Extended supersymmetry with central charges in Dirac action with curved extra dimensions

We discuss a new realization of $\mathcal{N}$-extended quantum-mechanical supersymmetry (QM SUSY) with central charges hidden in the four-dimensional (4D) mass spectrum of higher dimensional Dirac action with curved extra dimensions. We show that this $\mathcal{N}$-extended QM SUSY results from symmetries in extra dimensions, and the supermultiplets in this supersymmetry algebra correspond to the Bogomol'nyi--Prasad--Sommerfield states. Furthermore, we examine the model of the $S^2$ extra dimension with a magnetic monopole background and confirm that the $\mathcal{N}$-extended QM SUSY explains the degeneracy of the 4D mass spectrum.


N = QM SUSY in higher dimensional Dirac action
In this section, we show that the structure of N = 2 QM SUSY is always hidden in the 4D mass spectrum of the (4 + d)-dimensional Dirac action with curved extra dimensions.

13)
A = e ∆ −iγˆyeŷ y (∇ y + iqA y ) + W , (2.14) Then, by requiring that the KK mode functions satisfy the orthonormal relations we can obtain the following action: where ψ (n) α (x) = ψ (n) R,α (x)+ψ (n) L,α (x) indicate 4D Dirac spinors with mass m n and ψ (0) L/R,α (x) are massless 4D chiral spinors. The expression of the above effective 4D action coincides with the case of flat extra dimensions given in [45], although the effects of curved spaces and background fields appear as the mass spectrum through the definition of A, A † and (2.16).
Since we have assumed that the KK mode functions f (n) α and g (n) α form the complete set respectively, the orthonormal relations (2.16) lead to From the above relations, we can obtain , (2.19) where the supercharge Q, the Hamiltonian H and the "fermion" number operator (−1) F are defined by F yy ′ is the field strength for A y and R is the Ricci scalar defined on Ω. Then, we can find that the relations (2.19) realize the N = 2 supersymmetric quantum mechanics [1,48]. 5 In this model, the "bosonic" and "fermionic" states which form an N = 2 supermultiplet correspond to the KK mode functions ( f (n) α (y), 0) T and (0, g (n) α (y)) T . Before closing this section, we comment about the Hermiticity of the supercharge. From the action principle δS = 0, we obtain the following condition for the KK mode functions: for all m, n, α, β, where ∂Ω represents the boundary of Ω, and n y (y) is an orthonormal vector on ∂Ω. We can show that the above condition corresponds to the Hermiticity condition for the supercharge. Then, the supercharge Q is Hermitian as long as the action principle is required. Thus, we can conclude that the N = 2 QM SUSY is always realized in the 4D mass spectrum of the higher dimensional Dirac action and the doubly degenerate states ( f (n) α (y), 0) T and (0, g (n) α (y)) T are mutually related by the supercharge Q, except for zero energy states.

N-extended QM SUSY with central charges
In the previous section, we have described the N = 2 QM SUSY hidden in the doubly degeneracy of f (n) α and g (n) α (y). However, we can expect that further hidden structures exist in the 4D mass spectrum and this would lead to the extra degeneracy due to the index α in addition to the doubly one. 5 The N = 2 SUSY algebra {Q i , Q j } = 2Hδ i j (i, j = 1, 2) is obtained with Q 1 = Q and Q 2 = i(−1) F Q.
In this section, we show that the N-extended QM SUSY with central charges can be constructed from symmetries in the extra dimensions. This QM SUSY can explain the extra degeneracy in the 4D mass spectrum. Then, we clarify the representation of this algebra for the nonzero energy states and it will turn out that the eigenstates become BPS states. This section is devoted to the general discussion, and a concrete example will be given in the next section.

N-extended SUSY algebra with central charges
Here, we discuss a new realization of N-extended QM SUSY from symmetries. First, we consider sets of operators {â i (i = 1, 2, · · · , N a )}, {b i (i = 1, 2, · · · , N b )}, · · · , {α i (i = 1, 2, · · · , N α )}, {β i (i = 1, 2, · · · , N β )}, · · · , which are Hermitian and consistent with an imposed boundary condition for the mode functions f (n) α , 6 and commute with A † A Therefore, these operators do not change the mass eigenvalues and would be related to the symmetries in the extra dimensions. Furthermore, we require that these operators commute with the ones in the same sets and anticommute with the ones in the different sets for Roman and Greek letters 4) and the operators with the Roman letters and the ones with the Greek letters commute with each other Then, we define the following extended supercharges and obtain N = (N a + N b + · · · + N α + N β + · · · ) SUSY algebra with the central charges 7 where H denotes the Hamiltonian given by (2.21) and the central charges (3.11) 6 More precisely, we require that the functionsâ i f (n) , · · · also satisfy the imposed boundary condition. 7 Although we can also define the supercharges with the replacement of the operators with the Roman and the Greek letters, those are essentially same as the ones given in the above. Therefore, we can consider that the central charges in this SUSY algebra result from the symmetries in the extra dimensions. If we take the sets of operators as reflection operators and gamma matrices, this extended QM SUSY corresponds the one given in the previous papers [45].
For the existence of the extended SUSY with given sets of operators, the metric of curved spaces and the background fields are restricted to satisfy the condition (3.1). However, it seems difficult to find the constraints without any assumption for sets of operators. 8 Therefore, we will first prepare the geometry and the background fields, and then consider the sets of operators consistent with them when we see an example in section 4.
It should be mentioned that this central extension is given by direct sums of mutually (anti)commuting N = 2 SUSY algebras as well as the previous paper [45], with different "Hamiltonians" for each of them. Especially, the supercharges Q (A) This property is important for the discussion of the BPS states in the next subsection.

Representation of SUSY algebra
Then, let us clarify the representation of this algebra for the nonzero energy states. Since the Hamiltonian and the central charges commute with each other, we first look at the simultaneous eigenstates of them: where n and z indicate the labels of their eigenvalues m n and z (A) i j , 9 and s denotes the extra index to further classify the eigenstates in the following discussions. For these states, the algebra (3.8) is rewritten into (3.14) Since z (A) i j is the real symmetric matrix, it can be diagonalized by the orthogonal matrix U (A) i j . Then, by redefining the supercharges, we can obtain It should be noted that, in this algebra, at most one supercharge among Q ′(A) i (i = 1, · · · , N A ) can become nontrivial and the others must equal to 0 for any eigenstates. This is because the supercharges satisfy the relation due to (3.15) and the commutativity of Q (A) i and Q (A) j . Therefore, the number of nontrivial supercharges for the eigenstates are maximally given by the number of the sets of the operators related to symmetries, and the eigenstates necessarily become the BPS states if the sets have more than one operator.
Next, to construct the supermultiplets in this SUSY algebra, we introduce the following operators by means of the nontrivial redefined supercharges for the eigenstates: where A, B, C, and D are different with each other according to the above discussion. These operators commute with each other, and therefore, we can further classify the eigenfunctions by these operators. Since these operators satisfy for the eigenstates, we can parametrize their eigenvalues as 10 where s (AB) i j = ±. Here, we have described the index s as s (AB) i j s (CD) kl · · · . From the relation i j = ± , s (CD) kl = ± , · · · and we can explicitly construct the supermultiplet from Φ (n) ++··· ,z as

Example
In this section, we will confirm that the N-extended QM SUSY given in the previous section can be realized in higher dimnsional Dirac action with curved extra dimensions. As an example, we examine the S 2 -extra dimension with the Wu-Yang magnetic monopole background.

Spin-weighted spherical harmonics
As is well known, the mode functions on S 2 -can be expressed by the spin-weighted spherical harmonics [51][52][53]. Thus, we briefly review the Newman-Penrose ð (eth) formalism and this function. First, we consider the rotation of the orthogonal basis e θ , e φ defined in tangent space on S 2 (4.1) We call that a quantity η has spin weight s, if η transforms as follows under the above transformation: η → e isα η . Furthermore, we introduce the operators ð (eth) andð (eth bar) which act as follows for η with the spin weight s: We can show that ðη has spin weight s + 1 andðη has spin weight s − 1. Therefore, ð andð correspond to the spin weight raising and lowering operators, respectively.

From (4.3) and (4.4), we obtain
The spin-weighted spherical harmonics s Y jm with the spin weight s is given as the eigenfunction ofðð and ðð :ð or equivalently where the spin weight is given by s = 0 , ±1/2 , ±1 , ±3/2 · · · . The index j (= |s| , |s| + 1 , |s| + 2 , · · · ) denotes the main total angular momentum quantum number and the index m (= − j , − j + 1 , · · · , j − 1 , j) indicates the secondary total angular momentum quantum number. Since the spin-weighted spherical harmonics s Y jm form a complete set for the fixed spin weight s, any function on S 2 with the spin weight s can be decomposed into s Y jm . The explicit form of the normalized spin-weighted spherical harmonics is written as which satisfies the orthonormal relation Here, we have chosen the phase of this function in such a way that the function satisfies In the case of s = 0, the spin-weighted spherical harmonics corresponds to the spherical harmonic Y jm . As well as the ordinary spherical harmonics, this function corresponds to the representation of su(2) algebra where L 2 , L z and L ± are the angular momentum operators for quantities with spin weight s and satisfy Furthermore, this function satisfies the following properties:

KK mode functions and mass spectrum of S 2 -extra dimension with magnetic monopole
Then, we discuss the KK mode functions and the mass spectrum of S 2 -extra dimension with a magnetic monopole. We consider the space M 4 × S 2 with the radius a ds 2 = η µν dx µ dx ν + a 2 (dθ 2 + sin θ dφ 2 ) , (4.18) and we choose the basis of the vielbein as (1, 1, 1, 1, a, a sin θ) (K = 0, 1, 2, 3, θ, φ ,K =0,1,2,3,θ,φ) . Furthermore, we introduce the Wu-Yang magnetic monopole background where q is the gauge coupling constant. 11 The gauge fields A N and A S are defined on the north patch (0 ≤ θ < π , 0 ≤ φ < 2π) and the south patch (0 < θ ≤ π , 0 ≤ φ < 2π) on S 2 , respectively. 11 In the case of n = 0, the Einstein equation leads to a → ∞. However, in the case of n 0, the radius a is stabilized and given by a 2 = n 2 κ 2 /8q 2 where κ is the 6D gravitational coupling constant [54]. In this paper, we concentrate the structure of the mass spectrum of this model and we will not take into account the stability of a.
Then, we consider the 6D Dirac action with the monopole background and the bulk mass M: where we require that the Dirac fields on the north patch Ψ N (x, θ, φ) and on the south patch Ψ S (x, θ, φ) are related by gauge transformation Here, we define the internal chiral matrix γ in which satisfies and we decomposition Ψ N/S (x, θ, φ) as where the upper and lower signs in the exponential denote the ones for the north and south patches, respectively. (Here after, we use the same notation for the signs in the exponential if appear.) The operators ð s + andð s − represent the spin weight raising and lowering operators for the spin weight s ± = −(n ± 1)/2, and P ± = (1 ± γ in )/2 are the projection matrices for γ in . Then, A † A and AA † are given by Thus, we can find that the mode functions and the mass eigenvalues are obtained as follows from (2.19) and (4.6): where s ± , j and m are given by |s + | , |s + | + 1 , |s + | + 2 , · · · (for α = +) |s − | , |s − | + 1 , |s − | + 2 , · · · (for α = −) , m = − j , − j + 1 , · · · , j − 1 , j , (4.30) 12 The massive mode functions g (k)N/S α ∝ A f (k)N/S α are not the eigenfunction with γ in = α because A and γ in do not commute with each other. and we define e ± as the 2-component orthonormal vectors which satisfy γ in e ± = ±e ± , γθe ± = ie ∓ , γφe ± = ∓e ∓ . (4.31) Then, the degeneracy of j-th KK level is 2 j + 1 for each mode function.
We can see that the mode functions f This is also useful for the construction of the N-extended QM SUSY.

• properties of reflection operators
In the model without the magnetic monopole (n = 0), we can consider the reflection operators with gamma matrices γ in γθR θ and γ in γφR φ where R θ and R φ represent the reflections for the coordinates θ and φ In the case of S 2 with monopole (n 0), the above operators are ill defined and do not commute with A † A by the monopole background. However, the operator with the combination of the above reflections and the gauge transformation is Hermitian and well defined for the mode functions and satisfies We find that this operator connects the mode functions f Then, we can obtain various kinds of N-extended QM SUSYs from the above operators. As examples, we consider the following two N-extended QM SUSYs: • N = 6 extended QM SUSY with central charges from the angular momentum operator First, we consider the N = 6 extended QM SUSY with central charges which consist of the angular momentum operator L (n) z : where the supercharges Q k and the nonzero components of central charges Z kl are given by In this QM SUSY, Q 3 and Q 4 commute with each other and also Q 5 and Q 6 .
According to the discussion given in the section 3.2, we redefine the supercharges for the eigenfunctions with the eigenvalues H = m 2 j , Z 34 = m m 2 j and Z 33 = (−1 + m 2 ) m 2 j (where m indicates the angular momentum number): The SUSY algebra (4.40) is written into the following form for the redefined supercharges where z ′ k (k = 1, · · · , 6) is given by and also the eigenfunctions Φ ( jm) for n < 0 and j = s + . (4.53) Here z ′ indicates {z ′ i (i = 1, · · · , 6)} given in (4.51) and s 12 = ±, s 35 = ± denote the signs of the eigenvalues of Then we can show that the above eigenfunctions form the supermultiplets and satisfy the same relation as (3.21). In this extended QM SUSY, the four (two) -fold degenerated eigenfunctions with H M 2 (H = M 2 ) are related by the supercharges Q ′ k with k = 1, 2, 3, 5 (k = 3, 5). Furthermore, the extra (2 j + 1)-fold degeneracy exists in the eigenfunctions. The origin of this degeneracy comes from the angular momentum number m, which corresponds to the eigenvalues of central charges.
• N = 6 extended QM SUSY with central charges from the angular momentum operator and the reflection operator Next, we construct the N = 6 extended QM SUSY with central charges from the angular momentum operator and the reflection operator: where the supercharges Q k and the nonzero components of the central charges are taken to be the form of We can show that the above eigenfunctions form the supermultiplets and satisfy the same relation as (3.21). The supermultiplets for the BPS states can be constructed by use of the mode functions f ( jm)N/S α,r and g ( jm)N/S α,r in the same way. In this extended QM SUSY, the eight-fold degenerated non-BPS states is related by the six supercharges Q k (k = 1, · · · , 6). Therefore, we see that the additional two-fold degeneracy can be further explained by the supercharges compared with the previous extended QM SUSY (although the total degeneracy including the BPS states with m = 0 is not changed). This additional degeneracy corresponds to the parity even and odd for the reflection. Since the eigenfunctions with m = 0 are only equivalent for the parity even from the definition (4.59), they should become the BPS states.
Although we considered two examples, there are more extended QM SUSYs constructed from the symmetries. For instance, we can obtain them by using the reflection operators γ in γθR θ and γ in γφR φ in the case without the magnetic monopole, since those operators correspond to the symmetries in such model.

Summary and discussion
In this paper, we have constructed the new realization of the N-extended QM SUSY with central charges hidden in the higher dimensional Dirac action with curved extra dimensions. This extended QM SUSY results from symmetries in extra dimensions, and the supercharges and the central charges are obtained by use of them. We have also investigated the representation of the SUSY algebra and shown that the supermultiplets would become the BPS states. In addition, we considered the model of S 2 -extra dimension with the magnetic monopole as the concrete example. Then we have confirmed that the KK mode functions properly correspond to the representations in the two types of N-extended QM SUSYs which are obtained from the rotational and reflection symmetries in the extra dimensions.
The characteristic property of our extended QM SUSY is that certain supercharges commute with each other. Recently, a new generalization of supersymmetry is proposed, that is Z n 2 -graded supersymmetry whose supercharges have n degrees [55][56][57]. For their supercharges with different degrees, the algebra may not close in anticommutator but commutator. Therefore, it is interesting to investigate the relation between this supersymmetry and our QM SUSY.
Our analysis is not perfect. As we have seen in the section 4, there appear various extended QM SUSYs in a model, according to sets of symmetries. Thus, we should clarify what kinds of extended QM SUSYs can be obtained from a model. It is also important to reveal that what symmetries exist by the choice of boundary conditions, extra dimensional spaces and background fields since they would affect the structures of extended QM SUSYs. Furthermore, it is known that central charges are closely related to topological properties [40,58,59]. Although the central charges given in the section 4 might be related to the topology of S 2 and the magnetic monopole, the detailed structures are not unveiled. Thus, we should more investigate models with nontrivial topology. We can also expect the possibilities that the further structures are hidden in the 4D mass spectrum. Since our discussions are not completely general, other realizations of extended QM SUSY might be constructed.
Moreover, since we have obtained the new extended QM SUSY, it is fascinating to study new types of exactly solvable quantum-mechanical models. The issues mentioned in this section will be reported in a future work.