Unification of Conformal and Fuzzy Gravities with Internal Interactions—Study of Their Behaviour at Low Energies and Possible Signals in the Detection of Gravitational Waves

Abstract

In this work, we develop a unified framework for Conformal Gravity and Noncommutative (Fuzzy) Gravity incorporating internal interactions. Our approach relies on two fundamental observations: first, the dimensions of a curved manifold and those of its tangent group need not coincide, and second, both gravitational models can be formulated as gauge theories. We begin with a discussion of the gauge-theoretic formulation of gravitational dynamics, emphasizing the role of diffeomorphism invariance. We then outline the constructions of Conformal Gravity and Fuzzy Gravity within this formalism. Building on an extension of the four-dimensional tangent group, we propose a scheme that unifies the two theories while naturally incorporating internal degrees of freedom. We further investigate the low-energy limits that emerge after appropriate spontaneous symmetry-breaking mechanisms, and we comment on potential observational signatures—particularly those associated with cosmic strings and their imprint on gravitational-wave spectra.

1. Introduction

The unification of all fundamental interactions has been a central objective of theoretical physics for over a century. One of the earliest attempts was made by Kaluza and Klein [1,2], who proposed a framework unifying gravity and electromagnetism—the two well-established interactions at that time—by extending spacetime to five dimensions. Their idea was to reduce a purely gravitational theory defined in five dimensions to four, thereby generating a $U\left(1\right)$ gauge theory interpreted as electromagnetism, coupled with gravity. Although initially considered speculative due to its higher-dimensional nature, this approach gained renewed interest when it was realized that non-Abelian gauge theories could naturally emerge from similar settings [3,4,5] and be useful in the description of the Standard Model (SM) of particle physics. Specifically, it was found that a higher-dimensional spacetime of the form $M_D=M_4\times B$, with B a compact Riemannian manifold possessing non-Abelian isometry group S, leads, upon dimensional reduction, to a four-dimensional theory featuring Einstein gravity coupled with a Yang–Mills gauge theory with group S, along with scalar fields.

The primary appeal of this construction lies in its geometrical origin of gauge interactions, offering a natural explanation for the appearance of gauge symmetries. However, this framework faces notable obstacles, including the lack of a viable classical ground state with a simple direct-product structure and, more crucially for low-energy physics, its inability to yield chiral fermions in four dimensions after dimensional reduction [6]. A remarkable improvement arises when Yang–Mills fields are incorporated from the outset, albeit at the cost of abandoning a purely geometric unification.

In higher-dimensional Grand Unified Theory (GUT) frameworks that include both Yang–Mills fields and fermions [7,8], the emergence of chiral fermions in four dimensions requires the total spacetime dimension to be of the form $4n+2$ [9]. In the present work, we primarily explore approaches that move towards the opposite direction—approaches that treat all interactions, including gravity, as manifestations of gauge symmetries. Nevertheless, Superstring Theories have long dominated the investigation of higher-dimensional unification [10,11,12].

It is fair to say that Superstring Theories (SSTs) offer a consistent framework in higher dimensions, with the heterotic string theory [13]—formulated in ten dimensions—standing out as a particularly attractive case. This theory naturally accommodates Grand Unified Theory (GUT) gauge groups such as $E_8\times E_8$, whose dimensional reduction can, in principle, reproduce the SM. It must be emphasized, however, that experimental confirmation of these frameworks is still lacking.

Even prior to the emergence of SSTs, an alternative program was developed based on dimensionally reducing higher-dimensional gauge theories [14,15,16,17,18]. While less ambitious in scope—since it effectively neglects gravity as a dynamical interaction—this approach shared the overarching goal of unifying the remaining fundamental forces. Within this context, Forgacs and Manton (F-M) introduced the Coset Space Dimensional Reduction (CSDR) scheme [14,15,16,17], wherein chiral fermions can emerge naturally. In parallel, Scherk and Schwarz (S-S) developed the group manifold reduction approach [19], which, despite the fact that it cannot accommodate chiral fermions, essential for viable low-energy models, nevertheless inspired numerous developments in string-model building.

More recently, substantial efforts have been dedicated to constructing phenomenologically viable models within the CSDR framework, which, from the outset, appeared far more promising [20,21,22,23,24].

An important line of research that aligns closely with the framework that will be presented in this paper emerged within four-dimensional spacetime and builds on the natural link between gravity and gauge theories. The SM of particle physics, which is a highly successful gauge theory verified extensively in past and current experiments (notably at the LHC), exemplifies this paradigm. However, it has long been known that gravity too can be formulated as a gauge theory [25,26,27,28,29,30,31,32,33,34]. Interest in this formulation was reignited by developments in supergravity [35,36], which also rely on gauge principles. These ideas have since been extended to Noncommutative (NC) gravity [37,38,39,40,41,42,43,44].

Weyl [45,46] was the first to relate electromagnetism to local phase transformations of the electron field and introduced the vierbein formalism, which later became crucial in the gauge formulation of gravity. Utiyama [25] made the next, decisive step by showing that gravity could be regarded as a gauge theory of the Lorentz group $SO(1,3)$, though his introduction of the vierbein was somewhat ad hoc. This ‘weakness’ was resolved by Kibble [26] and Sciama [27], who advocated gauging the full Poincaré group. Further advances by Stelle and West [33,34] led to more elegant constructions based on the de Sitter ($SO(1,4)$) or anti-de Sitter ($SO(2,3)$) groups, incorporating spontaneous symmetry breaking (SSB) to recover Lorentz invariance. The conformal group $SO(2,4)$ also played a pivotal role in the formulation of Weyl Gravity (WG) [47,48], Fuzzy Gravity (FG) [38,39,40,41,42,43,44], and their supersymmetric extensions in $\mathcal{N}=1$ supergravity [35,47].

A more direct and ambitious unification strategy involves embedding gravity within a larger gauge group that also includes internal symmetries of particle physics on equal footing [49,50,51]. Recent (and current) work has revitalized this direction [43,48,52,53,54,55,56,57,58,59,60,61,62], capitalizing on the observation that the tangent group of a curved spacetime need not match the manifold’s dimensionality. This opens the door to using higher-dimensional tangent groups in four-dimensional spacetime, facilitating a unified gauge-theoretic treatment of gravity and internal interactions. In this context, methods developed for higher-dimensional theories, such as CSDR [9,14,15,16,17,18,19,20,21,22,23,24], can be adapted to these four-dimensional constructions. Furthermore, challenges such as implementing simultaneous Weyl and Majorana conditions to obtain realistic chiral spectra also reappear in this setting [9,17]. Recently, a unified gauge framework was constructed, combining conformal gravity and internal interactions [43,48,57,59,60,61,62] and extending to noncommutative (fuzzy) spaces [44].

In the present work, we begin by presenting the gauge-theoretic formulation of gravity. We then present the construction of Conformal Gravity and NC (Fuzzy) Gravity within this framework. Finally, we develop a unified description of conformal and fuzzy gravity with $SO\left(10\right)$ internal symmetries, based on the larger gauge group $SO(2,16)$. It is worth recalling that the spinor representation of $SO\left(10\right)$ naturally incorporates right-handed neutrinos, thus allowing the introduction of neutrino masses. We conclude with an estimation of the symmetry-breaking channels of $SO\left(10\right)$, evaluating their compatibility with proton lifetime bounds and their potential observability via gravitational signals from cosmic string production.

2. Conformal Gauge Gravity

As previously mentioned, Einstein Gravity (EG) has commonly been viewed through the lens of a gauge theory constructed from the Poincaré group. Nonetheless, a deeper and more precise formulation arises when one instead examines gauge theories associated with the de Sitter (dS), $SO(1,4)$, and anti-de Sitter (AdS), $SO(2,3)$, groups. As with the Poincaré group, these groups also contain 10 generators and can undergo spontaneous symmetry breaking towards the Lorentz group, $SO(1,3)$, via non-dynamical (auxiliary) scalar fields [33,34,40,48]. The dS and AdS groups are subgroups of the larger conformal group $SO(2,4)$, which has 15 generators and maintains the invariance of the null interval $d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=0$ under transformations in spacetime. In [63], the gravitational gauge theory paradigm was extended to incorporate this full conformal group, giving rise to what is known as Conformal Gravity (CG). Traditionally, transitions from CG to either EG or Weyl-invariant gravity have been implemented through constraint conditions (as seen in [63]). By contrast, the method introduced in [48] offers a new perspective by achieving spontaneous symmetry breaking of the conformal gauge group. This is realized by including a scalar field within the action, which attains a non-zero vacuum expectation value (vev) via the application of a Lagrange multiplier.

2.1. Spontaneous Symmetry Breaking by Introducing a Scalar in the Adjoint Representation

The conformal gauge group $SO(2,4)$ comprises fifteen generators. Within a four-dimensional setting, these are categorized into six Lorentz generators $M_{ab}$, four generators associated with spacetime translations $P_a$, four special conformal (or conformal boost) generators $K_a$, and one generator corresponding to dilatations D.

The gauge connection $A_{\mu }$, which resides in the Lie algebra of $SO(2,4)$, is constructed as a linear combination of these generators:

$$A_{\mu }={\displaystyle \frac{1}{2}}{{\omega }_{\mu }}^{ab}{M}_{ab}+{e_{\mu }}^a{P}_a+{b_{\mu }}^a{K}_a+{\tilde{a}}_{\mu }D,$$

(1)

where each component gauge field is associated with a specific generator. In particular, ${e_{\mu }}^a$ is interpreted as the vierbein (or tetrad), and ${{\omega }_{\mu }}^{ab}$ serves as the spin connection. The field strength tensor corresponding to $A_{\mu }$ is then defined as follows:

$$F_{\mu \nu }={\displaystyle \frac{1}{2}}{R_{\mu \nu }}^{ab}{M}_{ab}+{{\tilde{R}}_{\mu \nu }}^a{P}_a+{R_{\mu \nu }}^a{K}_a+R_{\mu \nu }D,$$

(2)

where

$$\begin{array}{cc}\hfill {R_{\mu \nu }}^{ab}& ={\partial }_{\mu }{{\omega }_{\nu }}^{ab}-{\partial }_{\nu }{{\omega }_{\mu }}^{ab}-{{\omega }_{\mu }}^{ac}{{\omega }_{\nu c}}^b+{{\omega }_{\nu }}^{ac}{{\omega }_{\mu c}}^b-8{e_{[\mu }}^{[a}{b_{\nu ]}}^{b]}\hfill \\ \hfill & =R_{\mu \nu }^{\left(0\right)ab}-8{e_{[\mu }}^{[a}{b_{\nu ]}}^{b]},\hfill \\ \hfill {{\tilde{R}}_{\mu \nu }}^a& ={\partial }_{\mu }{e_{\nu }}^a-{\partial }_{\nu }{e_{\mu }}^a+{{\omega }_{\mu }}^{ab}{e}_{\nu b}-{{\omega }_{\nu }}^{ab}{e}_{\mu b}-2{\tilde{a}}_{[\mu }{e_{\nu ]}}^a\hfill \\ \hfill & =T_{\mu \nu }^{\left(0\right)a}\left(e\right)-2{\tilde{a}}_{[\mu }{e_{\nu ]}}^a,\hfill \\ \hfill {R_{\mu \nu }}^a& ={\partial }_{\mu }{b_{\nu }}^a-{\partial }_{\nu }{b_{\mu }}^a+{{\omega }_{\mu }}^{ab}{b}_{\nu b}-{{\omega }_{\nu }}^{ab}{b}_{\mu b}+2{\tilde{a}}_{[\mu }{b_{\nu ]}}^a\hfill \\ \hfill & =T_{\mu \nu }^{\left(0\right)a}\left(b\right)+2{\tilde{a}}_{[\mu }{b_{\nu ]}}^a,\hfill \\ \hfill R_{\mu \nu }& ={\partial }_{\mu }{\tilde{a}}_{\nu }-{\partial }_{\nu }{\tilde{a}}_{\mu }+4{e_{[\mu }}^a{b}_{\nu ]a},\hfill \end{array}$$

(3)

where $T_{\mu \nu }^{\left(0\right)a}\left(e\right)$ and $R_{\mu \nu }^{\left(0\right)ab}$ represent the torsion and curvature in the standard vierbein formalism of General Relativity (GR), while $T_{\mu \nu }^{\left(0\right)a}\left(b\right)$ corresponds to the torsion related to the auxiliary gauge field ${b_{\mu }}^a$.

To proceed, we adopt a parity-preserving action that is quadratic in the field strength tensor (2). We also introduce an auxiliary scalar field transforming in the adjoint (15-dimensional) representation of $SO\left(6\right)\sim SO(2,4)$ and include a mass parameter m:

$$S_{SO(2,4)}=a_{CG}\int d^{4}x[tr{ϵ}^{\mu \nu \rho \sigma }m\varphi F_{\mu \nu }{F}_{\rho \sigma }+\lambda ({\varphi }^2-m^{-2}{\mathbb{1}}_4)],$$

(4)

where the trace is over the generators in accordance with their gauge representation.

This scalar auxiliary field $\varphi$, expanded on the generators of the gauge algebra, can be written as

$$\varphi ={\varphi }^{ab}{M}_{ab}+{\tilde{\varphi }}^a{P}_a+{\varphi }^a{K}_a+\tilde{\varphi }D,$$

(5)

Following the strategy used in [64], we fix the gauge so that $\varphi$ becomes diagonal and aligned solely along the dilatation generator D, specifically taking the form $diag(1,1,-1,-1)$. Under the constraint ${\varphi }^2=m^{-2}{\mathbb{1}}_4$, this reduces to

$$\varphi ={\varphi }^0=\tilde{\varphi }D\stackrel{{\varphi }^2=m^{-2}{\mathbb{1}}_4}{\to }\varphi =-2m^{-1}D.$$

(6)

Substituting this gauge choice into the action yields

$$S=-2a_{CG}\int d^{4}xtr{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }D.$$

(7)

Because of spontaneous symmetry breaking, the fields ${e_{\mu }}^a$, ${b_{\mu }}^a$, and ${\tilde{a}}_{\mu }$ must all be rescaled as $m{e_{\mu }}^a$, $m{b_{\mu }}^a$, and $m{\tilde{a}}_{\mu }$, respectively. Substituting the expanded field strength tensor from Equation (2), and applying the generator algebra, leads to

$$\begin{array}{c}S=-2a_{CG}\int d^{4}xtr{ϵ}^{\mu \nu \rho \sigma }[{\displaystyle \frac{1}{4}}{R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^{cd}{M}_{ab}{M}_{cd}D++i{ϵ}_{abcd}({R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^c{K}^{d}D-\\ -{R_{\mu \nu }}^{ab}{{\tilde{R}}_{\rho \sigma }}^c{P}^{d}D)+({\displaystyle \frac{1}{2}}{{\tilde{R}}_{\mu \nu }}^a{R}_{\rho \sigma }+2{{\tilde{R}}_{\mu \nu }}^a{R_{\rho \sigma }}^b)M_{ab}+({\displaystyle \frac{1}{4}}{R}_{\mu \nu }{R}_{\rho \sigma }-2{{\tilde{R}}_{\mu \nu }}^a{R}_{\rho \sigma a})D].\end{array}$$

(8)

When evaluating the trace, we use

$$tr\left[K^{d}D\right]=tr\left[P^{d}D\right]=tr\left[M_{ab}\right]=tr\left[D\right]=0,\mathrm{and}tr\left[M_{ab}{M}_{cd}D\right]=-{\displaystyle \frac{1}{2}}{ϵ}_{abcd},$$

(9)

This gives rise to the broken-symmetry action with only Lorentz invariance remaining:

$$S_{\mathrm{SO}(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^{cd},$$

(10)

Notably, ${\tilde{a}}_{\mu }$ no longer appears in the resulting expression. Consequently, we may set ${\tilde{a}}_{\mu }=0$, simplifying the expressions for ${{\tilde{R}}_{\mu \nu }}^a$ and ${R_{\mu \nu }}^a$:

$$\begin{array}{cc}\hfill {{\tilde{R}}_{\mu \nu }}^a& =m{T}_{\mu \nu }^{\left(0\right)a}\left(e\right)-2m^2{\tilde{a}}_{[\mu }{e_{\nu ]}}^a⟶m{T}_{\mu \nu }^{\left(0\right)a}\left(e\right),\hfill \\ \hfill {R_{\mu \nu }}^a& =m{T}_{\mu \nu }^{\left(0\right)a}\left(b\right)+2m^2{\tilde{a}}_{[\mu }{b_{\nu ]}}^a⟶m{T}_{\mu \nu }^{\left(0\right)a}\left(b\right).\hfill \end{array}$$

(11)

Since these components of the field strength tensor are not present in the action, we may set ${{\tilde{R}}_{\mu \nu }}^a={R_{\mu \nu }}^a=0$, choosing a torsion-free theory. Similarly, the absence of $R_{\mu \nu }$ in the action allows us to impose $R_{\mu \nu }=0$. Using its definition from (3), we then deduce a constraint linking the gauge fields e and b:

$${e_{\mu }}^a{b}_{\nu a}-{e_{\nu }}^a{b}_{\mu a}=0.$$

(12)

This constraint admits multiple solutions, two of which are examined below.

2.1.1. Case ${b_{\mu }}^a=a{e_{\mu }}^a$

This particular identification between b and e was first suggested in [65]. When this substitution is performed in Equation (10), it results in the following action:

$$\begin{array}{cc}\hfill S_{\mathrm{SO}(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}& \left[R_{\mu \nu }^{\left(0\right)ab}-4m^{2}a\left({e_{\mu }}^a{e_{\nu }}^b-{e_{\mu }}^b{e_{\nu }}^a\right)\right]\hfill \\ \hfill & \left[R_{\rho \sigma }^{\left(0\right)cd}-4m^{2}a\left({e_{\rho }}^c{e_{\sigma }}^d-{e_{\rho }}^d{e_{\sigma }}^c\right)\right]\hfill \end{array}$$

(13)

Through straightforward algebraic manipulation, this action becomes

$$\begin{array}{c}\hfill S_{\mathrm{SO}(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}[R_{\mu \nu }^{\left(0\right)ab}{R}_{\rho \sigma }^{\left(0\right)cd}-16m^{2}a{R}_{\mu \nu }^{\left(0\right)ab}{e_{\rho }}^c{e_{\sigma }}^d+\\ \hfill +64m^4{a}^2{e_{\mu }}^a{e_{\nu }}^b{e_{\rho }}^c{e_{\sigma }}^d].\end{array}$$

(14)

This resulting action consists of three terms: one topological Gauss–Bonnet invariant (which does not affect field equations), the Einstein–Hilbert (Palatini) action equivalent in the vierbein formalism, and a term that behaves as a cosmological constant. For $a<0$, the theory describes General Relativity in an Anti-de Sitter spacetime background.

2.1.2. Case ${b_{\mu }}^a=-{\displaystyle \frac{1}{4}}({R_{\mu }}^a-{\displaystyle \frac{1}{6}}R{e_{\mu }}^a)$

This relation between b and e is used in [35,63] and, when employed, leads to the following action:

$$\begin{array}{cc}\hfill S_W={\displaystyle \frac{a_{CG}}{4}}& \int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}\hfill \\ & \left[R_{\mu \nu }^{\left(0\right)ab}+{\displaystyle \frac{1}{2}}\left(m{e_{\mu }}^{[a}{R_{\nu }}^{b]}-m{e_{\nu }}^{[a}{R_{\mu }}^{b]}\right)-{\displaystyle \frac{1}{3}}{m}^{2}R{e_{\mu }}^{[a}{e_{\nu }}^{b]}\right]\hfill \\ \hfill & \left[R_{\rho \sigma }^{\left(0\right)cd}+{\displaystyle \frac{1}{2}}\left(m{e_{\rho }}^{[c}{R_{\sigma }}^{d]}-m{e_{\sigma }}^{[c}{R_{\rho }}^{d]}\right)-{\displaystyle \frac{1}{3}}{m}^{2}R{e_{\rho }}^{[c}{e_{\sigma }}^{d]}\right].\hfill \end{array}$$

(15)

By introducing the rescaled vierbein ${{\tilde{e}}_{\mu }}^a=m{e_{\mu }}^a$ and using the antisymmetric property of the curvature 2-form, $R_{\mu \nu }^{\left(0\right)ab}=-R_{\nu \mu }^{\left(0\right)ab}$, we obtain

$$\begin{array}{cc}\hfill S_W={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x& {ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}\hfill \\ & \left[R_{\mu \nu }^{\left(0\right)ab}-{\displaystyle \frac{1}{2}}\left({{\tilde{e}}_{\mu }}^{[a}{R_{\nu }}^{b]}-{{\tilde{e}}_{\nu }}^{[a}{R_{\mu }}^{b]}\right)+{\displaystyle \frac{1}{3}}R{{\tilde{e}}_{\mu }}^{[a}{{\tilde{e}}_{\nu }}^{b]}\right]\hfill \\ \hfill & \left[R_{\rho \sigma }^{\left(0\right)cd}-{\displaystyle \frac{1}{2}}\left({{\tilde{e}}_{\rho }}^{[c}{R_{\sigma }}^{d]}-{{\tilde{e}}_{\sigma }}^{[c}{R_{\rho }}^{d]}\right)+{\displaystyle \frac{1}{3}}R{{\tilde{e}}_{\rho }}^{[c}{{\tilde{e}}_{\sigma }}^{d]}\right].\hfill \end{array}$$

(16)

Each bracket matches the form of the Weyl conformal tensor ${C_{\mu \nu }}^{ab}$, which means that the above action is actually the following:

$$S_W={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{C_{\mu \nu }}^{ab}{C_{\rho \sigma }}^{cd},$$

(17)

This corresponds to the well-known scale-invariant Weyl action in four dimensions:

$$S_W=2a_{CG}\int {\mathrm{d}}^{4}x\left(R_{\mu \nu }{R}^{\nu \mu }-{\displaystyle \frac{1}{3}}{R}^2\right).$$

(18)

2.2. Spontaneous Symmetry Breaking by Introducing Two Scalars in Vector Representations

The spontaneous symmetry breaking (SSB) of the $SO(2,4)$ gauge symmetry can be realized by employing two scalar fields in the vector representation of $SO\left(6\right)$, noting that the Lie algebra of $SO(2,4)$ is isomorphic to those of $SU\left(4\right)$ and $SO\left(6\right)$. Consequently, the $SO(2,4)$ gauge group may be spontaneously broken into $S{p}_4=SO\left(5\right)$ by introducing a scalar field in the six-dimensional vector representation of $SO\left(6\right)$, which acquires a vacuum expectation value (vev) in the $⟨1⟩$ component, in accordance with the branching rules of the $SO\left(6\right)$ representation under its maximal subgroup, $SO\left(5\right)$:

$$\begin{array}{cc}\hfill SO\left(6\right)& \supset SO\left(5\right)\hfill \\ \hfill 6& =1+5.\hfill \end{array}$$

(19)

Next, the $SO\left(5\right)$ symmetry is spontaneously broken by another scalar field in the vector representation 5, with the branching rule

$$\begin{array}{cc}\hfill SO\left(5\right)& \supset SU\left(2\right)\times SU\left(2\right)\hfill \\ \hfill 5& =(1,1)+(2,2).\hfill \end{array}$$

(20)

Since the algebra of $SU\left(2\right)\times SU\left(2\right)$ is isomorphic to both $SO\left(4\right)$ and $SO(1,3)$, breaking the symmetry with a scalar field in the 5 component of $SO\left(5\right)$ which acquires a vev in the $⟨1,1⟩$ component of $SU\left(2\right)\times SU\left(2\right)$, will yield the desired $SO(1,3)$ gauge group. Let us now turn to the generators of the $SO(2,4)$ algebra:

$$\left[J_{AB},J_{CD}\right]={\eta }_{BC}{J}_{AD}+{\eta }_{AD}{J}_{BC}-{\eta }_{AC}{J}_{BD}-{\eta }_{BD}{J}_{AC}.$$

(21)

Here, $A,B=1,\dots ,6$, and the signature of the metric is ${\eta }_{AB}=(-1,1,1,1,-1,1)$. This choice aligns with the two-step symmetry-breaking procedure: first breaking a spacelike direction (the 6th component) and then a timelike one (the 5th component). A single gauge field ${A_{\mu }}^{AB}$ represents all generators and is written as $A_{\mu }={\displaystyle \frac{1}{2}}{A_{\mu }}^{AB}{J}_{AB}$. Thus, the field strength tensor $F_{\mu \nu }={\displaystyle \frac{1}{2}}{F_{\mu \nu }}^{AB}{J}_{AB}$ is given by

$$F_{\mu \nu }=[D_{\mu },D_{\nu }]={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]\to$$

(22)

$${F_{\mu \nu }}^{AB}={\partial }_{\mu }{A_{\nu }}^{AB}-{\partial }_{\nu }{A_{\mu }}^{AB}+{{A_{\mu }}^A}_C{A_{\nu }}^{CB}-{{A_{\nu }}^A}_C{A_{\mu }}^{CB}.$$

(23)

To build an $SO(2,4)$-invariant quadratic action, we introduce two scalar fields, ${\varphi }^E$ and ${\chi }^F$, each in the vector representation 6 of $SO(2,4)$, and two mass parameters, $m_{\varphi }$ and $m_{\chi }$:

$$\begin{array}{cc}\hfill S_{SO(2,4)}=a_{CG}\int d^{4}x& [{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{ABCDEF}{\varphi }^E{\chi }^F{m}_{\varphi }{m}_{\chi }{\displaystyle \frac{1}{4}}{F_{\mu \nu }}^{AB}{F_{\rho \sigma }}^{CD}+\hfill \\ \hfill & +{\lambda }_{\varphi }({\varphi }^E{\varphi }_E-m_{\varphi }^{-2})+{\lambda }_{\chi }({\chi }^F{\chi }_F+m_{\chi }^{-2})],\hfill \end{array}$$

(24)

where $a_{CG}$ is dimensionless, and $m_{\varphi }\ge m_{\chi }$. In our chosen gauge, the auxiliary scalar field ${\varphi }^E$ is

$${\varphi }^E={\varphi }^0=(0,0,0,0,0,m_{\varphi }^{-1}),$$

(25)

with $m_{\varphi }^2>0$ since ${\varphi }^0$ is spacelike. Then, the component gauge field for the five broken generators becomes

$${A_{\mu }}^{j6}=m_{\varphi }{f_{\mu }}^j,$$

(26)

where f denotes the rescaled gauge field.

The field strength tensor now reads as follows:

$${F_{\mu \nu }}^{jk}={\partial }_{\mu }{A_{\nu }}^{jk}-{\partial }_{\nu }{A_{\mu }}^{jk}+{{A_{\mu }}^j}_l{A_{\nu }}^{lk}-{{A_{\nu }}^j}_l{A_{\mu }}^{lk}-m_{\varphi }^2({f_{\mu }}^j{f_{\nu }}^k-{f_{\mu }}^k{f_{\nu }}^j),$$

(27)

for $j,k=1,\dots ,5$, and the resulting $SO(2,3)$-symmetric action becomes:

$$S_{SO(2,3)}=a_{CG}\int d^{4}x\left[{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{ijklm}{m}_{\chi }{\chi }^m{\displaystyle \frac{1}{4}}{F_{\mu \nu }}^{ij}{F_{\rho \sigma }}^{kl}+{\lambda }_{\chi }({\chi }^m{\chi }_m+m_{\chi }^{-2})\right].$$

(28)

Next, we perform the second SSB by fixing the second auxiliary scalar field in the gauge:

$${\chi }^m={\chi }^0=(0,0,0,0,m_{\chi }^{-1}),$$

(29)

with $m_{\chi }^2>0$, as ${\chi }^0$ is timelike.

Recall that the conformal group $SO(2,4)$ has 15 generators, interpreted in four dimensions as six Lorentz generators $M_{ab}$, four translations $P_a$, four conformal boosts $K_a$, and one dilatation D. The respective gauge fields are ${{\omega }_{\mu }}^{ab}$, ${e_{\mu }}^a$, ${b_{\mu }}^a$, and ${\tilde{a}}_{\mu }$. In contrast, $SO(2,3)$ includes only the six Lorentz and four translation generators.

Thus, the five broken generators in $SO(2,4)\to SO(2,3)$ are linear combinations of $P_a$, $K_a$, and D.

The field ${f_{\mu }}^j$ in 4D notation is

$${f_{\mu }}^j=\left({b_{\mu }}^a-{e_{\mu }}^a,-{\tilde{a}}_{\mu }\right),$$

(30)

for $j=1,\dots ,5$ and $a,b=1,\dots ,4$. The second symmetry-breaking instance yields

$${A_{\mu }}^{a5}=-m_{\chi }({b_{\mu }}^a+{e_{\mu }}^a).$$

(31)

Hence, four more generators are broken, leaving only the six Lorentz generators $M_{ab}$. The full set of gauge fields becomes:

$$\begin{array}{c}\hfill {A_{\mu }}^{j6}=m_{\varphi }{f_{\mu }}^j\Rightarrow \left\{\begin{array}{c}{A_{\mu }}^{a6}=m_{\varphi }({b_{\mu }}^a-{e_{\mu }}^a)\hfill \\ {A_{\mu }}^{56}=-m_{\varphi }{\tilde{a}}_{\mu }\hfill \end{array}\right.,{A_{\mu }}^{a5}=-m_{\chi }({b_{\mu }}^a+{e_{\mu }}^a),\end{array}$$

(32)

and the remaining unbroken gauge fields correspond to the spin connection:

$${A_{\mu }}^{ab}={{\omega }_{\mu }}^{ab}.$$

(33)

This leads us to

$$\begin{array}{cc}\hfill {F_{\mu \nu }}^{ab}=& {\partial }_{\mu }{{\omega }_{\nu }}^{ab}-{\partial }_{\nu }{{\omega }_{\mu }}^{ab}-{{\omega }_{\mu }}^{ac}{{\omega }_{\nu c}}^b+{{\omega }_{\nu }}^{ac}{{\omega }_{\mu c}}^b\hfill \\ \hfill & +({m_{\chi }}^2-{m_{\varphi }}^2)\left({e_{\mu }}^a{e_{\nu }}^b-{e_{\nu }}^a{e_{\mu }}^b+{b_{\mu }}^a{b_{\nu }}^b-{b_{\nu }}^a{b_{\mu }}^b\right)\hfill \\ \hfill & -({m_{\chi }}^2+{m_{\varphi }}^2)({b_{\mu }}^a{e_{\nu }}^b-{b_{\nu }}^a{e_{\mu }}^b+{e_{\mu }}^a{b_{\nu }}^b-{e_{\nu }}^a{b_{\mu }}^b)⟶\hfill \\ \hfill {F_{\mu \nu }}^{ab}=& {R_{\mu \nu }}^{ab}+({m_{\chi }}^2-{m_{\varphi }}^2)\left({e_{\mu }}^a{e_{\nu }}^b-{e_{\nu }}^a{e_{\mu }}^b+{b_{\mu }}^a{b_{\nu }}^b-{b_{\nu }}^a{b_{\mu }}^b\right)\hfill \\ \hfill & -({m_{\chi }}^2+{m_{\varphi }}^2)({b_{\mu }}^a{e_{\nu }}^b-{b_{\nu }}^a{e_{\mu }}^b+{e_{\mu }}^a{b_{\nu }}^b-{e_{\nu }}^a{b_{\mu }}^b),\hfill \end{array}$$

(34)

and

$$\begin{array}{cc}\hfill {F_{\mu \nu }}^{a5}& =-m_{\chi }({\partial }_{\mu }{e_{\nu }}^a-{\partial }_{\nu }{e_{\mu }}^a-{{\omega }_{\mu }}^{ab}{e}_{\nu b}+{{\omega }_{\nu }}^{ab}{e}_{\mu b}+{\partial }_{\mu }{b_{\nu }}^a-{\partial }_{\nu }{b_{\mu }}^a-{{\omega }_{\mu }}^{ab}{b}_{\nu b}+{{\omega }_{\nu }}^{ab}{b}_{\mu b})+\hfill \\ \hfill & +{m_{\varphi }}^2[{\tilde{a}}_{\mu }({e_{\nu }}^a-{b_{\nu }}^a)-{\tilde{a}}_{\nu }({e_{\mu }}^a-{b_{\mu }}^a)]⟶\hfill \\ \hfill {F_{\mu \nu }}^{a5}& =-m_{\chi }[{T_{\mu \nu }}^a\left(e\right)+{T_{\mu \nu }}^a\left(b\right)]+{m_{\varphi }}^2[{\tilde{a}}_{\mu }({e_{\nu }}^a-{b_{\nu }}^a)-{\tilde{a}}_{\nu }({e_{\mu }}^a-{b_{\mu }}^a)].\hfill \end{array}$$

(35)

Since ${F_{\mu \nu }}^{a5}$ does not appear in the final action, we can consistently set ${F_{\mu \nu }}^{a5}=0$, and hence ${\tilde{a}}_{\mu }=0$, which may imply a torsion-free theory.

The final broken symmetry-bearing action is

$$\begin{array}{cc}\hfill S_{SO(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int & d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{F_{\mu \nu }}^{ab}{F_{\rho \sigma }}^{cd}=\hfill \\ \hfill ={\displaystyle \frac{a_{CG}}{4}}\int & d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}[{R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^{cd}+\hfill \\ \hfill & +2({m_{\chi }}^2-{m_{\varphi }}^2){R_{\mu \nu }}^{ab}({e_{\rho }}^c{e_{\sigma }}^d-{e_{\sigma }}^c{e_{\rho }}^d+{b_{\rho }}^c{b_{\sigma }}^d-{b_{\sigma }}^c{b_{\rho }}^d)\hfill \\ \hfill & -2({m_{\chi }}^2+{m_{\varphi }}^2){R_{\mu \nu }}^{ab}({b_{\rho }}^c{e_{\sigma }}^d-{b_{\sigma }}^c{e_{\rho }}^d+{e_{\rho }}^c{b_{\sigma }}^d-{e_{\sigma }}^c{b_{\rho }}^d)\hfill \\ \hfill & -2({m_{\chi }}^4-{m_{\varphi }}^4)\left({e_{\mu }}^a{e_{\nu }}^b-{e_{\nu }}^a{e_{\mu }}^b+{b_{\mu }}^a{b_{\nu }}^b-{b_{\nu }}^a{b_{\mu }}^b\right)\hfill \\ \hfill & \times ({b_{\rho }}^c{e_{\sigma }}^d-{b_{\sigma }}^c{e_{\rho }}^d+{e_{\rho }}^c{b_{\sigma }}^d-{e_{\sigma }}^c{b_{\rho }}^d)\hfill \\ \hfill & +{({m_{\chi }}^2-{m_{\varphi }}^2)}^2\left({e_{\mu }}^a{e_{\nu }}^b-{e_{\nu }}^a{e_{\mu }}^b+{b_{\mu }}^a{b_{\nu }}^b-{b_{\nu }}^a{b_{\mu }}^b\right)\hfill \\ \hfill & \times ({e_{\rho }}^c{e_{\sigma }}^d-{e_{\sigma }}^c{e_{\rho }}^d+{b_{\rho }}^c{b_{\sigma }}^d-{b_{\sigma }}^c{b_{\rho }}^d)\hfill \\ \hfill & +{({m_{\chi }}^2+{m_{\varphi }}^2)}^2\left({b_{\mu }}^a{e_{\nu }}^b-{b_{\nu }}^a{e_{\mu }}^b+{e_{\mu }}^a{b_{\nu }}^b-{e_{\nu }}^a{b_{\mu }}^b\right)\hfill \\ \hfill & \times ({b_{\rho }}^c{e_{\sigma }}^d-{b_{\sigma }}^c{e_{\rho }}^d+{e_{\rho }}^c{b_{\sigma }}^d-{e_{\sigma }}^c{b_{\rho }}^d)].\hfill \end{array}$$

(36)

Furthermore, the component ${F_{\mu \nu }}^{56}$ is also absent, so we set

$${F_{\mu \nu }}^{56}=m_{\varphi }[{\partial }_{\mu }{\tilde{a}}_{\nu }-{\partial }_{\nu }{\tilde{a}}_{\mu }-m_{\chi }(e_{\mu a}{b_{\nu }}^a-e_{\nu a}{b_{\mu }}^a)]=0.$$

(37)

With ${\tilde{a}}_{\mu }=0$, this implies that

$$e_{\mu a}{b_{\nu }}^a-e_{\nu a}{b_{\mu }}^a=0.$$

(38)

2.2.1. Case ${b_{\mu }}^a=a{e_{\mu }}^a$

As shown previously, in this case, the final action becomes

$$\begin{array}{cc}\hfill S_{SO(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x& {ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{F_{\mu \nu }}^{ab}{F_{\rho \sigma }}^{cd}=\hfill \\ \hfill ={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}& [{R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^{cd}+4\left({m_{\chi }}^2{(1-a)}^2-{m_{\varphi }}^2{(1+a)}^2\right){R_{\mu \nu }}^{ab}{e_{\rho }}^c{e_{\sigma }}^d\hfill \\ \hfill & +4{\left({m_{\chi }}^2{(1-a)}^2-{m_{\varphi }}^2{(1+a)}^2\right)}^2{e_{\mu }}^a{e_{\nu }}^b{e_{\rho }}^c{e_{\sigma }}^d]\hfill \end{array}$$

(39)

As before, the first term of action (39) is a G-B topological term, the second term is the E-H (Palatini) action, and the last term is the cosmological constant. When ${m_{\chi }}^2/{m_{\varphi }}^2>{(1+a)}^2/{(1-a)}^2$, the above action describes GR in AdS space.

When $m_{\varphi }=m_{\chi }\equiv m$, we obtain

$$\begin{array}{cc}\hfill S_{SO(1,3)}={\displaystyle \frac{a_{CG}}{4}}\int & d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{F_{\mu \nu }}^{ab}{F_{\rho \sigma }}^{cd}=\hfill \\ \hfill ={\displaystyle \frac{a_{CG}}{4}}\int & d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}\left[{R_{\mu \nu }}^{ab}{R_{\rho \sigma }}^{cd}-16m^{2}a{R_{\mu \nu }}^{ab}{e_{\rho }}^c{e_{\sigma }}^d+64m^4{a}^2{e_{\mu }}^a{e_{\nu }}^b{e_{\rho }}^c{e_{\sigma }}^d\right],\hfill \end{array}$$

(40)

which, when $a<0$, describes GR in AdS space and is completely equivalent to the action resulting in the case of the SSB with a scalar in the adjoint rep, as analyzed above.

2.2.2. Case ${b_{\mu }}^a=-{\displaystyle \frac{1}{4}}({R_{\mu }}^a+{\displaystyle \frac{1}{6}}R{e_{\mu }}^a)$ and $m_{\varphi }=m_{\chi }$

By choosing this relation from among the gauge fields e and b, we obtain again the following action:

$$\begin{array}{cc}\hfill S={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}& \left[{R_{\mu \nu }}^{ab}+{\displaystyle \frac{1}{2}}\left(m{e_{\mu }}^{[a}{R_{\nu }}^{b]}-m{e_{\nu }}^{[a}{R_{\mu }}^{b]}\right)-{\displaystyle \frac{1}{3}}{m}^{2}R{e_{\mu }}^{[a}{e_{\nu }}^{b]}\right]\hfill \\ \hfill & \left[{R_{\rho \sigma }}^{cd}+{\displaystyle \frac{1}{2}}\left(m{e_{\rho }}^{[c}{R_{\sigma }}^{d]}-m{e_{\sigma }}^{[c}{R_{\rho }}^{d]}\right)-{\displaystyle \frac{1}{3}}{m}^{2}R{e_{\rho }}^{[c}{e_{\sigma }}^{d]}\right],\hfill \end{array}$$

(41)

where $m\equiv m_{\varphi }=m_{\chi }$. By applying the rescaled vierbein, ${{\tilde{e}}_{\mu }}^a=m{e_{\mu }}^a$, and the antisymmetric property of the curvature 2-form, ${R_{\mu \nu }}^{ab}=-{R_{\nu \mu }}^{ab}$, we again obtain

$$\begin{array}{cc}\hfill S={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}& \left[{R_{\mu \nu }}^{ab}-{\displaystyle \frac{1}{2}}\left({{\tilde{e}}_{\mu }}^{[a}{R_{\nu }}^{b]}-{{\tilde{e}}_{\nu }}^{[a}{R_{\mu }}^{b]}\right)+{\displaystyle \frac{1}{3}}R{{\tilde{e}}_{\mu }}^{[a}{{\tilde{e}}_{\nu }}^{b]}\right]\hfill \\ \hfill & \left[{R_{\rho \sigma }}^{cd}-{\displaystyle \frac{1}{2}}\left({{\tilde{e}}_{\rho }}^{[c}{R_{\sigma }}^{d]}-{{\tilde{e}}_{\sigma }}^{[c}{R_{\rho }}^{d]}\right)+{\displaystyle \frac{1}{3}}R{{\tilde{e}}_{\rho }}^{[c}{{\tilde{e}}_{\sigma }}^{d]}\right].\hfill \end{array}$$

(42)

The above action is equivalent to

$$S={\displaystyle \frac{a_{CG}}{4}}\int d^{4}x{ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}{C_{\mu \nu }}^{ab}{C_{\rho \sigma }}^{cd},$$

(43)

where ${C_{\mu \nu }}^{ab}$ is the Weyl conformal tensor. This action, as discussed before, actually leads to

$$S=2a_{CG}\int {\mathrm{d}}^{4}x\left(R_{\mu \nu }{R}^{\nu \mu }-{\displaystyle \frac{1}{3}}{R}^2\right),$$

(44)

which describes the four-dimensional scale-invariant Weyl theory of gravity.

The Weyl action of WG in the forms given in Equations (43) and (44), being scale-invariant, naturally does not contain a cosmological constant. WG is an attractive possibility for describing gravity at high scales (for some recent developments, see [66,67,68,69,70,71]), as CG does. However, in the case where WG is obtained after the SSB of the CG, as described above, a question remains: how can one obtain Einstein gravity from the SSB of WG? Our suggestion is the following. We start again from CG and introduce a scalar in the 15 rep of $SU\left(4\right)\sim SO\left(6\right)\sim SO(2,4)$, as described above, and by choosing the relation

$${b_{\mu }}^a=-{\displaystyle \frac{1}{4}}\left({R_{\mu }}^a-{\displaystyle \frac{1}{6}}R{e_{\mu }}^a\right),$$

(45)

we are left with, after the SSB of the scalar $\mathbf{15}$-plet, the Weyl action. In addition, we introduce a scalar in the second-rank antisymmetric representation of $SU\left(4\right)$, namely, $\mathbf{6}$, which, upon its own spontaneous breaking, yields the E-H action.

One can easily see the result of these successive breaking instances by examining the relevant branchings of $\mathbf{15}$ under $SU\left(2\right)\times SU\left(2\right)\times U\left(1\right)$, which describes the Lorentz and dilatation generators left after the SSB due to the $\mathbf{15}$-plet:

$$\begin{array}{cc}\hfill SU\left(4\right)& \stackrel{<\mathbf{15}>}{\to }SU\left(2\right)\times SU\left(2\right)\times U\left(1\right)\hfill \\ \hfill \mathbf{15}& =[{(\mathbf{3},\mathbf{1})}_0+{(\mathbf{1},\mathbf{3})}_0]+{(\mathbf{1},\mathbf{1})}_0+{(\mathbf{2},\mathbf{2})}_2+{(\mathbf{2},\mathbf{2})}_2,\hfill \end{array}$$

(46)

where $[{(\mathbf{3},\mathbf{1})}_0+{(\mathbf{1},\mathbf{3})}_0]$ corresponds to the Lorentz generators $M_{ab}$, ${(\mathbf{1},\mathbf{1})}_0$ corresponds to the dilatation generator D, and the two $(\mathbf{2},\mathbf{2})$ factors may be associated (with ambiguity in their assignment) with the translation generators $P_a$ and the conformal boost generators $K_a$. In this stage, $P_a$ and $K_a$ are broken by $⟨\mathbf{15}⟩$, while $M_{ab}$ and D remain unbroken.

Similarly, the decomposition of $\mathbf{15}$ under the $SO\left(5\right)$ subgroup to $SU\left(4\right)$, which breaks after the SSB due to the $\mathbf{6}$-plet, reads as follows:

$$\begin{array}{cc}\hfill SU\left(4\right)& \stackrel{<\mathbf{6}>}{\to }SO\left(5\right)\hfill \\ \hfill \mathbf{15}& =\mathbf{10}+\mathbf{5},\hfill \end{array}$$

(47)

and, under $SO\left(5\right)\supset SU\left(2\right)\times SU\left(2\right)$,

$$\begin{array}{cc}\hfill SO\left(5\right)& \supset SU\left(2\right)\times SU\left(2\right)\hfill \\ \hfill \mathbf{10}& =(\mathbf{3},\mathbf{1})+(\mathbf{1},\mathbf{3})+(\mathbf{1},\mathbf{1})+(\mathbf{2},\mathbf{2}),\hfill \\ \hfill \mathbf{5}& =(\mathbf{1},\mathbf{1})+(\mathbf{2},\mathbf{2}).\hfill \end{array}$$

(48)

From these branchings, one can readily recognize that the ten unbroken generators (from $\mathbf{10}$) can be identified with the Lorentz generators $M_{ab}$ and the translation generators $P_a$ (the latter being those previously broken by $⟨\mathbf{15}⟩$). The five broken generators contained in $\mathbf{5}$ are the dilatation singlet and the component $(\mathbf{2},\mathbf{2})$ that corresponds to the conformal boosts $K_a$.

In short, $⟨\mathbf{15}⟩$ breaks $P_a$ and $K_a$, leaving $M_{ab}$ and D unbroken, while $⟨\mathbf{6}⟩$ breaks D and contributes further to the breaking (and mass generation) of the $K_a$ gauge bosons.

It should be noted, as discussed explicitly in [62], that the on-shell vanishing of the torsion tensors ${{\tilde{R}}_{\mu \nu }}^a$ and ${R_{\mu \nu }}^a$ together with $F_{\mu \nu }=0$ guarantees the equivalence between diffeomorphisms and gauge transformations. Hence, the relation (45), which leads to Weyl gravity, is not necessary for this equivalence but rather constitutes a further (optional) constraint selecting a particular sector of solutions.

Finally, we remark that the gauge-fixing approach employed to implement and analyze the SSB pattern is a standard technical tool in gauge-theoretic constructions. Such fixings simply serve to make the physical degrees of freedom and the symmetry-breaking structure manifest and do not restrict the generality of the gauge-theoretic description of gravity.

3. Noncommutative (Fuzzy) Gravity

3.1. Gauge Theories on Noncommutative Spaces

We begin by reviewing the essential ingredients for constructing gauge theories on noncommutative (NC) spaces, which form the foundation of our approach. In the framework of NC geometry, gauge fields arise naturally and are closely associated with the notion of the covariant coordinate [72]. This concept plays a role analogous to that of the covariant derivative in conventional gauge theory, as we shall see.

Consider a scalar field $\varphi \left(X_a\right)$ defined in a fuzzy space, where the coordinates $X_a$ obey non-trivial commutation relations. The field $\varphi$ transforms under a representation of a gauge group G, and an infinitesimal gauge transformation with parameter $\lambda \left(X_a\right)$ acts as follows:

$$\delta \varphi \left(X\right)=\lambda \left(X\right)\varphi \left(X\right).$$

(49)

When the gauge parameter $\lambda \left(X\right)$ is a scalar function of the coordinates, the transformation is Abelian, and $G=U\left(1\right)$. If instead $\lambda \left(X\right)$ is a $P\times P$ Hermitian matrix function, then the gauge transformation is non-Abelian, corresponding to $G=U\left(P\right)$.

Importantly, the NC coordinates $X_{\alpha }$ are themselves taken to be invariant under gauge transformations:

$$\delta X_{\alpha }=0.$$

(50)

However, applying a gauge transformation to the product of a coordinate and a field yields

$$\delta \left(X_a\varphi \right)=X_a\lambda \left(X\right)\varphi ,$$

(51)

which is not covariant in general since

$$X_a\lambda \left(X\right)\varphi \ne \lambda \left(X\right)X_a\varphi .$$

(52)

To restore covariance, we adopt an approach analogous to that of conventional gauge theories, where covariant derivatives ensure the appropriate transformation behaviour. This motivates the introduction of the covariant coordinate ${\mathcal{X}}_a$, which transforms into

$$\delta \left({\mathcal{X}}_a\varphi \right)=\lambda {\mathcal{X}}_a\varphi ,$$

(53)

provided that

$$\delta {\mathcal{X}}_a=[\lambda ,{\mathcal{X}}_a].$$

(54)

This condition is satisfied by defining the covariant coordinate as

$${\mathcal{X}}_a\equiv X_a+A_a,$$

(55)

where $A_a$ is interpreted as the gauge connection of the theory. From Equations (53) and (54), the gauge transformation of the connection $A_a$ follows,

$$\delta A_a=-[X_a,\lambda ]+[\lambda ,A_a],$$

(56)

confirming its role as the gauge connection (For further discussion, see [73,74]).

The field strength associated with $A_a$ is accordingly defined as

$$F_{ab}\equiv [X_a,A_b]-[X_b,A_a]+[A_a,A_b]-{C_{ab}}^c{A}_c=[{\mathcal{X}}_a,{\mathcal{X}}_b]-{C_{ab}}^c{\mathcal{X}}_c,$$

(57)

and transforms covariantly under gauge transformations:

$$\delta F_{ab}=[\lambda ,F_{ab}].$$

(58)

3.2. The Background Space

Before developing the full gauge theory of fuzzy gravity (FG), we first have to define the background spacetime. In this work, we consider both four-dimensional de Sitter ($d{S}_4$) and anti-de Sitter ($Ad{S}_4$) spaces.

Both geometries can be realized as embedded in a five-dimensional flat space, subject to the constraint

$${\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }=s{R}^2,$$

(59)

where R denotes the curvature radius, ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1,s)$ denotes the ambient-space metric, and $s=\pm 1$ determines the signature: $s=+1$ corresponds to $d{S}_4$ with isometry group $SO(1,4)$, while $s=-1$ corresponds to $Ad{S}_4$ with isometry group $SO(2,3)$. Throughout this section, we address both cases simultaneously.

These isometry groups contain 10 generators, a number insufficient for fully covariantizing the five embedding coordinates as well as the local Lorentz symmetry. We thus consider the minimal covariant extensions of these groups: $SO(1,5)$ in the de Sitter case and $SO(2,4)$ for the anti-de Sitter case. Each of these has 15 generators, capable of accommodating the five coordinates as well as the 10 generators needed to form a Lorentz subgroup.

To further explore the underlying structure and further extend the algebra of observables, we consider an even larger group, $SO(1,6)$ or $SO(2,5)$ (This extension is inspired by the logic of Snyder and Yang, as discussed in Appendix A), containing 21 generators. This leads to the following symmetry-breaking chain:

$$SO(1,6)\supset SO(1,5)\supset SO(1,4)\supset SO(1,3),$$

(60)

$$\mathrm{or}SO(2,5)\supset SO(2,4)\supset SO(2,3)\supset SO(1,3).$$

(61)

Let us denote the generators of $SO(1,6)$ using $J_{MN}$, with $M,N=0,\dots ,6$. These obey the standard Lorentz-like algebra:

$$[J_{MN},J_{RS}]=i\left({\eta }_{MR}{J}_{NS}+{\eta }_{NS}{J}_{MR}-{\eta }_{NR}{J}_{MS}-{\eta }_{MS}{J}_{NR}\right),$$

(62)

with ${\eta }_{MN}=\mathrm{diag}(-1,1,1,1,s,1,1)$.

Following the above decompositions form the initial groups down to $SO(1,3)$, we obtain the following nontrivial commutation relations:

$$\begin{array}{cc}\hfill & [J_{ij},J_{kl}]=i({\eta }_{ik}{J}_{jl}+{\eta }_{jl}{J}_{ik}-{\eta }_{jk}{J}_{il}-{\eta }_{il}{J}_{jk}),\hfill \\ \hfill & [J_{ij},J_{k6}]=i({\eta }_{ik}{J}_{j6}-{\eta }_{jk}{J}_{i6}),[J_{ij},J_{k5}]=i({\eta }_{ik}{J}_{j5}-{\eta }_{jk}{J}_{i5}),\hfill \\ \hfill & [J_{ij},J_{k4}]=i({\eta }_{ik}{J}_{j4}-{\eta }_{jk}{J}_{i4}),[J_{i6},J_{j6}]=i{J}_{ij},\hfill \\ \hfill & [J_{i6},J_{j5}]=-i{\eta }_{ij}{J}_{56},[J_{i6},J_{j4}]=-i{\eta }_{ij}{J}_{46},\hfill \\ \hfill & [J_{i6},J_{56}]=i{J}_{i5},[J_{i6},J_{46}]=i{J}_{i4},\hfill \\ \hfill & [J_{i5},J_{j5}]=i{J}_{ij},[J_{i5},J_{j4}]=-i{\eta }_{ij}{J}_{45},\hfill \\ \hfill & [J_{i5},J_{56}]=-i{J}_{i6},[J_{i5},J_{45}]=i{J}_{i4},\hfill \\ \hfill & [J_{i4},J_{j4}]=is{J}_{ij},[J_{i4},J_{46}]=-is{J}_{i6},[J_{i4},J_{45}]=-is{J}_{i5},\hfill \\ \hfill & [J_{56},J_{46}]=-i{J}_{45},[J_{56},J_{45}]=i{J}_{46},[J_{46},J_{45}]=-is{J}_{56}.\hfill \end{array}$$

(63)

Here, the indices $i,j,k,l=0,\dots ,3$ denote four-dimensional spacetime directions, with ${\eta }_{ij}=\mathrm{diag}(-1,1,1,1)$.

We now define the relevant observables, i.e., the NC tensor, the coordinates, and the momenta, in terms of these generators,

$${\mathrm{\Theta }}_{ij}=\hslash J_{ij},X_i=\lambda J_{i5},P_i={\displaystyle \frac{\hslash }{\lambda }}{J}_{i4},$$

(64)

and the remaining operators as follows:

$$Q_i={\displaystyle \frac{\hslash }{\lambda }}{J}_{i6},q=J_{56},p=J_{46},h=J_{45}.$$

(65)

The identifications above lead to the following algebra:

$$\begin{array}{c}[{\mathrm{\Theta }}_{ij},{\mathrm{\Theta }}_{kl}]=i\hslash ({\eta }_{ik}{\mathrm{\Theta }}_{jl}+{\eta }_{jl}{\mathrm{\Theta }}_{ik}-{\eta }_{jk}{\mathrm{\Theta }}_{il}-{\eta }_{il}{\mathrm{\Theta }}_{jk}),\\ [{\mathrm{\Theta }}_{ij},X_k]={\displaystyle \frac{i}{\hslash }}({\eta }_{ik}{X}_j-{\eta }_{jk}{X}_i),[{\mathrm{\Theta }}_{ij},P_k]={\displaystyle \frac{i}{\hslash }}({\eta }_{ik}{P}_j-{\eta }_{jk}{P}_i),\\ [{\mathrm{\Theta }}_{ij},Q_k]={\displaystyle \frac{i}{\hslash }}({\eta }_{ik}{Q}_j-{\eta }_{jk}{Q}_i),[Q_i,Q_j]=i{\displaystyle \frac{\hslash }{{\lambda }^2}}{\mathrm{\Theta }}_{ij},[X_i,X_j]=i{\displaystyle \frac{{\lambda }^2}{\hslash }}{\mathrm{\Theta }}_{ij},\\ [P_i,P_j]=is{\displaystyle \frac{\hslash }{{\lambda }^2}}{\mathrm{\Theta }}_{ij},[Q_i,X_j]=-i{\displaystyle \frac{\hslash }{{\lambda }^2}}{\eta }_{ij}q,[Q_i,P_j]=-i{\displaystyle \frac{{\hslash }^2}{{\lambda }^2}}{\eta }_{ij}p,\\ [X_i,P_j]=-i\hslash {\eta }_{ij}h,[Q_i,q]=i{\displaystyle \frac{\hslash }{{\lambda }^2}}{X}_i,[Q_i,p]=i{P}_i,\\ [X_i,q]=-i{\displaystyle \frac{{\lambda }^2}{\hslash }}{Q}_i,[X_i,h]=i{\displaystyle \frac{{\lambda }^2}{\hslash }}{P}_i,\\ [P_i,p]=-is{Q}_i,[P_i,h]=-is{\displaystyle \frac{\hslash }{{\lambda }^2}}{X}_i,\\ [q,p]=-ih,[q,h]=ip,[p,h]=-isq.\end{array}$$

(66)

This algebra encodes more than just the noncommutativity of the coordinates (as in Snyder’s approach) and the momenta as well as Heisenberg-type relations among them (as in Yang’s framework). Moreover, it provides additional structure that will later be reflected in the choice of gauge group.

3.3. Gauge Group and Representation

To formulate a gauge theory of gravity in the spacetime we are considering, our first step is to pinpoint the appropriate symmetry group to be gauged. This will naturally be the isometry group of the background space, which is $SO(1,4)$ for de Sitter ($d{S}_4$) space or $SO(2,3)$ for Anti-de Sitter ($Ad{S}_4$) space.

However, when constructing gauge theories on noncommutative spaces, we must account for both the commutators and the anticommutators between various fields. To clarify this, let us take two arbitrary elements from a Lie algebra, $\epsilon \left(X\right)={\epsilon }^a\left(X\right)T_a$ and $\varphi \left(X\right)={\varphi }^a\left(X\right)T_a$, where $T_a$ are the algebra generators. Their commutator is expressed as

$$[\epsilon ,\varphi ]={\displaystyle \frac{1}{2}}\{{\epsilon }^a,{\varphi }^b\}\left[T_a,T_b\right]+{\displaystyle \frac{1}{2}}\left[{\epsilon }^a,{\varphi }^b\right]\{T_a,T_b\}.$$

(67)

In standard commutative cases, the second term vanishes because the component functions ${\epsilon }^a$ and ${\varphi }^b$ commute. But in a noncommutative space, this term persists due to the non-trivial commutation relations of these components, meaning the anticommutator $\{T_a,T_b\}$ remains in the expression.

Crucially, the anticommutator of two generators does not generally belong to the original algebra, which means the algebra fails to close. To address this, we have two main options: either explicitly include every anticommutator that falls outside the initial gauge group as an element of the algebra (which would lead to an infinite-dimensional, universal enveloping algebra) or, as we choose to do, select a specific representation for the generators and expand the original gauge group into a larger symmetry group where the algebra is closed under both commutators and anticommutators.

Consequently, the gauge group is expanded from $SO(1,4)$ to $SO(1,5)\times U\left(1\right)$ for $d{S}_4$ and from $SO(2,3)$ to $SO(2,4)\times U\left(1\right)$ for $Ad{S}_4$. It is worth noting that the algebra presented in Equation (66) already captures the structure of $SO(1,5)$ and $SO(2,4)$ (with $s=+1$ and $s=-1$, respectively), which is the group that ultimately must be gauged. This extended structure results from broadening the background isometry groups from $SO(1,5)$ to $SO(1,6)$ and from $SO(2,4)$ to $SO(2,5)$. The additional $U\left(1\right)$ symmetry is necessitated by the presence of anticommutators and is not a direct consequence of the background isometry group extensions discussed in Section 3.2.

Given the similar commutation relations for the generators of both algebras in (66) (they only differ by the sign of s), we will focus our subsequent discussion on the gauging of $SO(2,4)\times U\left(1\right)$.

We begin by establishing the representation of the algebra generators in the standard four-dimensional Dirac representation. In this framework, the generators of $SO(1,4)$ (and analogously $SO(2,3)$) are constructed from combinations of $\gamma$-matrices, which satisfy the following relation:

$$\{{\gamma }_a,{\gamma }_b\}=-2{\eta }_{ab}{\mathbb{1}}_4,$$

(68)

where $a,b=1,\dots ,4$, ${\eta }_{ab}$ is the (mostly positive) Minkowski metric, and ${\mathbb{1}}_4$ is the $4\times 4$ identity matrix.

For the $SO(2,4)\times U\left(1\right)$ case, we identify the generators in this representation as follows:

• Six Lorentz generators—$M_{ab}=-{\displaystyle \frac{i}{4}}[{\gamma }_a,{\gamma }_b]$;
• Four translation generators—$P_a=-{\displaystyle \frac{1}{2}}{\gamma }_a(1-{\gamma }_5)$;
• Four conformal boost generators—$K_a={\displaystyle \frac{1}{2}}{\gamma }_a(1+{\gamma }_5)$;
• One dilatation generator—$D=-{\displaystyle \frac{1}{2}}{\gamma }_5$;
• One $U\left(1\right)$ generator—${\mathbb{1}}_4$.

These generators fulfill the standard conformal algebra, with the commutation relations

$$\begin{array}{cc}\hfill [M_{ab},M_{cd}]& ={\eta }_{bc}{M}_{ad}+{\eta }_{ad}{M}_{bc}-{\eta }_{ac}{M}_{bd}-{\eta }_{bd}{M}_{ac},\hfill \\ \hfill [M_{ab},P_c]& ={\eta }_{bc}{P}_a-{\eta }_{ac}{P}_b,\hfill \\ \hfill [M_{ab},K_c]& ={\eta }_{bc}{K}_a-{\eta }_{ac}{K}_b,\hfill \\ \hfill [P_a,D]& =P_a,\hfill \\ \hfill [K_a,D]& =-K_a,\hfill \\ \hfill [K_a,P_b]& =-2({\eta }_{ab}D+M_{ab}),\hfill \end{array}$$

(69)

and the following anticommutation relations:

$$\begin{array}{cc}\hfill \{M_{ab},M_{cd}\}& ={\displaystyle \frac{1}{2}}({\eta }_{ac}{\eta }_{bd}-{\eta }_{bc}{\eta }_{ad})-i{ϵ}_{abcd}D,\hfill \\ \hfill \{M_{ab},P_c\}& =+i{ϵ}_{abcd}{P}^d,\hfill \\ \hfill \{M_{ab},K_c\}& =-i{ϵ}_{abcd}{K}^d,\hfill \\ \hfill \{M_{ab},D\}& =2M_{ab}D,\hfill \\ \hfill \{P_a,K_b\}& =4M_{ab}D+{\eta }_{ab},\hfill \\ \hfill \{K_a,K_b\}& =\{P_a,P_b\}=-{\eta }_{ab},\hfill \\ \hfill \{P_a,D\}& =\{K_a,D\}=0.\hfill \end{array}$$

(70)

3.4. Fuzzy Gravity

We have established that four-dimensional noncommutative gravity in $Ad{S}_4$ can be formulated as a gauge theory of the $SO(2,4)\times U\left(1\right)$ group (Noncommutative gravity on $d{S}_4$ can be handled similarly.). In order to derive the theory’s action, we will follow the approach outlined in [39]. We start by introducing the covariant coordinate ${\mathcal{X}}_{\mu }\equiv X_{\mu }+A_{\mu }$, where $A_{\mu }$ is the gauge connection, expanded on the gauge group generators as follows:

$$A_{\mu }=a_{\mu }\otimes {\mathbb{1}}_4+{{\omega }_{\mu }}^{ab}\otimes M_{ab}+{e_{\mu }}^a\otimes P_a+{b_{\mu }}^a\otimes K_a+{\tilde{a}}_{\mu }\otimes D.$$

(71)

Next, we define the appropriate covariant field strength tensor for this theory [39,75]:

$${\widehat{F}}_{\mu \nu }\equiv \left[{\mathcal{X}}_{\mu },{\mathcal{X}}_{\nu }\right]-{\kappa }^2{\widehat{\mathrm{\Theta }}}_{\mu \nu },$$

(72)

Here, the covariant noncommutativity tensor ${\widehat{\mathrm{\Theta }}}_{\mu \nu }\equiv {\mathrm{\Theta }}_{\mu \nu }+{\mathcal{B}}_{\mu \nu }$ has been introduced, with ${\mathcal{B}}_{\mu \nu }$ acting as a two-form field to ensure the correct transformation properties exist regarding $\mathrm{\Theta }$. Since the field strength is an element of the gauge algebra, it can also be expanded onto the generators of that algebra:

$${\widehat{F}}_{\mu \nu }=R_{\mu \nu }\otimes {\mathbb{1}}_4+{\displaystyle \frac{1}{2}}{R_{\mu \nu }}^{ab}\otimes M_{ab}+{{\tilde{R}}_{\mu \nu }}^a\otimes P_a+{R_{\mu \nu }}^a\otimes K_a+{\tilde{R}}_{\mu \nu }\otimes D.$$

(73)

To achieve the spontaneous breaking of the $SO(2,4)\times U\left(1\right)$ symmetry down to the Lorentz group, we introduce a scalar field $\mathrm{\Phi }\left(X\right)$ belonging in the adjoint (15-dimensional) representation of $SU\left(4\right)\sim SO(2,4)$ (or, equivalently the rank-2 antisymmetric of $SO(2,4)$ [39,41,48]). The scalar must also carry a $U\left(1\right)$ charge to ensure that the $U\left(1\right)$ symmetry is also broken and does not remain in the residual symmetry.

This action is given by

$$\mathcal{S}=Trtr\left[\lambda \mathrm{\Phi }\left(X\right){ϵ}^{\mu \nu \rho \sigma }{\widehat{F}}_{\mu \nu }{\widehat{F}}_{\rho \sigma }+\eta \left(\mathrm{\Phi }{\left(X\right)}^2-{\lambda }^{-2}{\mathbb{1}}_N\otimes {\mathbb{1}}_4\right)\right],$$

(74)

where Tr is the trace over the matrices representing the coordinates (playing the role of the integration of the commutative case), tr is the trace over the algebra generators, $\eta$ is a Lagrange multiplier that enforces the constraint on $\mathrm{\Phi }$, and $\lambda$ is a parameter with mass dimension. As an element of the gauge group, $\mathrm{\Phi }\left(X\right)$ can be expanded as follows:

$$\mathrm{\Phi }\left(X\right)=\varphi \left(X\right)\otimes {\mathbb{1}}_4+{\varphi }^{ab}\left(X\right)\otimes M_{ab}+{\tilde{\varphi }}^a\left(X\right)\otimes P_a+{\varphi }^a\left(X\right)\otimes K_a+\tilde{\varphi }\left(X\right)\otimes D.$$

(75)

Following the procedure from [39,41,48], we gauge-fix the scalar field along the dilatation generator:

$$\mathrm{\Phi }\left(X\right)={\left.\tilde{\varphi }\left(X\right)\otimes D\right|}_{\tilde{\varphi }=-2{\lambda }^{-1}}=-2{\lambda }^{-1}{\mathbb{1}}_N\otimes D.$$

(76)

On-shell, while this condition is satisfied, and after carefully considering the anticommutation relations among the generators and taking traces over their products, the action simplifies significantly to the expression

$${\mathcal{S}}_{br}=Tr\left({\displaystyle \frac{\sqrt{2}}{4}}{\epsilon }_{abcd}{R_{mn}}^{ab}{R_{rs}}^{cd}-4R_{mn}{\tilde{R}}_{rs}\right){\epsilon }^{mnrs},$$

(77)

in which all the additional terms, including the Lagrange multiplier, vanish because of the gauge fixing.

The resulting theory exhibits a residual $SO(1,3)$ gauge symmetry after the spontaneous symmetry breaking. In the commutative limit, where noncommutativity disappears (and with appropriate field redefinitions connecting noncommutative and commutative fields), this action reduces to the Palatini action. This is ultimately equivalent to the Einstein–Hilbert action with a cosmological constant term (as shown in [41]). In essence, we recover standard General Relativity with a cosmological constant.

4. Unification of Conformal and Fuzzy Gravities with Internal Interactions

The minimal way of unifying Conformal Gravity (CG) with internal interactions, specifically those governed by $SO\left(10\right)$, is realized by utilizing $SO(2,16)$ as the grand unification gauge group. This strategy, as previously noted, draws from the understanding that the dimension of the tangent group does not necessarily have to match that of the curved manifold itself [44,48,49,50,51,52,53,54,55,56,57,58,62].

The CG framework naturally arises from gauging $SO(2,4)\sim SU(2,2)\sim SO\left(6\right)\sim SU\left(4\right)$ (with $SO\left(6\right)$ and $SU\left(4\right)$ understood in terms of Euclidean signature). Therefore, starting with the $SO(2,16)$ gauge group, one first identifies the centralizer $C_{SO(2,16)}\left(SO(2,4)\right)=SO\left(12\right)$. Then, this $SO\left(12\right)$ is expected to further break down into $SO\left(10\right)$, which will serve as the symmetry group for the internal interactions.

To ensure simplicity in the analysis, a Euclidean signature is adopted (the implications of non-compact spaces are discussed in [48]). We begin with the group $SO\left(18\right)$, placing fermions in its 256-dimensional spinor representation. The SSB of $SO\left(18\right)$ proceeds initially to its maximal subgroup $SO\left(6\right)\times SO\left(12\right)$, and subsequently to $SO\left(6\right)\times SO\left(10\right)\times \left[U\right(1\left)\right]$, where the brackets on $U\left(1\right)$ are used to take into account both the local and global cases. For convenience, let us recall the decomposition of the relevant representations [64,76,77]:

$$\begin{array}{cccc}\hfill SO\left(18\right)& \supset SO\left(6\right)\times SO\left(12\right)\hfill & \hfill & \\ \hfill \mathbf{18}& =(\mathbf{6},\mathbf{1})+(\mathbf{1},\mathbf{12})\hfill & \hfill & \mathrm{vector}\hfill \\ \hfill \mathbf{153}& =(\mathbf{15},\mathbf{1})+(\mathbf{6},\mathbf{12})+(\mathbf{1},\mathbf{66})\hfill & \hfill & \mathrm{adjoint}\hfill \\ \hfill \mathbf{256}& =(\mathbf{4},\overline{\mathbf{32}})+(\overline{\mathbf{4}},\mathbf{32})\hfill & \hfill & \mathrm{spinor}\hfill \\ \hfill \mathbf{170}& =(\mathbf{1},\mathbf{1})+(\mathbf{6},\mathbf{12})+\left({\mathbf{20}}^{\prime },\mathbf{1}\right)+(\mathbf{1},\mathbf{77})\hfill & \hfill & 2\mathrm{nd}\mathrm{rank}\mathrm{symmetric}\hfill \end{array}$$

(78)

The SSB of $SO\left(18\right)$ to $SO\left(6\right)\times SO\left(12\right)$ is accomplished by assigning a vacuum expectation value (VEV) to the <$\mathbf{1},\mathbf{1}$> component of a scalar field belonging to the $\mathbf{170}$ representation. Regarding fermions, we begin with the $\mathbf{256}$ spinor representation.

In order to further break $SO\left(12\right)$ down into $SO\left(10\right)\times U\left(1\right)$ or $SO\left(10\right)\times U{\left(1\right)}_{\mathrm{global}}$, we can employ scalar fields from the $\mathbf{66}$ representation (contained within adjoint $\mathbf{153}$ of $SO\left(18\right)$) or the $\mathbf{77}$ representation (contained within the second-rank symmetric tensor representation $\mathbf{170}$ of $SO\left(18\right)$), respectively, given the following branching rules:

$$\begin{array}{cc}\hfill SO\left(12\right)& \supset SO\left(10\right)\times U\left(1\right)\hfill \\ \hfill \mathbf{66}& =\left(\mathbf{1}\right)\left(0\right)+\left(\mathbf{10}\right)\left(2\right)+\left(\mathbf{10}\right)(-2)+\left(\mathbf{45}\right)\left(0\right)\hfill \\ \hfill \mathbf{77}& =\left(\mathbf{1}\right)\left(4\right)+\left(\mathbf{1}\right)\left(0\right)+\left(\mathbf{1}\right)(-4)+\left(\mathbf{10}\right)\left(2\right)+\left(\mathbf{10}\right)(-2)+\left(\mathbf{54}\right)\left(0\right)\hfill \end{array}$$

(79)

Based on the information above, a VEV to the <$\left(\mathbf{1}\right)\left(0\right)$> component of the $\mathbf{66}$ representation leads to the gauge group $SO\left(10\right)\times U\left(1\right)$ after SSB. Correspondingly, a VEV to the <$\left(\mathbf{1}\right)\left(4\right)$> component of the $\mathbf{77}$ representation results in $SO\left(10\right)\times U{\left(1\right)}_{\mathrm{global}}$ after SSB.

Similarly, we can further break $SU\left(4\right)$ down into $SO\left(4\right)\sim SU\left(2\right)\times SU\left(2\right)$ in two stages. First, it breaks down into $SO(2,3)\sim SO\left(5\right)$ and then into $SO\left(4\right)$. For this, we can recall the following branching rules [76]:

$$\begin{array}{cc}\hfill SU\left(4\right)& \supset SO\left(5\right)\hfill \\ \hfill \mathbf{4}& =\mathbf{4}\hfill \\ \hfill \mathbf{6}& =\mathbf{1}+\mathbf{5}\hfill \end{array}$$

(80)

As an initial step, by assigning a VEV to the <$\mathbf{1}$> component of a scalar in the $\mathbf{6}$ representation of $SU\left(4\right)$, the latter breaks down into $SO\left(5\right)$. Then, according to the branching rules,

$$\begin{array}{cc}\hfill SO\left(5\right)& \supset SU\left(2\right)\times SU\left(2\right)\hfill \\ \hfill \mathbf{5}& =(\mathbf{1},\mathbf{1})+(\mathbf{2},\mathbf{2})\hfill \\ \hfill \mathbf{4}& =(\mathbf{2},\mathbf{1})+(\mathbf{1},\mathbf{2})\hfill \end{array}$$

(81)

By giving a VEV to the <$\mathbf{1},\mathbf{1}$> component of a scalar in the $\mathbf{5}$ representation of $SO\left(5\right)$, we ultimately obtain the Lorentz group $SU\left(2\right)\times SU\left(2\right)\sim SO\left(4\right)\sim SO(1,3)$. Additionally, it is notable that in this scenario, the $\mathbf{4}$ representation decomposes under $SU\left(2\right)\times SU\left(2\right)\sim SO(1,3)$ into the appropriate representations to describe two Weyl spinors.

One can also follow an alternative route to break $SU\left(4\right)$ down into $SU\left(2\right)\times SU\left(2\right)$, as discussed in Section 2.1. Specifically, to break the $SU\left(4\right)$ gauge group down into $SU\left(2\right)\times SU\left(2\right)$, we can use scalars in the adjoint $\mathbf{15}$ representation of $SU\left(4\right)$, which is contained in the adjoint $\mathbf{153}$ representation of $SO\left(18\right)$. In this case, we have

$$\begin{array}{cc}\hfill SU\left(4\right)\supset & SU\left(2\right)\times SU\left(2\right)\times U\left(1\right)\hfill \\ \hfill \mathbf{4}=& (\mathbf{2},\mathbf{1})\left(1\right)+(\mathbf{1},\mathbf{2})(-1)\hfill \\ \hfill \mathbf{15}=& (\mathbf{1},\mathbf{1})\left(0\right)+(\mathbf{2},\mathbf{2})\left(2\right)+(\mathbf{2},\mathbf{2})(-2)\hfill \\ \hfill & +(\mathbf{3},\mathbf{1})\left(0\right)+(\mathbf{1},\mathbf{3})\left(0\right)\hfill \end{array}$$

(82)

Then, by assigning a VEV to the <$\mathbf{1},\mathbf{1}$> direction of the adjoint representation $\mathbf{15}$, we obtain the known result [64] that $SU\left(4\right)$ spontaneously breaks down into $SU\left(2\right)\times SU\left(2\right)\times U\left(1\right)$. The method for eliminating the corresponding $U\left(1\right)$ gauge boson and retaining only $SU\left(2\right)\times SU\left(2\right)$ was already discussed in Section 2.1. Again, note that the $\mathbf{4}$ representation decomposes into the appropriate representations of $SU\left(2\right)\times SU\left(2\right)\sim SO(1,3)$ suitable for describing two Weyl spinors.

Having established the analysis of various symmetry breakings using branching rules under maximal subgroups, starting from the group $SO\left(18\right)$, one can readily consider instead the isomorphic algebras of the various groups. Specifically, instead of $SO\left(18\right)$, one can consider the isomorphic algebra of the non-compact groups $SO(2,16)\sim SO\left(18\right)$ and similarly $SO(2,4)\sim SO\left(6\right)\sim SU\left(4\right)$.

The Effects of Weyl and Majorana Conditions on Fermions

Having explored various SSB patterns in the previous sections, we now turn our attention to fermionic matter fields and examine the implications of imposing Weyl and Majorana conditions in different dimensions and under different signatures.

A Dirac spinor $\psi$ in D spacetime dimensions has $2^{D/2}$ independent components. The imposition of either the Weyl or Majorana conditions reduces this number by a factor of 2. The Weyl condition is only consistent in even-dimensional spacetimes; therefore, the simultaneous imposition of both conditions, when allowed, results in a Weyl–Majorana spinor with $2^{(D-2)/2}$ independent components.

The unitary representations of the Lorentz group $SO(1,D-1)$ are labelled by a continuous momentum vector k and a spin ‘projection’ corresponding to a representation of the compact subgroup $SO(D-2)$. The Dirac, Weyl, Majorana, and Weyl–Majorana spinors carry indices that transform as finite-dimensional non-unitary spinor reps of $SO(1,D-1)$.

It is well known that for non-compact groups $SO(p,q)$, the existence of Majorana–Weyl spinors with the signature $(p,q)$ depends on the difference $p-q$. Specifically, such spinors exist if $p-q=0mod8$. Given the above, the minimal group needed to construct a unified theory that includes both CG and internal interactions based on $SO\left(10\right)$ which also admits Majorana and Weyl fermions, would be $SO(1,17)$ [57].

To ensure clarity and fix notation, let us briefly recall the familiar case of four dimensions. The $SO(1,3)$ spinors in the usual $SU\left(2\right)\times SU\left(2\right)$ basis transform as $(2,1)$ and $(1,2)$, with the representations labeled by their dimensionality. The two-component Weyl spinors, ${\psi }_L$ and ${\psi }_R$, transform as the irreducible spinors ${\psi }_L\sim (2,1)$ and ${\psi }_R\sim (1,2)$ of $SU\left(2\right)\times SU\left(2\right)$, where ‘∼’ here denotes ‘transforms as’.

A Dirac spinor can be formed by combining the left- and right-handed Weyl components:

$$\psi \sim {\psi }_L+{\psi }_R\sim (2,1)+(1,2),$$

(83)

where, in the four-component Weyl basis, ${\psi }_L=({\psi }_L,0)$ and ${\psi }_R=(0,{\psi }_R)$, and these are eigenstates of ${\gamma }^5$ with eigenvalues of $-1$ and $+1$, respectively.

The standard Majorana condition for a Dirac spinor is expressed as $\psi =C{\overline{\psi }}^T$, with C denoting the charge–conjugation matrix. In four dimensions, C connects the $(2,1)$ and $(1,2)$ components; hence, it is off-diagonal in the Weyl basis. In general, in even dimensions, it is possible to choose a Weyl basis in which the ${\mathrm{\Gamma }}^{D+1}$ (comprising the product of all matrices in D dimensions) is diagonal:

$${\mathrm{\Gamma }}^{D+1}{\psi }_{\pm }=\pm {\psi }_{\pm }.$$

(84)

This operator can also be decomposed as ${\mathrm{\Gamma }}^{D+1}={\gamma }^5\otimes {\gamma }^{d+1}$, separating the four-dimensional chirality from that in the extra d dimensions, while their individual eigenvalues remain unconstrained.

Because ${\mathrm{\Gamma }}^{D+1}$ commutes with Lorentz transformations, each ${\psi }_{\pm }$ transforms into an irreducible spinor of $SO(1,D-1)$. In even dimensions, $SO(1,D-1)$ has two irreducible spinors, while for $D=4n$, there exist two self-conjugate spinors, ${\sigma }_D$ and ${\sigma }_D^{\prime }$, and for $D=4n+2$, there exist ${\sigma }_D$ (which, in this case, is not self-conjugate) and ${\overline{\sigma }}_D$.

Choosing ${\psi }_{-}\sim {\sigma }_D$ and ${\psi }_{+}\sim {\sigma }_D^{\prime }$ or ${\overline{\sigma }}_D$, the resulting Dirac spinor is expressed as the direct sum of the aforementioned Weyl spinors:

$$\psi ={\psi }_{+}\oplus {\psi }_{-}\sim \left\{\begin{array}{cc}{\sigma }_D\oplus {\sigma }_D^{\prime }\hfill & \mathrm{for}D=4n\hfill \\ {\sigma }_D\oplus {\overline{\sigma }}_D\hfill & \mathrm{for}D=4n+2.\hfill \end{array}\right.$$

(85)

Given that the Majorana condition can be imposed in $D=2,3,4+8n$, the Weyl–Majorana condition can only be imposed in the specific case where $D=4n+2$. Here, we will be focusing on the $D=4n+2$ case (for others, see refs. [9,17]). Imposing the Weyl–Majorana condition at higher dimensions essentially ensures that the representation $f_R$ is the charge conjugate of $f_L$ so that upon reduction to four dimensions, with fermions only in the $f_L$ representation.

Let us now consider our case, once again keeping the Euclidean signature. Starting with a Weyl spinor of $\mathrm{SO}\left(18\right)$, according to the breakings and branching rules discussed earlier in this Section, we have

$$\begin{array}{cc}\hfill SO\left(18\right)& \supset SU\left(4\right)\times SO\left(12\right)\hfill \\ \hfill \mathbf{256}& =(\mathbf{4},\overline{\mathbf{32}})+(\overline{\mathbf{4}},\mathbf{32}).\hfill \end{array}$$

(86)

Consequently, we have the following branching rule of 32 under $SO\left(10\right)\times \left[U\right(1\left)\right]$:

$$\begin{array}{cc}\hfill SO\left(12\right)& \supset SO\left(10\right)\times \left[U\left(1\right)\right]\hfill \\ \hfill \mathbf{32}& =\overline{\mathbf{16}}\left(1\right)+\mathbf{16}(-1).\hfill \end{array}$$

(87)

Recall that $\left[U\left(1\right)\right]$ above is there to take into account the fact that $U\left(1\right)$ either remains as a gauge symmetry or is broken, leaving a $U\left(1\right)$ as a residual global symmetry.

However, as noted earlier,

$$\begin{array}{cc}\hfill SU\left(4\right)& \supset SU\left(2\right)\times SU\left(2\right)\hfill \\ \hfill \mathbf{4}& =(\mathbf{2},\mathbf{1})+(\mathbf{1},\mathbf{2}).\hfill \end{array}$$

(88)

Therefore, after all the breakings, we obtain

$$\begin{array}{c}SU\left(2\right)\times SU\left(2\right)\times SO\left(10\right)\times \left[U\right(1\left)\right]\\ \{(\mathbf{2},\mathbf{1})+(\mathbf{1},\mathbf{2})\}\{\mathbf{16}(-1)+\overline{\mathbf{16}}\left(1\right)\}+\{(\mathbf{2},\mathbf{1})+(\mathbf{1},\mathbf{2})\}\{\overline{\mathbf{16}}\left(1\right)+\mathbf{16}(-1)\}\\ =2\times {\mathbf{16}}_L(-1)+2\times {\overline{\mathbf{16}}}_L\left(1\right)+2\times {\mathbf{16}}_R(-1)+2\times {\overline{\mathbf{16}}}_R\left(1\right),\end{array}$$

(89)

from which, given that ${\overline{\mathbf{16}}}_R\left(1\right)={\mathbf{16}}_L(-1)$ and ${\overline{\mathbf{16}}}_L\left(1\right)={\mathbf{16}}_R(-1)$ and by choosing to keep only the $-\mathbf{1}$ eigenvalue of ${\mathbf{\gamma }}^{\mathbf{5}}$, we obtain

$$\mathbf{4}\times {\mathbf{16}}_{\mathit{L}}(-\mathbf{1}).$$

(90)

Therefore, this construction yields a natural prediction of four fermion families, arising from the underlying group-theoretic structure. The flavour separation is left as an open problem for future work.

Let us recall once more that Weyl spinors can be defined in even dimensions; however, Weyl–Majorana spinors can be defined only for $\mathit{D}=\mathbf{2}\mathbf{mod}\mathbf{8}$ if $\mathit{F}$ is real and $\mathbf{6}\mathbf{mod}\mathbf{8}$ if $\mathit{F}$ is pseudoreal [9]. It is interesting that Weyl–Majorana spinors can be defined in $\mathit{SO}(\mathbf{1},\mathbf{9})$ and $\mathit{SO}(\mathbf{1},\mathbf{17})$ but not in $\mathit{SO}(\mathbf{2},\mathit{D}-\mathbf{2})$. The latter, if possible, could yield a further reduction in the resulting fermions in the present case (the real and pseudoreal spinors can be obtained from Table 2 of ref. [9]).

Finally, we briefly comment on the situation in the context of FG. As described in [44], when pursuing the unification of FG with internal interactions, in analogy to CG unification via $\mathbf{SO}\left(\mathbf{10}\right)$ [48], we encounter two primary challenges:

• The fermions must remain chiral in order to remain massless at low energies and avoid acquiring Planck-scale masses.
• The fermions must belong to matrix (tensorial) representations, since FG is defined as a matrix model.

One can construct a gauge theory with symmetry group $\mathbf{SO}\left(\mathbf{6}\right)\times \mathbf{SO}\left(\mathbf{12}\right)$ and fermions in the $(\mathbf{4},\overline{\mathbf{32}})+(\overline{\mathbf{4}},\mathbf{32})$ representation, thus satisfying both chirality and matrix representation requirements. When FG is interpreted as a gauge theory of gravity, the gravitational sector naturally aligns with the gauge group $\mathbf{SO}\left(\mathbf{6}\right)\times \mathbf{U}\left(\mathbf{1}\right)\sim \mathbf{SO}(\mathbf{2},\mathbf{4})\times \mathbf{U}\left(\mathbf{1}\right)$. Hence, from this perspective, the deviation from the CG framework is minimal, and a consistent embedding of chiral fermions into the fuzzy model is achieved.

5. From SO(2,16) to the Standard Model

Four different models that start from the $\mathit{SO}(\mathbf{2},\mathbf{16})\sim \mathit{SO}\left(\mathbf{18}\right)$ gauge group and lead to the SM are discussed in this section, along with the potential for their observation in experiments that look for proton decay and gravitational wave signals.

5.1. Field Content and Estimation of Symmetry-Breaking Scales

We begin by determining (following the approach in [59]) the full field content of the $\mathit{SO}\left(\mathbf{18}\right)$ gauge theory, from which we obtain EG and $\mathit{SO}\left(\mathbf{10}\right)\times {\left[\mathit{U}\left(\mathbf{1}\right)\right]}_{\mathit{global}}$. We use the breakings and field content of [78] in order to get the SM from the $\mathit{SO}\left(\mathbf{10}\right)$ GUT. In particular, $\mathit{SO}\left(\mathbf{10}\right)$ breaks down into an intermediate group, which consequently breaks into the SM group. The intermediate groups are the Pati–Salam, $\mathit{SU}{\left(\mathbf{4}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{L}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}$, possibly with a discrete left-right symmetry, $\mathcal{D}$, and the left-right group, $\mathit{SU}{\left(\mathbf{3}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{L}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}\times \mathit{U}{\left(\mathbf{1}\right)}_{\mathit{B}-\mathit{L}}$, with or without the $\mathcal{D}$ discrete symmetry. We will denote them as 422, 422D, 3221, and 3221D from now on. Thus, for each of the four low-energy cases, we have a distinct field content at the $\mathit{SO}\left(\mathbf{18}\right)$ level.

According to the previous section, $\mathit{SO}\left(\mathbf{18}\right)$ breaks down into $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$ via the $(\mathbf{1},\mathbf{1})$ of a scalar $\mathbf{170}$ rep, and we employ scalars in the $\mathbf{15}$ rep of $\mathit{SO}\left(\mathbf{6}\right)$ to break CG; these scalars are drawn from $\mathit{SO}\left(\mathbf{18}\right)$ rep $\mathbf{153}$:

$$\mathbf{153}=(\mathbf{15},\mathbf{1})+(\mathbf{6},\mathbf{12})+(\mathbf{1},\mathbf{66}).$$

(91)

The $\mathit{SO}\left(\mathbf{12}\right)$ gauge group is spontaneously broken by scalars in the $\mathbf{77}$ rep, which can result from the $\mathbf{170}$ rep of the $\mathit{SO}\left(\mathbf{18}\right)$ group. In $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$ notation, the scalars responsible for breaking this product group belong in $(\mathbf{15},\mathbf{1})$ and $(\mathbf{1},\mathbf{77})$. Fermions in the $\mathbf{16}$ rep of $\mathit{SO}\left(\mathbf{10}\right)$ are obtained from a $\mathbf{256}$ rep of $\mathit{SO}\left(\mathbf{18}\right)$ (which will result in the $\mathbf{16}$ through $(\overline{\mathbf{4}},\mathbf{32})$ in $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$ notation). The $\mathit{SO}\left(\mathbf{10}\right)$ GUT is then spontaneously broken by a scalar in the $\mathbf{210}$ rep into the 422 and 3221D gauge groups; by a scalar in the $\mathbf{54}$ rep into the 422D gauge group; and by a scalar in the $\mathbf{45}$ rep into the 3221 gauge group.

Each one of the intermediate gauge groups is spontaneously broken down into the SM by scalars in a $\overline{\mathbf{126}}$ rep, while the electroweak Higgs boson is accommodated in a $\mathbf{10}$ rep (in $\mathit{SO}\left(\mathbf{10}\right)$ language). From this point onward, the scale at which $\mathit{SO}\left(\mathbf{10}\right)$ is broken will be called GUT scale, ${\mathit{M}}_{\mathit{GUT}}$, since all gauge couplings unify at that scale. The scale at which the 422(D)/3221(D) groups break will be referred to as the intermediate scale, ${\mathit{M}}_{\mathit{I}}$. The consecutive breakings for each case are given below:

$$422:{\mathrm{SO}\left(10\right)|}_{M_{\mathit{GUT}}}\stackrel{⟨{\mathbf{210}}_{\mathbf{H}}⟩}{\to }\mathit{SU}{\left(\mathbf{4}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}{|}_{{\mathit{M}}_{\mathit{I}}}\stackrel{⟨{\overline{\mathbf{126}}}_{\mathbf{H}}⟩}{\to }\mathrm{SM};$$

(92)

$$422\mathrm{D}:{\mathrm{SO}\left(10\right)|}_{M_{\mathit{GUT}}}\stackrel{⟨{\mathbf{54}}_{\mathbf{H}}⟩}{\to }\mathit{SU}{\left(\mathbf{4}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}{\times \mathcal{D}|}_{{\mathit{M}}_{\mathit{I}}}\stackrel{⟨{\overline{\mathbf{126}}}_{\mathbf{H}}⟩}{\to }\mathrm{SM};$$

(93)

$$3221:{\mathrm{SO}\left(10\right)|}_{M_{\mathit{GUT}}}\stackrel{⟨{\mathbf{45}}_{\mathbf{H}}⟩}{\to }\mathit{SU}{\left(\mathbf{3}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{L}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}\times \mathit{U}{\left(\mathbf{1}\right)}_{\mathit{B}-\mathit{L}}{|}_{{\mathit{M}}_{\mathit{I}}}\stackrel{⟨{\overline{\mathbf{126}}}_{\mathbf{H}}⟩}{\to }\mathrm{SM};$$

(94)

$$3221\mathrm{D}:{\mathrm{SO}\left(10\right)|}_{M_{\mathit{GUT}}}\stackrel{⟨{\mathbf{210}}_{\mathbf{H}}⟩}{\to }\mathit{SU}{\left(\mathbf{3}\right)}_{\mathit{C}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{L}}\times \mathit{SU}{\left(\mathbf{2}\right)}_{\mathit{R}}\times \mathit{U}{\left(\mathbf{1}\right)}_{\mathit{B}-\mathit{L}}{\times \mathcal{D}|}_{{\mathit{M}}_{\mathit{I}}}\stackrel{⟨{\overline{\mathbf{126}}}_{\mathbf{H}}⟩}{\to }\mathrm{SM}.$$

(95)

According to the $\mathit{SO}\left(\mathbf{12}\right)$ branching rules,

$$\mathit{SO}\left(\mathbf{12}\right)\supset \mathit{SO}\left(\mathbf{10}\right)\times \mathit{U}{\left(\mathbf{1}\right)}_{\mathit{global}}$$

$$\mathbf{12}=\left(\mathbf{1}\right)\left(\mathbf{2}\right)+\left(\mathbf{1}\right)(-\mathbf{2})+\left(\mathbf{10}\right)\left(\mathbf{0}\right)$$

(96)

$$\mathbf{66}=\left(\mathbf{1}\right)\left(\mathbf{0}\right)+\left(\mathbf{10}\right)\left(\mathbf{2}\right)+\left(\mathbf{10}\right)(-\mathbf{2})+\left(\mathbf{45}\right)\left(\mathbf{0}\right)$$

(97)

$$\mathbf{77}=\left(\mathbf{1}\right)\left(\mathbf{4}\right)+\left(\mathbf{1}\right)\left(\mathbf{0}\right)+\left(\mathbf{1}\right)(-\mathbf{4})+\left(\mathbf{10}\right)\left(\mathbf{2}\right)+\left(\mathbf{10}\right)(-\mathbf{2})+\left(\mathbf{54}\right)\left(\mathbf{0}\right)$$

(98)

$$\mathbf{495}=\left(\mathbf{45}\right)\left(\mathbf{0}\right)+\left(\mathbf{120}\right)\left(\mathbf{2}\right)+\left(\mathbf{120}\right)(-\mathbf{2})+\left(\mathbf{210}\right)\left(\mathbf{0}\right)$$

(99)

$$\mathbf{792}=\left(\mathbf{120}\right)\left(\mathbf{0}\right)+\left(\mathbf{126}\right)\left(\mathbf{0}\right)+\left(\overline{\mathbf{126}}\right)\left(\mathbf{0}\right)+\left(\mathbf{210}\right)\left(\mathbf{2}\right)+\left(\mathbf{210}\right)(-\mathbf{2}),$$

(100)

we choose to incorporate the Higgs $\mathbf{10}$ rep into the $\mathbf{12}$ rep of $\mathit{SO}\left(\mathbf{12}\right)$, while the $\overline{\mathbf{126}}$ that breaks the intermediate gauge group comes from $\mathbf{792}$. As for the intermediate breakings, $\mathbf{45}$ comes from $\mathbf{66}$, $\mathbf{54}$ is from $\mathbf{77}$, and $\mathbf{210}$ is from $\mathbf{495}$. Considering the $\mathit{SO}\left(\mathbf{18}\right)$ branching rules,

$$\mathit{SO}\left(\mathbf{18}\right)\supset \mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$$

$$\mathbf{18}=(\mathbf{6},\mathbf{1})+(\mathbf{1},\mathbf{12})$$

(101)

$$\mathbf{3060}=(\mathbf{15},\mathbf{1})+(\mathbf{10},\mathbf{12})+(\overline{\mathbf{10}},\mathbf{12})+(\mathbf{15},\mathbf{66})+(\mathbf{6},\mathbf{220})+(\mathbf{1},\mathbf{495})$$

(102)

$$\begin{array}{cc}\hfill \mathbf{8568}=& (\mathbf{6},\mathbf{1})+(\mathbf{15},\mathbf{12})+(\mathbf{10},\mathbf{66})+(\overline{\mathbf{10}},\mathbf{66})+(\mathbf{15},\mathbf{220})+(\mathbf{6},\mathbf{495})+\hfill \\ \hfill & +(\mathbf{1},\mathbf{792}),\hfill \end{array}$$

(103)

together with (78) and (91), the $\mathbf{12}$ rep of $\mathit{SO}\left(\mathbf{12}\right)$ comes from the $\mathbf{18}$ rep of $\mathit{SO}\left(\mathbf{18}\right)$, $\mathbf{792}$ is from $\mathbf{8568}$, $\mathbf{66}$ is from $\mathbf{153}$, $\mathbf{495}$ is from $\mathbf{3060}$, and finally $\mathbf{77}$ comes from $\mathbf{170}$. The full field content with its reps under each gauge group is presented in Table 1.

Table 1. The full field content at each gauge group level.

$\mathit{SO}\left(10\right)$	$\mathit{SO}\left(6\right)\times \mathit{SO}\left(12\right)$	$\mathit{SO}\left(18\right)$	Type and Role
$\mathbf{16}$	$(\overline{\mathbf{4}},\mathbf{32})$	$\mathbf{256}$	fermion
-	$(\mathbf{15},\mathbf{1})$	$\mathbf{153}$	scalar, breaks $\mathit{SO}\left(\mathbf{6}\right)$
-	$(\mathbf{1},\mathbf{77})$	$\mathbf{170}$	scalar, breaks $\mathit{SO}\left(\mathbf{12}\right)$
$\mathbf{10}$	$(\mathbf{1},\mathbf{12})$	$\mathbf{18}$	scalar, breaks SM
$\overline{\mathbf{126}}$	$(\mathbf{1},\mathbf{792})$	$\mathbf{8568}$	scalar, breaks the intermediate groups into SM
$\mathbf{45}$	$(\mathbf{1},\mathbf{66})$	$\mathbf{153}$	scalar, breaks $\mathit{SO}\left(\mathbf{10}\right)$ into 3221
$\mathbf{210}$	$(\mathbf{1},\mathbf{495})$	$\mathbf{3060}$	scalar, breaks $\mathit{SO}\left(\mathbf{10}\right)$ into 422 and 3221D
$\mathbf{54}$	$(\mathbf{1},\mathbf{77})$	$\mathbf{170}$	scalar, breaks $\mathit{SO}\left(\mathbf{10}\right)$ into 422D

Next, we make an estimation of the scales at which each gauge-breaking instance occurs. As mentioned in the previous section, the initial gauge group features an even number of fermionic generations, and we chose to have four. This means there will be an extra fermionic generation that is expected to acquire masses and decouple just above the EW scale. However, we can still give a rough estimate of the various breaking scales of our model, based upon the 3gen, 1-loop RG run of [59] and assuming that the fourth generation does not significantly qualitatively alter the evolution of the gauge couplings. For all four cases, gauge unification is achieved, and the intermediate scale is estimated at ${\mathbf{10}}^{\mathbf{10}}\mathrm{GeV}\lesssim {\mathit{M}}_{\mathit{I}}\lesssim {\mathbf{10}}^{\mathbf{13}}\mathrm{GeV}$, while the GUT scale is estimated at ${\mathbf{10}}^{\mathbf{15}}\mathrm{GeV}\lesssim {\mathit{M}}_{\mathit{GUT}}\lesssim {\mathbf{10}}^{\mathbf{16}}\mathrm{GeV}$.

The breaking of CG (into EG) has a negative contribution to the cosmological constant. Thus, if this was the only contribution, the space would be AdS. However, there are positive contributions from the $\mathit{SO}\left(\mathbf{18}\right)$ and $\mathit{SO}\left(\mathbf{12}\right)$ breakings. By choosing to have either of these breakings take place at the same scale as the CG breaking, we can fine-tune the contributions to obtain a value of zero or a (slightly) positive value for the cosmological constant.

We will focus on three distinct scenarios regarding the breakings above ${\mathit{M}}_{\mathit{GUT}}$. In scenario A, the $\mathit{SO}\left(\mathbf{18}\right)$ group breaks into $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$, and they—in turn—break into EG and $\mathit{SO}\left(\mathbf{10}\right)$, all at the same scale, ${\mathit{M}}_{\mathit{X}}$. As such, the contribution from the breaking of $\mathit{SO}\left(\mathbf{18}\right)$ cancels the negative one coming from the CG breaking. In scenaria B and C, $\mathit{SO}\left(\mathbf{18}\right)$ breaks into $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$ at a scale of ${\mathit{M}}_{\mathit{B}}$, while $\mathit{SO}\left(\mathbf{6}\right)$ and $\mathit{SO}\left(\mathbf{12}\right)$ will break at a different scale, ${\mathit{M}}_{\mathit{X}}$, between ${\mathit{M}}_{\mathit{B}}$ and ${\mathit{M}}_{\mathit{GUT}}$. In both scenarios B and C, the contribution to the cosmological constant from the breaking of $\mathit{SO}\left(\mathbf{12}\right)$ cancels the negative one from the CG breaking.

While the RG running couplings below ${\mathit{M}}_{\mathit{GUT}}$ are straightforward, in the case of gauge theories based on non-compact groups, the situation is not that clear. Very intense calculations can be found for $\mathit{\beta }$-functions regarding Stelle’s ${\mathit{R}}^{\mathbf{2}}$ gravity, which has been proven to be renormalizable [79,80]. However, they are all achieved in Euclidean space [81,82,83,84,85,86]. Thus, strictly speaking, the $\mathit{\beta }$-functions of a gauge theory based on a non-compact group have not been calculated. We can speculate, though, that, at least at a one-loop level, these $\mathit{\beta }$-functions can be accurately approximated by the respective functions of their compact counterparts. This is supported by the suggestions made by Donoghue [87,88,89], which we adapt (see [59]).

• Scenario A: The $\mathit{SO}\left(\mathbf{18}\right)$ gauge coupling runs down to the ${\mathit{M}}_{\mathit{X}}$ scale, where it should match both the $\mathit{SO}\left(\mathbf{6}\right)$ and $\mathit{SO}\left(\mathbf{12}\right)$ gauge couplings:

  $${\mathit{\alpha }}_{\mathbf{10}}^{\left(\mathbf{1}\right)}\left({\mathit{M}}_{\mathit{X}}\right)={\mathit{\alpha }}_{\mathit{CG}}^{\left(\mathbf{1}\right)}\left({\mathit{M}}_{\mathit{X}}\right).$$

  (104)
  By substituting the above relation into (14) and focusing on its last term, we compare the term with the contributions to the cosmological constant that come from $\mathit{SO}\left(\mathbf{12}\right)$ and get an estimate of the scale:

  $${\mathit{M}}_{\mathit{X}}\sim {\mathbf{10}}^{\mathbf{18}}\mathrm{GeV}.$$

  (105)
  However, by running the $\mathit{SO}\left(\mathbf{18}\right)$ gauge coupling up to ${\mathit{M}}_{\mathit{Pl}}$, its steep $\mathit{\beta }$-function rapidly pushes it to the non-perturbative regime and it hits a Landau pole before it reaches the Planck scale.
• Scenarios B and C: In the first scenario, one the $\mathit{SO}\left(\mathbf{18}\right)$ gauge group breaks below the Planck scale, ${\mathit{M}}_{\mathit{B}}<{\mathit{M}}_{\mathit{Pl}}$, while in the second, the $\mathit{SO}\left(\mathbf{18}\right)$ group is broken at the Planck scale, ${\mathit{M}}_{\mathit{B}}={\mathit{M}}_{\mathit{Pl}}$. In both scenarios, the $\mathit{SO}\left(\mathbf{6}\right)\times \mathit{SO}\left(\mathbf{12}\right)$ gauge group runs down to ${\mathit{M}}_{\mathit{X}}$, below which we get $\mathit{SO}\left(\mathbf{10}\right)$ and EG (plus the global $\mathit{U}\left(\mathbf{1}\right)$s, which we can ignore throughout the study). Again, the $\mathit{SO}\left(\mathbf{6}\right)$ and $\mathit{SO}\left(\mathbf{12}\right)$ groups should be broken at the same scale to fine-tune the cosmological constant. Using a matching condition like (104) is out of the question in either case, since we now have ${\mathit{\alpha }}_{\mathbf{10}}^{\left(\mathbf{1}\right)}\left({\mathit{M}}_{\mathit{X}}\right)={\mathit{\alpha }}_{\mathbf{12}}^{\left(\mathbf{1}\right)}\left({\mathit{M}}_{\mathit{X}}\right)$. Here, by employing the $\mathit{SO}\left(\mathbf{10}\right)$ and $\mathit{SO}\left(\mathbf{12}\right)$ gauge $\mathit{\beta }$-functions and the approximative gauge $\mathit{\beta }$-function of $\mathit{SO}\left(\mathbf{6}\right)$, we make a rough estimate of the scale at ${\mathit{M}}_{\mathit{X}}\sim {\mathbf{10}}^{\mathbf{18}}$ GeV.

Above the ${\mathit{M}}_{\mathit{X}}$ scale, the $\mathit{SO}\left(\mathbf{12}\right)$ gauge coupling runs up to ${\mathit{M}}_{\mathit{B}}$ and, staying within the perturbative regime, as the reps were chosen in such a way that the scalars would always be singlets under the CG gauge group, thus avoiding multiplicities in the calculation of the gauge $\mathit{\beta }$-function of $\mathit{SO}\left(\mathbf{12}\right)$. In scenario B, the $\mathit{SO}\left(\mathbf{18}\right)$ gauge coupling should in principle be able to run up to ${\mathit{M}}_{\mathit{Pl}}$ while staying in the perturbative regime.

We close this subsection by recalling our previous comment on the FG case, as discussed in Section 4.

5.2. Cosmic Strings from Intermediate-Scale Symmetry Breaking and Constraints from Proton Decay

Although proton decay has not yet been observed, the lifetime of protons has been the focus of many experimental studies [90,91,92,93], which constrain it and, consequently, the GUT scale. Most of the breaking paths of $\mathit{SO}\left(\mathbf{10}\right)$ are tested at Super-Kamiokande (Super-K), while future experiments like Hyper-Kamiokande (Hyper-K) [94], DUNE [95], and JUNO [96] will improve the sensitivity by as much as one order of magnitude. These experiments could be getting very close to proton decay observation and, in turn, baryon number violation, a development that could in principle exclude many attempts of grand unification.

However, there are current and future experiments that focus on paths other than proton decay to probe GUTs.

When a gauge structure spontaneously breaks down into an SM, it produces topological defects. Numbering among the three most prominent topological defects, domain walls and monopoles dominate the energy density of the universe and are thus problematic, although this can be ameliorated under the assumption that inflation happens after their production, as it strongly suppresses their density. In contrast, cosmic strings are usually formed from the breaking of an abelian $\mathit{U}\left(\mathbf{1}\right)$ subgroup and do not feature such problems, since a cosmic string network has a scaling solution and consequently does not overclose the Universe, but it can survive and generate a source of gravitational radiation [97,98,99].

Gravitational waves (GWs) originating from cosmic strings have been recently identified as a way of probing high-energy models [100,101,102,103,104,105]. If inflation takes place before the formation of cosmic strings, when they intersect and form loops, their network becomes a GW source. When they transition between different states, they emit strong high-frequency GW beams, and the loops emit energy in a gravitational manner. This radiation is called stochastic gravitational wave background (SGWB). An approach to computing SGWB can be found in [106].

In our study, we demonstrate four breaking chains that lead from the $\mathit{SO}\left(\mathbf{10}\right)$ GUT to the SM, denoted as 422, 422D, 3221, and 3221D, respectively, and we estimate the intermediate scale ${\mathit{M}}_{\mathit{I}}$ and the unification scale ${\mathit{M}}_{\mathit{GUT}}$. The authors of [107] calculated, at a two-loop level, the intermediate and unification scales for all the breaking chains of $\mathit{SO}\left(\mathbf{10}\right)$, including the four cases at hand. Although our examples feature a fourth fermionic generation, we follow their analysis (which compares their numerical results to the experimental bounds on proton decay/lifetime of Super-K and Hyper-K) in an approximative manner, as above. Neither 422D nor 3221D satisfy the Super-K bounds, while 422 and 3221 are just above the lower bounds for ${\mathit{M}}_{\mathit{GUT}}$. However, if proton decay is not observed during the future Hyper-K run, 3221 would be excluded as well. As such, 422 is the candidate with the highest survivability.

Turning our attention to the production of topological defects from spontaneous symmetry breaking (see [107] for details), we can note that the breaking of the $\mathit{SO}\left(\mathbf{10}\right)$ gauge group leads to monopoles in all cases and additional cosmic strings in 422D and 3221D. Therefore, inflation should happen after this breaking and, since it washes them out, their gravitational signal is rendered undetectable. Considering the intermediate breaking, there is production of monopoles from the 422 gauge group, monopoles and domain walls from 422D, cosmic strings from 3221 and cosmic strings, and domain walls from 3221D. Following the above reasoning, inflation has to happen after the breaking of 422, 422D, and 3221D, and thus these cases cannot be probed through a GW background. However, in the 3221 case, inflation can strategically take place between the GUT and 3221 breakings, and therefore strings can, in principle, be observed through SGWB. The tension of cosmic strings that is generated from the 3221 breaking is estimated to be compatible with the Super-K bounds. Should proton decay not be detected in Hyper-K, this channel should also be excluded.

6. Conclusions

We have presented a comprehensive framework for a possible unification of gravity(ies) with internal interactions. This is primarily based on the recent suggestion made by [48] that such a unification can be achieved by gauging an enlarged tangent Lorentz group. This idea is motivated by the observation that the dimension of the tangent group need not coincide with that of the curved manifold. Moreover, since gravity can be described by gauge theories—similarly to the SM—a natural proposal emerges: both gravitational and internal interactions could originate from a common gauge structure, as captured in the four-dimensional unification scheme developed here.

The gravity theories considered in this work include CG and FG, both based on gauging the conformal group $\mathit{SO}(\mathbf{2},\mathbf{4})$ (FG also requires gauging an additional $\mathit{U}\left(\mathbf{1}\right)$). A significant result is that CG can be spontaneously broken down either into EG or WG, with WG also being capable of eventually breaking down into EG.

The unification of CG and FG with internal interactions in four dimensions was then achieved by gauging the higher-dimensional tangent group $\mathit{SO}(\mathbf{2},\mathbf{16})$, which is successively broken down into EG and the $\mathit{SO}\left(\mathbf{10}\right)$ GUT. Fermion inclusion, along with suitable application of the Weyl condition, resulted in a fully unified setting. A 1-loop analysis at low energies, exploring four possible breaking channels of $\mathit{SO}\left(\mathbf{10}\right)$ down to the SM, yielded estimates of all relevant breaking scales, ranging from the Planck to the EW scale.

Clearly, both CG and FG describe physics close to the Planck scale, as estimated in Section 5, while the GUT $\mathit{SO}\left(\mathbf{10}\right)$ and the Standard Model of Particle Physics appear at lower energies. At lower energies (see estimates in Section 5), the CG is broken down into EG, while FG first goes to its continuum limit and then behaves very similarly to CG. In both cases, the low-energy regime in which the GUT $\mathit{SO}\left(\mathbf{10}\right)$ and the SM are discussed is the usual framework of EG on the gravity side.

Finally, we estimated the experimental prospects of each breaking channel, focusing on gravitational wave signals and proton decay, following previous analyses. Two channels are excluded by proton lifetime bounds. Among the remaining two, one predicts the formation of cosmic strings that generate a stochastic gravitational wave background with detectable signal. Thus, the proposed framework is not only theoretically robust but also, in principle, potentially testable in experiments.

Although the present unified scheme belongs to the category that Stelle [79] has proven to be renormalizable, we do not claim that it is renormalizable. Similarly, we do not claim that is ghost-free, although we have been collaborating with [70] in an examination of the conditions for removing the ghost. We plan to return to these serious theoretical questions, as well as to further examination of cosmological consequences of the presented unified scheme, in future publications.