Deformed ${\cal N}{=}\,8$ mechanics of ${\bf(8,8,0)}$ multiplets

We construct new models of `curved' SU$(4|1)$ supersymmetric mechanics based on two versions of the off-shell multiplet ${\bf(8,8,0)}$ which are `mirror' to each other. The worldline realizations of the supergroup SU$(4|1)$ are treated as a deformation of flat ${\cal N}{=}\,8$, $d\,{=}\,1$ supersymmetry. Using SU$(4|1)$ chiral superfields, we derive invariant actions for the first-type ${\bf(8,8,0)}$ multiplet, which parametrizes special K\"ahler manifolds. Since we are not aware of a manifestly SU$(4|1)$ covariant superfield formalism for the second-type ${\bf(8,8,0)}$ multiplet, we perform a general construction of SU$(4|1)$ invariant actions for both multiplet types in terms of SU$(2|1)$ superfields. An important class of such actions enjoys superconformal OSp$(8|2)$ invariance. We also build off-shell actions for the SU$(4|1)$ multiplets ${\bf(6,8,2)}$ and ${\bf(7,8,1)}$ through appropriate substitutions for the component fields in the ${\bf(8,8,0)}$ actions. The ${\bf(6,8,2)}$ actions are shown to respect superconformal SU$(4|1,1)$ invariance.


Introduction
In recent years, mainly motivated by the study of higher-dimensional models with "curved" rigid supersymmetries (see e.g. [1]), there was a growth of activity in supersymmetric mechanics (SM) models underlain by some semi-simple superalgebras treated as deformations of flat one-dimensional supersymmetries with the same number of supercharges. The simplest superalgebra of this kind is su(2|1) (and its central-charge extension su(2|1)), which is a deformation of rigid N = 4, d = 1 supersymmetry by a mass-dimension parameter m.
It turns out that some admissible multiplets of flat N = 8 supersymmetry do not have SU(2|2) analogs, most importantly the so called "root" N = 8 multiplet (8,8,0). The significance of this root multiplet derives from the fact that all other flat N = 8 multiplets and their invariant actions can be obtained from the root one and its general actions through appropriate covariant substitution of the auxiliary fields (or Hamiltonian reductions, in the Hamiltonian formalism) [17] as a generalization of the phenomenon found in [5] at the linearized level. 3 Deforming the flat (8,8, 0) multiplet has remained an open problem.
In the present paper we show that the latter becomes possible within the alternative SU(4|1) deformation. Interestingly, there exist two such root SU(4|1) multiplets, which are complementary to each other in the sense that the SU(4) assignments of their fermionic and bosonic components are interchanged. Namely, in one multiplet, the bosonic d=1 fields are in 1 ⊕ 1 * ⊕ 6 of SU(4) (eight real fields) and the fermionic fields in 4 ⊕ 4 * (4 complex fields), while in the other multiplet the bosonic fields are in 4 ⊕ 4 * and the fermionic fields in 1 ⊕ 1 * ⊕ 6 . In the "flat" N = 8, d = 1 limit they go over to two different 8-dimensional multiplets of the SO(8) R-symmetry related by triality (see, e.g., [22], [23]). These two multiplets are analogs of the mutually "mirror" N = 4 multiplets (4, 4, 0), for which bosonic and fermionic components form doublets with respect to different SU(2) factors of the SO(4) R-symmetry. For this reason it is natural to treat the two root SU(4|1) (8,8, 0) multiplets as "mirror" to each other.
The main incentive of our paper is constructing invariant actions for both types of the (8,8,0) multiplets. To this end, we will use a manifestly SU(4|1) covariant superspace formalism along with the SU(2|1) superfield approach, in which the extra SU(4|1)/SU(2|1) transformations are realized in a hidden way. In some cases, it is simplest to use the component approach. The point is that SU(4|1) possesses many non-equivalent worldline supercosets, including the harmonic ones [24], and it is not easy to decide which superfield formalism is most adequate for one or another SU(4|1) multiplet. We utilize several versions of such an extended superfield approach for constructing invariant actions.
The paper is organized as follows. In Section 2 we present the superalgebra su(2|1) and describe the relevant worldline supercosets. In Section 3, on the example of flat N = 8, d = 1 supersymmetry, we discuss three possible (8,8,0) multiplets, which are not equivalent if the SO(8) R-symmetry is broken, and argue that only two of them can be extended to the deformed SU(4|1) case. The various superfield and component descriptions of the first version of the SU(4|1) (8,8,0) multiplet are the subject of Section 4. We find three different classes of invariant actions for this multiplet, including an OSp(8|2) invariant one, with an R-symmetry enhanced to SO (8). The analogous treatment of the second version of the multiplet (8,8,0) is given in Section 5. We show that its general invariant action is superconformal and equivalent to the superconformal action of the first version. Summary and outlook are given in Section 6. An Appendix A contains details of calculating the invariant actions in the appropriate harmonic SU(4|1) superspaces, and in Appendices B and C the off-shell actions for the SU(4|1) multiplets (6,8,2) and (7,8,1) are presented. The full set of (anti)commutation relations of the conformal superalgebra osp(8|2) is given in Appendix D.
Here, L I J are the generators of the R-symmetry group SU(4), and the capital indices I, J, K, L (I = 1, 2, 3, 4) refer to the SU(4) fundamental ("quark") representation and its conjugate. H is the U(1) generator. In the contraction limit m = 0 the above superalgebra goes over to the SU(4) covariant form of the flat N = 8, d = 1 superalgebra. This limiting superalgebra actually possesses an enhanced R-symmetry group SO(8) which mixes Q I withQ J (they are joined into SO(8) spinor). In what follows we will not need the explicit form of these enhanced SO(8)/SU(4) transformations, except for their realizations on the covariant "flat' spinor derivatives.
The basic real SU(4|1) , d = 1 superspace is defined as the coset superspace with the coset parameters being the superspace coordinates: One could define these coordinates within the standard exponential parametrization of the supercoset. However, it will be more convenient to use another parametrization, the one associated with the purely fermionic coset SU(n|1)/U(n) defined in [30] (see also [31]). We uplift the U(1) group from the stability subgroup U(4) into the numerator and consider an extension of the SU(4|1)/U(4) coordinate set by a time coordinate t. Thus this U(1) generator is associated with the Hamiltonian. Following to [30], one can then write generators of (2.1) acting on the extended coset (2.2) as Then, odd transformations corresponding to these supercharges are given by According to [30], one can define the integration measure as It is easily checked to be invariant under the transformations (2.5). Note that the Hamiltonian in (2.4) is not a pure time derivative. One could pass to the new parametrization of superspace as

Chiral superspaces
The supergroup SU(4|1) admits two mutually conjugated complex supercosets which can be identified with the left and right chiral subspaces: The left coordinate t L is related to the real time coordinate t via Then we check that the left chiral space ζ L is closed under the supersymmetry transformations The invariant left chiral measure is defined as One can consider reduction of the superspace (2.2) to the SU(2|1) superspace. It is performed on the superspace coordinates (2.3) as Limiting to the ǫ 1 and ǫ 2 transformations in (2.5), we obtain the reduced SU(2|1) supersymmetric transformations which coincide with those found in [6]: Respectively, the superalgebra (2.1) contains as a subalgebra the extended su(2|1)+ ⊃ u(1) superalgebra: (2.14) Here, SU(2) generators of SU(2|1) are defined as The combination H + m 2 F can be identified with the internal U(1) generator of SU(2|1), while F becomes an external R-symmetry U(1) generator.
The explicit expressions for the covariant spinor derivatives D k ,D k corresponding to the basic real coset of SU(2|1) defined in [8] and parametrized by the coordinates (2.12) with the transformation properties (2.13) are given by In what follows we will avoid using the explicit form of the SU(4|1) counterparts of these derivatives, though they can be straightforwardly constructed by applying the standard coset (super)space machinery.
3 SU(4) covariant formulations of (8, 8, 0) multiplet in flat N = 8 supersymmetry Prior to the discussion of the superfield description of the root (8, 8, 0) multiplets in SU(4|1) supersymmetry, we will consider SU(4) covariant form of its defining constraints in the standard flat N = 8 superspace, bearing in mind that the deformation to SU(4|1) mechanics must respect R-symmetry SU(4) . Such constraints can be written in the two superfield forms, both preserving not only SU(4) but also a non-manifest SO(8) R-symmetry. 4 In the first formulation one deals with a chiral superfield Φ and an antisymmetric tensor superfield Y IJ satisfying the constraints 5 In general, flat constraints defining the multiplet (8,8,0) can be given many equivalent forms. For instance, in [15], they were written in SU(2) × SU(2) × SU(2) × SU(2) covariant form. The common feature of all these formulations is the hidden covariance of the constraints under the full R-symmetry group of N = 8 superalgebra, the group SO (8). 5 For further use, we introduce the antisymmetric tensor ε IJ KL ≡ ε [IJ KL] , such that where the flat covariant derivatives are defined as It is straightforward to check that (3.1) is covariant under the non-manifest SO(8)/SU(4) symmetry transformations realized as where the antisymmetric complex 4 × 4 matrix accommodates just 12 real parameters of the coset SO(8)/U(4) and λ is the real U(1) ∼ SO(2) parameter. One can check that indeed Another form of the SU(4) covariant superfield description of the multiplet (8, 8, 0) involves the general superfield V I which is subject to the constraints The non-manifest SO(8)/SU(4) transformations of V I leaving covariant the system (3.7) are written this time as These transformations, together with the transformations of the covariant derivatives (3.3), preserve the constraints (3.7). One can also see that It is rather easy to check that the constraints (3.1) leave in the bosonic sector of Φ, Y IJ just the complex bosonic field φ(t) and tensorial field y IJ (t) which are first components of these superfields and transform as 1 and 6 of SU(4) . The physical fermions are defined as D I Φ| θ=0 and transform as 4 of SU(4). In the case of the constraints (3.9) the SU(4) assignment of the physical fields changes to the opposite: the physical bosons are the first components of V J and transform as 4 , while fermions are defined asD K V K | θ=0 , D KV K | θ=0 , D [I V J] | θ=0 and transform as 1 ⊕ 1 * ⊕ 6 . Thus, two (8, 8, 0) multiplets have "inverted" SU(4) contents: the contents of bosons and fermions of the first version coincide with those of fermions and bosons in the second one.
In order to better understand the interplay between the two forms of the (8, 8, 0) multiplet, we note that the fermionic superfield D I Φ transforms precisely as V I . It is easy to check that it satisfies the constraints (3.7) as a consequence of (3.1). Analogously, the fermionic superfields − 2 √ 2 D I V J and D KV K possess the same transformation properties as Y IJ andΦ , respectively. It is also straightforward to check that such fermionic superfields satisfy (3.1) as a consequence of (3.7). In other words, by the first multiplet one can construct the "derivative" fermionic multiplet satisfying the Grassmann-odd version of the second multiplet constraints (3.7). After establishing this correspondence, we could consider (3.7) for some new independent Grassmann-even superfield V I and so come to the system (3.7) as an alternative description of the (8, 8, 0) multiplet with the same Grassmann parities for the component fields as in the first version, but with "inverted" SU(4) assignments of these components. Its fermionic "derivative" satisfies the constraints (3.1).
This interplay between two (8, 8, 0) multiplets resembles a similar feature of "mirroring" of (4, 4, 0) multiplets in the standard (flat) N = 4 mechanics [25], [26]. The bosonic and fermionic components of the mutually mirror (4, 4, 0) multiplets form doublets with respect to different SU(2) factors of the full SO(4) R-symmetry group and are equivalent up to switching the roles of these two commuting SU(2) groups. However, there is an essential difference. In the N = 4 case the bosonic fields of the mutually mirror (4, 4, 0) multiplets are doublets of different R-symmetry SU(2) groups (the same is true for fermionic fields). As is seen from (3.6) and (3.9), in the N = 8 case the relevant fields form 8-dimensional irreps of the same full R-symmetry SO (8) and differ only in their assignments with respect to the fixed U(4) ⊂ SO (8). So these two descriptions are associated with different embeddings of U(4) into SO (8). The first version corresponds to splitting SO(8) → SO(2) × SO(6) and representing the SO(8)-multiplet of superfields as a sum of SO(2) and SO(6) vectors. Then SU(4) is identified with SO(6), the additional R-symmetry U(1) with SO(2), while Φ and Y IK with the corresponding SO(2) and SO(6) vectors. The second version corresponds to splitting SO(8) → SO(4) × SO(4) ′ and representing the relevant SO(8) superfield set as a sum of two 4-vectors. The diagonal SO(4) is identified with the "minimally embedded" SO(4) ⊂ SU(4), and two 4-vectors are joined into a complex fundamental spinor V I of SU (4).
Actually, the hidden SO(8) symmetry reveals the triality [22] between bosonic fields, fermionic fields and covariant derivatives. This triality interrelates the three irreducible fundamental representations of SO(8), viz. the vector representation and two spinorial ones. 6 All three representations can be written in the SU(4) × U(1) ∼ SO(6) × SO(2) notation [23] as where the subscript index refers to the U(1) charge. Comparing this with the U(4) assignments of the bosonic and fermionic fields of the (8, 8, 0) multiplets, as well as of the covariant derivatives, we observe that just these SO (8) representations are realized on the quantities in question. Supposing that the roles of two spinor representations can be switched, in flat N = 8, d = 1 supersymmetry we can introduce yet a third multiplet (8, 8, 0) living on a different superspace, with the covariant derivatives defined as 11) and so belonging to the vector representation. However, an SU(4) covariant formulation of this third (8,8, 0) multiplet is beyond our purpose because the SU(4|1) covariant derivatives are SU(4) spinors by definition. So, this third option does not admit a generalization to SU(4|1) supersymmetry, in contrast to the first two.
In the case of the constraints (3.1), the bosonic fields belong to the SO(8) vector representation, and the fermionic fields form SO (8) spinor. For the multiplet given by (3.7) the picture is reversed, that is, the bosonic fields form an SO(8) spinor and the fermionic fields are combined into SO(8) vector. So, from the standpoint of SO(8) R-symmetry, due to the triality property, both (8,8,0) multiplets can be considered as equivalent, once the spinorial representation of the covariant spinor derivatives has been fixed and one deals with SO(8) invariant actions for these multiplets (for more detail, see Section 5.4).
The crucial point of the equivalence just discussed is the hidden SO (8) where Φ is a chiral superfield and Y IJ is an antisymmetric tensor superfield. In what follows, we avoid calculation of the deformed covariant derivatives D I ,D J (they in general involve complicated U(4) connection terms) and consider the multiplet (8,8,0) in the chiral superspace description, harmonic superspace description and SU(2|1) superfield approach.

Chiral superfield
We consider the chiral superfield Φ given by the general θ-expansion The superfield Φ transforms as a singlet of the stability subgroup SU(4) , i.e. δ su(4) Φ = 0 . Taking into account (2.10), we can find the transformations of its components under the odd generators: The general supersymmetric action can be written as a sum of integrals over chiral subspaces [18], [13] as where the overall coefficient −1/4 is chosen for further convenience. The component form of this SU(4|1) invariant is found to be This invariant does not display the kinetic term of the fields in (4.2) and so must be treated as a kind of "pre-action" for the (8,8, 0) multiplet. The genuine action appears after imposing some extra SU(4|1) covariant conditions on the components in (4.2). Of course they should follow from the rest of the superfield constraints (4.1), but it is easier to guess their form directly at the component level, requiring the final field content to be (8,8,0) and resorting to the SU(4|1) covariance reasonings.
In this way we find that the components of the chiral superfield (4.2) must be subjected to the following additional constraints The odd SU(2|1) transformations are realized on this minimal set of fields as: They are consistent with the transformations (4.3) and leave invariant the constraints (4.6).

The final action
Substituting the constraints (4.6) into the pre-action (4.5), we find the correct component Lagrangian in the form We observe that the complex fields φ parametrizes a special Kähler (SK) manifold with the metric

Supercharges
The matrix models based on the multiplet under consideration, in the case of the simplest target space metric g = 1 (i.e for the free model), were studied in [28]. Here, we consider a one-particle model generalized to the general SK metric (4.9) and find the relevant classical SU(4|1) supercharges. Poisson (Dirac) brackets are written as Then the Noether supercharges are given by Taking into account the brackets (4.10), these supercharges close on the following bosonic generators 13) in full agreement with the superalgebra (2.1). The quantum version of these SU(4|1) (super)charges can be straightforwardly constructed and will be presented elsewhere.

Harmonic superspace description
We consider the harmonic coset of SU(4|1) with the harmonic part Defining the harmonic projections of the SU(4|1) Grassmann coordinates as one can find that they transform as where We observe the existence of the analytic subspace closed under the SU(4|1) supersymmetry Its integration measure is given by The only harmonic derivative D (+2)i a preserving the analytic subspace reads where The remaining harmonic covariant derivatives prove undeformed: One can check that and

Analytic harmonic superfield
The relevant analytic harmonic superfield is defined by the conditions and it transforms as It can be obtained by the "harmonization" of the superfield Y IJ satisfying the constraints These constraints in fact define the multiplet (6, 8, 2). On the other hand, they are part of the full set of the constraints (4.1) defining the multiplet (8, 8, 0) . Indeed, the solution of (4.25) reveals the field content (6,8,2), where The component fields transform as

Invariant action via harmonic superspace
Introducing the shifted superfield Calculation of all terms in harmonic superspace is rather complicated. We skip all these calculations and write the full component Lagrangian (4.50) in the next subsection by employing SU(2|1) superfields.
The second solution (4.48) gives the Lagrangian L SO(6) = g 2 φφ + 1 2ẏ The third solution (4.49) exhibits an invariance under the maximal R-symmetry group SO (8) and produces the component Lagrangian (4.52)

Superconformal symmetry
Redefining the component fields in (4.52) as we eliminate all the deformed terms proportional to m and write the Lagrangian in SO (8) invariant formulation: In the closure of these transformations, we obtain superconformal algebra osp(8|2) spanned by 16 supercharges and 31 bosonic generators (see Appendix D), 9 where the conformal Hamiltonian H conf is defined as The generators F IJ andF IJ produce SO(8)/U(4) transformations realized as To avoid calculation of the deformed covariant derivatives D I andD J , we instead consider harmonization of part of these constraints, viz.
with the rest of constraints being solved at the component level.

Harmonic superspace
The option for harmonic superspace relevant to the given case uses the harmonic variables on SU(4)/[SU(3) × U(1)] [32]. The set of these harmonic variables is given by u , where the index α = 1, 2, 3 refers to the SU(3) fundamental representation. The harmonics satisfy the following unitarity and unimodularity conditions: As in the previous case, we define the new coordinates is closed under the transformations (5.5). Its integration measure transforms as δ dζ The harmonic derivatives are found to be where The harmonic analytic superfieldV (+3) defined on (5.7) satisfies the harmonic constraints living on the full harmonic superspace (5.4) . Here we consider just the superfield V (+3) treated as an unconstrained deformed harmonic superfield satisfying the analyticity conditions (5.13) . The rest of constraints onV (+3) will be imposed below "by hand" at the component level, like in the previous cases.
The general expansion ofV (+3) reads where Taking into account the transformation rule the superfieldV (+3) transforms as This superfield transformation law amounts to the following component transformations From the transformation properties ofV (+3) one can draw the conclusion that the construction of a "pre-action" similar to (4.5) cannot be performed within the analytic harmonic superspace. We conjecture that such a construction could become possible after taking account of the additional set of c constraints defining the multiplet (8,8,0). Then the action can probably be constructed in the full harmonic superspace approach (see [19]). At the component level, the rest of the constraints (5.1) impose the relations The final form of the deformed transformations is

Invariant Lagrangian
The general SU(2|1) invariant action is written as Requiring it to be SU(4) invariant produces the following conditions: The unique solution of these equations is given by Here, the terms with the constants c 1 and c 2 do not affect the metric G, since it is a harmonic function. One can check that these terms drop out of the component Lagrangian, which is finally written as

Superconformal symmetry
By analogy with the Section 4.6, one can redefine the component fields as z I → z I e −imt/4 ,z I →z I e imt/4 , χ → χ e imt/2 ,χ →χ e −imt/2 , (5.37) after which the Lagrangian (5.35) becomes an even function of m. As a result, we obtain OSp(8|2) superconformal Lagrangian that is equivalent to (4.54): We see that the Lagrangians (4.54) and (5.38) have conformally flat metrics g 3 and G which both depend on the quadratic SO(8) invariants of the same power −3. The fields z I andz J can be reexpressed, by a linear transformation, through the bosonic fields y I ′ J ′ , φ andφ of the first multiplet (8,8,0), where I ′ and J ′ label the fundamental representation of a different SU(4) ′ subgroup of the SO(8) symmetry, such that it intersects with the first SU(4) in a common SU(3) subgroup. After an analogous linear transformation of the fermionic fields, the Lagrangian (5.38) will coincide with (4.54). So both superconformal Lagrangians are indeed equivalent. This feature of equivalence of (8, 8, 0) multiplets in the presence of exact SO(8) symmetry was already noted in the end of Section 3.
As for further applications of these results, the most appropriate arena might be provided by supersymmetric matrix models (see, e.g., [35], [36], [37]). These possess SU(4|2) invariance, hence multi-particle mechanics based on SU(2|2) ⊂ SU(4|2) or SU(4|1) ⊂ SU(4|2) may appear as some truncation of such matrix models. The matrix models studied so far lead to free worldline multiplets and actions. Our approach allows one to generate nontrivial interactions, which hopefully may be interpreted as effective actions with quantum corrections taken into account. An important ingredient of matrix models is a gauging of appropriate isometries by non-propagating gauge multiplets. To promote this to the SU(4|1) superfield language, one needs to define suitable gauge superfields generalizing those used in [38], [39] or [10].
Another problem for the future is finding an action including both types of deformed (8, 8, 0) multiplets and inquiring the ensuing target-space geometry.

A Some calculations
Here we collect the necessary identities for calculation of the function L (+4) in the invariant action (4.32). We represent it as an infinite series: All the identities below are given up to terms with a total harmonic derivative D (+2)i a . In addition, one must take into account the definitions (4.31). Each term in the variation of the series (A.1) also contains transformations compensating the measure transformations (4.19), i.e.
For calculating the component Lagrangians, we use the following identities (up to total harmonic derivatives): IJẏ The resulting Lagrangian is an even function of m and so is superconformal. Its form is typical for Lagrangians with the trigonometric realizations of superconformal groups. It can be shown that the superconformal group of (B. The supercharges Q I ,Q J together with the generators L I J and H = H conf + m 2 F form the subalgebra su(4|1) + ⊃ u(1) in osp(8|2), with F being an additional external R-symmetry U(1) generator. The second set of SU(4|1) supercharges S I ,S J extends this subalgebra to the full superconformal algebra osp(8|2). The latter involves twelve additional R-symmetry generators F IJ ≡ F [IJ] ,F IJ ≡F [IJ] which, together with the U(4) generators L I J , F , form the full R-symmetry algebra o (8). Additional conformal generators areT , T , such that three bosonic generators H conf ,T and T constitute the conformal d=1 subalgebra o(2, 1).
Actually, the parameter m drops out from the superconformal algebra after performing redefinitions similar to those made in [8] for the case of the N = 4, d = 1 superconformal algebra D(2, 1; α) .