Grand Symmetry, Spectral Action, and the Higgs mass

In the context of the spectral action and the noncommutative geometry approach to the standard model, we build a model based on a larger symmetry. With this"grand symmetry"it is natural to have the scalar field necessary to obtain the Higgs mass in the vicinity of 126 GeV. This larger symmetry mixes gauge and spin degrees of freedom without introducing extra fermions. Requiring the noncommutative space to be an almost commutative geometry (i.e. the product of manifold by a finite dimensional internal space) gives conditions for the breaking of this grand symmetry to the standard model.


Introduction
Noncommutative geometry [1][2][3][4] generalizes the concepts of ordinary geometry in an algebraic setting and enables powerful generalizations beyond the Riemannian paradigm. Its application to the standard model of fundamental interactions is a fascinating one [5][6][7][8][9]: the geometrical setting is that of an usual manifold (spacetime) described by the algebra of complex valued functions defined on it, tensor multiplied by a finite dimensional matrix algebra. This is usually called an "almost commutative geometry". The standard model is described as a particular almost commutative geometry, and the corresponding Lagrangian is built from the spectrum of a generalized Dirac operator. This approach to the standard model has a phenomenological predictive power and is approaching the level of maturity which enables it to confront with experiments.
Schematically the application of noncommutative geometry to the standard model has two sides. One is the mathematical request that a topological space is a manifold. This yields a set of algebraic requirements [10] involving the algebra of functions defined on the space, represented as bounded operators on a spinorial Hilbert space, and a (generalized) Dirac operator, plus two more operators representing charge conjugation and chirality. These requirements, being algebraic, can easily be generalized to noncommutative algebra [11]. In the almost commutative case some extra assumptions on the representation [12] single out the algebra corresponding to the standard model among a restricted number of cases [8,13] as the smallest algebra which satisfies the requirements.
The other side has to do with the spectral nature of the action. The spectral action principle [14] allows to derive from a unique noncommutative spacetime a Lagrangian for both general relativity (in Euclidean signature) and the standard model. The principle is purely spectral, based on the regularization of the eigenvalues of the Dirac operator 1 . In [8] (see also [13,20]) this noncommutative model was enhanced to include massive neutrinos and the seesaw mechanism. The most remarkable result is the possibility to predict the mass of the Higgs particle from the mass of the other fermions and the value of the unification scale. Earlier version of the model had a prediction around 170GeV, a value ruled out by Tevatron in 2008. Recently Connes and Chamseddine showed in [21] that the experimental value of the mass of Higgs at 126 GeV can be obtained introducing a new scalar field σ suitably coupled to the Higgs field. Such a field had previously been proposed from a completely different perspective by particle physicists (see for example [22]), to avoid an high energy instability [23][24][25] in the Higgs potential.
The idea of a new scalar field to lower the mass of the Higgs in the Connes approach is not new, and was already proposed by Stephan in [26]. However, he obtained it adding new fermions [27,28], whereas in [21] (as well as in the present paper) the fermionic contents of the standard model is not touched. In [21] the field σ is obtained by simply turning one of the elements of the internal Dirac operator into a field. As explained in section 2.4, this is somehow artificial because the usual NCG procedure to obtain scalar fields (the so called fluctuations of the metric) does not work for the field σ, because of one of the conditions on spectral triples (the first order condition). In §4.3 we show how to overcome this difficulty by considering a larger algebra. 2 This is the main result of the paper.
More precisely, in [12] it is shown that under minimal condition on the representation of the algebra, the smallest nontrivial almost commutative manifold corresponds to the standard model. Here we consider a larger algebra, that we term grand algebra.We show how to obtain the field σ by fluctuating the Majorana mass term of the Dirac operator, in a way compatible with the first order condition induced by this Majorana mass term. Then we show how the first order condition imposed by the free Dirac operator reduces the grand algebra to the one of the standard model. The field σ then appears as the Higgs-like field corresponding to this reduction. All this is possible because we intertwine in a non trivial way the Riemann-spin degrees of freedom (the components of spinors) with the internal degrees of freedom (the particles of the standard model). This puts in a new light also the phenomenon of fermion doubling [8,[31][32][33] present in the theory.
The paper is organized as follows. In section 2 we briefly recall the spectral triple construction ( §2.1) and introduce the Hilbert space ( §2.2) and the Dirac operator ( §2.3) of the standard model. We recall in §2.4 how the Higgs mass is obtained from the spectral action, and point out the difficulty in generating the field σ by fluctuation of the metric. Section 3 deals with the choice of the algebras and their representation. We first discuss the algebra of the standard model in §3.1, then introduce the grand algebra in §3.2. The reduction imposed by the grading condition are worked out in §3. 3. In section 4 we explain how the grand algebra allows to obtain the field σ: first we work out the most general Dirac operator D ν compatible with the grand algebra and containing a Majorana mass term for the neutrino ( §4.1), then we calculate the reduction imposed by the first order condition induced by D ν ( §4.2), finally we show that σ can be obtained by a fluctuation of D ν respecting this first order condition ( §4.3). Section 5 deals with the reduction of the grand algebra to the algebra of the standard model ( §5.1) and the issue of Lorentz invariance and the emergence of the spin structure ( §5.2). Possible physical interpretations are discussed in §5.3. A final section contains conclusions and some speculative comments.
2 The spectral triple of the standard model

Spectral triples
The basic device in the construction is a spectral triple (A, H, D) consisting of a *-algebra A of bounded operators in a Hilbert space H -containing the identity operator -and a non-necessarily bounded self-adjoint operator D. These three elements satisfy a set of properties allowing to prove Connes reconstruction theorem: given any spectral triple (A, H, D) with commutative A satisfying the required conditions, then A ≃ C ∞ (M) for some Riemannian spin manifold M. The required conditions can be found in [10], and their noncommutative generalization in [11]. In this work we will be interested only in -the grading condition: there is an operator Γ (called chirality) such that Γ 2 = I, -the order zero condition: there is an antilinear isometry J (called real structure) which implements an action of the opposite algebra 3 A • obtained by identifying b • = Jb * J −1 , and which commutes with the action of A: The operator J must obey 1) J 2 = ±I; 2) JD = ±DJ; 3) JΓ = ±ΓJ, with choice of signs dictated by the KO-dimension of the spectral triple.
-the first order condition

Hilbert space of the standard model
A particular form of noncommutative manifolds, suitable to describe the standard model of elementary particles [11], are the almost commutative geometries, given by the product of an ordinary manifold M (that from now on is assumed to have dimension 4) by a finite dimensional spectral triple. The algebra is where A F is a finite dimensional algebra, whose choice is dictated by the gauge group of the standard model and is discussed in section 3. For the Hilbert space a suitable choice is where sp(L 2 (M)) is the Hilbert space of square summable spinors on M and where J is the charge conjugation operator, γ 5 the product of the four γ matrices, and with cc the complex conjugation. Notice that right particles and left antiparticles have chirality +1, whereas left particles and right antiparticles have chirality −1.
The Hilbert space H defined in (2.5) is the tensor product of four dimensional spinors by the 96-dimensional elements of H F , thus (as a vector bundle over M) it has dimension 384, or 128 for a single generation. This redundancy of states is known as fermion doubling [31][32][33]. The problem is not only the over-counting, but the presence of states which do not have a definite parity, being left chiral in the inner indices and right chiral in the outer ones, or viceversa. Since the total chirality Γ is the product of γ F (which acts on the inner indices of H F ) by γ (which acts on the spin indices), the spurious states are the ones for which ΓΨ = −Ψ. Taking the functional integral of the fermionic action to be a Pfaffian [8] allows to project out these extra degrees of freedom. However, one cannot simply project out the extra states and work with a representation on a smaller Hilbert space, because in the bosonic spectral action all degrees of freedom are necessary [31] in order to obtain the proper action of the standard model coupled with gravity. We will see in the following that the fermion doubling may be in fact an essential feature of the model, by allowing to represent an algebra bigger than the one of the standard model.

Dirac operator
The operator D (still called Dirac operator) for the spectral triple of the standard model is (2.10) The matrix M contains the Yukawa couplings of the fermions and the mixing matrices (CKM for quarks and NPMS for neutrinos). It couples left with right particles. The matrix M R = M T R contains Majorana masses and couples right particles with right antiparticles.
The operators γ F , J F and D F are such that meaning that the finite part of the spectral triple has KO-dimension 6 [8,20]. The manifold part has KO-dimension 4, and the full spectral triple has KO-dimension 6 + 4 = 10 mod 8 = 2.

Spectral action, Higgs mass and the σ field
Given an almost commutative geometry (A, H, D), a fluctuation of the metric [11] 5 means the substitution of D by the gauge Dirac operator [35] is a generalized gauge potential. It is made of two parts: a scalar field on M with value in A F , and 1-form field on M with value in the group of unitaries of A F . In case A F = A sm is the algebra of the standard model (discussed in §3), the 1-form fields yield the vector bosons mediating the three fundamental interactions, and the scalar field is the Higgs field H.
The spectral action [14] is based on a regularization of the spectrum of D A . It reads: where χ is a cutoff function, usually the (smoothened) characteristic function on the interval [0, 1], and Λ is a renormalization scale. It has an expansion in power series of Λ −1 , lim Λ→∞ S B = n f n a n (D 2 A /Λ 2 ) (2.14) where the f n are the momenta of χ and the a n are the Seeley-de Witt coefficients [36,37].
Applied to the operator (2.9) fluctuated as in (2.12) with a i , b i ∈ C ∞ (M) ⊗ A sm , the expansion (2.14) yields the bosonic part of the Lagrangian of the standard model coupled with gravity [8, § 4.1] 6 . Furthermore the parameters related to the Higgs come out to be function of the parameters in D F , i.e. the Yukawa couplings, which are in turn dominated by the top quark coupling. In this sense the model predicts the Higgs mass as a function of the other gauge couplings, the Yukawa top mass and the scale Λ which appears also as the scale in which the three gauge couplings coincide. This last point is known to be true only in an approximate sense. If one takes the unification scale to be Λ = 10 17 GeV then one finds -assuming the big desert hypothesis -a Higgs mass of the order of 170 GeV. This value is not in agreement with the recent LHC experiments [45].
One can think of extending the model to solve this. There have been several proposals in this sense, and some of them are reviewed in [7]. In particular C. Stephan has proposed in [26] that the presence of an extra scalar field, corresponding to the breaking of a extra U(1) symmetry, can bring down the mass of the Higgs to 126 GeV. This model however contains extra fermions. Earlier examples of extensions are in [27,28,[46][47][48][49][50].
Recently, in [21] the noncommutative geometry model was enhanced to also overcome the high energies instability of a Higgs boson with mass around 126 GeV, in addition to predicting the correct mass. This is done ruling out the hypothesis of the "big desert" and considering an additional scalar field σ that lives at high energies and gives mass to the Majorana neutrinos. Explicitly σ is obtained in [21] by turning (inside the finite dimensional part D F of the Dirac operator) the constant-entry k R of the Majorana matrix M R into a field: However, the origin of the field σ is quite different from the Higgs. The latter, like the other bosons, are components of the gauge potential A. They are obtained from the commutator of D F with the algebra: D F has constant components, that is without space dependence, but when these numbers are commuted with elements of the algebra they give rise to the desired bosonic fields. One could hope to obtain σ in a similar way, by considering k R as a Yukawa coupling. As explained in appendix B, the problem is that in taking the commutator with elements of the algebra A sm , the coefficient k R does not contribute to the potential because of the first order condition. This forced the authors of [21] to "promote to a field" only the entry k R , in a somewhat arbitrary way. Indeed the components of D F cannot all be fields to start with, otherwise the model would loose its predictive power, in that all Yukawa couplings would be fields, and the masses of all fermions would run independently, thus making any prediction impossible. In the following (sections 3 and 4) we show that there is a way to obtain the field σ from k R by a fluctuation of the metric, provided one starts with an algebra larger than the one of the standard model.

Algebras and representations
Under assumptions on the representation (irreducibility, existence of a separating vector), the most general finite algebra in (2.4) that satisfies all conditions for the noncommutative space to be a manifold is This algebra acts on an Hilbert space of dimension 2(2a) 2 [12,51].

The algebra of the standard model
To have a non trivial grading on M a (H) the integer a must be at least 2, meaning the simplest possibility is Hence an Hilbert space of dimension 2(2 · 2) 2 = 32, that is the dimension of H F for one generation. The grading condition [a, Γ] = 0 reduces the algebra to the left-right algebra: This is basically a Pati-Salam model [52], one of the not many models allowed by the spectral action [53]. The order one condition reduces further the algebra to [8] (for a review see also [9]) where H are the quaternions, which we represent as 2 × 2 matrices, and M 3 (C) are 3 × 3 complex valued matrices. A sm is the algebra of the standard model, that is the one whose unimodular group is U(1)×SU(2)×U(3). The details of these reductions are given in appendix A.
These algebras -tensorized by C ∞ (M) -are represented on the Hilbert space (2.5), whose elements are 384 components vectors. The number 384 comes from degrees of freedom which have different physical meaning. Some of them refer to "internal" degrees of freedom, like colour, some refer to the Riemannian-spin structure, and have a spacetime meaning. We denote a generic fermion, i.e. an element of H by The position of the indices, whose meaning is described below, is a matter of convention, Ψ is a C 384 -vector valued function on M, we write some of them as upper indices and some as lower to avoid having six indices in a row. Note the difference between H F and H F : the latter is a 96 dimensional space and its vectors are to be multiplied by spinors, while the former is the larger 384 dimensional space which exhibits explicitly the fermion doubling over-counting. So far in the literature the Hilbert space has been considered always in its factorized form involving H F . One of the novelties of this work is to use the factorized form involving H F . This allows us in section 3.2 to consider algebras which do not act separately on spinors and the internal part. This means that in addition of the internal degrees of freedom used in [13], our tensorial notation also includes spin indices s,ṡ.
The meaning and range of the various indices of Ψ CIm sṡα (x) is the following: are the spinor indices. They are not internal indices in the sense that the algebra A F acts diagonally on it. They take two values each, and together they make the four indices on an ordinary Dirac spinor. The index s = r , l indicates chirality and runs over the right, left part of the spinor, whileṡ differentiates particles from antiparticles. In the chiral basis one thus has 7 where for µ = 0, 1, 2, 3 one defines with σ i , i = 1, 2, 3 the Pauli matrices, namely σ 0 = I 2 , For the remainder of this paper the generation index m does not play any role. We will therefore suppress it and work with one generation, thus effectively considering H F and H F having dimension 32 and 128 respectively.
) acts as a matrix on vectors of H F with index structure (3.5), it is therefore a matrix with twice as many indices 8 : the finite dimensional algebra A F acts only on the internal indices. The two terms in the bracket act only on particles and antiparticles respectively, as signified by δ C 0 and δ C 1 . They are such that the order zero condition hold. Note in fact that for particles the action is trivial on the I, J indices, and for antiparticles is trivial on the α, β indices. Since the real structure J exchanges particles with antiparticles the two A and JBJ −1 will commute. There is no room for the representation of a larger algebra satisfying the order 0 condition, unless more fermions are added, or one renounces to the trivial action on the spin indices. The second possibility is the one we will use for the grand algebra in the following sections.

The grand algebra
The case a = 3 in (3.1) would require a 72-dimensional Hilbert space, and there is no obvious way to build it from the particle content of the standard model. The next case, a = 4, requires the Hilbert space to have dimension 128, which is the dimension of H F . Said in an other way, considering together the spin and internal degrees of freedom as part of the "grand Hilbert space" H F gives precisely the number of dimension to represent the grand algebra This means that C ∞ (M) ⊗ A G can be represented on the same Hilbert space H as It is a remarkable "coincidence" that the passage from the standard model to the grand algebra, namely from a = 2 to a ′ = 4 = 2a, requires to multiply the dimension of the internal Hilbert space by 4 (for 2(2a ′ ) 2 = 2(4a) 2 = 4(2(2a) 2 )) which is precisely the dimension of spinors in a spacetime of dimension 4. Once more we stress that no new particles are introduced: The representation of the grand algebra A G on H F is more involved than the one of A F on H F in the previous section. In analogy with what was done earlier we consider an element of A G as two 8 × 8 matrices, and see both of them having a block structure of four 4 × 4 matrices. Thus the component Q ∈ M 4 (H) of the grand algebra gets two new extra indices with respect to the quaternionic component of A F , and the same is true for M ∈ M 8 (C). For the quaternions we choose to identify these two new indices with the spinor (anti)-particles indices0,1; and for the complex matrices with the spinor left-right indices r, l introduced in §3. This choice is not unique, and we leave a full investigation of the possible alternatives for future work. Having both sectors diagonal on different indices ensures that the order zero condition is satisfied, as explained below.
We therefore have where, for anyṡ,ṫ ∈ 0 ,1 and s, t ∈ {l, r}, the matrices Q˙t β sα ∈ M 2 (H), M tI sJ ∈ M 4 (C) (3.11) have the index structure defined below (3.8). This means that the representation of the element A = (Q, M) ∈ A G is 9 : This representation is to be compared with (3.8). As before the quaternionic part acts on the particle sector of the internal indices (δ C 0 ) and the complex part on the antiparticle sector (δ C 1 ). The difference is that the grand algebra acts in a nondiagonal way not only on the flavour and lepto-colour indices α, I, but also on the s andṡ indices. The novelty is in this mixing of internal and spacetime indices: at the grand algebra level, the spin structure is somehow hidden. Specifically, the representation (3.12) is not invariant under the action of the Lorentz group (or rather of Spin(4) since we are dealing with spin representation, in Euclidean signature). This point is adressed in section 5.2.
The representation of C ∞ (M) ⊗ A G is given by (3.12) where the entries of Q and M are now functions on M. Since the total Hilbert space H is unchanged, there is not reason to change the real structure and the grading. In particular one easily checks that the order zero condition holds true for the grand algebra This is because the real structure J in (2.7) acts as the charge conjugation operator on the spinor indices, and as J F in H F (where it exchanges the two blocks corresponding to particles and antiparticles). In tensorial notations one has where we use Einstein summation and define Hence J preserves the indices structure in (3.12), apart from the exchange δ C 0 ↔ δ C 1 : since Q and M act on different indices, the commutation (3.13) is assured. Notice that without the enlargement of the action of the finite dimensional algebra to the spinorial indices, it would have been impossible to find a representation of A G which satisfies the order zero condition, unless one adds more fermions. In this respect the grand algebra is not anymore an internal algebra.

Reduction due to grading
In a way similar to the reduction A F → A LR explained in appendix A, the grading condition imposes a reduction To see it, recall that the chirality Γ in (2.7) acts as γ 5 = η t s δ˙t˙s on the spin indices, and as γ F on the internal indices: where η C D and η β α are defined as in (3.16). Changing the order of the indices so that to match (3.12)), one has Since the representation of A G is diagonal in the C index, the grading condition is satisfied if and only if it is satisfied by both sectors -quaternionic and complex -independently.
For quaternions, one asks [η t where for anyṡ,ṫ ∈ 0 ,1 one has Q˙s β tα = q Rṡ t 0 2 0 2 q Lṡ t αβ with q Rṡ t , q Lṡ t ∈ H. Hence the reduction of the grand algebra to A ′ G . Notice that the grading causes a reduction not only in the quaternionic sector, as in the case of A F , but also in the complex matrix part. This is because A G is not anymore acting only on internal indices.

The Majorana coupling and the σ field
In this section we see how the grand algebra makes possible to have a Majorana mass giving rise to the field σ. Although the calculations are quite involved, the principle is quite simple. Since we have a larger algebra, the Majorana Dirac operator needs not be diagonal in the spin indices. This added degree of freedom enables the possibility to satisfy the order one condition in a non trivial way, namely to still have a one form which commutes with the opposite algebra, but that at the same time gives rise to a field. In the following we will show this analytically, all calculations have also been performed with a symbolic manipulation programme, leading to the same results.
We first work out in §4.1 the most general Dirac operator D ν with Majorana coupling compatible with the grading condition and the KO dimension of the spectral triple of the standard model. Then we study the first order condition induced by D ν and the subsequent reduction A G → A ′′ G of the grand algebra (4.2). Finally we show in 4.3 that D ν can be fluctuated by A ′′ G so that to generate the field σ as required by (2.15).

Dirac operator with Majorana mass term
We will consider a Majorana-like mass only for the right handed neutrinos. This choice is dictated by physics, and elsewhere we will investigate the more general case. The natural mass scale of this matrix is very high, so that it provides a natural see-saw mechanism (although in realistic scheme the right handed neutrino mass is somewhat lower than the Planck scale). The standard model can be considered as a low energy limit of the theory we present in this section. We will assume therefore that all the quantities involved in the internal Dirac operator D F but the Majorana coupling are small compared to the scale of the breaking described here. Moreover we work with one generation only, meaning that D F = D R is given by (B.1). We take advantage of the flexibility introduced by the grand algebra and we do not assume a priori that the Majorana coupling is diagonal on the spin indices. This means that instead of γ 5 ⊗ D F as in (2.9) we consider a finite dimensional matrix D ν containing a Majorana mass term with non trivial action on the spin indices. Right handed neutrinos have indices I = 0 and α = 1, so that the most general Majorana coupling matrix is where R is -at this stage -an arbitrary 4 × 4 complex matrix while Ξ is the projector on the first component The constraints on the matrix R come from the grading condition and the real structure. Remembering (3.19), one has that Ξ I J and Ξ β α commute with δ I J and η β α , while the r.h.s. of (4.1) as a matrix in CD anticommutes with η C D . So D ν anticommutes with Γ if and only if R commutes with γ 5 , meaning that R is block diagonal The requirement to have KO-dimension 2 mod 8 means that JD ν = D ν J. Remembering (3.15), this is equivalent to i.e. rτ = τ r T and lτ = τ l T , whose solution is r˙l˙s = k r δ˙l˙s, l˙l˙s = k l δ˙l˙s k r , k l ∈ C. (4.7) The second equation in (4.5) is then satisfied as well.
Eq. (4.7), (4.3) and (4.1) give the most general Dirac operator D ν on L 2 (R 4 ) ⊗ H F , with Majorana mass term, coupling the right neutrino with its anti-particle. In tensorial notations, one has By choosing k r = −k l = 1, one gets R = γ 5 and one retrieves the Majorana coupling D ν = γ 5 ⊗ D R of the standard model. However, at this stage nothing forces us to make this choice.

First order condition for Majorana Dirac operator
We aim at obtaining the field σ as a fluctuation of D ν , compatible with the first order condition. By (4.9) 10 To lighten notation, for any pairs of indices x, y and u, v we write δ xu yv = δ x y δ u v .
As well, with B = (R, N) ∈ A G , a generic element of the opposite algebra is where we defineR˙t β sα = (τRτ )˙t β sα ,Ñ tI sJ = −(ηNη) tI sJ = −N tI sJ . (4.11) The first order condition for D ν means that (4.12) We look for solutions that satisfy the grading condition, i.e. in A ′ G . Inspired by the first order condition for A LR and D F described in appendix A, we also impose the reductions as well as (4.14) The reduction (4.13) is obtained assuming that the components in The reduction (4.14) is obtained imposing that the off-diagonal part of Q in (3.20) is zero: Finally we impose that q R and q ′ R are diagonal quaternions, that is We thus look for solutions of (4.12) in Notice that we do not claim there is no solution of (4.12) outside A ′′ G . But for our purposes, it turns out that it is sufficient to work with A ′′ G . Under these conditions, the first equation coming from (4.12), namely (4.20) has explicit components with m r , m l the components of M and c R , c ′ R the one of Q. Similarly we define the matrices n r,l from the components n l,r of N, and the matrices d, d ′ R from the components d R , d ′ R of R. The matrix Ξ carries the indices I, J in the second equation, and α, β in the third. In each equation, to pass from the first to the second lines one uses (4.11).
Collecting the components and assuming that both k r and k l are non zero, one finds that (4.20) is equivalent to A similar calculation for the second components of (4.12) yields the same system of equations. Thus one solution to the first order condition induced by D ν is to impose c R = m r = m l and d R = n r = n l , (4.24) meaning the reduction of A ′′ G to

The σ field as a 1-form
We now consider the set of 1-forms i B i [D ν , A i ] generated by the Majorana Dirac operator and the algebra A ′′′ G above. We are interested in showing that this set is non empty, and it is enough to consider the simplest 1-form We begin with A = (Q, M) in A ′ G . With notations of the precedent section, one has

By the reduction
does not. This is the crucial difference with the algebra of the standard model: the grand algebra allows to generates a non-vanishing 1-form associated to the Majorana Dirac operator D ν , which satisfies the first order condition.
Restoring the order sṡIα of the indices, the matrix above is R = R tṫ sṡ Ξ J I Ξ β α with The conjugate action of the real structure J yields where the charge conjugation J acts only on the spin indices. Explicitly, omitting the factor Ξ J I Ξ β α in the expression of R, one gets that is −J R † J is obtained by permuting the components in the blocksṡṫ of R. As well is obtained from −R † by permuting the components inṡṫ. Consequently, Now, considering that A is in C ∞ (M) ⊗ A ′′ G , the coefficients m r and c ′ R becomes functions on the manifold M. Taking k l = −k r = k R , one obtains R tṫ sṡ = k R σγ 5 where is now a field on M. In other terms, the fluctuation of D ν by A G yields the substitution (2.15). The grand algebra gives a justification for the presence of the field σ, necessary to obtain the mass of the Higgs in agreement with experiment.

Reduction to the standard model
Starting with the grand algebra A G reduced to A ′ G by the grading condition, we have shown how to generate the field σ by a fluctuation of the Majorana-Dirac operator D ν , in a way satisfying the first order condition imposed by D ν . As explained below (4.8), one can choose in particular D ν = γ 5 ⊗ D R , where D R is the internal Dirac operator D F of the standard model in which only the dominant term (i.e. the Majorana mass) is taken into account. In other words, the field σ is generated by fluctuating the second term in the Dirac operator (2.9) of the standard model. We now show that the first order condition of the first term in (2.9), i.e. the free Dirac operator, yields the reduction of the grand algebra to the standard model.

First order condition for the free Dirac operator
The first term in (2.9) is the Euclidean free Dirac operator, extended trivially to the internal space of one generation. In tensorial notation it reads has components (omitting the non relevant δ) where P˙t β sα = (σ µu s ∂ µ Q˙t β uα ), T˙t β,μ sα = σ µṫṡ , Q˙t β sα (5.4) and similar definitions forP andT withσ instead of σ; and with JBJ given in (4.10) is a block diagonal matrix in CD with components Omitting the indices (and noticing that the P, T,P ,T all commute withÑ r r ,Ñ l l ), the first components is a diagonal matrix with first entry The vanishing of the differential operator part implies either T µ = 0 orÑ l l =Ñ r r . But the expression should be zero in particular for non-constant fields, that is for P = 0. So in case one imposes T µ = 0, the vanishing of the term in P impliesÑ l =Ñ r . In case one imposesÑ l l =Ñ r r , the vanishing of the remaining term implies either T µ = 0, orÑ l l =Ñ r r = cst. The last solution is unacceptable, it would mean that spacetime is reduced to a point, hence in any case one has both conditions: T µ = 0 andÑ l l =Ñ r r . One then checks that the other components of (5.7) vanish as well.
The only matrix that commutes with all the Pauli matrices is the identity, therefore where representation of the r.h.s. algebra is now diagonal on the spinorial indicesṡ, s.
To summarize, the grand algebra A G is broken by the chirality and the first order condition of the free Dirac operator to the left-right symmetric algebra A LR of the standard model.

Emergence of spin
In noncommutative geometry the topological information is encoded in the algebra, while the geometry (e.g. the metric 11 ) is in the D operator. In particular the Riemann-spin structure is encoded in the way this operator, which contains the gamma matrices, acts on the Hilbert space. Without this operator there is just an algebra which acts in an highly reducible way on a 128 dimensional Hilbert space. This is conceptually what distinguishes H F from H F in (3.5): on C ∞ (M) ⊗ H F , the free Dirac operator (trivially extended to the internal indices) is On C ∞ (M) ⊗ H F the same operator writes 5.14) and the spin structure, carried by the γ matrices, is hidden among the internal degrees of freedom. In this sense the first order condition, which governs the passage from (5.14) to (5.13), corresponds to the emergence of the spin structure.
Alternatively, a spin structure means that the vectors in the Hilbert space transform in a particular representation under the "Lorentz" group. Since we are dealing with spinors in the Euclidean case, the group is actually Spin(4). It is generated by the commutators of the Dirac matrices, that act on H as .
(5.15) Let us distinguish between an element a of C ∞ (M)⊗A G and its representation π(a) := A given in (3.12). For any Λ = λ µν S µν ∈ Spin(4) and A ∈ π(C ∞ (M) ⊗ A G ), let The representation (3.12) of the grand algebra is not invariant under the adjoint action (5.16) of Spin(4) since α Λ π(a) is not in π(C ∞ (M) ⊗ A G ). In this sense the representation of the grand algebra is not Lorentz invariant, unlike its reduction to A LR which is diagonal in the spin indices. However, at the abstract level the algebra is preserved under Lorentz transformations since the latter are implemented by unitary operators: for any Λ one has that α Λ (π(C ∞ (M) ⊗ A G )) is isomorphic to C ∞ (M) ⊗ A G . This suggests to view the grand algebra as a phase of the universe in which the spin and rotation structure of spacetime has not yet emerged, only the topology (i.e. the abstract algebra) is fixed.

Fiat neutrino
The grand algebra together with the Majorana Dirac operator D ν generates the field σ at the right position (i.e. as required in (2.15)), respecting the first order condition induced by D ν . However, by (4.35) one has that σ becomes constant when one takes into account the first order condition imposed by the free Dirac operator, because (5.9) implies that c ′ R = c R = m r . This suggests a scenario in which the neutrino Majorana mass is the first field to appear and fluctuate, before the geometric structure of spacetime emerges through the breaking described in §5.2. In this picture, the field σ is viewed as a fluctuation of a vacuum that satisfies the first order condition of the free Dirac operator. This scenario is supported by some preliminary calculations, which indicate that the first order condition of the free Dirac operator can be equivalently obtained as a minimum of the spectral action. In this way, the geometrical breaking imposed by the mathematical requirement of the theory becomes a dynamical breaking, and the field σ would appear as the "Higgs field" associated to it. This idea has been investigated, in the case of the standard model algebra, in the recent papers [29,30]. The case of the grand algebra is in progress.

Conclusions and outlook
It is known that, although the spectral action requires the unification of interactions at a single scale, the usual grand unified theories, such as SU (5) or SO(10), do not fit in the noncommutative geometry framework, and are possible only renouncing to associativity [55,56]. In this paper we pointed out there is a "next level" in noncommutative geometry, but that it is intertwined with the Riemannian and spin structure of spacetime, and therefore it naturally appears at a high scale. The added degrees of freedom are related to the Riemann-spin structure of spacetime, which emerges as a symmetry breaking very similar in nature to the Higgs mechanism. In addition, the higher symmetry explains the presence of the σ field necessary for a correct fit of the mass of the Higgs. The results presented here, as is common in this model, are crucially depending on the Euclidean structure of the theory. This is particularly important as far as the role of chirality and the doubling of the degrees of freedom is concerned. A Wick rotation is far from simple in this context, and the construction of a Minkowskian noncommutative geometry is yet to come (for recent works see [57,58]).
The presence of this grand symmetry will have also phenomenological consequences which should be investigated. The breaking mechanisms described in this paper are just barely sketched, we only looked at the group structure. A more punctual analysis should reveal more structure, and possibly alter the running of the constants at high energy.
For the moment we can only speculate. One of the problems of the spectral action in its present form is that it requires unification of the three gauge couplings at a single scale, Λ, and physical predictions are based on the value of this scale. It is known experimentally that in the absence of new physics the three constants do not meet in a single point, but the three lines form an elongated triangle spanning nearly three orders of magnitude. On the other side in the spectral action is not clear what would happen after this point, if one consider scales higher that Λ, i.e. earlier epochs. For a theory dealing with the unification of gauge theory and gravity a more natural scale is the Planck scale. An unification of the coupling constants at the Planck scale in the form of a pole has been considered [59,60], but it requires new fermions. In the case at hand the "new physics" is in the form of a different structure which mixes spacetime spin and gauge degrees of freedom. This might have consequences for the interactions, and hence for the running of the various quantities, as well.
Other mathematical issues should be investigated. In particular the choice of the action of the grand algebra on the spin indices reflects how much the algebra is not diagonal in the "interaction" encoded by the free Dirac operator. Hopefully this could be interpreted at the light of Connes unitary invariant in Riemannian geometry [61]. In this basis the internal Dirac operator is where To guarantee that A commutes with Γ, the factor η β α in (3.18) requires Q to be diagonal in the α index, which reduces this part of the algebra to H L ⊕ H R . This means that Q α β in (3.8) acts separately on the left and right handed doublets. No such breaking occurs in complex part, because δ I J in (3.18) does not put any constraints on M 4 (C). Likewise, η t s does not produce any breaking because the action of both quaternions and complex matrices is diagonal on the spin indices. Thus we have A F → A LR .
Let us now examine the first order condition. For any B = (R, N) ∈ A LR one has Assuming there is no neutrino Majorana mass (i.e D 1 2 = 0), the first order condition for the finite dimensional spectral triple yields  can also be obtained directly from D R alone, without reducing first to the algebra (A.12).
Consequently, starting from C ∞ (M) ⊗ A F reduced to C ∞ (M) ⊗ A LR by the grading condition, there is no way to fluctuate the Dirac operator -respecting the first order condition -so that to obtain the field σ as required by eq. (2.15).