New Model of N=8 Superconformal Mechanics

Using an N=4, d=1 superfield approach, we construct an N=8 supersymmetric action of the self-interacting off-shell N=8 multiplet {\bf (1, 8, 7)}. This action is found to be invariant under the exceptional N=8, d=1 superconformal group F(4) with the R-symmetry subgroup SO(7). The general N=8 supersymmetric {\bf (1, 8, 7)} action is a sum of the superconformal action and the previously known free bilinear action. We show that the general action is also superconformal, but with respect to redefined superfield transformation laws. The scalar potential can be generated by two Fayet-Iliopoulos N=4 superfield terms which preserve N=8 supersymmetry but break the superconformal and SO(7) symmetries.

The N =8 superconformal actions of the off-shell multiplets (5,8,3) and (3,8,5) were explicitly given in [14] in terms of the properly constrained N =4, d=1 superfields. It was found that in both cases the underlying superconformal symmetry is OSp(4 ⋆ |4) . It is interesting to construct superconformal models associated with other N =8, d=1 superconformal groups. An example of such a system is presented here. It is the N =8 supersymmetric mechanics model associated with the off-shell multiplet (1,8,7) from the list of [17]. The underlying N =8 superconformal symmetry is the exceptional supergroup F (4) with the R-symmetry subgroup SO (7). In the N =4 superfield approach which we use throughout the paper, the "manifest" superconformal group is D(2, 1; −1/3) ⊂ F (4) .
In terms of N =4 superfields, the multiplet in question amounts to a sum While for the multiplet (1, 4, 3) one can write manifestly N =4 supersymmetric actions in the ordinary N =4 superspace [3,18], actions of the fermionic N =4 multiplet (0, 4, 4) are naturally written in the analytic harmonic N =4 superspace [19,14,17]. In order to construct the N =8 supersymmetric actions of the multiplet (1,8,7) we use the N =4, d=1 harmonic superspace description for both N =4 multiplets in (1.1). A sum of the free (1,8,7) action [17] and the superconformal action constructed here yields the most general (1,8,7) action. We present the relevant component off-shell action and show that it agrees with that found in [20] by a different method. Surprisingly, the general action is also superconformal, though with respect to redefined superfield transformation laws. For both N =4 multiplets one can construct N =8 supersymmetric Fayet-Iliopoulos terms which, however, break superconformal symmetry.
2 Preliminaries: N =4, d=1 harmonic superspace The harmonic analytic N =4 superspace [21,19,22,23] is parametrized by the coordinates They are related to the standard N =4 superspace (central basis) coordinates z = (t, θ i ,θ i ) as The N =4 covariant spinor derivatives and their harmonic projections are defined by

4)
In the analytic basis, the derivatives D + andD + are short, The analyticity-preserving harmonic derivative D ++ and its conjugate D −− are given by and become the pure partial derivatives ∂ ±± in the central basis. They satisfy the relations where D 0 is the operator counting external harmonic U(1) charges. The integration measures in the full harmonic superspace (HSS) and its analytic subspace are defined as (2.9) 3 The multiplets (1,4,3) and (0, 4, 4)

(1, 4, 3)
The off-shell multiplet (1, 4, 3) is described by a real N =4 superfield v(z) obeying the con- The same constraints in HSS read [17] The extra harmonic constraint guarantees the harmonic independence of v in the central basis.
Recently, it was shown [23] that this multiplet can be also described in terms of the real analytic gauge superfield V(ζ, u) subjected to the abelian gauge transformation In the Wess-Zumino gauge just the irreducible (1, 4, 3) content remains The constraints (3.1) are recovered as a consequence of the harmonic analyticity of V We shall need a "bridge" representation of V through the superfields v(z) and V −− (z, u) The term v(z) is just given by the expression (3.5). The analyticity conditions (3.6) imply Below are some useful corollaries of (3.8), (3.6) and (2.5) The general invariant action of the multiplet (1, 4, 3) reads The free action corresponds to the quadratic Lagrangian where, for the correct d=1 field theory interpretation, one must assume that v develops a nonzero background value, v = 1 + . . . . The transformation properties of some relevant objects under the conformal N =4 supersymmetry ⊂ D(2, 1; α) are as follows [19,23] and ε ± = ε i u ± i ,ε ± =ε i u ± i , ε i ,ε i being mutually conjugated Grassmann transformation parameters. Using (3.16), (3.17) and (3.18), (3.19), it is easy to check the D(2, 1; α) invariance of the action (3.14) and the covariance of the relation (3.5).
One can also construct an N =4 supersymmetric Fayet-Iliopoulos (FI) term which produces a scalar potential after elimination of the auxiliary field A (ik) in the sum of (3.12) and (3.20). This term is superconformal only for the special choice α=0 [23].

N =8 supersymmetry
As shown in [17], one can define the hidden N =4 supersymmetry 3 It commutes with the explicit N =4 supersymmetry and so forms, together with the latter, N =8, d=1 Poincaré supersymmetry. It is easy to check the compatibility of (4.1) with the constraints (3.1), (3.24). The same transformations, being rewritten in HSS, read where η ±A = η iA u ± i . The appropriate transformation of the analytic prepotential V is As expected, (4.3) closes on the time derivative of V only modulo a gauge transformation: where we used the identities (3.9)-(3.11) and anticommutation relations (2.5). Now we wish to construct the most general action of the multiplets (1, 4, 3) and (0, 4, 4) which would enjoy the hidden supersymmetry (4.1), (4.2) and so present the N =4 superfield form of the general N =8 supersymmetric action of the multiplet (1, 8, 7) .
A convenient starting point of such a construction is offered by the Ψ-actions (3.25) and (3.26) in view of their uniqueness. The N =8 completion of the free action (3.25) was found in [17]. The variation of (3.25) under (4.2) can be written as where we used the relation (2.9), constraint (3.21) and the harmonic independence of v in the central basis. This variation is cancelled by that of the free v action (3.13), so the action is N =8 supersymmetric. It breaks superconformal symmetry, since its first term is invariant under D(2, 1; α= − 1/2) (see (3.14)), while the second one is invariant under D(2, 1; α=0) . Now let us promote the interaction action (3.26) to an N =8 invariant. To calculate the variation δ η S (ψ) sc , we firstly note that it is fully specified by the variation δ η Ψ +A , since δ η V does not contribute because of the nilpotency property (Ψ +A ) 3 = 0 . Then, using (3.9), we represent restore the full superspace integration measure in δ η S (ψ) sc and rewrite this variation as where the bridge representation (3.7) for V was used. Taking into account the harmonic constraint (3.21) and the properties D ++ η −A = η +A and D ++ η +A = 0 , we observe that all terms in (4.8) except for the first one are reduced to a total harmonic derivative, whence This is cancelled out by the variation of −1/3 dtd 4 θ v 3 , so the second N =8 supersymmetric action is given by Since the first term in (4.10) is D(2, 1; α) invariant at any α, while the second one is invariant under D(2, 1; α=−1/3) (see eq. (3.14)), we conclude that the full action (4.10) is invariant under the N =4 superconformal group D(2, 1; α= − 1/3) . Since it is also invariant under the rigid N =8, d=1 supersymmetry, it is invariant under some N =8 superconformal group. Therefore the action (4.10), provided that V and v start with a constant, V = 1 +Ṽ , v = 1 +ṽ , defines a new model of N =8 superconformal mechanics associated with the N =8 multiplet (1, 8, 7) . It is easy to recognize which N =8, d=1 superconformal group we are facing in the present case. As follows from [16], the only such supergroup in which one can embed D(2, 1; α= − 1/3) (to be more exact, an equivalent supergroup D(2, 1; β) , β= − 1+α α =2) is the exceptional N =8, d=1 superconformal group F (4) , with the R-symmetry subgroup SO (7).
. The transformations of the hidden N =4 supersymmetry remain unaltered. There is no way to make superconformal the free action (4.6), while (4.11) is N =8 superconformal at any γ ′ = 0 .
One can also check that each of the two FI terms, (3.20) and (3.27), is invariant under N =8 supersymmetry (4.2), (4.3). However, they both break superconformal symmetry and SO (7) . The same symmetry properties are exhibited by the on-shell potential terms arising upon eliminating the auxiliary fields A ik , a m andā m from the sum of (4.11) and (3.20), (3.27).

Component actions
Using the explicit component expansions (3.4) and (3.21), it is straightforward to find the component form of the superfield actions (3.13) and (3.14) To interpret (5.13) as the superconformal action, one needs to make the shift x = 1 +x . We observe that the general off-shell action (4.11) is specified by the linear function f (x) = γ +γ ′ x , and its derivative f x = γ ′ , in agreement with the result of [20] where the component off-shell action of the multiplet (1,8,7) was constructed by a different method. Here we reproduced this result in a manifestly N =4 supersymmetric off-shell superfield formalism. We took advantage of the latter to show the F (4) superconformal invariance of the action (5.13), as well as of a sum of (5.12) and (5.13) (with respect to modified F (4) transformations). Also, we showed that the FI terms (3.20), (3.27) preserve N =8 supersymmetry (although they break F (4) ).
The component expressions for (3.20), (3.27) can be easily found After eliminating the auxiliary fields A ik , a A ,ā A in the sum S (N =8) F I by their algebraic equations of motion, there appears a scalar potential ∼ (γ + γ ′ x) −1 (plus some accompanying fermionic terms) which is not conformal. Perhaps, conformal potentials could be generated by coupling the multiplet (1,8,7) to some additional N =8 multiplets.
For completeness, we present the component form of the transformations (4.1) It is straightforward to check the invariance of (5.12) and (5.13) under these transformations. Let us also give the R-symmetry transformations belonging to the coset SO(7)/[SU(2)] 3 , where [SU(2)] 3 is the product of three SU(2) symmetries: SU(2) P G acting on the indices A and commuting with the manifest N =4 supersymmetry, manifest R-symmetry SU(2) R acting on the doublet indices i, k, . . . , and one more hidden SU(2) R ′ which rotates θ i throughθ i , D i throughD i , ω i throughω i and a A throughā A . These transformations read Here λ (ik)B ,λ (ik)B are 6 complex parameters, which, together with 9 real parameters of [SU(2)] 3 , are the 21 real parameters of the group SO(7) . The bosonic field x is an SO(7) singlet.

Conclusions
In this article, using the manifestly N =4 supersymmetric language of the N =4, d=1 harmonic superspace, we constructed a new N =8 superconformal model associated with the off-shell multiplet (1,8,7) and showed that the corresponding N =8 superconformal group is the exceptional supergroup F (4) . We also found that the generic sigma-model type off-shell action of this multiplet is given by a sum of the superconformal action which is trilinear in the involved N =4 superfields and the free bilinear action. The generic action is also superconformal, but with respect to modified F (4) transformations. The component action is in agreement with the one derived in [20]. The N =8 supersymmetric potential terms can be generated by two superfield FI terms which break both superconformal and SO (7) symmetries. An interesting problem for further study is to see whether superconformal potentials could be generated by coupling this model to some other known N =8 multiplets. Also it would be of interest to find out possible implications of this new superconformal model in the brane, black holes and AdS 2 /CFT 1 domains, e.g. along the lines of refs.