Supersymmetry and Cotangent Bundle over Non-compact Exceptional Hermitian Symmetric Space

We construct N=2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces M=E_{6(-14)}/SO(10)xU(1) and E_{7(-25)}/E_6xU(1). In order to construct them we use the projective superspace formalism which is an N=2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N=2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N=1 superfields, once the Kahler potentials of the base manifolds are obtained. We derive the N=1 supersymmetric nonlinear sigma models on the Kahler manifolds M. Then we extend them into the N=2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the N=2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces M. In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.


Introduction
Supersymmetry (SUSY) has been studied in particle physics for a long time as a promising candidate of the theory beyond the Standard Model. On the other hand, it has been revealed that SUSY has an intimate relation to complex geometry in mathematics. Indeed, it is well known that target spaces of N = 1 and N = 2 SUSY nonlinear sigma models (NLSMs) must be Kähler [1] and hyperkähler manifolds [2], respectively. Meanwhile many complex manifolds have been studied through construction of SUSY NLSMs. formulation in four-dimensional space-time 1 . In this formalism, N = 2 SUSY NLSMs on cotangent bundles over Kähelr manifolds have been constructed [9,10,11,12,13,14,15].
A key observation in the developments is that once a certain N = 1 SUSY NLSM is obtained, this model can be extended into the N = 2 SUSY NLSM with use of the projective superspace formalism. If we have the N = 1 SUSY NLSM on the Kähelr manifold, we can obtain N = 2 SUSY NLSM on the cotangent bundle over the Kähelr manifold. The target space of the N = 2 SUSY NLSM is shown to be an open domain of the zero section of the cotangent bundle [9,10]. Namely it is hyperkähler. There is also a similar development in mathematics: It is proved that for a Kähler manifold M a kähler structure on an neighborhood of the zero section of the cotangent bundle over M exists [17,18,19]. The proofs in [9,10] and [17,18,19] were perfomed independently. Based on the observation in [9,10], the N = 2 SUSY NLSMs on the cotangent bundles over the Hermitian symmetric spaces (HSSs) have been constructed [9,10,11,12,13,14,15]. The irreducible HSSs classified by Cartan [16] consist of compact type and non-compact type. They are summarized in Table 1.
Let us see more in detail about the recent developments of construction of N = 2 SUSY NLSM. The projective superspace consists of the standard N = 2 superspace and the so-called projective coordinate ζ parameterizing SU(2) R /U(1), where the SU(2) R is an internal symmetry of the N = 2 SUSY algebra. Superfields are defined on a subspace of the projective superspace. While there are several superfields on the subspace, a relevant one for constructing SUSY NLSMs is called the polar multiplet which represents hypermultiplet. Starting with an N = 1 SUSY NLSM on a Kähler manifold M in terms of chiral superfields, the N = 2 SUSY extension is obtained by replacing the chiral superfields with the polar multiplets and integrating over the subspace of the projective superspace [9,10]. The model obtained is written by fields representing the base manifold coordinates (chiral superfields) and tangent vectors (complex linear superfields) along with an infinite set of unconstrained auxiliary superfields. After eliminating the auxiliary fields the model represents the N = 2 SUSY model on the tangent bundle over M, which we write S tb . Exchanging the tangent vectors in S tb to the cotangent vectors with aid of the generalized Legendre transformation [5], we obtain the N = 2 SUSY NLSM on the cotangent bundle S ctb over M.
One of the main problems to obtain N = 2 SUSY NLSMs in the above method is to solve the equations of motion for the auxiliary fields. While various ways to solve them have been proposed in constructing the N = 2 SUSY NLSMs on cotangent bundle over classical type exceptional type compact type  [9,10,11,12,13,14,15], the useful one is to use the properties of the HSS and SUSY [13]. This is applicable to all the HSSs. The other problem is to perform the generalized Legendre transformation. It is easy to implement for all the compact and noncompact classical types of HSSs [9,10,12] and the compact exceptional type E 6 /SO(10)× U(1) [13]. However, it is difficult for the compact exceptional type E 7 /SO(10) × U(1). Accordingly another method has been developed [15] based on the result in [14]. In [14], the general form the cotangent bundle over the HSS is derived. Afterwards, a different form is obtained in [15] and it has applied to derivation of the N = 2 SUSY NLSM on the cotangent bundle over E 7 /SO(10) × U(1).
In this paper we construct the N = 2 SUSY NLSM on cotangent bundle over the noncompact exceptional HSS, M = E 6(−14) /SO(10)×U(1) 2 . We first derive the N = 1 SUSY NLSM on M. Second we use the general expression of the action for the tangent bundle obtained in [13] to obtain the action for the tangent bundle over M. After obtaining the tangent bundle action, with the use of the generalized Legendre transformation, we obtain the N = 2 SUSY NLSM on the cotangent bundle action over M.
The paper is organized as follows. In Section 2, we give a brief review of the N = 2 NLSMs in the projective superspace formalism. In Section 3, we explain the E 6(−14) algebra and then derive N = 1 SUSY NLSM on M. Methods to derive the models of the tangent bundle and the cotangent bundle are explained in Sections 4 and 5, respectively. In Section 4, the N = 2 SUSY model on the tangent bundle over M is obtained by using the projective superspace formalism. Applied the generalized Legendre transformation to the result in Section 4, the N = 2 SUSY NLSM on the cotangent bundle over the M is constructed in Section 5. Section 6 is devoted to conclusion. In Appendix A, we briefly summarize the Clifford algebra. In Appendix B, detailed calculation to obtain (3.22) from (3.21) is summarized.
2 N = 2 sigma models and the projective superspace 2.1 General case Projective superspace is described as (x µ , θ αi ,θ iα , ζ), where µ = 0, 1, 2, 3 is a space-time index, α,α = 1, 2 are spinor indices, i = 1, 2 is an SU(2) R index and ζ is the projective coordinate. Superfields are functions on its subspace, which are defined by the so-called projective condition [3] similar to the chiral condition in the four-dimensional N = 1 SUSY field theory. This condition makes a number of the Grassmann coordinates be half and integration measure for SUSY invariant action reduces to one on the full N = 1 superspace z M = (x µ , θ α ,θα) with the projective coordinate ζ. A certain class of fourdimensional N = 2 NLSM is described in terms of N = 1 language as [20,9,10] where I, J are indices of fields 3 . The contour encircles the origin of the ζ-plane in anticlockwise direction. The action is written by the function of the superfields representing the polar multiplets Υ andΥ, which are called an arctic superfield and an antarctic superfield respectively. They are expanded with respect to ζ as where Υ 0 ≡ Φ is a chiral superfield (DαΦ = 0) and Υ 1 = Σ is a complex linear superfield (D 2 Σ = 0), whereDα = − ∂ ∂θα − iθ α σ µ αα ∂ ∂x µ is the covariant derivative in the N = 1 superspace. An infinite set of unconstrained auxiliary fields is expressed as A which contains terms with an order higher than ζ 2 . The antarctic superfieldΥ is a conjugate of Υ, which is the combination of the ordinary complex conjugate and the antipodal map ζ → −1/ζ on the Riemann sphere.
The action (2.1) is an N = 2 extension of the general N = 1 SUSY NLSM [1] where K(Φ I ,ΦJ ) is the Kähler potential of a Kähler manifold. Indeed, expanding (2.1) with respect to ζ, we can see the Kähler potential in (2.3) is included [9]: and respectively. From the transformation (2.8), we find the transformation law for Σ I as It is seen that Σ I transforms as a tangent vector.
In order to represent the action (2.1) in terms of physical fields (Φ I , Σ J ) only, we need to eliminate the auxiliary fields by using their equations of motion For a general Kähler manifold, it is possible to eliminate Υ n (n ≥ 2) and their conjugates by solving (2.10) perturbatively [21]. After eliminating all the auxiliary fields, the following form of the action is obtained where L(Φ,Φ, Σ,Σ) is the part describing the tangent space: Here L IJ = −g IJ (Φ,Φ) while the tensors L I 1 ···InJ 1 ···Jn (n ≥ 2) are functions of the Riemann tensor R IJKL (Φ,Φ) and its covariant derivative. Each term of the action contains equal powers of Σ andΣ because the action (2.1) possesses the U(1) invariance [9] 14) The action (2.12) is written by the base manifold coordinate Φ and the tangent vector Σ. Therefore this action represents the N = 2 SUSY model on the tangent bundle over the Kähler manifold.
The rest of the work is to derive the Kähler potential of the cotangent bundle over the Kähler manifold. It is carried out by changing the tangent vectors Σ's in (2.12) into chiral one-forms, cotangent vectors Ψ's. It can be performed by the generalized Legendre transform [5] as follows.
where U is a complex unconstrained superfield and Ψ is a chiral superfield satisfyinḡ DαΨ = 0. This action goes back to the tangent bundle action (2.12) after eliminating the chiral superfields Ψ andΨ by their equations of motion. Indeed, the equations of motion for Ψ andΨ give the constraint D 2 U =D 2 U = 0 which is exactly the condition for a complex chiral superfield. On the other hand, eliminating U andŪ with the aid of their equations of motion, the action is written only in terms of Φ, Ψ and their conjugates: Here g IJ is the inverse metric of g IJ . The variables (Φ I , Ψ J ) parameterize the cotangent bundle over the Kähler manifold and therefore the action gives the Kähler potential of the cotangent bundle over the Kähler manifold.

HSS case
The explicit form of L is obtained for the case that the base manifold is a HSS [13].
Since the tangent space part (2.13) hides the second SUSY invariance after eliminating the auxiliary fields perturbatively, we first derive the condition so that the L is invariant under the second SUSY transformation. Here we take into account that the base manifold is the HSS. Second we solve the condition and obtain the explicit formula for L.
By construction in the projective superspace formalism, the action is invariant under the N = 2 SUSY transformation [5,6] δΥ where ε α i ,ε iα are transformation parameters and Q i α ,Qα i are supercharges. The action (2.12) is written in terms of N = 1 superfield and therefore N = 1 SUSY is only manifest. In the following we investigate the invariance of the action under the second SUSY transformation (2.18) for i = 2. It is shown that the second SUSY acts on Φ and Σ as [7] δΦ I =ε .
αD . α Σ I , The condition for a Riemann tensor R I 1J1 I 2J2 of the HSS where ∇ L and∇L are covariant derivatives with respect to Φ andΦ, and Γ I JK is the Christoffel symbol. We impose the tangent bundle action (2.12) to be invariant under (2.21), and then find [13] The equation (2.23) is rewritten as by using This yields the explicit form for L On the other hand, acting (2.21) on L leads to the following equation It is also the condition for invariance under the second SUSY transformation. Indeed The cotangent bundle action (2.16) has to be invariant under the following second SUSY transformations [13] The requirement of invariance under such transformations can be shown to be equivalent to the following nonlinear equation [13]: This equation also follows from (2.27) using the definition of the Ψ, or it is possible to obtain it by the generalized Legendre transformation. Detailed discussion is given in [13].
3 Kähler potential of E 6(−14) /SO(10) × U (1) In this section we derive the N = 1 SUSY NLSM on the non-compact exceptional HSS E 6(−14) /SO(10) × U(1). To do that, we need a transformation law of the field parameterizing the space and construct the model so that it is invariant under the transformation.

Kähler potential
Now we construct the N = 1 SUSY NLSM whose Kähler potential is of M = E 6(−14) /SO(10)× U(1). The fields in the NLSM can be interpreted as the Nambu-Goldstone boson when the symmetry E 6(−14) is broken down to SO(10) × U(1). The unbroken generators are E α and its complex conjugateĒ α . Corresponding representation is of the 16 complex Weyl spinor of SO (10). We write it with Greek index appeared in the algebra as The Kähler potential is constructed so that it is invariant under the transformation of Φ α . The transformation law for Φ α is read from the following commutation relations 5 : where satisfying the properties: The closure of the E 6(−14) algebra on Φ α can be checked by using the Jacobi identities involving two generators of E 6(−14) algebra and one Φ α . For instance, one can check that the following non-trivial identity is satisfied ] + (cyclic permutations) = 0, (3.14) by using (3.13). The commutators (3.7)-(3.10) lead to the infinitesimal transformation law for Φ α as where ǫ α andǭ β are transformation parameters.
where P + is the projection operator defined by (A.7). Similarly, we embed the transformation parameter ǫ α into the 32-component spinor ε a The generator σ AB in the Weyl representation forms the Dirac representation as where Γ A is a 32 × 32 gamma matrix defined by (A.14). With use of (3.16)-(3.18), (3.15) is rewritten by Defining the charge conjugation where C is given by (A.16), (3.19) turns out to be with the projection operator P − defined by (A.7). Here we have used (A.10). By using the Fierz identity we exchangeε a and φ a in (3.21) and obtain where Γ ± A ≡ Γ A P ± . The detailed calculation to obtain (3.22) from (3.21) is summarized in Appendix B.
Let us derive the Kähler potential invariant under the nonlinear transformation law (3.22). In order to do that, first we consider the SO(10) × U(1) invariants:

25)
In deriving (3.26) we have used the transformation law for the 10-dimensional representation of the SO(10) × U(1) from the product of spinors 16 × 16, φ c Γ + A φ. It is given by It can be checked by starting with Applying (A.1) to the last term in the right-hand side and using (B.7), we obtain (3.27).
Using the relations (3.25) and (3.26) one can check that the following function The right-hand side consists of the holomorphic and anti-holomorphic parts with respect to φ. It means that (3.29) is invariant under (3.19) up to the Kähler transformation (3.30). Therefore we conclude that (3.29) is the Kähler potential of the non-compact HSS E 6(−14) /SO(10) × U(1).
Finally for later use we shall go back to the Weyl representation. Using the representation of the gamma matrix (A.14), the Kähler potential is written by where C is the charge conjugation matrix in the Weyl representation defined by (A.16). The former sign is not determined by using the transformation law (3.22) but it is chosen to ensure positivity of the metric. The Kähler potential (3.31) is very similar to one of the compact HSS E 6 /SO(10) × U(1) given by [24,25,26] 6 . Only the differences are the sign in front of the logarithm and in the second term inside the logarithm.
4 Tangent bundle over E 6(−14) /SO(10) × U (1) In this section, we derive the tangent bundle action for the non-compact exceptional HSS E 6(−14) /SO(10) × U(1) by using the general formula (2.26). First we need to calculate the first-order differential operator (2.25) which is for the E 6(−14) /SO(10) × U(1) case: Since a symmetric space is homogeneous, it is sufficient to perform the calculations of our interest at the origin, Φ = 0. The metric and Riemann tensor at Φ = 0 are derived from the Kähler potential (3.31) as The operator (4.1) is then written by Here we have used the identity This follows from the Fierz identity (B.1). From the expression (4.5) we find the following form for the tangent space part One can extend this expression to one at an arbitrary point Φ of the base manifold by making the replacement Then we obtain the action of the tangent space part This is the correct result for L. Indeed, one can check that (4.9) satisfies the equation (2.27) which reads in the present case, Let us briefly prove that (4.9) satisfies this equation. It is again sufficient to consider at Φ = 0. In this case, the first term of the left-hand side (4.10) becomes where we have used (4.6). This exactly cancels against other terms in (4.10).
5 Cotangent bundle over E 6(−14) /SO(10) × U (1) Let us derive the N = 2 SUSY NLSM on the cotangent bundle over E 6(−14) /SO(10)×U(1) from (4.9). In order to do that, we consider the first order action (2.15), which in the present case is written by Let us eliminate U andŪ . As in the case of the tangent bundle, we again consider the action at Φ = 0 since the base manifold is homogeneous. Then, the action (5.1) is where ψ is a cotangent vector at Φ = 0 and with The equations of motion for U andŪ are These equations yieldψ From these equations we have 3) it is seen that the correspondence between the tangent and cotangent vectors should be such that U → 0 ⇔ ψ → 0. This means that we have to choose the following solution of (5.11).
From the above result we obtain Ω in terms of ψ and its conjugate. By definition of Ω (5.3), we have This is equivalent to Since we should have Ω → 1 when ψ → 0, it is necessary to choose the following solution for (5.14): The above result is of one at the origin Φ = 0 of the base manifold. In order to extend to one at an arbitrary point Φ = 0, we have to make the following replacement: Substituting (5.6), (5.7) and (5.16) into (5.1), we obtain the cotangent bundle action S ctb S ctb = d 8 zH(Φ,Φ, Ψ,Ψ), (5.18) where and Finally let us check that the cotangent bundle action (5.19) satisfies the equation (2.31). In the present case, the equation takes the following form: To prove this, we again set Φ = 0. Then, the left-hand side in (5.21) becomes where in the second equality we have used (5.7) to express ψ in terms of Σ. Taking (5.3) into account, we see that the expression obtained is exactlyΨ β at Φ = 0.

Conclusion
We have constructed the N = 2 SUSY NLSM on the cotangent bundle over the noncompact exceptional HSS M = E 6(−14) /SO(10) × U(1) by using the method elaborated in [13]. The point is to use the projective superspace formalism which is an N = 2 off-shell superfield formulation. Once an N = 1 SUSY NLSM on a certain Kähler manifold is obtained, it is possible to extend it to the N = 2 SUSY model containing the corresponding N = 1 SUSY NLSM. We first derived the transformation law of the field parameterizing M and constructed the N = 1 SUSY NLSM on M invariant under the derived transformation law. Second we extended the N = 1 SUSY NLSM to one with the N = 2 SUSY model by using the projective superspace formalism. The resultant model turns out to be the action of the tangent bundle S tb over M. This is written in terms of the chiral superfields (base manifold coordinates) and the complex linear superfields (tangent vectors). Finally we exchanged the tangent vectors in S tb to the cotangent vectors by using the generalized Legendre transformation and obtained the N = 2 SUSY NLSM on the cotangent bundle S ctb over M.
We comment on future directions. The N = 2 SUSY NLSM on the cotangent bundle over all the HSSs except the non-compact exceptional HSS E 7(−25) /E 6 × U(1) (see Table  1) have been completed constructing. The rest of the work is to derive the N = 2 SUSY NLSM on the cotangent bundle over E 7(−25) /E 6 × U(1). Since the Kähler potential for E 7(−25) /E 6 × U(1) has not been known yet, we first need to derive it. Once the Kähler potential is obtained, it is easy to obtain the N = 2 SUSY model over the tangent bundle over E 7(−25) /E 6 × U(1) as in the case studied in this paper. However, it would be difficult to perform the generalized Legendre transformation. We had this problem for the compact exceptional HSS E 7 /E 6 × U(1). To overcome the problem in construction of the N = 2 SUSY NLSM over the cotangent bundle over E 7 /E 6 × U(1), we needed to derive the general expression of the Kähler potential for the cotangent bundles over all the compact HSSs [15] based on the result in [14]. We are now constructing the general expression of the Kähler potential for the cotangent bundles over all the non-compact HSSs and are planning to apply it to the case for E 7(−25) /E 6 × U(1).
by which we define the projection operator There exists a charge conjugation matrix which relates Γ A to Γ A * . The latter forms an equivalent representation of the Clifford algebra: The matrix C has the following properties: For Γ 11 and P ± , we have Let us give representation of the gamma matrix. We take the Γ 11 to be diagonal: In this case, the gamma matrices are block off-diagonal. They are constructed by a tensor product of the gamma matrices of SO(6) and SO(4) (for instance, see [28]): , · · · , 10, α, β = 1, · · · , 16, (A.14) where the σ A 's are the gamma matrices on the Weyl spinor basis. The generators of the SO(10) group are defined by The charge conjugation matrix C takes the form where C is the 16 × 16 matrix given by  To derive (3.22) from (3.21), we use the Fierz identity (for instance, see [27]). In the Dirac representation of SO(10), the Fierz identity is described bȳ ψ a λ b = 1 32 5 n=0 a n n! (Γ A 1 ···An ) a b (λΓ A 1 ···Anψ ), a, b = 1, · · · 32, (B.1) where the indices A i (i = 1, · · · 5) run from 1 to 10 andψ, λ are the Dirac spinors of SO (10). In this Appendix, we express the summation over the indices A i by contraction with upper and lower indices. The coefficients a n are given by a 0 = a 1 = a 4 = 2, a 2 = a 3 = −2, a 5 = 1.