Beyond the standard model with six-dimensional spacetime
BBeyond the standard model with six-dimensional spacetime
David Chester ∗ Quantum Gravity Research,Los Angeles, CA, USA
Alessio Marrani † Dipartimento di Fisica e Astronomia ’Galileo Galilei’, Univ. di Padova,andINFN, Sez. di Padova,Via Marzolo 8, I-35131 Padova, Italy
Michael Rios ‡ Dyonica ICMQG,Los Angeles, CA, USA
6D spacetime with SO (3 ,
3) symmetry is utilized to efficiently encode three generations ofmatter. A graviGUT model is generalized to a class of models that all contain three generationsand Higgs boson candidates. Pati-Salam, SU (5), and SO (10) grand unified theories are found whena single generation is isolated. For example, a SO (4 ,
2) spacetime group may be used for conformalsymmetry,
AdS → dS , or simply broken to SO (3 ,
1) of Minkowski space. Another class of modelsfinds SO (2 ,
2) and can give
AdS . Keywords:
Beyond the standard model, graviGUT, 6D spacetime ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ phy s i c s . g e n - ph ] F e b CONTENTS
I. Introduction 2A. A review of various grand unification models 3II. Three mass generations from 6D spacetime 4A. Intuition from the magic star 4B. Three momenta with different mass 5III. High energy theories from four space-like dimensions 6A. SO (10) GUT with spacetime 7B. Pati-Salam GUT with spacetime 8C. SU (5) GUT with spacetime 9IV. High energy theories from four time-like dimensions 12A. An attempt for SO (3 ,
1) 12B. Spacetime from
AdS I. INTRODUCTION
Besides the Pati-Salam su ⊕ su ⊕ su model [1–6], the e -series of Lie algebras ( e ∼ su [7–14], e ∼ so [15–31], e [32–39], e [40–42], and e [43–61]) is used to describe various grand unification theories (GUTs). Additionally,string theory proposes an E × E heterotic theory [62–66]. Beyond the standard model (BSM) physics is also studiedwith the infinite-dimensional exceptional Lie algebras, such as e [67–71] and e [72–74]. This work investigatesthe role of e − in unification physics, not as a GUT, but as a single algebra that breaks to GUTs and spacetimealgebras with their corresponding representations of a Higgs and fermions.6D spacetime via so , is a feature of all of the models discussed within this work. 6D physics has been successfulfor additional mass terms and unitarity methods [75–77], standard model physics [78, 79], whereas three-time physicshas been discussed in supersymmetric models with multiple superparticles [80, 81], and various graviweak/graviGUTproposals [82–91] have used so , for the standard model [61, 92–94]. In previous work, the authors have exploredbranes given by exceptional periodicity [95], and all of the e ( n )8( − ( n ∈ Z + ) algebras allow reductions along 3-branesor 4-branes with dual magnetic brane cohomologies encoding spinors. The n = 3 case in D = 27 + 3 was used topropose a worldvolume intepretation for M-theory [96]. Here, we build on the result for n = 1 that e − allows fora 3-brane (with (3,3) worldvolume) to be found by breaking to so , → so , .The double copy finds a relationship between Yang-Mills and gravity [97–104]. Given the recent developments on thedouble copy and heterotic theories [105–107], this work is complementary with these developments and the graviGUTmodels by finding an internal charge space and external spacetime. For e − , breaking so , → so , ⊕ so , allowsfor the identification of 6D spacetime and 10D charge space . Removing the charge space lightcone gives so , andrelates to branes found in three-time supersymmetry models [80, 81, 108, 109]. The benefit of three-times allows forthree superparticles, which can be interpreted to yield three generations of fermions.Given the difficulties of finding UV-finite quantum gravity and its small coupling constant, GUTs were proposedto unify all of the fundamental forces besides gravity. Recently, it has been suggested that torsion allows for UV-complete fermions [110–112], while the Gauss-Bonnet term allows for two-loop graviton scattering [113], both ofwhich are related to the MacDowell-Mansouri formalism [114–117] studied in various graviGUT models [61, 118]. This charge space is found to have the same signature of superstring theory without spacetime. While so , is relevant as a charge spacefor the standard model, the scope of this work is to simply demonstrate the connection of e − to so GUT.
Unsurprisingly, grand unified theory has been studied in a supersymmetry or supergravity context [6, 18, 21, 24, 29,36, 45, 47, 48, 52, 54].Using so , spacetime allows for e − to describe three generations of matter with only 128 degrees of freedominstead of 192, which corrects aspects of the sl , C model [61]. Complaints with E for unified theory [119] do notapply to 6D spacetime, as only 128 fermions are needed for three generations [120]. While E does not containcomplex representations, the algebra can be broken to smaller algebras with complex representations, such as so .While typical GUTs study an algebra and add representations, the models discussed here break e − alone intoGUT algebras, spinors, spacetime algebras, and Higgs representations. E gauge theory cannot be used as a GUT inthe conventional sense; nevertheless, E can be used to encapsulate GUTs with spacetime.The manuscript is ordered as follows. Next, SO (10), SU (5), Pati-Salam, and E GUTs are introduced. Section (II)explains how E GUT fits into E for one generation and how the magic star projection of E (also called the g decomposition by Mukai [121]) motivates three generations [122, 123]. It also introduces so , spacetime via a toy modelfrom f that contains various spinors. Section (III) discusses extensions of previous (and new) graviGUT modelsby breaking so , with four space-like dimensions to the standard model spectrum. Section (IV) explores additionalmodels from so , with four time-like dimensions. Concluding remarks are given in Section (V). Appendix (A) brieflydiscuss possibilities for removing ghosts with three time dimensions. A. A review of various grand unification models
The su ⊕ su ⊕ u gauge algebra of the standard model can be found from the symmetry breaking of the Pati-SalamGUT with su ⊕ su ⊕ su gauge symmetry [1] and the Georgi-Glashow GUT with su [7]. Pati-Salam GUT allowsfor a fermionic unification of the quarks and the leptons into ( , , ) and ( , , ) representations by treating theleptons as a fourth color. Alternatively, su unifies the bosons into a single gauge group. The fermions are placed inthe and representations.Both of these GUT algebras can be unified into so with ⊕ chiral spinors for fermions, since so → su ,c ⊕ su ,L ⊕ su ,R ↓ ↓ (1.1) su → su ,c ⊕ su ,L ⊕ u ,Y → su ,c ⊕ u ,e , where su .c ⊕ su ,L ⊕ u ,Y is the algebra associated with the standard model and u ,e describes electromagnetism.The commutative diagram In Eq. (1.1) [124] denotes that the same standard model gauge group is found from so via su ⊕ u or su ⊕ su ⊕ su . Finding u ,Y in su or su ⊕ u differentiates between SU (5) or flipped SU (5)GUT [125–133]. For so → su ⊕ u , a right-handed neutrino is required, since the contains − , so → su ⊕ u ,X → su ,c ⊕ su ,L ⊕ u ,X ⊕ u ,Z → su ,c ⊕ su ,L ⊕ u ,Y , (1.2) = ⊕ ⊕ − = ( , ) , ⊕ ( , ) , − ⊕ ( , ) − , ⊕ ( , ) − , − ⊕ ( , ) − , ⊕ ( , ) − , , = ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) , = ⊕ ⊕ − ⊕ = ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) − , − ⊕ ( , ) − , − ⊕ ( , ) − , − ⊕ ( , ) − , = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) − ⊕ ( , ) − ⊕ ( , ) , = ⊕ − = ( , ) , − ⊕ ( , ) , ⊕ ( , ) − , ⊕ ( , ) − , − = ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − . For standard SU (5) GUT, the U (1) Y charge Q Y is proportional to Q Z (as shown above), while flipped SU (5) GUTfinds Q Y proportional to Q X − Q Z . The so GUT allows for either a , , or a Higgs [134]. Here, the Higgs of so is shown to break to a Higgs of su . Singularities from E in F-theory have been argued to lead toflipped SU (5) GUTs [132]. Earlier investigations of flipped SU (5) found a hidden SO (10) × SO (6) gauge group [129],which may naturally fit in SO (16) ⊂ E .Breaking from so to the su ⊕ su ⊕ su of Pati-Salam GUT gives so → su ⊕ su ⊕ su → su ⊕ su ⊕ u ,R → su ,c ⊕ su ,L ⊕ u ,R ⊕ u ,B − L → su ,c ⊕ su ,L ⊕ u ,Y , = ( , , ) ⊕ ( , , ) = ( , ) ⊕ ( , ) ⊕ ( , ) − (1.3)= ( , ) , − ⊕ ( , ) , ⊕ ( , ) − , ⊕ ( , ) − , − ⊕ ( , ) , ⊕ ( , ) , − = ( , ) ⊕ ( , ) − ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) , = ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , )= ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − = ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) − , ( , ) , ⊕ ( , ) , − ⊕ ( , ) − , ⊕ ( , ) − , − = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) − , = ( , , ) ⊕ ( , , ) = ( , ) ⊕ ( , ) ⊕ ( , ) − = ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) − , = ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − , where two algebras u ,R and u ,B − L can be mixed to give Q Y proportional to Q R + Q B − L . The electroweak Higgscomes from ( , , ), which fits into the of so . Note that other VEVs are also required for each symmetry breaking,such as ( , , ) for su → su ,c ⊕ u ,B − L , ( , , ) for su ,R → u ,R , and ( , , ) for u ,R ⊕ u ,B − L → u ,Y [3]. Inour interpretation, we will find the that the off-shell fermionic degrees of freedom allow for the symmetry breaking to u ,Y .The flipped SO (10) GUT uses the algebra so ⊕ u [135, 136]. The flipped so GUT uses the entire fordescribing fermions, which is similar to other work in this manner [137–140]. We rule out the possibility with e − ,as only of the degrees of freedom are fermionic, which will become more clear below. While the ⊕ spinorsof so describes the 32 on-shell degrees of freedom for a single generation, so GUT does not provide any mechanismfor naturally describing three generations.The e GUT has three embeddings of so ⊕ u . Breaking e to so ⊕ u gives the following branchings, → ⊕ ⊕ − ⊕ , → ⊕ − ⊕ , (1.4)where and are the fundamental and adjoint of e , respectively. The trinification GUT uses su ⊕ su ⊕ su , amaximal and non-symmetric subalgebra of e [141–150]. Also, e contains su ⊕ su , which can give su GUT [151–154]with an additional su [155]. Next, e and e are shown to encapsulate e GUT.
II. THREE MASS GENERATIONS FROM 6D SPACETIMEA. Intuition from the magic star
The only exceptional GUT algebra is e . However, e and e contain representations of e GUT. The adjointrepresentation of E contains bosons and a single generation of fermions via e → e ⊕ u , (2.1) = ⊕ ⊕ ⊕ . From the perspective of GUTs, the utility of E is not to generalize E GUT to E GUT, but to simply place all ofthe content of E GUT for one generation within E . The magic star projection of e [122] breaks to the maximalsubalgebra su ⊕ e to naturally give three generations, e → e ⊕ su , (2.2) = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) , which can be visualized in Fig. (1). Three distinct embeddings of e are within e and overlap by e to give threegenerations. Breaking e → e gives the five-grading e → e ⊕ su → e ⊕ u , (2.3) = ( , ) + ( , ) + ( , ) = − + − + ( + ) + + , FIG. 1. The exceptional Lie algebra e is projected onto a g -like root lattice, which places e in the center with threefundamental and three anti-fundamental representations of e , and , respectively. where the magic star projected along one of three axes gives the five-grading above.Unsurprisingly, e ⊕ su has been embedded inside e to give a way to extend e GUT to include a family unification su ,F [50, 156]. Also, e contains su ⊕ su ⊕ su ⊕ su , suggesting a quadrification model as trinification with a globalfamily su ,F [149]. However, we take a different approach in this work. The only real form of e that has a chance ofobtaining so and so , is e − . While this can’t give su ,F , so ,F is suggested via three time-like dimensions togive so , spacetime. The e − algebra is natural for three generations of matter and the e GUT algebra.The magic star projection of e in Fig. (1) [157] isolates six exceptional Jordan algebras surrounding e . ThePeirce decomposition of the exceptional Jordan algebra occurs when breaking from e to so , as shown in Eq. (1.4)Three ’s and three ’s emerge from e , which are representations of so and give the 96 on-shell fermionic degreesof freedom. Following the breaking of e → so shown in Eq. (1.4), an additional ⊕ is contained within theadjoint of e . The ⊕ inside e represents additional off-shell fermionic degrees of freedom, as e contains 128spinorial roots, 96 of which are outside of e . The magic star of e , shown in Fig. (1), allows for three embeddings of e within e . Each embedding of e contains the same central e . The magic star projection of e provides a simplegeometric way to see how three mass generations of fermions fit inside the inside e .It has been stated that all GUTs besides su and so require mirror fermions [120], unless supersymmetry isintroduced. Mirror fermions must have weak hypercharge for the right-handed chirality states, instead of the left-handed. The relationship between three generations and mirror fermions is discussed in the next section. B. Three momenta with different mass
In this section, so , spacetime is shown to efficiently encode three Dirac spinors of different mass in 16 off-shelldegrees of freedom, provided that they have the same charge. A 4D Dirac spinor is given by eight off-shell degreesof freedom, implying that three generations of a particle requires 24 off-shell degrees of freedom. The scatteringamplitudes community utilizes massless so , spinors to encode massive 4D spinors for computational simplicity [75].Other work also utilized so , or six dimensions for three generations [35, 78]. Spacetime from so , more clearlyallows for three so , spacetimes as a subalgebra. The intuition for multiple time dimensions is to encode multiplemass generations. While so for a single time is a trivial algebra with no generators, so for three times allows for agauge-theoretic explanation for three generations [158–161].To briefly show how so , encodes three masses, consider a 6D massive vector p ¯ µ , where ¯ µ = − , − , − , , ,
3. Itis clear that we can find three 4D momenta p µi for i = 1 , , or 3, where time is taken from ¯ µ = − , −
2, or −
3. Given Strictly speaking, only the noncompact real forms e − and e admit real forms of exceptional Jordan algebras as fundamentalrepresentations. Since e − , e − , and e have complex fundamental representations, exceptional Jordan algebras cannot befound, but the decompositions are similar. that p ¯ µ p ¯ µ = − m with a positive signature for space, m = − p ,µ p µ = m − p − − p − ,m = − p ,µ p µ = m − p − − p − , (2.4) m = − p ,µ p µ = m − p − − p − . This demonstrates that a 6D momentum can be projected to three generations of 4D momenta with different masses m , m , and m . To obtain 4D spinors, each momentum p µi can be decomposed into spinors via the isomorphism so , ∼ sl , C . Another subtle point about these models is the utilization of so , , despite three generations. Similarto how Pati and Salam treated leptons as a fourth color, additional off-shell fermionic degrees of freedom are treatedas a fourth generation, which has been found in heterotic string theory [63].Before diving into e − and the eight charges associated with fermions found in the standard model, we demon-strate how f can be used to efficiently contain spacetime symmetry with fermions and anti-fermions with theirmirrors in a single algebra. The fermions are contained in the of so , inside e − . The subalgebra f ⊕ g − is found within e − to give the following representations, e − → f ⊕ g − , (2.5) = ( , ) ⊕ ( , ) ⊕ ( , ) . Note that f contains so , as a maximal subalgebra. The of so , can be found inside the of f . Since f also contains a , the fermions as of so , are contained in one inside f and seven inside ( , ) of f ⊕ g − . The standard model contains eight charge configurations of the electron, three up quarks, three downquarks, and neutrino. Focusing on f allows for the isolation of a singe charge configuration, such as the electron,giving instead of . As shown above, the so , inside f allows for three distinct so , spacetime algebras withdifferent time-like projections that can lead to different masses in 4D spacetime.As hinted at above, focusing on f allows for a simple demonstration of how three generations of fermions arecontained inside e − before diving into all of the different charge configurations. The f algebra is broken in thefollowing manner: f → so , → so , → so , ⊕ so , → so , ⊕ so , ⊕ u . = ⊕ = ⊕ v ⊕ s ⊕ c (2.6)= ⊕ ⊕ ⊕ − ⊕ ⊕ ⊕ − ⊕ ⊕ (cid:48)− ⊕ (cid:48) ⊕ − = ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) − , ⊕ ( , ) − , ⊕ ( , ) − , − ⊕ ( , ) , ⊕ ( , ) − , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) − , ⊕ ( , ) − , − ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) − , ⊕ ( , ) − , − . Mirror fermions are identified when breaking so , spacetime to so , ⊕ u and thinking of the u as a dummy (electric)charge. The weight of so , identifies the mirror fermions with a −
1. Spinors of so , combine the fermion of onechirality with the anti-fermion of the opposite chirality.While not experimentally measured, mirror fermions preserve symmetry with the weak force, which must acquire alarge mass if physical. Mirror fermions typically require additional fermionic degrees of freedom. Given that we willuse e − to give three generations of matter in 128 degrees of freedom, it appears as if the mirror fermions in so , spacetime for a single generation are created from the on-shell degrees of freedom from the other generations. Thissubtlety leads to the description of 3 generations with mirror fermions all inside 128 degrees of freedom. It is clearthat we should not assign e roots to degrees of freedom at the beginning, but rather symmetry break and see whatparticles arise at lower energies, or more appropriately described as different phases. In generalizing to e − , u willbe replaced by so . The u can be thought as providing a charge, which helps identify fermions vs antifermions.The use of so , also may combine the notion of so , as the massive little group of so , with so , as a gaugesymmetry for comprehensive family unification proposed by Wilczek et al. [158–161]. Ghosts are often thought tomake multiple time dimensions problematic, but a time-like so , gauge symmetry may help remove ghosts. SeeAppendix (A) for additional details. III. HIGH ENERGY THEORIES FROM FOUR SPACE-LIKE DIMENSIONS
This section looks to generalize the work of Nesti and Percacci, who used so , ⊕
64 to describe SO (10) GUT withspacetime for one generation [92, 93]. With the utilization of so , and the intuitive picture provided by the magicstar projection of e − , we look to establish how the standard model and spacetime can fit into various high energytheories. Additionally, new routes that directly lead to SU (5) and Pati-Salam GUTs that bypass so are found. Theutilization of e − is more similar to a Lie group cosmology model [61] than Ref. [55], yet we differ on the fermionicinterpretations and demonstrate how this noncompact real form connects to various GUTs.The following breaking could be taken to isolate so , spacetime, e − → so , → so , ⊕ so , . (3.1)While this breaking isolates a spacetime of interest, it introduces a Lorentzian so , charge algebra, which is not idealfor connecting with high energy theory. As it turns out, the ⊕ (cid:48) spinors of so , contain the same physicalcontent as ⊕ of so . While so spinors separates degrees of freedom into left and right chiralities, so , spinorsseparates degrees of freedom into particles and antiparticles. Adding multiple so , lightcones allows for multiple massgenerations, and including additional off-shell degrees of freedom can be thought about as a fourth lightcone, giving so , . As we will show, chiral so spinors can be found from e − .The notion of so , charge space will be pursued in more detail in future work, but the primary goal of this workis to establish the high energy theory inside e − . We briefly note that in addition to high energy GUTs, e − also contains a dual Lorentz symmetry, which was sought after in an attempt to understand the origins of the doublecopy [97] and the low-energy nonsupersymmetric field theory limit of the KLT relations in string theory [162]. It isquite curious to find the signature of superstring theory, however, this work does not look to find spacetime inside so , . Isolating so for Pati-Salam GUT and the strong force would lead a commuting so , . This seems to be aninternal symmetry that mirrors spacetime and allows for a dual Lorentz symmetry different than the one found inpure gravity [163, 164].Similar to how there are three distinct e subalgebras in e , there are also three distinct so ’s inside e and three so ’s inside f . These three distinct so ’s can be found inside e − , which simultaneously isolates three distict so , ’s, which are presumed to fit inside so , . In addition to so , spacetime, so , is found, which can either beused as a conformal symmetry or allow for the introduction of AdS = SO (4 , /SO (4 ,
1) for a single generation. Wepropose a three-time generalization of
AdS , which is SO (4 , /SO (4 ,
3) and gives three copies of dS , generalizingwhat was found in Ref. [61]. Isolating so , in so , leaves behind a commutative so . However, in order to connectwith so in e − , one must isolate a single generation. A. SO (10) GUT with spacetime
Using so for GUT is the most popular model, as it unifies the bosons into a single gauge group and a singlegeneration of fermions into a single chiral spinor . Three related ways to break to so and include so , forspacetime, so , ⊕ so , (cid:37) (cid:38) e − → so , → so , ⊕ so , → so , ⊕ so ⊕ so , . (3.2) (cid:38) (cid:37) so , ⊕ so In generalizing Nesti and Percacci’s so , model [92, 93] to e − with so , , the following path of symmetrybreaking is taken: e − → so , → so , ⊕ so , → so , ⊕ so ⊕ so , , = ⊕ = ⊕ ⊕ ⊕ − ⊕ ⊕ (cid:48)− (3.3)= ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) − . where breaking to so , ⊕ so , gives a five-grading of e − mentioned in Ref. [60]. While typical so GUT analysisrefers to on-shell degrees of freedom via ⊕ , including so , allows for the off-shell degrees of freedom to beaccounted for, introducing ( , ) for left chiralities and ( , ) for right chiralities. The so , weight here is +1 forstandard model fermions and antifermions, while − so , and itsspinors only gives one generation and a mirror fermion. However, the algebraic structure is richer and contains three e ’s, which have a fourth “generation” shared amongst the others, giving three on-shell generations.A Higgs candidate is found in ( , , ) , which was not found in Nesti and Percacci’s model [92, 93]. Variousbosonic vectors are also found. These additional degrees of freedom will be explored in subsequent work and areoutside the scope of this paper.Next, spacetime is isolated from the beginning and shown to give the same result when breaking so , , e − → so , → so , ⊕ so , → so , ⊕ so ⊕ so , = ⊕ (3.4)= ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , )= ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − . Removing so , isolates so , , which happens to be the spacetime signature of F-theory. The difference here is that so , is used for a type of Lorentzian charge space, rather than spacetime. Breaking off the charge space lightcone so , isolates so and splits the ( , , ) of so , into a left-handed spinor of so with its mirror , given by( , , ) and ( , , ) − , respectively.Alternatively, one may also immediately isolate so to give so , , e − → so , → so , ⊕ so → so , ⊕ so ⊕ so , = ⊕ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) (3.5)= ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − . This approach gives so , , which may be either the conformal symmetry of so , or the algebra of isometries of AdS (which can be broken to dS [165]). While so , is useful for 4D physics, dS = SO (4 , /SO (3 ,
1) is applicable foran expanding universe with a positive cosmological constant. While typical GUT refers to the on-shell fermionicdegrees of freedom only, we find that including so , ∼ sl , C allows for the identification of off-shell fermions, such as( , , ) .Since so GUT can be embedded inside e − and e − , it is also possible to break e − to either real formof e and obtain so , e − → e − ⊕ su , ↓ ↓ e − ⊕ su , → so ⊕ su , ⊕ u , (3.6)However, this does not allow for the isolation of so , spacetime. Nevertheless, e may isolate three on-shell generationsfrom the “fourth” additional off-shell generation, e − → e − ⊕ su , → so ⊕ su , ⊕ u = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) (3.7)= ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − , (3.8)as the of so , gets separated to ⊕ of su , . It appears that e doesn’t directly refer to mirror fermions, while so does. It’s also worth noting that this approach to e − is dissimilar to E GUT, as the typical E GUT introducesadditional fermions into the of E , while this approach only assigns fermions to the of the Peirce decompositionof . Furthermore, the interpretation of as an exceptional Jordan algebra only occurs with e − and e ,which contains so , ⊕ so , and so , ⊕ so , as a subalgebra, respectively, rather than so ⊕ u . Since the completecomparison of so and so , is saved for future work, we will not develop a full-fledged E GUT model, as there aremany possibilities to consider.
B. Pati-Salam GUT with spacetime
Since it is already understood that su ⊕ su ⊕ su GUT can be found from so , obtaining Pati-Salam GUT from e is trivial, since Section (III A) found so from e . Now, we focus on including reference to spacetime to explicitlyconfirm chiralities.To start, we break to Pati-Salam with so , and then to so , to ensure the appropriate chiralities and confirm thatthe so , weight refers to nonmirror or mirror fermions. Breaking e − to so and su ⊕ su ⊕ su gives e − → so , → so , ⊕ so → so , ⊕ su ⊕ su ⊕ su → so , ⊕ su ⊕ su ⊕ su ⊕ so , = ⊕ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) (3.9)= ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , )= ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − . The fermions with so , weight 1 correspond to one generation, while -1 weights correspond to mirror fermions. Next,a different approach to get Pati-Salam GUT that bypasses so is discussed.The unification of the three forces via GUT seems computationally motivated by the almost unification of thecoupling constants of the strong and electroweak forces [166]. This with the combination of difficulty of treatinggeneral relativity as a quantum field theory tends to unify the strong force with the electroweak force before gravity.However, treating gravity as a gauge theory may help. In particular, the frame field can be used for a Higgs-likemechanism [167, 168] for breaking from higher to lower dimensions [169]. Also, the dilaton relates to conformalsymmetry breaking and has been proposed as a Higgs candidate [170]. Furthermore, the electroweak Higgs bosonprovides mass, which is a charge of gravity.Since it appears that trivially combining the strong force with the electroweak force under a single gauge groupleads to proton decay, we demonstrate a way to unify spacetime with the electroweak force. Starting from e − , anew path to break to Pati-Salam GUT is found that bypasses SO (10) GUT. e − → so , → so , ⊕ so , → so , ⊕ su ⊕ su ⊕ so , → so , ⊕ su ⊕ su ⊕ su ⊕ so , = ⊕ = ( , ) ⊕ ( , ) ⊕ ( v , v ) ⊕ ( s , c ) ⊕ ( c , s ) (3.10)= ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , v ) ⊕ ( , , , , v ) ⊕ ( , , , , c ) ⊕ ( , , , , c ) ⊕ ( , , , , s ) ⊕ ( , , , , s )= ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − ⊕ ( , , , , ) ⊕ ( , , , , ) − . As shown above, this route avoids so , which is known to give proton decay. It turns out that the mirror fermionshave weight +1 this time. While further work is needed to systematically determine if this path avoids proton decay,this approach presents a candidate, although there are nontrivial alternatives with su [13, 14]. C. SU (5) GUT with spacetime
Next, we demonstrate that there are three different ways to symmetry break and obtain su GUT with spacetime.The most straightforward one breaks from so , since Section (III A) found so inside e − . Since e − has su , as a subalgebra, su with spacetime can be recovered, as su , ∼ so , . The time-like su allows for the cohomologydescription of the fermions [96, 171]. Additionally, su , ⊕ su is also a subalgebra of e − . This provides at leastthree distinct ways to break from e − to su ⊕ so , , (cid:37) : so , → so ⊕ su , (cid:38) e − → : su ⊕ su , → su ⊕ so , ⊕ u (cid:38) : su , (cid:37) (3.11)0Breaking from e − through so and to su gives : e − → so , → so ⊕ so , → su ⊕ so , ⊕ u , (3.12) = ⊕ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , )= ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ⊕ ( , ) − ⊕ ( , ) . As shown in Eq. (1.3), ( , ) − of su ⊕ so , contains left-handed quarks, anti-up quarks, and the positron, while( , ) contains left-handed anti-down quarks and leptons (of a single generation with their mirror fermions). Once su , ∼ so , is obtained, this may be used as the isometry of AdS and broken to dS or simply broken to so , ⊕ so , .Let us start and consider four different paths from su ⊕ su , , : e − → ( su ⊕ su , ) I → su ⊕ su , ⊕ u → su ⊕ so , ,a ⊕ u ⊕ so , , su ⊕ so , ,b ⊕ u ⊕ so , , ( su ⊕ su , ) II → su ⊕ su , ⊕ u → su ⊕ so , ,a ⊕ u ⊕ so , , su ⊕ so , ,b ⊕ u ⊕ so , , (3.13)where I : (cid:26) e − → su ⊕ su , , = ( , ) + ( , ) + ( , ) + (cid:0) , (cid:1) + (cid:0) , (cid:1) + (cid:0) , (cid:1) , (3.14) II : (cid:26) e − → su ⊕ su , , = ( , ) + ( , ) + ( , ) + (cid:0) , (cid:1) + (cid:0) , (cid:1) + (cid:0) , (cid:1) . (3.15)The two a ∼ su in e are not equivalent, thus their embedding in e is not symmetric under the exchange of them.Moreover, a : su , → so , ⊕ so , , = ( , ) ⊕ ( , ) − , = ( , ) − ⊕ ( , ) , = ( , ) ⊕ ( , ) ⊕ ( , ) − , (3.16) b : su , → so , ⊕ so , , = ( , ) − ⊕ ( , ) , = ( , ) ⊕ ( , ) − , = ( , ) ⊕ ( , ) ⊕ ( , ) − . (3.17)The two a inside su are not equivalent, if one consider the charge with respect to T ; thus, the embedding of a + a + T into a is not symmetric under the exchange of the two a ’s. Note that the exchange a ↔ b is equivalentto multiplying the weight associated with so , by − of e − goes as follows, .I.a : = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) = ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ ( , ) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) = ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ (cid:0) , , (cid:1) , − ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) − , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) − , − ⊕ ( , , ) − , ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , − . (3.18)1 .II.a : = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) = ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ ( , ) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ ( , ) − ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) ⊕ ( , ) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − = ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , − ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) − , ⊕ ( , , ) − , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ ( , , ) , − . (3.19)The same calculations were also worked out for .I.b and .II.b , which found the same results above except withopposite weights. Thus, it holds that = .I = .II | ↔ , ↔ , (3.20)where the subscript ↔ , ↔ refers to the representations of su .Next, we consider su , , : e − → su , → su ⊕ su , ⊕ u → su ⊕ so , ,a ⊕ u ⊕ so , ; su ⊕ so , ,b ⊕ u ⊕ so , , (3.21)where, as above, there are two non-equivalent embeddings of so , ⊕ so , into su , . The branching of the of e − goes as follows, .a : = ⊕ ⊕ = ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ ( , ) − ⊕ ( , ) = ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) − , − ⊕ ( , , ) − , ⊕ ( , , ) , ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) − , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , − ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) − , − ⊕ ( , , ) − , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , − ⊕ ( , , ) , . (3.22) .b : = ⊕ ⊕ = ( , ) ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ ( , ) ⊕ ( , ) − ⊕ ( , ) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) − ⊕ ( , ) − ⊕ ( , ) = ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ ( , , ) , − ⊕ ( , , ) , ⊕ ( , , ) − , ⊕ (cid:0) , , (cid:1) − , ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , ⊕ (cid:0) , , (cid:1) , − ⊕ (cid:0) , , (cid:1) − , − ⊕ (cid:0) , , (cid:1) − , ⊕ ( , , ) − , ⊕ ( , , ) − , ⊕ ( , , ) − , − ⊕ ( , , ) , ⊕ ( , , ) , − . (3.23)Thus, it holds .a | Q u → Q u / = .I.b | ↔ , ↔ = .II.b, (3.24) .b | Q u → Q u / = .I.a | ↔ , ↔ = .II.a, (3.25)2where “ Q u → Q u /
3” denotes a rescaling of the charge of u by a factor 1 / To recap:
As shown above, so ⊕ so , , su , , and su ⊕ su , all can lead to the same su ⊕ so , ⊕ u , which canfurther be broken to su ⊕ so , ⊕ u ⊕ so , . Up to a rescaling of the u charges, all of the representations from thesethree paths coincide with the same result. IV. HIGH ENERGY THEORIES FROM FOUR TIME-LIKE DIMENSIONS
Given the reluctancy to study additional time-like dimensions, there may be additional reluctancy to pursue twelvetime-like dimensions. Therefore, in this section, we look to study e − → so , with twelve space-like and fourtime-like dimensions. In this section, we explore two possibilities to work with e − with four times. First, weattempt to connect to graviweak unification with so , spacetime, which demonstrates some internal consistency butmost likely violates the Coleman-Mandula theorem, since we work with a real form of e . Then, we explore so , spacetime, which can be utilized for AdS and have no issues with the Coleman-Mandula theorem. Rather thanfocusing on various GUTs, we primarily focus on so and Pati-Salam GUTs in this section, as this allows for theeasiest comparison with so , ⊕ so , . A. An attempt for SO (3 , When working with so , , it is tempting to isolate so , for spacetime. This, however, leaves behind so , , whichdoes not allow for so or Pati-Salam GUTs. While it could be conceivable that so could overlap with so , maximallyby so or so , this would appear to violate the Coleman-Mandula theorem, as this implies that the spacetime andinternal symmetries would not be a direct product. However, studying gravity as a gauge theory has allowed forclever ways to get around the Coleman-Mandula theorem [82, 167, 172–175]. While it is impossible to break so , to so ⊕ so , , we compare two paths of symmetry breaking that go through so , ⊕ so and so , ⊕ so , and look tosee if there is at least self-consistency with respect to chirality of the so spinors. Since graviweak unification workswith complex so , [82], it’s clear that we can’t recover the full graviweak unification with a real form of e . However,we can take two different paths of symmetry breaking and show how so , spacetime and su ,L ⊕ su ,R of Pati-Salammay overlap.A double gauge theory that acts on spinors from the left and right to allow for gravity on one side and the electroweaksymmetry on the other can be used in a Clifford/geometric algebra formalism, [172, 174, 175] ψ → ψ (cid:48) = LψU, L = e B , U = e iσ z χ , (4.1)where L generates Lorentz transformations in terms of a bivector generator B , σ z is an SU (2) Pauli matrix of weakisospace, and χ is a U (1) hypercharge gauge parameter such that A µ (cid:48) = A µ − ∂ µ χ . This notion of two gauge theoriesacting from different sides bypasses the assumptions of Coleman-Mandula [174]. While it is unclear if the two paths ofsymmetry breaking below are related precisely to graviweak unification or the Clifford/geometric algebra approaches,it is plausible that something similar allows for the Coleman-Mandula theorem to not be violated.The two paths explored are e − ↓ so , (cid:46) (cid:38) so , ⊕ so so , ⊕ so , ↓ ↓ so , ⊕ so ⊕ su L ⊕ su R so , ⊕ so ⊕ so , c ↓ ↓ so , ⊕ so ⊕ su L ⊕ u R so , ⊕ so ⊕ so , c ⊕ so , c ↓ ↓ so , ⊕ so ⊕ u L ⊕ u R so , ⊕ so ⊕ so , c ⊕ so , c ⊕ so , s ↓ so , ⊕ so ⊕ so , c (cid:48) ⊕ u L ⊕ u R (4.2)To clarify, we break su L ⊕ su R to u L ⊕ u R to compare with so , c ⊕ so , s , where the subscripts c and s referto charge space and spacetime. We also establish that so , c = so , c (cid:48) , which allows for the bottom of the left chainabove to be related to the bottom of the right chain above.3Starting with the left chain in Eq. (4.2) and omitting, here and below, intermediate breakings for brevity, e − → so , → so , ⊕ so → so , ⊕ su ⊕ su L ⊕ su R → so , ⊕ su ⊕ su L ⊕ u R → so , ⊕ su ⊕ u L ⊕ u R → so , ⊕ su ⊕ so , c (cid:48) ⊕ u L ⊕ u R , (4.3) = ⊕ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , )= ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) ⊕ ( , , , ) L ⊕ ( , , , ) L ⊕ ( , , , ) R ⊕ ( , , , ) R = ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) − , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , − , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , , ⊕ ( , , ) , − , ⊕ ( , , ) , , − ⊕ ( , , ) , − , − ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) − , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) − , , ⊕ ( , , ) , − , ⊕ ( , , ) , − , ⊕ ( , , ) − , − , ⊕ ( , , ) , , − ⊕ ( , , ) , , − ⊕ ( , , ) − , , − ⊕ ( , , ) , − , − ⊕ ( , , ) , − , − ⊕ ( , , ) − , − , − ⊕ ( , , ) L , , ⊕ ( , , ) LM − , , ⊕ ( , , ) L , − , ⊕ ( , , ) LM − , − , ⊕ ( , , ) L , , ⊕ ( , , ) LM − , , ⊕ ( , , ) L , , − ⊕ ( , , ) LM − , , − ⊕ ( , , ) R , , ⊕ ( , , ) RM − , , ⊕ ( , , ) R , − , ⊕ ( , , ) RM − , − , ⊕ ( , , ) R , , ⊕ ( , , ) RM − , , ⊕ ( , , ) R , , − ⊕ ( , , ) RM − , , − , where the superscripts L, L, R, and R denote ψ L , ψ L , ψ R , and ψ R of Pati-Salam, respectively, and M corresponds tomirror fermions.Next, we focus on the right chain in Eq. (4.2), e − → so , → so , ⊕ so , → so , ⊕ so ⊕ so , c → so , ⊕ so ⊕ so , c ⊕ so , c → so , ⊕ so ⊕ so , c ⊕ so , c ⊕ so , s , (4.4) = ⊕ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( (cid:48) , (cid:48) )= ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , − , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , − , ⊕ ( , , ) − , , ⊕ ( , , ) − , , ⊕ ( , , ) − , − , ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , − , ⊕ ( , , ) , − , ⊕ ( , , ) , − , − ⊕ ( , , ) , , ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) − , , ⊕ ( , , ) − , , ⊕ ( , , ) − , , − ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , − , ⊕ ( , , ) , − , − ⊕ ( , , ) − , , ⊕ ( , , ) − , , − ⊕ ( , , ) − , − , ⊕ ( , , ) − , − , − ⊕ ( , , ) , , ⊕ ( , , ) , , − ⊕ ( , , ) , − , ⊕ ( , , ) , − , − ⊕ ( , , ) − , , ⊕ ( , , ) − , , − ⊕ ( , , ) − , − , ⊕ ( , , ) − , − , − Comparing Eqs. (4.3) and (4.4), the representations are identical, besides the charges. However, they can be relatedto each other by Q L = 12 ( Q c − Q s ) , Q R = 12 ( Q c + Q s ) , (4.5)where Q L and Q R are the charges from Eq. (4.3), while Q c and Q s are the charges from Eq. (4.4). Furthermore,the weights associated with so , c and so , c (cid:48) are identical. This demonstrates that the two paths from Eq. (4.2) areidentical up to rescaling of the charges shown above.Next, we would like to ensure that the fermionic spinors have a self-consistent chirality with respect to so and so , . Since it was impossible to isolate so ⊕ so , , the right path in Eq. (4.4) broke so , spacetime to so , ⊕ so , s ,which allowed for us to ensure that the representations matched with those found in Eq. (4.3). This so , is notspacetime, however, so we look to break so , in two paths to understand the appropriate chiralities with respect to so , , ( A ) : so , ⊕ so , c ⊕ so , so , ⊕ so , c (cid:37)(cid:38) (cid:38)(cid:37) so , c ⊕ so , s ⊕ so , ⊕ so , s ( B ) : so , ⊕ so , c ⊕ so , s (4.6)4The so , c factor is included to help keep track of mirror fermions. Disregarding so and its associated reps, we focuson a subclass of fermions that corresponds to the ones studied in Section (II), giving32 = ⊕ ⊕ ⊕ M − ⊕ (cid:48) ⊕ (cid:48) M − (4.7)where these representations are for so , ⊕ so , c . Note that 32 is not a formal representation for any algebra, but weuse it as a compact way to refer to these 32 degrees of freedom.Focusing on the top path ( A ),( A ) : so , ⊕ so , c ⊕ so , → so , c ⊕ so , s ⊕ so , ⊕ so , s , (4.8)32 = ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) L , ⊕ ( , ) R , − ⊕ ( , ) LM − , ⊕ ( , ) RM − , − ⊕ ( , ) R , ⊕ ( , ) L , − ⊕ ( , ) RM − , ⊕ ( , ) LM − , − = , , , ⊕ , , , ⊕ , − , , ⊕ , , , ⊕ , , , ⊕ , , , − ⊕ , , , ⊕ , , , ⊕ , , , ⊕ , , , − ⊕ , − , , ⊕ , − , , − ⊕ , , − , ⊕ , , − , − ⊕ , − , − , ⊕ , − , − , − ⊕ L , , , ⊕ L , − , , ⊕ R , , − , ⊕ R , , − , − ⊕ LM − , , , ⊕ LM − , − , , ⊕ RM − , , − , ⊕ RM − , , − , − ⊕ R , , , ⊕ R , , , − ⊕ L , , − , ⊕ L , − , − , ⊕ RM − , , , ⊕ RM − , , , − ⊕ LM − , , − , ⊕ LM − , − , − , . This allows us to ensure which fermionic degrees of freedom are associated with left and right chiralities, which arelabelled by the superscripts L and R .Focusing on the bottom path ( B ),( B ) : so , ⊕ so , c ⊕ so , s → so , c ⊕ so , s ⊕ so , ⊕ so , s , (4.9)32 = ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , ⊕ ( , ) , − ⊕ ( , ) , ⊕ ( , ) L , ⊕ ( , ) R , − ⊕ ( , ) LM − , ⊕ ( , ) RM − , − ⊕ ( , ) R , ⊕ ( , ) L , − ⊕ ( , ) RM − , ⊕ ( , ) LM − , − = , , , ⊕ , , , ⊕ , , − , ⊕ , , , ⊕ , , , ⊕ , , , − ⊕ , , , ⊕ , , , ⊕ , , , ⊕ , , , − ⊕ , , − , ⊕ , , − , − ⊕ , − , , ⊕ , − , , − ⊕ , − , − , ⊕ , − , − , − ⊕ L , , , ⊕ L , , − , ⊕ R , − , , ⊕ R , − , , − ⊕ LM − , , , ⊕ LM − , , − , ⊕ RM − , − , , ⊕ RM − , − , , − ⊕ R , , , ⊕ R , , , − ⊕ L , − , , ⊕ L , − , − , ⊕ RM − , , , ⊕ RM − , , , − ⊕ LM − , − , , ⊕ LM − , − , − , . Since the two paths ( A ) and ( B ) lead to the same representations (so long as the two u charges are swapped),we can understand how to label the fermionic roots with L and R superscripts for chirality of so , , even thoughrepresentations of so , are found. As it turns out, ( , ) of so , corresponds to a left-handed chirality.Now, we look back at Eq. (4.3) and remember that the of so with positive so , c (cid:48) weight are left-handed,while the negative weight gives mirrors that would be right-handed. Furthermore, we look at Eq. (4.4) and see thatrepresentations of so , allow for us to identify chiralities of the fermions. Comparing the representations from bothpaths allows us to determine that the chiralities of the fermions are self-consistent in the sense that the fermionicrepresentations of so have the desired chiralities as would be found with so , .In closing of this subsection, we do not make any claims whether e − can be utilized in this way to give arealistic model that bypasses the Coleman-Mandula theorem, but figured it was worthwhile to at least demonstratesimilarities with graviweak unification [82–91]. Graviweak unification does allow for a nontrivial way to have theweak force overlapping with spacetime without violating the Coleman-Mandula theorem, so perhaps this work willbe inspirational for future work to address if e − can be used in this manner. B. Spacetime from
AdS To refer to the gauge symmetry via Pati-Salam or so GUT, so , must be broken to so , , as a time-like AdS for three generations, or so , spacetime for a single generation with so , charge algebra internally, which can givean AdS × AdS dual symmetry. Alternatively, SO (10 ,
2) can be thought of as the conformal symmetry of SO (9 , so is a high energy GUT,considering this in 3D spacetime may allow for low-energy physics in 4D.5Breaking from e − to so , ⊕ so ⊕ u gives e − → so , → so , ⊕ so , → so , ⊕ so ⊕ u = ⊕ = ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , (cid:48) ) (4.10)= ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − ⊕ ( , , ) ⊕ ( , , ) − . From here, breaking to SU (5) or Pati-Salam GUTs can be pursued to lead to the standard model. While AdS seemsbizarre in this case, AdS is a possibility given its relation to D = 3 gravity with a CFT boundary theory. C. Branes and GUT symmetry breaking
Next, Pati-Salam GUT and the standard model emerging from SO (10) is placed in a modern string perspective.The D3-brane in D = 9 + 1 type-IIB supergravity, which comes from F-theory, has a near-horizon geometry of AdS × S [165]. It was shown by Sezgin, Rudychev, and Sundell that the D = 11 + 3, (1 ,
0) superalgebra can reduceto the D = 9 + 1 type IIA, IIB and heterotic superalgebras, as well as the N = 1 superalgebras for D = 11 + 1 and10+1 [108, 109]. The D = 11 + 3, (1 ,
0) superalgebra supports a 3-brane and 7-brane, where the 3-brane can reduceto the 3-brane of F-theory in D = 11 + 1 along a single time projection. In D = 11 + 1, the 3-brane near horizongeometry is AdS × S , while the 7-brane has AdS × S near horizon geometry. We can see these geometries frombreaking SO (12 , → SO (4 , × SO (8) or breaking SO (12 , → SO (8 , × SO (4), respectively. Projecting to a1-brane slice of the 3-brane, one recovers AdS × S near horizon geometry, which can be recovered from breaking SO (12 , → SO (2 , × SO (10). This projection can be done three different ways along each spatial direction of the3-brane.If one reduces S of AdS × S with respect to S of the 7-brane near horizon geometry, one has the isometrybreaking SO (10) → SO (6) × SO (4), giving the sphere decomposition S → S × S . From here, projecting S → CP induces SO (6) ∼ SU (4) → SU (3) × U (1) B − L , while projecting S → CP induces SO (4) → SU (2) × U (1). Both ofthese have corresponding fibrations S S −−→ CP , (4.11) S S −−→ S ∼ CP , over S fibers, as SU (3) = Isom( CP ), SU (2) = Isom( CP ), and U (1) = Isom( S ). This provides a consistentgeometric justification for the breaking of so GUT symmetry down to the standard model.
V. CONCLUSIONS
In summary, we have demonstrated that the most exceptional Lie algebra e has precisely one non-compact realform, namely the quaternionic form e − , that allows for the combination of spacetime with various GUTs. Inparticular, so , could be used as a conformal symmetry or for AdS with twelve-time models, while so , can befound as the isometry of AdS with four-time models. Both pictures may allow for a holographic description, leadingto AdS /CF T and AdS /CF T holography. The twelve-time models may naturally reduce AdS to dS to relateto our physical universe. The four-time models may allow for a computationally tractable way to stitch together 3Dgravity results to obtain 4D physics, as a 4D Riemannian manifold with local affine charts can be regarded as affinetransformations of copies of CP . Vertex operator algebras may be useful for stitching together multiple copies of CP to obtain 4D gravity from 3D.We obtained SO (10), SU (5), and Pati-Salam GUTs in both time conventions, giving a class of generalizd graviGUTmodels [92]. One of which was previously mentioned [60], and we expand on this by demonstrating how a Higgs scalarwith three generations can be found. A new path of unification that bypasses SO (10) and goes directly to Pati-SalamGUT was proposed, which may allow for a high energy theory that has no proton decay. Another new path for SU (5)GUT with spacetime was also found that starts from su , or su ⊕ su , .While E and exceptional Jordan algebras are found in these models, these seem to differ from various approaches,such as E GUT [32, 33, 140] and recent attempts to connect J , O to the standard model [137–139, 177–179]. Instead,6we find the Peirce decomposition to give bosons and fermions, rather than only fermions . This is intuitive from theperspective of string theory. It still remains an open question if these class of e − models suggest a new e GUT,or if the e algebra allows for a convenient packaging of GUTs, similar to e − .In future work, we look to establish a Lagrangian formalism for at least one of these models. While it may appearthat these models contain many additional bosonic degrees of freedom outside the standard model and spacetime,this may not be the case. Note that these models account for all of the off-shell degrees of freedom of the fermions,yet the symmetry-breaking analysis of GUTs merely counts the adjoint degrees of freedom, not the bosonic off-shelldegrees of freedom. It may turn out that e − nontrivially accounts for off-shell bosonic degrees of the standardmodel as well, which warrants more careful study in future work.Various phenomenological aspects such as neutrino masses and mass/flavor mixing also warrant additional study.Since the mirror electron was identified as borrowing on-shell degrees of freedom from the muon and tau, it isconceivable that e − may also allow for mass and flavor eigenstates.Additionally, the exploration of charge space and its relevance for the origins of the double copy [181] and KLTrelations [162] is warranted, as a dual Lorentz symmetry [163, 164] is found between spacetime and charge space. Thenotion of so triality [182] and so , charge space is suggestive of a new type of supersymmetry. If these models do allowfor supersymmetry, it is clear that it is a type of charge space supersymmetry, rather than spacetime supersymmetry.This seems to differ, as additional unphysical superparticles are not needed to be introduced. This may suggest away to break spacetime supersymmetry while preserving a charge space (i.e. internal) supersymmetry. However, itstill remains unclear if these models actually contain supersymmetry or not, which should be investigated further.Finally, the magic star algebras occurring in exceptional periodicity allow for a natural way to generalize e − that is distinct from the infinite-dimensional Kac-Moody algebras [157, 183]. Given their apparent ability to describea monstrous M-theory that adds fermions to bosonic M-theory [96], further work is warranted to study BSM physicsin relation to brane dynamics similar to those studied in generalizations of M-theory, such as F-theory and beyond [81,95, 109, 184–186]. ACKNOWLEDGEMENTS
Thanks to Piero Truini for encouraging discussions.
Appendix A: On removing ghosts
Multiple time dimensions are often thought to introduce problematic ghosts. However, Bars et al. has extensivelyshown how so , ∼ sp , R can be used for two times with signature so , via a conformal shadow [187–193]. Thesymplectic groups are associated with Poisson brackets, and this conformal shadow treats position and momentum asbeing indistinguishable, warranting the study of phase space.For example, Bars introduced an Sp (2 , R ) gauge field A ij as a 2 × Q ij . The simplest two-time worldline action is given by [190, 192] S = (cid:90) dτ (cid:18) ∂ τ X µ P µ ( τ ) − A ij ( τ ) Q ij ( X ( τ ) , P ( τ )) (cid:19) , (A1)where µ runs over the six dimensions associated with so , . The field Q ij must satisfy the constraints of the sp , R Liealgebra, which for gravity takes the form [192] Q ij = (cid:18) W ( x ) V µ ( x ) P µ V µ ( x ) P µ G µν P µ P ν (cid:19) . (A2)This leads to the introduction of an auxiliary scalar W ( x ) that helps remove the ghosts in two-time physics, whilethe constraints lead to a solution for V µ ( x ) = ∂ µ W ( x ) /
2. The auxiliary field Q ij in flat spacetime is found to be Q ij = (cid:18) X µ X µ X µ P µ X µ P µ P µ P µ (cid:19) . (A3) Recent work by and private communications with Dubois-Violette and Todorov [180] suggest that the appropriate utilization of so inRefs. [137–139, 177, 178] is similar to the so inside so , discussed in Eq. (4.2). δ ( W ) to the Lagrangiansto remove the ghosts [192, 193].Generalizing Bars to three times via two conformal shadows suggests the study of so , spacetime with so , ∼ sp , R .While working out these details is outside the scope of this current work, it is conjectured that adding an additionalconformal shadow warrants the introduction of δ ( W ) δ ( W ) into the Lagrangians. An Sp (4 , R )-invariant auxiliaryfield for gravity in curved spacetime is suggested.This approach to three-time physics seems plausible, but may lead to unwanted complexities, as Sp (4 , R ) requiresthe solution of 45 equations, rather than 3 found with Sp (2 , R ). To further simplify, it may be possible to use so ,F asa gauge group rather than so , , substituting so , (studied in Section (II)) for so , . This so ,F [158–161, 194] seemsto be associated with three times and three mass generations.The use of so , spacetime for three generations seems to allow for the ability to remove ghosts via a time-like so ,F family unification as a gauge theory. Also, supersymmetry has been used by Sezgin for three-time physics to removeghosts [80]. If so , is needed for conformal shadows, then it is also conceivable to use the recently introduced EPalgebra f (2)4 , which contains so as a maximal subalgebra and could extend the models in Section (III) by one spatialdimension. [1] J. C. Pati and A. Salam, Phys. Rev. D10 , 275 (1974), [Erratum: Phys. Rev.D11,703(1975)].[2] J. C. Pati, Int. J. Mod. Phys.
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