Study Notes on Gravito-Electroweak Unification using E8 Algebra, Notas de estudo de Física

Baixe Study Notes on Gravito-Electroweak Unification using E8 Algebra e outras Notas de estudo em PDF para Física, somente na Docsity! Preprint typeset in JHEP style - HYPER VERSION An Exceptionally Simple Theory of Everything A. Garrett Lisi SLRI, 722 Tyner Way, Incline Village, NV 89451 E-mail: alisi@hawaii.edu Abstract: All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold. Keywords: ToE. ar X iv :0 71 1. 07 70 v1 [ he p- th ] 6 N ov 2 00 7 Contents 1. Introduction 1 1.1 A connection with everything 2 2. The Standard Model Polytope 4 2.1 Strong G2 5 2.2 Graviweak F4 8 2.2.1 Gravitational D2 8 2.2.2 Electroweak D2 10 2.2.3 Graviweak D4 11 2.2.4 F4 13 2.3 F4 and G2 together 14 2.4 E8 16 2.4.1 New particles 21 2.4.2 E8 triality 22 3. Dynamics 23 3.1 Curvature 23 3.2 Action 25 3.2.1 Gravity 25 3.2.2 Other bosons 26 3.2.3 Fermions 27 4. Summary 28 5. Discussion and Conclusion 28 1. Introduction We exist in a universe described by mathematics. But which math? Although it is inter- esting to consider that the universe may be the physical instantiation of all mathematics,[1] there is a classic principle for restricting the possibilities: The mathematics of the universe should be beautiful. A successful description of nature should be a concise, elegant, unified mathematical structure consistent with experience. Hundreds of years of theoretical and experimental work have produced an extremely successful pair of mathematical theories describing our world. The standard model of parti- cles and interactions described by quantum field theory is a paragon of predictive excellence. – 1 – 2. The Standard Model Polytope The structure of a simple Lie algebra is described by its root system. An N dimensional Lie algebra, considered as a vector space, contains an R dimensional subspace, a Cartan subalgebra, spanned by a maximal set of R inter-commuting generators, Ta, [Ta, Tb] = TaTb − TbTa = 0 ∀ 1 ≤ a, b ≤ R (R is the rank of the Lie algebra) Every element of the Cartan subalgebra, C = CaTa, acts linearly on the rest of the Lie algebra via the Lie bracket (the adjoint action). The Lie algebra is spanned by the eigenvectors of this action, the root vectors, Vβ, with each corresponding to an eigenvalue, [C, Vβ] = αβVβ = ∑ a iCaαaβVβ Each of the (N−R) non-zero eigenvalues, αβ, (imaginary for real compact groups) is linearly dependent on the coefficients of C and corresponds to a point, a root, αaβ, in the space dual to the Cartan subalgebra. The pattern of roots in R dimensions uniquely characterizes the algebra and is independent of the choice of Cartan subalgebra and rotations of the constituent generators. Since the root vectors, Vβ, and Cartan subalgebra generators, Ta, span the Lie algebra, they may be used as convenient generators — the Cartan-Weyl basis of the Lie algebra, A = ABTB = AaTa +AβVβ The Lie bracket between root vectors corresponds to vector addition between their roots, and to interactions between particles, [Vβ, Vγ ] = Vδ ⇔ αβ + αγ = αδ ⇔
γ β δ (2.1) Elements of the Lie algebra and Cartan subalgebra can also act on vectors in the various representation spaces of the group. In these cases the eigenvectors of the Cartan subalge- bra (called weight vectors) have eigenvalues corresponding to the generalized roots (called weights) describing the representation. From this more general point of view, the roots are the weights of the Lie algebra elements in the adjoint representation space. Each weight vector, Vβ, corresponds to a type of elementary particle. The R coordinates of each weight are the quantum numbers of the relevant particle with respect to the chosen Cartan subalgebra generators. – 4 – G2 Vβ g3 g8 •◦ grḡ (T2 − iT1) 1 0 •◦ gr̄g (−T2 − iT1) −1 0 •◦ grb̄ (T5 − iT4) 1/2 √3/2 •◦ gr̄b (−T5 − iT4) −1/2 −√3/2 •◦ gḡb (−T7 − iT6) 1/2 −√3/2 •◦ ggb̄ (T7 − iT6) −1/2 √3/2 NM qr [1, 0, 0] 1/2 1/2√3 HO q̄r [1, 0, 0] −1/2 −1/2√3 NM qg [0, 1, 0] −1/2 1/2√3 HO q̄g [0, 1, 0] 1/2 −1/2√3 NM qb [0, 0, 1] 0 −1/√3 HO q̄b [0, 0, 1] 0 1/√3 g3 g8 Table 1: The su(3) weight vectors and weight coordinates of the gluon, quark, and anti-quark weights form the G2 root system. 2.1 Strong G2 The gluons, g ∈ su(3), in the special unitary group of degree three may be represented using the eight Gell-Mann matrices as generators, g = gATA = gA i2λA = C + g βVβ = = i2  g 3+ 1√ 3 g8 g1−ig2 g4−ig5 g1+ig2 −g3+ 1√ 3 g8 g6−ig7 g4+ig5 g6+ig7 −2√ 3 g8  =  i 2g 3+ i 2 √ 3 g8 grḡ grb̄ gr̄g −i2 g 3+ i 2 √ 3 g8 ggb̄ gr̄b gḡb −i√ 3 g8  (2.2) The Cartan subalgebra, C = g3T3 + g8T8, is identified with the diagonal. This gives root vectors — particle types — corresponding to the six non-zero roots, such as [C, Vggb̄ ] = i ( g3 (−1/2) + g8( √ 3/2) ) Vggb̄ Vggb̄ = (T7 − iT6) =  0 0 00 0 1 0 0 0  ggb̄ = ggb̄Vggb̄ = i2(g6 − ig7)Vggb̄ =  0 0 00 0 ggb̄ 0 0 0  for the green anti-blue gluon. (By an abuse of notation, the coefficient, such as ggb̄, has the same label as the particle eigenvector containing the coefficient, and as the root — the usage is clear from context.) Since the Cartan subalgebra matrix in the standard representation acting on 3, and its dual acting on 3̄, are diagonal, the weight vectors, Vβ and V̄ β, satisfying CVβ = ∑ a igaαaβVβ and C̄V̄ β = −CT V̄ β = ∑ a igaαaβV̄ β – 5 – are the canonical unit vectors of the 3 and 3̄. The weights for these — the su(3) quantum numbers of the quarks and anti-quarks — can be read off the diagonals of C and C̄ = −CT = −C. The set of weights for su(3), the defining 3, and its dual 3̄, are shown in Table 1. These weights are precisely the 12 roots of the rank two simple exceptional Lie group, G2. The weight vectors and weights of the 3 and 3̄ are identified as root vectors and roots of G2. The G2 Lie algebra breaks up as g2 = su(3) + 3 + 3̄ allowing a connection to be separated into the su(3) gluons, g, and the 3 and 3̄ quarks and anti-quarks, . q and .̄ q, related by Lie algebra duality. All interactions (2.1) between gluons and quarks correspond to vector addition of the roots of G2, such as [Vgrḡ , Vqg ] = Vqr ⇔ grḡ + qg = qr ⇔ (1, 0) + (−12 , 1 2 √ 3 ) = (12 , 1 2 √ 3 ) ⇔
qg grḡ qr We are including these quarks in a simple exceptional Lie algebra, g2, and not merely acting on them with su(3) in some representation. The necessity of specifying a representation for the quarks has been removed — a significant simplification of mathematical structure. And we will see that this simplification does not occur only for the quarks in g2, but for all fermions of the standard model. Just as we represented the gluons in the (3× 3) matrix representation (2.2) of su(3), we may choose to represent the gluons and quarks using the smallest irreducible, (7× 7), matrix representation of g2,[6] g + q + q̄ =  0 −1√ 2 q̄b −1√ 2 qb −1√ 2 qr −1√ 2 q̄r −1√ 2 qg −1√ 2 q̄g 1√ 2 qb i√ 3 g8 0 1√ 2 q̄g gr̄b −1√ 2 q̄r −gḡb 1√ 2 q̄b 0 −i√ 3 g8 grb̄ 1√ 2 qg −ggb̄ −1√ 2 qr 1√ 2 q̄r −1√ 2 qg −gr̄b i2g 3+ i 2 √ 3 g8 0 gr̄g 1√ 2 qb 1√ 2 qr −grb̄ −1√ 2 q̄g 0 −i2 g 3− i 2 √ 3 g8 1√ 2 q̄b grḡ 1√ 2 q̄g 1√ 2 qr gḡb −grḡ −1√ 2 qb −i2 g 3+ i 2 √ 3 g8 0 1√ 2 qg ggb̄ 1√ 2 q̄r −1√ 2 q̄b −gr̄g 0 i2g 3− i 2 √ 3 g8  (2.3) Squaring this matrix gives all interactions between gluons and quarks, equivalent to su(3) acting on quarks and anti-quarks in the fundamental representation spaces. The G2 root system may also be described in three dimensions as the 12 midpoints of the edges of a cube — the vertices of a cuboctahedron. These roots are labeled g and qIII in Table 2, with their (x, y, z) coordinates shown. These points may be rotated and scaled,B2g3 g8  = 1√ 2  −1√ 3 −1√ 3 −1√ 3 −1√ 2 1√ 2 0 −1√ 6 −1√ 6 √ 2√ 3   xy z  (2.4) – 6 – The d2 = so(3, 1) = Cl2(3, 1) valued gravitational spin connection is written using the six Clifford bivector generators, γµν = 12 [γµ, γν ], as ω = 12ω µνγµν = [ (12ω επ τεπ −iωτ4)iστ 0 0 (12ω επ τεπ +iω τ4)iστ ] = = [ (ωτS−iωτT )iστ 0 0 (ωτS+iω τ T )iστ ] = [ ωL 0 0 ωR ] (2.8) with six real coefficients redefined into the spatial rotation D2G 12iω 3 T 1 2ω 3 S 1 2ω 3 L 1 2ω 3 R •◦ ω∧L 1 1 1 0•◦ ω∨L −1 −1 −1 0•◦ ω∧R −1 1 0 1•◦ ω∨R 1 −1 0 −1  e∧S 0 1 1/2 1/2  e∨S 0 −1 −1/2 −1/2  e∧T 1 0 −1/2 1/2  e∨T −1 0 1/2 −1/2 NM f∧L 1/2 1/2 1/2 0 NM f∨L −1/2 −1/2 −1/2 0 NM f∧R −1/2 1/2 0 1/2 NM f∨R 1/2 −1/2 0 −1/2 Table 3: Gravitational D2 weights for the spin connection, frame, and fermions, in two coor- dinate systems. and temporal boost parts, ωτS = 1 2ω επ τεπ ω τ T = ω τ4 These relate to the left and right-chiral (selfdual and anti- selfdual) parts of the spin connection, ωL/R = (ω τ L/R)iστ = ωS ∓ iωT which are sl(2,C) valued but not independent, ωτR = ωτ∗L . The Cartan subalgebra of gravity, in several different coordinates, is C = ω12γ12 + ω34γ34 = ω3Sγ12 + ω 3 Tγ34 = = [ (ω3S−iω3T )iσ3 0 0 (ω3S+iω 3 T )iσ3 ] = [ ω3Liσ3 0 0 ω3Riσ3 ] Taking the Lie bracket with C gives root vectors and roots for the spin connection, such as[ C, 14(−γ13 + γ14 − iγ23 + iγ24) ] = i ( ω3S(2) + 1 iω 3 T (2) ) 1 4(−γ13+γ14−iγ23+iγ24) for ω∧L, and weight vectors and weights for the frame, such as[ C, i2(γ3 − γ4) ] = i ( 1 iω 3 T (2) ) i 2(γ3 − γ4) for e∧T . The fermions, such as the left-chiral spin-up up quark, .u ∧ L, are in the 4 of the spinor representation space (2.8) with weight vectors, such as [1, 0, 0, 0], equal to the canonical unit vectors, and weights read off the diagonal of C. The collection of fields and their weights are shown in Table 3. The two coordinate systems in the table are related by a π4 rotation and scaling, [ ω3L ω3R ] = 1√ 2 [ 1√ 2 1√ 2 −1√ 2 1√ 2 ][ 1 iω 3 T ω3S ] = [ 1 2( 1 iω 3 T + ω 3 S) 1 2(− 1 iω 3 T + ω 3 S) ] (2.9) – 9 – Unlike other standard model roots, the roots of so(3, 1) are not all imaginary — the coordinates along the ω3T axis are real. The Spin +(3, 1) Lie group of gravity, with Lie algebra so(3, 1), is neither simple nor compact — it is isomorphic to SL(2,C) = SL(2,R)×SL(2,R). According to the ADE classification of Lie groups it is still labeled D2 — the same as Spin(4) = SU(2)× SU(2) — since it has the same root system, albeit with one real axis. 2.2.2 Electroweak D2 The electroweak gauge field, W ∈ su(2)L, acts on left-chiral doublets, such as [ .uL, .dL]. The Pati-Salam GUT introduces a partner to this field, B1 ∈ su(2)R, acting on all right-chiral fermion doublets. Part of this field, B31 i 2σ3 ∈ u(1)R, joins with the u(1)B−L complement, B2, of the strong su(3) to give the electroweak B ∈ u(1)Y . The left-right electroweak partner fields may be joined in a d2 partner to gravity, so(4) = su(2)L + su(2)R Since both W and B1 act on the Higgs doublet, [φ+, φ0], it is sensible to consider the 4 real fields of this Higgs doublet to be components of a vector acted on by the so(4). This suggests we proceed as we did for gravity, using a complementary chiral matrix representation for the four orthonormal basis vectors of Cl(4), γ′1 = σ1 ⊗ σ1 γ′2 = σ1 ⊗ σ2 γ′3 = σ1 ⊗ σ3 γ′4 = σ2 ⊗ 1 These allow the Higgs vector field to be written as φ = φµγ′µ = [ 0 −iφ4+φεσε iφ4+φεσε 0 ] =  0 0 −φ1 φ+ 0 0 φ− φ0 −φ0 φ+ 0 0 φ− φ1 0 0  ∈ Cl1(4) with coefficients equal to those of the Higgs doublet, φ+ = φ1 − iφ2 φ− = φ1 + iφ2 φ0 = −φ3 − iφ4 φ1 = −φ3 + iφ4 The d2 = so(4) = Cl2(4) valued electroweak connection breaks up into two su(2) parts, wew = 12w µν ewγ ′ µν = [ (V τ+U τ ) i2στ 0 0 (V τ−U τ ) i2στ ] = [ W τ i2στ 0 0 Bτ1 i 2στ ] The U and V fields are analogous to the 1iωT and ωS of gravity, and are related to the electroweak W and B1, analogous to the ωL and ωR, by the same π4 rotation and scaling (2.9). The Cartan subalgebra, C = 14(W 3 +B31)γ ′ 12 + 1 4(W 3 −B31)γ′34 = i 2  W 3 0 0 0 0 −W 3 0 0 0 0 B31 0 0 0 0 −B31  – 10 – gives root vectors and roots for the electroweak fields, D2ew W 3 B31 √ 2√ 3 B2 1 2Y Q •◦ W+ 1 0 0 0 1 •◦ W− −1 0 0 0 −1 •◦ B+1 0 1 0 1 1•◦ B−1 0 −1 0 −1 −1  φ+ 1/2 1/2 0 1/2 1 ♦ φ− −1/2 −1/2 0 −1/2 −1  φ0 −1/2 1/2 0 1/2 0 ♦ φ1 1/2 −1/2 0 −1/2 0 NM νL 1/2 0 1/2 −1/2 0 NM eL −1/2 0 1/2 −1/2 −1 NM νR 0 1/2 1/2 0 0 NM eR 0 −1/2 1/2 −1 −1 NMNMNM uL 1/2 0 −1/6 1/6 2/3 NMNMNM dL −1/2 0 −1/6 1/6 −1/3 NMNMNM uR 0 1/2 −1/6 2/3 2/3 NMNMNM dR 0 −1/2 −1/6 −1/3 −1/3 Table 4: Weights for electroweak D2, for B2 from Table 2, and electroweak hypercharge and charge. such as W±, and weight vectors and weights for the Higgs, such as[ C, 12(−γ ′ 3+iγ ′ 4) ] = i ( W 3(−1/2)+B31(1/2) ) 1 2(−γ ′ 3+iγ ′ 4) for φ0. The fermions are acted on in the standard 4, equivalent to the independent su(2)L and su(2)R action on left and right-chiral Weyl doublets, such as [uL, dL] and [uR, dR]. The electroweak D2 weights for various fields are shown in Table 4. The two right-chiral gauge fields, B±1 , are not part of the standard model. They are a necessary part of the Pati-Salam GUT, and presumably have large masses or some other mechanism breaking left-right symmetry and impeding their detection. As in the Pati-Salam GUT, the B2 weights from Table 2 and the B31 weights may be scaled and rotated ((2.4) and (2.9)) into two new coordinates, including the weak hypercharge, 1 2Y = B 3 1 − √ 2√ 3 B2 This scaling implies a weak hypercharge coupling constant of g1 = √ 3/5 and Weinberg angle satisfying sin2 θW = 3/8, typical of almost all grand unified theories. There is also a new quantum number partner to the hypercharge, X, corresponding to the positive combination of quantum numbers B31 and B2. The hypercharge may be scaled and rotated with W 3 to give the electric charge, Q = W 3 + 12Y These weights, shown in Table 4, are in agreement with the known standard model quantum numbers, and justify our use of the corresponding particle labels. 2.2.3 Graviweak D4 The electroweak d2 = so(4) and gravitational d2 = so(3, 1) combine as commuting parts of a graviweak d4 = so(7, 1). The 4 Higgs fields, φ, a vector of the electroweak so(4), combine with the 4 gravitational so(3, 1) vectors of the frame, e, into 16 bivector valued fields, eφ, of the graviweak D4 gauge group. This combination is achieved by adding the weights of Table 3 with those of Table 4 to obtain the weights of D4 in four dimensions, as shown in Table 5. The weights of the fermions also add to give their D4 weights. The fermion weights correspond to the fundamental positive-chiral spinor representation space, 8S+, of D4. To construct this explicitly, we use Trayling’s model,[7] and combine – 11 – 8S− 12ω 3 L 1 2ω 3 R W 3 B31 tri B31 1 2ω 3 L W 3 1 2ω 3 R NM ν∧/∨µL 0 ±1/2 1/2 0 NM µ∧/∨L 0 ±1/2 −1/2 0 NM ν∧/∨µR 1/2 0 0 ±1/2 NM µ∧/∨R −1/2 0 0 ±1/2 8V 12ω 3 L 1 2ω 3 R W 3 B31 tri 12ω 3 R B 3 1 W 3 1 2ω 3 L NM ν∧/∨τL 0 0 1/2 ±1/2 NM τ∧/∨L 0 0 −1/2 ±1/2 NM ν∧/∨τR ±1/2 1/2 0 0 NM τ∧/∨R ±1/2 −1/2 0 0 Table 6: The 8S− and 8V weights complete the D4 and 8S+ weight system of Table 5 to form the F4 root system. The 48 roots are projected from four dimensions to two and plotted, with lines shown between triality partners. Guided by this triality symmetry, we will continue to label the triality partners with generation labels — though this should be understood as an idealization of a more complex and as yet unclear relationship between physical particles and triality partners. If we wished, we could write the constituent particles of F4 as matrix elements of its smallest irreducible, 26 dimensional representation,[11] as we did for the gluons and quarks (2.3) in G2. We can also compute particle interactions by adding the roots in Tables 5 and 6, such as e∧L + e ∧ Tφ+ = ν ∧ eR between an electron, a frame-Higgs, and an electron neutrino. These graviweak interactions, described by the structure of F4, do not involve anti-fermions or color; to include all standard model interactions we will have to combine F4 and G2. 2.3 F4 and G2 together The coordinate axes chosen in Tables 5 and 6 are a good choice for expressing the quan- tum numbers for gravity and the electroweak fields, but they are not the standard axes for – 14 – describing the F4 root system. We can rotate to our other coordinate system, { 12iω 3 T , 1 2ω 3 S , U 3, V 3} using a pair of π4 rotations (2.9) and thereby express the 48 roots of F4 in standard coordi- nates, shown in Table 7. These coordinate values are described by various permutations of ±1, ±1/2, and 0; and a similar description of the G2 and U(1) weights from Table 2 is also presented. F4 12iω 3 T 1 2ω 3 S U 3 V 3 perms # •◦ so(7, 1) ±1 ±1 all 24 NM 8S+ ±1/2 ±1/2 ±1/2 ±1/2 even# > 0 8 NM 8S− ±1/2 ±1/2 ±1/2 ±1/2 odd# > 0 8 NM 8V ±1 all 8 G2 + U(1) x y z perms # •◦ su(3) 1 −1 all 6 NMNMNM qI ±1/2 ±1/2 ±1/2 two > 0 3 HOHOHO q̄I ±1/2 ±1/2 ±1/2 one > 0 3̄ NM l −1/2 −1/2 −1/2 one 1 HO l̄ 1/2 1/2 1/2 one 1̄ NMNMNM qII −1 all 3 HOHOHO q̄II 1 all 3̄  qIII 1 1 all 3 ♦ q̄III −1 −1 all 3̄ Table 7: Roots of F4 and weights of Table 2 described with allowed permutations of coordinate values. To completely describe every field in the standard model and gravity we need to combine these two sets of quantum numbers. The graviweak F4 root system includes the two quantum numbers of so(3, 1) gravity and the two of the su(2)L and su(2)R electroweak fields, with three generations of fermions related through so(7, 1) triality (2.11). The G2 weight system includes the three quantum numbers of the su(3) strong fields and a u(1)B−L contributing to hypercharge, with fermions and anti-fermions related through duality (2.6). To match the quantum numbers of all known standard model and gravitational fields, the so(7, 1) of F4 and su(3) and u(1)B−L of G2 + U(1) must act on three generations of 8 fermions for each of the 3 colors of quark, 1 uncolored lepton, and their anti-particles, so(7, 1) + (su(3) + u(1)) + (8 + 8 + 8)× (3 + 3̄ + 1 + 1̄) (2.12) as depicted in the periodic table, Figure 1. The weights of these 222 elements — corresponding to the quantum numbers of all gravitational and standard model fields — exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8. – 15 – 2.4 E8 E8 x1 x2 x3 x4 x5 x6 x7 x8 # so(16) ±1 ±1 all perms 112 16S+ ±1/2 ... even# > 0 128 Table 8: The 240 roots of E8. “E8 is perhaps the most beautiful structure in all of mathematics, but it’s very complex.” — Hermann Nicolai Just as we joined the weights of D2G and D2ew to form the F4 graviweak root system, the weights of F4 and G2 may be joined to form the roots of E8 — the vertices of the E8 polytope — shown in Table 8. Combining these weights in eight dimensions requires the introduction of a new quantum number, w, with values determined by the F4 and G2 numbers. These quantum numbers uniquely identify each root of E8 as an elementary particle — Table 9. E8 12iω 3 T 1 2ω 3 S U 3 V 3 w x y z F4 G2 # •◦ •◦ ω∧/∨L ω∧/∨R ±1 ±1 0 0 0 D2G 1 4•◦ •◦ W± B±1 0 ±1 ±1 0 0 D2ew 1 4 ♦♦ eφ+ eφ− eφ1 eφ0 ±1 ±1 0 0 4× 4 1 16 NMNMNMNM νeL eL νeR eR ±1/2 ... even#>0 −1/2 −1/2 −1/2 −1/2 8S+ l 8 HOHOHOHO ν̄eL ēL ν̄eR ēR ±1/2 ... even#>0 1/2 1/2 1/2 1/2 8S+ l̄ 8 NM NM NM NM uL dL uR dR ±1/2 ... even#>0 −1/2 ±1/2 ... two>0 8S+ qI 24 HO HO HO HO ūL d̄L ūR d̄R ±1/2 ... even#>0 1/2 ±1/2 ... one>0 8S+ q̄I 24 NMNMNMNM νµL µL νµR µR ±1/2 ... odd#>0 −1/2 1/2 1/2 1/2 8S− l 8 HOHOHOHO ν̄µL µ̄L ν̄µR µ̄R ±1/2 ... odd#>0 1/2 −1/2 −1/2 −1/2 8S− l̄ 8 NMNMNMNMNMNMNMNMNMNMNMNM cL sL cR sR ±1/2 ... odd#>0 1/2 ±1/2 ... two>0 8S− qI 24 HOHOHOHOHOHOHOHOHOHOHOHO c̄L s̄L c̄R s̄R ±1/2 ... odd#>0 −1/2 ±1/2 ... one>0 8S− q̄I 24 NMNMNMNM ντL τL ντR τR ±1 1 0 8V 1 8 HOHOHOHO ν̄τL τ̄L ν̄τR τ̄R ±1 −1 0 8V 1 8 NMNMNMNMNMNMNMNMNMNMNMNM tL bL tR bR ±1 0 −1 8V qII 24 HOHOHOHOHOHOHOHOHOHOHOHO t̄L b̄L t̄R b̄R ±1 0 1 8V q̄II 24 •◦ g 0 0 1 −1 1 A2 6  ♦ x1Φ 0 −1 ±1 1 qII 6  ♦ x2Φ 0 1 ±1 1 qII 6  ♦♦♦ x3Φ 0 0 ±(1 1) 1 qIII 6 Table 9: The 240 roots of E8 assigned elementary particle labels according to F4 and G2 subgroups. The E8 root system was first described as a polytope by Thorold Gosset in 1900,[12] and the triacontagonal projection plotted by hand in 1964. This plot,[13] now with elementary particle symbols assigned to their associated roots according to Table 9, is shown in Figure 2, with lines drawn between triality partners. – 16 – Figure 3: The E8 root system, rotated a little from F4 towards G2. – 19 – Figure 4: The E8 root system, rotated a little from G2 towards F4, showing E6. – 20 – Inspecting the e6 subalgebra of e8 reveals how the fermions and anti-fermions — up to now described as living in real representations — are combined in complex representations.[14] The e6 decomposes to graviweak so(7, 1) acting on three complex generations of fermions as e6 = f4 + (8 + 8 + 8)× 1̄ + u(1) + u(1) = so(7, 1) + (8 + 8 + 8)× (1 + 1̄) + u(1) + u(1) = so(9, 1) + u(1) + 16SC in which the final u(1) is the complex structure, i, related to the w quantum number, and the 16SC is a complex spinor acted on by the so(9, 1). Although considering its e6, f4, and g2 subalgebras is useful, the E8 Lie algebra may be broken down to the standard model via a more direct route,[8] e8 = so(7, 1) + so(8) + (8S+× 8S+) + (8S−× 8S−) + (8V × 8V ) = so(7, 1) + (su(3) + u(1) + u(1) + 3× (3 + 3̄)) + (8 + 8 + 8)× (3 + 3̄ + 1 + 1̄) This decomposition is directly visible Figure 5: A periodic table of E8. in Table 9, in which the first four co- ordinate axes are of so(7, 1) and the last four are of so(8). The so(7, 1) decomposes into the graviweak fields, and the so(8) decomposes into strong su(3), u(1)B−L, and new fields via the embedding of su(4) in so(8). A matched triality rotation of so(7, 1) and so(8) relates the three genera- tions of fermions. The Lie algebra structure of E8, and its relation to the structure of the standard model, is depicted in Figure 5 — a periodic table of E8. A comparison of this structure with Figure 1 shows the extremely close fit to the standard model, with only a handful of new particles suggested by the structure of E8. 2.4.1 New particles After all algebraic elements of the standard model have been fit to the E8 Lie algebra there are a few e8 elements remaining, representing new, non-standard particles. There are two new quantum numbers, X and w, representing the Pati-Salam partner to weak hypercharge – 21 – The curvature of this connection, an e8 valued collection of 2-forms and Grassmann 1-forms, =·F = d .A+ 1 2 [ .A, .A] = d .A+ .A .A = = F 1 + = F 2 +D .ΨI +D .ΨII +D .ΨIII (3.2) may be computed and broken up into standard model parts. The so(7, 1) part of the curvature, = F 1 = = FG + = F gw + = F ew includes the gravitational so(3, 1) part, the mixed graviweak 4 × (2 + 2̄) part, and the elec- troweak su(2)L + su(2)R part. The gravitational so(3, 1) part of the curvature is = FG = 12((dω + 1 2ωω) + 1 8eφeφ) = 1 2(=R− 1 8eeφ 2) (3.3) in which = R is the Riemann curvature 2-form, ee is the spacetime area bivector, and φ2 is the amplitude of the Higgs squared. The mixed graviweak 4× (2 + 2̄) part is = F gw = (de+ 12 [ω, e])φ− e(dφ+ [W +B1, φ]) = =Tφ− eDφ (3.4) in which = T is the gravitational torsion and D is the covariant derivative. And the electroweak su(2)L + su(2)R part of the curvature is = F ew = (dW +WW ) + (dB1 +B1B1) = =F W + = FB1 (3.5) The so(8) part of the curvature, = F 2 = = Fw + = FB2 + = F x + = F g + xΦxΦ (3.6) includes the u(1) and u(1)B−L parts, the mixed 3× (3 + 3̄) part, and the strong su(3) part. The last term does not easily separate — xΦxΦ contributes to all three parts of = F 2. The u(1) and u(1)B−L parts are = Fw = dw = FB2 = dB2 The mixed 3× (3 + 3̄) part is = F x = (dx+[w+B2, x])Φ− x(dΦ+[g,Φ]) = (Dx)Φ−xDΦ And the strong su(3) part is = F g = dg + gg Due to the exceptional structure of e8, the fermionic part of the curvature for the first generation is D .Ψ = d .Ψ + [H1 +H2, .Ψ] = (d+ 12ω + 1 4eφ) .Ψ +W .ΨL +B1 .ΨR − .Ψ(w +B2 + xΦ)− .Ψq g with D the covariant massive Dirac derivative in curved spacetime. The second and third fermionic generation parts of this curvature are similar. – 24 – 3.2 Action The most conservative approach to specifying the dynamics is to write down an action agree- ing with the known standard model and gravitational action while satisfying our desire for minimalism. With these two motivations in mind, an action for everything can be economi- cally expressed as a modified BF theory action over a four dimensional base manifold, S = ∫ 〈 = · B =·F + πG 4 =B G = BGγ − = B′ = ∗B′〉 (3.7) in which =·F is the curvature (3.2), = · B = = B + ≡ · B is an e8 valued collection of 2-form and anti- Grassmann 3-form Lagrange multiplier fields, = BG is the so(3, 1) part of = B, = B′ is the rest of = B, γ = Γ1Γ2Γ3Γ4 is the Clifford algebra volume element, ∗ is the Hodge star, and <> takes the scalar part (the trace). After varying = B and plugging it back in (3.7), this action — up to a boundary term — is S = ∫ 〈 ≡ · BD .Ψ +∼e 1 16πGφ 2(R− 32φ 2)− 14 =F ′ = ∗F ′〉 (3.8) in which ∼e is the spacetime volume 4-form, R is the gravitational scalar curvature, and =F ′ is the non-so(3, 1) part of = F1 and =F2. This is recognizable as the action for the standard model and gravity, with a cosmological constant related to the Higgs vacuum expectation value, Λ = 34φ 2 The details of the action, and its agreement with the standard model and general relativity, can be worked out for each sector of the E8 Lie algebra. 3.2.1 Gravity The modified BF action for gravity was discovered by MacDowell and Mansouri in 1977, [16] and revived by Smolin, Starodubtsev, and Freidel during their work on loop quantum gravity.[17, 18] The remarkable and surprising fact that gravity, described by the spin con- nection, ω, and frame, e, can be described purely in terms of a unified connection, ω+ e, was the seed idea that led to the unification of all fields in a single connection.[2] The gravitational part of the action (3.7) is SG = ∫ 〈 = BG = FG + πG4 =B G = BGγ〉 in which the gravitational part of the curvature (3.3) is = FG = 12(=R− 1 8eeφ 2) ∈ = so(3, 1) Extremizing the action under variation of the gravitational part of the Lagrange multiplier, δ = BG, requires = BG = 2πG=F Gγ = 1πG(=R− 1 8eeφ 2)γ – 25 – and plugging this back into the action gives SG = 1πG ∫ 〈 = FG = FGγ〉 = 14πG ∫ 〈( = R− 18eeφ 2)( = R− 18eeφ 2)γ〉 Multiplying this out gives three terms. The term quadratic in the Riemann curvature is the Chern-Simons boundary term, 〈 = R = Rγ〉 = d〈(ωdω + 13ωωω)γ〉 Dropping this, the other two terms give the Palatini action for gravity, SG = 116πG ∫ { 1 12〈eeeeγ〉φ 4 − 〈 = Reeγ〉φ2 } = 116πG ∫ ∼e φ 2 ( R− 32φ 2 ) equal to the Einstein-Hilbert action with cosmological constant, Λ = 34φ 2. The magnitude of the Higgs, √ φ2, is a conformal factor that can be absorbed into the magnitude of the frame. The vacuum solution to Einstein’s equation with positive cosmological constant is de Sitter spacetime ( = R = Λ6 ee and R = 4Λ), which should be considered the background vacuum spacetime for particle interactions in this theory. Since the symmetry of this spacetime is so(4, 1) and not the Poincaré group, the Coleman-Mandula theorem does not apply to restrict the unification of gravity within the larger group. It should be emphasized that the connection (3.1) comprises all fields over the four dimensional base manifold. There are no other fields required to match the fields of the standard model and gravity. The gravitational metric and connection have been supplanted by the frame and spin connection parts of .A. The Riemannian geometry of general relativity has been subsumed by principal bundle geometry — a significant mathematical unification. Devotees of geometry should not despair at this development, as principal bundle geometry is even more natural than Riemannian geometry. A principal bundle with connection can be described purely in terms of a mapping between tangent vector fields (diffeomorphisms) on a manifold, without the ab initio introduction of a metric. 3.2.2 Other bosons The part of the action (3.8) for the bosonic, non-so(3, 1) parts of the connection is S′ = − ∫ 1 4〈 =F ′ = ∗F ′〉 = Sgw + Sew + S2 in which the relevant parts of the curvature (3.2) are the mixed graviweak part (3.4), the electroweak part (3.5), and the so(8) part (3.6). The mixed graviweak part of the action is Sgw = − ∫ 1 4〈=F gw = ∗F gw〉 – 26 – curvature and the covariant Dirac derivative in curved spacetime. The fermions fit together perfectly in chiral representations under graviweak so(7, 1), and the frame-Higgs has all the correct interactions. This frame-Higgs naturally gets a φ4 potential and produces a positive cosmological constant. Finally, and most impressively, the fit of all fields of the standard model and gravity to E8 is very tight. The structure of E8 determines exactly the spinor multiplet structure of the known fermions. There are also aspects of this theory that are poorly understood. The relationship be- tween fermion generations and triality is suggested by the structure of E8 but is not perfectly clear — a better description may follow from an improved understanding of the new w+ xΦ fields and their relation to ω + eφ. This relationship may also shed light on how and why nature has chosen a non-compact form, E IX, of E8. Currently, the symmetry breaking and action for the theory are chosen by hand to match the standard model — this needs a mathematical justification. Quantum E8 theory follows the methods of quantum field theory and loop quantum gravity — though the details await future work. One enticing possibility is that the grav- itational and cosmological constants run from large values at an ultraviolet fixed point to the tiny values we encounter at low energies.[19, 20] At the foundational level, a quantum description of the standard model in E8 may be compatible with a spin foam description in terms of braided ribbon networks[21] through the identification of the corresponding finite groups. And there is a more speculative possibility: if the universe is described by an excep- tional mathematical structure, this suggests quantum E8 theory may relate to an exceptional Kac-Moody algebra.[22] The theory proposed in this paper represents a comprehensive unification program, de- scribing all fields of the standard model and gravity as parts of a uniquely beautiful math- ematical structure. The principal bundle connection and its curvature describe how the E8 manifold twists and turns over spacetime, reproducing all known fields and dynamics through pure geometry. Some aspects of this theory are not yet completely understood, and until they are it should be treated with appropriate skepticism. However, the current match to the stan- dard model and gravity is very good. Future work will either strengthen the correlation to known physics and produce successful predictions for the LHC, or the theory will encounter a fatal contradiction with nature. The lack of extraneous structures and free parameters ensures testable predictions, so it will either succeed or fail spectacularly. If E8 theory is fully successful as a theory of everything, our universe is an exceptionally beautiful shape. Acknowledgments The author wishes to thank Peter Woit, Sergei Winitzki, Lee Smolin, Tony Smith, David Richter, Fabrizio Nesti, Sabine Hossenfelder, Laurent Freidel, David Finkelstein, Michael Edwards, James Bjorken, Sundance Bilson-Thompson, John Baez, and Stephon Alexander for valuable discussions and encouragement. Some of the work was carried out under the – 29 – wonderful hospitality of the Perimeter Institute for Theoretical Physics. This research was supported by grant RFP1-06-07 from The Foundational Questions Institute (fqxi.org). References [1] M. Tegmark, “Is ‘the theory of everything’ merely the ultimate ensemble theory?” gr-qc/9704009. [2] A.G. 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Richter, “Triacontagonal coordinates for the E(8) root system,” arXiv:0704.3091. [14] F. Gürsey, “Symmetry Breaking Patterns In E6,” Invited talk given at First Workshop on Grand Unified Theories, Durham, N.H., Apr 10-12 (1980). [15] A.G. Lisi, “Quantum mechanics from a universal action reservoir,” physics/0605068. [16] S.W. MacDowell and F. Mansouri, “Unified Geometric Theory of Gravity and Supergravity,” Phys. Rev. Lett. 38 (1977) 739. [17] L. Smolin and A. Starodubtsev, “General relativity with a topological phase: An Action principle,” hep-th/0311163. [18] L. Freidel and A. Starodubtsev, “Quantum gravity in terms of topological observables,” hep-th/0501191. [19] M. Reuter, “Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity,” arXiv:0708.1317. [20] R. Percacci, “Asymptotic Safety,” arXiv:0709.3851. [21] S. O. Bilson-Thompson et al., “Quantum gravity and the standard model,” hep-th/0603022. [22] V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press (1990). – 30 –