Supersymmetric Dirac Operator on the Noncommutative Geometry

We extend naturally the spectral triple which define noncommutative geometry (NCG) in order to incorporate supersymmetry and obtain supersymmetric Dirac operator D_M which acts on Minkowskian manifold. Inversely, we can consider the projection which reducts D_M to $\not{D}$, the Dirac operator of the original spectral triple. We investigate properties of the Dirac operator, some of which are inherited from the original Dirac operator. Z/2 grading and real structure are also supersymmetrically extended. Using supersymmetric invariant product, the kinematic terms of chiral and antichiral supermultiplets which represent the wave functions of matter particles and their superpartners are provided by D_M. Considering the fluctuation given by elements of the algebra to the extended Dirac operator, we can expect to obtain vector supermultiplet which includes gauge field and to obtain super Yang-Mills theory according to the supersymmetric version of spectral action principle.


Introduction
The new boson with the mass about 125 GeV/c 2 discovered by the LHC experiments turned out to be the long sought Higgs boson of the electroweak theory [1,2,3]. In the Lagrangian of gauge theory, the gauge principle gives the term which describes each gauge boson of SU (3), SU (2), U (1) internal symmetry, while in the standard model(SM) the Higgs mechanism fixes the terms which describe weak bosons and photon in the spontaneaously symmetry broken theory which is independent of the principle. So, the SM has many free parameters and its predictve power is spoiled.
The noncommutative geometry (NCG) was applied to construct the standard model of the elementary particles by many authors [4,5,6,7]. A remarkable feature of this geometric approach is that both the gauge field and the Higgs field are introduced by internal fluctuations of the metric of NCG. The advantage of this approach is that the Higgs field emerges on the same footing as the gauge field and the Higgs coupling constants are related to the gauge coupling constants.
The NCG standard model, extended to include neutrino masses, was constructed [8,9]. Although the standard model provides a remarkably successful description of presently known phenomena, there are some unanswered questions, which suggest the existence of new physics beyond the standard model: The gauge couplings unification in the renormalization group equation is not viable phenomenologically in the framework of the minimal standard model. There is also the infamous "hierarchy problem", which is that the Higgs squared mass parameter m 2 H receives enormous quantum corrections from the virtual effects of every particle which couples to the Higgs field [10]. In addition, to answer the questions of the origin of the dark matter discovered by the astronomical observation is outside the standard model [11].
It is known that these shortcomings may be remedied by introducing supersymmetry into the standard model. In order for the NCG standard model to be phenomenologically viable, it is quite desirable to incorporate supersymmetry in the model. The purpose of this paper is to investigate how to introduce supersymmetry into NCG and to derive the supersymmetric Dirac operator on the Riemannian manifold whose fluctuation induces vector supermultiplet, while we will leave the discussion on the part of supersymmetric Dirac operator whose fluctuation induces Higgs supermultiplet to our future papers. In section 2, we extend the Hilbert space L 2 (M, S) to H M which includes not only spinor wave function but also its superpartner and auxiliary field. It constitutes of the space of chiral supermultiplet and that of antichiral supermultiplet. Then we obtain algebra which acts on H M . We also look for Dirac operator D M which acts on H M and verify that it is supersymmetric and gives the kinetic term of chiral and antichiral supermultiplet of matter particles and its superpartners. In section 3, we investigate properties of the supersymmetric Dirac operator. We can extend the element of the spectral triple to that of the extended triple and vice versa so that the supersymmetric Dirac operator has inherited some properties from that of the spectral triple. Z/2 grading operator and antilinear operator of real structure are also extended to their supersymmetric version. In section 4, we describe the conclusions and a discussion on how to introduce vector supermultiplet as internal fluctuation of the Dirac operator.

Supersymmetry and Dirac operator
The basic element of NCG consists of an involutive algebra A of operators in a Hilbert space H and of a self-adjoint unbounded operator D on H with compact resolvent such that the commutator [D, a] is bounded for all elements in A. A set of (A, H, D) is named a spectral triple or a K-cycle [12].
Here and in what follows we use the notation and convention of Wess-Bagger [13].
In order to introduce supersymmetry we need two complex scalar functions ϕ + (x) and F + (x) which are superpartners of a chiral spinor ψ +α (x). Here ϕ + (x) and F + (x) have mass dimension 1 and 2, respectively, and ψ +α (x) have mass dimension 3 2 . These functions are supposed to obey the following supersymmetry transformation: (2. 2) The set of functions is called a chiral supermultiplet, which is denoted here in the vector notation. For the antichiral spinor ψα − (x), we introduce ϕ − (x) and F − (x) which obey the supersymmetry transformation given by and the set of functions is called an antichiral supermultiplet. Then H M is the direct sum of two subsets, H + and H − : where H + is the space of chiral supermultiplets and H − is the space of antichiral supermultiplets and the element of H M is given by A supersymmetric invariant product of wave functions Ψ, Ψ ′ is defined by Now let us construct an algebra A M of operators in H M . For the basis of H M given by Eq.(2.7), the element of A M is expressed by the following matrix form: where u ij is given by the following triangular matrix, Here the scalar functions ϕ η (x), F η (x) and the spinor functions η α (x) are supposed to form a chiral supermultiplet (ϕ η , η α , F η ) and obey the same type of the supersymmetry transformation given by Eq.(2.2). In Eq.(2.11), m 0 stands for the mass parameter which was inserted to adjust the mass dimension. Since (ϕ η , η α , F η ) is the chiral supermultiplet, the triangular matrix u ij given by Eq.(2.11) obeys the following multiplication rule: As a matter of fact, ϕ η3 , η 3α and F η3 transform again as Eq.(2.2) and form a chiral supermultiplet.
For the antichiral sector,vīj in Eq.(2.10) is given by where ϕ * χ (x), F * χ (x) andχα(x) are chosen to form an antichiral supermultiplet and they obey the supersymmetry transformation given by Eq. (2.4). The multiplication rule forvīj is given by Taking into account Eq.(2.12) and Eq.(2.15), we obtain the multiplication rule for the element of A M given by Eq.(2.10) as The multiplication rule of Eq.(2.12) with Eq.(2.13) is easily understood by the help of the superfield notation. In order to do this, we introduce anticommuting parameters θ α ,θα. Then u ij of Eq.(2.11) can be expressed by the following chiral superfield, Since the product of two chiral superfields is again a chiral superfield such that U 3 = U 1 U 2 , we can deduce the relation given by Eq.(2.13).
For the antichiral supermultiplet expressed by Eq.(2.14), the corresponding antichiral superfield is given byV The multiplication formula for the antichiral supermultiplets given by Eq.(2.16) is again obtained from V 3 =V 1V2 . From Eq.(2.10), we see that the element of A M is expressed by and where A + is the subspace which acts on H + and A − is the subspace which acts on H − so that A M is the direct sum of A + and A − : In the rest of this section we construct the supersymmetrically extended Dirac operator D M in H M , which is supersymmetric and reduces to the usual Dirac operator / D of the noncommutative geometry if H M is restricted to the spinorial subspace L 2 (M, S). For the basis of H M expressed by Eq. (2.7), the γ matrices in the Weyl representation amount to where 0 n×m stands for the n × m null matrix. The Dirac operator in H M is now given by Although / D is not supersymmetric, we can show that the modified operator D M defined by and = ∂ µ ∂ µ is supersymmetric.
In order to verify that iD M is supersymmetric, let us express the supersymmetry transformation δ ξ given by Eq.(2.2) and Eq.(2.4) on the basis of H M as follows: In what follows we shall call D M the "supersymmetrically extended Dirac operator" or simply "extended Dirac operator". D M is written in the matrix form as follows: The operator Dī j operates on (Ψ + ) j and generates the element in H − in the following way, Using the definition of supersymmetric invariant product (2.8), the kinetic term in the action constructed by chiral and antichiral supermultiplets of matter fields with their superpartners is expressed by

Noncommutative geometry and Supersymmetry
Since models based on NCG in the flat space-time is constructed in the Euclidean space-time, if we see the correspondence between "the triple" and the spectral triple which defines a NCG, we must transform the variables in Minkowskian coordinates to Euclidean ones. The space-time variables are transformed by Wick rotation as follows: The algebra of SL(2, C) turns out the algebra of SU (2) ⊗ SU (2) under the rotation. The Weyl spinors which transform as ( 1 2 , 0), (0, 1 2 ) of SL(2, C) are to be replaced by ( 1 2 , 0) and (0, 1 2 ) representations of SU (2) ⊗ SU (2),respectively. The spinors which have appeared in H M are replaced as follows: where spinors with indices α transform as ( 1 2 , 0) and those with indicesα transform as (0, 1 2 ) of SU (2) ⊗ SU (2),respectively. The upper index is related to the complex conjugate of the lower index by ρ 1 = ρ * 2 , The metric and Pauli matrices which have appeared in the extended Dirac operator are to be replaced by Embedding these expressions (3.2)∼ (3.5), the triple is rewritten in the Euclidean signature. The basis of H M is denoted by the same form as (2.7), but now Ψ + and Ψ − are given by The elements of A M which correspond to (2.11) and (2.14) is now given by where η Eaα , χα Ea are ( 1 2 , 0) and (0, 1 2 ) of SU (2) ⊗ SU (2),respectively. For the extended Dirac operator on the Minkowskian manifold in (2.29), the transformed form is given by The invariant product under Euclidean supersymmetry transformation is given by the same form as (2.8), but Γ 0 should be replaced by The triple (A M , H M , D M ) is supersymmetrically extended from the spectral triple which specify a NCG. Inversely, we can consider the operator P which project the triple to the original spectral triple embedded in it. On the basis of (3.6),(3.7), P is given by (3.14) In fact, the actions of P for the elements of the triple are expressed by Under the supersymmetric invariant product (2.8), the definition of self-adjoint is given by This condition is rewritten to and For large |λ n |, there are about (π 2 /2)|λ n | 4 eigenvalues inside the four dimensional ball with the radius |λ n |. If we arrange |λ n | in an increasing sequence, we obtain for large n. So ds = D −1 M is an infinitesimal of order 1/d.
As a matter of fact, for any ε > 0 with sufficiently large N the norm of the resolvent obeys the following relation: so that it is given by where C is the following charge conjugation matrix; and * is the complex conjugation (Hermitian conjugation for matrices). The operator J M obeys the following relation: The real structure J M is now expressed on the basis (Ψ, Ψ c ) T in the following matrix form: On the same basis, the Dirac operator D M and the Z/2 grading Γ M is expressed by The real structure defined by Eq.(3.36) satisfies the following relations: (3.41) In the Minkowskian signature, we shall define J M by the same relation as Eq. (3.32). In this case, however, the charge conjugation is defined for Dirac spinors. A Dirac spinor ψ is composed of two Weyl spinors. The state Ψ in (2.7) and its charge conjugate state Ψ c in H M is denoted by and The charge conjugation matrix in Eq.(3.33) is now given by The Z/2 grading in the Minkowskian signature is defined by

Discussions and Conclusions
We have considered the supersymmetric extension of the spectral triple which define NCG in order to incorporate supersymmetry and obtained the supersymmetrically extended Dirac operator D M . It acts on H M that is also extended from the Hilbert space C ∞ (M ) to include not only spinor wave functions which represents matter fields but also their superpartners and auxiliary fields. Transferring from Minkowskian signature to Euclidean signature by the Wick rotation and applying the projection operator P which transforms H M to the subspace of spinors L 2 (M, S) embedded in H M , we can make the triple go back to the original spectral triple. The extended Dirac operator D M inherits self-adjoint characteristics, compactness of its resolvent. Z/2 grading operator and antilinear operator which gives the real structure of H M are also introduced.
But, [D M , a] is not bounded for all a ∈ A M , because D M contains second rank derivatives of spacetime variables. So, we shall note that the triple does not define NCG extended from the original one. Our goal is not to construct supersymmetric new NCG, but to extend the theories of particles and interactions based on NCG to those incorporating supersymmetry. It includes the prescriptions to obtain the super Yang-Mills theory and minimum supersymmetric standard model such as supersymmetric version of spectral action principle. Indeed, as for the case without vector supermultiplets with gauge degrees of freedom, we have shown that D M provides the kinetic terms of matter particles and their superpartners in Eq.(2.35).
In our next paper, we will introduce the "triple" in the finite space denoted by (A F , H F , D F ), where H F is the space of labels which denotes quantum numbers of matter particles and A F is the algebra represented on H F . We will show that the supersymmetrically extended Dirac operator D F provides mass terms of matter particles and their superpartners. We will also in our next paper discuss the vector supermultiplet with gauge degrees of freedom. In the NCG theory without supersymmetry, gauge fields are introduced by the fluctuations of the Dirac operator / D. The matrix form of the Dirac operator / D and elements of algebra are given by where ϕ 1j , ϕ 2j , · · · are complex functions with internal degrees of freedom. The internal fluctuation to / D is given by If the induced field A µ is gauge field, it obeys the transformation law of the gauge field and satisfies the condition: Authors choose a simple solution [6] to the conditions which is given by

7)
A µ = j iϕ * 1j (∂ µ ϕ 1j ). (4.8) In the supersymmetrically extended theory, vector supermultiplet which includes gauge field will be also introduced by the internal fluctuation due to elements of A M to the Dirac operator D M and will be given by the following form: where a iī;jj ,b iī;jj is an element of A M in the form of (2.10). As for the elements in (2.10), however, since u ij is constructed by elements of chiral supermultiplet and yetvīj is constructed by those of antichiral supermultiplet, u ij does not equalvīj identically. So, we can not extend the solution (4.8) supersymmetrically. We must discover the solution in which each component of vector supermultiplet, gauge field, gaugino and auxiliary field D satisfies the adequate transformation laws. We will show that the fluctuation of the form of (4.9) indeed produces vector supermultiplet with U (N ) gauge degrees of freedom and the heat kernel expansion of squared D M modified by the fluctuation will give the action of super Yang-Mills theory [14,15].