Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

On the gauge Potts model and the plaquette percolation problem

It is shown that the ar 1, 0 limit of the Potts gauge model describes plaquette percolation as the analogous limit of the spin model describes bond percolation. These results further strengthen the connection between gauge theories and random surfaces. Moreover, further generalizations to other types of gauge theories are presented.     

Spontaneous symmetry breakings in Z2 gauge theories for doped quantum dimer and eight-vertex models

Behavior of doped fermions in Z₂ gauge theories for the quantum dimer and eight-vertex models is studied. Fermions carry charge and spin degrees of freedom. In the confinement phase of the Z₂ gauge theories, these internal symmetries are spontaneously broken and a superconducting or Neél state appears. On the other hand in the deconfinement-topologically ordered state, all symmetries are respected. From the view point of the quantum dimer and eight-vertex models, this result indicates interplay of the phase structure of the doped fermions and background configuration of the dimer or the eight-vertex groundstate. At the quantum phase transitions in these systems, structure of the doped fermions groundstate and also that of the background dimer or eight-vertex groundstate both change. Translational symmetry breaking induces a superconducting or antiferromagnetic state of the doped fermions.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

Compactification of Einstein-Yang-Mills theories on CPn

We study compactification of Einstein-Yang-Mills theories in 2n + 4 dimensions on the manifold CP_n, with a classical gauge field that is equal to the spin connection. The complete boson fluctuation spectrum is calculated and no tachyons, ghosts or massless scalars are found for the minimal Yang-Mills group SU(n) × U(1). For larger groups, tachyons or massless scalars may appear.     

Spin Hall effect induced by a gravitational field

The experiment by Collela et al. (1975) [1] evidenced in a striking manner how the gravitational field appears in quantum mechanics. Within the modern framework of gauge theories, one can ascribe such effect as due to gauge fields originated from fundamental symmetries of spacetime: local transformations of the Lorentz-Poincaré group. When this gauge principle is applied to the Dirac equation, we obtain kinematical correlations between the gravitational field and the spin of the particles. The phenomenon is similar to the spin Hall effect found in condensed matter systems, although much smaller in magnitude. Actual measurements may require highly precision interferometric techniques with spin-polarized neutrons.     

Analytic approximations to Hamiltonian lattice field theories (II). The weak-coupling behavior of U(1) theories

It is shown that at weak coupling physical quantities in hamiltonian U(1) lattive gauge (or global symmetric) theories of arbitrary dimension are provided as expectation values in a d ? 1 dimensional lagrangian Z(2) gauge (or spin) theory with calculable long-range interactions.Confinement and the existence of a magnetic mass gap are equivalent to the existence of infinite-range plaquette-plaquette (or link-link) correlations in the spin field. The existence of infinite range correlations is simply related to the dimension of the lattice and the transformation property of the order parameter. As expected, only the d = 2+1 U(1) gauge theory confines electric charges at all non-vanishing coupling.     

Critical behaviour in random field gauge theory

We study the thermodynamic behaviour of spin and gauge systems in the presence of a quenched external random field. In particular, we show that forZ(2) andSU (2) gauge theory in two space dimensions, the random field destroys the ordered phase and thus leads to a shift in the lower critical dimension, just as found for the corresponding Ising model.     

Gauge invariant unified field structures,quantizable dynamical systems,charge, and spin structures

Gauge invariant unified field structures on a manifold B are introduced. Necessary and sufficient conditions for their existence are studied. The connection with charge is studied; it is shown that such gauge invariant structures, e.g. quantizable dynamical systems, over simply connected manifolds B are completely classified by charge. Complex analytic gauge invariant unified field structures are studied. These structures over a complex analytic manifold B whose square is the canonical line bundle are in bijective correspondence with the spin structures on B. Finally, a class of homogeneous quantizable dynamical systems are shown not to carry spin structure.     

BRS gauge and ghost field supersymmetry in gravity and supergravity

Becchi-Rouet-Stora transformations are obtained for the following systems: (i) Pure Einstein gravity in first order form with vierbein and spin connection as independent fields. (ii) First order Einstein gravity coupled to Yang-Mills fields. (iii) Pure supergravity. For the first two systems the results are as in Yang-Mills theory. But for conventional supergravity the BRS transformations leave the effective action invariant only if the classical equations of motion are satisfied. New transformations of the gauge fields of supergravity have been proposed under which the supersymmetry algebra closes. The corresponding BRS transformations do leave the effective action invariant without the need to use the classical equation of motion; moreover, as in Yang-Mills theories, they are nilpotent and have unit Jacobian.     

Duality for Z(N) gauge theories

The duality properties of simple Z(N) gauge theories are discussed. For N ? 4 we find self duality in four dimensions and we give the transition points. For N > 4 these systems are not self dual. Also, the order parameter is discussed. The general Z(N) gauge theory is found to be self dual for all N.     

Gauge theory approach for diffusive and precessional spin dynamics in a two-dimensional electron gas

We develop a gauge theory for diffusive and precessional spin dynamics in a two-dimensional electron gas. Our approach reveals a direct connection between the absence of the equilibrium spin current and a strong anisotropy in the spin relaxation: both effects arise if spin-orbit coupling is reduced to a pure gauge SU(2) field. In this case, the spin-orbit coupling can be removed by a gauge transformation in the form of a local SU(2) spin rotation. The resulting spin dynamics is exactly described in terms of two kinetic coefficients: the spin diffusion and electron mobility. After the inverse transformation, full diffusive and precessional spin density dynamics, including the anisotropic spin relaxation, formation of stable spin structures, and spin precession induced by a macroscopic current are restored. Explicit solutions of the spin evolution equations are found for the initially uniform spin density and for stable, nonuniform structures. Our analysis demonstrates a universal relation between the spin relaxation rate and spin-diffusion coefficient.     

Universally coupled massive gravity, II: Densitized tetrad and cotetrad theories

Einstein’s equations in a tetrad formulation are derived from a linear theory in flat spacetime with an asymmetric potential using free field gauge invariance, local Lorentz invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. These results are adapted to produce universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The theories derived, upon fixing the local Lorentz gauge freedom, are seen to be a subset of those found by Ogievetsky and Polubarinov some time ago using a spin limitation principle. In view of the stability question for massive gravities, the proven non-necessity of positive energy for stability in applied mathematics in some contexts is recalled. Massive tetrad gravities permit the mass of the spin 0 to be heavier than that of the spin 2, as well as lighter than or equal to it, and so provide phenomenological flexibility that might be of astrophysical or cosmological use.     

Renormalization group approach to QCD phase transitions

Effective scalar theories for QCD are proposed to investigate the deconfining and chiral phase transitions. The orders of the phase transitions are determined by infrared stabilities of the fixed points. It is found that the transitions inSU (3) gauge theories are of 1st order for any number of massless flavors. The cases ofSU (2) andSU (4) gauge theories are also discussed.     

Duality and the θ parameter in Abelian lattice models

The phase diagrams of abelian lattice gauge theories in four dimensions, and spin models in two dimensions, in the presence of a θ-parameter, are constructed using duality arguments. A rich structure involving an infinite number of phases emerges, together with a mechanism for spontaneous CP violation at θ = π.     

Spin Hall effect in noncommutative coordinates

A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter θ. The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and θ deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting θ different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.     

Comparing O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions to SU(N) gauge theories in 3 + 1 dimensions

We have computed the scale breaking Λ parameters of the euclidean and hamiltonian formulations of the lattice regulated O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions in terms of the Λ_PV parameters of the Pauli-Villars regulated continuum models. Using lattice perturbation theory, the renormalized mass gap has been determined in terms of Λ_PV for each model. These results are compared to analogous calculations in SU(N) gauge theories.     

Understanding correlated electron systems by a classification of Mott insulators

This paper surveys the physics of systems proximate to Mott insulators, and presents a classification using conventional and topological order parameters. This classification offers a valuable perspective on a variety of conducting correlated electron systems, from the cuprate superconductors to the heavy fermion compounds. Connections are drawn, and distinctions made, between collinear/non-collinear magnetic order, bond order, neutral spin 1/2 excitations in insulators, electron Fermi surfaces which violate Luttinger’s theorem, fractionalization of the electron, and the fractionalization of bosonic collective modes. Two distinct categories of Z₂ gauge theories are used to describe the interplay of these orders. Experimental implications for the cuprates are briefly noted, but these appear in more detail in a companion review paper (S. Sachdev, cond-mat/0211005).     

Deconfining and chiral phase transitions in quantum chromodynamics at finite temperature

We give further arguments to support the claim of Svetitsky and Yaffe that the finite-temperature transition in 4-dimensional SU(N) gauge theories is in the universality class of 3-dimensional Z_N spin models. We show that this implies a smoothing out of the transition when quarks are added to the system as long as N ≠ 3. For N = 3 the pure gauge transition is expected to be first order and will be smoothed by quarks only if the quark contribution to the internal energy is larger than the latent heat of transition.