Two-dimensional Internal Space Research Articles

Diffraction of a model set with complex windows

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals.

Gauged supergravities and their symmetry-breaking vacua in F-theory

We first derive a class of six-dimensional (1,0) gauged supergravities arising from threefold compactifications of F-theory with background fluxes. The derivation proceeds via the M-theory dual reduction on an SU(3)-structure manifold with four-form G_4-flux. We then show that vacuum solutions of these six-dimensional theories describes four-dimensional flat space times a compact two-dimensional internal space with additional localized sources. This induces a spontaneous compactification to four space-time dimensions and breaks the supersymmetry from N=2 to N=1, which allows the reduced theory to have a four-dimensional chiral spectrum. We perform the reduction explicitly and derive the N=1 characteristic data of the four-dimensional effective theory. The match with fourfold reductions of F-theory is discussed and many of the characteristic features are compared. We comment, in particular, on warping effects and one-loop Chern-Simons terms generically present in four-dimensional F-theory reductions.

Can (electric-magnetic) duality be gauged?

There exists a formulation of the Maxwell theory in terms of two vector potentials, one electric and one magnetic. The action is then manifestly invariant under electric-magnetic duality transformations, which are rotations in the two-dimensional internal space of the two potentials, and local. We ask the question: can duality be gauged? The only known and battled-tested method of accomplishing the gauging is the Noether procedure. In its decanted form, it amounts to turn on the coupling by deforming the abelian gauge group of the free theory, out of whose curvatures the action is built, into a non-abelian group which becomes the gauge group of the resulting theory. In this article, we show that the method cannot be successfully implemented for electric-magnetic duality. We thus conclude that, unless a radically new idea is introduced, electric-magnetic duality cannot be gauged. The implication of this result for supergravity is briefly discussed.

Elastic properties of two-dimensional quasicrystals

Quasicrystals (QC) with two-dimensional quasiperiodic and one-dimensional periodic structure are considered. Their symmetry can be described by embedding the three-dimensional physical space V(E) in a five-dimensional superspace V, which is the direct sum of V(E) and a two-dimensional internal space V(I). A displacement v in V can be written as v = u + w, where u in V(E) and w in V(I). If the QC has a point group P in V(E) that is crystallographic, it is assumed that w and a vector u' in V(E) lying in the plane in which the crystal is quasiperiodic transform under equivalent representations of P, inequivalent ones if the point group is 5-, 8-, 10- or 12-gonal. From the Neumann principle follow restrictions on the form of the phonon, phason and phonon-phason coupling contributions to the elastic stiffness matrix that can be determined by combining the restrictions obtained for a set of elements generating the point group of interest. For the phonon part, the restrictions obtained for the generating elements do not depend on the system to which the point group belongs. This remains true for the phason and coupling parts in the case of crystallographic point groups but, in general, breaks down for the non-crystallographic ones. The form of the symmetric 12 x 12 matrix giving the phonon, phason and phonon-phason coupling contributions to the elastic stiffness is presented in graphic notation.

Instability of brane cosmological solutions with flux compactifications

We discuss the stability of the higher-dimensional de Sitter (dS) brane solutions with two-dimensional internal space in the Einstein–Maxwell theory. We show that an instability appears in the scalar-type perturbations with respect to the dS spacetime. We derive a differential relation which has a very similar structure to the ordinary laws of thermodynamics as an extension of the work for the six-dimensional model (Kinoshita et al 2007 J. Cosmol. Astropart. Phys. JCAP05(2007)018). In this relation, the area of dS horizon (integrated over the two internal dimensions) behaves exactly as the thermodynamical entropy. The dynamically unstable solutions are in the thermodynamically unstable branch. An unstable dS compactification either evolves toward a stable configuration or a two-dimensional internal space is decompactified. These dS brane solutions are equivalent to the accelerating cosmological solutions in the six-dimensional Einstein–Maxwell–dilaton theory via dimensional reduction. Thus, if the seed higher-dimensional solution is unstable, the corresponding six-dimensional solution is also unstable. From the effective four-dimensional point of view, a cosmological evolution from an unstable cosmological solution in higher dimensions may be seen as a process of the transition from the initial cosmological inflation to the current dark-energy-dominated universe.

Stationary vs. singular points in an accelerating FRW cosmology derived from six-dimensional Einstein–Gauss–Bonnet gravity

Abstract Six-dimensional Einstein–Gauss–Bonnet gravity (with a linear Gauss–Bonnet term) is investigated. This theory is inspired by basic features of results coming from string and M-theory. Dynamical compactification is carried out and it is seen that a four-dimensional accelerating FRW universe is recovered, when the two-dimensional internal space radius shrinks. A non-perturbative structure of the corresponding theory is identified which has either three or one stable fixed points, depending on the Gauss–Bonnet coupling being positive or negative. A much richer structure than in the case of the perturbative regime of the dynamical compactification recently studied by Andrew, Bolen, and Middleton is exhibited.

3-branes and uniqueness of the Salam–Sezgin vacuum

We prove the uniqueness of the supersymmetric Salam–Sezgin (Minkowski)4×S2 ground state among all non-singular solutions with a four-dimensional Poincaré, de Sitter or anti-de Sitter symmetry. We construct the most general solutions with an axial symmetry in the two-dimensional internal space, and show that included amongst these is a family that is non-singular away from a conical defect at one pole of a distorted 2-sphere. These solutions admit the interpretation of 3-branes with negative tension.

Spontaneous symmetry breaking in six-dimensional Einstein-Yang-Mills theory

We investigate all possible solutions of the six-dimensional Einstein-Yang-Mills equations with four-dimensional Poincaré invariance and geometry M 4 × B. (B is a two-dimensional internal space with finite volume.) Excluding the trivial case of B being flat, we find that for all solutions the internal space is a sphere and the gauge fields are in a monopole configuration on S 2. Therefore, the isometry group SO(3) is not spontaneously broken.

Reinterpretation of the Two-Dimensional Internal Space Associated with Dual-Resonance Models. II

A possible connection is suggested between the conventional Lagrangian field theory for spinors in the 2-dimensional Minkowski space and the dual-resonance field theory based on the anti-commutation relations. The former is the one detailed ever since Thirring and the latter contains the theory of Neveu and Schwarz's field and the theory of Ramond's field. It is in partieular argued that the Virasoro algebra for spinor fields is also reproduced within the framework of the Lagrangian field theory, and the transformations generated by the Gliozzi-Virasoro operators are in response to the conformal group of the 2-dimensional Min· kowski space.

Reinterpretation of the Two-Dimensional Internal Space Associated with Dual-Resonance Models. I

Reinterpretation of the Two-Dimensional Internal Space Associated with Dual-Resonance Models. I

Symmetrical structure and conservation relations of four-component neutrino field

A four-component spinor field with zero rest-mass ψ is represented as the field defined by its modulus B, an orthogonal frame in two-dimensional internal space, and a Lorentz frame in 4-dimensional internal space. This representation manifests the perfect isotropic property of the field in both internal spaces. In fact, the neutrino equation is invariant under arbitrary rotations in those internal spa es which are conversely expressible by a certain four-component spinor ζ As the result of these new invariance properties the neutrino field is shown to satisfy several conservation relations that are vector or tensor equations with respect to both internal spaces. They contain, besides the conventional ones, a number of new conservation laws.