Minimum Supersymmetric Standard Model on the Noncommutative Geometry

We have obtained the supersymmetric extension of spectral triple which specify a noncommutative geometry(NCG). We assume that the functional space H constitutes of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model(MSSM). We introduce the internal fluctuations to the Dirac operator on the manifold as well as on the finite space by elements of the algebra A in the triple. So, we obtain not only the vector supermultiplets which meditate SU(3)xSU(2)xU(1)_Y gauge degrees of freedom but also Higgs supermultiplets which appear in MSSM on the same standpoint. Accoding to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley-DeWitt coeffients give the correct action of MSSM. We also verify that the relation between coupling constants of $SU(3)$,$SU(2)$ and $U(1)_Y$ is same as that of SU(5) unification theory.


INTRODUCTION
Yang-Mills gauge theory which provides the basis of the standard model of high energy physics and general relativity theory greatly have succeeded in describing the basic interactions in our Universe. But in the quantum level, they are difficult to be unified due to the unrenormalizability of gravity.
The standard model coupled to gravity was derived on the basis of noncommutative geometry(NCG) by Connes and his co-workers [1,2,3]. The framework of NCG is specified by a set called spectral triple (H, A, D) [4]. Here, A is a noncommutative complex algebra, acting on the Hilbert space H, whose elements correspond to spinorial wave functions of physical matter fields, while the Dirac operator D is a self adjoint operator with compact resolvent which play the role of metric of the geometry. Higgs, gauge and gravity fields which meditate all basic interactions that we know are introduced in the same standpoint, the fluctuations of Dirac operator.
The internal fluctuation of the Dirac operator is given as follows: (1.1) The functional space H M on the the Minkowskian space-time manifold is the direct sum of two subsets, H + and H − : where H + is the space of chiral supermultiplets which constitutes of Weyl spinors which transform as the ( 1 2 , 0)of the Lorentz group SL(2, C) and their superpartners and H − is the space of antichiral supermultiplets. The elements are expressed by A supersymmetric invariant product of wave functions Ψ,Ψ ′ is defined by where Ψ is expressed on the base by

7)
Γ 0 is given by The algebra A M = A + ⊕ A − which acts on H M constitutes of elements expressed by where {ϕ a (ϕ * a ), ψ aα (ψα a ), F a (F * a )} is a chiral(antichiral) supermultiplet. We note that A ± includes the constant functions expressed by The Dirac operator which acts on the H M is given by The algebra A F on the finite space H F is given by (2.14) where H is the space of quaternions and M 3 (C) is the space of 3 × 3 complex matrices. The representation space of A F , H F is the space of labels which denote quantum numbers of quarks and leptons as follows: and labels of their antiparticles are also elements of H F which are expressed by where A denotes indices on which 3 × 3 complex matrices act and I denotes those on which quaternions act. L and R denote that the eigenvalue of Z/2 grading γ F is −1 and 1, respectively. We define the action of a ∈ A F for the quark sector as follows: where λ ∈ C, q ∈ H and m ∈ M 3 (C). When we denote the base of quark sector as a is represented by a matrix given by where 1 3 denotes the unit matrix of M 3 (C), 1 2 is that of H and m i are integers.
For the lepton sector, we take the base expressed by and define a as follows: For an element a ∈ A F , we define the real structure as a antilinear isometry of H F as follows: J F satisfies the following conditions: J F define the right action of A F in H F as follows: From (2.26), we see that the left action of a for the antiparticle determine the right action for the particle and can be moved into the left action through J F . For an example, let us consider Q AI in (2.15). From (2.20), the left and right actions of A F on Q are summarized by The Dirac operator D F on H F is given for each label of s = u, d, e by where m s is the mass matrix with respect to the family index. In order to evade fermion doubling, we impose the following condition and extract the physical wave functions as follows: for quarks Table 1: The list of matter fields and the action of the algebra for them: The third column denotes the actions of elements of the algebra moved from the antiparticle part through the real structure J. and for leptons  a are elements of A − in the form of (2.10) and they are related to u (s) a as follows: Constant elements of A ± ⊗ A F are also given by ± denotes the matrix form of (2.11). If we replace Q → l and u R (d R ) → e R in (2.32),(2.33),(2.35), we obtain formulas for the lepton sectors. Hereafter, we discuss the quark sector principally and obtain formulas for the lepton sector through this replacement.
On the basis, the total Dirac operators for the quark sector is given as follows: where M (u(d)) represents the doublet term which includes mass of quarks expressed by where y u and y d are Yukawa matrices with regard to generation. For the antiparticle sector, the total Dirac operator is given by where J is the real structure of H = H M ⊗ H F . We note that under the definition of supersymmetric invariant product (2.6), iD tot has the following hermiticity:

INTERNAL FLUCTUATION, VECTOR AND HIGGS SUPERMULTIPLET
In the theory of noncommutative geometry without supersymmetry, gauge fields and Higgs fields are derived through internal fluctuation of Dirac operator in the form expressed by The second term of (3.1) is the fluctuation given by the elements of algebra which act on the antiparticle part and transferred to the particle part through the real structure J. And gauge fields are derived from the fluctuations for the Dirac operator which acts on the Riemann manifold , while the Higgs fields are given by those for the Dirac operator on the finite space. We will consider to extend these constructions to supersymmetric version. The major differences from the non-supersymmetric case is due to the fact that the algebra which is the source of the fluctuation is the direct sum of the spaces A ± ⊗ A F . Using (2.32),(2.33),(2.35),(2.36), we obtain the following expressions: where s = u R (d R ). Eq. (3.2) and (3.3) give fluctuations only to iD M ⊗ 1 F , so we will see later that this type of fluctuation produce vector superfields, while Eq.(3.4) and (3.5) give contribution only to γ M ⊗ D F so that this type of fluctuation will give Higgs superfields.

VECTOR SUPERMULTIPLET
We introduce a set of elements of A in (2.32),(2.33) expressed by a ; a = 1, 2, · · · , n r } ⊂ A + ⊗ A F , (3.6) where r = 1, 2, 3 and u a are in the matrix form of (2.9),(2.10) and given in the Table 1. The elements of Π + and Π − are chosen such that the products of u ′ a s andū ′ a s do not belong to Π + , Π − any more.
We can define elements (A ) of a vector supermultiplet as follows: 14) The condition for the fields in (3.11)∼ (3.15) to vanish is Wess-Zumino condition and given by aīk Dk ℓ u a . In the Table 1, we see that the algebra which act on Q α field is the space of functions in H ⊕ M 3 (C). Let us assume that we choose elements of A in (2.32) in the quark sector as follows: The fluctuation for D, D which produces vector supermultiplet is given by Hereafter, for simplicity, we abbreviate the upper index [r]of c and The algebra which acts on u R (d R ) field is the space of functions in C ⊕ M 3 (C). If we choose the following elements of A in (3.21): instead of (3.23),(3.24), we obtain the following expressions: and We show the detail of the above operations in Appendix A. We also show in Appendix A that (A ) is a vector supermultiplet which becomes to the adjoint representation of U (r) gauge symmetry.
As forD s ,D s in the lepton sector, we again the same form as (3.28), but the vector supermultiplet meditates U (2), U (1) internal degrees of freedom for s = l, s = e R , respectively.
The internal symmetry corresponding to elements of (M 3 (C), H, C) in the Table 1 amounts to U (3) × U (2)×U (1). In order to construct unified theory, we relate U (1) ′ s included in the U (2), U (3) to the U (1)'s induced by the fluctuations which correspond to u a , j = 1, 2, 3 in Table 1. We choose appropriate u This operation is to rewrite the notation of the vector supermultiplets as follows: The derived internal symmetry U (2) becomes SU (2) ⊗ U (1) and U (3) becomes SU (3) ⊗ U (1). These U (1)'s combined with the other U (1)'s shall become to that of weak hypercharge.
Using (3.31), we can rewrite (A.13) which is the fluctuation of D induced by u [3] a as follows: In the same way, using (3.32), we can rewrite the fluctuation (A.42) into Through the above operations, the vector supermultiplet of the Q-sector is described as follows: The vector supermultiplets in the other sector can be written as well. The quantum number Y which describes the strength of coupling between {B µ , λ 1α , D 1 } and quark, lepton is just weak hypercharge. The matrix form of weak hypercharge is given by The matrix elements shall be decided such that the electric charge of the particle is given by The solution of (3.39) is given by Here we can verify Tr Y = 0.

Higgs Supermultiplet
The fluctuation for γ M ⊗ D F is given by (3.4) and (3.5). Taking the commutativity of M (u(d)) and u [3] a into account, non-vanishing elements for the quark sector are given by The total Dirac operator modified by the fluctuations which have appeared up to the present in the quark sector is given by and The concrete form of Higgs supermultiplet is given by (3.47) so that which constructs a isospin doublet. In the same way, we obtain that The elements of H u , H d construct chiral multiplets as follows:

Spectral Action
Here, using the modified total Dirac operator (3.44), we express the action of the kinetic terms of chiral, antichiral supermultiplets and their minimum interaction with vector supermultiplets including gauge fields and Higgs supermultiplets. Let Ψ s+ = (ϕ s , ψ sα , F s ) and Ψ s− = (ϕ * s ,ψα s , F * s ) denote chiral and antichiral supermultiplets which represent wave functions of matter particles and their superpartners. With the definition of supersymmetric invariant product (2.6), the action with regard to the chiral and anti-chiral supermultiplets is given by where Φ is the wave function defined in (2.30),(2.31). L kinetic , L mass are expressed,respectively, by On the other hand, in the noncommutative geometric approach, the action of the vector and Higgs supermultiplets are given by the supersymmetric version of the Seeley-DeWitt coefficients of heat kernel expansion of the elliptic operator P : c n a n (P ), (4.4) where f (x) is an auxiliary smooth function on a smooth compact Riemannian manifold without boundary of dimension 4 similar to the non-supersymmetric case. Since the contribution to P from the antiparticles is the same as that of the particles, we consider only the contribution from the particles. Then the elliptic operator P in our case is given by the square of the Wick rotated Euclidean Dirac operatorD tot [9]. Nonvanishing a n 's for n in the flat space are given by where E and the bundle curvature Ω µν in the flat space are defined as follows: We note that the trace for the spin degrees of freedom is the supertrace [9] defined by The square of total Dirac operator for the quark sector (3.44) is given by where 14) When we decompose ω = 1 2 A µ describes gauge connection and does not have non-diagonal elements of (4.12), because it does not act on the finite space H F . So the non-diagonal elements which include differential operators belong to B. So, we obtain the following expressions: We note that the field strength Ω µν also does not have non-diagonal element.
In the previous paper [9], we have already obtained the heat kernel expansion coefficients a n (P ) due to (iD M ) 2 ⊗ 1 F . Let a (M) n denotes these coefficients. a and is given by where s runs over the elements of H F , s = Q, u R , d R , l, e R and F µν s is the field strength of gauge field of the matter particle that s indicates.
Here, we rescale the vector supermultiplet with SU(3) and SU(2) gauge degrees of freedom as follows: where λ p , τ p denotes Gell-Mann matrix and Pauli matrix, respectively. For the vector supermultiplet with reference to weak hypercharge, we rescale those as The field strength is expressed by and Tr(λ p λ q ) = Tr(τ p τ q ) = 2δ pq , the contribution of F Qµν to the action is given by In the r.h.s. of (4.24), the factor 2, 3, 6 is from the fact that the left-handed quark Q transforms as (3,2) under the gauge group SU (3) × SU (2). In the same way, we obtain that s=uR,dR Tr(F (s) µν F (s)µν ) = 2 The sum of (4.24),(4.25) and (4.26) amounts to Normalizing the Yang-Mills terms to − 1 4 F (j) nµν F (j)µν n gives: Known already in [7], this expression is same as that of SU (5) grand unification theory. The non-diagonal elements of (4.12) produce Higgs Lagrangian. Let us calculate tr V ((E 2 ) higgs ) = Str( i,j E ij E ji ). Using (4.13)∼(4.16), we obtain the result as follows: where s = u(d), and diag y † s y s = diag y s y † s = |(y s ) 11 | 2 , |(y s ) 22 | 2 , |(y s ) 33 | 2 , (4.32) Tr y † s y s = Tr y s y † Taking tr V ((E 2 ) higgs ) due to the lepton sector into account, the supersymmetric action of Higgs fields which interact with vector superfields is given by where we use (4.28). Then we have obtained all the terms of correct Lagrangian which give MSSM.

CONCLUSIONS
We defined the "triple" extended from the spectral triple which was to specify NCG. As the functional space, we take chiral and antichiral supermultiplets which correspond to matter fields and their superpartners in the supersymmetric standard model. we introduced algebra A and total Dirac operator iD tot = iD M ⊗ 1 + γ M ⊗ D F in the flat space-time which acted on the functional space H = H M ⊗ H F . We considered the internal fluctuations induced by the elements of A. The fluctuation for iD M ⊗ 1 generated vector supermultiplets. The vector supermultiplets meditate U (3)⊗U (2)⊗U (1) gauge degrees of freedom. We took out U (1)'s from U (3), U (2) and combined the other U (1)'s to obtain U (1) of weak hypercharge so that the gauge degrees of freedom amounted to those of MSSM, SU (3) × SU (2) × U (1) Y and each matter particle was distributed to adequate quantum numbers. On the other hand, the fluctuation for γ M ⊗ D F generated supermultiplets which transformed as Higgs fields of MSSM.
Following the supersymmetric version of spectral action principle, we calculated the square of the total Dirac operator and Seeley-Dewitt coefficients of heat kernel expansion. From the coefficient a 4 (P ) (M) , we obtained the action of vector supermultiplets of MSSM. Normalizing the coefficients of the squared field strength of each of SU (3),SU (2),U (1) Y gauge field to the same value, we found the relations between coupling constants which was same as that of SU(5) grand unification theory. We also verified that the coefficient due to non-diagonal elements of total Dirac operator gave the action for the Higgs supermultiplets and that we arrived at the correct whole action of MSSM.
All formulae in this paper were established in the Minkowskian space. We are now preparing the theory in which the total Dirac operator in the curved Riemannian space is assumed in order to take gravity into account. It will give the supersymmetric version of NC geometric view of unifying the gravity and gauge,Higgs fields.

Appendix A Vector supermultiplet with U(r) gauge symmetry
In this appendix, we will see that choosing suitable components of (2.32),(2.33), we can construct the vector supermultiplet in (3.8)∼(3.10) which meditates the U (r) gauge degrees of freedom.

A.2 U(3)
In this case, the elements of A F ⊗ A M are 3 × 3 matrix-valued functions (u [3] a (x)) B A , A, B = 1, 2, 3 which act on the internal degrees of freedom of quarks A = 1, 2, 3. The 3 × 3 matrix-valued functions (u [3] a ) B A ∈ A F ⊗ A + and (u [3] a ) B A ∈ A F ⊗ A − can be expressed in the matrix form as follows: where λ k are Gell-Mann matrices. The internal fluctuation due to (u [3] a ) B A and (u a ) D C is given by Substituting (A.1), (A.2), the fluctuation due to the part of (u [3] a ) B A is obtained by where we use the following formulae: From (2.9),(2.10),(2.13), the product ofū a ,D,u a is given bȳ In the way same as (A.7), we obtain the following formula: , where we use the Wess-Zumino condition Substituting (A.8) into (A.4)Cwe obtain the following formulae: where and NextCwe deal with the fluctuation due to the part of u a ) D C . As for the representation of SU (3)Cit is to extract (1 ⊕ 8) × (1 ⊕ 8) from irreducible representations into which the tensor product (8 ⊗ 8) ⊗ (8 ⊗ 8) is decomposed, where × denote the product of representation matrix. So we contract the tensor product as shown in the second term of (A.3). We let V Continuing the similar calculations, we see that the matrix elements vanish except for V is given as follows: Here, using (A.14) and (A.17), A [3] µ is given by W Z )ī j = V [3] ij + V Repeating the similar calculation, the fluctuation for the opposite chirality sector is given by Here, we calculate internal fluctuation due to quaternion-valued function. Quaternion-valued chiral multiplet (u [2] a ) I J ∈ A F ⊗ A + is expressed into matrix form as follows: The matrix form of (u [2] a ) IJ ∈ A F ⊗ A − is also given by a with the aid of (A.33), (A.35), we obtain We also obtain the following expressions . (A.41) Substituting these into (A.37), we obtain In the same way, we have  W Z )ī j = V [2] ij + V [2] − i∂ µ A [2] µ − 2iA [2] µ ∂ µ − A [2] µ A [2] (1) In Eq. (3.22), in addition to the elements in V [3] W Z ,V [3] W Z ,V [2] W Z ,V [2] W Z , there exist elements which correspond to the second and forth term for (r, r ′ )=(2, 3), (3,2). Under the Wess-Zumino gauge condition, we calculate these terms as follows: In (A.50), the contraction of indices of internal degrees of freedom corresponds to extracting (3,8)   c a c b − 1 2 (ψ [2] aσ µ ψ [2] a ) J I (ψ [3] bσ µ ψ [3] b ) B A − i (ψ [2] aσ µ ψ [2] a ) J I (ϕ where the last equation is due to the commutativity of (A