Four-dimensional Torus Research Articles

Vacuum structure of an eight-dimensional SU(3) gauge theory on a magnetized torus

We analyze the vacuum structure of an eight-dimensional non-Abelian gauge theory with a compactified four-dimensional torus as the extra dimensions. As a nontrivial background configuration of the gauge field of an SU(n) gauge group, we suppose a magnetic flux in two extra dimensions, and continuous Wilson line phases are also involved. We introduce matter fields and calculate the mass spectrum of low-energy modes appearing in a four-dimensional effective theory in an SU(3) model as an explicit example. As expected, potentially tachyonic states in four-dimensional modes appear from extra-dimensional gauge fields that couple to the flux background since the gauge group is simply connected. The Wilson line phases give a nonvanishing contribution to their masses, and we have a low-energy mass spectrum without tachyonic states, given that these phases take an appropriate value. To verify the validity of the values of the Wilson line phases, we examine the one-loop effective potential for these phases and explicitly show the contribution from each type of field present in our model. It is clarified that, although there seems to be no local minimum in the potential for the Wilson line phases in the pure Yang-Mills case, by including matter fields, we could find a vacuum configuration where tachyonic states disappear. Published by the American Physical Society 2024

Higher-group structure in lattice Abelian gauge theory under instanton-sum modification

We consider the U(1) gauge theory on a four-dimensional torus, where the instanton number is restricted to an integral multiple of p. This theory possesses the nontrivial higher-group structure, which can be regarded as a generalization of the Green–Schwarz mechanism, between mathbb {Z}_q 1-form and mathbb {Z}_p 3-form symmetries. Here, mathbb {Z}_q is a subgroup of the center of U(1). Following the recent study of the lattice construction of the U(1)/mathbb {Z}_q principal bundle, we examine how such a structure is realized on the basis of lattice regularization.

A Liouville-Type Result for a Fourth Order Equation in Conformal Geometry

We show that there are no conformal metrics $g=e^{2u}g_{\mathbb {R}^{4}}$ on $\mathbb {R}^{4}$ induced by a smooth function u ≤ C with Δu(x) → 0 as $|x|\to \infty $ having finite volume and finite total Q-curvature, when Q(x) = 1 + A(x) with a negatively definite symmetric 4-linear form A(x) = A(x,x,x,x). Thus, in particular, for suitable smooth, non-constant $f_{0}\le {\max \limits } f_{0}=0$ on a four-dimensional torus any “bubbles” arising in the limit λ↓ 0 from solutions to the problem of prescribed Q-curvature Q = f0 + λ blowing up at a point p0 with dkf0(p0) = 0 for $k=0,\dots ,3$ and with d4f0(p0) < 0 are spherical, similar to the two-dimensional case.

Conical singular points and vector fields

In this article, several examples of mechanical systems which configuration spaces are smooth manifolds with a unique singular point are considered. Configuration spaces are the following: two smooth curves with a common point (or tangent) on the two-dimensional torus, four smooth curves on the four-dimensional torus with a common point, twodimensional cone (cusp) in the space R6. The main problem in the article is the calculation of (co)tangent space at a singular point by using different theoretical approaches. Outside of the singular point, the motion could be described in the frames of classical mechanics. But in the neighborhood of the singular points the terms like “tangent vector” and “cotangent vector” must have new conceptual definitions. In this article, the approach of differential spaces is used. Two differential structures for the modeling conical singular point are studied in order to construct (co)tangent space at singular points: locally-constants functions near to the cone vertex and the algebra of the restrictions of smooth functions in the comprehensive Euclidean space on the cone. In the first case, tangent and cotangent spaces at the singular points are zero. In the second case, the value of the functions on the cotangent bundle is constant on the cotangent layer under the singular point.

Meromorphic flux compactification

We present exact solutions of four-dimensional Einstein’s equations related to Minkoswki vacuum constructed from Type IIB string theory with non-trivial fluxes. Following [1, 2] we study a non-trivial flux compactification on a fibered product by a four-dimensional torus and a two-dimensional sphere punctured by 5- and 7-branes. By considering only 3-form fluxes and the dilaton, as functions on the internal sphere coordinates, we show that these solutions correspond to a family of supersymmetric solutions constructed by the use of G-theory. Meromorphicity on functions constructed in terms of fluxes and warping factors guarantees that flux and 5-brane contributions to the scalar curvature vanish while fulfilling stringent constraints as tadpole cancelation and Bianchi identities. Different Einstein’s solutions are shown to be related by U-dualities. We present three supersymmetric non-trivial Minkowski vacuum solutions and compute the corresponding soft terms. We also construct a non-supersymmetric solution and study its stability.

On infrared problems of effective Lagrangians of massive spin 2 fields coupled to gauge fields

In this paper we analyze the interactions of massive spin-2 particles charged under both Abelian and non-Abelian group using the Porrati–Rahman Lagrangian. This theory is valid up to an intrinsic cutoff scale. Phenomenologically a theory valid up to a cutoff scale is sensible as all known higher spin particles are non-fundamental and it is shown that indeed this action can be used to estimate some relevant cross section. Such action necessarily includes Stückelberg field and therefore it is necessary to fix the corresponding gauge symmetry. We show that this theory, when the Stückelberg symmetry is gauge-fixed, possesses a non-trivial infrared problem. A gauge fixing ambiguity arises which is akin to the Gribov problem in QCD in the Abelian case as well. In some cases (such as when the space–time is the four-dimensional torus) the vacuum copies can be found analytically. A similar phenomenon also appears in the case of Proca fields. A very interesting feature of these copies is that they arise only for “large enough” gauge potentials. This opens the possibility to avoid the appearance of such gauge fixing ambiguities by using a Gribov–Zwanziger like approach.

Bifurcation analysis of four-frequency quasi-periodic oscillations in a three-coupled delayed logistic map

We present herein an extensive analysis of the bifurcation structures of quasi-periodic oscillations generated by a three-coupled delayed logistic map. Oscillations generate an invariant three-torus, which corresponds to a four-dimensional torus in vector fields. We illustrate detailed two-parameter Lyapunov diagrams, which reveal a complex bifurcation structure called an Arnol'd resonance web. Our major concern in this study is to demonstrate that quasi-periodic saddle-node bifurcations from an invariant two-torus to an intermittent invariant three-torus occur because of a saddle-node bifurcation of a stable invariant two-torus and a saddle invariant two-torus. In addition, with some assumptions, we derive a bifurcation boundary between a stable invariant two-torus and a stable invariant three-torus due to a quasi-periodic Hopf bifurcation with a precision of 10−5.

Heterotic-Type II duality and wrapping rules

We show how the brane wrapping rules, recently discovered in closed oriented string theories compactified on tori, are extended to the case of the Type IIA string compactified on K3. To this aim, a crucial role is played by the duality between this theory and the Heterotic string compactified on a four-dimensional torus T^4. We first show how the wrapping rules are applied to the T^4/Z_N orbifold limits of K3 by relating the D0 branes, obtained as D2 branes wrapping two-cycles, to the perturbative BPS states of the Heterotic theory on T^4. The wrapping rules are then extended to the solitonic branes of the Type IIA string, finding agreement with the analogous Heterotic states. Finally, the geometric Type IIA orbifolds are mapped, via T-duality, to non-geometric Type IIB orbifolds, where the wrapping rules are also at work and consistent with string dualities.

TRANSITION TO CHAOS FROM QUASIPERIODIC MOTIONS ON A FOUR-DIMENSIONAL TORUS PERTURBED BY EXTERNAL NOISE

We propose a new autonomous dynamical system of dimension N = 4 that demonstrates the regime of stable two-frequency motions. It is shown that the system of two generators of quasiperiodic oscillations with symmetric coupling can realize motions on four-dimensional torus with resonant structures on it in the form of three- and two-dimensional torus. We show that with increase of noise intensity when the dimension of torus is higher it is destroyed faster.

Dynamics on a torus

The dynamics of atoms on the surface of a torus is considered. The simple illustration of motion with regard to rotating and fixed space gives a model of a four-dimensional (4D) torus. Two different schemes including rotation and shear in angular frame are used to take into account shears of the surface. In general, a variable-cell-shape molecular dynamics method analogous to the Parrinello-Raman one is developed. The six dynamical variables, the three radiuses and the three angles, specifying the deformations of the surface describe the cell dynamics. The new equations of motion contain no vectors of translations of the cell making its shape irrelevant for the structural and thermodynamical description of the system. The new method was tested on two problems concerning structure transformations of two-dimensional lattices.

About uniform distribution of four-dimensional torus

Results on the uniform distribution of ergodic endomorphisms on the four-dimensional torus are obtained. They extend previous results [1] for the case when the characteristic polynomial of matrix V may be reducible.

A comparison of optimal FFTs on torus and hypercube multicomputers

In this paper we compare the optimum performance of the fast Fourier transform (FFT) on torus and hypercube multicomputers. The optimal number of floating point operations is architecturally invariant so the relative performance is determined by communication time. We compare the performance of known multiport communication algorithms that utilize the full bandwidth of the interconnection network on every cycle. Furthermore, data is routed unblocked over paths with minimum length. Specifically we compare FFT performance on the hypercube as well as two-, three- and four-dimensional torus architectures. On the hypercube we observe the somewhat surprising result that, in the limit, communication is ultimately negligible compared to computation. While the opposite is true of the torus, it is nevertheless possible to obtain comparable performance over a broad range of processors and problem sizes. As computers are built with increasing numbers of processors, torus performance can still be made comparable to the hypercube by gradually increasing the dimension of the interconnect. Any number of processors can be used to compute the FFT with efficiency that is theoretically bounded from zero.

Perturbative construction of self-dual configurations on the torus

We develop a perturbative expansion which allows the construction of non-abelian self-dual SU(2) Yang-Mills field configurations on the four-dimensional torus with topological charge 1/2. The expansion is performed around the constant field strength abelian solutions found by 't Hooft. Next to leading order calculations are compared with numerical results obtained with lattice gauge theory techniques.

Bihermitian Structures on Complex Surfaces

Bihermitian complex surfaces are oriented conformal four-manifolds admitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on R4 and the four-dimensional torus T4 have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti-canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and S1 × S3. Further properties of compact bihermitian surfaces are also investigated. 1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

On Nahm's transformation with twisted boundary conditions

Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the four-dimensional torus T 4 with non-zero twist tensor. The transform is again a self-dual gauge field defined on a new torus T ^ 4 and with non-zero twist tensor. It preserves the property of being an involution.

Pair correlation of four-dimensional flat tori

Pair correlation of four-dimensional flat tori

Explicit doubly-hermitian metrics

We construct explicit examples of 4-dimensional Riemannian metrics which admit precisely two independent hermitian structures with the same orientation. It is shown that metrics of this type exist on four-dimensional tori.

On the Wilson criterion

U(1) gauge theory with the Villain action on a cubic lattice approximation of three- and four-dimensional torus is considered. The naturally chosen correlation functions converge to the correlation functions of the R-gauge electrodynamics on three- and four-dimensional torus as the lattice spacing approaches zero only for the special scaling. This special scaling depends on a choice of a correlation function system. Another scalings give the degenerate continuum limits. The Wilson criterion for the confinement is ambiguous. The asymptotics of the smeared Wilson loop integral for the large loop perimeters is defined by the density of the loop smearing over a torus which is transversal to the loop plane. When the initial torus radius tends to infinity the correlation functions converge to the correlation functions of the R-gauge Euclidean electrodynamics.

Triangulation du tore de dimension 4

We build a combinatorial four-dimensional torus with 31 vertices, different from that obtained by Kuhnel and Lassman. Our torus has also a vertex transitive group action but the symmetry group is of order 2 × 3 × 31, different from 2 × 5 × 31 for the symmetry group of the Kuhnel–Lassmann torus. Furthermore, we give a geometrical description of this torus, as well as various results concerning triangulations of tori of dimensions 2 and 3.

Chaos with Few Degrees of Freedom

As a mathematician, astronomer, and physicist, Poincaré was the first to study the phenomenon of chaos in classical mechanics, in particular the three-body-problem of the Moon orbiting around the Earth under the perturbation of the Sun. His work helps us to understand atomic and molecular physics, because our intuition in these areas is always based on a classical picture for the motion of the nuclei and electrons. The Moon's motion was found to be multi-periodic by the early observers; Newton explained the main frequencies and calculated the lowest terms in the corresponding Fourier expansion. But inspite of ingenious efforts, like the work of Hill involving as a starting point a periodic orbit of the full problem, Poincaré showed that perturbation theory is unable to guarantee the convergence of the Fourier expansion. In this traditional representation, the lunar trajectory would be restricted to a four-dimensional invariant torus although the available phase-space has eight dimensions. The invariant tori are destroyed because of increasing phase-lock due to perturbations, whenever the system is close to a resonance. The KAM-theorem gives a mathematical criterion for this loss, which is best seen in a surface of section for such simple systems as the Anisotropic and the Diamagnetic Kepler Problem (AKP and DKP). Instead of invariant tori, the flow in phase space involves a double foliation where each trajectory is the intersection of one leaf from the unstable foliation with a leaf from the stable foliation. Neighboring trajectories drift exponentially away from each other in the future along the unstable leaf, and approach each another in the stable leaf coming from the past. This situation, called “hard chaos”, also allows for a symbolic description of each trajectory in terms of a simple code like binary sequences in the AKP. Einstein showed that the usual connection with quantum mechanics is valid only for the case of “regular behavior”, i.e. the presence of invariant tori. For classically chaotic systems, VanVleck's expression for the propagator seems to be the best starting point, provided it is summed over all the classical trajectories that go from the source to the defector. Its trace becomes a sum over periodic orbits, which lends additional weight to Poincaré's emphasis on their importance in classical mechanics. In trying to understand the trace of an operator as a function of time, or the scattering phase-shift as a function of momentum, one runs into almost-periodic functions. They are able to mimic locally any arbitrary smooth function provided the spectrum of frequencies, although discrete, is rich enough. This unexpected, but smooth behavior can be seen as a symptom of quantum chaos, in contrast to the fractal nature of classical chaos.