Pseudo-Riemannian structures in Pati-Salam models

We discuss the role of the pseudo-Riemannian structure of the finite spectral triple for the family of Pati-Salam models. We argue that its existence is a very restrictive condition that separates leptons from quarks, and restricts the whole family of Pati-Salam models into the class of generalized Left-Right Symmetric Models.


Introduction
The Standard Model of Particle Physics is the preeminent theory of fundamental constituents and their interactions. It has been experimentally tested and verified to be accurate with a very high degree of precision. Nevertheless, there are several open questions which, to date, the Standard Model has left unresolved. Among them, several are related to the masses of neutrinos and the seesaw mechanism [1], and others to baryon asymmetry [2]. Nowadays, there is also a tremendous set of cosmological data [3] which suggest the existence of dark matter. Several attempts to explain the aforementioned questions have already been proposed. Due to the proven success of the Standard Model, most of them are extensions thereof and are known as theories which go Beyond the Standard Model. An interesting one, which has already been under consideration for several years and intensively studied by several physicists, is the model introduced by J.C. Pati and A. Salam [4].

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The Pati-Salam model is a Yang-Mills-type model based on the SU(2) R ×SU(2) L ×SU(4) gauge group. It extends the usual Standard Model by e.g. introducing leptoquark symmetry and left-right symmetry. It was also considered to explain the origin of parity symmetry breaking [5,6].
Models used to describe theories of Particle Physics are traditionally constructed in the Lagrangian formalism, i.e. the form of the action is postulated based upon the desired symmetries of the resulting model. A possible geometrical explanation for the structure of such theories is provided by spectral geometry. The Standard Model has been studied for several years in the framework of almost-commutative geometry (see e.g. [7,8] and [9]) but some puzzles remain unsolved, not only for the product spectral triple, but also for its finite part. Recently, in [10], it was proposed to consider the finite spectral triple for the Standard Model as the shadow of some pseudo-Riemannian triple in such a way that the pseudo-Riemannian structure leads to the existence of some nontrivial grading on the Riemannian triple.
The Pati-Salam model was also considered as a noncommutative geometry by several authors, see e.g. [11][12][13][14] and [15]. We also highly recommend [16] as a survey of the historical development of these methods and their applications in Particle Physics. Reduced versions of the Pati-Salam models, i.e. the Left-Right Symmetric Models, were also considered in the framework of noncommutative geometry, firstly as potentially interesting examples for the Connes-Lott scheme of spectral geometry, but then also from the point of view of possible physical applications -see e.g. [17][18][19] and [20]. However, since some of the fundamental axioms of noncommutative geometry were not satisfied, such models were not satisfactory. Later on, due to the trend of relaxing some of the axioms, e.g. the first order condition, and further development of the spectral theory in their absence (see e.g. [21]), the family of Pati-Salam models was analyzed.
In this paper we propose to take into account the pseudo-Riemannian structure of the finite triple for the Pati-Salam model, in the sense introduced in [10]. Herein, we analyze the finite spectral triples for such models, discuss possible pseudo-Riemannian structures, relate them to the grading that distinguishes leptons from quarks, and argue that the existence of such a pseudo-Riemannian structure restricts the whole family of Pati-Salam models to models of the Left-Right Symmetric type.

Finite spectral triples for Pati-Salam models
In this section we consider the finite spectral triple for the family of Pati-Salam models. We discuss algebras and their commutants, different choices of chiral structures and possible Dirac operators.

Spectral data
The algebra for the Pati-Salam model is of the form (2.1)

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Consider F = M 4 (C) with the inner product v, w = Tr(v * w) (where v * denotes the Hermitian conjugate of v ∈ F ). The elements v of F can be presented in the following form Let H = F ⊕ F * be the Hilbert space for the model we consider here, where F * is the dual representation to F . Following [22] we can identify where the matrix algebra is represented on H as and for all α, β, v, w ∈ M 4 (C) and a, b, c, d ∈ C. Therefore, we have to represent all operators acting on H as elements of this tensor product space, and also find the form of the representation π : A → End C (H) in this language.
Let e ij be the matrix that has 1 in the entry (i, j) and zero otherwise. Then the grading γ has the following matrix representation There is also another possible choice of grading [23], for which left-handed leptons have the same parity as right-handed quarks, and vice versa for the opposite chirality: With respect to the decomposition H = F ⊕ F * , let J be the real structure, i.e.
It is used to define the opposite representation [22]. For ξ = π α ⊗ a b c d ⊗ β we take From now on we will omit the π symbol for the representation of an endomorphism.

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The elements of the algebra A = H R ⊕ H L ⊗ M 4 (C) are represented on H as where q 1 ∈ H R , q 2 ∈ H L and m ∈ M 4 (C). Notice that γ does not commute with this representation of the algebra A unless the symmetry following from M 4 (C) is broken into C ⊕ M 3 (C).
Therefore, here we are considering two algebras. The first one being H R ⊕ H L ⊕ M 4 (C) which we refer to as corresponding to an unreduced Pati-Salam model, and the second one , which we will call reduced.
Since the Dirac operator D ∈ End C (H), it is of the form , for some natural number K. From now on, summations will be understood to be over the entire range of all indices unless explicitly stated otherwise.

Commutants
We now consider the commutants of several algebras related to the unreduced and reduced Pati-Salam models. These results will be crucial to the discussion in section 3.3.
Notice first, that for any (real or complex) matrix algebra A contained in M N (C) for some N , the commutant A is the same as A C , where A C denotes the complexification of A.
By a straightforward computation we can check that the commutant of the algebra of elements q 1 q 2 , with q 1 , q 2 ∈ H is the algebra with elements α1 2 β1 2 , where α, β ∈ C.
We denote this algebra by C 1 .
In a similar manner, the commutant of the algebra of elements λ n , with λ ∈ C, n ∈ M 3 (C) is the algebra with elements α β1 3 , where α, β ∈ C. We denote this algebra by C 2 .
Furthermore, notice that M 4 (C) ∼ = C. Therefore, we can describe the commutants of the Pati-Salam algebra for the unreduced (i.e. with A = H R ⊕ H L ⊕ M 4 (C)) case, and the reduced one (i.e. with A = H R ⊕ H L ⊕ C ⊕ M 3 (C)).
is the algebra generated by elements of the form where A 1 ∈ C 1 and E 1 , E 2 ∈ M 4 (C).
Proof. Any element of the considered algebra may be represented as

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with q 1 , q 2 ∈ H and m ∈ M 4 (C). It is enough to find which elements of the form commute with q 1 q 2 ⊗ e 11 + m ⊗ e 22 for all q 1 , q 2 and m.
The only possible solutions are with A 1 ∈ C 1 , A 4 ∼ 1 4 and A 2 = A 3 = 0.
In a perfectly similar way, we get the following is the algebra generated by elements of the form 14) where A 1 ∈ C 1 , A 2 ∈ C 2 and E 1 , E 2 ∈ M 4 (C).

Dirac operators and reality
Let us now consider self-adjoint Dirac operators that commute with the real structure J.
Because DJ = JD, we have D = JDJ −1 = (D * ) • , but since D = D * , the necessary condition that has to be satisfied is D = D • . Notice that this is a weaker condition than the original two together. Therefore, we will consider them separately. Observe first, that the Dirac operator D = D ijklrs e kl ⊗ e ij ⊗ e rs is self-adjoint if and only if D ijklrs =D jilksr for all indices.
Suppose now that DJ = JD, i.e. D = JDJ −1 . We may write the Dirac operator as as previously described in (2.10). Notice that for an operator X of the form A ⊗ a b c d ⊗ B we have for all k, l, r, s.

Dirac operators for γ
Since we are interested in Dirac operators that commute with the real structure J we can write D = D 0 + JD 0 J −1 . We first consider all Dirac operators D that anticommute with γ, given by (2.6), as a grading. It is enough to restrict ourselves to the D 0 part. We have the following Proof. Notice that and hence by direct inspection we see that the Dirac operator has to be of the claimed form. JHEP07(2020)072

Dirac operators for γ
Again, we are considering all Dirac operators which commute with J, and therefore of the form D = D 0 + JD 0 J −1 , but now which anticommute with γ , given by (2.7), as a grading. This time we have, Proof. As before, let us first write the grading in a more convenient form, and likewise Therefore, a straightforward comparison shows that D anticommutes with γ if and only if it is of the claimed form.

JHEP07(2020)072 3 Pseudo-Riemannian structures
Let us recall that a pseudo-Riemannian spectral triple (A, H, D, γ, J, β), of signature (p, q), is a system consisting of an algebra A, Hilbert space H, Dirac operator D, Z/2Z-grading γ, real structure J and an additional grading β ∈ End(H) such that β * = β, β 2 = 1 and which commutes with the representation of A and defines a Krein structure on the Hilbert space. These objects are supposed to satisfy several conditions that are collected in [10], section II. For our purposes it is enough to recall that β has to satisfy βγ = (−1) p γβ and p Jβ, where DJ = JD and = ±1 depending on the KO-dimension of the triple. Furthermore, we assume that D is β-selfadjoint, i.e. D * = (−1) p βDβ. We say that the triple is time-oriented if β can be presented as the image of a certain Hochschild p-cycle.
Out of the pseudo-Riemannian spectral triple (A, H, D, γ, J, β) one can construct its Riemannian restriction, i.e. a triple (A, H, D + , γ, J, β) with D + = 1 2 (D + D * ) which is a self-adjoint operator and βD + = (−1) p D + β. This spectral triple is of the same KOdimension as the one we started with.
As was noticed in [10] the spectral triple for the Standard Model can be treated as a Riemannian restriction of some pseudo-Riemannian triple, with an additional grading originating from the time-orientation on the pseudo-Riemannian level. The existence of such a grading results in a restriction on the number of possible Dirac operators, compatible with the other elements of the triple.
Here we are looking for similar effects in the case of Pati-Salam models. From now on we will denote D + by D, and since the spectral triple for the Pati-Salam models has to be of KO-dimension 6, we take the signature to be (0, 2). We are looking for all possible βs and (self-adjoint) Dirac operators D such that βD = Dβ.
Therefore we are looking for β of the form with q 1 , q 2 , q 1 , q 2 ∈ H and m, m ∈ M 4 (C) for the unreduced case, and β = π(q 1 , q 2 , λ, n)Jπ(q 1 , q 2 , λ , n ) * J −1 , with q 1 , q 2 , q 1 , q 2 ∈ H, λ, λ ∈ C and n, n ∈ M 3 (C) in the reduced case. For simplicity we have assumed here that β is 0-cycle containing only one summand. We postpone the discussion about the more general situation until section 3.3.
Moreover, we require that

The unreduced Pati-Salam model
Let us start with this case first. Then β can be represented as (3.4)

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Since β is a 0-cycle, it commutes with the grading by construction. Notice the fact that β commutes both with the algebra and the opposite algebra (since βJ = Jβ and the order zero condition holds) fixes all matrices q 1 , q 2 , q 1 , q 2 and m, m to be proportional to the identity, and moreover, it enforces m = m t and q i = q t i for i = 1, 2. The condition β 2 = 1 fixes all these proportionality factors to be a sign.
Therefore, the only possible pseudo-Riemannian structures are So, up to the trivial rescaling by a factor of −1 there are only two such possible operators: (3.6) The first of these is the identity operator, so it commutes with everything. This is the trivial case in which we are not interested.

Compatible Dirac operators for the unreduced Pati-Salam model
We are looking for all possible generic Dirac operators D (not necessarily anticommuting with a grading) such that Dβ = βD. Moreover, we already assume that D commutes with J, so it is of the form D 0 + JD 0 J −1 . For such a D, we get the following

7)
if and only if D 0 is of the following form Proof. Following a similar argument as that used to prove Propositions 2.5 and 2.6, but with β replacing γ or γ , we first write

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Noticing that for D = D ijklrs we have and similarly The result follows from a straightforward comparison of these expressions.

Physical consequences of the unreduced Pati-Salam model with γ
Since D anticommutes with γ, we see that the only possibility for such a D to commute with the nontrivial β discussed above is that terms of the form · · · ⊗ e 11 ⊗ · · · and · · · ⊗ e 22 ⊗ · · · must vanish. Here we are looking for possible extensions of the Standard Model, for which the Dirac operator contains terms of the form which encode the Yukawa parameters for leptons and quarks. Therefore, physically acceptable extensions of the Standard Model, i.e. those that can be reduced to the Standard Model after imposing additional conditions on terms of the Dirac operator, also have to contain these terms. Therefore, no Pati-Salam model with the algebra H R ⊕ H L ⊕ M 4 (C), grading γ, and with the pseudo-Riemannian structure β is physically acceptable.

Compatible Dirac operators for the reduced Pati-Salam model
Now, we will discuss restrictions on a generic Dirac operator (not necessarily anticommuting with a grading) which follow from commutation with the nontrivial βs allowed in the case of the reduced Pati-Salam model. As before, we assume that D commutes with the real structure so that it is of the form D 0 + JD 0 J −1 . For the two remaining cases in (3.14) we get the following results. Firstly, we have if and only if D 0 is of the form , and α lk , β lk , γ k , δ lk , σ lk , µ lk , ν lk ∈ C.
Proof. It follows from a by now familiar computation that, for D = D ijklrs ,  Comparing these two expressions we get the claimed result.

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Similarly, if and only if D 0 is of the form Proof. In a similar manner to the previous cases we compute The result follows from a straightforward comparison of these terms.

Physical consequences of the reduced Pati-Salam model with γ
Notice that since it is required of the Dirac operator to anticommute with the grading γ , and moreover any physically interesting model should be an extension of the Standard Model, we conclude that the only possibility is therefore the β from Proposition 3.3, in which case the freedom of possible Dirac operators D = D 0 + JD 0 J −1 is reduced to those JHEP07(2020)072 with D 0 of the form We can treat this model as an extension of the Standard Model with modified chiralities, i.e. in which left-handed (resp. right-handed) leptons have the same parity as right-handed (resp. left-handed) quarks. Therefore, the only compatible extension beyond the Standard Model and contained within the family of Pati-Salam models which have γ as a grading, and possesses a pseudo-Riemannian structure in the sense of the existence of a one-term 0-cycle β, is precisely the reduced Pati-Salam model with exactly the same pseudo-Riemannian structure which was uniquely possible in the case of the Standard Model [10]. Since γ explicitly breaks the SU(4)-symmetry into U(1)×U(3), it is not surprising that the resulting class of models also has this property. Nevertheless, it is worth noting that there is exactly one (up to an irrelevant global sign in β) such possibility, and moreover that it is consistent with the one that is known to be the only possibility in the case of the Standard Model. Furthermore, we observe that the pseudo-Riemannian structure still allows for the existence of SU(2)-doublets of right-handed particles. This is an interesting feature. Notice also, that there are further restrictions on the entries of the compatible Dirac operators which follow from the assumption of self-adjointness. These restrictions are summarized at the beginning of section 2.3.

Physical consequences of the reduced Pati-Salam model with γ
Since the grading γ is compatible with the unreduced Pati-Salam model, it is also compatible with the reduced model. Therefore we can consider Dirac operators that anticommute with γ and commute with β for this case. For the nontrivial βs we see that only one of them, i.e. β from Proposition 3.3, is compatible with the requirement of being an extension of the Standard Model. For this choice we have the following

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Moreover, notice that the only β which is admissible in this case is exactly the same one that prevented the existence of leptoquarks in the Standard Model [10]. Therefore, the only possible extension of the Standard Model contained within the family of Pati-Salam models, which takes into account the pseudo-Riemannian structure for finite triples in the sense defined in [10], has to be of the reduced form. That is, the SU(4)-symmetry is broken into a U(1) × U(3)-symmetry. Therefore, instead of the full SU(2) R × SU(2) L × SU(4) Pati-Salam gauge group we must reduce to the case with SU(2) R × SU(2) L × U(1) × U(3). In this extension the right particles are doublets under the SU(2)-symmetry, and leptons are separated from quarks, i.e. they are not the fourth color, so there are no leptoquarks. In a similar manner to the previous case, there are further restrictions on the entries of the compatible Dirac operators which follow from the assumption of self-adjointness -see section 2.3.

Generic β-structures
Here we are looking for all possible βs that are 0-cycles and which satisfy all required conditions but we do not assume these operators to consist of only one term.

The unreduced Pati-Salam model
Since we require that β commutes with the representation of the unreduced Pati-Salam algebra it follows from Proposition 2.1 that where E 1 , E 2 , F ∈ M 4 (C).
Since β 2 = 1, we get hence E 2 1 = E 2 2 = 1 4 . Since β commutes with J, this implies that Let us write F = F 11 F 12 F 21 F 22 , then (3.26) is equivalent to the following set of conditions
Notice that all such βs are 0-cycles, and there are (up to a trivial rescaling) only two possibilities: (3.30) These are exactly the same as under the assumption of β being only a one-term 0-cycle, i.e. the conclusion of section 3.1 remains valid for more general βs.

The reduced Pati-Salam model
In a similar manner, since we require that β commutes with the representation of the reduced Pati-Salam algebra, it follows from Proposition 2.2 that where E 1 , E 2 , F 1 , F 2 ∈ M 4 (C). Since β 2 = 1, we infer that Now, from the condition βJ = Jβ, repeating the previously used argument, we end up with the following form of β: where η i = ±1 for i = 1, . . . , 4. There are only eight independent (i.e. up to a global sign) possibilities. They are listed below: (3.37) (3.38) (3.41) All of the above are 0-cycles (more precisely: , but only four of them are images of all non-zero elements of the algebra: β 1 , β 3 , β 6 and β 7 . These are exactly the same cases we had in the case of the single term 0-cycles. Moreover, an analogous computation to before shows that βs which are not of the one term type do not allow for physically acceptable Dirac operators in the sense explained in section 3.1.2. The reason is that the restrictions on the · · · ⊗ e 11 ⊗ · · · and · · · ⊗ e 22 ⊗ · · · terms that follow from the fact that the coexistance of β and γ (or γ ) requires that such terms contain matrices that are simultaneously blockdiagonal and anti-diagonal. To sum up, the conclusion in section 3.2 remains valid when more general βs are considered.

The Standard Model
Let us now discuss the generic case for the Standard Model. In [10] the one term case was discussed. Now, mirroring the above computation we can get the following family of possible βs:

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where η i = ±1, for i = 1, . . . , 6. This straightforward check shows that the only case in which β is a 0-cycle and allows for an extension of the Standard Model (in the previously discussed sense) is of the one term type and is exactly the same β that prevented the existence of leptoquarks in [10]. Therefore, the validity of the conclusion therein is maintained when generalizing to allow, from the outset, for more general βs.

Conclusions and outlook
We have discussed a role which may be played by the existence of pseudo-Riemannian structures for the finite spectral triples associated with the family of Pati-Salam models.
We have shown that their existence reduces the algebra to H R ⊕ H L ⊕ C ⊕ M 3 (C). Despite the fact that the existence of the additional grading as the shadow of a pseudo-Riemannian structure does not determine the Dirac operator uniquely, we have a huge reduction of the possible choices.
The Left-Right Symmetric models were also considered from the point of view of noncommutative geometry, initially as possibly interesting examples for the Connes-Lott scheme, but later on also as possible extensions of the Standard Model -see e.g. [17][18][19] and [20]. The main interest was in the determination of whether, in this framework, such models provide a mechanism to explain the origin of parity symmetry breaking. In [17] it was argued that in the almost-commutative Yang-Mills-Higgs models, parity cannot be spontaneously broken. This followed from the requirement that Poincaré duality must be satisfied.
The family of reduced Pati-Salam models generalizes both the Left-Right Symmetric Models and also the Chiral Electromagnetism Model [17]. The latter contains the U(1) R × U(1) L gauge group instead of the SU(2) ones. This theory played the role of a toy model for the application of the Connes-Lott scheme to LRS theories.
Due to the trend of relaxing some of the axioms of spectral geometry, it may be interesting to revisit the previous results for these classes of models in such a framework. We postpone this for future research.