On $OSp(N|2)$ superconformal mechanics

Superparticle models with $OSp(N|2)$ supersymmetry group are studied. We first consider the $N=4$ case and construct the models with $\kappa$-symmetry on the coset spaces of the $OSp(4|2)$ supergroup. In addition, within the canonical formalism we present an $OSp(4|2)$ superparticle model with semi-dynamical angular variables. For generic $N$ we construct a superparticle model on $AdS_2\times S^{N-1}$ with the reduced $\kappa$-symmetry. It is demonstrated that the Hamiltonian of this model has the same structure as the one for the $N=4$ case because additional fermions contribute to the second-class constraints only.


Introduction
One-dimensional superconformal systems can be thought of as the limiting case of the higher-dimensional analogues which capture silent features of the latter in a simpler context. Typically one is concerned with one-dimensional (super)conformal mechanics because they provide dual theories for two-dimensional gravity [1,2,3,4,5]. On the other hand, they are of interest on their own. For instance, some d = 1 supermultiplets cannot be obtained by dimensional reduction from higher dimensions. d = 1, N = 4 superconformal algebras and their mechanical realizations attracted particular interest due to the proposal that they might be relevant for a microscopic description of extreme black holes [6,7]. This proposal inspired extensive studies [8]- [24], which proved useful for understanding the structure of interactions of N = 4 supermultiplets of su(1, 1|2), osp(4|2) and D(2, 1; α). Some of these models are described by the κ-symmetric actions [11,15,26].
It is natural to wonder what happens beyond the N = 4 case. The most powerful approach for constructing the superconformal mechanics is the superfield formalism. However, in practical applications it may turn out to be difficult to single out irreducible supermultiplets by imposing constraints on superfields [31]. Another way is the direct construction of the superconformal mechanics within the canonical formalism [16,23,24,32]. This method relies upon the structure relations of the superalgebra, and an going beyond N = 4 may turn out to be problematic [32] 1 . Superconformal mechanics can also be constructed by higher-dimensional reduction [27,28]. Finally, by using the method of nonlinear realizations one can directly build the action, whose form can be restricted by requiring the κ-symmetry [11,12,15,25,26,29,30]. In a recent paper [26], this method was applied for the construction of superpaticle models on the coset spaces of the SU(1, 1|N) supergroup. It was shown that the κ-symmetry for the general N-case is broken down to a one-parametric fermionic gauge symmetry. This work gives an explanation for the problem encountered in [32], in which the authors vainly attempted to construct an SU(1, 1|N) superparticle system with angular variables by a straightforward generalization of the SU(1, 1|2) case.
In the present paper, we extend the analysis of [26] to the osp(N|2) superalgebra. The isntance N = 4 is special. We show that it is the only nontrivial case when one is able to construct an action on the corresponding coset space with unbroken κ-symmetry. Remarkably enough, the coset of OSp(4|2) supergroup with the bosonic part AdS 2 × S 2 is also of particular interest from the superfield standpoint, as it represents the only superextension of AdS 2 ×S 2 which admits a superconformally flat supervielbein and superconnections [33]. For generic N the κ-symmetry is broken down to a one-parametric gauge symmetry in analogy with [26]. A remarkable fact, however, is that the Hamiltonian structure remains the same as in the N = 4 case, because the additional fermionic degrees of freedom show up in second class constraints only.
The organization of the work is the following. We remind the structure relations of the osp(N|2) superlagebra in the next section. In sect. 3 we investigate the N = 4 case and construct models on two different coset spaces reproducing some of the models in [24]. In addition, we construct an analogue of the model [21,33] with semi-dynamical angular degrees of freedom. Sect. 4 is devoted to the general N case, where we construct the action with the reduced κ-symmetry and analyze the model within the canonical formalism. We conclude and summarize in sect. 5. Appendices contain some technical details regarding the explicit form of the Maurer-Cartan one-forms.

Superalgebra
We start with the standard form of the osp(N|2) superalgebra: where condensed notation was introduces for the so(N) generators J jk → J a satisfying the algebra [J ij , J kl ] = δ jk J il + (3 terms).
All generators are real. The bosonic subalgebra is the direct sum so(1, 2) ⊕ so(N), where the d = 1 conformal algebra so(1, 2) is presented by {H, K, D}. Fermionic part of the superalgebra is generated by the real supersymmetry charge Q i and the superconformal partner S i with i =, 1 . . . , N. The antisymmetric matrices λ ij kl → λ a kl define N-dimensional representation of the rotation algebra where f abc are the so(N) structure constants (2). These matrices are supposed to satisfy a Fierz-like identity λ a ij λ a kl = δ ik δ jl − δ il δ jk .
For convenience we make a standard decomposition of the so(N) generators into so(N − 1) subalgebra and the set of operators, which will be used to generate the cosets J mn := M mn , J mN := P m , m, n = 1, . . . , N − 1.
In these notations commutation relations of so(N) read [M mn , P q ] = δ qm P n − δ qn P m , [P m , P n ] = M mn , We define the Maurer-Cartan (MC) one-forms for a coset elementG in the standard where summation over the rotation indices carried by fermions is assumed. In what follows, we will need the bosonic one-forms only. When analyzing the κ-symmetry properties of the actions, the special form of variations of these one-forms will be needed, which can be retrieved from the MC equations (see e.g. [39]) where we introduced the notation for a MC one-form L A = L A M dZ M . The κ-symmetry is characterized by the vanishing of the bosonic variations on the coset with respect to [39] (see also [25]) 3 Special case: N = 4 Before turning to generic N, we discuss the instance of N = 4. In this section we reproduce the known models of OSp(4|2) superconformal mechanics and construct some novel ones.

Internal degrees of freedom
Technically the issue of finding a κ-symmetric action amounts to using the properties of the matrices λ m corresponding to the coset generators P m , which depend on the way the subalgebra SO(N − 1) was extracted from SO(N). In this sense the case of N = 4 is subtle, because the algebra so(4) is the direct sum of two copies of so (3) [P m ± , P n ± ] = δ ±,± ǫ mnq P q ± .
As the superalgebra osp(4|2) involves two copies of so(3) in a symmetric manner, the coset space can be generated by any of these. In accordance with the structure relations of so(4), we introduce a set of matrices which satisfy the (anti)commutation relations and the Fierz-like identities Here σ i are the Pauli matrices.

Angular variables parametrizing S 3
Now we turn to the construction of the coset on which an invariant action can be build. The building blocks for the action are the MC one-forms and their invariant quadratic combinations. First of all, let us note that the stability subgroup is allowed to be generated by the bosonic operators only. It proves to be impossible to construct quadratic combinations of the MC one-forms invariant under the left action of the group otherwise. In addition, the κ-symmetry also imposes strong constraints on the structure of the action. One may see that the coset of a maximal dimension which allows one to construct a κ-invariant supersymmetric action is OSp(4|2)/SO (3). The bosonic part of this space is AdS 3 × S 3 . As it has a larger supersymmetry group, we refrain from considering it in this work (see e.g. [38]). The next case is with the bosonic part AdS 2 × S 3 . We construct the action on this coset space where the MC one-forms L m + correspond to the generators P m + and a, m are constant parameters. In the pure bosonic background the first term under the root represents a metric on AdS 2 , while the second term corresponds to S 3 . As the stability subgroup is generated by the bosonic operators only, the invariants built in terms of the MC one-forms in the pure bosonic case, are invariants of the whole group as well. Linear term proportional to L D is transformed as an Abelian connection (see e.g. [42]) and can be thought of as the Wess-Zumino term [25]. Taking into account these observations, one concludes that the action (16) is invariant under the Osp(4|2) supergroup.
To see that the action is also invariant under the κ-symmetry, we use the standard method [39] which relies upon a technically convenient representation for variations of the MC one-forms. For the case at hand they are given in (8) and should be supplemented by (10). The requirement that the action is invariant under the κ-symmetry transformation leads to the set of linear algebraic equations Let us now construct the action explicitly. The gauge-fixed MC one-forms on the coset space (15) are given in Appendix A. Making the change of the coordinates one brings the action (16) to the form To establish a relation between our models and those constructed previously, we turn to the canonical formalism. Introducing the momenta (p z , p θ , p φ , p χ ), one finds the Hamiltonian where are the bosonic parts of the rotation generators P m + = J m + + 1 2 (ψλ m + ψ). Fermionic momenta give rise to the second class constraints To put the Hamiltonian into the standard form, one makes the canonical transformation which leads to As was shown above, one copy of so (3) is realized on the bosonic and fermionic variables by P m + , while the second one acts upon the fermionic variables only P m − = 1 2 (ψλ m − ψ). This model is invariant under the additional symmetry transformations, originating from the right action of SO(3) on itself Summarizing the discussion above, the model is invariant under OSp(4|2) × SO(3)transformations. At the Hamiltonian level, it is easy to see that the functions J m + can be arbitrary generators of the rotation algebra [23,24]. In particular, we may set p χ to be a constant, thus breaking the symmetry group down to OSp(4|2) supergroup. In this case the model describes an OSp(4|2) supersymmetric extension of the particle propagating in the near horizon region of an electrically and magnetically charged black hole. Taking into account the isomorphism osp(4|2) ≃ D(2, 1; − 1 2 ), one may verify that the Hamiltonian (25) exactly reproduces D(2, 1; α) superconformal mechanics studied in [24] for α = −1/2.

S 2 and magnetic monopole
The next coset we consider is We introduce a superparticle action on this coset by where the summation over m = 1, 2 is assumed. The bosonic part of this action describes a particle on the AdS 2 × S 2 spacetime with the two-from flux and the corresponding background can be thought of as the near horizon region of a nonrotating black hole, where the parameters a and b are proportional to electric and magnetic charges of the black hole, respectively. It represents an OSp(4|2) counterpart of the SU(1|2) superparticle of [11,12]. Due to the isomorphism osp(4|2) ≃ D(2, 1; − 1 2 ), the action (28) describes exactly the κ-symmetric model constructed in [26] on the coset space of the D(2, 1; α) group for α = −1/2.
As S 3 is a bundle over S 2 with U(1) fiber, the action (28) can be obtained from (16) by the dimensional reduction. At the classical level the only effect of this reduction is the fixation of the momentum conjugate to the coordinate parametrizing U(1), thus producing the model of the previous section with constant p χ . However, the requirement that the action (28) is invariant under the κ-symmetry transformations yileds the constraint Note that this expression explicitly involves the magnetic charge b, which is the momentum p χ fixed by the dimensional reduction. Hence, it differs from the model in the previous section with constant p χ and (28).

Constrained angular sector
This section we consider the model with constrained angular sector, which is governed by the action where n = 1, 2 for the coset (27) and n = 1, 2, 3 for (15). One can see that this action is invariant under the κ-symmetry transformations provided the parameters a, b, m are subject to the constraint (29). In addition to the κ-symmetry this action enjoys the reparametrization invariance. Using explicit form of the MC one-forms and imposing the static gauge as in the previous section, one finds the Hamiltonian and the first-class constraint where J m + are given in (22) for the coset (15), while for the coset (27) are the same but with p χ = 0. Implementing the canonical transformation (24) results in the Hamiltonian (25) on the constraint surface (36). One recognizes in this model an OSp(4|2) analogue of the SU(1, 1|2) superparticle constructed in [15]. Note also that this model could be constructed starting from (25) and using a projection on the constraint surface (36). However, there is a subtlety at the quantum level because b 2 should be quantized. In contrast to the model obtained by the projection of (25), the requirement for the original action (30) to enjoy the κ-symmetry leads to the relation (29), thus providing quantization of the mass parameter m.

Semi-dynamical angular variables
Although the model in the previous section has the same symmetry group and the same number of physical degrees of freedom as the OSp(4|2)-model with spin variables in [33] (see also [21,17,37]), its physical content is different. In this section we pay special attention to the semi-dynamical angular variables.
Let us introduce a set of real variables α i , β i , i = 1, . . . , 4 which obey the bracket One can use the variables in order to construct the so(4)-generators Using the Fierz-like identities (14) for the matrices λ m + , we can find the Casimir element J m To construct an analogue of the superparticle [33], it seems natural to restrict the phase space variables to the surface provided that they commute with the generators J m ± . These are the first class constraints. They eliminate (2 + 2) degrees of freedom leaving four physical degrees of freedom. Next, we can define the supercharges which give rise to the Hamiltonian on the constraint surface At the Lagrangian level the bracket (33) and the constraints (36) can be presented by the term where A and B are the Lagrange multipliers. The kinetic term is of the first order giving rise to the Dirac bracket (33). Clearly, the model can be considered without a projection on the constraint surface, thus having eight phase space degrees of freedom on which the algebra so(4) is realized. Finally, it should be noted that the model at hand can be straightforwardly generalized to accommodate the D(2, 1; α) symmetry group [24].

The general case
We now turn to the general case of osp(N|2) superalgebras and their superparticle realizations with the κ-symmetry. As will be demonstrated below, the κ-symmetry is broken down to the one-parametric fermionic gauge symmetry similar to the SU(1, 1|N) superparticle [26]. We construct our model on the coset whose bosonic part is AdS 2 ×S N −1 . In what follows we consider the action (16) but now with n = 1, . . . , N − 1. The issue of the κ-symmetry can be addressed within the approach used in the previous sections. The algebraic equations which come from the requirement that the action is invariant under the κ-transformations coincide with (17) but again n = 1, . . . , N −1.
These equations can be put in the form provided the relation (18) holds, where the curly brackets denote the matrix anticommutator.
To compute it explicitly, it is convenient to define the matrices λ jk in bra-ket notations Now, it is easy to find the anticommutator encountered above which implies the one-parametric solution to the equations (41) where κ is an arbitrary Grassmann-valued parameter. Note that the equations (41) hold for any [δη] only in trivial case N = 2. As it was explained in Sect. 2, the case N = 4 is special because the algebra so(4) is the direct sum of two copies of so(3) and can be presented by the matrices (12) with the properties (13) and (14). In the general case, the only decomposition of the so(N) algebra on so(N − 1) and the set of operators generating coset is given by (5), which leads to the anticommutation property (43) and to the result (44).
Let us now construct the action explicitly. Taking into account (B.2), the action can be written in the form where all the differentials in L H and L m are replaced by the velocities with respect to t. We may fix the gauge by setting the N'th component of η to be vanishing, but it does not lead to simplification of the action. Note that the fermionic contribute quartic terms, which is the general feature of the superparticle and superstring actions constructed with the use of the superalgebra in the standard superconformal basis [41].
Let us analyze the model in the canonical formalism. A lengthy but straightforward computation gives the Hamiltonian where we have used the fact that e µ m O am is a set of Killing vectors (see e.g. [40]) and thus p µ e µ m O am := J a is a set of phase space generators of so(N) algebra. Fermionic momenta give rise to the constraints We put the Hamiltonian and the constraints into the standard conformal mechanics form by implementing the canonical transformation (24) Note that the Hamiltonian has the same structure as in the N = 4 case (25) 2 , but now we have additional fermionic variables and the constraint (49). The Dirac bracket constructed with respect to the fermionic constraints has a very complicated form.

Conclusion
To summarize, in this work we have constructed superconformal mechanics with the OSp(N|2) supersymmetry group. First, we considered the instance of N = 4. Two different coset spaces were considered, the corresponding bosonic parts being AdS 2 × S 2 and AdS 2 × S 3 , on which the κ-symmetric actions have been constructed. Some of these models were shown to reproduce those previously known. In particular, the gauge-fixed version of the model (16) on AdS 2 × S 3 is linked by a suitable canonical transformation to the work in [24], while the model (28) is exactly the κ-symmetric superparticle of [26] in view of the isomorphism osp(4|2) ≃ D(2, 1; − 1 2 ). For arbitrary N we have shown that the κ-symmetry is broken down to a one-parametric fermionic gauge symmetry, in agreement with a recent analysis of the su(1, 1|N) superparticle in [25]. Interestingly enough, the Hamiltonian for this model has exactly the same structure as the one for the N = 4 case. The additional fermionic variables, which are not gauged away due to the breakdown of the κ-symmetry, give rise to second-class constraints and modify the canonical bracket. We have also constructed an analogue of the model with semi-dynamical angular variables [33,21].
Turning to possible future developments, it is of interest to generalize the classical treatment and consider our models at the quantum level. It is worth studying whether one can construct superparicle model with semi-dynamical variables in the superfield formalism. Another open problem is to deform this model in the spirit of [35] and to consider its quantum version [36].

Acknowledgements
We thank A. Galajinsky for posing the problem and reading the manuscript. This work was supported by the Tomsk Polytechnic University competitiveness enhancement program.
A MC one-forms for the case N = 4 As is known, the κ-symmetry reduces the number of fermionic degrees of freedom by half. For the case at hand, this is achieved by imposing the gauge fixing condition, setting the fermions corresponding to the generator S to be vanishing. The gauge-fixed coset element readsG = e tH e zK e ψQ u, u = e P 1 + φ e P 2 + (θ−π/2) e P 3 + χ .