On the renormalized coupling constant and the susceptibility in φ44 field theory and the Ising model in four dimensions

We discuss the euclidean φ₄⁴ field theory, and the critical behavior in ferromagnetic systems in four dimensions. It is rigorously shown that there are at most logarithmic corrections to the mean field law in the behavior of the magnetic susceptibility _X = ΣS₂(0, x). Furthermore, if any such corrections are present in a continuum limit which is used to construct a φ₄⁴ field theory, the limiting theory would be non-interacting. Our analysis extends to ferromagnetic systems of variables which belong to the Griffiths-Simon class.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Gauged N = 8 supergravity in five dimensions

We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6,R), specially SO(p, 6?p) for p = 0, 1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E₆₍₆₎ symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.     

The geometry of N = 2 Maxwell-Einstein supergravity and Jordan algebras

We construct the general coupling of nN = 2 Maxwell super-multiplets to N = 2 supergravity in five spacetime dimensions. In the case that the scalar field manifold is symmetric we find a complete classification based on Jordan algebras. Apart from the generic case there are also four “exceptional” cases associated with the Jordan algebras J₃^$\text{A}$ of 3 × 3 hermitian matrices over the division algebras $\text{A}\text{=}\text{R}\text{,}\text{C}\text{,}\text{H}\text{,}\text{O}$. Similar results follow for four dimensions, by dimensional reduction.     

Supersymmetric AdS_4 vacua in N=3 gauged supergravity

We study the maximally supersymmetric AdS backgrounds of matter-coupled N=3 gauged supergravity in four dimensions. We find that to admit supersymmetric AdS vacua, the gauge group can only be of the form G_0×H?SO(3,n) with G_0 =SO(3),SO(3,1) or SL(3,R) and H a compact group of dimension n+3-dim(G_0). We also show that these AdS vacua have no moduli, namely they correspond to critical points in field space.     

A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

On the triviality of λϕd4 theories and the approach to the critical point in d(−) > 4 dimensions

It is shown that one- and two-component λ|?|⁴ theories and non-linear σ-models in five or more dimensions approach free or generalized free fields in the continuum (scaling) limit, and that in four dimensions the same result holds, provided there is infinite field strength renormalization, as expected. Some critical exponents for the lattice theories in five or more dimensions are shown to be mean field. The main tools are Symanzik's polymer representation of scalar field theories and correlation inequalities.     

AgTh2(PO4)3 — A new case of the non-hydrogen bonded phosphate ferroelectric

The basic crystallographic data of AgTh₂(PO₄)₃ single crystals have been determined for the first time by means of X-ray diffraction methods. The crystals are monoclinic, space group isCc with four formula units per unit cell. The dimensions of the unit cell are:a=17·385 Å,b=6·815 Å,c=8·148 Å,β=101·10°. Using the Sawyer and Tower method it has been proved that the crystals possess ferroelectric properties. Performing the measurements at room temperature the values of spontaneous polarization and coercive field in the direction normal to (100) face have been determined.     

Chiral ordering in the four-dimensional planar spin glass

The chiral glass behaviour of the nearest-neighbour random-bond XY spin glass in four dimensions is studied by Monte Carlo simulations. A chiral glass transition at T _cg = 0:90 ± 0:05 is found by a finite-size scaling analysis of the results. The associated chiral correlation-length exponent is estimated to be gv _cg = 0:6±0:1. Both values are very similar to those reported recently for the spin glass transition in this model. The results strongly suggest a simultaneous ordering of spin and chirality in four dimensions.     

A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

The problem of confinement: From two to four dimensions

The Coulomb law of “electrodynamics” in two space time dimensions in extended to four dimensions. The result is that antisymmetric tensor potentials A_μν? subject to the gauge transformation δA_μν? = ?_μΛ_ν? + ?_?Λ_?μ + ?_?Λ_μν can be effectively used as confining potentials.     

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.     

Axiomatic renormalization of the stress tensor of a conformally invariant field in conformally flat spacetimes

Using only the general properties which the renormalized stress-energy tensor T_μν should satisfy—and not relying on any assumptions associated with specific renormalization techniques—we derive the expression for T_μν for conformally invariant fields in conformally flat spacetimes of two and four dimensions. In two dimensions, these arguments rederive the Davies-Fulling-Unruh expression for the stress tensor of a scalar field; in four dimensions the results agree with those of Brown and Cassidy, except that we exclude the local curvature term depending on fourth-order derivatives of the metric. The dynamics of a k = 0 Robertson-Walker universe filled with radiation of the conformally invariant field is investigated and it is found that the equations cease to admit a solution when the Planck density is reached.     

Small E8 instantons and tensionless non-critical strings

T-duality is used to extract information on an instanton of zero size in the E₈ × E₈ heterotic string. We discuss the possibility of the appearance of a tensionless anti-self-dual non-critical string through an implementation of the mechanism suggested by Strominger of two coincident 5-branes. It is argued that when an instanton shrinks to zero size a tensionless non-critical string appears at the core of the instanton. It is further conjectured that the appearance of tensionless strings in the spectrum leads to new phase transitions in six dimensions in much the same way as massless particles do in four dimensions.     

The pure gauge sector of compactified superstrings

We study the classical field equations of the pure gauge sector of superstrings compactified to four dimensions. Instantons no longer turn out to be solutions of the field equations. We also determine the dynamical behavior of the gauge coupling g in the background of a monopole field, with the result that g and the strong CP-violating angle θ tend to a constant far from the monopole. The energy of such a field configuration is finite.     

Nonlocal current algebra and N = 2 superconformal field theory in two dimensions

We present a new construction of the unitary highest weight representations of the N = 2 superconformal algebras in two dimensions. This construction is based on the non-local current in the Z_N conformal theory and a free scalar field. It provides a physical realization of all the unitary N = 2 superconformal field theories by critical systems. The correlation functions of the theories can be calculated through this construction.     

On clusters and critical dimensions in spin glasses

We discuss the problem of critical dimensions in the Sherrington-Kirkpatrick model of a spin glass, using the presence of clusters above T_c. It is argued that these clusters are ramified. This fact accounts for the different (upper) critical dimensions of the spherical and Ising models. The special role of dimension four is pointed out.     

Geometries with Killing Spinors and Supersymmetric <Emphasis Type="Italic">AdS</Emphasis> Solutions

The seven and nine dimensional geometries associated with certain classes of supersymmetric AdS ₃ and AdS ₂ solutions of type IIB and D = 11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in 2n + 2 dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for n ≥ 3, we show that when the geometry in 2n + 2 dimensions is a cone we obtain a class of geometries in 2n + 1 dimensions, specified by a Riemannian metric, a scalar field and a closed two-form, which includes the seven and nine-dimensional geometries mentioned above when n = 3, 4, respectively. We also consider various ansätze for the geometries and construct infinite classes of explicit examples for all n.     

Optimal heat kernel estimates for schrödinger operators with magnetic fields in two dimensions

Sharp smoothing estimates are proven for magnetic Schrödinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constantB ₀. As a consequence theL∞ norm of the associated integral kernel is bounded by theL∞ norm of the Mehler kernel of the Schrödinger semigroup with the constant magnetic fieldB ₀.     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.