Real Projective Plane Research Articles

Dark Matter in a twisted bottle

Abstract The real projective plane is a compact, non-orientable orbifold of Euler characteristic 1 without boundaries, which can be described as a twisted Klein bottle. We shortly review the motivations for choosing such a geometry among all possible two-dimensional orbifolds, while the main part of the study will be devoted to dark matter study and limits in Universal Extra Dimensional (UED) models based on this peculiar geometry. In the following we consider such a UED construction based on the direct product of the real projective plane with the standard four-dimensional Minkowski space-time and discuss its relevance as a model of a weakly interacting Dark Matter candidate. One important difference with other typical UED models is the origin of the symmetry leading to the stability of the dark matter particle. This symmetry in our case is a remnant of the six-dimensional Minkowski space-time symmetry partially broken by the compactification. Another important difference is the very small mass splitting between the particles of a given Kaluza-Klein tier, which gives a very important role to co-annihilation effects. Finally the role of higher Kaluza-Klein tiers is also important and is discussed together with a detailed numerical description of the influence of the resonances.

Minimal generating and normally generating sets for the braid and mapping class groups of $$\mathbb{D }^{2}$$ , $$\mathbb S ^{2}$$ and $$\mathbb{R P^2}$$

We consider the (pure) braid groups \(B_{n}(M)\) and \(P_{n}(M)\), where \(M\) is the \(2\)-sphere \(\mathbb S ^{2}\) or the real projective plane \(\mathbb R P^2\). We determine the minimal cardinality of (normal) generating sets \(X\) of these groups, first when there is no restriction on \(X\), and secondly when \(X\) consists of elements of finite order. This improves on results of Berrick and Matthey in the case of \(\mathbb S ^{2}\), and extends them in the case of \(\mathbb R P^2\). We begin by recalling the situation for the Artin braid groups (\(M=\mathbb{D }^{2}\)). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for \(M=\mathbb S ^{2}\) or \(\mathbb R P^2\), the induced action of \(B_n(M)\) on \(H_3(\widetilde{F_n(M)};\mathbb{Z })\) is trivial, \(F_{n}(M)\) being the \(n^\mathrm{th}\) configuration space of \(M\).

Separability and the genus of a partial dual

Partial duality generalizes the fundamental concept of the geometric dual of an embedded graph. A partial dual is obtained by forming the geometric dual with respect to only a subset of edges. While geometric duality preserves the genus of an embedded graph, partial duality does not. Here we are interested in the problem of determining which edge sets of an embedded graph give rise to a partial dual of a given genus. This problem turns out to be intimately connected to the separability of the embedded graph. We determine how separability is related to the genus of a partial dual. We use this to characterize partial duals of graphs embedded in the plane, and in the real projective plane, in terms of a particular type of separation of an embedded graph. These characterizations are then used to determine a local move relating all partially dual graphs in the plane and in the real projective plane.

Bundles over Quantum RealWeighted Projective Spaces

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

Simplicial Arrangements with up to 27 Lines

We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Grunbaum’s catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines, respectively. As a byproduct we classify simplicial arrangements of pseudolines with up to 27 lines. In particular, we disprove Grunbaum’s conjecture about unstretchable arrangements with at most 16 lines, and prove the conjecture that any simplicial arrangement with at most 14 pseudolines is stretchable.

Surface braid groups and coverings

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of B_{n} due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear.

Maps into projective spaces: Liquid crystal and conformal energies

Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.

The universal Real Projective Plane: LHC phenomenology at one loop

The Real Projective Plane is the lowest dimensional orbifold which, when combined with the usual Minkowski space-time, gives rise to a unique model in six flat dimensions possessing an exact Kaluza Klein (KK) parity as a relic symmetry of the broken six dimensional Lorentz group. As a consequence of this property, any model formulated on this background will include a stable Dark Matter candidate. Loop corrections play a crucial role because they remove mass degeneracy in the tiers of KK modes and induce new couplings which mediate decays. We study the full one loop structure of the corrections by means of counter-terms localised on the two singular points. As an application, the phenomenology of the (2,0) and (0,2) tiers is discussed at the LHC. We identify promising signatures with single and di-lepton, top antitop and 4 tops: in the dilepton channel, present data from CMS and ATLAS may already exclude KK masses up to 250 GeV, while by next year they may cover the whole mass range preferred by WMAP data.

Four tops on the real projective plane at LHC

We explore the four top signal $ t\bar{t}t\bar{t} $ at the 7TeV Large Hadron Collider as a probe of physics beyond the standard model. Enhancement of the corresponding cross-section with respect to the Standard Model value can probe the electroweak symmetry breaking sector or test extra dimensional models with heavy Kaluza-Klein gluons and quarks. We perform a detailed analysis including background and detector simulation in the specific case of a universal extra-dimensional model with two extra dimensions compactified using the geometry of the real projective plane. For masses around 600 GeV, a discovery is possible for an effective cross section above 210 fb (36 fb) for 1/fb (10/fb) of integrated luminosity. This implies a branching ratio in tops of the (1, 1) heavy photon above 13% (5%). Furthermore, the 4-top signal from the (2, 0) and (0, 2) tiers can be discovered with an integrated luminosity of 3.5/fb. The results of our simulation can be easily adapted to other models since the background processes are identical. Concerning the signal, typical production mechanisms for the $ t\bar{t}t\bar{t} $ signal are similar even if cross-section values may vary considerably depending on the model and the spectrum of the new particles.

Minimal fields of definition for simplicial arrangements in the real projective plane

For each simplicial arrangement in the real projective plane of the catalogue of Grunbaum, we determine the minimal extension of the rationals over which there exists a realization of its incidence structure. For the infinite families we use the symmetries of the incidence. For the sporadic arrangements we give an algorithm that uses Grobner bases.

A simplification of the proof of Bol’s conjecture on sextactic points

In a previous work with Thorbergsson, it was proved that a simple closed curve in the real projective plane $\mathbf{P}^{2}$ that is not null-homotopic has at least three sextactic points. This assertion was conjectured by Gerrit Bol. That proof used an axiomatic approach called ‘intrinsic conic system’. The purpose of this paper is to give a more elementary proof.

On the Zone Complexity of a Vertex

Let be a set of lines in the real projective plane in general position. We show that there exists a vertex such that is positioned in a face of size at most 5 in the arrangement obtained by removing the two lines passing through .

A CHARACTERIZATION OF THE REAL PROJECTIVE PLANE

It is proved in this article, that in the framework of Riemannian geometry, the existence of large sets of antipodes (i.e. farthest points) for diametral points of a smooth surface has very strong consequences on the topology and the metric of this surface. Roughly speaking, if the sets of antipodes of diametral points are closed curves, then the surface is nothing but the real projective plane.

Into how many regions do n lines divide the plane if at most n − k of them are concurrent?

The number of connected components of the complement in the real projective plane to a family of n ≥2 different lines such that any point belongs to at most n − k of them is estimated. If \( n \geqslant \frac{{k^2 + k}} {2} + 3 \), then the number of regions is at least (k+1)(n−k). Thus, a new proof of N. Martinov’s theorem is obtained. This theorem determines all pairs of integers (n, f) such that there is an arrangement of n lines dividing the projective plane into f regions.

Non-euclidean geometries: the Cayley-Klein approach

A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begrundeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces over arbitrary fields and of arbitrary dimensions requires two steps, the introduction of a metric in a pappian projective space and the definition of substructures as Cayley-Klein geometries. While the first step is taken in H. Struve and R. Struve (J Geom 81:155–167, 2004), the second step is made in this article. We show that the concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of non-euclidean geometries including the main geometries studied in the foundations of geometry by D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al.

Universal extra dimensions on real projective plane

We propose a six-dimensional Universal Extra Dimensions (UED) model compactified on a real projective plane RP2, a two-sphere with its antipodal points being identified. We utilize the Randjbar-Daemi–Salam–Strathdee spontaneous sphere compactification with a monopole configuration of an extra U(1)X gauge field that leads to a spontaneous radius stabilization. Unlike the sphere and the S2/Z2 orbifold compactifications, the massless U(1)X gauge boson is safely projected out. We show how a compactification on a non-orientable manifold results in a chiral four-dimensional gauge theory by utilizing 6D chiral gauge and Yukawa interactions. The resultant Kaluza–Klein mass spectra are distinct from the ordinary UED models compactified on torus. We briefly comment on the anomaly cancellation and also on a possible dark matter candidate in our model.

Recursive Formulas for Welschinger Invariants of the Projective Plane

Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we establish a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.

On the scalar curvature of hypersurfaces in spaces with a Killing field

Abstract We consider compact hypersurfaces in an (n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when n = 2 and is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ≥ 3 and is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product 𝕄 × ℝ1.

The Borsuk–Ulam theorem for maps into a surface

Let ( X , τ , S ) be a triple, where S is a compact, connected surface without boundary, and τ is a free cellular involution on a CW-complex X. The triple ( X , τ , S ) is said to satisfy the Borsuk–Ulam property if for every continuous map f : X → S , there exists a point x ∈ X satisfying f ( τ ( x ) ) = f ( x ) . In this paper, we formulate this property in terms of a relation in the 2-string braid group B 2 ( S ) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples ( X , τ , S ) for which the Borsuk–Ulam property holds. We also consider various cases where X is not necessarily a surface without boundary, but has the property that π 1 ( X / τ ) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S 2 and the real projective plane R P 2 , then we show that the Borsuk–Ulam property does not hold for ( X , τ , S ) unless either π 1 ( X / τ ) ≅ π 1 ( R P 2 ) , or π 1 ( X / τ ) is isomorphic to the fundamental group of a compact, connected non-orientable surface of genus 2 or 3 and S is non-orientable. In the latter case, the veracity of the Borsuk–Ulam property depends further on the choice of involution τ; we give a necessary and sufficient condition for it to hold in terms of the surjective homomorphism π 1 ( X / τ ) → Z 2 induced by the double covering X → X / τ . The cases S = S 2 , R P 2 are treated separately.

Nielsen Type Numbers of Self-Maps on the Real Projective Plane

Employing the induced endomorphism of the fundamental group and using the homotopy classification of self-maps of real projective plane , we compute completely two Nielsen type numbers, (f) and (f), which estimate the number of periodic points of f and the number of fixed points of the iterates of map f.