${\cal N}{=}4$ Supersymmetric $d=1$ Sigma Models on Group Manifolds

We construct manifestly ${\cal N}=4$ supersymmetric off-shell superfield actions for the HKT $d=1$ sigma models on the group manifolds U(2) and SU(3), using the harmonic $d=1$ approach. The underlying $({\bf 4, 4, 0})$ and $({\bf 4, 4, 0})\oplus({\bf 4, 4, 0})$ multiplets are described, respectively, by one and two harmonic analytic superfields $q^+$ satisfying the appropriate nonlinear harmonic constraints. The invariant actions in both cases are bilinear in the superfields. We present the corresponding superfield realizations of the U(2) and SU(3) isometries and show that in fact they are enlarged to the products U(2)$\times$SU(2) and SU(3)$\times$U(2). We prove the corresponding invariances at both the superfield and component levels and present the bosonic $d=1$ sigma model actions, as integral over $t$ in the U(2) case and over $t$ and $SU(2)$ harmonics in the SU(3) case. In the U(2) case we also give a detailed comparison with the general harmonic approach to HKT models and establish a correspondence with a particular action of the off-shell nonlinear multiplet $({\bf 3, 4, 1})$. A possible way of generalizing U(2) model to the matrix U(2n) case is suggested.


Introduction
The N = 4 supersymmetric d = 1 sigma models based on the multiplets with the off-shell content (4, 4, 0) 1 are known to lead to the HKT ("Hyper-Kähler with torsion") geometry in the bosonic target space and, as a particular case, the HK ("Hyper-Kähler") geometry [2] - [17]. The intrinsic geometries of supersymmetric sigma models are displayed in the clearest way in the appropriate superfield formulations, with all supersymmetries being manifest and off-shell [18] - [21]. For HKT d = 1 sigma models with a target space of real dimension 4n such a general formulation was proposed in our paper [22], where it was shown that they are described by 2n analytic harmonic superfields q +a , a = 1, . . . 2n, (with the appropriate reality conditions) subjected to the nonlinear harmonic constraint 2 D ++ q +a = L +3a (q + , u ± ) , (1.1) where L +3a is an arbitrary analytic function carrying the harmonic charge +3. The general superfield action is the following integral over the total N = 4, d = 1 harmonic superspace, The analytic function L +3a (q + , u) and the general function L(q + , q − , u ± ) are two independent HKT "prepotentials" fully characterizing the given HKT geometry. The general expressions for the target metric and torsion were given and a few interesting particular cases were discussed. It was shown that the general HK geometries arise for the simplest choice L = q +a q −b Ω ab , L +3a (q + , u) = ∂L +4 (q + , u) ∂q + a , (1.3) where Ω ab = −Ω ba , Ω ab Ω bc = δ c a , is a constant symplectic metric, while L +4 is the renowned Hyper-Kähler potential [20], [19]. If L +3a (q + , u) in (1.3) is arbitrary, i.e. is not subject to the second condition but L remains quadratic, the relevant geometries are strong HKT, that is, such that the corresponding torsion is closed. This class of HKT geometries is the same as in the d = 2 N = (4, 0) heterotic sigma models and the analytic prepotential L +3a (q + , u) coincides with the one introduced in [13]. The case of general HKT potentials L +3a and L amounts to the "weak" HKT geometry, in which there is no closedness condition on the torsion two-form. A more detailed geometric analysis of the two-potential formulation of the HKT geometry was undertaken in a recent paper [25].
While for the HK manifolds a few concrete examples of N = 4, d = 1 sigma models were presented (see, e.g., [24], [26] - [28]), not too many explicit examples of this kind are known for the HKT case. It still remains to find the precise form of the potentials for the HKT manifolds known in the literature, e.g., for the inhomogeneous example given in [10]. On the other hand, there exists a wide class of homogeneous group-manifold strong HKT metrics associated with those groups which admit a quaternionic structure [7,8]. The full list of such groups was given 2 SU(2)×U(1) group manifold 2

(2.4)
The invariant sigma-model type action is written as an integral over the whole harmonic N = 4 superspace (the particular normalization factor was chosen for further convenience). Using the fact that the integral of an analytic superfield over the full superspace is vanishing, as well as the constraints (2.1) together with the evident relation D −− λ −− = 0, it is easy to check that the superfield action (2.5) is indeed invariant under the SU(2) transformations (2.2) up to a total harmonic derivative in the integrand. Though the action is bilinear in the involved superfields, in components it yields a nontrivial nonlinear SU(2)×U(1) group manifold sigma-model action.
Combining the superfields ω and N ++ into yet another analytic superfield q +a as the constraints (2.1) and the action (2.5) can be cast into the standard form [22] D ++ q +a = L +3a , L +3a := u −a (u + · q + ) 2 , (2.8) Based on the results of [22], this reformulation immediately implies that the bosonic target space of the system under consideration is "strong HKT", that is a HKT manifold with a closed torsion 3-form. At the component level, this property is well known and is common to all group manifold HKT models [7]. In the present case we deal with the simplest non-trivial example of such sigma model corresponding to the four-dimensional target space SU(2)×U (1).
In what follows, it will be more convenient to deal with the superfields ω and N ++ . We also note that one more evident symmetry of the constraints (2.1) and the actions (2.5),(2.9) is the standard harmonic SU(2) H realized on the harmonic variables as It induces linear SU(2) rotations of all component fields with respect to their doublet indices.
We will see later that its presence ensures U(1)×SU(2)×SU(2) symmetry of the nonlinear sigma model appearing in the bosonic sector of the action (2.5).
In fact, in the present case SU(2) H is a combination of the standard automorphism group of N = 4 supersymmetry in the q +a representation and of the so called Pauli-Gürsey SU(2) P G , which commutes with supersymmetry and acts on the doublet index of the superfield q +a . Such a non-uniqueness of the automorphism SU(2) group was mentioned in [21]. It is with respect to such "shifted" SU(2) H that the superfield projections defined in (2.7) are singlets. Though the action (2.9) (and (2.5), up to a total harmonic derivative) is formally invariant under SU(2) P G taken alone, the constraints (2.8) (or (2.1)) are covariant only with respect to the "shifted" SU(2) H .

Bosonic component action and its symmetries
In the limit where all fermionic fields are suppressed, the analytic superfields N ++ and ω have the following θ expansion where all component fields are defined on the coordinate set (t, u ± i ). The harmonic superfield constraints (2.1) imply the following ones for the component fields (2.12) After rather simple manipulations which make use of the completeness condition for harmonics, one solves all these equations, except for the last one, as , (2.13) where the fields b, c and a (ik) depend only on t, but not on harmonics. After substituting these expressions into the remaining equation for σ −2 and taking the harmonic integral of both sides, we obtain a relation between the fields b(t) and c(t) (2.14) Taking into account the last equation, we are left with a set of four d = 1 fields b(t), a (ik) (t) parametrizing (as it will become clear soon) the target SU(2)×U(1) group manifold. The harmonic integrals entering (2.14) can be explicitly carried out, giving us a simple expression for the field c(t) in terms of these basic fields. We dropped in (2.14) the term which is vanishing upon integration over harmonics. The bosonic component action following from the superfield one (2.5) is After substituting the solutions (2.13) into L(t, u), integrating by parts with respect to ∂ t and ∂ ++ , as well as using the constraint for σ −2 , the Lagrangian L(t, u) can be presented in the following form (2.17) It remains to calculate the harmonic integrals in (2.14) and (2.17). Using the formulas (B.2) -(B.5), after some algebra (2.14) can be reduced to Next, as explained in Appendix A, the first term in the square bracket in (2.17) gives zero contribution to the harmonic integral in (2.16). The harmonic integral of the second term can be represented as which is the most general structure compatible with the SU(2) covariance. Substituting the second expression into the first one, we immediately find that the latter is also vanishing. Thus only the first line in (2.17) contributes. Using the formulas (B.12) -(B.14), it is easy to compute the remaining harmonic integrals and to present the final answer for the Lagrangian L(t) in (2.16) as Here, the fieldb is invariant under the nonlinear realization of SU(2) acting on the second (sigma-model) piece of (2.19), while a (ik) are just Goldstone fields supporting this nonlinear realization. Thus (2.19) describes a nonlinear d = 1 sigma model on the group manifold SU(2)×U(1), parametrized by the d = 1 fields a (ik) (t) andb(t) . The U(1) factor acts as constant shifts ofb(t) , while nonlinear SU(2) transformations of a ik can be found by considering the component-field realization of the SU(2) symmetry originally defined on the superfields ω and N ++ .
Starting from the transformation of n ++ , 20) and using the first constraint in (2.12), one obtains an equation which determines δa (ik) : After some algebra, making use of the completeness relation for harmonics, one finds It is easy to check that the Lie bracket of these transformations is again of the same form, i.e. we deal with a realization of SU(2) on itself by left (or right) multiplications. The fields a (ik) (t) just provide a particular parametrization of the SU(2) group element. Analogously, starting from the SU(2) transformation of ω 0 , and rewriting this variation in terms of the solution for ω 0 in (2.13), where the signs chosen ensure that the relevant Lie bracket parameter is composed in the same way as in the cases (2.22) and (2.27). We denoted the transformation parameter in (2.28) by the same letter as in (2.27), hoping that this will not give rise to confusion. The mutually commuting SU(2) transformations (2.22) and (2.28) in fact amount to the left and right shifts on the same SU(2) group manifold. It is easy to see that the diagonal SU(2) subgroup in the product of these two commuting nonlinearly realized SU(2) is just SU(2) H (2.27), so the bosonic sigma-model in (2.19) describes a nonlinear realization of this product on the coset manifold SU(2)× SU(2)/ SU(2) H , i.e., the 3-sphere S 3 . The field b transforms under the transformations (2.28) as while the redefined fieldb is inert as in the case of λ-transformations. At the superfield level, the τ transformations are generated by transformations of the same form as (2.2), modulo the common sign minus before their r.h.s. and the appropriate addition of the linear harmonicinduced SU(2) H transformations of N ++ and ω. Thus the full symmetry of the model under consideration, both on the superfield and the component levels, is the direct product U(1)×SU(2)×SU(2) 4 . It is worth pointing out a crucial difference between the left and right SU(2) symmetries. The left one (2.2) (as well as the constant shift (2.6)) commutes with supersymmetry and so defines the triholomorphic (or "translational" ) isometry of the model under consideration. It is realized as a subgroup of general analytic reparametrizations of the superfields q +a . The right SU(2) involves the R-symmetry SU(2) H as its essential part and so does not commute with supersymmetry. For what follows it is instructive to give the realization of the triholomorphic (left) SU(2) in terms of q +a : with ∂ +b := ∂ ∂q +b . At this point, let us recall the existence of two distinguished connections on S 3 as a textbook example of a parallelizable manifold (see, e.g., [29]). One of them is the standard Levi-Civita torsionless connection and it corresponds to treating S 3 as a symmetric Riemannian coset space SO(4)/SO(3). Another connection involves a closed torsion and has zero curvature, which corresponds to the identification of S 3 with the non-symmetric SU(2) group manifold itself. This torsionful connection is what is called a "Bismut connection" (see [25] and references therein) and it is the one ensuring the SU(2)×U(1) manifold to be HKT. Just with respect to the Bismut connection is the triplet of quaternionic complex structures covariantly constant. In more detail, these geometric issues are discussed in section 3.

Reduction to nonlinear (3, 4, 1) multiplet
The last topic of the present section is the gauging of the U(1) symmetry (2.6) along the line of ref. [24]. We promote the constant parameter λ to an arbitrary analytic superfield parameter, λ → λ(ζ, u), ω ′ = ω + λ , N ++′ = N ++ , and introduce two abelian harmonic connections V ±± , (2.32) The constraints (2.1) and the superfield action (2.5) are covariantized as The modified constraints and the action are, respectively, covariant and invariant under all rigid SU(2) transformations discussed in this section. While checking this, an essential use of the flatness condition (2.32) is needed. Lets us show that the system so constructed describes the appropriate SU(2) invariant sigma model of the single nonlinear multiplet N ++ . To this end, we choose the N = 4 supersymmetric gauge In this gauge, the second constraint in (2.33) just amounts to identifying while the action (2.34) becomes Using (2.32), V −− can be expressed through V ++ = N ++ by the well-know formula [21] involving the harmonic distributions where V −− is taken at the harmonic "point" u ± i , while N ++ in the r.h.s. is taken at the point u ± 1i . Then the action (2.37) can be written solely in terms of N ++ (in the central basis) as (2.39) The component bosonic action can be obtained in a simple form with the alternative choice of the Wess-Zumino gauge for V ±± (2.40) In this gauge, the fourth of the component constraints (2.12) is modified as After some simple algebra, one finds that the only modification of the bosonic Lagrangian (2.19) is the replacementḃ →ḃ − A . (2.43) Keeping in mind that under local U(1) transformationb ′ =b + λ(t), one can always choose the gaugeb = 0, after which the relevant bosonic Lagrangian will reduce to The field A proves to be just the auxiliary field of the nonlinear (3, 4, 1) multiplet described by the superfield N ++ . It fully decouples in the bosonic action (2.44) leaving us with the S 3 non-linear sigma model for 3 physical bosonic fields. So, applying the general gauging procedure of ref. [24] to the superfield Lagrangian (2.5) (or (2.9)) of the U(2) HKT model, we recovered a particular Lagrangian of the nonlinear (3, 4, 1) multiplet, with the S 3 nonlinear sigma model in the bosonic sector. 5

U(2) model as an example of HKT geometry in the harmonic approach
Here we recover the metric and the torsion associated with the U(2) group manifold sigma model directly from the general geometric formalism of nonlinear (4, 4, 0) multiplets pioneered in [22] and further elaborated on in [25]. Since we will be interested in the bosonic target geometry, we omit fermions and put q +a = f +a altogether (in particular, in (2.30), (2.31)). The bosonic fields σ and σ −2 defined in (2.11) are of use only while deriving the bosonic sigma model action from the superfield one. In the general formalism of ref. [22] these additional quantities play no role, as they are eliminated there, from the very beginning, by the harmonic superfield constraint. The basic object of the geometric formalism is the "bridge" from the target λ frame parametrized by the coordinates f +a (t, u) and f −a (t, u) := ∂ −− f +a and harmonics u ±i to the τ frame parametrized by the harmonic-independent coordinates, in our case the d = 1 fields b(t) and a ik (t) defined in the previous subsection. The bridge is a 2 × 2 complex matrix M b a subject to the equation The underlined doublet indices refer to the τ frame, and on them some harmonic-independent τ group acts. In the present case the latter is realized as SU (2) rotations with the parameters λ a b , λ a a = 0 (these parameters are the same as in (2.2), (2.30) and (2.31)), accompanied by some rescaling, see eqs. (3.7), (3.8) below. On the non-underlined indices another realization of the same SU(2) is defined, as a particular isometry subgroup of general harmonic analyticitypreserving target space reparametrizations.
Eq. (3.1) also implies that where the non-analytic connection E −2c a is related to E +2c a by the "harmonic flatness condition" Fortunately, eq. (3.1) has the simple solution can be easily restored from (3.3): This expression can be checked to satisfy eq. (3.3) 6 . It is also straightforward to check that under the isometry (2.2), (2.30) and (2.31) the bridge M b a has the correct geometric transformation law (as anticipated above) where ∂ +a δ λ q +c is given in (2.31) and are the harmonic-independent parameters of the τ group the generical definition of which is given in [22].
Having bridges at hand, we can calculate some "semi-vierbeins" which are building blocks of all geometric quantities in the approach of [22]. They transform the tangent space objects to the world objects and, being specialized to the case under consideration, are defined by 7 Simple calculations yield (3.10) 6 In fact, the same expression for E −2c a can be directly derived by solving (3.3). 7 We changed some signs as compared to [22].
The vierbein coefficients satisfy the standard orthogonality conditions. It can be checked that, under the triholomorphic SU(2) isometry, they transform as covariant and contravariant tensors with respect to non-underlined indices, with the infinitesimal parameters ∂ (ik) δa (jl) , and as spinors with the parameters (3.8) with respect to the underlined doublet indices a, b, . . .. On the indices i, k, . . . the standard automorphism SU(2) group acts. Now we are prepared to compute the basic τ frame geometric objects of the U(2) model by specializing the general expressions of ref. [22], [25] to this case.
Metric. The target metric components are calculated by the general formula Using the harmonic integral formulas from Appendix B, it is easy to find (3.13) Substituting this into (3.11), we find (3.14) which precisely matches with the sigma-model Lagrangian (2.19). From now on, it will be convenient to pass to the 3-vector notation, All world tensor indices (ia) are replaced by the 3-vector ones by attaching the matrix factors i √ 2 (σ m ) ia . Then the metric takes the form Torsion. The torsion in the tangent space notation is expressed as and In our case G [cb] := ε cb G and F [c d] = ε cd , so (3.18) is simplified to (3.20) Next, E +d ab = 2u −d u + a u + b and, by solving the harmonic equation in (3.19), we find It is also easy to compute det M = (1 + a +− ) 2 .

(3.24)
It is easy to check the full antisymmetry of this expression with respect to permutation of the pairs of indices. Next we project this expression to the world-index representation by contracting it with the "semi-vierbeins" (3.10). We find that all its components containing the index 4 vanish and only the projections on the 3-vector subspace are non-zero. Substituting the triplet indices by the vector ones by the rule (3.15), we finally obtain the only non-zero torsion component as It is evidently closed as it should be for the group-manifold HKT models [7].
Bismut connection. The Bismut connection on 4q dimensional HKT manifold is defined aŝ 26) where Γ N M S is the standard symmetric Levi-Civita connection and C P M S is the torsion tensor. With respect to it, the triplet of the corresponding quaternionic complex structures is covariantly constant. Since in our case (with q = 1) C 4mn = 0, the only non-vanishing components of the Bismut connection areΓ n ms = Γ n ms + (3.27) The connection Γ n ms for the metric defined in (3.16) is easily calculated to be Γ n ms = − 1 2 (3.28) The triplet of complex structures in the tangent space representation is given by where σ A , A = 1, 2, 3 , are Pauli matrices. Transforming it to the world indices by contracting with the vierbeins (3.10) and then passing to the indices 4, m through the correspondence (3.15), we explicitly find Despite the somewhat involved form of these expressions, it can be checked that they form the algebra of imaginary quaternions and possess the correct tensor transformation rules under the SU(2) isometry After some effort it can be also checked that they are covariantly constant with respect toΓ m ns : One more important property of the Bismut connectionΓ m ns (specific just for the groupmanifold HKT models [7]) is that its Riemann curvature vanishes (the property pertinent to the parallelized manifolds [29]), Obata connection. Besides the Bismut connection, one more important geometric object of HKT models is the Obata connection [31]. It is a symmetric connection with respect to which the triplet of complex structures is still covariantly constant (but not the metric, as distinct from the Levi-Civita connection). For HK manifolds it coincides with the Levi-Civita connection, but for HKT manifolds it does not.
In the harmonic approach to HKT geometry, Obata connection in the tangent space representation was defined in [25]. It is the following deformation of the Bismut connectioñ Γ ia kb lc = Γ ia kb lc + 1 2 C ia kb lc + ∆C ia kb lc =Γ ia kb lc + 1 2 ∆C ia kb lc , The full symmetry ofΓ ia kb lc in the last two pairs of indices can be proved using the cyclic identity [22] ∇ kc G [a b] + cycle a, b, c = 0 .
It is also of interest to calculate the curvature of the Obata connection. Surprisingly, it turns out to vanish: According to ref. [6], the vanishing of the curvature of the Obata connection for some HKT manifold signals that the three corresponding complex structures are simultaneously integrable and so there exists a coordinate frame where all these can be chosen constant. In the harmonic approach, this amounts to the conjecture [25] that there exists a redefinition of the original q + superfields such that they satisfy a linear harmonic constraint 8 . This interesting issue will be studied elsewhere.

SU(3) group manifold 4.1 Superfield consideration
In this case, beside the superfields N ++ , ω with four physical bosonic fields parametrizing SU(2)×U(1), we need one more analytic superfield in order to accommodate four extra bosonic fields which complete the SU(2)×U(1) group manifold to that of SU(3). The natural choice is the complex analytic superfield (q + ,q + ), (q + ) = −q + where "tilde" stands for the generalized conjugation defined in [19], [21] and becoming the ordinary conjugation on the component fields. Thus in the SU(3) case we operate with the following set of analytic N = 4, d = 1 superfields N ++ , ω , q + ,q + . The extra fields appear as first terms in the harmonic expansion of q + ,q + , q + = f i u + i + . . . ,q + =f i u + i + . . . ,f i = (f i ) , i.e. they are doublets with respect to the automorphism SU (2). The corresponding new constant group parameters are defined in a similar way Using the trial and error method, we have eventually found the following self-consistent set of the coset SU(3)/U(2) transformations for q + ,q + , N ++ : Here, α = α 0 + iα 1 and α 1 is, for the time being, an undetermined free real parameter. These transformations close on the SU(2)×U(1) ones: Note the necessary modification of the SU(2) transformation rule for N ++ as compared to the pure SU(2)×U(1) case. Also note the presence of a non-trivial U(1) phase transformation in the closure on q + (the closure transformations ofq + can be obtained by conjugation of δ br q + ). The non-zero real parameter α 0 can be fixed at any value via a simultaneous rescaling of q + and of the coset parameters ξ i . It remains to quote the corresponding transformation properties of the superfield ω. Its coset SU(3)/U(2) transformation reads where γ 1 is yet another undetermined real parameter. The closure of these transformations is the same as on the superfields q + ,q + , N ++ , with The parameters α 1 and γ 1 cannot be fixed from considering the closure of the above transformations: they form an su(2) algebra irrespective of the choice of these parameters. Surprisingly, they are fixed when constructing the invariant action. But before discussing this issue, let us write down the relevant set of the harmonic constraints generalizing and extending (2.1). This set is as follows . It is interesting that the set of constraints (4.8) for the choice α 1 = −γ 1 can be cast in a simpler suggestive form. Introducing Φ ++ := N ++ − α q +q+ ,Φ ++ = N ++ +ᾱq +q+ , (4.9) we can rewrite (4.8) as (4.12) The constraints (4.10) supplemented by the condition can be treated as the basic ones. The constraint (4.12) serves just for expressing the superfield ω (up to the harmonic-independent part) from the known Φ ++ , q + and the conjugated superfields. Note that the set q + , Φ ++ is closed under the SU(3)/U(2) transformations (4.3) and hence under the U(2) ones as well: Thus the set (q + , Φ ++ , u ± i ) can be interpreted as a kind of complex analytic subspace in the harmonic extension of the target SU(3) group manifold (likewise, in the U(2) case the set (N ++ , u ± i ) can be treated as some analytic subspace of the harmonic extension of the U(2) group manifold).
The invariant action should be an extension of the action (2.5) and, following the general reasoning of ref. [22], admit a formulation in the full harmonic superspace as an expression bilinear with respect to the superfields involved and linear in the harmonic derivative D −− . Combining (2.5) withq + D −− q + , we find that requiring it to be invariant, up to a total harmonic derivative, under the transformations (4.3) and (4.6) (and, hence, under their closure), uniquely fixes the ratio of these two terms in such a way that 16) and, what is even more surprising, fixes the constants α 1 and γ 1 in terms of the single normalization constant α 0 as While checking the invariance, an essential use of the harmonic constraints (4.8) was made.
It is instructive to give some technical details of the proof of the SU(3) invariance of the action (4.16). It will be enough to prove the invariance under the ξ transformations, as the rest of SU(3) is contained in their closure. Moreover, it suffices to consider the holomorphic ξ i parts of these transformations as the antiholomorphic ones ∼ξ i are obtained through the tilde-conjugation. So we start from the transformations 19) and the following ansatz for the superfield Lagrangian where κ is some real parameter. The variations of L 1 and L 2 , up to total time and harmonic derivatives and upon using the harmonic constraints (4.8) along with the property that the full superspace integral of an analytic expression vanishes, are reduced to The superfield structures in these variations are independent, so the conditions for the vanishing of δL are The separate vanishing of the real and imaginary parts of these equations uniquely yields that precisely matches with (4.16) -(4.18). It is also easy to directly prove the invariance of (4.16) under the U(2) ⊂ SU(3) transformations and under the transformations of the second U(2) commuting with SU(3) (the second SU(2) transformations are a fixed combination of the first SU(2) ones and of those of the harmonic SU(2), while the second U(1) factor acts just as a constant shift of ω, see eqs. (4.49) -(4.51) below).

Bosonic limit and its symmetries
Here we solve the purely bosonic part of the constraints (4.8) and find the realization of the SU(3) transformations on the physical bosonic variables.
The bosonic core of the involved superfields is as follows N ++ = n ++ +θ +θ+ σ , ω = ω 0 +θ +θ+ σ −2 , q + = f + +θ +θ+ µ − ,q + =f + −θ +θ+μ− , (4.24) where all fields in the r.h.s are defined on the manifold (t, u ± i ), e.g., f + = f + (t, u), etc. The basic bosonic constraints are obtained from the superfield ones by the direct replacement (q + ,q + , N ++ , ω) → (f + ,f + , n ++ , ω 0 ), (4.25) The constraints on these fields literally mimic (4.8), in which one should just put θ = 0 and make the replacements (4.25). It will be more convenient to start from the equivalent constraints (4.10) -(4.12), in which case we obtain where φ ++ = Φ ++ θ=0 . The φ ++ -constraints in (4.26) and (4.27) can be solved as whence (4.31) Then the constraints for f + andf + in (4.26) and (4.27) are solved as The complex field A ik can be divided into real and imaginary parts The real field a ik is just an analog of the field a ik of the U(2) model and it parametrizes the SU(2) part of the SU(3) manifold. The doublet fields f k ,f k , together with the appropriate analog of the singlet field b, complement this SU(2) manifold to the whole SU(3). Using the algebraic constraint (4.28), b ik can be expressed in terms of a ik and f i ,f i : The constraint (4.28), in terms of the harmonic projections, can be also written as In terms of the ordinary fields A ik ,Ā ik , it amounts to Note also the relation (and the analogous relation for the complex-conjugated fields). Some useful relations following from the constraint (4.37) are Using the superfield transformation laws (4.14) and (4.15), we can find the SU(3)/U(2) transformations of the "central basis" fields A ik and f i ,f i : as well as their U (2) transformations (4.48) An interesting point is that, like in the previously considered U(2) case, there exists another SU(2) which commutes with the SU(3) transformations given above. It is realized by the transformations So we come to the conclusion that the invariance group of the system we are considering is the product SU(3)×U (2). Only SU(3)×U(1) in this product commutes with N = 4 supersymmetry and so defines the triholomorphic isometries 9 .
In analogy with the SU(2)×U (1) where X is a 3 × 3 matrix which should be chosen as X = diag(κ, κ, χ) in order to preserve the SU(3) L ×U(2) R symmetry. Here κ and χ are some numerical parameters. At κ = χ, i.e. for X = κ I (3×3) , the symmetry is enhanced to the product SU(3) L ×SU(3) R , while for κ = χ SU(3) R is broken to U(2) R . Just the latter option is expected to be valid in our N = 4 SU(3) model, with the ratio of the parameters χ and κ strictly fixed. In order to check all this, we need the explicit expression for the bosonic invariant action as an integral over t. This form of the bosonic action (with the harmonic integrals explicitly done) will be presented elsewhere.

Solving further constraints
Our ultimate purpose is to find the bosonic component Lagrangian. Besides the constraints on the fields ω 0 , f i ,f i and n ++ , we now need also those for the remaining bosonic components in the θ expansions (4.24), because all such components are involved in the bosonic action.
Using (4.31), it is easy to solve eq. (4.55): is a real harmonic-independent d = 1 field. Also, the general solution of eqs. (4.54) isσ , and c.c. , (4.60) where c = c(t) is a complex harmonic-independent d = 1 field. Note that (4.60) and the definition (4.58) allow one to find Further, taking the harmonic integral of eq. (4.56) and making use of the formulas from Appendix, we find one relation between the fields c and b: It is convenient to redefine After some work, the solution is found to bê where Note the useful formulȧ Now, using the explicit expressions for µ − ,μ − given above and the relation (4.61), we can establish one more relation between fields c(t) and b(t), in addition to (4.62) where The set of eqs. (4.62) and (4.69) can be solved for c andc as In what follows, we will also usê To close this subsection, we present the SU(3) transformation laws of the field b(t). They can be found by comparing the θ = 0 part of the transformations (4.6), (4.7) with the direct variation of ω 0 in (4.59): It is easy to construct the covariant derivative D t b. We first notice that under the ξ i and λ transformations: where B − was defined in (4.42). Then From this definition it follows: Like in the U(2) case, the covariant derivative D t b is simplified after the redefinitioñ Finally, we note that the extra U(1) symmetry (4.51) acts as a constant shift of the field b (or ofb) The group U(1) ⊂ SU(3) also acts as a shift of b, but it acts as well on f i ,f i as a phase transformation. The extra U(1) affects only the field b(t) 11 . It is also worth noting that under the extra SU(2) symmetry (4.49), (4.50) the field b transforms as The covariant derivative D t b is invariant under the τ transformations too, δ (τ ) D t b = 0 .

The bosonic invariant action
The invariant action (4.16) consists of two parts, which can be written as S su(3) = S(N, ω) + α 0 S(q). (4.80) After integrating over Grassmann coordinates and passing to the bosonic limit, we obtain and so S bos su(3) = S 1 + α 0 S 2 . (4.82) In Appendix C we will present the proof that (4.82) is indeed SU(3) invariant like its full superfield prototype (4.16).
To compute the bosonic action as a t-integral, one needs to substitute the explicit expressions of the harmonic fields that were found in the previous subsection and then do the harmonic integrals. The latter task is rather difficult, because, as opposed to the U(2) case, we face here integrals involving in the denominator products of the harmonic factors (1 + A +− ) and (1 +Ā +− ) with A ik =Ā ik . Some basic integrals of this kind are calculated in Appendix B. We postpone the purely technical task of restoring the full bosonic component action to the next publication. Here we limit our consideration only to the b-dependent part of the bosonic action (4.82).
To find this part, we should collect those terms in (4.82) which involve the field b. The full set of such terms is as follows . (4.83) Here, the first two lines are the contribution from S 1 , while the third line comes from S 2 . The explicit expressions for the relevant harmonic integrals are collected in Appendix B. Using them and integrating by parts in the last line of (4.83), we finally obtain in accordance with the formula (4.77). The b-independent terms completing S(b) to the total expression (4.77) should come from the remainder of the bosonic action (4.82).

Summary and outlook
The basic aim of the present paper was to start the construction of the harmonic superfield actions for the group manifolds with quaternionic structure as nice explicit examples of the "strong" HKT N = 4 supersymmetric d = 1 sigma models the general formulation of which was given in [22]. We limited our attention to the simple cases of the 4-dimensional group manifold U(2) and the 8-dimensional one SU (3). For both cases we have found the relevant nonlinear harmonic constraints and shown that the relevant invariant actions are bilinear in the superfields involved, as it should be for the strong HKT models in agreement with the reasoning of [22]. It was found that the full internal symmetry of the U(2) and SU(3) models are, respectively, U(2)×SU(2) and SU(3)×U (2), with the standard harmonic SU(2) H symmetry forming the diagonals in these products. For the U(2) case we computed the full bosonic action, which is none other than the S 3 ×S 1 one, and showed that the gauging of its U(1) isometry, both at superfield and at component levels, yields a particular case of the nonlinear (3, 4, 1) multiplet action, with pure S 3 bosonic target. For the SU(3) case we presented the bosonic action in the space (t, u ± i ), independently checked its SU(3)×U(2) invariance and gave explicitly an important part of it by doing the integration over harmonic variables. For the U(2) case we also performed the detailed comparison with the general harmonic approach to HKT geometries (of refs. [22] and [25]) and found excellent agreement with this approach. We explicitly constructed the closed torsion for this case, as well as the Bismut and Obata connections, with respect to which the corresponding triplet of quaternionic complex structures is covariantly constant. The Riemann curvatures of both these connections were checked to vanish.
It may be possible to generalize the considerations of section 2 to bigger groups. One could consider harmonic superfields N ++ and Ω which are n by n matrices, satisfying the constraints D ++ Ω + N ++ Ω = 0, D ++ N ++ + (N ++ ) 2 = 0. The bosonic part of the superfields N ++ and Ω should parametrize the group SU(2n)×U(1), and the transformations in (5.2) correspond to infinitesimal SU(2n)×U(1) translations. These fields generalize those in section 2. However, we have not been able as yet to write an invariant action generalizing (2.5). We leave this problem for further study. It remains as well to explicitly derive the complete component action for the SU(3) model, and to recover the basic geometric objects of this model from the general harmonic HKT formalism of refs. [22] and [25], as it was done for the U(2) model in section 3.
It is direct to check that the conditions (A.5b) and (A.5c) are identically satisfied as a consequence of the single condition (A.5a), αᾱ − 4α 2 0 = 0 ⇐⇒ α 2 1 = 3α 2 0 , (A. 6) which is just the relation already found in (4.18). Thus we have independently proved that the bosonic action (4.82) is SU(3) invariant under the condition (A.6) and so is guaranteed to define some d = 1 sigma model on the SU(3) group manifold. It is also straightforward to show its invariance under the transformations of the U(2) group commuting with SU(3). It is worth pointing out that checking all these invariances does not require doing explicitly the harmonic integrations in (4.82).
Keeping in mind that SU(2) acting on the doublet indices cannot be broken in the process of harmonic integration, these integrals should be of the form I (ik) = a (ik) f (a 2 ) , I ′ (ik) = a (ik) f ′ (a 2 ) . (B.9) Contracting I (ik) with a (ik) and using (B.4), we find This expression is obviously non-singular at a 2 = 0 . Using (B.9), it is easy to show, e.g., that a ilȧk l I (ik) = 0 , which just means vanishing of the expression (2.15) and of the first term in the square bracket in (2.17). The explicit form of the integral I ′ (ik) can also be easily found It is non-singular at a 2 = 0 , like I (ik) .