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We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in

Conical space-time defects were first introduced by Staruszkiewicz [

In

A somewhat surprising feature of the description of a point particle in

One might wonder whether conical defects in higher dimensional (Minkowski) spaces possess properties similar to the three-dimensional case and, in particular, if their motion can be parametrized by some Lie group related to deformations of relativistic symmetries. To this end, let us recall that the best studied example of deformed relativistic symmetries in four dimensions is given by the

A massless conical defect can be obtained by boosting a (timelike) massive defect to the speed of light. In order to achieve a nontrivial limit, one usually applies the boost following a prescription of Aichelburg and Sexl, which was first proposed to derive the gravitational field of a photon from the Schwarzschild metric [

In this work, we analyze how momenta of massless defects in

An intuitive picture of a conical space-time defect is obtained by considering Minkowski space with a wedge removed and the faces of the wedge “glued together" to form a cone. The gluing is realized by identifying the opposite faces via a rotation by the

As we will discuss below, conical defects are curvature singularities. Thus, a given defect carries some mass/energy on its infinitely thin hyperplane. In

The metric of a conical defect can be written in terms of cylindrical coordinates, in which its geometric properties are most transparent. Here, we focus on a defect in

A generalization of holonomy (

The metric of a moving defect can be obtained [

One can also notice that, due to the nontrivial parallel transport of frames, the presence of a conical singularity in the otherwise flat space-time introduces an ambiguity in the direction of a boost as perceived by different observers. To resolve this issue, we have to consistently choose one frame for defining boosts (e.g., such one that the defect’s wedge lies symmetrically behind the boost’s direction). On the other hand, the whole problem can be avoided by directly boosting holonomy (

So far, we have dealt with massive defects. Not surprisingly, it is also possible to consider

An alternative construction of the metric of a massless defect, free from the “ambiguity” associated with performing a boost in the conical space-time, was given in [

In the next section, we will focus on the group theoretic structure needed to characterize the motion of a massless defect. This will lead us to a suggestive connection with the momentum space of certain widely studied models of

As discussed in the previous section, the motion of a conical defect in Minkowski space (with a given number of dimensions) is completely characterized by the Lorentz holonomy associated with a loop encircling the defect. Thus, the space of all possible holonomies can be thought of as energy-momentum space of the defect. In order to describe this momentum space in more detail in the case of massless defects, we will now parametrize such a defect using space-time vectors, as it is customarily done for a moving point particle. To get an intuitive picture, we start from familiar defects in

Let us first consider a massless cosmic string in four-dimensional Minkowski space that is oriented along the spacelike direction given by the vector

The full space of momenta described above can be restricted in two simple ways. Firstly, we may fix the spatial orientation of a string

To obtain the holonomy of a defect with a fixed

Let us now notice that the generators

Finally, we notice that the generator

The generalization of the above picture to any number of space-time dimensions is conceptually straightforward. In

Quite interestingly, the Lie algebra

In more than

The most interesting is the

In the section below, we extend our discussion to (massless) defects in

We look here at a generalization of the derivation of a massless conical defect as it was done in

Our goal is to derive a massless defect. Therefore, for convenience, we first rescale the radial coordinate

The string can be better visualized in a different coordinate system, defined in analogy with [

The form of (

In this work, we have provided an exploration of the relation between the holonomies of conical defects in more than three space-time dimensions and group-valued momenta, which appear in scenarios of deformed relativistic symmetries. Our motivation was the well established fact that momentum space of point particles in

For the purpose of illustration, we write down here the three independent

For completeness, let us also discuss conical defects in anti-de Sitter space, with

For brevity, we set

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank M. van de Meent for the very useful correspondence and J. Kowalski-Glikman for comments on the manuscript. The work of Michele Arzano is supported by a Marie Curie Career Integration Grant within the 7th European Community Framework Programme and in part by a grant from the John Templeton Foundation. Tomasz Trześniewski acknowledges the support by the Foundation for Polish Science International PhD Projects Programme cofinanced by the EU European Regional Development Fund and the additional funds provided by the National Science Center under Agreements nos. DEC-2011/02/A/ST2/00294 and 2014/13/B/ST2/04043.