Polyvector-valued Extensions Of (supersymmetric) Gauge Theories

ON CHERN-SIMONS (SUPER) GRAVITY, E8 YANG-MILLS AND POLYVECTOR-VALUED GAUGE THEORIES IN CLIFFORD SPACES Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, GA. 30314, [email protected] January 2006, Revised June, 2006 Abstract It is shown how the E8 Yang-Mills theory is a small sector of a Cl(16) algebra Gauge Theory and why the 11D Chern-Simons (Super) Gravity theory can be embedded into a Cl(11) algebra Gauge theory. These results may shed some light into the origins behind the hidden E8 symmetry of 11D Supergravity. To finalize, we explain how the Clifford algebra gauge theory (that contains the Chern-Simons gravity action in D = 11, for example ) can itself be embedded into a more fundamental polyvector-valued gauge theory in Clifford spaces involving tensorial coordinates xµ1 µ2 , xµ1 µ2 µ3 , ...., xµ1 µ2 ....µD in addition to antisymmetric tensor gauge fields Aµ1 µ2 , Aµ1 µ2 µ3 , ...., Aµ1 µ2 .....µD . The polyvector-valued supersymmetric extension of this polyvector valued bosonic gauge theory in Clifford spaces may reveal more important features of a Cliffordalgebraic structure underlying M, F theory. 1 INTRODUCTION : WHY CLIFFORD ALGEBRAS Ever since the discovery [1] that 11D supergravity, when dimensionally reduced to an n-dim torus led to maximal supergravity theories with hidden exceptional symmetries En for n ≤ 8, it has prompted intensive research to explain the higher dimensional origins of these hidden exceptional En symmetries [2, 6] . More recently, there has been a lot of interest in the infinite-dim hyperbolic Kac-Moody E10 and non-linearly realized E11 algebras arising in the asymptotic chaotic oscillatory solutions of Supergravity fields close to cosmological singularities [1,2]. The classification of symmetric spaces associated with the scalars of N extended Supergravity theories (emerging from compactifications of 11D supergravity to lower dimensions), and the construction of the U -duality groups as spectrum-generating symmetries for four-dimensional BPS black-holes [6] also involved exceptional symmetries associated with the Jordan algebras J3 [R, C, H, O]. The discovery of the anomaly free 10-dim heterotic string for the algebra E8 × E8 was another hallmark of the importance of Exceptional Lie groups in Physics. Exceptional, Jordan, Division and Clifford algebras are deeply related and essential tools in many aspects in Physics [3, 5, 8, 9,14,15,16,17,18,19,20]. In this work we will focus mainly on the Clifford algebraic structures and show how the E8 Yang-Mills theory can naturally be embedded into a Cl(16) algebra Gauge Theory and why the 11D Chern-Simons (Super) Gravity [4] is a very small sector of a more fundamental theory based on the Cl(11) algebra Gauge theory. Polyvector-valued Supersymmetries [11] in Clifford-spaces [3] turned out to be more fundamental than the supersymmetries associated with M, F theory superalgebras [7,10]. For this reason we believe that Clifford structures may shed some light into the origins behind the hidden E8 symmetry of 11D Supergravity and reveal more important features underlying M, F theory. In the remaining of this introduction a very brief overview of the basic features of the Extended Relativity in Clifford spaces is presented along with the basic formulas involving the polyvector valued generalized supersymmety algebra in Clfford spaces. In section 2 we show how the E8 Yang-Mills theory can be obtained from a gauge theory based on the Clif f ord (16) algebra. In section 3 the Chern-Simons Gravity in 11-dim is embedded into a Clifford algebra gauge theory. This in itself is no much different than constructing a Chern-Simons gravity theory based on a gl(N, R) algebra, for example. However, the fundamental difference is shown in section 4 where we explain how the Clifford algebra gauge theory ( that contains the ChernSimons gravity action in D = 11 ) is itself embedded into a more fundamental polyvector-valued gauge theory in Clifford spaces involving tensorial coordinates xµ1 µ2 , xµ1 µ2 µ3 , .... in addition to antisymmetric tensor gauge fields. Aµ1 µ2 , Aµ1 µ2 µ3 , .... . We leave for future work the explicit construction of the polyvectorvalued generalized supersymmetric extension of our polyvector valued bosonic gauge theory in Clifford spaces and its implications for the further developments of M, F theory. 1

1.1 The Extended Relativity in Clifford Spaces The Extended Relativity theory in Clifford-spaces ( C-spaces ) is a natural extension of the ordinary Relativity theory . A natural generalization of the notion of a space-time interval in Minkwoski space to C-space is : dX 2 = dσ 2 + dxµ dxµ + dxµν dxµν + ...

(1.1)

The Clifford valued poly-vector: X = X M EM = σ 1 + xµ γµ + xµν γµ ∧ γν + ...xµ1 µ2 ....µD γµ1 ∧ γµ2 .... ∧ γµD .

(1.2a)

denotes the position of a polyparticle in a manifold, called Clifford space or C-space. The series of terms in (1.2a) terminates at a f inite value depending on the dimension D. A Clifford algebra Cl(r, q) with r +q = D has 2D basis elements. For simplicity, the gammas γ µ correspond to a Clifford algebra associated with a flat spacetime : 1 µ ν {γ , γ } = η µν 1. (1.2b) 2 but in general one could extend this formulation to curved spacetimes with metric g µν . The multi-graded basis elements EM of the Clifford-valued poly-vectors are EM ≡ 1,

γµ,

γ µ1 ∧ γ µ2 ,

γ µ1 ∧ γ µ2 ∧ γ µ3 ,

γ µ1 ∧ γ µ2 ∧ γ µ3 ∧ ..... ∧ γ µD .

(1.2c)

It is convenient to order the collective M indices as µ1 < µ2 < µ3 < ...... < µD . The connection to strings and p-branes can be seen as follows. In the case of a closed string (a 1loop) embedded in a target flat spacetime background of D-dimensions, one represents the projections of the closed string (1-loop) onto the embedding spacetime coordinate-planes by the variables xµν . These variables represent the respective areas enclosed by the projections of the closed string (1-loop) onto the corresponding embedding spacetime planes. Similary, one can embed a closed membrane (a 2-loop) onto a D-dim flat spacetime, where the projections given by the antisymmetric variables xµνρ represent the corresponding volumes enclosed by the projections of the 2-loop along the hyperplanes of the flat target spacetimr background. The connection to strings and p-branes can be seen as follows. In the case of a closed string (a 1loop) embedded in a target flat spacetime background of D-dimensions, one represents the projections of the closed string (1-loop) onto the embedding spacetime coordinate-planes by the variables xµν . These variables represent the respective areas enclosed by the projections of the closed string (1-loop) onto the corresponding embedding spacetime planes. Similary, one can embed a closed membrane (a 2-loop) onto a D-dim flat spacetime, where the projections given by the antisymmetric variables xµνρ represent the corresponding volumes enclosed by the projections of the 2-loop along the hyperplanes of the flat target spacetimr background. This procedure can be carried to all closed p-branes ( p-loops ) where the values of p are p = 0, 1, 2, 3, ....D − 2. The p = 0 value represents the center of mass and the coordinates xµν , xµνρ .... have been coined in the string-brane literature [21] as the holographic areas, volumes, ...projections of the nested family of p-loops ( closed p-branes ) onto the embedding spacetime coordinate planes/hyperplanes. The classification of Clifford algebras Cl(r, q) in D = r + q dimensions ( modulo 8 ) for different values of the spacetime signature r, q is discussed, for example, in the book of Porteous [22]. All Clifford algebras can be understood in terms of CL(8) and the CL(k) for k less than 8 due to the modulo 8 Periodicity theorem CL(n) = CL(8) × Cl(n − 8) where Cl(r, q) is a matrix algebra for even n = r + q or the sum of two matrix algebras for odd n = r + q. Depending on the signature, the matrix algebras may be real, complex, or quaternionic. For furher details we refer to [22] . If we take the differential dX and compute the scalar product among two polyvectors < dX T dX >scalar we obtain the C-space extension of the particles proper time in Minkwoski space. The symbol X T denotes the reversion operation and involves reversing the order of all the basis γ µ elements in the expansion of X . The C-space proper time associated with a polyparticle motion is then : 2

< dX T dX >scalar = dΣ2 = (dσ)2 + Λ2D−2 dxµ dxµ + Λ2D−4 dxµν dxµν + ..

(1.3)

Here we have explicitly introduced the Planck scale Λ since a length parameter is needed in order to tie objects of different dimensionality together: 0-loops, 1-loops,..., p-loops. Einstein introduced the speed of light as a universal absolute invariant in order to “unite” space with time (to match units) in the Minkwoski space interval: ds2 = c2 dt2 − dxi dxi . (1.4) A similar unification is needed here to “unite” objects of different dimensions, such as xµ , xµν , etc... The Planck scale then emerges as another universal invariant in constructing an extended scale relativity theory in C-spaces [3]. To continue along the same path, we consider the analog of Lorentz transformations in C-spaces which transform a poly-vector X into another poly-vector X 0 given by X 0 = RXR−1 with R = eω

A

EA

= exp [(ω1 + ω µ γµ + ω µ1 µ2 γµ1 ∧ γµ2 .....)].

(1.5)

and R−1 = e−ω

A

EA

= exp [−(ω1 + ω ν γν + ω ν1 ν2 γν1 ∧ γν2 .....)].

(1.6)

where the ω parameters also belong to a Clifford-valued quantity ω; ω µ ; ω µν ; ....

(1.7)

they are the C-space version of the Lorentz rotations/boosts parameters. Since a Clifford algebra admits a matrix representation, one can write the norm of a poly-vectors in terms of the trace operation as: ||X||2 = T race X 2 Hence under C-space Lorentz transformation the norms of poly-vectors behave like follows: 2

T race X 0 = T race [RX 2 R−1 ] = T race [RR−1 X 2 ] = T race X 2 .

(1.8)

These norms are invariant under C-space Lorentz transformations due to the cyclic property of the trace operation and RR−1 = 1. Another way of rewriting the inner product of polyvectors is by means of the reversal operation that reverses the order of the Clifford basis generators : (γ µ ∧ γ ν )T = γ ν ∧ γ µ , etc... Hence the inner product can be rewritten as the scalar part of the geometric product < X T X >s . The analog of an orthogonal matrix in Clifford spaces is RT = R−1 such that < X 0T X 0 >s =< (R−1 )T X T RT RXR−1 >s =< RX T XR−1 >s =< X T X >s = invariant.

(1.9a)

This condition RT = R−1 , of course, will restrict (constrain ) the type of terms allowed inside the exponential defining the rotor R in eq-(1-6) because the reversal of a p-vector obeys (γµ1 ∧ γµ2 ..... ∧ γµp )T = γµp ∧ γµp−1 ..... ∧ γµ2 ∧ γµ1 = (−1)p(p−1)/2 γµ1 ∧ γµ2 ..... ∧ γµp

(1.9b)

Hence only those terms that change sign ( under the reversal operation ) are permitted in the exponential defining R = exp[ω A EA ]. Another possibility is to complexif y the C-space polyvector valued coordinates = Z = Z A EA = A X EA + iY A EA (which is not the same as complexif ying the Clifford algebra) as well as the boost/rotation parameters ω A in order to allow the unitarity condition U † = U −1 to hold . The generalized Clifford unitary transformations Z 0 = U ZU † = U ZU −1 associated with the complexified polyvector Z = Z A EA must be such so the interval < dZ † dZ >s = d¯ σ dσ + d¯ z µ dzµ + d¯ z µν dzµν + d¯ z µνρ dzµνρ + ..... remains invariant under these unitary transformations above (upon setting the Planck scale Λ = 1). 3

(1.9c)

The unitary condition U † = U −1 , under the combined reversal and complex-conjugate operation, will constrain the form of the complexified boosts/rotation parameters ω A appearing in : U = exp[ ω A EA ]. The parameters ω A must be either purely real, or purely imaginary, depending if the reversal EA T = ±EA , to ensure that an overall change of sign occurs in the terms ω A EA inside the exponential defining U so that U † = U −1 actually holds, and the norm < Z † Z >s remains invariant under the analog of unitary transformations in complexif ied C-spaces. These techniques are not very different from Penrose Twistor spaces. As far as we know a Clifford-Twistor space construction of C-spaces has not been performed so far. Another alternative is to define the unitary transformations by U = exp (ΩAB [EA , EB ]) where the C commutator [EA , EB ] = FAB EC is the C-space analog of the i[γµ , γν ] commutator which is the generator of the Lorentz algebra, and the parameters ΩAB are the C-space analogs of the rotation/boots parameters. The diverse parameters ΩAB are purely real or purely imaginary depending whether the reversal [EA , EB ]T = ±[EA , EB ] to ensure that U † = U −1 such that the scalar part < Z † Z >s remains invariant under the transformations Z 0 = U ZU −1 . This last alternative seems to be more physical because a polyrotation should map the EA direction into the EB direction in C-spaces, hence the meaning of the generator [EA , EB ] which is the extension of the i[γµ , γν ] Lorentz generator. We refer to the review [3] for further details about the Extended Relativity Theory in Clifford spaces. In particular, why Relativity in curved Clifford-spaces is equivalent to a higher derivative gravity with torsion associated with the underlying spacetime [3]. 1.2 Polyvector-valued Super Poincare Algebras and Clifford-space Supersymmetry Polyvector Super-Poincare algebras as extensions of ordinary Super-Poincare algebras have been studied by [10]. The former Lie superalgebras (involving commutators and anti-commutators) should not be confused with the Z2 -graded extensions of ordinary Lie algebras, in particular with Z2 -graded extensions of Clifford algebras involving only commutators. The Polyvector Super Poincare algebras have the form of g = g0 + g1 , where the even sector is g0 = so(V ) + W0 and the odd sector g1 = W1 consists of a spinorial representation of so(V ) = so(p, q); i.e. W1 is an so(p, q)-spinorial-module where V is a vector space of signature p, q. The algebra of generalized translations W = W0 + W1 is the maximal solvable ideal of g. W0 is generated by W1 : [W1 , W1 ] ⊆ W0 and [W0 , W1 ] = 0; [W0 , W0 ] = 0. For example, in the ordinary ¯ ∼ P Super-Poincare algebra, the translations are generated by the supersymmetry generators : {Q, Q} and [Q, P ] = [P, P ] = 0. Choosing W1 to be a spinorial so(V )-module consisting of a sum of spinors and semispinors (chiral spinors) the authors [10] proved that W0 consists of polyvectors. They provided the classification of all polyvector Lie superalgebras, for all dimensions and signatures, after analysing all the so(V )-invariant polyvector-valued bilinear forms that can be defined on the spinor modules. N -extended polyvector super Poincare algebras were also classified in [10]. The anti-commutator is : X (k) {Sα , Sβ } = (CΓµ1 µ2 ....µk )αβ W0 µ2 µ2 ....µk (1.10) k

where α, β denote spinor indices and the summation over k must obey certain crucial restrictions to match degrees of freedom with the terms in the l.h.s. The matrix C is the charge conjugation matrix. Depending on the given spacetime and its signature there are at most two charge conjugation matrices CS , CA given by the product of all symmetric and all antisymmetric gamma matrices, respectively . In special spacetime signatures they collapse into a single matrix. These charge conjugation matrix C are essential in order to satisfy the nontrivial graded super Jacobi identities. A Chern-Simons Supergravity (CS-SUGRA) in D = 11 involves the symplectic supergroup OSp(32|1) and the connection [4] ¯α Aµ = eaµ Γa + ωµab Γab + Aaµ1 a2 ....a5 Γa1 a2 ....a5 + Ψ (1.11) µ Qα . whereas the M theory superalgebra involve 32-component spinorial supercharges Qα whose anticommutators are [7,10] {Qα , Qβ } = (AΓµ )αβ P µ + (AΓµ1 µ2 )αβ Z µ1 µ2 + (AΓµ1 µ2 ....µ5 )αβ Z µ1 µ2 .....µ5 .

(1.12)

there are 32 × 32 symmetric real matrices with at most 21 (32 × 33) = 528 independent components that match the number of degrees of freedom associated with the translations P µ and the antisymmetric rank 4

2, 5 abelian tensorial central charges Z µ1 µ2 , Z µ1 µ2 .....µ5 in the r.h.s since 11 + 55 + 462 = 528. The matrix A plays the role of the timelike γ 0 matrix in Minkowski spacetime and is used to introduce barred-spinors [7,10] The F theory 12D super-algebra involves the Majorana-Weyl spinors with 32 components whose anticommutators are [7] (1.13) {Qα , Qβ } = (AΓµν )αβ Z µν + (AΓµ1 µ2 ....µ6 )αβ Z µ1 µ2 ....µ6 . and the counting of components in D = 12 yields also 32×33 = 528 = 66 + 462. In 13D it requires the 2 superalgebra OSp(64|1) which is connected to a membrane, a 3-brane and a 6-brane, respectively, since = 78 + 286 + 1716 = 2080 components. antisymmetric tensors of ranks 2, 3, 6 in 13D have a total of 64×65 2 Therefore, by studying the Polyvector Super Poincare algebras, the M and F theory superalgebras (1.10, 1.11, 1.12) one concludes that these cannot be incorporated into Clifford-superspaces because one cannot have a restricted summation in the k rank of the terms appearing in the {Qα , Qβ } (anti)commutators. Unless one adds further spinorial degrees of freedom by introducing multi-spinor valued quantities θα1 α2 , θα1 α2 α3 , ... which are the fermionic partners of xµ1 µ2 , xµ1 µ2 µ3 , .., or by recurring to N extended supersymmetries, one will not be able to match the number of degrees of freedom in a satisfactory manner. N extended Polyvector Super Lie Algebras which were also studied by [10] . This means that the odd sector W1 consists of N copies of the irreducible spinor module S. There are cases where there are two inequivalent copies ( complex even dimensional, or real with spatial signatures s = 0, 4 ) involving N+ chiral generators and N− anti-chiral ones. For further details we refer to [10] . Hence, by introducing a judicious number of extra spinorial degrees of freedom θα1 α2 , θα1 α2 α3 , ... in ”Clifford-Superspace”, depending on the dimensions and spacetime signatures, one can acommodate for the larger number of polyvector coordinates associated with C-spaces. For this reason we believe that Polyvectorvalued Supersymmetries in Clifford-Superspaces [11] deserve to be investigated further since they are more fundamental than the supersymmetries associated with M, F theory superalgebras. Nevertheless, there are instances, in particular when D = 4 = 3 + 1, that the {Qα , Qβ } is a symmetric matrix in α, β with 10 independent components and which matches exactly the degrees of freedom of the momentum vector and bi-vector P µ , P µν given by 4 + 6 = 10. Therefore, in D = 4, one may have the anticommutator written in terms of the charge conjugation matrix C as {Qα , Qβ } =

1 1 µ Cγ Pµ + Cγ µν Pµν 2 2

(1.14)

and the Jacobi indentities { [Mµ1 µ2 , Qα ], Qβ } + { [Mµ1 µ2 , Qβ ], Qα } = [ Mµ1 µ2 , {Qα , Qβ } ].

{ [Mµ1 µ2 µ3 µ4 , Qα ], Qβ } + { [Mµ1 µ2 µ3 µ4 , Qβ ], Qα } = [ Mµ1 µ2 µ3 µ2 , {Qα , Qβ } ].

(1.15a)

(1.15b)

with the commutators [Mµ1 µ2 , Pρ1 ρ2 ] = −ηµ1 ρ1 Pµ2 ρ2 + ηµ2 ρ1 Pµ1 ρ2 ± .....; where {Qα , Qβ } =

1 [Mµ1 µ2 , Qα ] = − (γµ1 µ2 )αδ Qδ . 2

1 µ 1 Cγ Pµ + Cγ µν Pµν 2 2

(1.15c)

(1.15d)

the spinorial charges Qα behave under poly-rotations as follows 1 [Mµ1 µ2 µ3 µ4 , Qα ] = − (γµ1 µ2 µ3 µ4 )αδ Qδ . 2

(1.15e)

and the remaining commutators are [Mµ1 µ2 µ3 µ4 , Pν1 ν2 ] = ηµ1 µ2 ν1 ν2 Pµ3 µ4 + ηµ3 µ4 ν1 ν2 Pµ1 µ2 ± ...........; 5

[Mµ1 µ2 µ3 µ4 , Pν1 ] = 0.

(1.15f )

[Mµ1 µ2 , Mν1 ν2 ] = −ηµ1 ν1 Mµ2 ν2 + ηµ2 ν1 Mµ1 ν2 ± ...

(1.15g)

[Mµ1 µ2 µ3 µ4 , Mν1 ν2 ν3 ν4 ] = ηµ1 µ2 ν1 ν2 Mµ3 µ4 ν3 ν4 ± ......

(1.15h)

[Mµ1 µ2 , Mν1 ν2 ν3 ν4 ] = −ηµ1 ν1 Mµ2 ν2 ν3 ν4 ± ........

(1.15i)

In the Appendix we will prove that this algebra eqs-(1.15) closes and satisfies the Jacobi identities. GM N is the f lat C-space generalized metric ηµ1 ν1 µ2 ν2 .....µn νn given by the determinant of the N × N matrix Υmn whose entries are ηµm νn . For instance : ηµ1 ν1 µ2 ν2 .....µn νn = det Υmn =

1 i1 i2 ....in j1 j2 ....jn   ηµi1 νj1 ηµi2 νj2 .......ηµin νjn . N!

(1.16)

so that ηµ1 ν1 µ2 ν2 = ηµ1 ν1 ηµ2 ν2 − ηµ1 ν2 ηµ2 ν1

etc....

(1.17)

Similar results apply to the definition of ηi1 j1 ....in jn . The graded super Jacobi identities (nontrivial matter ) in C-space due to the nontrivial algebraic relations obtained from the (geometric) product of two polyvector basis elements ΓM ΓN that involves a sum of terms with polyvectors of mixed grade : < ΓM ΓN >m+n

< ΓM ΓN >m+n−2

< ΓM ΓN >m+n−4

..........

< ΓM ΓN >|m−n| .

(1.18)

Using the standard notation γ ν1 ν2 ........νp ≡ γ µ1 ∧ γ µ2 ∧ .... ∧ γ µp .

(1.17)

where the anti-symmetrization of indices is performed with unit weight, one has for example : γµ γν =

1 1 1 µ ν {γ , γ } + [γ µ , γ ν ] = η µν 1 + γ µν . 2 2 2

(1.19)

γ µ1 µ2 ....µp γ µp+1 = γ µ1 µ2 .....µp µp+1 + p γ [µ1 µ2 .....µp−1 η µp ]µp+1 .

(1.20)

γ µ γ ν1 ν2 ........νp − (−1)p γ ν1 ν2 ........νp γ µ = 2p η µ[ν1 γ ν2 ν3 .....νp ] .

(1.21)

Having outlined the basic features of the Extended Relativity theory in Clifford spaces and Polyvectorvalued supersymmetries we proceed with the main bulk of this work. 2. THE E8 YANG-MILLS FROM A Cl(16) ALGEBRA GAUGE THEORY It is well known among the 120 ⊕ 128. The E8 admits also admits the vector 8v and spinor spinor representations decompose

experts that the E8 algebra admits the SO(16) decomposition 248 → a SL(8, R) decomposition [6]. Due to the triality property , the SO(8) representations 8s , 8c . After a triality rotation, the SO(16) vector and as [6] 16 → 8s ⊕ 8c .

(2.1a)

128s → 8v ⊕ 56v ⊕ 1 ⊕ 28 ⊕ 35v .

(2.1b)

128c → 8s ⊕ 56s ⊕ 8c ⊕ 56c .

(2.1c)

To connect with (real) Clifford algebras [8], i.e. how to fit E8 into a Clifford structure , start with the 248-dim fundamental representation E8 that admits a SO(16) decomposition given by the 120-dim bivector representation plus the 128-dim chiral-spinor representations of SO(16). From the modulo 8 periodicity of Clifford algebras one has Cl(16) = Cl(2 × 8) = Cl(8) ⊗ Cl(8), meaning, roughly, that the 216 = 256 × 256 6

Cl(16)-algebra matrices can be obtained effectively by replacing each single one of the entries of the 28 = 256 = 16 × 16 Cl(8)-algebra matrices by the 16 × 16 matrices of the second copy of the Cl(8) algebra. In particular, 120 = 1 × 28 + 8 × 8 + 28 × 1 and 128 = 8 + 56 + 8 + 56 , hence the 248-dim E8 algebra decomposes into a 120 + 128 dim structure such that E8 can be represented indeed within a tensor product of Cl(8) algebras. At the E8 Lie algebra level, the E8 gauge connection decomposes into the SO(16) vector I, J = 1, 2, ...16 and (chiral) spinor A = 1, 2, ...128 indices as follows A Aµ = AIJ µ XIJ + Aµ YA .

XIJ = −XJI .

I, J = 1, 2, 3, ...., 16.

A = 1, 2, ....., 128.

(2.3)

where XIJ , YA are the E8 generators. The Clifford algebra (Cl(8) ⊗ Cl(8) ) structure behind the SO(16) A decomposition of the E8 gauge field AIJ µ XIJ + Aµ YA can be deduced from the expansion of the generators XIJ , YA in terms of the Cl(16) algebra generators. The Cl(16) bivector basis admits the decomposition IJ IJ X IJ = aIJ ij (γij ⊗ 1) + bij (1 ⊗ γij ) + cij (γi ⊗ γj ).

(2.4)

where γi , are the Clifford algebra generators of the Cl(8) algebra present in Cl(16) = Cl(8) ⊗ Cl(8); 1 is the unit Cl(8) algebra element that can be represented by a unit 16 × 16 diagonal matrix. The tensor products ⊗ of the 16 × 16 Cl(8)-algebra matrices, like γi ⊗ 1, γi ⊗ γj , ...... furnish a 256 × 256 Cl(16)-algebra matrix, as expected. The Cl(8) algebra basis elements are γM = 1,

γi ,

γi1 i2 = γi1 ∧ γi2 ,

γi1 i2 i3 = γi1 ∧ γi2 ∧ γi3 , ......., γi1 i2 ....i8 = γi1 ∧ γi2 ∧ .... ∧ γi8

(2.5)

Therefore, the decomposition in (2.4) yields the 28+28+8×8 = 56+64 = 120-dim bivector representation of SO(16); i.e. for each f ixed values of IJ there are 120 terms in the r.h.s of (2.4), that match the number of independent components of the E8 generators X IJ = −X JI , given by 21 (16 × 15) = 120 . The decomposition of YA is more subtle. A spinor Ψ in 16D has 28 = 256 components and can be decomposed into a 128 ˙ component left-handed spinor ΨA and a 128 component right-handed spinor ΨA ; The 256 spinor indices are ˙ ˙ ˙ ˙ α = A, A; β = B, B, ...... with A, B = 1, 2, ....128 and A.B = 1, 2, ..., 128, respectively. Spinors are elements of right (left) ideals of the Cl(16) algebra and admit the expansion Ψ = Ψα ξ α in a 256-dim spinor basis ξ α which in turn can be expanded as sums of Clifford polyvectors of mixed grade; i.e. into a sum of scalars, vectors, bivectors, trivectors, ..... . The chiral ( left handed, right-handed ) 128-component spinors Ψ± are obtained via the projection operators Ψ± =

1 (1 ± Γ17 )Ψ. Γ17 = Γ1 ∧ Γ2 ∧ ......... ∧ Γ16 . 2

(2.6)

˙

α α ≡ ξ A , so the left-handed (right-handed) spinor basis ξ± can be represented by a such that ξ+ ≡ ξ A ; ξ− column matrix (an element of the left ideal) with 128 non-vanishing upper ( lower ) components in the Weyl representation as α ξ± =(

1 ± Γ17 αβ ) [ (1 ⊗ 1)βδ Aδ + (γi ⊗ 1)βδ Aδi + (γi1 i2 ⊗ 1)βδ Aδi1 i2 + ......... 2 (γi1 i2 .....i7 ⊗ 1)βδ Aδi1 i2 .....i7 + (γi1 i2 .....i8 ⊗ 1)βδ Aδi1 i2 .....i8 ]

(2.7)

where the numerical tensor-spinorial coefficients in the r.h.s of (2.7) are constrained to satisfy all the conditions imposed by the definition of an ideal element of the Cl(16) algebra; namely that any element of the ideal upon a multiplication from the left by any Clifford algebra element yields another element of the left ideal. Similar definitions apply to the right ideal elements upon multiplication from the right by any Clifford algebra element. The row matrix (an element of the right ideal) with 128 non-vanishing components is just given by (ξ ± )† . The rigorous procedure to construct spinors as elements of right/left ideals of Clifford algebras using primitive idempotents can be found in [5] and references therein. The final outcome is the same as performing the expansion (2.7) and solving for the coefficients. In this fashion one can construct the 128-dim left handed 7

(right handed ) chiral spinor representations of SO(16) that match the number of 128 generators YA . Hence, the total number of E8 generators is then 120 + 128 = 248. What remains to be done is to enforce the E8 commutation relations that in conjunction with the defining relations of a primitive ideal element of the Cl(16) algebra will fix the values of the coefficients appearing in eqs-(2.4, 2.7) . Based on the fact that the Clifford algebra commutators of even and odd grade satisfy the relations [Even , Even] = Even. [Odd , Odd] = Even. [Even , Odd] = [Odd , Even] = Odd.

(2.8)

which are similar to the E8 commutation relations described below, one can immediately choose to expand the spinor basis elements in (2.7) as sums of Polyvectors of odd grade only, meaning that for each fixed value of δ, there are only 128 terms in the r.h.s of (2.7) given by the number of odd-grade elements of the Cl(8) algebra 8 + 56 + 56 + 8 = 128. This is consistent with the fact that a chiral spinor in 16D has 128 ˙ non-vanishing components in a Weyl representation. Therefore, the generators Y A ≡ Y+α ; Y A = Y−α must involve odd grade elements of the form 1 ± Γ17 αβ ) [(γi ⊗1)βδ Aδi +(γi1 i2 i3 ⊗1)βδ Aδi1 i2 i3 +(γi1 i2 ....i5 ⊗1)βδ Aδi1 i2 ...i5 +(γi1 i2 .....i7 ⊗1)βδ Aδi1 i2 .....i7 ] 2 (2.9) The commutation relations of E8 are [6]

Y±α = (

[X IJ , X KL ] = 4(δ IK X LJ − δ IL X KJ + δ JK X IL − δ JL X IK ) 1 1 Y B; [Y A , Y B ] = ΓIJ X IJ (2.10) [X IJ , Y A ] = − ΓIJ 2 AB 4 AB The combined E8 indices are denoted by A ≡ [IJ], A ( 120 + 128 = 248 indices in total ) that yield the Killing metric and the structure constants η AB =

1 A BCD 1 T r T A T B = − fCD f 60 60

(2.11a)

.

LJ f IJ,KL,M N = −8δ IK δM N + permutations;

1 f IJ,A,B = − ΓIJ ; 2 AB

η IJKL = −

1 IJ KL,CD f f 60 CD

(2.11b)

Therefore, the odd grade expansion in (2.9) and the bivector grade expansion in (2.4) is consistent with the commutation relations of E8 . We shall proceed with the construction of a novel Cl(16) gauge theory that encodes the exceptional Lie algebra E8 symmetry from the start. The E8 gauge theory in D = 4 is based on the E8 -valued field strengths IJ IJ KL N B Fµν XIJ = (∂µ AIJ AM [XKL , XM N ] + AA ν − ∂ν Aµ ) XIJ + Aµ ν µ Aν [YA , YB ].

(2.12)

A A A IJ Fµν YA = (∂µ AA ν − ∂ν Aµ ) YA + Aµ Aν [YA , XIJ ].

(2.13)

The E8 actions are Z ST opological [E8 ] = Z

1 A B d x T r [ Fµν Fρτ TA TB ] µνρτ = 60 4

Z

A B d4 x Fµν Fρτ ηAB µνρτ =

IJ KL A B IJ B d4 x [ Fµν Fρτ ηIJKL + Fµν Fρτ ηAB + 2Fµν Fρτ ηIJB ] µνρτ .

where µνρτ is the covariantized permutation symbol and Z Z 1 √ 4 √ A B µρ ντ A B SY M [E8 ] = d x g T r [ Fµν Fρτ TA TB ] g g = d4 x g Fµν Fρτ ηAB g µρ g ντ = 60 8

(2.14)

Z

d4 x

√

IJ KL A B IJ B g [ Fµν Fρτ ηIJKL + Fµν Fρτ ηAB + 2Fµν Fρτ ηIJB ] g µρ g ντ .

(2.15)

The above E8 actions (are part of ) can be embedded onto more general Cl(16) actions with a much larger number of terms given by Z Z 4 M N µνρτ M N ST opological [Cl(16)] = d x < Fµν Fρτ ΓM ΓN >  = d4 x Fµν Fρτ GMN µνρτ . (2.16) and Z SY M [Cl(16)] =

d4 x

√

M N g < Fµν Fρτ ΓM ΓN > g µρ g ντ =

Z

d4 x

√

M N g Fµν Fρτ GMN g µρ g ντ .

(2.17)

where < ΓM ΓN > = GMN 1 denotes the scalar part of the Clifford geometric product of the gammas. Notice that there are a total of 65536 terms in M N I I I 1 I2 I 1 I 2 I1 I2 .......I16 I1 I2 ......I16 Fµν Fρτ GMN = Fµν Fρτ + Fµν Fρτ + Fµν Fρτ + .......... + Fµν Fρτ .

(2.18)

where the indices run as I = 1, 2, .....16. The Clifford algebra Cl(16) has the graded structure ( scalars, bivectors, trivectors,....., pseudoscalar ) given by 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1.

(2.19)

16

consistent with the dimension of the Cl(16) algebra 2 = 256 × 256 = 65536. The possibility that one can acommodate another copy of the E8 algebra within the Cl(16) algebraic structure warrants further investigation by working with the duals of the bivectors XIJ and recurring to the remaining YA˙ generators. The motivation is to understand the full symmetry of the E8 ×E8 heterotic string from this Clifford algebraic perspective. A clear embedding is, of course, the following E8 × E8 ⊂ Cl(8) ⊗ Cl(8) ⊗ Cl(8) ⊗ Cl(8) ⊂ Cl(16) ⊗ Cl(16) = Cl(32).

(2.20)

where SO(32) ⊂ Cl(32) and SO(32) is also an anomaly free group of the heterotic string that has the same dimension and rank as E8 × E8 . 3 CHERN-SIMONS-GRAVITY IN 11D FROM A CLIFFORD ALGEBRA GAUGE THEORY The 11D Chern-Simons Supergravity action is based on the smallest Anti de Sitter OSp(32|1) superalgebra. The Anti de Sitter group SO(10, 2) must be embedded into a larger group Sp(32, R) to accomodate the fermionic degrees of freedom associated with the superalgebra OSp(32|1). The bosonic sector involves the connection [4] a1 a2 ....a5 Aµ = Aaµ Γa + Aab Γa1 a2 ....a5 = eaµ Γa + ωµab Γab + Aaµ1 a2 ....a5 Γa1 a2 ....a5 µ Γab + Aµ

(3.1) 2( 32×31 2 )

= with 11 + 55 + 462 = 528 generators. A Hermitian complex 32 × 32 matrix has a total of 32 + 992 + 32 = 1024 = 322 = 210 independent real components (parameters), the same number as the real parameters of the anti-symmetric and symmetric real 32 × 32 matrices 496 + 528 = 1024. The dimension of Sp(32) = (1/2)(32 × 33) = 528. Notice that 210 = 1024 is also the number of independent generators of the Cl(11) algebra since out of the 211 generators, only half of them 210 , are truly independent due to the duality conditions valid in odd dimensions only : a1 a2 .....a2n+1 Γa1 ∧ Γa2 ∧ ..... ∧ Γap ∼ Γap+1 ∧ Γap+2 ∧ ..... ∧ Γa2n+1 .

(3.2)

This counting of components is the underlying reason why the Cl(11) algebra appears in this section. The generators of the Cl(11) algebra {Γa , Γb } = 2η ab 1 and the unit element 1 generate the Clifford polyvectors (including a scalar, pseudoscalar ) of different grading ΓA = 1, Γa , Γa1 ∧ Γa2 , Γa1 ∧ Γa2 ∧ Γa3 , ......., Γa1 ∧ Γa2 ∧ ........ ∧ Γa11 . 9

(3.3)

obeying the conditions (3.2). The commutation relations (see eqs-(3.4) below) involving the generators Γa , Γab , Γa1 a2 ....a5 do in fact close due to the duality conditions (3.2). The Cl(11) algebra commutators, up to numerical factors, are [Γa , Γb ] = Γab . [Γa , Γbc ] = 2η ab Γc − 2η ac Γb

(3.4a)

[Γa1 a2 , Γb1 b2 ] = −η a1 b1 Γa2 b2 + η a1 b2 Γa2 b1 − ....

(3.4b)

[Γa1 a2 a3 , Γb1 b2 b3 ] = Γa1 a2 a3 b1 b2 b3 − (η a1 b1 a2 b2 Γa3 b3 + ....).

(3.4c)

[Γa1 a2 a3 a4 , Γb1 b2 b3 b4 ] = −(η a1 b1 Γa2 a3 a4 b2 b3 b4 + ....) − (η a1 b1 a2 b2 a3 b3 Γa4 b4 + ....).

(3.4d)

[Γa1 a2 , Γb1 b2 b3 b4 ] = −η a1 b1 Γa2 b2 b3 b4 + ....

(3.4e)

[Γa1 , Γb1 b2 b3 ] = Γa1 b1 b2 b3 .

[Γa1 a2 , Γb1 b2 b3 ] = −2η a1 b1 Γa2 b2 b3 + ....

(3.4f )

[Γa1 , Γb1 b2 b3 b4 ] = −η a1 b1 Γb2 b3 b4 + .....

(3.4g)

[Γa1 a2 ....a5 , Γb1 b2 ....b5 ] = Γa1 a2 ...a5 b1 b2 ....b5 + (η a1 b1 a2 b2 Γa3 a4 a5 b3 b4 b5 + .....) + (η a1 b1 a2 b2 a3 b3 a4 b4 Γa5 b5 + .....) = a1 a2 ...a5 b1 b2 ....b5 c Γc + (η a1 b1 a2 b2 a3 a4 a5 b3 b4 b5 c1 c2 .....c5 Γc1 c2 ....c5 + .....) + (η a1 b1 a2 b2 a3 b3 a4 b4 Γa5 b5 + .....). (3.4h) etc....... with ηa1 b1 a2 b2 = ηa1 b1 ηa2 b2 − ηa2 b1 ηa1 b2

(3.5a)

ηa1 b1 a2 b2 a3 b3 = ηa1 b1 ηa2 b2 ηa3 b3 − ηa1 b2 ηa2 b1 ηa3 b3 + .......

(3.5b)

ηa1 b1 a2 b2 ......an bn =

1 i i ......in j1 j2 ......jn ηai1 bj1 ηai2 bj2 ......... ηain bjn . n! 1 2

(3.5c)

The Cl(11) algebra gauge field is

a a1 a2 Γa1 a2 + Aaµ1 a2 a3 Γa1 a2 a3 + ......... + Aaµ1 a2 ....a11 Γa1 a2 .......a11 . Aµ = AA µ = A µ 1 + A µ Γa + A µ

(3.6)

and the Cl(11)-algebra-valued field strength b1 a A Fµν ΓA = ∂[µ Aν] 1 + [ ∂[µ Aaν] + Ab[µ2 Aν] ηb1 b2 + ..... ] Γa + a1 a b1 b a1 a2 a b1 b2 b a b [ ∂[µ Aab Aν] ηa1 b1 a2 b2 − Aa[µ1 a2 a3 a Aaν]1 b2 b3 b ηa1 b1 a2 b2 a3 b3 + ..... ] Γab + ν] + A[µ Aν] − A[µ Aν] ηa1 b1 − A[µ b1 bcd a1 a b1 bc abcd − Aa[µ1 a Aν] ηa1 b1 + ...... ] Γabcd + ......... [ ∂[µ Aabc ν] + A[µ Aν] ηa1 b1 + ...... ] Γabc + [ ∂[µ Aν]

[ ∂[µ Aaν]1 a2 ....a5 b1 b2 .....b5 + Aa[µ1 a2 ...a5 Abν]1 b2 ....b5 + ...... ] Γa1 a2 ....a5 b1 b2 .....b5 + .... The Chern-Simons actions rely on Stokes theorem Z Z µ1 µ2 ....µ11 µ12 ∂µ12 (Aµ1 µ2 ....µ11 ) = M 12

∂M 12 =Σ11

µ1 µ2 ....µ11 µ12 Aµ1 µ2 ....µ11 dΣ11 µ12 .

(3.7)

(3.8)

which in our case reads d (LClif f ord ) =< F ∧ F ∧ .......... ∧ F > = < F A1 ∧ F A2 ∧ ......... ∧ F A6 ΓA1 ΓA2 ....ΓA6 > 10

(3.9)

where the bracket < ...... > means taking the scalar part of the Clifford geometric product among the gammas. It involves products of the dABC , fABC structure constants corresponding to the ( anti ) commutators {ΓA , ΓB } = dABC ΓC and [ΓA , ΓB ] = fABC ΓC . One of the main results of this work is that the Cl(11) algebra based action (3.9) contains a vast number of terms among which is the Chern-Simons action of [4] L11 CS (e, ω, A5 ) 11 LClif f ord (AA µ ΓA ) = LCS (ω, e, A5 ) + EXT RA T ERM S.

Z SCS (ω, e, A5 ) = ∂M 12

L11 CS =

Z Σ11

L11 CS .

(3.10) (3.11)

11 11 11 L11 CS (ω, e, A5 ) = LLovelock (ω, e) + LP ontryagin (ω, e) + L (A5 , ω, e)

(3.12)

In odd dimensions D = 2n − 1, the Lanczos-Lovelock Lagrangian is LD Lovelock =

n−1 X

ap Lp (D).

p=0

ap = κ

(±1)p+1 l2p−D n−1 Cp ; (D − 2p)

p = 1, 2, ....., n − 1

(3.13)

Cpn−1 is the binomial coefficient. The constants κ, l are related to the Newton’s constant G and to the cosmological constant Λ through κ−1 = 2(D − 2)ΩD−2 G where ΩD−2 is the area of the D − 2-dim unit sphere and Λ = ±(D − 1)(D − 2)/2l2 for de Sitter ( Anti de Sitter ) spaces [4] . A derivation of the vacuum energy density of Anti de Sitter space (de Sitter ) as the geometric mean between an upper and lower scale was obtained in [17] based on a BF-Chern-Simons-Higgs theory. Upon setting the lower scale to the Planck scale LP and the upper scale to the Hubble radius (today) RH , it yields the observed value of the −2 −4 2 −120 cosmological constant ρ = L−2 MP4 . P RH = LP (LP /RH ) ∼ 10 The terms inside the summand of (3.13) are Lp (D) = a1 a2 .......aD Ra1 a2 Ra3 a4 ....Ra2p−1 a2p ea2p+1 .......eaD

(3.14)

where we have omitted the space-time indices µ1 , µ2 , ......... Despite the higher powers of the curvature ( after eliminating the spin connection ωµab in terms of the eaµ field ) the LD Lovelock furnishes equations of motion for the eaµ field containing at most derivatives of second order, and not higher, due to the Topological property of the Lovelock terms a1 a2 d (L11 + Lovelock ) = a1 a2 .....a11 (R

ea1 ea2 ea9 ea10 )......(Ra9 a10 + ) T a11 = Euler density in 12D. 2 l l2

(3.15)

The exterior derivative of the Lovelock terms can be rewritten compactly as A1 A2 d (L11 ......F A11 A12 Lovelock ) = A1 A2 ....A12 F

(3.16)

1 A2 where F A1 A2 is the curvature field strength associated with the SO(10, 2) connection ΩA in 12D and µ a ab which can be decomposed in terms of the fields eµ , ωµ , a, b = 1, 2, ...., 11 by identifying ΩaD = 1l eaµ and µ ab ab Ωµ = ωµ so that the Torsion and Lorenz curvature 2-forms are

T a (ω, e) = F aD = dΩaD + Ωab ∧ ΩbD =

1 a (de − ωba ∧ eb ). l

1 a e ∧ eb . Rab (ω) = dω ab + ωca ∧ ω cb (3.17) l2 where a length parameter l must be introduced to match dimensions since the connection has units of 1/l. This l parameter is related to the cosmological constant. L11 P ontryagin (ω, e) is the Chern-Simons 11-form whose exterior derivative F ab = (dΩab + Ωac ∧ Ωcb ) + (ΩaD ∧ ΩDb ) = Rab (ω) +

d (LP ontryagin ) = FAA21 FAA32 .......FAA65 FAA16 11

(3.18)

is the (one of the many) Pontryagin 12-form (up to numerical factors) for the SO(10, 2) connection in 12D. As mentioned above, the SO(10, 2) connection ΩAB can be broken into the eaµ field and the SO(10, 1) spin µ ab connection ωµ such that the number of components is 11 + 21 (11 × 10) = 66 = 12 (12 × 11). Finally, the exterior derivative dL11 (A5 , ω, e) is the 12-form (we are omitting space-time indices µ1 , µ2 , ......, µ12 ) that is comprised of terms of the form (Ra1 a2 .....a5 Ra1 a2 ....a5 )(Rb1 b2 Rb1 b2 )(T c Tc ); (Ra1 a2 .....a5 Ra1 a2 ....a5 )2 (Rb1 b2 Rb1 b2 ); (Ra1 a2 .....a5 Ra1 a2 ....a5 )3 ; ....... (3.19) the curvature 2-form associated with the field Acµ1 c2 .....c5 is given by c1 c2 .....c5 5 Rµν = ∂[µ Acν]1 c2 .....c5 + Aa[µ1 a2 .....a5 Abν]1 b2 .....b5 fac11ac22....c ...a5 b1 b2 ....b5 .

(3.20)

where the structure constants fABC in (3.18) are obtained from the Cl(11) algebra commutation relations in (3.4h). 4. CONCLUSIONS : GENERALIZED CHERN-SIMONS GRAVITY IN CLIFFORD SPACES The Cl(11) algebra based action (3.9, 3.10) can in turn be embedded into a more general expression in C-space (Clifford Space) which is a generalized tensorial spacetime of coordinates X = σ, xµ , xµν , xµνρ .... [3] involving a scalar Φ(X) and antisymetric tensor gauge fields Aµ (X), Aµν (X), Aµνρ (X)..... of higher rank (higher spin theories) [13]. The most general action onto which the action (3.9,3.10) itself can be embedded requires a tensorial gauge field theory [13] (Generalized Yang-Mills theories) and an integration w.r.t all the Clifford-valued coordinates X = X M ΓM corresponding to the 2D -dim C-space associated with the underlying Cl(2n)-algebra in D = 2n dimensions Z S=

n

n

[d2 X] < (F ∧ F ∧ ..... ∧ F) > . [d2 X] = (dσ)(dxµ )(dxµν )(dxµνρ )......

(4.1)

A different sort of Generalized Yang-Mills theories have been studied by [12] without the Clifford algebraic structure. Given a Lie algebra G with generators Ta for a = 1, 2, 3, ....dim G, it has for commutators [Ta , Tb ] = c fab Tc and whose structure constants fabc are fully antisymmetric in their indices. The Lie-algebra valued one-form is A = (AaM (X)Ta )dX M and its generalized Lie-algebra valued field strength c M F = [FM ∧ dX N = N (X) Tc ] dX c [ ∂[M AcN ] (X)Tc + g AaM (X)AbN (X) fab Tc ] dX M ∧ dX N .

(4.2)

has for components F c[

[µ1 µ2 ...µm ] [ν1 ν2 .....νn ] ]

=

c ∂x[µ1 µ2 ...µm ] Ac[ν1 ν2 ....νn ] − ∂x[ν1 ν2 ...νn ] Ac[µ1 µ2 ....µm ] + g Aa[µ1 µ2 ...µm ] Ab[ν1 ν2 ....νn ] fab .

(4.3)

The remaining components are of the form c c F[0N ] =F [

0 [ν1 ν2 .....νn ] ]

c = ∂σ Ac[ν1 ν2 ....νn ] − ∂x[ν1 ν2 ...νn ] Ac0 + g Aa0 Ab[ν1 ν2 ....νn ] fab .

(4.4)

where Ac0 is the Clifford-scalar part Φ(X) of the Lie-algebra valued Clifford-polyvector and in general we must consider the m = n and m 6= n cases resulting from the mixing of different grades ( ranks ). The antisymmetry with respect the collective indices M N is explicit. In order to raise, lower and contract polyvector indices in C-space it requires a generalized metric GM N . In flat C-space it is defined by the components : Gµν = η µν .

Gµ1 µ2

ν1 ν2

= η µ1 ν1 η µ2 ν2 − η µ1 ν2 η µ2 ν1 12

etc..

(4.5)

in addition to the scalar-scalar component Gσσ = 1. It can be recast as : Gµ1 µ2 ....µm

ν1 ν2 ....νm

= det GµI νJ =

1 i i ...i j j ....j η µi1 νj1 η µi2 νj2 ......η µim νjm . m! 1 2 m 1 2 m

(4.6)

where GµI νJ is an m × m matrix whose entries are η µi νj for i, j = 1, 2, 3, ......m ≤ D and µ, ν = 1, 2, 3, ......D. As a result of the expression for the flat C-space metric, given by sums of antisymmetrized products of η µν , the Clifford-space generalized Yang-Mills action is of the form Z X 1 SY M = − [DX] trace [ F a[ [µ1 µ2 ...µm ] [ν1 ν2 .....νm ] ] F [ [µ1 µ2 ...µm ] [ν1 ν2 ......νm ] ] b Ta Tb ] + 2 Z X 1 − [DX] trace [ F a[ 0 [ν1 ν2 .....νm ] ] F [ [ 0 [ν1 ν2 ......νm ] ] b Ta Tb ] (4.7) 2 where the C-space 2D -dim measure associated with a Clifford algebra in D-dim is [DX] = [dσ] [Π dxµ ] [Π dxµ1 µ2 ] [Π dxµ1 µ2 µ3 ].... [dxµ1 µ2 .....µd ]

(4.8)

and the indices are ordered as µ1 < µ2 < µ3 ....... < µm , etc... The action (4.7) is invariant under the infinitesimal gauge transformations c δξ AcM = ∂M ξ c + gfab AaM ξ b ;

c δξ Acµ1 µ2 ....µn = ∂xµ1 µ2 ....µn ξ c + gfab Aaµ1 µ2 ....µn ξ b .

(4.9)

associated with a Lie-algebra valued Clifford-scalar parameter ξ(X) = ξ a (X)Ta . In section 1.1 it was explained why another alternative to define the transformations in C-space was by C EC writing the generators of polyrotations as R = exp (ΩAB [EA , EB ]) where the commutator [EA , EB ] = FAB is the C-space analog of the i[γµ , γν ] commutator which is the generator of the Lorentz algebra, and the parameters ΩAB are the C-space analogs of the rotation/boots parameters. This last alternative seems to be more physical because a polyrotation should map the EA direction into the EB direction in C-spaces, hence the meaning of the generator [EA , EB ] which is the generalization of the ordinary i[γµ , γν ] Lorentz generator. Therefore, when we recast the generators of polyrotations as JAB = [ΓA , ΓB ], an action of the form Z A B2d−1 A1 B 1 A2 B2 S(Cspace ) = [DX] FM FM ...... FM2d−1 A1 B1 A2 B2 ......A2d−1 B2d−1 M1 N1 M2 N2 .....M2d−1 N2d−1 . 1 N1 2 N2 d−1 N d−1 2

2

(4.10) is the natural generalization of the Euler density types of the D-dim ( D = 2n) actions given by eq-(3.16) in C-space. This action S(Cspace ) (4.10) is more general than the action SClif f ord (AA µ ΓA ) of eq-(3.10), and which in turn, is more general than the Chern-Simons gravitational action SCS (ω, e, A5 ) given by eq-(3.12). Therefore, we have the inclusions µ AB µ µ1 µ2 SCS (ω, e, A5 ) ⊂ SClif f ord [ AA , xµ1 µ2 µ3 , ....) JAB ]. µ (x ) ΓA ] ⊂ S(Cspace ) [ AM (σ, x , x

(4.11)

which should be very relevant in future developments of M, F theory upon the introduction of Polyvectorvalued Supersymmetries in C-spaces [11] . These generalized supersymmetries deserve to be investigated further since they are more fundamental than the supersymmetries associated with M, F theory superalgebras and also span well beyond the N -extended Supersymmetric Field Theories involving super-algebras, like OSp(32|N ) for example, which are related to a SO(N ) gauge theory coupled to matter fermions (besides the gravitinos). It is these Polyvector-valued Supersymmetries in C-spaces [11] that will permit the supersymmetrization of the most general action in C-spaces S(Cspace ) given by (4.10). Finally, the results of this work may shed some light into the origins behind the hidden E8 symmetry of 11D Supergravity , the hyperbolic Kac-Moody algebra E10 and the non-linearly realized E11 algebra related to Chaos in M theory and oscillatory solutions close to cosmological singularities [1,2,6]. 13

Acknowledgments We are indebted to Frank (Tony) Smith for numerous discussions and to M. Bowers for her hospitality.

APPENDIX : CLOSURE OF THE CLIFFORD SPACE SUPERSYMMETRY The classification of the family of symmetric matrices (Cγ µ1 µ2 ....µn )αβ is what restricts the type of terms that appear in the {Qα , Qβ } anticommutator and depends on the number of space time dimensions D, the signatures (s, t) and the rank n. A table of the allowed values of D, s, t, n can be found in [34] . In particular, when D = 4 = 3 + 1, the {Qα , Qβ } is a symmetric matrix in α, β with 10 independent components and which matches the degrees of freedom in P µ , P µν given by 4 + 6 = 10. Let us study the closure of { [Mµ1 µ2 µ3 µ4 , Qα ], Qβ } + { [Mµ1 µ2 µ3 µ4 , Qβ ], Qα } = [ Mµ1 µ2 µ3 µ2 , {Qα , Qβ } ]. where {Qα , Qβ } =

1 1 µ Cγ Pµ + Cγ µν Pµν 2 2

(A.1) (A.2)

In D = 4 , with signatures −, +, +, + one can find a charge conjugation matrix C and its transpose C T obeying the properties (Cγ µ )T = (Cγ µ ). (Cγ µν )T = (Cγ µν ) C T = −C,

Cγµ C −1 = −γµT .

C † C = CC † = 1,

T C −1 γµν C = −γµν .

(A.3) (A.4)

It is convenient to use a Majorana representation where the charge conjugation matrix is given by C = γ0 and γ5T = −γ5 is a hermitian matrix that has zero entries along the diagonal and −iσ1 , iσ1 off the diagonal. We must verify that (A-1) is obeyed. This requires that the spinorial charges Qα behave under polyrotations as follows 1 [Mµ1 µ2 µ3 µ4 , Qα ] = − (γµ1 µ2 µ3 µ4 )αδ Qδ . 2

(A.5)

[Mµ1 µ2 µ3 µ4 , Pν1 ν2 ] = ηµ1 µ2 ν1 ν2 Pµ3 µ4 + ηµ3 µ4 ν1 ν2 Pµ1 µ2 ± ............

(A.6)

and the ± signs in the r.h.s of (A-6) depend on the permutation of indices w.r.t to the initial combination µ1 µ2 µ3 µ4 , ν1 ν2 . There are 6 terms in ( A.6 ). The l.h.s of ( A-1) is 1 1 − γ5 (Cγ µ Pµ + Cγ µν Pµν ) − [ γ5 (Cγ µ Pµ + Cγ µν Pµν ) ]T = 4 4 1 1 − γ5 (Cγ µ Pµ + Cγ µν Pµν ) − [(Cγ µ )T γ5T Pµ + (Cγ µν )T γ5T Pµν ] = 4 4 1 1 − γ5 (Cγ µ Pµ + Cγ µν Pµν ) + (Cγ µ Pµ + Cγ µν Pµν )γ5 . 4 4 where we have used the conditions (A-3 ) and γ5T = −γ5 . Multiplying ( A-7) from the left by C −1 and using C −1 γ5 C = −γ5 yields 1 1 (γ5 γ µ + γ µ γ5 )Pµ + (γ5 γ µν + γ µν γ5 )Pµν = 4 4 1 1 1 (γ5 γ µν + γ µν γ5 )Pµν = γ5 γ µν Pµν = γ[µ1 γµ2 γµ3 γµ4 ] γ ν1 ν2 Pν1 ν2 = 4 2 2 1 [ γ[µ1 µ2 ] ηµν13νµ24 + ................... ]Pν1 ν2 = 2 14

(A.7)

1 [ γ[µ1 µ2 ] Pµ3 µ4 + γ[µ3 µ4 ] Pµ1 µ2 ± ................... ]. 2

(A.8)

one may notice that due to the condition {γ5 , γµ } = 0 there are no Pµ terms in (A-8). The r.h.s of (A-1 ) is 1 [ Mµ1 µ2 µ3 µ4 , (Cγ ν1 Pν1 ) + (Cγ ν1 ν2 )Pν1 ν2 ] = 2 1 (Cγ ν1 ν2 ) [ ηµ1 µ2 ν1 ν2 Pµ3 µ4 + ηµ3 µ4 ν1 ν2 Pµ1 µ2 + ............ ]. 2

(A.9)

[ Mµ1 µ2 µ3 µ4 , (Cγ ν1 Pν1 )] = Cγ ν1 [ Mµ1 µ2 µ3 µ4 , Pν1 ] = 0.

(A.10)

where Multiplying (A-9) on the left by C −1 yields 1 ν1 ν2 1 γ [ ηµ1 µ2 ν1 ν2 Pµ3 µ4 + ηµ3 µ4 ν1 ν2 Pµ1 µ2 + .......... ] = [ γ[µ1 µ2 ] Pµ3 µ4 + γ[µ3 µ4 ] Pµ1 µ2 ± ...... ]. 2 2

(A.11)

We have seen that a left-multiplication of the r.h.s and l.h.s of (A-1) by C −1 , leads to the equality of (A-8) with ( A-11 ), which implies that (A-1) is indeed satisfied . The Jacobi identity { [Mµ1 µ2 , Qα ], Qβ } + { [Mµ1 µ2 , Qβ ], Qα } = [ Mµ1 µ2 , {Qα , Qβ } ].

(A.12)

when 1 1 1 [Mµ1 µ2 , Pρ1 ρ2 ] = −ηµ1 ρ1 Pµ2 ρ2 ± ... ; [Mµ1 µ2 , Qα ] = − (γµ1 µ2 )αδ Qδ . {Qα , Qβ } = Cγ ν Pν + Cγ ν1 ν2 Pν1 ν2 2 2 2 (A.13) involves terms containing Pµ and Pµν . We know that the Jacobi identity is satisfied for the Pµ terms since this is what the ordinary supersymmetry algebra entails. The Pµν terms involve the commutator −[γµ1 µ2 , γν1 ν2 ]P ν1 ν2 = (ηµ1 ν1 γµ2 ν2 ± ........ )P ν1 ν2 .

(A.14)

Each one of the four terms in ( A-14) , for example, like the term ηµ1 ν1 γµ2 ν2 P ν1 ν2 can be rewritten as : ηµ1 ν1 γµ2 ν2 P ν1 ν2 = ηµ1 ν1 γ ρ1 ρ2 ηρ1 ρ2 µ2 ν2 η ν1 ν2 µ2 ρ2 Pµ2 ρ2 = −ηµ1 ν1 δρν11 γ ρ1 ρ2 Pµ2 ρ2 = −ηµ1 ρ1 γ ρ1 ρ2 Pµ2 ρ2 . (A.15) and similarly one can rewrite the other three terms of (A-14 ), so that the Jacobi idenity (A-12) is satisfied due to the equality in (A-15) γ ρ1 ρ2 [Mµ1 µ2 , Pρ1 ρ2 ] = γ ρ1 ρ2 (−ηµ1 ρ1 Pµ2 ρ2 ± ...) = −[γµ1 µ2 , γν1 ν2 ]P ν1 ν2 = P ν1 ν2 (ηµ1 ν1 γµ2 ν2 ± ...). (A.16) i.e, the equality among the terms of (A-16) can be seen effectively as exchanging γ ↔ P and (ν1 , ν2 ) ↔ (ρ1 , ρ2 ). One must have as well : [Qα , Pµ ] = [Qα , Pµν ] = 0. [Pµ , Pν ] = [Pµ1 µ2 , Pν1 ν2 ] = 0....

(A.17)

This example in D = 4 should be valid in other dimensions and signatures provided we have the appropriate list of symmetric (Cγ µ1 µ2 ...µn )αβ matrices. One has the remaining commutators : [Mµ1 µ2 , Mν1 ν2 ] = −ηµ1 ν1 Mµ2 ν2 + ηµ2 ν1 Mµ1 ν2 ± ...

(A.18)

[Mµ1 µ2 µ3 µ4 , Mν1 ν2 ν3 ν4 ] = ηµ1 µ2 ν1 ν2 Mµ3 µ4 ν3 ν4 ± ......

(A.19)

[Mµ1 µ2 , Mν1 ν2 ν3 ν4 ] = −ηµ1 ν1 Mµ2 ν2 ν3 ν4 ± ......

(A.20)

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