SU(2|2) supersymmetric mechanics

We introduce a new kind of non-relativistic ${\cal N}{=}\,8$ supersymmetric mechanics, associated with worldline realizations of the supergroup $SU(2|2)$ treated as a deformation of flat ${\cal N}{=}\,8$, $d{=}1$ supersymmetry. Various worldline $SU(2|2)$ superspaces are constructed as coset manifolds of this supergroup, and the corresponding superfield techniques are developed. For the off-shell $SU(2|2)$ multiplets $({\bf 3,8,5})$, $({\bf 4,8,4})$ and $({\bf 5,8,3})$, we construct and analyze the most general superfield and component actions. Common features are mass oscillator-type terms proportional to the deformation parameter and a trigonometric realization of the superconformal group $OSp(4^*|4)$ in the conformal cases. For the simplest $({\bf 5, 8, 3})$ model the quantization is performed.


Introduction
In recent years, interest has grown in theories invariant under some "curved" analogs of rigid Poincaré supersymmetry in diverse dimensions [1,2,3]. The main motivation was to check general gauge/gravity correspondences in concrete field-theoretical examples, classically as well as quantum mechanically. One construction of such theories is by the localization method [4], which proceeds from the relevant supergravity theories in component formulation. Alternatively, one can start from the supergroup of the corresponding "curved" supersymmetry, list its various coset superspaces and develop appropriate superfield techniques. These permit the derivation of invariant actions as superspace Berezin integrals, with Lagrangians being functions of superfields and their covariant derivatives. This second approach was used in [5,6,7] and goes back to [8] where superfield techniques for OSp(1|4) supersymmetry in four dimension were fully developed for the first time.
Supersymmetric mechanics [9,10] represents the extreme d=1 case of Poincaré-supersymmetric field theory. In the underlying d=1 "Poincaré superalgebra" the supercharges square to the Hamiltonian (and perhaps some constant or operator-valued central charges). Mechanical analogs of higher-dimensional curved rigidly supersymmetric theories can be based on semi-simple supergroups which yield flat d=1 supersymmetries through some contraction. In other words, mechanical models on such supergroups can be treated as deformations of standard supersymmetric mechanics. The main difference between the two types of supersymmetric mechanics models lies in the closure of the supercharges: In the deformed case it contains not only the Hamiltonian but also generators of some nontrivial internal symmetry. As a consequence, the corresponding Hilbert spaces and spectra essentially differ from each other. In particular, in the deformed case an energy level may carry unequal numbers of bosonic and fermionic states.
The plan of the paper is as follows. In Section 2 we describe coset superspaces of SU (2|2), to be used in the following sections for defining superfields carrying various irreducible SU (2|2) multiplets. Besides the standard real SU (2|2) superspace we introduce the chiral superspace, the harmonic superspace and the biharmonic superspace. We define the necessary elements of the corresponding superfield technique: covariant derivatives, transformation laws, and invariant integration measures. In Sections 3, 4 and 5 we present the models associated with the off-shell SU (2|2) multiplets (3,8,5), (4,8,4) and (5,8,3). We give both the superfield and component-field actions for all cases. Some of these actions reveal enhanced superconformal-type symmetries, some do not. Common features of most actions are an oscillator-type mass term for the fields and a trigonometric realization of the superconformal symmetries. As an example of a quantum model, in Subsection 5.3 we discuss SU (2|2) quantum mechanics based on a free (5,8,3) multiplet. In the concluding Section 6 we mention links with other models and outline some directions for further study. We also adduce arguments why certain flat N = 8, d=1 multiplets (in particular the "root" multiplet (8,8,0)) seem not to admit a deformation to SU (2|2) mechanics. We transferred into three Appendices some technical points, including the calculation of various harmonic integrals, the embedding of the superalgebra su(2|2) into the N = 8, d=1 superconformal algebra osp(4 * |4), the realization of the latter on the SU (2|2) multiplets considered, as well as a short account of the off-shell SU (2|1) multiplet (3,4,1). The latter is an important constituent of our SU (2|2) multiplets but was not properly treated in previous papers on SU (2|1) mechanics.
2 Deformed N = 8, d=1 superspaces In this section, we formulate a deformed real N = 8, d=1 superspace where worldline realizations of the supergroup SU (2|2) can be defined. Then we construct the corresponding chiral, analytic harmonic and analytic biharmonic SU (2|2) superspaces. In the following sections, these types of superspaces will be used for defining different types of superfields and for setting up SU (2|2) invariant actions of the latter, generalizing those constructed in [29,31,33,34] in the presence of flat N = 8, d=1 supersymmetry.

Superalgebra
Our starting point is the superalgebra su(2|2) with three central charges: All other (anti)commutators are vanishing. The superalgebra su(2|2) contains in general three central charges C, C 1 and H. The generators L ij = L ji , R ab = R ba form two mutually commuting su(2) algebras, su(2) L and su(2) R . The conjugation rules are as follows: 2 (2. 2) The mass dimension parameter m plays the same role as in the SU (2|1) case: by contraction m → 0 the relations (2.1) are reduced to those of the flat N = 8, d=1 "Poincaré" superalgebra extended by central charges C, C 1 and possessing a restricted R-symmetry group SO(4) ∼ SU (2) L × SU (2) R . 3 Correspondingly, (2.1) can be considered as a deformation of the flat N = 8, d=1 supersymmetry, with m as a deformation parameter.
To understand the origin of the central charge operators in (2.1), let us note that these relations in fact coincide with those defining a deformation of the flat N =(4, 4), d=2 Poincaré superalgebra. Indeed, in the m = 0 limit (2.1) can be identified with a sum of two independent N =4 , d=2 algebras in the left and right sectors of d=2 Minkowski space-time in the light-cone parametrization, with H+ C 1 and H −C 1 being the mutually commuting translation operators along two light-cone directions. Moreover, one can realize the d=2 Lorentz group SO(1, 1) as an additional automorphism group of (2.1) acting as real rescalings of the mutually (anti)commuting sets (Q ia , H + C 1 ) and (S ia , H − C 1 ) (with the weights (1/2, 1) and (−1/2, −1), respectively). In such an interpretation, the generator C in (2.1) is SO(1, 1) singlet and so it is the central charge from the d=2 perspective as well, while C 1 generates the translation along the spatial d=2 direction. The natural and simplest reduction from d=2 to d=1 proceeds by eliminating altogether the dependence on the spatial coordinate, i.e. just by putting to zero the generator C 1 . In what follows we will deal with such a limited su(2|2) superalgebra, corresponding to the choice C 1 = 0 in (2.1). In principle, it is easy to construct the SU (2|2), d=1 superfield formalism with C 1 = 0, 4 but in all examples considered below there is no need to activate this central charge. It is not the case for the "genuine" central charge C which defines an actual symmetry, e.g., in the models based on the multiplet (4, 8, 4) (Sect. 4).
One can rewrite the superalgebra (2.1) (hereafter with C 1 = 0) in a different form by defining the complex supercharges In the complex basis, the (anti)commutators of (2.1) become The doublet indices are raised and lowered in the standard way by the ε symbols, e.g., The supergroup SU (2|2) contains a few SU (2|1) subgroups. One of them has the bosonic subgroup SU (2) L ×U (1) R with U (1) R ⊂ SU (2) R , while another has the bosonic subgroup SU (2) R × U (1) L with U (1) L ⊂ SU (2) L . These supergroups are equivalent up to switching SU (2) R ↔ SU (2) L . In what follows, we will mainly deal with the first choice, where SU (2|1) generators [16] are singled out as 5 The second basic su(2|1) subalgebra is formed by the generators Π 1a ,Π 1b , R ab , L 12 , H. Actually, the generators (2.5) form the centrally extended superalgebraŝu(2|1) with the central charge H, and the same is true for the second SU (2|1). The central charge H in (2.5) is a difference of external and internal U (1) generators in the extended superalgebra su(2|1) ⊕ u(1) ext [17]. If the generator H − mR 12 is chosen as the full internal U (1) generator of su(2|1) (such a choice is admissible since H commutes with everything), then R 12 decouples and becomes a generator of the external U (1) ext R-symmetry, such that u(1) ext ⊂ su(2) R .

Basic SU(2|2) supercoset, Cartan forms and covariant derivatives
We will be first interested in the realization of SU (2|2) supersymmetry in a real N = 8, d=1 superspace identified with the following supercoset of the supergroup with the superalgebra (2.1): Here, the supergroup P SU (2|2) is a corresponding supergroup SU (2|2) without central charges. Further, we will use the notation SU (2|2) as a supergroup with central extensions. An element of this supercoset is defined as 8) and the supercoset parameters are treated as a set of superspace coordinates The central charge generator H is associated with a translation generator along R 2 /R 1 ∼ R 1 parametrized by the time coordinate t. Before presenting the realization of SU (2|2) on these coordinates induced by the left shifts of the element (2.8), it will be convenient to calculate the left-covariant Cartan 1-forms defined by (2.10) The explicit expressions for these forms are , The other pair of supercharges Π i2 ,Πj2 also form an su(2|1) superalgebra, with the same set of bosonic generators.
The covariant derivatives of some superfield Φ A t, θ ia ,θ jb can be found from the general expression for its covariant differential Here,L ij andR ab are "matrix" parts of the full SU (2) generators (realized as well on Grassmann coordinates), which act on the external indices of covariant derivatives as In the same way, they act on the external SU (2) L × SU (2) R indices of superfields. The rule of complex conjugation for these matrix parts is as follows (2.14) Explicitly, the covariant derivatives are given by the following expressions They satisfy the anticommutation relations By rewriting the covariant derivative as we obtain that

Transformation properties
The transformation properties of the N = 8 superspace coordinates under the left shifts with the parameters ǫ ia andǫ ia , as well as the induced stability subgroup infinitesimal transformations, can be found from the general formula The explicit calculations yield the following transformations: The induced elements in (2.20) are It is straightforward to check that the coset-space Cartan forms undergo SU (2) L × SU (2) R induced transformations under the coordinate transformations (2.21): Superfields are assumed to transform according to the general law where an external index A of the superfield Φ A specifies the SU (2) L ×SU (2) R matrix representation by which this superfield is transformed (and that ofC). The SU (2|2) invariant N = 8, d=1 superspace integration measure is given by

Chiral SU(2|2) superspace
We introduce the complex coordinates which are related to those defined in (2.9) as It will be also convenient to pass to the new infinitesimal parameters in terms of which the transformation properties of the superspace coordinates in (2.26) are as follows The measure of integration over (2.26) can be checked to be invariant under these transformations: Hereafter, we use the notation: Now it is easy to show the existence of a left chiral subspace parametrized by the coordinates Indeed, the set (2.33) is closed under the SU (2|2) transformations Actually, the set (2.33) can be identified with the following complex coset superspace of SU (2|2): The invariant measure of integration over (2.33), dζ L , is defined by In the coordinates (2.33), the covariant derivatives (2.31) are written as From the structure of the covariant derivativeD ia we observe that the general covariantly chiral can be made explicitly chiral after the appropriate ϑ,θ-dependent SU (2) L and SU (2) R rotation of Φ A with respect to the external indices. For instance, if Φ A has the SU (2) L × SU (2) R matrix assignment (1/2, 1/2), this additional redefinition is given by On the other hand, it is not possible to eliminateC fromD ia in a similar way. In fact,C should always be vanishing on chiral superfields, as follows from the anticommutation relations (2.19), which are just the integrability conditions for the chirality constraint (2.39) and its anti-chirality counterpart:D

Harmonic superspace
We perform a harmonization of the SU (2) L indices and define the analytic subspace It is closed under the following SU (2|2) transformations Note that the transformation properties of the harmonic variables w ± i , as well as the precise relation between the "central basis" coordinates (t, θ ia ,θ ia ) and the "analytic basis" coordinates (t (A) , θ + a ,θ + a ) , are uniquely fixed just by requiring (2.43) to be closed under the SU (2|2) transformations.
For further calculations, it will be convenient to pass to another set of harmonic variables in the harmonic superspace (2.43), With this choice, the realization of the fermionic SU (2|2) transformations in the analytic subspace is as follows, where The SU (2|2) covariant harmonic derivative D ++ preserving analyticity is uniquely defined by requiring it to transform as It reads This harmonic derivative reveals some unusual properties to be used below: The analytic subspace integration measure It is easy to check that

Biharmonic superspace
One can extend the superspace (2.9) by biharmonic coordinates w (2.56) The relation between the coordinates (2.56) and the original coordinates (2.9) is given by the following substitutions, Now it is straightforward to explicitly find the relevant coordinate SU (2|2) transformations leaving closed the analytic coordinate set (2.56) Here The analytic subspace has an invariant integration measure, Now we can define the covariant harmonic derivatives (2.63) They possess the standard transformation laws One can check that

Kinematics
On one hand, the multiplet (3,8,5) can be described by a superfield V ij satisfying the constraints According to (2.24), the "passive" transformation law of V ij is On the other hand, one can define the harmonic superfield which lives on the analytic harmonic superspace (2.43). The Grassmann analyticity conditions for V ++ amount just to the constraints (3.1) for V ij in the original central basis. After passing to the new harmonic variables (2.46), the transformation of V ++ can be written through the parameter Λ defined in (2.54) as The analytic superfield V ++ satisfies the harmonic condition which can be proved using (3.3) expressed in terms of the harmonic u ± i , as well as the explicit expression for D ++ , eq. (2.50). In fact at this step one can forget about the relation (3.3) and deal with the real analytic harmonic superfield V ++ ( V ++ = V ++ ) subjected to (3.5). The harmonic constraint implies the following component structure of V ++ : where the fields satisfy the reality conditions: Thus, we are left with three physical bosonic fields v ij (t), eight fermionic fields ξ i a (t),ξ i a (t) and five bosonic auxiliary fields A 0 (t), C 0 (t), C ab (t), i.e., just with the (3, 8, 5) content. A new constant µ , [µ] = 1 , came out in the course of solving (3.5). It survives in the flat limit m = 0.
We also present here the transformation properties of the component fields, Note that the SU (2|2) covariant constraint (3.5) and the transformation law (3.4) can be generalized to an arbitrary analytic superfield q +n of the harmonic U (1) charge n: This is similar to the analogous phenomenon observed in the flat N = 4, d=1 harmonic superspace [20]. For even n one can impose the reality condition on q (+n) . The difference from the N = 4, d=1 case is that for n = 1 the constraint in (3.9) implies the equations of motion for the physical fields and is similar in this respect to the harmonic equation of motion for the analytic hypermultiplet superfield in N = 2, 4D case [50,51]. For n = 2 , this constraint remains purely kinematic and defines the d=1 analog of the N = 2, d=4 tensor multiplet, with the constant µ appearing as a solution of the d=1 reduction of the well-known "notoph" condition ∂ µ A µ = 0 in 4D. All these features are retained in the flat limit m = 0 .
Confronting the transformation law of V ++ (3.4) with the transformation of the analytic measure (2.53), one concludes that it is impossible to construct any invariant Lagrangian out of V ++ (even the free one), with harmonic U (1) charge +4 (needed to cancel the negative charge −4 of the measure). To evade this difficulty, we will proceed by analogy with the construction of the superconformal actions in [52,20].
The procedure is as follows. We introduce an auxiliary constant triplet c ij . Its harmonic that follows from the completeness relation for the harmonics. Without loss of generality, we choose c ij c ij = 1 . Next, we define the "shifted" superfieldV ++ as The tripletc ++ satisfies the condition and so The appearance of an additional term inc ++ is related to the properties (2.51) of D ++ . Note also the useful relation 14) The component structure of the shifted analytic superfieldV ++ related to V ++ by (3.11) is obtained The newly defined quantities are transformed as where Λ ++ = −D ++ Λ (recall eq. (2.55)). Using these relations, one can construct invariant actions (see Appendix A.1) with the superfield Lagrangian To find the component form of the action we use the normalization The main technical problem is to do the relevant harmonic integrals. This can be accomplished using the formulas listed in Appendix A.2. The component Lagrangian finally reads . Here, |v| := v ij v ij . The expression within the square brackets can basically be obtained by a dimensional reduction d=4 → d=1 from the d=4 Lagrangian of [52]. The new terms are those ∼ µ (they survive in the m = 0 limit), the fermionic "mass" mixed term ∼ m and the bosonic potential term ∼ m 2 . The last term ∼ µ is a special WZ term for v ik known as a Lorentz-force type coupling to Dirac magnetic monopole [20,29].

Duality transformations
In [52], duality transformations of the tensor multiplet was shown to lead to the free hypermultiplet action. Here, we define in the same way duality transformations for the d=1 multiplet (3,8,5).
We can rewrite the action (3.17) as where f ++ is an analytic superfield related to V ++ andV ++ by In view of this one-to-one correspondence, the harmonic constraint (3.13) implies a nonlinear constraint on the superfield f ++ . The transformations of f ++ can be found from (3.21) Next, we add to the action (3.20) an additional term with the Lagrange multiplier ω, and thereby get rid of the condition (3.13), ending up with two independent analytic superfields, ω and f ++ . The requirement of invariance of this action implies ω to transform as Integrating by parts the last term in (3.23), we obtain By analogy to [52], we can cast the Lagrangian (3.25) in the form of the free action where From this relation, one can establish that We observe that the external doublet index i of q +i is inert with respect to the whole SU (2|2) , including the SO(4) transformations. So it a sort of Pauli-Gürsey index and it is convenient to replace it by another letter, e.g. as q +i −→ q +A . (3.30) The action (3.26) respects an additional invariance under an extra SU (2) PG rotating the doublet index A.
The superfield q +A has the following θ-expansion: Here, all fields are defined on the extended bosonic space t (A) , u ±i , i.e., their harmonic expansions produce infinite towers of fields [51]. Eliminating auxiliary fields by the relevant part of the equation of motion for (3.26), we obtain the on-shell superfield q +A containing a finite set of physical fields, The constraint (3.32) puts the residual component fields on-shell: They can be re-derived from the on-shell component Lagrangian

Kinematics
The multiplet (4,8,4) can be described by the superfield q ia , with (q ia ) = q ia . The proper constraints are imposed as The SU (2|2) covariance of these constraints requires that According to (2.24), the odd transformations of q i can be written as δq ia = 2im ǫ jb θ ib q ja +ǫ ja θ jb q ib . Now one can define the analytic biharmonic superfield living on the analytic subspace (2.56) and transforming as where Λ (0,0) was defined in (2.66). While the Grassmann constraints (4.1) are automatically satisfied for q (1,1) in the analytic basis, the restricted harmonic dependence in (4.4) amounts to the harmonic constraints Taking into account the transformation laws of D (2,0) and D (0,2) , eqs. (2.64), as well as the definitions (2.65) and (2.66), it is easy to establish the SU (2|2) covariance of (4.6). The solution of (4.6) is given by the undeformed superfield (4.7) With taking into account (4.5) and (2.58), its components transformations are found to read Since the superfield q (1,1) in itself is not deformed (only its transformation properties prove to be deformed), we can realize on it the supersymmetry SU (2|2) in parallel with the standard flat N = 8, d=1 Poincaré supersymmetry, or even with another SU (2|2) involving the flipped-sign deformation parameter −m. The closure of all these symmetries including the original SU (2|2) turns out to constitute an extended superalgebra introduced in [33]: [I ij , I kl ] = ε il I kj + ε jk I il , [I ab , I cd ] = ε ad I bc + ε bc I ad , In fact, the superalgebra (4.9) -(4.11) contains four SU (2) subalgebras with the generators I ab , I ij , J ab , J ij . These generators differently act on the indices of the component fields f ia , χ a b ,χ i j , F ia . The generators I ab , I ij rotate only the upper-case indices i and a, while J ab , J ij act only the lowercase ones (though denoted by the same characters). Thus, the two types of SU (2) indices of the component fields can actually be split into four types.
The SU (2|2) generators of (2.1) can be identified with the following linear combinations of the generators of the extended superalgebra (4.9) -(4.11): Hence, the superalgebra (2.1) can be viewed as a subalgebra of the extended superalgebra (4.9) -(4.11), with the central charge C = − imZ. (4.13) The second SU (2|2) supergroup is generated by the supercharges (4.14) The integration measure (2.61) is invariant under the transformations of both SU (2|2) supergroups, with the parameters m and −m , i.e. it is also invariant under the transformations produced by all generators of (4.9) -(4.11). The generators appearing in (4.12) are realized on the biharmonic superspace (2.56) as While applying these operators to the superfield q (1,1) , one is led to put Zq ia = q ia , in accord with (4.2) and (4.13). The algebra of the generators (4.15) can be extended by the generator

Invariant actions
Let us define the new "shifted" superfield where c ia is a constant satisfying c ia c ia = 1. It is enough to consider the ǫ-transformations Such transformations are similar to the "superconformal" transformations [33]. Then it follows that an SU (2|2) invariant action can be constructed in the same way: Since the superfieldq (1,1) is not deformed, this action coincides with the one given in [33] and so it is invariant under the full hidden supersymmetry with the algebra (4.9) -(4.11) and the additional transformations with the generator (4.16). The central charge generator Z acts as a dilatation generator in the target space, δ Z q (1,1) = ωq (1,1) , where ω is a constant parameter. Note that (4.20) is not invariant under the standard dilatations which affect not only q (1,1) , but also the time coordinate t (B) . Despite the transformations (4.8) are mass-deformed, the component Lagrangian of (4.20) contains no terms with the parameter m. In particular, the bosonic core of this Lagrangian is as follows On the other hand, from the SU (2|1) standpoint, the multiplet (4, 8, 4) is a direct sum of two SU (2|1) multiplets, (4,8,4) = (4, 4, 0) ⊕ (0, 4, 4) , and it is known [19] that the Lagrangian of the multiplet (4, 4, 0) in the general case explicitly involves the deformation parameter m. In particular, its bosonic core is The only option for which the mass term becomes a constant and so fully decouples is just G(f ) = 1/f 2 required by the SU (2|2) invariance. 6 It can be shown that for this special choice the parameter m disappears also from all other terms in the (4, 4, 0) Lagrangian. Note that the Lagrangian (4.21) is invariant under the Z 'dilatations", δ Z f ia = ωf ia , δF ia = ωF ia and, up to a total derivative, under the transformations generated by the operator K defined in (4.16), 5 The multiplet (5,8,3)
We consider the complex superfield Ψ satisfying the standard chiral constraints This superfield lives as unconstrained on the chiral subspace (2.33). It means that the solution of (5.1) is given by the general ϑ-expansion The passive transformation law δΨ = 0 implies the following component transformations: Indeed, their Lie brackets are easily checked to form SU (2|2) symmetry. The chiral superfield (5.2) contains 16 bosonic and 16 fermionic fields and so is reducible. To single out the multiplet (5,8,3), we impose the extra SU (2|2) covariant constraints where V ij is an additional deformed N = 8 superfield. Solving the constraints, we find that This field content now corresponds to the multiplet (5,8,3), and the deformed transformations (5.3) are rewritten for the involved fields as

Invariant actions
The N = 8 invariant deformed action can be written as an integral over chiral subspaces, like in the case of flat N = 8 supersymmetry [32]: The component Lagrangian reads Here, g is a special Kähler metric defined as As compared to the undeformed case, we observe the appearance of the oscillator-type fermionic (∼ m) and bosonic (∼ m 2 ) potential terms, as well as the internal bosonic WZ term accompanied by some new Yukawa-type couplings. The simplest free action S free (5,8,3) corresponds to the choice f (Ψ) = Ψ 2 /4 . Its component off-shell Lagrangian reads L free (5,8,3) In [47], SU (2|2) supersymmetry was shown to underlie N = 8 massive quantum mechanics of type I inspired by some super Yang-Mills theory. One can show that the relevant Lagrangian in the abelian case with U (1) as a gauge symmetry coincides with the on-shell Lagrangian obtained from (5.10). It would be interesting to inquire to which higher-dimensional system the general Lagrangian (5.8) could correspond.
The SU (2|1) superspace coordinates are defined in the basis (2.26) as t , ϑ i1 ,θ i1 =: t , θ i ,θ i and are transformed under SU (2|1) according to Here the parameters ǫ i ,ǭ i are related to the parameters in (2.29) as the ε-transformations being associated with the hidden supersymmetry which extends SU (2|1) to SU (2|2). The ǫ-transformations in (5.6) are split into SU (2|1) transformations corresponding to the chiral multiplet (2,4,2) [16] with the U (1) charge κ = 0 and the multiplet (3, 4, 1) (see Appendix C): Generally, the SU (2|2) invariant Lagrangian can be written in terms of these SU (2|1) superfields as where F is an arbitrary real scalar function of SU (2|1) superfields satisfying the five-dimensional Laplace equation [28,30]: The metric g := g (z,z, v ij ) of the target space is expressed as One can explicitly check that (5.15) is the only condition which is required for the invariance under the second subgroup SU (2) R of SU (2|2) in the terms quadratic and quartic in fermions. Since the closure of SU (2|1) and SU (2) R transformations necessarily yields the supersymmetry SU (2|2), the equations (5.15) is none other than the conditions of the SU (2|2) supersymmetry. One can treat the invariant Lagrangian (5.14) as a Lagrangian constructed in terms of harmonic superfields associated with Ψ,Ψ and V ij . This way of obtaining (5.14) can presumably be figured out from the harmonic formalism elaborated in [30]. As a solution of (5.15), the Lagrangian (5.8) can be rewritten in terms of SU (2|1) superfields as Here, the function f is related to (5.7) as 18) and the relevant metric (5.16) coincides with that defined in (5.9). The metric (5.9) corresponds to the most general solution of (5.15) for F restricted to the 2-dimensional target space as F ≡ F (z,z) , g ≡ g (z,z). One can consider more general solutions involving some extra dependence on the triplet v ij . For instance, the most general solution with g ≡ g (v ij ) yields the Lagrangian In the component form it reads .
One can explicitly check that this Lagrangian is invariant under (5.6). There can be many other solutions of (5.15) depending on all five fields. An example of such a solution producing a superconformal model is given in Appendix B.3.

The free quantum model
As an example, here we present quantization of the simplest free model corresponding to (5.10). Eliminating auxiliary fields, we obtain After performing Legendre transformations we obtain the canonical Hamiltonian Other Noether charges are given by The Poisson and Dirac brackets are imposed as and they are quantized in the standard way We will use the operators where In terms of the so defined creation and annihilation operators, the quantum version of the generators of (2.4) takes the form As follows from the definition (5.28), the quantum generator L i j in fact does not involve the parameter m. So the latter appears only in the supercharges and the Hamiltonian.
To construct the Hilbert space of wave functions, we use the creation operators∇z, ∇ +ij , ψ ia and the annihilation operators ∇ z , ∇ −ij ,ψ ia . Then, the energy spectrum of H is found to be where Ω (ℓ) is a wave function at the Landau level ℓ. The ground state corresponds to ℓ = 0 and the first excited level to ℓ = 1. The relevant wave functions are given by the expressions: , The coefficients a (0) , a (1) , b ij and c (1) ia are some arbitrary antiholomorphic functions. This infinite degeneracy is caused by action of the additional generators ∇ z + imz and∇ − imz (magnetic translations) which commute with all quantum generators (5.29). All the higher levels ℓ > 1 have wave functions of more complex structure and we will not consider them here.
A few words about SU (2|2) representations are to the point. The ground state Ω (0) is annihilated by all quantum generators (5.29), i.e., it is just a singlet. According to [54], the level ℓ = 1 corresponds to the atypical SU (2|2) representation 1, 0; 1/2, 0, 0 , with the overall dimension 8. All the higher ℓ wave functions can be also classified based on the analysis of [54].

Conclusions
Using powerful d=1 superfield coset techniques, we have constructed and studied several models of SU (2|2) supersymmetric mechanics based on the off-shell multiplets (3,8,5), (4,8,4) and (5,8,3). This new kind of supersymmetric mechanics is a deformation of flat N = 8 supersymmetric mechanics. The corresponding actions were presented, both in terms of superfields and of component fields. The extended symmetries of these actions were revealed, and quantization was explicitly performed in one simple case.
In [47], two types of N = 8 massive super Yang-Mills quantum mechanics provided matrix descriptions of supermembranes. Type I is based on the supergroup SU (2|2), while type II uses the product supergroup SU (2|1) × SU (2|1). In Section 5, we noticed that the type I model of [47] reduced to the simplest U (1) gauge symmetry corresponds to the free Lagrangian (5.10) of the multiplet (5,8,3). Our superfield approach gives the more general SU (2|2) supersymmetric Lagrangian (5.8). It will be interesting to explicitly consider deformations yielding the supergroups SU (2|1) × SU (2|1) [47] or SU (1|4) [46]. Further off-shell deformed N = 8 multiplets and the associated mechanics models may be constructed in this way. It is especially interesting to inspect worldline realizations of the supergroup SU (2|4) [44,45], as they should bear a direct relation to the matrix models of [42] (see also [46]). Such models can be studied directly in an SU (2|2) mechanics language, proceeding from the fact that SU (2|2) is a subgroup of SU (2|4) and representing the multiplets of the latter as direct sums of the appropriate SU (2|2) multiplets. For instance, the SU (2|4) on-shell multiplet (10,16) can hopefully be organized from two copies of the SU (2|2) multiplet (5,8,3).
Appendix A Details of invariant action for the multiplet (3,8,5) A

.1 Calculation of the superfield action
The idea is to construct the invariant Lagrangian as a power series inV ++ , Using the fact thatV ++ is transformed inhomogeneously in (3.15), we require that the variations of the adjacent terms in the sum cancel each other modulo a total derivative, which will impose strict relations between the coefficients b n and, finally, fix the form of the above series. We will properly employ the freedom in normalizing (A.1). The physical normalization of the kinetic term of the boson triplet fixes b 2 = 1/2. We include the transformation of the integration measure (2.53) into the variations of various terms in the Lagrangian (A.1). Then such a generalized variation of the first term in the sum in (A.1) is reduced to (up to a total harmonic derivative) The first piece of this variation is going to be compensated from the variation of the term ∼ V ++ 3 in (A.1), while the second piece is canceled with the variation of the term ∼ ρ in (A.1). Indeed, it is easy to show that, up to a total derivative, δ ρc ++V ++ = 4Λ ρc ++V ++ , (A. 3) and the choice ρ = 3/2 ensures the cancelation needed. Next, we consider the cubic term b 3 c −− V ++ 3 with the variation In this term, the variation ofV ++ yields a vanishing contribution due to the presence of the highest-order θ monomial and the fact that both Λ and Λ ++ involve at least one power of the Grassmann-odd coordinates. So its full variation is exclusively defined by the variations of the explicit θ s. Proceeding further, we find that After some effort, using the general formula c −− n c ++ = n 2n + 1 c −− n−1 + 1 2 (n + 1) (2n + 1) D ++ 2 c −− n+1 , n 1 , c ij c ij = 1 , we find a recurrence relation for the coefficients a n and b n : a n = − (n + 1) (2n + 1) Then, using the property that, up to a total derivative, we represent the total Lagrangian L (+4) as It remains to learn to which functions these series sum up. Using the Taylor expansions it is straightforward to write (A.10) as Then, the SU (2|2) invariant Lagrangian for the multiplet (3,8,5) is given by

A.2 Harmonic integrals
For the calculation of the component Lagrangian of (A.13), we take as input the known harmonic integrals [52,20] du (1 + 2 c −−v++ ) 3/2 = 1 1 + 2 c ijv ij +v ijv ij , (A.14) After some algebraic manipulations involving integration by parts, the component Lagrangian is reduced to a few terms containing the expressions (all taken at θ = 0) They appear with the specific combinations of harmonics and the corresponding harmonic integrals are computed as There is also the harmonic integral (A.18) responsible for the potential term ∼ m 2 . It is not immediately obvious how to compute it. It is easier to calculate this integral by considering its series expansion Using the identities we transform (A.19) to the form The final result is given by the integral (A.14) as This redefinition just means that we passed to the equivalent basis where, instead of the su(2) algebra with the generators T αβ 2 , we deal with the diagonal su(2) in the direct sum of the former su(2) and the one with the generators T ab 1 . Then, we define the m-deformed supercharges and the bosonic generators In terms of these redefined generators the superalgebra osp(4 * |4) takes the form: L ij , L kl = ε il L kj + ε jk L il , R ab ,R cd = ε adRbc + ε bcRad , R ab , R cd = ε ad R bc + ε bc R ad , R ab ,R cd = ε ad R bc + ε bc R ad , R ab , U (cd) = ε ad U (bc) + ε bc U (ad) , R ab , U (cd) = − ε ac ε bd + ε ad ε bc U, The component fields of (3.6) can be rewritten in a complex notation as C ab C ab . (B.12) The relevant transformations (3.8) leaving this Lagrangian invariant (modulo a total derivative), become δv ij = − η a (jψi)a e − i 2 mt +η a (j ψ i)a e i 2 mt , δψ ia = 2iη j av ij − 2m η j a v ij − iη b i C ab + 2µ η ia e − i 2 mt + 2iη ia A e i 2 mt , δψ ia = − 2iη j av ij + 2mη j a v ij + iη b i C ab + 2µη ia e Exploiting the property that the Lagrangian (B.12) depends only on m 2 , we can define additional SU (2|2) transformations with m → −m: 14) The closure of these two types of SU (2|2) transformations gives a trigonometric realization of the full superconformal symmetry OSp(4 * |4). Thus, the Lagrangian (B.12) is superconformal and is recognized as a deformation of the parabolic Lagrangian given in [31] by the oscillator mass term ∼ m 2 . Making the change m → −m in these transformations, we define additional SU (2|2) transformations. In the same way as in the previous case, the two types of SU (2|2) transformations close on the superconformal symmetry OSp(4 * |4) in the trigonometric realization. Superconformal Lagrangian admits construction in terms of SU (2|1) superfields corresponding to the multiplets (2, 4, 2) and (3, 4, 1) as SU (1, 1|2) ⊂ OSp(4 * |4) superconformal trigonometric Lagrangian. Superfield Lagrangian satisfying (5.15) is given by L conf. (5,8,3) In contrast to Sec. 5.3, the chiral SU (2|1) superfield Φ describing the multiplet (2, 4, 2) has a central charge 9 b = −1 [18]. By analogy with (B.12) and previously constructed trigonometric superconformal Lagrangians [57,18], the relevant (5,8,3) superconformal Lagrangian is a deformation of the parabolic superconformal Lagrangian [29,31] by oscillator term. The bosonic truncation of superconformal Lagrangian reads L conf. (5,8,3) | bos = v ij v ij + zz −3/2 żż +v ijv ij + 1 2 B ab B ab − m 2 v ij v ij + zz . (B.18) One can see that this Lagrangian is SO(5) × SU (2) invariant, where dynamical bosonic fields form SO(5) vector and auxiliary fields are combined into SU (2) triplet.