aa r X i v : . [ h e p - t h ] J u l Standard Model Fermions and K (E ) Axel Kleinschmidt , and Hermann Nicolai Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)M¨uhlenberg 1, DE-14476 Potsdam, Germany International Solvay Institutes, ULB-Campus Plaine,CP 231, BE-1050 Bruxelles, Belgium
In recent work [1] it was shown how to rectify Gell-Mann’s proposal for identifying the 48 quarksand leptons of the Standard Model with the 48 spin- fermions of maximal SO(8) gauged super-gravity remaining after the removal of eight Goldstinos, by deforming the residual U(1) symmetryat the SU(3) × U(1) stationary point of N = 8 supergravity, so as to also achieve agreement ofthe electric charge assignments. In this Letter we show that the required deformation, while not inSU(8), does belong to K (E ), the ‘maximal compact’ subgroup of E which is a possible candidatesymmetry underlying M theory. The incorporation of infinite-dimensional Kac–Moody symmetriesof hyperbolic type, apparently unavoidable for the present scheme to work, opens up completely newperspectives on embedding Standard Model physics into a Planck scale theory of quantum gravity. The question whether or not the maximally extended N = 8 supergravity theory [2, 3] can be related to Stan-dard Model physics has been under debate for a longtime. Very recent work [1] has taken up an old proposalof Gell-Mann’s [4] on how to match the 48 quarks andleptons (including right-chiral neutrinos) of the StandardModel with the 48 spin- fermions of maximal SO(8)gauged supergravity that remain after the removal ofeight Goldstinos (as required by the complete breakingof N = 8 supersymmetry). This scheme, which was sub-sequently shown to be realised at the SU(3) × U(1) sta-tionary point of maximal gauged SO(8) supergravity [5],relies on identifying the residual SU(3) of supergravitywith the diagonal subgroup of the colour group SU(3) c and a new family symmetry SU(3) f . Intriguingly, in thisway complete agreement is found in the SU(3) charge as-signments of quarks and leptons and the spin- fermionsof N = 8 supergravity, but there remained a systematicmismatch in the electric charges by a spurion charge of q = ± . The main advance reported in [1] was to iden-tify the ‘missing’ U(1) q that rectifies this mismatch, andthat was found to take a surprisingly simple form. How-ever, this deformation cannot be explained from ‘within’ N = 8 supergravity (nor from a hypothetical embeddingof maximal gauged supergravity into the known super-string theories), as U(1) q is not contained in its R sym-metry group SU(8). In this Letter we show that the re-quired deformation is, however, contained in an infinitedimensional extension of SU(8), namely the involutory‘maximal compact’ subgroup K (E ) of the hyperbolicKac–Moody group E , which has been proposed as apossible candidate symmetry of M theory [6].[24] This,we believe, places the question stated above, and also theeventual incorporation of the chiral electroweak gauge in-teractions (not considered in [4, 5]), in an entirely newcontext, by embedding (at least a subset of) the StandardModel symmetries into an infinite-dimensional extensionof the exceptional duality symmetries of maximal super-gravity. This approach, never tried before to the best ofour knowledge, offers completely new perspectives on thepossible Planck scale origin of Standard Model physics. For the rest of this text we will concentrate on thefermionic sector of N = 8 supergravity, which consistsof eight gravitinos ψ iµ transforming in the , and a tri-spinor of spin- fermions χ ijk transforming in the ofSU(8), whence χ ijk is fully antisymmetric in the SU(8)indices i, j, k , with (positive and negative) chirality cor-responding to (upper and lower) position of the indices,and χ ijk = ( χ ijk ) ∗ . Here we will, however, restrict at-tention to the vector-like SO(8) subgroup of SU(8), forwhich the distinction between upper and lower indices isimmaterial, whence we will not distinguish between χ ijk and χ ijk in the remainder. The residual vector-like SO(8)transformations act as χ ijk → U il U jm U kn χ lmn with U ∈ SO(8) . (1)In order to obtain the correct electric charge assign-ments of the quarks and leptons it was found in [1] thatthe U(1) subgroup of SU(3) × U(1) must be deformed bya new (still vector-like) U(1) q whose action on the tri-spinor χ ijk is generated by the following 56-by-56 matrix I := 12 (cid:16) T ∧ ∧ + ∧ T ∧ + ∧ ∧ T + T ∧ T ∧ T (cid:17) (2)acting in the ∧ ∧ representation of SO(8). Here T = − − − − , (3)represents the imaginary unit in the breaking of SO(8)to SU(3) × U(1). We note that, from T = − we have I = − , whence (2) can be trivially exponentiatedto a U(1) q phase rotation. The combination (2) dif-fers from the usual co-product obtained from (1) with U = exp( ωT ) by the 56-by-56 matrix T ∧ T ∧ T . Im-portantly, the latter is not in SU(8), although it doescommute with the SU(3) × U(1) subgroup of SO(8), andhence merely deforms this subgroup, but does not en-large it. We will now show how to accommodate thetriple wedge product T ∧ T ∧ T by enlarging the R sym-metry SU(8) of N = 8 supergravity to the bigger, andin fact, infinite-dimensional R symmetry K (E ), in ac-cordance with the anticipated enlargement of the finite-dimensional exceptional dualities of maximal supergrav-ities to infinite-dimensional groups.To proceed we recall how the fermions of D = 11 su-pergravity [8] are related to those of N = 8 supergravity[2, 9]. Denoting the (spatial) D = 11 gravitino compo-nents by Ψ aA (with a, b, ... = 1 , ...,
10 and D = 11 spinorindices A, B, ... = 1 , ...,
32) and adopting the temporalsupersymmetry gauge Ψ A = (Γ Γ a ) AB Ψ aB as in [10, 11],we split the D = 11 gravitino into four-dimensional spa-tial and internal components as followsΨ aA = (cid:0) Ψ ˆ aαi , Ψ ¯ aαi (cid:1) (4)with flat spatial indices ˆ a, ˆ b, ... = 1 , , a, ¯ b, ... = 4 , . . . ,
10, whose position again does notmatter as they are pulled up and down with δ ab . The D = 11 spinor indices A, B, ... are split as A ≡ ( α, i )into D = 4 spinor indices α, β, ... = 1 , ..., i, j, ... = 1 , ..., ψ i ˆ aα ∝ Ψ i ˆ aα − X ¯ c =4 Γ ¯ cij (cid:16) γ γ ˆ a Ψ j ¯ c (cid:17) α (5) χ ijkα ∝ X ¯ a =4 Γ ¯ a [ ij Ψ ¯ ak ] α (6)where we temporarily suspend the summation conven-tion for the indices a, b, ... (the summation conventionremains, however, in force for all other indices). For theimplementation of the action of T ∧ T ∧ T we also needthe following redefinition of the D = 11 gravitino [12]:Φ a A = Γ aAB Ψ aB (no sum on a !) (7)Because there is no summation on the spatial index a ,manifest SO(10) covariance is lost. To emphasise thispoint we adopt a different font ( a , b , ... ) although theseindices have the same range as a, b, .... before [13]. Im-portantly, however, the position of the indices a , b , ... now does matter, as they are to be raised and lowered withthe (Lorentzian) DeWitt metric and its inverse G ab = δ ab − ⇔ G ab = δ ab − . (8)With the redefinition (7) the formula (6) becomes χ ijk α ∝ X a =4 Γ a [ ij Γ a kl ] Φ a lα . (9) The action of T ∧ T ∧ T is therefore realised via (nowsuppressing D = 4 spinor indices) χ ijk → T il T jm T kn χ lmn ∝ X a =4 ( T Γ a T ) [ ij ( T Γ a T ) kl ] T lm Φ a m (10)where we have inserted a factor T T = − and used theantisymmetry of T . Next we recall that there is a repre-sentation of the SO(7) Γ-matrices where T ij = Γ ij (11)(see e.g. appendix E of [15]); it is then easy to see that( T Γ a T ) [ ij ( T Γ a T ) kl ] = Γ a [ ij Γ a kl ] (12)even without summation over a . Using this formula weconclude from (10) that the desired action takes a verysimple form on the redefined spinors (7), to wit,Φ a iα −→ T ij Φ a jα (13)which leaves the D = 4 spinor indices unaffected. Ofcourse, one could also (though less elegantly) express thisaction in terms of the original spinors Ψ aA . We stress thatin order to preserve the relation (5), (13) must hold forall a = 1 , ...,
10. From this follows the action of the newgenerator on the D = 4 gravitino, an insight that thearguments in [1] could not provide. Observe that theredefinition ψ i ˆ a → γ ˆ a ψ i ˆ a implied by (7) does not affectthis conclusion, as γ ˆ a commutes with T .We now want to show that the action (13) is containedin K (E ), the supposed R symmetry of M theory. Werefer to our previous work [10, 11, 13, 14] for detailed ex-planations on K (E ), and here simply summarise somesalient results (see also [16, 17] for related work). Thegroup K (E ) is the involutory subgroup of E which isleft invariant by the Cartan-Chevalley involution definedon E in terms of its Chevalley-Serre presentation. Assuch, it contains the R symmetries of all D ≥ chiral trans-formations for even D ); more specifically, we haveSU(8) ⊂ SO(16) ⊂ K (E ) ⊂ K (E ) (14)The fermions transform in spinorial (double-valued) rep-resentations of K (E ). A remarkable property of thealgebra K (E ) is that, though infinite-dimensional, itadmits finite-dimensional , hence unfaithful representa-tions [11, 16]. These are the Dirac [17] and vector-spinor representations [11, 16], which can be directlydeduced from D = 11 supergravity (in addition, two‘higher spin’ realisations are known [13]). As a con-sequence, K (E ) is not simple, because it has non-trivial (finite codimension) ideals J which are associ-ated with the unfaithful representations in the way ex-plained in [11]. Accordingly, the quotient K (E ) / J isa finite-dimensional group; more specifically, denotingthe vector-spinor ideal by J vs , evidence was presentedin [13] that K (E ) (cid:14) J vs = Spin(288 , . The fact thatthe ‘compact’ subgroup K (E ) ⊂ E in this way givesrise to a non-compact quotient group is another unusualfeature of K (E ).A convenient realization of the K (E ) Lie algebra gen-erators in the vector-spinor representation was found in[13, 14] (following earlier work on K (AE ) in [12, 18, 19]).Like the generators of E , the generators k rα of K (E )can be labeled by E roots α and the associated multi-plicity index r , but such that [14]k rα = − k r − α , for all E roots α . (15)As shown in [13], for the vector spinor representationthere is a concrete realization of these generators in termsof 320-by-320 matrices. For all real roots α of E (forwhich the multiplicity label r is not needed) we have(k α ) a A, b B = 12 X ab ( α )˜Γ( α ) AB (16)where the symmetric matrix X ab is given byX ab ( α ) = − α a α b + 14 G ab (17)in terms of the root components α a in the ‘wall basis’ usedin [13]; indices a , b are raised and lowered by means of(8). As explained in [13] there is a map from the E rootlattice into the SO(10) Clifford algebra that associates toeach root α of E a particular element ˜Γ( α ) = − ˜Γ( − α )of the Clifford algebra; furthermore the matrices ˜Γ( α )are anti-symmetric for α ∈ Z + 2 and symmetric for α ∈ Z . Because the SO(10) Clifford algebra is finite-dimensional, and because there are infinitely many realand imaginary roots of E , it follows that infinitely manyE roots α are mapped to the same element of the Clif-ford algebra.To prove that (16) indeed generates the algebra K (E ), one substitutes the ten simple roots of E into(16) and verifies the defining relations for K (E ) [13](the latter characterise the involutory subalgebra in amanner analogous to the Chevalley–Serre presentationfor general Kac–Moody algebras [20, 21]). The Lie al-gebra K (E ) in the vector spinor representation is thusgenerated by taking commutators of the above real rootgenerators in all possible ways. In this way one ‘reaches’all imaginary root spaces with α ≤
0. However, dueto the unfaithfulness of the representation the image ofthe root space elements consists of linear combinations offinitely many basis elements. The generating elements,and thus K (E ), leave invariant the Lorentzian bilinearform( V, W ) ≡ G ab V a A W b A (of signature (288 , rα generated in this way are antisymmetric under interchange of the index pairs ( a A ) and ( b B ), thatis, (k rα ) a A, b B = − (k rα ) b B, a A . (19)and can thus be written as a linear combination of ma-trices of the form (16), with either X ab symmetric in ( ab )and ˜Γ( α ) AB anti-symmetric in [ AB ], or antisymmetric in[ ab ] and symmetric in ( AB ). Because all such matricesleave invariant the Lorentzian bilinear form (18) they allbelong to the Lie algebra of so (288 ,
32) [13].Although we do not have a general formula for arbi-trary imaginary roots, explicit formulas do exist for nullroots δ , and for certain time-like roots Λ [14]. For nullroots δ , we have(k rδ ) a A, b B = ε r [ a δ b ] ˜Γ( δ ) AB (20)with eight transversal polarisation vectors ε r . For time-like roots Λ with Λ = 2 − n (for n ≥ r Λ can be realised in the form (16) by choos-ing a decomposition Λ = α + β with α = β = 2 and α · β = − (2 n + 1); this givesX ( α ) ab (Λ) = − α a α b − β a β b − (2 n +1) α ( a β b ) + 14 G ab (21)Taking n = 1 (that is, Λ = −
2) as an example andletting the decomposition range over all pairs of real roots( α, β ) with Λ = α + β one thus re-constructs the fullroot space, of dimension mult(Λ) = 44. For larger n themultiplicity of Λ increases rapidly[25], and one can nolonger exhaust the full root space with the X ( α ) ab (Λ).Returning to our initial problem we note that k a A, b B = G ab T AB ≡ G ab δ αβ T ij ∈ so (288 ,
32) (22)whence this matrix can be generated by a linear com-bination of matrices obtained by multiple commutationof the basic K (E ) generators (because a linear com-bination may be required, we omit the root and multi-plicity labels on k ). To see how one can arrive at therequisite linear combination we note that there are in-finitely many roots α (both real and imaginary) that sat-isfy ˜Γ( α ) = T = Γ . The task of finding a K (E ) gen-erator that implements T ∧ T ∧ T of (13) is then reducedto finding a combination of tensors X ab that equals G ab .We are not aware of a single root that achieves this butestablishing the existence of a linear combination can beachieved as follows. In accordance with (21) one consid-ers the set of all X ab that can arise from the commutationof two real root generators X ab ( α ) and X cd ( β ) (given asin (17)) such that α + β = Λ is an imaginary root thatsatisfies ˜Γ(Λ) = Γ . Similarly, one can perform the sameanalysis for odd multiples of Λ given by (2 k + 1)Λ sincethen ˜Γ (cid:0) (2 k + 1)Λ (cid:1) = ˜Γ(Λ) = Γ . We have shown byan explicit computer analysis that one can find a linearcombination of the generated X ab (cid:0) (2 k + 1)Λ (cid:1) that equals G ab and therefore the desired realization of T ∧ T ∧ T onthe spinors of D = 11 supergravity within K (E ). Thegenerator just constructed only extends the R symmetrySU(8) ⊂ SO(3) × SU(8) ⊂ K (E ) and thus leaves thespatial rotation SO(3) symmetry untouched.The above argument demonstrates the existence of anelement of K (E ) that acts according to (13), but thecombination identified above does not necessarily have asimple algebraic interpretation. Because the spinors φ a A form an unfaithful representation of K (E ) there are in-finitely many elements that act in this way, and it is thuspossible that an alternative realization of T ∧ T ∧ T ex-ists that has a simple physical origin. For the realisationfound here one already has to go up to level ℓ = 18 in alevel decomposition of K (E ) (that follows directly fromthe corresponding tables for E given in [22]); there isthus no easy way of reproducing this result by simple iter-ation of the low level K (E ) transformation rules givenin [11]. The explicit realization of the charge shifting U (1) q generator above relies on the existence of time-likeimaginary roots and their integer multiples, but theremay be other possibilities, in particular, using only realroots. In any case, it does not appear possible to con-struct the requisite element without use of the ‘hyper-bolic’ over-extended root of E , since the structure of theroot system of the affine subalgebra e is too restricted.In this sense, the extension to the full hyperbolic Kac–Moody algebra and its involutory subalgebra could beessential for linking N = 8 supergravity to the real world.We note that the embedding of T ∧ T ∧ T into K (E )in principle also allows for a realisation of this transfor-mation on the bosonic fields of the spinning E /K (E )model studied in [11], although the ambiguities related tothe unfaithfulness of the fermionic realisation of K (E )remain to be resolved. More precisely, while there areinfinitely many combinations of K (E ) generators thatact in the same way on the fermions, these will act dif- ferently on the bosonic coset variables on which K (E )is realised faithfully. The bosonic variables can thus inprinciple be used to remove all ambiguities.The results of [1] and this Letter represent a signif-icant shift away from the standard paradigm of howto understand the possible emergence of the StandardModel fermions from a Planck scale unified theory, asfor instance embodied in currently popular superstringinspired scenarios of low energy ( N = 1) supergravity.There one starts from a finite-dimensional compact Yang-Mills gauge group (such as E × E ), with the fermionstransforming in a standard representation. This symme-try is assumed to be present as a space-time-based sym-metry already at the Planck scale, and then assumed tobe broken in a cascade of symmetry reductions as one de-scends to the electroweak scale. By contrast, the presentscheme proceeds from an infinite-dimensional group thatcan be fully present as a symmetry only in a phase of thetheory prior to the emergence of classical space and time ,in accord with the proposal of [6], and crucially relies onthe infinite-dimensionality of this group (and the associ-ated Kac–Moody algebra).[26] We emphasise once againthat K (E ) does possess chirality, offering new perspec-tives for the incorporation of chiral gauge symmetries,such that the electroweak sector of the Standard Modelmay eventually be understood in a way very differentfrom currently prevailing views. Acknowledgments:
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