theory duality

In this paper we study the parallel phase and the coincident phase of D-brane systems with the compactification of one closed modulus. D-brane systems with two phases are described by different 4-folds in terms of Type-Ⅱ/F-theory duality, and the phase transitions are related by the blow-up from a 4-fold with singularities to a 4-fold without. In terms of gauge theory, the phase transition corresponds to the enhancement of gauge group U(1)×U(1)→U(2) connecting the Coulomb branch and the Higgs branch. For the sextic and octic with two D-branes,using mirror symmetry and Type-Ⅱ/F theory duality, A-model superpotentials are obtained from the B-model side for the two phases, and the U(1) Ooguri-Vafa invariants for the parallel phase and U(2) Ooguri-Vafa invariants for the coincident phase are extracted from the A-model superpotential. The difference between the invariants of the two phases is evidence of the phase transition between the Coulomb branch and the Higgs branch.