Supergravity on the noncommutative geometry

Two years ago, we found the supersymmetric counterpart of the spectral triple which specified noncommutative geometry. Based on the triple, we derived gauge vector supermultiplets, Higgs supermultiplets of the minimum supersymmetric standard model and its action. However, unlike the famous theories of Connes and his co-workers, the action does not couple to gravity. In this paper, we obtain the supersymmetric Dirac operator $\mathcal{D}_M^{(SG)}$ on the Riemann-Cartan curved space replacing derivatives which appear in that of the triple with the covariant derivatives of general coordinate transformation. We apply the supersymmetric version of the spectral action principle and investigate the heat kernel expansion on the square of the Dirac operator. As a result, we obtain a new supergravity action which does not include the Ricci curvature tensor.


Introduction
The standard model of high energy physics has some defects. It can not include gravity theory, can not solve hierarchy problem, has many free parameters including coupling constants of gauge groups in the theory which must be decided by experiments. More essentially, it cannot explain why the gauge group is SU (3) × SU (2) × U (1). Connes and his co-workers derived the standard model coupled to gravity on the basis of noncommutative geometry(NCG) [1][2][3][4]. Their result is that if the space-time is a product of a continuous Riemannian manifold M and a finite space F of KO-dimension 6, gauge theories of the standard model are uniquely derived [5,6]. In the model, three coupling constants of SU (3), SU (2), U (1) are unified by the same relation as that of SU (5) grand unified theory(GUT). The Weinberg angle is also fixed at that of the GUT.
On the other hand, the most powerful candidate of new physics to solve the hierarchy problem is supersymmetric theory [7]. One loop correction to squared Higgs mass m 2 H from a Dirac spinor contains square of ultraviolet cut off, Λ 2 U V . If Λ U V is order of Plank scale, this correction is 30 orders of magnitude larger than the value of m 2 H . Introducing supersymmetry brings about one more loop correction from a boson which is the superpartner of the fermion. It has the same absolute value as that of the fermion loop but with opposite sign, so that the hierarchy problem is systematically removed.
Unfortunately, it is difficult, perhaps impossible to extend the NCG itself to new one which produce supersymetric particle models. The framework of NCG is specified by so-called the spectral triple (H 0 , A 0 , D 0 ), where H 0 is a Hilbert space which consists of spinorial wave functions of matter fields in the standard model, A 0 and D 0 are algebra and Dirac operator which act on H 0 . Let us consider to extend the spectral triple to supersymmetric counterpart, (H, A, D), where H is the space which consists of not only spinorial wave functions but also of bosonic wave functions of superpartners of the matter fields and elements of A and D are operators which act on H. Supersymmetric theories are mostly formulated on the Minkowskian space-time, while the standard model constructed from the NCG is formulated on the space-time with Euclidean signature. So the space H will not be a Hilbert space. In addition, since the extended Dirac operator D will include d'Alembertian which appears in the Klein-Gordon equation, [D, a] will not be bounded for an arbitrary element a ∈ A, so that D −1 can not play the role of infinitesimal length element ds of a geometry. These facts do not obey axioms of NCG.
Nevertheless, if not only supersymmetry but also NCG have important meaning in particle physics, these successful theories must coexist. Recently, we arrived at the minimum supersymmetric standard model(MSSM) based on NCG [8][9][10]. We have found the supersymmetric counterpart of the process to construct the standard model action from the NCG one by one. At first, we obtained "the triple" (H, A, D) extended from the spectral triple and verified the supersymmetry of it. The space H is the product of the functional space H M on the Minkowskian space-time manifold and the finite space H F which is the space of labels denoting the matter particles. According to the constitution of H, the algebra A/the Dirac operator D consists of A M /D M which acts on the manifold and A F /D F which acts on the finite space, respectively. The above construction was performed in the Minkowskian signature in order to incorporate supersymmetry. As mentioned earlier, the spectral triple corresponds to the NCG, but the triple does not define a new NCG. However, projecting H M to the fermionic part and changing the signature from the Minkowskian one to the Euclidean one by the Wick rotation, we found that it reduced to the theory constructed on the original spectral triple. We derived internal fluctuation of the Dirac operator, D, which induced vector supermultiplets of gauge degrees of freedom and Higgs supermultiplets.
Secondarily, we obtained the action of the NCG model in terms of the supersymmetric version of spectral action principle [11] which was expressed by where Φ k denoted the wave functions which described the chiral or antichiral supermultiplet, P = (iD) 2 and f (x) was an auxiliary smooth function on a 4D compact Riemannian manifold without boundary [11]. We calculated the Seeley-Dewitt coefficients due to the second term of (1), which gave the action of the Non-Abelian gauge fields and Higgs fields. However, since the triple is constructed on the Minkowskian space-time,i.e. flat spacetime, it does not give the action of gravity. In this paper, We replace the derivative i∂ µ which appears in D M with the covariant derivative i∇ µ which includes spin connection with contorsions generated by gravitino [12,13]. Then we obtain the supersymmetric version of Dirac operator, D SG M on the curved space-time. We investigate the square of the Dirac operator P = (iD SG M ) 2 and the Seeley-Dewitt coefficients due to P in order to obtain the action of supergravity.

2/14
2. Supersymmetrically extended triple on the flat space-time In our previous papers, we introduced the triple for the supersymmetric theory which was extended from the spectral triple of the NCG on the flat Riemannian manifold. In this section, let us review it. The functional space H M on the Minkowskian space-time manifold is the direct sum of two subsets, H + and H − : The element of H M is given by Here, Ψ + , Ψ − are denoted by and in the vector notation. Here, ϕ + and F + of Ψ + are complex scalar functions with mass dimension one and two, respectively, and ψ +α , α = 1, 2 are the Weyl spinors on the spacetime M which have mass dimension 3 2 and transform as the ( 1 2 , 0) representation of the Lorentz group, SL(2, C). Ψ + (x) obey the following chiral supersymmetry transformation and form a chiral supermultiplet.
On the other hand,ψα transform as the (0, 1 2 ) of SL(2, C) and Ψ − (x) form an antichiral supermultiplet which obey the antichiral supersymmetry transformation as follows: The Z/2 grading of the functional space H M is given by an operator which is defined by In this basis, we have γ M (Ψ + ) = −i and γ M (Ψ − ) = i. Hereafter, we suitably abbreviate unit matrices or subscripts which denote sizes of unit and zero matrices.

3/14
For the state Ψ ∈ H M , the charge conjugate state Ψ c is given by The antilinear operator J M is defined by so that it is given by where C is the following charge conjugation matrix: and * is the complex conjugation. The operator J M obeys the following relation: The real structure J M is now expressed for the basis of the Hilbert space (Φ, Φ c ) T in the following form: The Z/2 grading Γ M on the basis is expressed by Corresponding to the construction of the functional space (2), the algebra A represented by them are expressed as Here an element u a of A + , which acts on H + , and an elementū a of A − , which acts on H − are given by where {ϕ a (ϕ * a ), ψ aα (ψα a ), F a (F * a )} are chiral(antichiral) multiplets. Note that these multiplets are not related to the multiplets in the functional space in Eqs. (5) and (6). The elements of A, u a andū a together with the Dirac operator are the 4/14 origin of the gauge and Higgs supermultiplets, while the elements (5) and (6) of the functional space are the origin of matter fields.
On the basis (Φ, Φ c ) T , the Dirac operator D M on the manifold is given by and where We verified in Ref. [8] that the Dirac operator and the supersymmetric transformation expressed by Eq. (7) and (8) were commutative. When we change the order of elements in the basis (3), (4), (5), (6) to the Dirac operator (21) is replaced with We note again that the above formalism was given in the framework of Minkowskian signature in order to incorporate supersymmetry. When we restrict the functional space H M to its fermionic part H 0 and transfer to the Euclidean signature, we recover the original spectral triple which gives the framework of NCG.

Dirac operator on the curved space-time
In order to obtain the supersymmetric Dirac operator on a curved space-time, we must consider torsion tensor. The torsion consists of gravitino ψ µ which is a majorana spinor vector.
It is the antisymmetric part of affine connection expressed by The affine connection with the torsion is a sum of Christoffel symbol Γ λ µν and contorsion Y λ µν :Γ The relation between the contorsion and the torsion is given by

5/14
The spin connection is separated to the contorsion and the part without contorsion [14]: where e a µ is vielbein which connects general coordinates denoted by subscript of the Greek letter to local inertial coordinates denoted by that of Roman letter. The covariant derivative for a spinor in the curved space is described bỹ whereω µ is a sum of products of the spin connection and commutator of γ matrices γ ab = 1 2 [γ a , γ b ] which is expressed bỹ On the curved space, for the (2,2)-th entry of the matrix (24), we replace the partial derivative ∂ µ with the covariant derivative and for the (1,3)-th and (3,1)-th entries which act on bosonic wave functions, we adopt the operator which appears in the equation given by the action of the Klein-Gordon field in the curved space [15]. So, the Dirac operator on the curved space is expressed by whereR is curvature with the torsion and υ is an unknown constant.

Supergravity action
In our noncommutative geometric approach to supersymmetry, the action for supergravity will be obtained by the coefficients of heat kernel expansion of the operator P = (D (SG) M ) 2 [16]. The prescription to obtain these coefficients on the curved space with the torsion for the spinorial part of D (SG) M , i.e. (2,2)-th entry of the matrix (33) and its result are given by [13]. We want to obtain the coefficients for D (SG) M including (1,3)-th and (3,1)-th entries. At first, we expand the operator P into the following form: We define a vector S µ as follows: where Q µ is torsion trace T α µα . In our theory, trace over the vector bundle are replaced with supertrace. When a matrix M in the basis (23) is given by 6/14 the supertrace is expressed by Since the bosonic degrees of freedom equals that of fermionic states, the supertrace of I vanishes. Then the coefficients a n (P ) are given by where Ω µν is the bundle curvature that we will describe later and Z is a function defined as follows: After some algebra in terms of the Riemann curvature tensorR λ ρµν with torsion in appendix A, the square of (33) is given by where D (ϕ) On the basis (23), the functionsÃ,B in (34) and S µ in (35) are expressed by the matrix form as follows: Then the matrix form of the function Z is given by Using Eq.(48) and Eq.(49), a 2 (P ) of (39) can be converted into In the basis (23), the bundle curvature Ω µν in Eq.(40) is also given by where Ω (ϕ) µν is given by and using (A.10), Ω (ψ) µν is also obtained by After long and tedious algebra using the supertraces in appendix B, the coefficient a 4 (P ) of (40) is converted into In general, the coefficients a n (P ) vanish for odd values of n [16]. In addition, the mass dimension of each term of the integrand in a n (P ) is n so that in order to conserve the renormalizabilty, a n (n > 4) should not appear in the action. Then, we have obtained the all renormalizable terms of our supergravity action.

Conclusions and discussions
In this paper, we have derived in Eq.(33) the supersymmtric Dirac operator D supertrace and calculated the Seeley-Dewitt coefficients of the heat kernel expansion. The coefficient a 0 (P ) in Eq.(38) cancels out. It means that the cosmological constant vanishes in the supersymmetric theory. So the supergravity action of our theory is given by where α and β are some constants, the coefficients a 2 (P ) and a 4 (P ) are given in Eq. (50) and (54). It is a modified Einstein-Hilbert action.
In the action (62), there is no term with the Ricci curvature tensor. Therefore, when we construct based on NCG a gravity theory which possesses physically important property such as conformal invariance, renormalizability, if we want the theory able to be extended supersymmetrically, we should build its action not to include Ricci curvature tensor. For example, one of the simplest theory which possesses local conformal symmetry and renormalizabilty consists of the Weyl action term d 4 x √ −gC 2 and the Gauss-Bonnet topological term and surface terms d 4 x √ −g(ηG b + τ R) [17] , where C 2 and G b terms are given by The linear combination whose terms with Ricci tensor cancel out is given by Then we obtain υ = − 1 6 . The coefficient a 2 (P ) in the Eq.(50) is replaced with The coefficient a 4 (P ) is also replaced with a 4 (P ) When we reduce the coefficient a 4 (P ) in Eq.(68) to non-supersymmetric part, i.e. the part without terms including torsion tensor, it has captured the local conformal symmetry and 10/14 renormalizability. Indeed, the coefficient a 4 (p) includes a new type of non-supersymmetric gravity action S 1 which is given by The variation of S 1 due to the conformal transformation δg µν = −ǫg µν is given by where G µν is the Einstein tensor, G µν = R µν − 1 2 g µν R. Therefore, we can verify the conformal invariance of S 1 by the Bianchi's identity ∇ µ G µν = 0. We also provide some else equations about the affine connection, the spin connection and the vielbein as follows: We also note some equations on traces of gamma matrices and their product with curvature and torsion tensors.