# Space-Time Defects and Group Momentum Space

Authors: Michele Arzano; Tomasz Trześniewski
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

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 inn+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.


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## 1. Introduction

Conical space-time defects were first introduced by Staruszkiewicz [1] as point particles coupled to gravity in 2+1 space-time dimensions. Several years later, Deser et al. [2] generalized this result to configurations of many arbitrary particles. The peculiarity of such systems is that particles are represented by topological defects, due to the lack of local degrees of freedom for gravity in three space-time dimensions, and thus strictly speaking they do not interact gravitationally. The same kind of particle solution was also shown to exist for nonzero cosmological constant, corresponding to 2+1-dimensional de Sitter or anti-de Sitter space [3].In3+1 dimensions, a conical defect can be obtained simply by replacing the point-like singularity with a singular one-dimensional object. Such linear defects are known under the name ofcosmic strings and first turned out to be possibly generated during a spontaneous gauge symmetry breaking in the early universe [4]. They were then studied in cosmology as a possible source of primordial density fluctuations [5], where their contribution was subsequently constrained by observations of the cosmic microwave background (see, e.g., [6]). Cosmic “superstrings" can be also produced in string inflation models [7]. At a more abstract level, ’t Hooft has recently employed cosmic strings as ingredients of hispiecewise flat model of gravity [8]. In full analogy with a point particle in 2+1 dimensions, the strings he considers are infinitely thin and straight. The generalization of the concept of a codimension two conical defect to an arbitrary number of space-time dimensions is then completely straightforward: a straight line is replaced by a hyperplane. Such idealized defects are also the focus of our paper.A somewhat surprising feature of the description of a point particle in2+1 gravity (with vanishing cosmological constant) is that its extended momentum space is actually a Lie group, SO(2,1), which as a manifold has the form of three-dimensional anti-de Sitter space [9]. Interestingly, a nontrivial geometry of momentum space is also one of the effects of (quantum) deformations of relativistic symmetries in four space-time dimensions, which are at the basis of the so-calleddoubly special relativity framework [10, 11] and of its most recent incarnation: therelative locality program [12–14]. In this context, the role of deformed symmetries within 2+1 gravity has been studied in various approaches since the late 1990s [15–21].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 theκ-Poincaré (Hopf) algebra [22], associated with the noncommutative κ-Minkowski space [23]. The momentum space corresponding to such a deformation is given by the Lie group AN(3), which is a subgroup of the five-dimensional Lorentz group SO(4,1) [24, 25]. Therefore, it is of interest to investigate whether this group plays any role in the characterization of space-time defects. We argue here that, at least to a certain extent, this is indeed the case when one considerslight-like ormassless defects.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 [26]. This method was subsequently extended to other singular null objects, in particular massless cosmic strings [27–29]. Some of the null sources coexist with impulsive gravitational waves but this is not the case for the straight strings [29]. Moreover, since 3+1-dimensional (anti-)de Sitter space can be represented as a hyperboloid embedded in the 4+1-dimensional flat space-time, the Aichelburg-Sexl boost can also be utilized [30, 31] to obtain massless particle solutions in (anti-)de Sitter space from the Schwarzschild (anti-)de Sitter solution (with particles generating impulsive gravitational waves). The same approach was applied to derive null particle solutions in 2+1-dimensional (anti-)de Sitter space [32]. Since in 2+1 dimensions there are no propagating degrees of freedom, in this case, particles are not accompanied by gravitational waves.In this work, we analyze how momenta of massless defects inn+1-dimensional Minkowski space can be described by null rotations belonging to the AN(n-1) group, the momentum space corresponding to the n-dimensional noncommutative κ-Minkowski space. We also discuss the relation between massless defects in Minkowski space and the same defects in de Sitter space of one dimension less. The structure of the paper is as follows. In Section 2, we begin with a discussion of massive conical defects generalized to 4+1-dimensional Minkowski space. Static defects are presented first and then we turn to moving defects. In particular, we describe how such defects are characterized by holonomies of the metric connection, which encode their energy and momentum. In Section 3, we explain how massless defects in n+1 dimensions are parametrized by elements of the AN(n-1) group, especially for a fixed direction of propagation. In Section 4, we discuss both massive and massless defects in de Sitter space, focusing on the 3+1-dimensional case. We conclude with a summary of our analysis.


## 2. Conical Defects in Minkowski Space

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 thedeficit angle characterizing the defect. One can generalize this construction to include identifications of the two faces (not necessarily forming a wedge) via a general Poincaré transformation, thus producing defects known as “dislocations” and “disclinations,” adopting the terminology used in the classification of defects in solid media (see [33] for an extended discussion).As we will discuss below, conical defects are curvature singularities. Thus, a given defect carries some mass/energy on its infinitely thin hyperplane. In2+1-dimensional gravity, when one works with units c=1, Newton’s constant G has the dimension of inverse mass (it is the inverse Planck mass) and hence provides a natural mass scale for the theory. The quantity μ≔Gm is the dimensionless “rest energy per point," which is proportional to the deficit angle of a particle at rest α=8πGm. If we generalize this picture to n+1 dimensions, G will have the dimension of length to the power n-2 times inverse mass. Similar to the three-dimensional case, we can then define the dimensionless energy density μ≔Gρ, where ρ is the mass per unit of volume of the defect’s hyperplane (e.g., length of the string in 3+1 dimensions).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 in4+1 space-time dimensions, for reasons which will become clear in the following, but an extension to any number of dimensions is straightforward. In analogy with a conical defect in 2+1 [2] and 3+1 dimensions [29], in the five-dimensional case, the metric has the form(1)ds2=-dt2+dz2+dw2+dr2+1-4μ2r2dϕ2.This metric describes an infinite, flat “cosmic brane" identified with the zw-plane. The brane can be thought of as the “tip" of a conical defect whose deficit angle α=8πμ is determined by the (dimensionless)rest energy density of the brane μ∈(0,1/4). The surrounding space-time is locally isometric to Minkowski space. Indeed, as one can verify by a straightforward calculation, the Riemann curvature vanishes everywhere outside the brane’s world volume, which is the tzw-hyperplane. Consequently, the result of the parallel transport of a vector along an arbitrary loop encircling the defect depends only on the conical curvature singularity and the loop’s winding number. The parallel transport around a loop γ(λ) is measured by theholonomy of the Levi-Civita connection given by(2)hγμν=Pexp⁡-∫γΓμσνdγσdλdλ,where P denotes the path ordering and Γμσν are Christoffel symbols associated with metric (1). One may notice that in four space dimensions it is possible to take a loop around a two-dimensional plane due to the fact that closing a path around a given hyperplane requires at least two directions orthogonal to it. Holonomy (2) of a loop containing the defect is a nontrivial Lorentz transformation, which can be understood as a result of the presence of a delta-peaked curvature on the defect’s world volume [34]. Not surprisingly, this Lorentz transformation is also gluing the faces of the defect’s wedge [9].A generalization of holonomy (2) can be used to characterize both position and momentum of a (moving) defect. This is achieved by considering the parallel transport of a coordinate frame instead of just a vector. If the defect is displaced from the frame’s origin, the resultingPoincaré holonomy in addition to a Lorentz transformation contains also a translation, carrying information about the defect’s position [8, 35]. However, here we will restrict our considerations to Lorentz holonomies (2), which completely describe the energy-momentum density of a moving defect, as it was first shown for 2+1 dimensions [9]. Let us start by explaining how the rest energy density μ appearing in the conical metric is encoded in the Lorentz holonomy of a static defect. We do this by deriving the explicit form of holonomy (2) corresponding to metric (1). The calculation is most easily carried out by picking a simple loop, for example, a circle parametrized by r=(1-4μ)-1, ϕ=2πλ, and t,z,w=0. The result written in terms of Cartesian coordinates x=rcos⁡ϕ and y=rsin⁡ϕ is given by(3)hRα=100000cos⁡α-sin⁡α000sin⁡αcos⁡α000001000001.Such a holonomy is simply a rotation (anelliptic Lorentz transformation) by the deficit angle α=8πμ around the origin in the xy-plane.The metric of a moving defect can be obtained [29] by “boosting” the static metric (1) in Cartesian coordinates using an ordinary boost in a direction which lies in the xy-plane. Indeed, metric (1) is obviously invariant under boosts in the directions z and w and therefore the defect can only move in the plane perpendicular to itself, like a point particle in 2+1 dimensions. For simplicity, we may take a boost in the x direction:(4)t⟼tcosh⁡χ+xsinh⁡χ,x⟼xcosh⁡χ+tsinh⁡χ,where χ is the rapidity parameter. Introducing light-cone coordinates u, v via t=v-u/2, x=v+u/2, we obtain the metric(5)ds2=2dudv+dy2+dz2+dw2-1-1-4μ2eχudy-ydu+e-χvdy-ydv2e2χu2+e-2χv2+2uv+y2,describing a massive brane traveling with velocity V=tanh⁡χ in the direction x. It can be shown after a simple calculation [35] that the deficit angle α′ of a moving defect becomes wider than the deficit angle α at rest and that they are related by the formula tan⁡(α′/2)=tan⁡α/2cosh⁡χ. Here, we observe that, in the limit of small α and α′, the latter simplifies to(6)α′=αcosh⁡χ,which can be interpreted as the familiar expression for the relativistic energy density. Namely, as we already explained, the rest energy density E0=μ~α, while the factor cosh⁡χ=(1-V2)-1/2 and hence the total energy density E0(1-V2)-1/2=E~α′.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 (3). This is achieved by acting on the rotation hR(α) by a boost b(χ) via the conjugation b-1(χ)hR(α)b(χ). The resulting Lorentz holonomy(7)hRα,χ=1+2sinh2⁡χsin2⁡α/2-sinh⁡2χsin2⁡α/2-sinh⁡χsin⁡α00sinh⁡2χsin2⁡α/21-2cosh2⁡χsin2⁡α/2-cosh⁡χsin⁡α00-sinh⁡χsin⁡αcosh⁡χsin⁡αcos⁡α000001000001is thus an element of theconjugacy class of rotations by the angle α, which fully characterizes the motion of a defect described at rest by metric (1).


### 2.1. Massless Defects

So far, we have dealt with massive defects. Not surprisingly, it is also possible to considermassless (light-like) defects, moving with the speed of light. This can be achieved by deriving the theoretical limiting case of the calculation described above, which consists in performing an “infinite" boost via the prescription first introduced by Aichelburg and Sexl [26] in the case of the Schwarzschild metric. The trick is to keep the quantity ϱ≡8πμcosh⁡χ fixed (which is the laboratory energy density, as can be shown by a simple calculation [29]) while taking the limit of the rapidity χ→∞. If we use the well known distributional identity(8)limb→0⁡ba2+b2=πδa,metric (5) in the Aichelburg-Sexl limit becomes(9)ds2=2dudv+dy2+dz2+dw2-2ϱyδudu2and it describes the geometry of a massless cosmic brane. One might be worried about the distributional nature of this metric. However, solutions to Einstein equations involving distributions have a long history and have been extensively studied (see, e.g., [36]). Metric (9) corresponds to a zw-plane moving along the null direction v. Space-time is still flat outside the defect’s world volume and hence the holonomy of a loop around the defect reflects the presence of a curvature singularity at the hyperplane u,y=0. In the considered coordinates, it is convenient to take a square loop with u,y∈[-1,1] and transform the associated holonomy to Cartesian coordinates, which gives(10)hPϱ=1+ϱ22-ϱ22-ϱ00ϱ221-ϱ22-ϱ00-ϱϱ1000001000001.This holonomy is anull rotation by the deficit angle ϱ with respect to the null axis v in the uy-plane. Such Lorentz transformations are calledparabolic since their orbits are parabolas on a given light-like plane (i.e., open curves) in contrast to the circular orbits of usual rotations. Null rotations can be expressed by the homogeneous combinations of boosts and usual rotations, as we will discuss in detail in the next section.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 [29]. As mentioned above, one can perform a boost b(χ) of the holonomy describing a static defect (3) via the conjugation b-1(χ)hR(α)b(χ). If we take the Aichelburg-Sexl limit, the resulting Lorentz group element will be a null rotation (10). The metric leading to such a holonomy can be then reconstructed following the example of a massive defect (1). The geometries will be similar but now the deficit angle has to be cut out from a null plane instead of a spacelike one. To this end, we need cylindrical-like light-cone coordinates, which can be derived starting from usual light-cone coordinates and making a transformation to q=v+1/2y2u-1, φ=yu-1. However, the “angular" coordinate φ has an infinite range and therefore the cut cannot be introduced by a simple rescaling of it, like it is done for ϕ in (1). One rescales φ only in the region u>0, obtaining the metric of the form(11)ds2=2dudq+dz2+dw2+1-fφΘu2u2dφ2,∫-∞∞fφdφ=2ϱ,where a smooth function f(φ)<1 with the compact support is the analog of the rescaling factor 4μ in (1). As one can verify, this metric has the same holonomy as (9) and thus they are equivalent. Furthermore, (11) has the advantage of being smooth away from the u=0 hyperplane and it shows that a massless conical defect is not accompanied by gravitational waves (see [29] for details).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 ofdeformed relativistic symmetries.


## 2.1. Massless Defects

So far, we have dealt with massive defects. Not surprisingly, it is also possible to considermassless (light-like) defects, moving with the speed of light. This can be achieved by deriving the theoretical limiting case of the calculation described above, which consists in performing an “infinite" boost via the prescription first introduced by Aichelburg and Sexl [26] in the case of the Schwarzschild metric. The trick is to keep the quantity ϱ≡8πμcosh⁡χ fixed (which is the laboratory energy density, as can be shown by a simple calculation [29]) while taking the limit of the rapidity χ→∞. If we use the well known distributional identity(8)limb→0⁡ba2+b2=πδa,metric (5) in the Aichelburg-Sexl limit becomes(9)ds2=2dudv+dy2+dz2+dw2-2ϱyδudu2and it describes the geometry of a massless cosmic brane. One might be worried about the distributional nature of this metric. However, solutions to Einstein equations involving distributions have a long history and have been extensively studied (see, e.g., [36]). Metric (9) corresponds to a zw-plane moving along the null direction v. Space-time is still flat outside the defect’s world volume and hence the holonomy of a loop around the defect reflects the presence of a curvature singularity at the hyperplane u,y=0. In the considered coordinates, it is convenient to take a square loop with u,y∈[-1,1] and transform the associated holonomy to Cartesian coordinates, which gives(10)hPϱ=1+ϱ22-ϱ22-ϱ00ϱ221-ϱ22-ϱ00-ϱϱ1000001000001.This holonomy is anull rotation by the deficit angle ϱ with respect to the null axis v in the uy-plane. Such Lorentz transformations are calledparabolic since their orbits are parabolas on a given light-like plane (i.e., open curves) in contrast to the circular orbits of usual rotations. Null rotations can be expressed by the homogeneous combinations of boosts and usual rotations, as we will discuss in detail in the next section.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 [29]. As mentioned above, one can perform a boost b(χ) of the holonomy describing a static defect (3) via the conjugation b-1(χ)hR(α)b(χ). If we take the Aichelburg-Sexl limit, the resulting Lorentz group element will be a null rotation (10). The metric leading to such a holonomy can be then reconstructed following the example of a massive defect (1). The geometries will be similar but now the deficit angle has to be cut out from a null plane instead of a spacelike one. To this end, we need cylindrical-like light-cone coordinates, which can be derived starting from usual light-cone coordinates and making a transformation to q=v+1/2y2u-1, φ=yu-1. However, the “angular" coordinate φ has an infinite range and therefore the cut cannot be introduced by a simple rescaling of it, like it is done for ϕ in (1). One rescales φ only in the region u>0, obtaining the metric of the form(11)ds2=2dudq+dz2+dw2+1-fφΘu2u2dφ2,∫-∞∞fφdφ=2ϱ,where a smooth function f(φ)<1 with the compact support is the analog of the rescaling factor 4μ in (1). As one can verify, this metric has the same holonomy as (9) and thus they are equivalent. Furthermore, (11) has the advantage of being smooth away from the u=0 hyperplane and it shows that a massless conical defect is not accompanied by gravitational waves (see [29] for details).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 ofdeformed relativistic symmetries.


## 3. Momentum Space of Massless Defects and theAN(n) Lie Group

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 in3+1 dimensions and then generalize the discussion to higher dimensional cases and in particular to 4 + 1 dimensions, which we are especially interested in.Let us first consider a massless cosmic string in four-dimensional Minkowski space that is oriented along the spacelike direction given by the vectord=(0,0,0,1) and moving in the null direction n=(1,1,0,0) (in Cartesian coordinates t,x,y,z). Its motion is completely captured by the holonomy (cf. (10))(12)hPϱ=1+ϱ22-ϱ22-ϱ0ϱ221-ϱ22-ϱ0-ϱϱ100001.The Lorentz group element (12) is a null rotation by the angle ϱ with respect to the null axis n, in the plane spanned by the light-like vector (1,-1,0,0) and spacelike vector (0,0,1,0). In general, a massless string in 3+1 dimensions can be completely characterized by 4 parameters [35]. Indeed, one needs to specify a parabolic angle ϱ, carrying the defect’s energy density, a light-like vector n, along which the defect propagates, and a spacelike vector d, which is the direction of the spatial extension of the defect, with the orthogonality condition n·d=0 (a defect is invariant under boosts acting along d). The overall scaling of n and d is irrelevant. In the end, one has 2+2-1+1=4 independent coefficients, which parametrize all possible holonomies. Thus, the space of holonomies/momenta of a massless string is bigger than momentum space of a massless particle in 3+1 dimensions, which is only three-dimensional. The reason is that conical defects in space-time with more than three dimensions are extended objects and the holonomies must account for their nontrivial orientation in space.The full space of momenta described above can be restricted in two simple ways. Firstly, we may fix the spatial orientation of a stringd. Then, however, the defect effectively behaves like a point particle in 2+1 dimensions, since it can only move in two directions perpendicular to itself. Secondly, we may choose to fix the direction of motion n. It turns out that this case is more interesting. To be specific, let us introduce a complete orthogonal set of null vectors n(x)=(1,1,0,0), n(y)=(1,0,1,0), and n(z)=(1,0,0,1), as well as a set of orthonormal spacelike vectors d(x)=(0,1,0,0), d(y)=(0,0,1,0), and d(z)=(0,0,0,1). Suppose that we fix the direction of defect’s velocity as n=n(a),a∈{x,y,z}. Then, the spatial orientation of a defect is orthogonal to n(a) and is a linear combination of two vectors d(j),j=1,2, which can be chosen to be d(b),b≠a. For example, for n(x), we may take d(1)≡d(z) and d(2)≡d(y). We observe that, for a given n(a), the space of holonomies of a massless defect is a subgroup of the Lorentz group SO(3,1) determined by the generators of null rotations Xj(a)≔Jbt+Jba, with each j corresponding to one of the spatial indices b≠a (see also (20)). The holonomy of a defect characterized by n(a) and d(j) is obtained by exponentiating such a generator to(13)hjakj=eikjXja,a=x,y,z,j=1,2(no summation in the exponent), while the parabolic angle ϱ≡kj parametrizes the family of holonomies. For example, if the defect moves along n(x), we have two generators X1(x)=Jyt-Jyx and X2(x)=Jzt+Jzx, which can be written in a four-dimensional matrix representation as(14)X1x=i00-1000-10-11000000,X2x=i000-1000-10000-1100.Holonomy (12) in this picture is given by the group element h1(x)=eik1X1(x), where ϱ≡k1, as can be verified by a direct calculation.To obtain the holonomy of a defect with a fixedn(a) but an arbitrary d, we just take the product(15)h1ak1h2ak2=eikjXjaand then the (normalized) spacelike vector is a linear combination d=k12+k22-1(-k2d(1)+k1d(2)), while the parabolic angle is given by ϱ≡k12+k22, as one can straightforwardly calculate. Therefore, the defect’s energy density is naturally expressed in terms of group coordinates k1 and k2, which also specify the defect’s orientation. This agrees with the counting of degrees of freedom discussed above since the choice of n fixes two parameters and another two remain free.Let us now notice that the generatorsX1(a) and X2(a) commute and thus span amaximal Abelian subalgebra of the Lorentz algebra so(3,1). To complete the picture, we may additionally consider the boost generator X0(a)≔J0a, which in our matrix representation for a=x is given by(16)X0x=-i0100100000000000.(The representations of generators for other choices of n(a) are collected in Appendix A.) Group elements generated by a given X0(a), namely,(17)g0a=eik0X0a,a=x,y,z,form a one-dimensional subgroup of hyperbolic Lorentz transformations in the direction of the spatial component of n(a), which can be seen as complementary with respect to null rotations generated by X1(a) and X2(a). Acting on a holonomy hj(a)(kj) via the conjugation(18)g0ahjakjg0a-1=hjaek0kj,one rescales the parabolic angle ϱ≡kj by the factor of ek0, while preserving the orientation of the defect. Thus, the effect of a boost is to red- or blue-shift the defect’s energy density. We should stress that the boost, analogously to the case of massive defects, has to act via conjugation since such an action preserves the trace of a Lorentz group element and consequently the property that the holonomy belongs to the parabolic conjugacy class.Finally, we notice that the generatorX0(a) does not commute with X1(a) and X2(a), and the respective commutators are given by(19)X0a,Xja=-iXja,j=1,2.The Lie algebra generated by X0(a), X1(a), and X2(a) (with a trivial commutator between X1(a) and X2(a)) is usually denoted by an(2) and called the (three-dimensional)Abelian nilpotent algebra, since it possesses an Abelian subalgebra spanned by the generators X1(a) and X2(a), which are also nilpotent (e.g., (Xj(a))3=0) in a four-dimensional matrix representation.The generalization of the above picture to any number of space-time dimensions is conceptually straightforward. Inn+1 dimensions, we have n independent null directions, corresponding to n families of generators of null rotations and complementary boosts, labeled by a=1,…,n:(20)X0a=J0a,Xja=Jj0+Jja,j≠a=1,…,n,where Jμν are generators of the Lorentz group SO(n,1). More precisely, J0a generates boosts in the xa-direction, Jj0 boosts in one of the directions xj orthogonal to xa, and Jja rotations in the plane spanned by xa and xj. Similar to the generators appearing in (19), families (20) are actually representations of the generators X0 and Xj (with nilpotent generators satisfying (Xj)3=0 in an (n+1)×(n+1) matrix representation) of the an(n-1) algebra, a subalgebra of the Lorentz algebra so(n,1).Quite interestingly, the Lie algebraan(n-1) is very popular in the noncommutative geometry community, where it is known as the κ-Minkowski space-time (first introduced in the four-dimensional version in [23]) and the generators X0(a) and Xj(a), after the appropriate rescaling by a dimensionful constant, are identified with time and space coordinates, respectively. Furthermore, the group AN(n-1) generated by an(n-1) can be obtained from the (local) Iwasawa decomposition of the Lorentz group [37] and, in the field theoretic models on κ-Minkowski space, coordinates on this group aremomentum variables determined by aquantum group Fourier transform [20, 38, 39] of functions of noncommuting space-time coordinates. These momentum coordinates transform under the nonlinear boosts and correspond to translation generators of a quantum deformation of the Poincaré algebra known as the κ-Poincaré Hopf algebra [22, 40]. From the geometric point of view, the Lie group manifold AN(n-1) is given by half of the n-dimensional de Sitter space [24, 25] and thus the momentum space of such a model is curved, which is a characteristic feature of the recently proposed “relative locality” approach to modeling phenomenological aspects of quantum gravity [12, 13].In more than3+1 dimensions, to completely characterize a given massless defect, it is enough to take a parabolic deficit angle, carrying its energy density, and two vectors that determine a two-form normal to the defect’s world volume (which is of codimension 2) [35]. The vectors (one light-like and one spacelike) are mutually orthogonal, and their scaling is irrelevant, and thus in n+1 dimensions we deal in total with 2n-2 parameters, which should be encoded in the defect’s holonomy. If we now consider the restriction imposed on a defect by fixing its direction of motion, as we did in the 3+1-dimensional case, then the holonomy will have n-1 parameters less and hence only n-1 parameters will remain. Therefore, momentum space of a restricted defect is always given by the maximal Abelian subgroup of AN(n-1), formed by null rotations.The most interesting is the4+1-dimensional case, corresponding to (a subgroup of) the AN(3) group, which is the momentum space associated with the κ-Poincaré algebra in 3+1 space-time dimensions. The holonomy of an arbitrary massless cosmic brane with the fixed velocity vector n (we drop here the superscript (a)) can be written as(21)hk1,k2,k3=h1k1h2k2h3k3,where hj,j=1,2,3, are defined analogously to their four-dimensional counterparts (13) and (17). As can be shown after some calculations, the energy density of a brane characterized by this holonomy is given by ϱ=k12+k22+k32. The brane’s spatial plane is spanned by two perpendicular vectors, which can be put in the form dI=nd(1)+n′d(2)+d(3), dII=n′′d(1)-(1+nn′′)(n′)-1d(2)+d(3), where the coefficients n=(k2-k3)/(k1-k2), n′=(k3-k1)/(k1-k2), and n′′=(k2(k2-k1)+k3(k3-k1))/(k1(k1-k3)+k2(k2-k3)) and the vectors d(j),j=1,2,3, form an orthonormal set orthogonal to n(a). Thus, there are three parameters n,n′,n′′ specifying the brane as an extended physical object but one of them is irrelevant since dI and dII give a purely geometrical internal construction, while it is enough to know a single spacelike vector that is orthogonal to the defect’s world volume.In the section below, we extend our discussion to (massless) defects in3+1-dimensional de Sitter space, with cosmological constant Λ>0, showing how their space-time metric is strictly related to the metric of defects in 4+1-dimensional Minkowski space.


## 4. Conical Defects in de Sitter Space

We look here at a generalization of the derivation of a massless conical defect as it was done in2+1-dimensional de Sitter space [32]. (In Appendix B, we have included a similar discussion for defects in anti-de Sitter space.) We begin with a static massive conical defect in 3+1 dimensions [41, 42], analogous to a point particle in 2+1 dimensions [3], whose metric in static de Sitter coordinates has the form(22)ds2=-1-λr2dτ2+1-λr2-1dr2+r2dθ2+1-4μ2sin2⁡θdϕ2,(where we denote λ≡Λ/3) and describes a cosmic string of rest energy density μ, corresponding to the deficit angle 8πμ. Let us observe that the above metric can be written in terms of embedding Cartesian coordinates of the 4+1-dimensional Minkowski space(23)t=λ-1-r2sinh⁡λτ,w=±λ-1-r2cosh⁡λτ,x=rcos⁡ϕsin⁡θ,y=rsin⁡ϕsin⁡θ,z=rcos⁡θ,subject to the (4,1)-hyperboloid condition λ-1=-t2+x2+y2+z2+w2. Transformed (22) turns out to have an identical form to metric (1) of a defect in 4 + 1-dimensional Minkowski space. There are only two obvious subtleties in the de Sitter case: space-time coordinates are constrained by the hyperboloid condition and the interpretation of μ has to be adjusted according to the number of dimensions, since it now represents thelinear energy density rather than energy per surface. A consequence of this situation is that we could indirectly apply the derivation of light-like defects from Section 2 to the de Sitter case. Nevertheless, it may be more illuminating to stick to the approach of [32] and see whether we will obtain in this way the same metric as (9).Our goal is to derive a massless defect. Therefore, for convenience, we first rescale the radial coordinater to r/(1-4μ) and expand the metric to the first order in μ, which gives(24)ds2≈dsdS2+8μλr2dτ2+1-λr2-2dr2+r2dθ2,where dsdS2 is the pure de Sitter metric dsdS2=-(1-λr2)dτ2+(1-λr2)-1dr2+r2dθ2+sin2⁡θdϕ2. We subsequently transform (24) to embedding coordinates (23) and perform a boost t↦tcosh⁡χ+xsinh⁡χ, x↦xcosh⁡χ+tsinh⁡χ, with the rapidity parameter χ, obtaining(25)ds2=dsdS2+8μT2-w22λ-1+T2-w2wdT-Tdw2+λ-2TdT-wdw2λ-1+T2-w2+8μz-TdT+zdz+wdw+λ-1+T2-z2-w2dz2λ-1+T2-w2λ-1+T2-z2-w2,where T≡tcosh⁡χ-xsinh⁡χ (and dsdS2 is now given in embedding coordinates). The metric dsdS2 is preserved by the boost since the latter belongs to the isometry group of de Sitter space. To proceed further, we introduce light-cone coordinates u=(x-t)/2 and v=(x+t)/2 and employ the Aichelburg-Sexl boost prescription, taking χ→∞ and keeping the laboratory energy density ϱ≔8πμcosh⁡χ constant. Using the distributional identity [30](26)limχ→∞⁡fT2cosh⁡χ=12δu∫-∞+∞fT2dT,we finally obtain(27)ds2=2dudv+dy2+dz2+dw2-2ϱyδudu2.Metric (27) corresponds to a conical defect at the hypersurface u,y=0, z2+w2=λ-1, that is, a light-like, circular string on a meridian of the cosmological horizon of de Sitter space (for an observer on the worldline y,z,w=0, x=x(t)>λ-1, the future/past horizon is the sphere y2+z2+w2=λ-1, x±t=0). Such a result is qualitatively different from 2+1 dimensions, where there is a pair of massless particles (similar to the massive solution [3]) on the opposite points of the horizon [32]. However, it can simply be seen as the dimensional reduction of a circle to the degenerate case of a pair of points.The string can be better visualized in a different coordinate system, defined in analogy with [31]. To this end, we first make the following transformation of coordinates:(28)t=12ηλ-1-η2+X+λ-1/22+Y2+Z2,x=1ληX+λ-1/2,y=1ληY,z=1ληZ,w=12ηλ-1+η2-X+λ-1/22-Y2-Z2,and then we set X=ρcos⁡ϕsin⁡θ, Y=ρsin⁡ϕsin⁡θ, and Z=ρcos⁡θ, so that metric (27) takes the form(29)ds2=1λη2-dη2+dρ2+ρ2dθ2+sin2⁡θdϕ2-ϱλsin⁡ϕsin⁡θδη-ρdη-dρ2+δη+ρdη+dρ2.As one can observe, this metric describes a light-like circular string lying at the great circle ϕ=0,π, θ∈[0,π] on the cosmological horizon ρ=|η| of de Sitter space, whose metric is in the first line. The first term containing the Dirac delta is responsible for the string at times η≥0 and the other at times η≤0.The form of (27) is exactly the same as metric (9) of a light-like cosmic brane in 4+1-dimensional Minkowski space, as could be expected from the analogous situation for (22). This result easily generalizes to an arbitrary number of dimensions and thus we might say that a conical defect in the n+1-dimensional de Sitter space is a “projection” of the corresponding defect in the embedding (n+1)+1-dimensional Minkowski space. The projection is understood in simple geometric terms: the n-1-dimensional hyperplane of a Minkowski defect is cutting through the n-dimensional sphere of a de Sitter spatial slice, determining the n-2-dimensional sphere of the projected defect.


## 5. Summary

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 in2+1 gravity, treated as conical defects, is a Lie group. We started by recalling how the motion of massive conical defects can be characterized by their holonomies, belonging to the Lorentz group. Generalizing the discussion to massless defects, we observed that holonomies describing such defects in n+1-dimensional Minkowski space, given by null rotations, are elements of the maximal Abelian subgroup of the AN(n-1) group. In particular, for a fixed direction of propagation, the space of all possible holonomies of a massless defect is exactly the above subgroup. Since the full AN(n-1) group plays the role of momentum space associated with n-dimensional κ-Minkowski space, this fortunate coincidence provides a partial “physical" rendition of the group-valued momenta emerging in the context of deformed relativistic symmetries, which were originally introduced as rather formal mathematical structures. At the same time, group momentum space (or, more broadly, momentum space with a nontrivial geometry) is a crucial ingredient of doubly special relativity and relative locality approaches to the problem of quantum gravity. Both of these frameworks are based on heuristic assumptions and often use classical and quantum 2+1 gravity as a toy model for their concepts. Thus, our analysis of massless defects offers a practical setting to explore such ideas in a higher number of dimensions and in the context of the well studied κ-Minkowski space, though only to a certain extent (a subgroup of AN(n-1) will correspond to a subset of κ-Minkowski space). Furthermore, similar group momentum spaces are most likely associated with defects coupled to the BF theory [43], a topological field analog of gravity in more than three dimensions. It remains to be investigated how the space of holonomies generalizes when one considers spinning defects, which is a somewhat problematic case, leading to the occurrence of closed timelike curves [35]. In this paper, we also discussed the case of massless conical defects in 3+1-dimensional de Sitter space, showing how they can be seen as closed light-like strings and how their metric is strictly related to the one of massless defects in 4+1-dimensional Minkowski space. This led us to conjecture that the same holonomies, belonging to the AN(3) group, which characterize the latter, could completely capture the motion of massless defects in de Sitter space and that such a statement could be generalized to more space-time dimensions. A proof of this conjecture is postponed to future studies.



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