Narrow Results

In this present article, we study the local features of the worldsheet in the case when probe bosonic string moves in antisymmetric background field. We generalize the geometry of surfaces embedded in spacetime to the case when the torsion is present. We define the mean extrinsic curvature for spaces with Minkowski signature and introduce the concept of mean torsion. Its orthogonal projection defines the dual mean extrinsic curvature. In this language, the field equation is just the equality of mean extrinsic curvature and extrinsic mean torsion, which we call CT-duality. To the worldsheet described by this relation we will refer as CT-dual surface.

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

In this paper, the singularities of the geometry for four classes of worldsheets, which are respectively, located in three-dimensional hyperbolic space and three-dimensional de Sitter space–time are considered. Under the theoretical frame of geometry of space–time and as applications of singularity theory, it is shown that these worldsheets have two classes of singularities, that is, in the local sense, these four classes of worldsheets are, respectively, diffeomorphic to the cuspidal edge and the swallowtail. The first hyperbolic worldsheet and the second hyperbolic worldsheet are ${\mathrm{\Delta }}_1$-dual to the tangent curves of spacelike curves. Moreover, it is also revealed that there is a close relationship between the types of singularities of worldsheets and a geometric invariant ${\sigma }_d(s)$, depending on whether ${\sigma }_d(s)\ne 0$ or ${\sigma }_d(s)=0$ and ${\sigma }_d^{\prime }(s)\ne 0$, the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.

In physics, the application of the regular cases of spacelike fronts has been studied extensively; however, under the gravitational influence, the spacelike fronts can be singular. In this paper, we will consider the traveling trajectories of a set of parallel light rays as spacelike fronts with singularities in null de Sitter sphere. It is of great significance to investigate and explore the spacelike fronts which may have singularities in null de Sitter sphere. We also characterize the singularities of the nullcone dual worldsheets generated by spacelike fronts via the spherical curvature. Then we define the nullcone caustics and analyze the geometric properties of the nullcone caustics at singular points. Moreover, we establish the dual relationship between spacelike fronts and the nullcone dual worldsheets.