Couplings in D(2,1;a) superconformal mechanics from the SU(2) perspective

Dynamical realizations of the most general N=4 superconformal group in one dimension D(2,1;a) are reconsidered from the perspective of the R-symmetry subgroup SU(2). It is shown that any realization of the R-symmetry subalgebra in some phase space can be extended to a representation of the Lie superalgebra corresponding to D(2,1;a). Novel couplings of arbitrary number of supermultiplets of the type (1,4,3) and (0,4,4) with a single supermultiplet of either the type (3,4,1), or (4,4,0) are constructed. D(2,1;a) superconformal mechanics describing superparticles propagating near the horizon of the extreme Reissner-Nordstrom-AdS-dS black hole in four and five dimensions is considered. The parameter a is linked to the cosmological constant.


Introduction
The exceptional supergroup D(2, 1; α) describes the most general N = 4 supersymmetric extension of the conformal group in one dimension SO (2,1). It is parametrized by one real number α. As far as realizations in superspace are concerned, the generators of the corresponding Lie superalgebra are associated with time translations, dilatations, special conformal transformations, supersymmetry transformations and their superconformal partners, as well as with two variants of su(2)-transformations. One su (2) is interpreted as the R-symmetry subalgebra, while the other affects only fermions. Recent interest in D(2, 1; α) and specifically in SU(1, 1|2) which arises at α = −1 was motivated by a possible link to a microscopic description of the near horizon extreme Reissner-Nordström black hole and the desire to better understand peculiar features of extended supersymmetry in d = 1 which are absent in higher dimensions 1 . It is curious that none of the D(2, 1; α) superconformal mechanics models considered thus far assigned any physical meaning to the parameter α (for geometric interpretations see [3]).
A related line of research is the construction of superconformal particles propagating on near horizon extreme black hole backgrounds. Such systems can be linked to the conventional superconformal mechanics by applying a proper coordinate transformation [4,5]. It is believed that they will help to establish a precise relation between supergravity Killing spinors and supersymmetry charges of superparticles on curved backgrounds.
Conventional means of building superconformal mechanics include the superfield approach, the method of nonlinear realizations, and the direct construction of a representation of the desired superconformal algebra within the Hamiltonian framework. While the superfield technique is definitely more powerful, the Hamiltonian approach is more efficient in analyzing the dynamical content and the structure of interactions because non-dynamical auxiliary fields are absent.
The goal of this work is to extend the Hamiltonian analysis in [1] to the case of the exceptional supergroup D(2, 1; α). In doing so, we recover the results in a recent work [2] and further extend them by constructing a D(2, 1; α)-invariant model which describes coupling of arbitrary number of supermultiplets of the type (1,4,3) and (0, 4, 4) to a single supermultiplet of either the type (3, 4, 1), or (4,4,0). We also discuss D(2, 1; α) superconformal mechanics in the so called AdS basis [4] and connect the systems based on (3, 4, 1)-, and (4, 4, 0)supermultiplets to superparticles propagating near the horizon of the extreme Reissner-Nordström-AdS-dS black hole in four and five dimensions. In that context, the parameter α is linked to the cosmological constant and thus, for the first time in the literature, it is given a clear physical interpretation.
The work is organized as follows. In the next section it is argued that any representation of the R-symmetry subalgebra su(2) in terms of phase space functions can be automatically extended to a representation of the Lie superalgebra associated with D(2, 1; α). In Sect. 3, based on the earlier work [6], we construct a novel coupling of an arbitrary number of supermultiplets of the type (1,4,3) and (0, 4, 4) to a single supermultiplet of either the type (3, 4, 1), or (4, 4, 0). D(2, 1; α)-superparticles on black hole backgrounds are considered in Sect. 4. Models associated with the near horizon geometry of the extreme Reissner-Nordström-AdS-dS black hole in four and five dimensions are linked to D(2, 1; α) superconformal mechanics based on supermultiplets of the type (3, 4, 1) and (4, 4, 0), respectively. The parameter α is linked to the cosmological constant. Our spinor conventions are gathered in Appendix. Throughout the paper summation over repeated indices is understood.

Extending
Consider a representation of su(2) in terms of functions on some phase space where a = 1, 2, 3 and ǫ abc is the totally antisymmetric symbol with ǫ 123 = 1. In what follows three realizations we will of interest. The first is given by the angular momentum of a free particle moving on two-dimensional sphere one-forms with (Θ, p Θ ), (Φ, p Φ ) and (Ψ, p Ψ ) forming the canonical pairs. Note that (3) follows from (2) by introducing the coupling to the external vector field potential p a → p a + A a (Θ, Φ) and imposing the structure relations of su(2), while (4) results from (3) by implementing the oxidation q → p Ψ with respect to the constant q. Focusing on the Casimir element J a J a , it is important to stress that all the su(2)-realizations exhibited above are characterized by a non-degenerate metric which accompanies terms quadratic in momenta. Direct sums of J a in (2), (3), (4) yield degenerate metrics which prove to be unsuitable for the applications to follow.
Each realization of su(2) in a phase space can be extended to a representation of the Lie superalgebra corresponding to D(2, 1; α). It suffices to introduce an extra bosonic canonical pair (x, p) along with a fermionic SU(2)-spinor ψ α , α = 1, 2, and its complex conjugate (ψ α ) * =ψ α , and impose the brackets 2 Then it is straightforward to verify that the functions where σ a are the Pauli matrices (for our spinor conventions see Appendix), do obey the 2 Within the Hamiltonian formalism the canonical bracket {ψ α ,ψ β } = −iδ α β is conventionally understood as the Dirac bracket {A, Here (p ψ α , pψ α ) stand for the momenta canonically conjugate to the variables (ψ α ,ψ α ), respectively. Choosing the right derivative for the fermionic degrees of freedom, the action functional, which reproduces the Dirac bracket for the fermionic pair, reads S = dt i 2ψ αψ α − i 2ψ α ψ α . Similar consideration applies to the fermionic pair (χ α ,χ α ) which appears in Sect. 3. structure relations of the Lie superalgebra corresponding to D(2, 1; α) When verifying the structure relations (7), the properties of the Pauli matrices and the spinor identities gathered in Appendix were extensively used. As far as dynamical realizations are concerned, H is interpreted as the Hamiltonian. D and K are treated as the generators of dilatations and special conformal transformations. Q α are the supersymmetry generators and S α are their superconformal partners. J a generate the R-symmetry subalgebra su (2). So do also I ± , I 3 for which the Cartan basis is chosen. The extra su (2), which is realized on the fermions, makes the main difference with the su(1, 1|2) superconformal algebra, which arises at α = −1.
It should be mentioned that a representation similar to (6) was first considered in [3]. Yet, the functions J a were assigned quite a different meaning. In [3] they involved non-dynamical harmonic variables which represented spin degrees of freedom. In this work, we suggest to realize J a in terms of the fully fledged dynamical variables as displayed in Eqs. (2), (3), (4) above. Then Eqs. (6) provide a Hamiltonian description of D(2, 1; α)-supermultiples of the type (3, 4, 1) (two on-shell versions) or (4,4,0). Worth mentioning is also the work in [7] where it was demonstrated that the angular part of a generic conformal mechanics can be lifted to a D(2, 1; α)-invariant system. D(2, 1; α) supermultiplets from the su(2) perspective
Consider a set of canonical pairs which involves bosons (x i , p i ) and fermions (ψ i α ,ψ iα ), (χ A α ,χ Aα ), with i = 1, . . . , M + 1, A = 1, . . . , N, α = 1, 2, obeying the brackets Guided by our previous study of the su(1, 1|2) superconformal mechanics [8], on such a phase space we introduce ansatze for the D(2, 1; α)-generators which involve two scalar prepotentials . It is assumed that J a is one of the su(2) realizations exposed in Eqs. (2), (3), (4) above. The structure relations (7) impose the following constraints on the prepotentials: When verifying (7), the spinor algebra and the properties of the Pauli matrices given in Appendix were extensively used. Note that for a nonzero value of the parameter α the system (12) does not allow the prepotential V to vanish. The restrictions (12) have been obtained under the assumption that the su(2) generators J a in Eqs. (9), (11) are nontrivial. The choice J a = 0, which is also compatible with (1), would have altered the leftmost equation entering the first line in (12) [6,8]. Inspired by the earlier work [3], a representation similar to (11) has been constructed in [6] 3 . In particular, a plenty of interesting solutions to the master equations (12) have been found, which relied upon the root systems and their deformations. Yet, like in [3], the functionsJ a were realized in terms of non-dynamical spin degrees of freedom and the possibility to include into the consideration N copies of (0, 4, 4)-supermultiplet described by (χ A α ,χ Aα ) remained unnoticed.
Completing this section, we exhibit the on-shell Lagrangian formulations associated with the Hamiltonian description (11) and assume α = 0 which excludes V = 0. Given J a in (3), let us introduce the 3-vector λ a parameterizing a point on the unit sphere λ a = (cos Φ sin Θ, sin Φ sin Θ, cos Θ), λ a λ a = 1, which has the components Then the on-shell Lagrangian which describes a coupling of a single supermultiplet of the type (3,4,1) to and arbitrary number of (1, 4, 3)-, and (0, 4, 4)-supermultiplets reads In obtaining (16), one has to rewrite the phase space functions (3) within the Lagrangian framework where . Note that the kinetic terms for the fermions correlate with the form of the canonical (Dirac) bracket chosen above (see the footnote on page 3). Alternatively, the system (16) can be viewed as describing an interaction of M +1 copies of (3, 4, 1)-supermultiplet, in which angular degrees of freedom are identified, with N supermultiplets of the type (0, 4, 4).
The Lagrangian system based on the realization of su(2) in (4) is constructed likewise.
which has the components and implementing the inverse Legendre transformation to the Hamiltonian in (11), one gets When shuffling between the Lagrangian and Hamiltonian formulations, it proves helpful to use the identity which relates J a in (4) and L a in (19). A possible alternative interpretation of the action (20) is that it describes a coupling of M + 1 copies of (4, 4, 0)-supermultiplet, in which angular degrees of freedom are identified, to N supermultiplets of the type (0, 4, 4).

D(2, 1; α) superparticles on near horizon black hole backgrounds
By applying an appropriate canonical transformation, (3, 4, 1)-supermultiplet of the supergroup SU(1, 1|2) can be linked to the model of a massive relativistic superparticle propagating near the horizon of the extreme Reissner-Nordström black hole carrying the electric charge [4,5] or both the electric and magnetic charges [9]. Likewise, the near horizon geometry of the d = 5, N = 2 supergravity interacting with one vector multiplet turns out to be a proper background in the case of (4, 4, 0)-supermultiplet of SU(1, 1|2) [1]. The two coordinate systems are referred to as the conformal and AdS bases [4]. In this section, we generalize the previous studies in [1,5,9] to the case of the exceptional superconformal group D(2, 1; α) and link the parameter α to the cosmological constant.
Consider the canonical transformation where M and b are real constants, α is the parameter entering D(2, 1; α), and the prime designates coordinates in the conformal basis. Being rewritten in the AdS basis, the phase space functions (6) read where ϕ is a scalar field, and It is straightforward to verify that the set of fields where M, Q, and α are constants, does solve (30) provided the constraints hold. Because a charged massive particle couples only to the electromagnetic field and gravity, the static gauge action functional reads where m and e are the mass and electric charge of a particle probe. The corresponding canonical Hamiltonian takes the form where (p θ , p φ , p ψ ) denote momenta canonically conjugate to (θ, φ, ψ). As follows from the last line in Eq. (3) and the first line in Eq. (23), the model (34) is amenable to D(2, 1; α) superconformal extension provided the BPS-like condition on the particle parameters holds. We thus conclude that (4, 4, 0)-supermultiplet of D(2, 1; α) based on the realization of su(2) in (4) can be linked to a near horizon BPS-superparticle minimally coupled to fields of the d = 5, N = 2 supergravity interacting with one vector multiplet in spacetime with cosmological constant. As in the preceding case, the parameter α turns out to be related to the cosmological constant.

Conclusion
To summarize, in this work we generalized the analysis in [1] to the case of the most general N = 4 superconformal group in one dimension D(2, 1; α). It was shown that any realization of the R-symmetry subalgebra su(2) in terms of phase space functions can be extended to a representation of the Lie superalgebra corresponding to D(2, 1; α). Novel coupling of arbitrary number of supermultiplets of the type (1,4,3) and (0, 4, 4) to a single supermultiplet of either the type (3, 4, 1), or (4, 4, 0) has been constructed by arranging the su(2)-generators so as to include both bosons and fermions. Alternatively, this system can be viewed as describing an interaction of M + 1 copies of either (3, 4, 1)-, or (4, 4, 0)supermultiplet, in which angular degrees of freedom are identified, with N supermultiplets of the type (0, 4, 4). A canonical transformation which relates D(2, 1; α) superconformal mechanics based on supermultiplets of the type (3, 4, 1) and (4, 4, 0) to BPS-superparticles propagating near the horizon of the extreme Reissner-Nordström-AdS-dS black hole in four and five dimensions was found. The parameter α was linked to the cosmological constant.
There are several directions in which the present work can be continued. First of all, it would be interesting to construct an off-shell superfield Lagrangian formulation for the component Hamiltonian framework presented in Sect. 3. Interacting systems in Sect. 3 were interpreted as describing a coupling of arbitrary number of supermultiplets of the type (1,4,3) and (0, 4, 4) to a single supermultiplet of either the type (3, 4, 1), or (4, 4, 0). As was mentioned above, in principle, an alternative interpretation is possible in which several copies of (3, 4, 1)-, or (4, 4, 0)-supermultiplets are first identified along angular degrees of freedom and then they are coupled to (0, 4, 4)-supermultiplets. It is interesting to understand whether superfield constrains leading to such an identification along the angular degrees of freedom can be formulated in superspace. A κ-symmetric Lagrangian formulation for the BPS-superparticles in Sect. 4 and a possible connection between the supersymmetry charges and the Killing spinors characterizing the background geometry are worth studying as well. Finally, it is of interest to study the models in this work from the perspective of the Kirillov-Kostant-Souriau method (see a recent work [11] and references therein). ψ αψβ =