Warm Dark Matter from Higher-Dimensional Gauge Theories

Warm dark matter particles with masses in the keV range have been linked with the large group representations in gauge theories through a high number of species at decoupling. In this paper, we address WDM fermionic degrees of freedom from such representations. Bridging higher-dimensional particle physics theories with cosmology studies and astrophysical observations, our approach is two-folded, i.e., it includes realistic models from higher-dimensional representations and constraints from simulations tested against observations. Starting with superalgebras in exceptional periodicity theories, we discuss several symmetry reductions and we consider several representations that accommodate a high number of degrees of freedom. We isolate a model that naturally accommodates both the standard model representation and the fermionic dark matter in agreement with both large and small-scale constraints. This model considers an intersection of branes in $D=27+3$ in a manner that provides the degrees of freedom for the standard model on one hand and 2048 fermionic degrees of freedom for dark matter, corresponding to a $\sim$2 keV particle mass, on the other. In this context, we discuss the theoretical implications and the observable predictions.


Introduction
In the past decade, having been thoroughly tested against the data from observations, the cold dark matter (CDM) model has encountered difficulties in explaining these data and has proved to be inconsistent with the observations under several aspects. Besides the two initial concerns-the large cores in galaxies and the small number of satellites-that may or may not be reconciled by introducing finely tuned baryonic processes in the simulations, there are several other points of concern at both small and large scales, e.g., the fact that the population of dwarf galaxies in voids is in contradiction with the predictions from CDM simulations [1][2][3]; the failure to replicate, in CDM simulations, the high number of pure bulgeless galaxies observed and the low number of galaxy mergers [4,5]; the formation and growth of supermassive black holes in agreement with observed distributions [6]; and the recent detection of high-redshift quasars [7].
WDM scenarios with particles in the keV range are gaining ground in explaining some of these discrepancies and many studies show the advantages that a keV dark matter model presents in explaining the observations at both small and large scales [8][9][10][11][12][13][14][15][16][17][18]. A keV WDM particle, for example, can explain the existence of cores in galaxies of varied masses, from dwarf galaxies to spirals, when considering the quantum Fermi pressure of the particles [12][13][14]. Moreover, while, for spiral galaxies, in an attempt to reconcile the CDM model with observations, the presence of the cores has been attributed to the stars and gas concentration at the for particles in the eV-hundreds keV range that do not decay. More complex models that consider decay [37] and re-thermalization [38] have also been considered as well as quantum semi-degenerate particles with a non-negligible chemical potential [16].
However, in numerical simulations, the varied models are oversimplified, often using a 'cutoff' in the power spectrum as the only parameter that distinguishes WDM from CDM particles, as reviewed in [39]. The model most favored by simulation studies for a thermal relic WDM particle, which provides this 'cutoff' mass-velocity, is the one described in [23]. Below, we discuss this model, its assumptions and its implications.

WDM Degrees of Freedom
The particle assumed in [23] is a thermal relic particle that decouples while still relativistic, but, by equality, is nonrelativistic. The streaming speed at equality v s /c ∝ T X /m X , where the effective temperature T X varies inversely with the scale factor. In order to have a streaming speed that is low enough to keep up with the CDM success at large scales-meaning, the formation of structures at an early enough redshift and within the observable mass scales for halos-the decrease in the effective temperature is suggested [23]. If the particle decouples with thermal abundance, ρ X ∝ T 3 X m X must be held fixed. Therefore, to obtain a reasonable reduction in velocity, one needs to decrease the temperature and increase the mass. One way the reduction in temperature can be realized is through increasing the number of the degrees of freedom at decoupling. The example given in [23] assumes a reduction in T X of ∼4 and an increase in mass of 4 3 , thus corresponding to 688 degrees of freedom. However, for 1 keV, the number of degrees of freedom is higher and this is usually what is indirectly used in simulations when the velocity of the WDM particle is introduced, as shown in [16,17].
Below, we remind the derivation in [23] and the discussion in [16]. As recalled in [23] (see [23] Appendix A for a detailed derivation and [16] for a detailed assessment of that derivation), for a thermal relict particle X that decouples when relativistic, the abundance n X relative to photons is n X n γ = 43/4 g dec 4 11 where g dec is the number of relativistic species present at decoupling-degrees of freedomand g X is the number of spin states of the particle. Assuming that the distribution function scales as the non-thermal distribution (exp(v/v 0 ) + 1) −1 , for a redshift z, the velocity v 0 (in Equation (A3) [23]) is given, The relation between the cosmological parameters and the particle's density ρ x is for the Hubble constant H ≡ 100h km s −1 kpc −1 . Therefore, for an m X = 1 keV particle, considering Ω X = 0.3 as in [23], we have the following formula for the velocity: that gives g dec ≈ 1000 (g X /1.5) 1/3 for a 1 keV particle.
This large value for the number of degrees of freedom at decoupling is much higher than the value for the standard model, ∼107, and even the supersymmetric extension, ∼229 [24]. Other studies arbitrarily choose, in their analyses, a much more conservative value for the number of degrees of freedom of the WDM particle candidates, i.e., ∼150 for a massive neutrino [24]; ∼100 for right-handed neutrinos that would decouple before the electro-weak phase transition [40]. The number of degrees of freedom scales with the mass of the particle, therefore a higher mass would correspond to a higher number of degrees of freedom (and a stronger decrease in temperature). Since this value is correlated to the mass-velocity correspondence that dictates the behavior of the WDM particle and its influence on the structure formation and evolution, we study the theoretical models where such a large number of degrees of freedom can be achieved.
While the authors in [23] do hint at larger group representations with extra dimensions in relation to the high g dec number, they also mention that entropy-producing processes that take place after the decoupling of the particle can mimic an increase in g dec . Indeed, trying to avoid such large group representations, [22,[41][42][43] discuss the entropy generation by the out-ofequilibrium decay of heavier particles, which would allow the number of degrees of freedom of the keV particle to be similar to that used for active neutrinos [41]. Self-interactions as a possible way to reduce the number densities by self-annihilations are also discussed in [44]. However, in this paper, we look at the large gauge group representations that would naturally produce this high number of degrees of freedom, starting with the symmetry reductions in algebras with higher-dimensional spinors in exceptional periodicity theories.

Algebras with Higher-Dimensional Spinors
A class of algebras involving higher-dimensional spinors is studied in the so-called exceptional super Yang-Mills (ESYM) theories, firstly discussed in [30,31], in which higherdimensional theories [45][46][47][48][49][50] beyond string/M-theory [51,52] have been considered and extended, also in relation to exceptional periodicity (EP) algebras [53][54][55]. In this section, we provide various branchings of these algebras with a number of spinor degrees of freedom of O(10 3 ). Here, we discuss the symmetry reductions along the worldvolume (WV) of p-branes, which centrally extend the underlying chiral superalgebra and the reductions that follow the so-called Magic Star structure-named so in [53] after the star polygon geometric shape-of the corresponding EP algebra.
3.1. Superalgebra in 17 + 1 and 9-Brane WV reduction One of the simplest examples is based on the (1, 0) superalgebra obtained by setting n = 1 in Equation (3.5) of [30], namely, the chiral superalgebra of ESYM in D = s + t = 17 + 1 space-time dimensions, whose central extensions are given by the r.h.s. of the anticommutator namely, by a 1-brane (fundamental string), an electric 5-brane and its dual, a magnetic 9brane. The corresponding EP 3-graded algebra, whose semisimple, 0-graded part yields the homogeneous Lorentz algebra of 17 + 1 space-time, is obtained by setting n = 1 in (II.14) of [55], namely, where 256 and 256 , respectively, denote the Majorana-Weyl (MW) semispinor in 17 + 1 dimensions and its conjugate (note that the EP level n is understood with +1 in [30], namely, the first non-trivial level of EP is given by n = 2 therein). It should be remarked that, when the spinors are interpreted as Abelian translational generators, the non-negatively graded part of the EP algebra e 6(−26) (6) yields the Lie algebra of the global isometry group of the homogeneous non-symmetric real special manifold L(16, 1, 0) in the classification cited in [56,57]. Symmetry reduction along the WV of the highest-dimensional central extension occurring in the r.h.s. of (5), namely, along the 9 + 1 dimensional WV of the 9-brane, yields the breaking thus yielding an so 8 inner/fiber symmetry, whose only peculiar polarization is shown in the branchings (7)-(9). Let us recall that, in the models for family unification beyond the SM, such as the ones discussed in [58,59], one posits a further breaking of the inner symmetry, where so 3 is used for family symmetry [58].

Superalgebra in 20 + 4 and 12-Brane WV Reduction
Another example is provided by the quasi-conformal level of the above symmetry, namely, by the the (1, 0) superalgebra obtained by setting n = 1 in Equation (3.8) of [30], the chiral superalgebra of ESYM in D = 20 + 4 space-time dimensions, whose central extensions are given by the r.h.s. of the anticommutator namely, by a 0-brane, a 4-brane, an electric 8-brane and its dual, a magnetic 12-brane. The corresponding 5-graded EP algebra is obtained by setting n = 1 in (II.16) of [55] e (1) where 2048 and 1024 denote the MW semispinors in 20 + 4 and 19 + 3 dimensions, respectively. Again, when the spinors and vectors are interpreted as Abelian translational generators, the non-negatively graded part of the EP algebra e  (14) corresponds to the Lie algebra of the global isometry group of the homogeneous non-symmetric quaternionic Kähler manifold given by the image of the aforementioned space L(16, 1, 0) under the composite c·R-map [56,57].
The symmetry reduction along the WV of the highest-dimensional central extension occurring in the r.h.s. of (12), namely the 12 + 4 dimensional WV of the 12-brane, yields the breaking thus again yielding an so 8 inner/fiber symmetry, where 128 and 128 stand for the MW semispinor in 12 + 4 dimensions and its conjugate, respectively.

Further Reduction in Inner/Fiber Symmetry
Through a disentangling mechanism discussed in Section 4, one copy of the 128 encodes SM matter and the rest provides thousands of fermionic degrees of freedom.
A possible way to do this is to further break the inner/fiber symmetry so 8 down to so 7 and/or g 2 : or While this isolates a 128 as a singlet of g 2 and has additional fermions, breaking so 12,4 further to identify the standard model and electromagnetism would lead to the additional fermions to have the same charge, therefore not be dark matter. Nevertheless, the utilization of the 2048 spinor in e (1) 8(−24) is later used to identify a candidate for WDM.

Magic Star Reduction in
Another possible symmetry reduction on the same ESYM, which is useful in order to discuss family unification beyond the SM, concerns the exploitation of the so-called Magic Star [53] decomposition of the EP algebra e as depicted in Figure 1, where T 8,1 3 denotes the Hermitian part of the special Vinberg cubic T-algebra [60,61] with elements realized as formal 3 × 3 Hermitian matrices, whose manifestly so 16 -covariant structure reads where 128 and 128 , respectively, denote the MW semispinor in 16 dimensions and its conjugate (the case q = 8 and n = 1 of Equation (III.1) of [55]). In addition, it is here worth recalling that the Hermitian matrix-like structure (22) corresponds to the manifestly so 17,1 -covariant Peirce decomposition where the so 17,1 -singlet corresponds to the so 16 -singlet 1 I I I in (22).

Symmetry Reduction to the Standard Model and Warm Dark Matter Disentanglement
By studying E 8 × E 8 heterotic string theory on the boundary of an 11-dimensional supergravity bulk, two E 8 gauge theories on the endpoints of an orbifold from R 10 × S 1 /Z 2 have been found [62,63]. The anomaly cancellation requires a twelve-form, which is factorized into a 4-form and 8-form, yet these are invoked in 10-and 11-dimensional theories, implying a complete description from larger dimensions. Heterotic string theory must have an origin from D = 25 + 1 or higher. While bosonic string theory exists in D = 25 + 1, heterotic string theory places 16 dimensions on an even, self-dual lattice and includes fermions. D = 26 + 1 bosonic M-theory and D = 26 + 2 target space-time for 2D gravity has been introduced [64,65]. How D = 11 + 3 unifies the superalgebras of type I, type IIA, type IIB and heterotic theories has been articulated in [46][47][48]. The superalgebras of M-theory and F-theory can arise from nested braneworlds in D = 27 + 3, as the exceptional Yang-Mills superalgebra contains an 11-brane worldvolume with spinors [31]. Dimensional reduction also finds a 10-brane worldvolume with a 16D transverse space and includes spinors.
The Horava-Witten theory [62,63] shows how M-theory on R 9 × S 1 × S 1 /Z 2 , when compactified on S 1 , leads to type IIA and E 8 × E 8 heterotic string theories, which are T-dual to type I and Spin(16) × Spin(16) ⊂ Spin(32) superstring theories. This implies that the type I superstring on R 9 × S 1 naturally has a Spin(16) × Spin(16) vacuum. Dark matter has been discussed in relation to nonsupersymmetric Spin(16) × Spin(16) string theory [25,26,29]. While heterotic string theory leading to the supersymmetric standard model allows for supersymmetric light dark matter candidates such as neutralinos [66], light supersymmetric partners of the standard model below 1.5 TeV have not been observed [67]. The exceptional Yang-Mills superalgebras facilitate an understanding of various worldvolume constructions for M-theory and beyond, which allows for supersymmetric forms of matter beyond the standard model that are different from the superpartners or the nonsupersymmetric dark matter models.
Since the membranes discussed in this section are more complicated than those contained within M-theory itself, a brief pedagogical introduction to membranes and their relationship to gauge theory is warranted. To start, let us consider the worldline formulation of a free point particle described by the following action: Varying this action leads to the standard geodesic equation in general relativity. Extended objects in supergravity lead to "p-brane" for arbitrary-dimensional membranes, which spans a (p + 1)-dimensional worldvolume and generalizes the worldline formulation above [68]. While strings are 1-branes on a worldsheet and M-theory has a 2-brane worldvolume, 3brane worldvolumes were found in type IIB supergravity [69]. F-theory admits 3-brane phenomenology that can include the standard model with gravity [70]. Furthermore, 3-branes are intimately related to AdS 5 , whose isometry group is the conformal group Spin(4, 2). Additionally, AdS 5 supports a 4D CFT with a 3-brane on the boundary, such that dS 4 can be obtained within the Randall-Sundrum model [71,72]. Before the full significance of the Yang-Mills theory for the standard model was established, the nonlinear sigma model was introduced for pions, which is a nonrenormalizable theory similar to general relativity. General relativity is equivalent to a nonlinear sigma model Lagrangian [73], which is similarly used for the bosonic sector of p-branes: where C M 1 ...M p+1 is a p + 1-form potential sourced by the p-brane and X M is a scalar field with an index with respect to the target space, such as D = 10 + 1 in M-theory. The bosonic p-brane Lagrangian in the static gauge leads to a Born-Infeld action, where T = 1/(2πl P ) p is the brane tension. The Born-Infeld action was introduced as an electrodynamical theory; it is the 1-loop renormalization of QED. The historical developments of transitioning from the non-renormalizable nonlinear sigma model to the renormalizable Yang-Mills theory suggest that the Yang-Mills gauge gravity theories, such as the MacDowell-Mansouri theory, will remain important for the unification of the standard model and gravity [74]. As braneworlds are intimately related to the study of gravity and beyond-the-standard-model physics is necessary for dark matter, we propose that braneworld spinors allow us to consider new dark matter candidates. We briefly review the representation theory of the standard model as well as the most common high-energy GUT models, recalling that SU(3) c × U(1) e ⊂ SU(5) ⊂ Spin(10) ⊂ E 6(−78) ⊂ E 8(−24) . E 6(−78) allows us to consider three charts of Spin(10) × U(1), such that E 8(−24) contains three charts of Spin(10) × Spin(2, 4) for a combination of GUT and conformal symmetries. The D = 12 + 4 (1,0) chiral super-Poincaré algebra also exists and contains a 128 spinor. When breaking to Spin(2, 4) × Spin(10), the three conformal charts span Spin(4, 4) (associated to a conformal braneworld symmetry), such that three distinct 16 representations of Spin(10) are found, each leading to the standard model fermion representations with respect to SU(3) c × SU(2) L × U(1) Y , Once the bosons are unified into a single gauge group, such as SU (5) or Spin(10), the vector bosons are described by the Yang-Mills action At high energies, an SU(2, 2) × U(1) conformal gauge group, axial U(1) and SU(5) GUT leads to SU (7,2), which is also a maximal subalgebra of E 8(−24) . An E 8(−24) gauge theory does not need to be considered, but rather gauge theories from subalgebras. For instance, E 6 GUT contains a complex 27 and 27 in combination with the adjoint representation 78 [75].
Combined with another U(1), the total of all of these states leads to E 7 by combining the adjoint bosonic sector with the complex fermionic sector, As shown above, the representations found within E 6 GUT stem from a single adjoint representation of E 7 ; similarly, E 8 . Recently, using E 6 to obtain the standard model spectrum has been explored [76] without using E 6 GUT. E 8 for three generations of matter has also been explored [77], although conformal spacetime symmetry was not considered. This nonsupersymmetric E 8 GUT model and the supersymmetric E 8 × E 8 heterotic string theory use this to make contact with three generations of matter. Since two of the authors recently found EP algebras that generalize E 8 [53], we expand the exploration of the representations associated with these algebras. Dark matter is an immediate natural candidate, as the EP algebras contain spinors for matter beyond the standard model. Superalgebras are explored to motivate braneworld intersections that naturally lead to the standard model spectrum as described above, along with additional dark matter candidates.  (16), just as the superstring has an 8 s spinor of Spin (8). This can be compared to the Horava-Witten theory [62,63] with R 10 × S 1 /Z 2 , by orbifolding a spatial direction of the 10-brane. The 18brane gives a D = 18 + 1 "M-theory" that reduces to a D = 17 + 1 string theory as studied in relation to Kac-Moody algebras [79]. Quasiconformal extensions of the 2-brane, 10-brane, 18-brane and global space-time motivate Spin(4, 4), Spin (12,4), Spin (20,4) and Spin (28,4) symmetries and their EP extensions. Spin (12,4) is the conformal symmetry of Spin (11,3), whose exceptional Yang-Mills superalgebra admits a Hodge duality between a 3-brane and 7-brane when the 3-brane is treated as a worldvolume for space-time, generalizing the 2-brane and 5-brane. The isometry group of the 3-brane (3,3) worldvolume has Spin(4, 4) conformal symmetry, which describes three generations of matter [80,81]. A transverse Spin(8) is used for visible matter within Spin (12,4), where fermionic degrees of freedom can be encoded by a 128 spinor of Spin (12,4), leading to e 8(−24) . Going to Spin (20,4) or Spin(28, 4) allows us to consider transverse Spin (16) or Spin(24) for possible dark matter D = 28 + 4 entails the following braneworld twistors for three generations: Taking the 11-brane and 19-brane in D = 27 + 3, they are forced to intersect spatially via a 3-brane. The 11-brane isolates a D = 11 + 3 worldvolume with a 3-brane and dual 7-brane, while the 19-brane isolates a D = 19 + 3 worldvolume with a 3-brane and dual 15-brane. Our proposal is that the D = 11 + 3 worldvolume theory encodes baryonic matter, while the D = 19 + 3 theory encodes a dark matter sector.
EP algebras provide new dark matter models via intersected or nested higher membrane worldvolumes. In order to combine dark matter and the standard model within the same space-time worldvolume, e 8 also contains so 16 ⊕ so 16 , which has been studied in relation to dark matter in a different context [29].

Braneworld Intersections for Warm Dark Matter beyond the Standard Model
One candidate for describing the standard model from string theory is the utilization of D3 and D7 branes from type IIB supergravity or F-theory [82]. The central extension of D = 11 + 3 minimal chiral (1, 0) superalgebra naturally contains a 3-brane and dual 7-brane, as well as the type IIB superalgebra [47,48]. The role of E 8(−24) with Spin (12,4) in the context of background-independent two-time M-theory and heterotic string theory with N = 2 or 1 has been mentioned in [83]. A single generation of fermions can be found via 64 of Spin(11, 3) as off-shell degrees of freedom [84]. While breaking Spin(12, 4) → Spin(2, 4) × Spin(10) is tempting, this makes it difficult to identify the three generations of matter with less than 2048 spinors without another consideration. Some of the authors have explored how SU(3, 2) × SU(5) (maximal and non-symmetric subgroup of E 8(−24) ) naturally leads to three conformal charts, in a manner similar to how quasiconformal groups with four times are the conformal group for three generations of matter [81]. The corresponding (1,0) chiral super-Poincaré algebra in D = 12 + 4 entails three generations of the standard model within 128 of Spin (12,4) instead of 192 off-shell degrees of freedom.
Throughout this section, it is assumed that three generations of the standard model stem from a 3-brane and dual 7-brane, immersed in an 11-brane worldvolume. To go beyond the standard model and include dark matter candidates, a 15-brane is also considered. The 3brane and 15-brane exist inside a 19-brane worldvolume. The possible p-branes in D = 27 + 3 are shown in Figure 2. In particular, visible matter and dark matter are isolated on the 11brane and 19-brane, such that they intersect on the 11 + 19 − 27 = 3-brane worldvolume for space-time. In this manner, the braneworld intersection identifies visible and dark matter with different forces, yet they exist within the same space-time. The D = 27 + 3 theory may be isolated to D = 19 + 3 for dark matter, yet reduced to D = 11 + 3 for visible matter.
By restricting to the 11-brane worldvolume, the 3-brane is dual to a 7-brane. Isolating the conformal group of the 11-brane worldvolume within e Once the 3-brane worldvolume is isolated within the 11-brane worldvolume, Spin(8) is found as being transverse to Spin(4, 4) as the conformal worldvolume symmetry. The 3-brane within the 11-brane within D = 27 + 3 has the effect of making the Spin(16) transverse to the 11-brane hidden with respect to the matter isolated to the 3-brane and dual 7-brane within D = 11 + 3.
On the other hand, by restricting to the 19-brane worldvolume, the 3-brane is dual to a 15-brane. Isolating the conformal group of the 19-brane worldvolume within e Once the 3-brane worldvolume is isolated within the 19-brane worldvolume, Spin(16) is transverse to Spin(4, 4). The 3-brane within the 19-brane in a bulk D = 27 + 3 leaves the Spin(8) transverse to the 19-brane hidden with respect to matter isolated to the 3-brane and its dual 15-brane within D = 19 + 3.
While isolating the 3-brane in the 19-brane versus 11-brane symmetries within e (2) 8 (−24) leads to different representations, not all representations of the fermions are identifiable. For instance, the 11-brane cannot access a 15-brane in its transverse space, such that the so 16 representations are unobservable, while the 19-brane cannot access a 7-brane in its transverse space, such that the so 8 representations are unobservable. We assume that the so 8 transverse to the 19-brane contains the electromagnetic u 1 of the standard model [85]. With the 11-brane and 19-brane intersection, the 3-brane effectively sees the following fermionic representations with respect to Spin(4, 4) × Spin(8) × Spin (16): , 1, 128), (38) where the singlets effectively arise at low energies, since a 3-brane in an 11-brane cannot probe a transverse 15-brane immersed in D = 27 + 3, for instance. As such, since the 3-brane is the intersection of two worldvolumes, 128 visible and 2048 dark fermions are found effectively at low energy from an intersecting braneworld model that contains worldvolumes that support the commonly studied supergravities and string theories. The 2048 dark fermions come from (8 s , 1, 128 ) ⊕ (8 c , 1, 128), which cannot be probed by the so 8 containing the electromagnetic u 1 . This number of degrees of freedom corresponds to a ∼2 keV particle that is in agreement with observations at both large scales-galaxy formation and distribution-and small scales-cores [12,13].

Conclusions
In this paper, we discuss possible large gauge group representations that allow us to obtain a high number of fermionic degrees of freedom using superalgebras in the context of exceptional periodicity models and M-theory braneworlds. We find that an 11-brane and 19-brane in D = 27 + 3 intersect by a 3-brane for space-time such that respective dual 7-branes and 15-branes provide both the degrees of freedom for the standard model and the ones for fermionic WDM, while preserving the 'separation' and the lack of interaction between SM and WDM particles. Since so 12,4 combines conformal symmetry with so 10 , the so 8 contains su 3 ⊕ u 1 of the strong and electromagnetic forces, such that the 19-brane does not interact with light; therefore, it is dark.
In this realization, the intersecting braneworlds span D = 27 + 3, but individually span D = 11 + 3 and D = 19 + 3. For D = 11 + 3, the 3-brane is dual to a 7-brane for light matter in 128 off-shell degrees of freedom. For D = 19 + 3, the 3-brane is dual to a 15-brane for dark matter. We find 2048 fermionic degrees of freedom in this representation, which correspond to ∼2 keV in the model cited in [23]. This particle spectrum is in agreement with constraints from structure formation. Moreover, a 2 keV particle influences the formation of cores in galaxies [12,13]. Several other representations have been discussed and analyzed. However, this representation has the advantage of naturally allowing for a WDM candidate that is interesting from the point of view of astrophysical constraints and also implying worldvolumes that support supergravities and string theories. The evolution of a particle produced in this manner can be discussed further. While beyond the scope of this paper, further investigations of what happens to these species once the particle decouples would provide additional constraints.
Author Contributions: S.P. and M.R. initiated this research; S.P. contributed the WDM insights and discussion, M.R. the superalgebras and brane reduction; A.M. contributed to the superalgebras and symmetry breaking; D.C. contributed the braneworld intersection and matter-multiplet phenomenology. All authors contributed to the writing of the paper. All authors have read and agreed to the published version of the manuscript.
Funding: This research study received no external funding.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Acknowledgments: S.P. would like to thank N.G. Sanchez for the invitation to submit to this special issue and for useful suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: