Four-dimensional Manifold Research Articles

Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold

There is a ( ) ( ) 4' 4 UU × -bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particles and fields in Standard Model (SM) of particle physics and Einstein gravity are found in square root metric and the connections of bundle. The interactions between particles/fields are described by Lagrangian explicitly. There are few new physics in this model. The gravity theory is Einstein-Cartan kind with torsion. There are new particles, right handed neutrinos, dark photon, Fiona, c X ± and 01212 ** ,, ,, Y YY YY

Chebyshev spectral methods for computing center manifolds

<p style='text-indent:20px;'>We propose a numerical approach for computing center manifolds of equilibria in ordinary differential equations. Near the equilibria, the center manifolds are represented as graphs of functions satisfying certain partial differential equations (PDEs). We use a Chebyshev spectral method for solving the PDEs numerically to compute the center manifolds. We illustrate our approach for three examples: A two-dimensional system, the Hénon-Heiles system (a two-degree-of-freedom Hamiltonian system) and a three-degree-of-freedom Hamiltonian system which have one-, two- and four-dimensional center manifolds, respectively. The obtained results are compared with polynomial approximations and other numerical computations.

Spectral action in matrix form

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang–Mills theory. We illustrate the procedure for the toy electroweak model.

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Three-Dimensional Manifolds

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Four-Dimensional Manifolds

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Four-Dimensional Manifolds

Four-dimensional homogeneous manifolds satisfying some Einstein-like conditions

We classify four-dimensional homogeneous manifolds satisfying some generalized Einstein conditions.

Derivation of Lighthill’s Eighth Power Law of an Aeroacoustic Quadrupole in Acoustic Spacetime

Acoustic spacetime is a four-dimensional manifold analogue to the relativistic spacetime with the reference speed of light replaced by the speed of sound. It has been established primarily for the indirect studies of relativistic phenomena by means of their better understood acoustic analogues. More recently, it has also been used for the analytical treatment of sound propagation in various uniform and non-uniform flows of the background fluid. In this paper the analogy is extended and utilized to derive Lighthill’s eight power law for sound generation of an aeroacoustic quadrupole. Adding to the existing analogue theory, propagating sound waves are described in terms of a weak perturbation of the background acoustic spacetime metric. The obtained result proves that the acoustic analogy can be extended to cover both weak perturbation of the fluid due to the sound waves and certain sound generation mechanisms, at least in incompressible low Mach number flows.

Cohomological localization of mathcal{N} = 2 gauge theories with matter

We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general mathcal{N} = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

A geometric algebraic approach to fluid dynamics

Geometric algebra plays a major role in merging the physical and mathematical ideas in the context of various physical systems. In this paper, we explore certain properties associated with barotropic and non-barotropic fluid flows with the help of geometric algebra over a four-dimensional Euclidean space time manifold. We introduce the concepts of multivectors associated with vorticity, helicity, and parity, which evolve from a four-velocity field. In this context, the fluid dynamical analogs of the Poynting theorem, Lorentz force, and Maxwell’s equations are derived. The fluid Maxwell’s equations can be extracted from a single equation.

Parallel and totally geodesic hypersurfaces of non‐reductive homogeneous four‐manifolds

AbstractWe classify totally geodesic and parallel hypersurfaces of four‐dimensional non‐reductive homogeneous pseudo‐Riemannian manifolds.

Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant. II.

We consider general relativity with a cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional spacetime manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature (case C). These solutions are invariant with respect to the Lorentz $\mathbb{S}\mathbb{O}(1,2)$ or Poincare $\mathbb{I}\mathbb{O}(1,1)$ groups acting on the Lorentzian surface.

A note on rigidity of Einstein four-manifolds with positive sectional curvature

In this paper, we first prove a topological obstruction for a four-dimensional manifold carrying an Einstein metric. More precisely, assume (M, g) is a closed Einstein four-manifold with $$Ric=\rho g$$ . Denote by K the sectional curvature of M. If $$K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}$$ (or $$K\le \delta \le \frac{2\rho +\sqrt{5}}{6}$$ ) for some constant $$\delta $$ , then the Euler characteristic $$\chi $$ and the signature $$\tau $$ of M satisfy $$\begin{aligned}\chi \ge \left( \dfrac{3}{8\left( 1-3\delta /\rho \right) ^2} +\dfrac{3}{2}\right) \left|\tau \right|.\end{aligned}$$ Our second result is a rigidity theorem for closed oriented Einstein four-manifolds with positive sectional curvature. Assume $$\lambda _1$$ is the first eigenvalue of the Laplacian of an oriented closed Einstein four-manifold (M, g) with $$Ric = g$$ . We show that M must be isometric to a round 4-sphere or $$\mathbb {CP}^2$$ with the (normalized) Fubini-Study metric if the sectional curvature bounded above by $$1-\frac{4}{9\lambda _1+12}$$ (or bounded below by $$\frac{2}{9\lambda _1+12}$$ ).

Lévy Laplacians and instantons on manifolds

The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group [Formula: see text] is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang–Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modified Lévy Laplacians from this class.

From spacetime to space and time: a reply to Markosian

AbstractIn a recent article, Ned Markosian gives an argument against four-dimensionalism understood as the view that time is one of four similar dimensions that constitute a single four-dimensional manifold. In this paper, I show that Markosian attacks a straw man as his argument targets a theory known to be false on empirical grounds. Four-dimensionalism rightly conceived in no way entails that time is identical to space. I then address two objections raised by Markosian against four-dimensionalism rightly conceived.

FOUR-MANIFOLDS WITH POSITIVE CURVATURE

AbstractIn this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

Adiabatic limit in Ginzburg—Landau and Seiberg—Witten equations

We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ℝ1+2 and Seiberg—Witten equations on four-dimensional symplectic manifolds. We show that the Seiberg—Witten equations can be regarded as a complex version of the Ginzburg—Landau equations.

Quantum fields and gravity: Expanding space-times

We discuss a proposal for a somewhat new formulation of quantum field theory (set in a four-dimensional manifold, the space-time) that includes an analysis of its implications for the evolution of Einstein-Friedmann cosmological models. The proposed theory displays two peculiar features: (i) a local Hilbert-Fock space is associated with each space-time point: we are dealing with a vector bundle whose fibers are Hilbert spaces; the operator-valued sections of the bundle are the quantum fields; (ii) the vacuum energy density is finite, being regularized in a space-time curvature dependent way, independently at each point. In fact everything is finite: self-masses, self-charges, quantum fluctuations: they depend on the space-time curvature and diverge only for a flat metric. In an Einstein-Friedmann model the vacuum (zero-point) energy density is consequently time-dependent and in general not negligible. Then it is shown that, for some choices of the parameters of the theory, the big-bang singularity is resolved and replaced by a bounce driven by the vacuum energy density, which becomes (very) large and negative near the bounce (negative by the contribution of the Fermi fields). But for large times (now, say) the Bose fields’ positive vacuum energy eventually overcomes the negative one and we are finally left with the present vacuum energy: positive and reasonably small.

Four-dimensional almost Einstein manifolds with skew-circulant stuctures

We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.

Smooth Invariants of Four-Dimensional Manifolds and Quantum Field Theory

Smooth Invariants of Four-Dimensional Manifolds and Quantum Field Theory

New cases of integrable ninth-order systems with dissipation

The paper shows the integrability of certain classes of ninth-order dynamic systems that are homogeneous in part, in which the system on the tangent bundle to four-dimensional manifolds is distinguished. In this case, the force fields have various dissipation and generalize the previously considered ones.