Hervé Audren

Abderrahmane Kheddar

Model-predictive control in multi-contact based on stability polyhedrons

We propose a new linear model-predictive control scheme in multi-contact based on the center of mass reduced model. In order to linearize the dynamics of the CoM, we exploit the notion of stability polyhedrons associated to given contact stances, inside which the existence of contact forces is guaranteed. To compute stability polyhedrons, we have first to specify a convex polytope inside which the center of mass's acceleration lies. We then generate a minimum jerk trajectory inside these successive stability polyhedrons that also yields contact transitions timings, and integrate it with our quadratic programming whole-body controller as part of the multi-objective tasks.

I. INTRODUCTION

In this paper we introduce a model-predictive control (MPC) approach to generate multi-contact motion. Typical MPC adopt an hierarchical approach, see e.g. in [START_REF] Wensing | Generation of dynamic humanoid behaviors through task-space control with conic optimization[END_REF], [START_REF] Audren | Model preview control in multi-contact motion-application to a humanoid robot[END_REF], [START_REF] Brasseur | A robust linear MPC approach to online generation of 3d biped walking motion[END_REF], [START_REF] Caron | Multi-contact walking pattern generation based on model preview control of 3d com accelerations[END_REF]: they compute trajectories to be best tracked by the underlying local whole-body controller -see e.g. in [START_REF] Righetti | Optimal Distribution of Contact Forces with Inverse-dynamics Control[END_REF], [START_REF] Vaillant | Multi-contact vertical ladder climbing with an hrp-2 humanoid[END_REF]. Later controller decisions (in terms of desired instant posture position or torque) may impact balance in future motions. The most part of the MPC computed trajectories are discarded and only their newly computed head part (e.g. identified by a time interval) is fed to the wholebody controller. Indeed, we compensate for uncertainties and perturbations by recomputing very often the trajectory from the real state of the robot (and its interaction with the environment). Thus, the computation time must be small with respect to the trajectory length. A lot of approaches from earlier cited works use a simplified linear model that yields convex optimization problems.

In this work, we first recall the notion of robust stability [START_REF] Audren | 3d robust stability polyhedron in multicontact[END_REF] that is used in our MPC: it consists of defining a convex region of acceptable center of mass (CoM) acceleration G, from which we can deduce an acceptable polyhedron of CoM positions P. This allows us to formulate a linear minimum jerk problem in multi-contact and in 3 dimensions without further assumptions. Moreover, as the resulting problem can be solved fast, we extend the formulation to search for the best contact transition timings.

In the context of legged robotics, perhaps the most wellknown application of MPC is [START_REF] Kajita | Biped walking pattern generation by using preview control of zero-moment point[END_REF] in which the authors optimized a Zero-Moment-Point (ZMP) trajectory. As it relied on the ZMP, it is however not applicable to multicontact motion. This approach was extended in [START_REF] Wieber | Trajectory free linear model predictive control for stable walking in the presence of strong perturbations[END_REF] where This work is supported by H2020 European project COMANOID RIA No: 645097 and by JSPS Grant-in-Aid for Scientific Research (B) Number 16H02886 ("Cutting-Edge multi-contact behaviors" H. Audren and A. Kheddar are with CNRS-AIST JRL UMI3218/RL, Tsukuba, Japan and the CNRS-UM LIRMM, Interactive Digital Human group, Montpellier, France the authors optimized a jerk trajectory under constraints. Similarly, it was limited to walking on flat surfaces.

Since then, we have proposed a multi-contact MPC formulation [START_REF] Audren | Model preview control in multi-contact motion-application to a humanoid robot[END_REF]. However, in order to linearize the CoM dynamics, we had to specify the motion along one axis, leave one component of the momentum uncontrolled and specify stance transition timings in a heuristic (non satisfactory) way. In later work [START_REF] Herzog | Trajectory generation for multi-contact momentum-control[END_REF], the above restrictions were lifted in favor of generating momentum trajectories, but the algorithm no longer performed well enough for real-time operation.

In other approaches [START_REF] Erez | An integrated system for real-time model predictive control of humanoid robots[END_REF], [START_REF] Chrétien | Gpu robot motion planning using semi-infinite nonlinear programming[END_REF] parallelism and general purpose GPU were exploited to solve very quickly a whole-body non-linear MPC problem giving a set of contact transitions. Unfortunately, it is hard to generate long enough whole-body trajectories in real-time, even on powerful computers. Also non-linear optimization solvers are often hard to tune and gradient-based ones suffer from local minima.

Finally, some MPC aim at generating spatial trajectories by solving non-linear problems, and then using Time Optimal Path Parametrization [START_REF] Hauser | Fast interpolation and time-optimization on implicit contact submanifolds[END_REF], [START_REF] Naveau | A reactive walking pattern generator based on nonlinear model predictive control[END_REF], [START_REF] Caron | Dynamic walking over rough terrains by nonlinear predictive control of the floating-base inverted pendulum[END_REF] to determine at which speed to execute the trajectory. Performing TOPP is fast, but solving the non-linear spatial problem can be costly unless specific assumptions or simplifications are made. Moreover, optimizing separately the kinematics from the dynamics can yield suboptimal solutions in terms of execution times. In our formulation, once the slightly expensive robust stability regions have been determined, each subsequent optimization is very fast, which makes it a great fit for MPC formulations.

We first recall in section II how we define and compute the stability regions in which CoM trajectories are generated. Then our contributions stands as follows:

• We introduce a linear, minimum jerk MPC formulation in section III, that aims at optimizing the spatial trajectory and finding stances transition timings;

• this MPC trajectory generator is then coupled with our whole-body, tasks-based controller in section IV;

• finally, simulation results are discussed in section V.

II. ROBUST STABILITY

We recall briefly few results of our work in [START_REF] Audren | 3d robust stability polyhedron in multicontact[END_REF] in which we extended the notion of static equilibrium in multi-contact to the notion of robust equilibrium with respect to a predefined acceleration polytope. To do so, we used a flywheel model where the robot is reduced to a single rigid body mass (m): Definition 1. A CoM position c is said to be in robust equilibrium w.r.t. to a convex acceleration polytope G if and only if there exists contact forces f i applied at points r i that satisfy:

∀c ∈ G, ∃f s.t.        f i = m(c -g) f i × (c -r i ) = 0 Bf ≤ u T f (1)
where B, u define the non-linear friction cones relation, g is the gravity vector, u stacks the contacts' normals.

Considering that the contact points r i are fixed, the above equations define a system of convex equalities and inequalities. It entails that there exists a convex shape, P such that: Definition 2. ∀c ∈ P, c is in robust equilibrium w.r.t G.

We then show that by adapting the recursive projection algorithm [START_REF] Bretl | Testing static equilibrium for legged robots[END_REF] it is possible to obtain an explicit representation of P, that is to say a set of inequalities H P , b P such that:

c ∈ R 3 |H P c ≤ b P = P (2)
Note that the contact forces f do not appear in the above criteria: by restricting the possible accelerations to a known convex polytope G, we have transformed the original nonlinear problem eq. ( 1) into a linear one, eq. ( 2).

III. POLYHEDRON MPC

From the previous section, we know, given a convex acceleration polytope G, how to compute a convex CoM polyhedron P such that: if c ∈ P and c ∈ G, there exist contact forces that realize that acceleration. We now see how to exploit that fact to generate the CoM trajectories online.

We define that a motion is in robust equilibrium if and only if (iff) for every time instant t:

∀t c(t) ∈ P(t) and c(t) ∈ G(t) (3)
Given that P and G are entirely determined by the contact geometry and friction cones, we can denote C(t) the active contacts at the time t. The above condition thus write:

∀t c(t) ∈ P (C(t)) and c ∈ G (C(t)) (4)
Thus, to generate a trajectory we need to not only optimize the CoM trajectory itself (c(t), c(t)) but the instants at which the contacts change (C). As changing contacts is a discrete event, we parametrize those changes with integer variables.

A. Formulation

We consider a number of stances, i.e. consecutive sets of contacts, C i . We discretize the times into t k . We can thus write that:

∀k c k ∈ P C i(k) and c ∈ G C i(k) (5)
where i(k) represents the active contact set index at instant k. It is a function over finite sets:

i : 0 . . K → 0 . . N (6)
k → i(k) (7)
Our problem is formulated as a minimum jerk cost. Thus let us first form the evolution equation:

  c k+1 ċk+1 ck+1   =   1 dt dt 2 2 0 1 dt 0 0 1     c k ċk ck   +   dt 3 6 dt 2 2 dt   ... c k (8)
x k+1 = Ax k + B ... c k (9)
We can thus repeatedly integrate the above with initial condition x 0 to obtain the whole trajectory:

X = x T 1 x T 2 . . . x T K T (10) =      A A 2 . . . A K      x 0 +      B AB B . . . A K-1 B A K-2 B . . . B           ... c 0 ... c 1 . . . ... c n      (11) = Φx 0 + ΨU (12)
We can thus build a quadratic cost that is the distance to a reference and the weighted norm of the command:

X -X 2 + W u U 2 = U T QU + 2c T U + γ (13)
We introduce the positive, diagonal matrix W x to weight the diffent parts of X independently:

Q = Ψ T W x Ψ + W u (14)
c T = x T 0 Φ T -XT W x Ψ (15)
We can thus formulate the problem as:

min U,i(k) 1 2 U T QU + c T U (16) s.t. ∀k ∈ 0 . . K c k ∈ P C i(k) ck ∈ G C i(k) (17)
We obtained a quadratic objective, but the optimization is over a function, and the constraints are non-linear. The next sections present how to properly select the function i(k).

B. Choosing a timing function

The family of functions i(k) are defined over finite integer intervals. Thus, one could exhaustively search all (N + 1) K of them. But this number is considerably large even for small values of N and K. Moreover, not any function can fit: we should go from one polyhedron to the next, without skipping one and without going backwards.

Hence, instead of looking at all possible i(k), we consider a specific shape of i(k): a strictly increasing by one function. That is to say, a stair-shaped function. It is defined by the τ i that indicate at which time we switch from P i to P i+1 . Instead of looking for the value of K variables in 0 . . N , we have to find the value of N variables in 0 . . K , but those variables are known to be strictly increasing, which drastically reduces the size of the search space. See fig. 1 for a simple example of such a function. The actual number of possible functions is given by:

N +1 i=1 (p i (K)i!) (18)
Where p stands for the partition function (see e.g. [START_REF] Hardy | An introduction to the theory of numbers[END_REF]). Indeed the partition function tells us how many tuples of integers of a certain size sum to K. Those integers represent the duration of a step. As we consider N switching times, we are looking for the number of partitions of size N + 1. We then account for the fact that we might not switch, and thus sum over all partitions of size less or equal than N + 1. Finally, we consider that there are i! possible orderings of a partition of size i.

A numerical application for a typical scenario where K = 15 and N = 2 illustrates the huge reduction, 5 orders of magnitude, in the size of the search space:

• For the complete space search: (N + 1) K = 14348907.

• For the function search eq. ( 18): 129. Whenever the transition times τ i are fixed, the problem is reduced to a simple QP in few variables (typically a few dozens) and a lot of constraints (a few thousands).

Indeed, as c k and ck , are part of our state, eq. ( 5) can be written as linear constraints in U :

H P i(k) c k ≤ b P i(k) ( 19
)
⇔H P i(k) S c k (Φx 0 + ΨU ) ≤ b P i(k) ( 20
)
⇔H P i(k) S c k ΨU ≤ b P i(k) -Φx 0 (21)
And:

H G i(k) ck ≤ b G i(k) ( 22
)
⇔H G i(k) S ck ΨU ≤ b G i(k) -Φx 0 (23)
With S c k and S ck selection matrices that extract c k and ck from X respectively. This can be solved in milliseconds (using the quadratic program QL solver [START_REF] Schittkowski | QL: A fortran code for convex quadratic programming -user's guide[END_REF]), and thus exploring all the possible τ i for a small number of transitions and instants, i.e. N = 2 and K = 15, can be done in about 500 ms.

To further reduce the search space, we apply the following strategy: for each choice of τ 0 , we test if τ 1 = N is feasible.

If it is, we explore all possible τ 1 starting from N and working backwards towards τ 0 until the problem is no longer feasible. This amounts to discarding all solutions where it is not possible to stop in the intermediate polyhedron. As long as we consider a symmetric G that contains the gravity g, this is not highly restrictive. Indeed, by symmetry, at any point, decelerating as much as we previously accelerated is feasible while containing g means that stopping is a solution.

To confirm this fact, we solved a hundred MPCs. In 97 % of the cases, there was no loss of optimality and in none of them did the reduction of the search space result in infeasibility.

We then select the solution (X, τ 0 , τ 1 ) that yields the smallest cost. This easily extends to more τ i but the combinatorial nature of the problem makes it quickly intractable. This method is sumarized in algorithm 1.

Algorithm 1 Algorithm MINTIM to determine the optimal timings

for τ 0 ∈ 0 . . K do for τ 1 ∈ K . . τ 0 + 1 do res ← solveQP(τ 0 , τ 1 ) if res then X, cost ← res else break end if end for end for return argmin τ0,τ1,X (cost) Although exhaustive search is typically not the most efficient solution, it is the best that we have implemented. Hill-climbing [START_REF] Russell | Artificial Intelligence: a Modern Approach[END_REF] is not adapted to our problem as the objective function (the optimal cost of our QP) is not convex. Simulated annealing [START_REF] Dowsland | Simulated annealing in Modern Heuristic Techniques for Combinatorial Problems[END_REF] is not well adapted to small search spaces, and requires tuning of the temperature and acceptance function. Finally, we can recast our problem as a Mixed-Integer Quadratic Program (MIQP). Indeed, we would like to express the following relation between constraints:

τ i < j ≤ τ i+1 =⇒ c j ∈ P i ∧ cj ∈ G i (24)
Which can be transformed into linear constraints by introducing binary slack variables. However, our tentative implementation did not show any improvement in terms of computation time and failed to converge in some instances.

IV. INTEGRATION FOR MULTI-CONTACT CONTROL

In this section, we present how we integrated the trajectory generation of the previous section within our multi-contact planning and control framework. First, we perform multi-contact planning [START_REF] Escande | Planning contact points for humanoid robots[END_REF], [START_REF] Bouyarmane | Humanoid robot locomotion and manipulation step planning[END_REF], [START_REF] Vaillant | Multi-contact vertical ladder climbing with an hrp-2 humanoid[END_REF] on our environment in order to obtain a sequence of stances. Each stance is composed of a set of contacts, a robot posture and an action. This action tells us if we are adding or removing a contact in this stance.

Then, from the contacts positions, we compute the polyhedrons P i by appropriately setting the G i . We then set the number of stances to look ahead N = 2 and a length of the preview window K = 10. In this setting, we solve algorithm 1 in about 50 ms.

We can now compute a CoM trajectory and timings by providing a starting state and a reference: (X, τ 0 , τ 1 ) = MINTIM(x 0 , X). To do so, we connect the current state of the QP controller to the x 0 input. The X reference was chosen to be the simplest: it is a reference position (the endpoint) with zero velocity and acceleration. This means that we are only specifying where to go, but not giving any hints as to how to get there.

As for the gains tuning, we simply set W u to 1e -4 and W x to 1 from gathered experiments. Indeed, our primary objective is to reach the objective, while minimizing the jerk should have a lower priority.

Our tasks-base controller [START_REF] Bouyarmane | Humanoid robot locomotion and manipulation step planning[END_REF], [START_REF] Vaillant | Multi-contact vertical ladder climbing with an hrp-2 humanoid[END_REF] is built around solving a quadratic program in the generalized coordinates acceleration and contact forces, regrouped in the decision vector y:

min. y 1 2 y T Qy + c T y def = i β i T i (y) s.t. Ay ≤ B ; Cy = D (25)
The task functions T allow us to express a variety of objectives in a quadratic form, and are weighted according to the β i . In this paper, we only use the posture objective that minimizes the distance to a given posture and the CoM objective which tracks a CoM trajectory.

As for the constraints, they allow us to express a variety of physical limitations of the robot. Amongst them we are interested in:

• Joint limits constraints either in position, velocity or torque

• Discretized friction cones constraints • Collision constraints
The controller is supervised by a three state finite state machine (FSM), illustrated in fig. 2. In each state, we always track the posture generated by the planner for the current stance, with a low weight. The states are:

1) Initial state: we simply track the posture of the initial stance. 2) Remove state: as we are removing a contact, the only additional task is to track the CoM trajectory while remaining in the current stance's polyhedron. 3) Add state: we are adding a contact, we insert two tasks to both track the CoM trajectory and track a spline for the end-effector, while remaining in the current polyhedron. For the add state, the control points of this spline are determined by the geometry of the contacts while the time parametrization comes from the timings computed by the MPC. The spline passes through the initial contact position, with zero initial velocity and acceleration, a waypoint that is 10 % above the highest of the start and end points, and arrives at the final contact point with zero velocity and acceleration. From the MPC, we have the whole duration of the add phase, and thus specify that we pass through the waypoint at 30 % of the total duration. More complex settings would of course be necessary as the environment becomes more cluttered. Now, we specify when to shift from one stance to the next: 1) Initial: whenever the posture error is low.

2) Remove: whenever we are inside the next stance's polyhedron. 3) Add: whenever we are inside the next stance's polyhedron and we are done executing the end-effector trajectory. For each transition, the type and the contacts of the next stance are given by the plan.

Finally, the current FSM stance, N 0 , is sent back to the MPC component as shown in fig. 3. This value is used as the starting stance for the MPC: instead of looking for values of i(k) in 0 . . N , we look for them in N 0 . . N 0 + N . This Collision avoidance constraints means we only look for a small number of transition timings, and only advance the starting polyhedron / polytope, when the whole-body controller has realized the current action.

The MPC thus waits until the whole-body controller has performed the action before advancing to the next stance.

V. RESULTS

The following sections present two illustrations of the framework presented above: crossing a ridge and climbing stairs. In both cases, HRP2-Kai is able to perform the motion if we do not enforce real-torque limitations. This means that indeed, there exists contact forces that allow the controller to track the MPC CoM trajectory. However, those contact forces are not always realizable by the whole-body system. Indeed, during trajectory generation, we only relied on a reduced model, that does not take into account joint torques as those depend non-linearly on the joint angles.

The controller is also unable to perform the end-effector trajectory in time, but this is not problematic: the CoM remains in the previous polyhedron as long as the new contact has not been established. The MPC then computes new timings based on the current state.

In both scenarii, we used a single G for all stances, that is the convex hull of lateral accelerations with a magnitude of a = 0.5 m s -2 : G = {{a, 0, 0}, {-a, 0, 0}, {0, a, 0}, {0, -a, 0}}

We also set the duration of each time-step to dt = 0.15 s, which yields a total duration of 1.5 s for the trajectories.

A. Ridge crossing

In this simulation, we want to cross a narrow ridge, while pushing on a flat, non-horizontal, support for added stability. This is the same environment as in [START_REF] Audren | Model preview control in multi-contact motion-application to a humanoid robot[END_REF], but the multi-contact plan we used is newly generated. A few keyframes are available in fig. 4. At most, we have three unilateral contacts with the environment.

Compared to our previous work [START_REF] Audren | Model preview control in multi-contact motion-application to a humanoid robot[END_REF], the CoM trajectory that we obtain is much more conservative. This is most evident in the first stance: HRP2-Kai leans quite heavily backwards with its CoM above the right foot. That stance is composed of a single unilateral contact, thus the region of stability is very small at that time. Indeed, as G contains the gravity, we are rather statically stable at all times. For this particular stance, it means that the CoM has to remain above the right foot, at a low altitude. However, we no longer need to use a carefully tuned timing heuristic and the timings automatically adjust based on the current state. Moreover, we no longer specify the z motion, nor do we have to carefully choose a X.

B. Stairs climbing

In this scenario, HRP2-Kai is tasked with climbing a flight of stairs, using the handrail. Halfway up, it grasp again the handrail while using its other hand as unilateral support on the top platform. Thus, some stances have three unilateral contacts and a bilateral one. The resulting motion is illustrated in fig. 5.

We first remark that this motion appear much more plausible as compared to the previous scenario. The CoM trajectory is less constrained because the robot establishes more than one contact with the environment at all times: we do not produce the 'non-expected' poses of the previous scenario. However, as research on how humans decide to make or break contacts is scarce, it is difficult to quantify that feeling. It would seem coherent that humans use more contacts when confronted with a difficult and/or uncertain terrain. When walking under such conditions, humans indeed adopt stiffer, more static gaits [START_REF] Cappellini | Motor Patterns During Walking on a Slippery Walkway[END_REF].

Still, the motion is executed much faster than in the quasistatic case: we climb the whole flight in about 30 s whereas it took about 4 min in our previous work [START_REF] Audren | Stability polygons reshaping and morphing for smooth multi-contact transitions and force control of humanoid robots[END_REF]. Indeed, we gain substantial time by not waiting for the CoM to reach a predetermined position. Yet, as this motion generates hightorques, we would probably have to slow it down to execute it on a real hardware, especially when approaching contact surfaces so as not to damage the robot in case of unpredicted contact.

VI. CONCLUSIONS AND FUTURE WORK

We have shown in this paper how the notion of stability regions allows us to generate a linear CoM MPC in multicontact, without making any assumptions outside of restrictions on the possible CoM accelerations. Furthermore, we exploit the linearity to solve many such problems at once, and find the optimal stance transitions timings. Finally, this approach was validated by simulation.

Yet, we have many avenues of improvement left open. Once all of the following points are solved, we implement our formulation on the real HRP2-Kai.

Firstly, we would like to improve even further the computation times. To do so, we could parallelize our algorithm in order to solve multiple QPs at the same time. We would probably gain the most by finding a better way to explore the possible timings than exhaustive search, for example by using MIQP. Our problem also has quite a specific form, and there is possibly a way to solve it faster.

Secondly, we need to make the trajectory generation trackable by the whole-body controller: this entails integrating part or whole of the whole-body constraints in either the trajectory generation or the polyhedron computation.

Finally, we have used only one G to constrain the accelerations. We would like to have it vary, for example through alternate optimizations, to obtain solutions that are no longer statically stable.

Fig. 1 .

 1 Fig. 1. Schema of the polyhedron CoM preview control. Left shows the spatial trajectory going through three different polygons. Right shows the evolution of the integer variable with time.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Representation of the FSM that drives the whole-body controller

Fig. 4 .

 4 Fig. 4. Keyframes of HRP2-Kai crossing a ridge: the blue ribbon represents the current CoM trajectory being tracked.

Fig. 5 .

 5 Fig. 5. Keyframes of HRP2-Kai climbing the stairs: the blue ribbon represents the current CoM trajectory.

TABLE I TASKS

 I PART OF THE COST AND TASKS PART OF THE CONSTRAINTS FOR

		THE QP CONTROLLER
	Cost Tasks CoM Trajectory Posture Task	Constraint Tasks CoM in polyhedron Contact constraints Joint limits constraints (position and velocity)


F Delduc

E Ivanov
email: eivanov@theor.jinr.ru

The Common Origin of Linear and Nonlinear Chiral Multiplets in N =4 Mechanics

Keywords: 11.30.Pb, 11.15.-q, 11.10.Kk, 03.65.-w Keywords: Supersymmetry, gauging, isometry, superfield

Elaborating on previous work (hep-th/0605211, 0611247), we show how the linear and nonlinear chiral multiplets of N =4 supersymmetric mechanics with the off-shell content (2,4,2) can be obtained by gauging three distinct two-parameter isometries of the "root" (4,4,0) multiplet actions. In particular, two different gauge groups, one abelian and one non-abelian, lead, albeit in a disguised form in the second case, to the same (unique) nonlinear chiral multiplet. This provides an evidence that no other nonlinear chiral N =4 multiplets exist. General sigma model type actions are discussed, together with the restricted potential terms coming from the Fayet-Iliopoulos terms associated with abelian gauge superfields. As in our previous work, we use the manifestly supersymmetric language of N =4, d=1 harmonic superspace. A novel point is the necessity to use in parallel the λ and τ gauge frames, with the "bridges" between these two frames playing a crucial role. It is the N =4 harmonic analyticity which, though being non-manifest in the τ frame, gives rise to both linear and nonlinear chirality constraints.

Introduction

The one-dimensional supersymmetry and related models of supersymmetric (quantum) mechanics reveal many specific surprising features and, at the same time, have a lot of links with higher-dimensional theories of current interest (see e.g. [START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF][START_REF] Bellucci | A Journey Through Garden Algebras[END_REF] and refs. therein). This motivates many research groups towards thorough study and further advancing of this subject (see e.g. [START_REF] Bellucci | A Journey Through Garden Algebras[END_REF][START_REF] Kuznetsova | [END_REF]).

Recently, we argued [START_REF] Delduc | [END_REF]5] that the plethora of relationships between various d=1 supermultiplets with the same number of fermionic fields, but different divisions of the bosonic fields into physical and auxiliary ones (so called "d=1 automorphic dualities" [START_REF] Gates | On Extended Supersymmetric Quantum Mechanics[END_REF]), can be adequately understood in the approach based on the gauging of isometries of the invariant actions of some basic ("root") multiplets by non-propagating ("topological") gauge multiplets (these isometries should commute with supersymmetry, i.e. be triholomorphic). The key merit of our approach is the possibility to study these relationships in a manifestly supersymmetric superfield manner, including the choice of supersymmetry-preserving gauges. Previous analysis [START_REF] Gates | On Extended Supersymmetric Quantum Mechanics[END_REF]7] was basically limited to the component level and used some "ad-hoc" substitutions of the auxiliary fields. In the framework of the gauging procedure, doing this way corresponds to making use of the Wess-Zumino-type gauges.

In [START_REF] Delduc | [END_REF]5] we focused on the case of N =4 supersymmetric mechanics and showed that the actions of the N =4 multiplets with the off-shell contents [START_REF] Kuznetsova | [END_REF][START_REF] Delduc | [END_REF][START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF], [START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF][START_REF] Delduc | [END_REF][START_REF] Kuznetsova | [END_REF] and (0, 4, 4) can be obtained by gauging certain isometries of the general actions of the "root" multiplet (4, 4, 0) in the N =4, d=1 harmonic superspace. Based on this, we argued that the latter is the underlying superspace for all N =4 mechanics models. Here we confirm this by studying, along the same line, the remaining N =4 multiplets, the chiral and nonlinear chiral off-shell multiplets with the field content [START_REF] Bellucci | A Journey Through Garden Algebras[END_REF][START_REF] Delduc | [END_REF][START_REF] Bellucci | A Journey Through Garden Algebras[END_REF] . They prove to naturally arise as a result of gauging some two-parameter isometry groups admitting a realization on the harmonic analytic superfield q +a which describes the multiplet [START_REF] Delduc | [END_REF][START_REF] Delduc | [END_REF]0). The origin of the difference between these two versions of the (2, 4, 2) multiplet is attributed to the fact that they emerge from gauging two essentially different isometries: the linear multiplet is associated with gauging of some purely shift isometries, while the nonlinear one corresponds to gauging the product of two "rotational" isometries, viz. the target space rescalings and a U (1) subgroup of the SU (2) Pauli-Gürsey group. We also recover the same nonlinear version of this multiplet, though in disguise, by gauging a mixture of the rescaling and shift isometries. It is the last possible two-parameter symmetry implementable on q +a . Thus we show that two known off-shell forms of the N =4, d=1 multiplet (2, 4, 2), the linear and nonlinear ones, in fact exhaust all possibilities.

It should be emphasized that the existence of non-linear cousins of the basic d=1 supermultiplets is one of the most amazing features of extended d=1 supersymmetry. In superspace they are described by superfields satisfying some nonlinear versions of the standard constraints (e.g. of chirality constraints). As a result, the realization of the corresponding off-shell d=1 supersymmetry on the component fields of such supermultiplets is intrinsically nonlinear. The list of such multiplets known to date includes the nonlinear analogs of the N =4 multiplets (4, 4, 0) [START_REF] Hellerman | Supersymmetric quantum mechanics from light cone quantization[END_REF][START_REF] Delduc | [END_REF][START_REF] Burdik | [END_REF][START_REF] Delduc | [END_REF]11], [START_REF] Kuznetsova | [END_REF][START_REF] Delduc | [END_REF][START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF] [12,13], (2, 4, 2) [13], as well as of the N =8 multiplets [START_REF] Delduc | [END_REF][START_REF] Hellerman | Supersymmetric quantum mechanics from light cone quantization[END_REF][START_REF] Delduc | [END_REF] [14] and [START_REF] Bellucci | A Journey Through Garden Algebras[END_REF][START_REF] Hellerman | Supersymmetric quantum mechanics from light cone quantization[END_REF][START_REF] Gates | On Extended Supersymmetric Quantum Mechanics[END_REF] [15]. As shown in [START_REF] Delduc | [END_REF], within the approach based on the gauging procedure the difference between linear and nonlinear (3, 4, 1) multiplets originates from the fact that the first multiplet is related to the gauging of the shift or rotational U (1) symmetries, and the second one to the gauging of the target space rescalings. In the case of the (2, 4, 2) multiplets we find an analogous intimate relation between the type of multiplet and the q +a two-parameter symmetry group which has to be gauged to generate it.

Brief preliminaries

Throughout the paper we use the same N =4, d=1 harmonic superspace (HSS) techniques, conventions and notation as in [12,[START_REF] Delduc | [END_REF]5]. The N =4 "root" multiplet (4, 4, 0) is described by the harmonic analytic superfield q +a (ζ, u) which is subjected to the Grassmann harmonic and bosonic harmonic constraints (a) D + q +a = D+ q +a = 0 , (b) D ++ q +a = 0 .

(2.1)

Here (ζ, u) are coordinates of the harmonic analytic N =4 superspace [12], (ζ, u)

= (t A , θ + , θ+ , u ± i ) , u +i u - i = 1 , they are related to the standard N =4 superspace (central basis) coordinates z = (t, θ i , θi ) as t A = t -i(θ + θ-+ θ -θ+ ), θ ± = θ i u ± i , θ± = θi u ± i . (2.2)
Respectively, the N =4 covariant spinor derivatives and their harmonic projections are defined by

D i = ∂ ∂θ i + i θi ∂ t , Di = ∂ ∂ θi + iθ i ∂ t , (D i ) = -Di , {D i , Dk } = 2i δ i k ∂ t , (2.3)
D ± = u ± i D i , D± = u ± i Di , {D + , D-} = -{D -, D+ } = 2i ∂ t A . (2.4)
In the analytic basis, the derivatives D + and D+ are short,

D + = ∂ ∂θ -, D+ = - ∂ ∂ θ-, (2.5)
so the conditions (2.1a) become the harmonic Grassmann Cauchy-Riemann conditions stating that q +a does not depend on the coordinates θ -, θin this basis. The analyticity-preserving harmonic derivative D ++ and its conjugate D --in the analytic basis are given by

D ++ = ∂ ++ -2iθ + θ+ ∂ t A + θ + ∂ ∂θ -+ θ+ ∂ ∂ θ-, D --= ∂ ---2iθ -θ-∂ t A + θ -∂ ∂θ + + θ-∂ ∂ θ+ , ∂ ±± = u ± i ∂ ∂u ∓ i , (2.6)
and are reduced to the pure harmonic partial derivatives ∂ ±± in the central basis. They satisfy the commutation relations

[D ++ , D --] = D 0 , [D 0 , D ±± ] = ±2D ±± , (2.7)
where D 0 is the operator counting external harmonic U (1) charges. In the analytic basis it is given by

D 0 = u + i ∂ ∂u + i -u - i ∂ ∂u - i + θ + ∂ ∂θ + + θ+ ∂ ∂ θ+ -θ -∂ ∂θ --θ-∂ ∂ θ-, (2.8)
while in the central basis it coincides with its pure harmonic part. On the extra doublet index a of the superfield q +a the so-called Pauli-Gürsey group SU (2) P G is realized. It commutes with the N =4 supersymmetry generators, as distinct from the R-symmetry SU (2) R group which acts on the doublet indices i, k of the Grassmann and harmonic coordinates, spinor derivatives and N =4 supercharges. The free action of q +a can be written either in the analytic, or the central superspace

S free q = - 1 4 µ H (q +a q - a ) = i 2 µ (-2) A (q +a ∂ t q + a ) , (2.9)
where q -a ≡ D --q +a and the integration measures are defined as

µ H = dudtd 4 θ = dudt A (D -D-)(D + D+ ) = µ (-2) A (D + D+ ), µ (-2) A = dudζ (-2) = dudt A dθ + d θ+ = dudt A (D -D-) .
(2.10)

The general sigma model-type action of q +a (with a non-trivial bosonic target space metric) is given by S q = µ H L(q +a , q -b , u ± ) .

(2.11)

The constraint (2.1) possesses a seven-parameter group of rigid symmetries commuting with supersymmetry (it includes SU (2) P G as a subgroup) [5]. One can single out the appropriate subclasses of the general action (2.11) (including the free action (2.9)) which are invariant with respect to one or another symmetry of this sort. For further use we give here the full list of non-equivalent two-parameter symmetries.

Abelian symmetries

(a) δq +a = λ 1 u +a + λ 2 c (ab) u + b ; (b) δq +a = λ 1 q +a -λ 2 c (ab) q + b . (2.12)
Nonabelian symmetry δq +a = λ 1 q +a + λ 2 u +a .

(2.13)

Here the constant triplet c ab is normalized as

1 c 2 = c ab c ab = 2 . (2.14)
The algebra of the transformations (2.13) provides an example of two-generator solvable algebra. All other possible two-parameter symmetry groups listed in [5] can be reduced to (2.12), (2.13) by a redefinition of q +a . In what follows we shall gauge these symmetries and show that this gauging gives rise to three versions of the N =4 multiplet (2, 4, 2), with the corresponding general actions arising from the appropriate invariant subclasses of the general q +a action (2.11). It turns out that the standard linear chiral N =4 multiplet emerges as the result of gauging purely shift isometry (2.12a) while the two alternative gaugings give rise to two nonlinear versions of this multiplet. The nonlinear multiplet obtained from gauging the group (2.12b) is identical to the one discovered in [13]. The multiplet obtained from gauging (2.13), although looking different, can be identified with the previous one after suitable redefinitions.

Chiral multiplet

Our gauging prescriptions are basically the same as in other cases [START_REF] Delduc | [END_REF]5].

We start with the gauged version of the transformations (2.12a)

δq +a = Λ 1 u +a + Λ 2 c ab u + b , c ab = c ba , (3.1)
where Λ 1 and Λ 2 are now charge-zero analytic superfields. The gauge covariantization of the q +a harmonic constraint (2.1b) is given by

D ++ q +a -V ++ 1 u +a -V ++ 2 c ab u + b = 0 , (3.2)
where V ++ 1 and V ++ 2 are analytic gauge superfield transforming as

δV ++ 1 = D ++ Λ 1 , δV ++ 2 = D ++ Λ 2 . (3.3)
In the cases considered here (as distinct from the cases treated in [START_REF] Delduc | [END_REF]5]), it is convenient to make use of the "bridge" representation of the gauge superfields [16,[START_REF] Galperin | Harmonic Superspace[END_REF].

The analytic superfields V ++ 1 and V ++ 2 may be represented as

V ++ 1 = D ++ v 1 , V ++ 2 = D ++ v 2 , (3.4)
where v 1 (t, θ, θ, u) and v 2 (t, θ, θ, u) are non-analytic harmonic superfields which may be interpreted as "bridges" between the analytic and central superspace gauge groups (called the λ and τ gauge groups, see [START_REF] Galperin | Harmonic Superspace[END_REF]). They transform under local shifts as

δv 1 = Λ 1 (ζ, u) -τ 1 (t, θ, θ) , δv 2 = Λ 2 (ζ, u) -τ 2 (t, θ, θ) , (3.5)
where τ 1 (t, θ, θ) and τ 2 (t, θ, θ) are non-analytic gauge superparameters bearing no dependence on the harmonic variables

D ++ τ 1 (t, θ, θ) = 0 , D ++ τ 2 (t, θ, θ) = 0 . (3.6)
We now define the non-analytic doublet superfield Q +a by

Q +a = q +a -v 1 u +a -v 2 c ab u + b . (3.7)
As a consequence of (3.2), (3.4), it satisfies the simple harmonic constraint

D ++ Q +a = 0 , (3.8)
which implies

Q +a (t, θ, θ, u) = Q ba (t, θ, θ)u + b , Q ba = Q ba , (3.9)
where the superfields Q ba (t, θ, θ) are independent of the harmonic variables and form a real quartet. We also have

Q -a ≡ D --Q +a = Q ba (t, θ, θ)u - b = q -a -v 1 u -a -v 2 c ab u - b , (3.10)
where q -a ≡ D --q +a -V --

1 u -a -V -- 2 c ab u - b (3.11)
and

V -- 1 = D --v 1 , V -- 2 = D --v 2 , δV -- 1 = D --Λ 1 , δV -- 2 = D --Λ 2 .
(3.12)

From (3.10) and (3.12) one can find the gauge transformation law of the non-analytic harmonic superfield q -a :

δq -a = Λ 1 u -a + Λ 2 c ab u - b . (3.13)
Now it is easy to determine how the superfields Q ±a introduced in (3.7), (3.10) transform under the local shift symmetries. They are inert under the λ gauge transformations and have the following τ gauge transformation law

δQ ±a = τ 1 u ±a + τ 2 c ab u ± b ⇔ δQ ba = τ 2 c ab -τ 1 ǫ ba . (3.14)
As a consequence of the Grassmann analyticity constraints (2.1), the superfield Q +a satisfies the following fermionic constraints

D + q +a = 0 ⇔ D + Q +a + (D + v 1 )u +a + (D + v 2 )c ab u + b = 0, D+ q +a = 0 ⇔ D+ Q +a + ( D+ v 1 )u +a + ( D+ v 2 )c ab u + b = 0 . (3.15)
It is important that, due to the analyticity of the gauge superfields V ++ I , I = 1, 2, the fermionic connections D + v I , D+ v I depend linearly on the harmonic variables. We shall use the notations

D + v I (t, θ, u) = A a I (t, θ, θ)u + a , D+ v I (t, θ, u) = -Āa I (t, θ, θ)u + a . (3.16)
Using (3.9) and (3.16), one can rewrite (3.15) in the following equivalent form with no harmonic dependence at all:

D (a Q b)c -A (a 1 ǫ b)c + A (a 2 c b)c = 0 , D(a Q b)c + Ā(a 1 ǫ b)c - Ā(a 2 c b)c = 0 . (3.17)
Now let us more closely inspect the transformation laws (3.14). We start by choosing a frame in which the matrix c ab has only one non-vanishing component:

c 12 = c 21 = i , c 11 = c 22 = 0 . (3.18)
In this frame the transformations (3.14) look as

δQ ±1 = (τ 1 -iτ 2 ) u ±1 , δQ ±2 = (τ 1 + iτ 2 ) u ±2 , (3.19)
or, in terms of the N =4 superfields Q ab (t, θ, θ) defined in (3.9),

δQ 12 = (τ 1 + iτ 2 ) , δQ 21 = -(τ 1 -iτ 2 ) , δQ 11 = δQ 22 = 0 . (3.20)
It is then convenient to choose the unitary-type gauge

Q 12 = Q 21 = 0 . (3.21)
Now the constraints (3.17) determine the spinor connection superfields A a I and their conjugate Āa I in terms of the remaining superfield Q 11 ≡ Φ and its complex conjugate

Q 22 = Φ A 1 1 = 1 2 D 2 Φ , A 2 1 = - 1 2 D 1 Φ , A 1 2 = i 2 D 2 Φ , A 2 2 = i 2 D 1 Φ , (and c.c.) , (3.22)
simultaneously with imposing the constraints on these superfields

D 1 Φ = D1 Φ = 0 , D 2 Φ = D2 Φ = 0 . (3.23)
These constraints may be interpreted as twisted chirality conditions. They can be given the standard form of the chirality conditions by relabelling the spinor derivative D i , Di in such a way that the R-symmetry SU (2) acting on the indices i gets hidden, while another SU (2) (which rotates D i through Di ), gets manifest2 . Thus we have succeeded in deriving the linear chiral N =4, d=1 multiplet (2, 4, 2) from the analytic multiplet (4, 4, 0) by gauging two independent shift isometries realized on the latter.

Let us now examine this correspondence on the level of the invariant actions. We start with the gauge covariantization of the free action (2.9) of the analytic superfield q +a . In the full superspace the covariantized action reads

S f ree cov = µ H [ q +a D --q + a -2V -- 1 q +a u + a -2V -- 2 q +a c b a u + b +2(V ++ 1 V -- 2 -V ++ 2 V -- 1 )c +-] . (3.24)
It is gauge invariant up to a total harmonic derivative in the integrand, i.e. it is of the Chern-Simons type. It may be equivalently rewritten in terms of the superfield Q +a and the bridges

v 1 , v 2 S f ree cov = µ H [ Q +a D --Q + a +2v 1 Q +a u - a +2v 2 Q +a c b a u - b +(v 1 V ++ 2 -v 2 V ++ 1 )c --+v 2 1 +v 2 2 ] . (3.25)
In this form it is invariant under both the λ and τ gauge transformations. In fact, the action can be written as a sum of two terms, the first of which transforms only under the τ gauge group, and the second only under the λ group

S f ree cov = S τ + S λ , S τ = µ H Q +a D --Q + a - 1 4 (Q +a u - a -Q -a u + a ) 2 - 1 4 (Q +a c a b u - b -Q -a c a b u + b ) 2 ,(3.26) S λ = µ H 1 4 (q +a u - a -q -a u + a ) 2 + 1 4 (q +a c a b u - b -q -a c a b u + b ) 2 -V ++ 2 q -a c a b u - b -V ++ 1 q -a u - a ) . (3.27)
To check the invariance of S τ we use the following transformation laws :

δ(Q +a u - a -Q -a u + a ) = 2τ 1 , δ(Q +a c a b u - b -Q -a c a b u + b ) = 2τ 2 , δ(Q +a Q - a ) = τ 1 (Q +a u - a -Q -a u + a ) + τ 2 (Q +a c a b u - b -Q -a c a b u + b ) . (3.28)
It is worthwhile to note that all three terms in the action S τ are SU (2) singlets (independent of harmonic variables), so that the harmonic integral du is in fact not necessary. After some simple algebra, making use of the integration by parts with respect to the harmonic derivatives, the constraint (3.2) and the definitions (3.4), (3.11) and (3.12), one can show that, up to a total harmonic derivative,

S λ = 1 2 S f ree cov , (3.29)
whence one obtains the representation of the gauge-covariantized q + action (3.24) solely in terms of the superfields Q ±a :

S f ree cov = 2S τ = 2 µ H Q +a Q - a - 1 4 (Q +a u - a -Q -a u + a ) 2 - 1 4 (Q +a c a b u - b -Q -a c a b u + b ) 2 . (3.30)
Now, the gauge condition (3.21) simply amounts to

Q +a u - a -Q -a u + a = 0 , Q +a c a b u - b -Q -a c a b u + b = 0 , (3.31)
and in this gauge the action (3.30) takes the standard form of the free action of the (twisted) chiral (2, 4, 2) multiplet

S f ree cov = 2 µ H Q +a Q - a = 2 dtd 4 θ Φ Φ . (3.32)
Thus the free action of the chiral N =4 multiplet arises as a particular gauge of the properly gauge-covariantized free action of the analytic multiplet q +a . Note that this equivalence, like in other cases [START_REF] Delduc | [END_REF]5], was shown here in a manifestly N =4 supersymmetric superfield approach, without any need to pass to the components. It is interesting to note that there exist two more equivalent useful forms of the action (3.24) in terms of the original superfields q ±a :

S f ree cov = µ H q +a q - a -V ++ 1 (q -a u - a ) -V ++ 2 (q -a c b a u - b ) = µ H (D ++ q -a )q - a . (3.33)
Checking the gauge invariance of the action in the second form is especially simple: one uses the transformation law (3.13) and the fact that D ++ q -a is analytic in virtue of the relation

D ++ q -a = q +a + V ++ 1 u -a + V ++ 2 c ab u - b . (3.34)
Let us now comment on the general sigma-model type action. The only invariant of the λ gauge transformations (3.1), (3.13) which one can construct from q +a and q -a is the quantity X defined as follows

X = (q +a u + a )c ---(q -a u - a )c ++ , D ±± X = ±2(q ±a u ± a )c +-∓ (q +a u - a + q -a u + a )c ±± , D ++ D --X = 2X , (D ++ ) 2 X = (D --) 2 X = 0 , (3.35)
where c ±± = c ab u ± a u ± b , etc . It admits the equivalent representation, in which its τ gauge invariance becomes manifest

X = (Q +a u + a )c ---(Q -a u - a )c ++ = 2Q (ad) c b d u + (a u - b) (3.36)
(the remaining relations in (3.35) preserve their form modulo the replacements q ±a ⇒ Q ±a ). The subclass of the general sigma-model type q + action (2.11) invariant under the gauge transformations (3.1), (3.13) is then defined as follows

S cov = µ H L(X, D ++ X, D --X, u ± ) . (3.37)
In the gauge (3.21) and in the frame (3.18) we have

X = 2i Φu + 1 u - 1 -Φu + 2 u - 2 (3.38)
and, after integration over harmonics, the action (3.37) is reduced to the most general action of the twisted chiral superfields Φ, Φ. Note the relation

Φ = i X u - 2 u + 2 - 1 2 D ++ X u - 2 u - 2 - 1 2 D --X u + 2 u + 2 .
(3.39)

The free action (3.24) can also be expressed through the universal invariant X:

S f ree cov = - 3 2 µ H X 2 . (3.40)
This relation can be proved, starting from the τ form of the free action (3.30) and integrating by parts with respect to the harmonic derivatives. Another way to see this is to compare both sides in the original λ frame representation (i.e. in terms of q ±a ) by choosing the Wess-Zumino gauge for the superfields V ±±

1 and V ±± 2 V ±± I = θ ± θ± B I .
Finally, we address the issue of the Fayet-Iliopoulos (FI) terms. In the present case one can define two independent gauge invariant FI terms

S F I 1 = iξ 1 dudζ (-2) V ++ 1 , S F I 2 = iξ 2 dudζ (-2) V ++ 2 . (3.41)
Let us consider the first one. Rewriting it in the full harmonic superspace

S F I 1 = iξ 1 µ H θ -θ-D ++ v 1 = -iξ 1 µ H (θ + θ-+ θ -θ+ )v 1 , (3.42)
expressing v 1 from the relation (3.7), integrating by parts with using the analyticity property of q +a and V ++ 1 , V ++ 2 and performing in the end the integration over harmonics, this term in the gauge (3.21) and frame (3.18) can be transformed to the expression

S F I 1 = - i 2 ξ 1 dtd 4 θ θ 1 θ2 Φ -θ 2 θ1 Φ . (3.43)
It is a particular case of twisted chiral superpotential term. Analogously,

S F I 2 = 1 2 ξ 2 dtd 4 θ θ 1 θ2 Φ + θ 2 θ1 Φ . (3.44)
It is unclear whether a general chiral superpotential can be generated from some gauge invariant q +a action.

Nonlinear chiral multiplet

Let us now consider the gauging of the two-parameter abelian symmetry (2.12b)

δq +a = Λ 1 q +a + Λ 2 c a b q +b , Λ I = Λ I (ζ, u) , I = 1, 2 . (4.1)
The harmonic constraint (2.1b) is now covariantized as

D ++ q +a -V ++ 1 q +a -V ++ 2 c a b q +b = 0 . (4.
2)

The analytic potentials V ++ I , I = 1, 2 , possess the same gauge transformation laws (3.3) and are expressed through the bridges v I with the mixed transformation rules (3.5) by the same relations (3.4). However, since now q +a transforms homogeneously under the gauge transformations, the relation (3.7) between the λ and τ world objects has to be modified:

q +a = e v 1 cos v 2 Q +a + sin v 2 c a b Q +b , Q +a = e -v 1 cos v 2 q +a -sin v 2 c a b q +b , (4.3)
or, in another form,

q +a + ic a b q +b = e v 1 -iv 2 Q +a + ic a b Q +b , q +a -ic a b q +b = e v 1 +iv 2 Q +a -ic a b Q +b . (4.4)
A direct calculation shows that the constraint (4.2) entails, for Q +a ,

D ++ Q +a = 0 , (4.5)
whence, in the central basis,

Q +a = Q ia (t, θ, θ)u + i (4.6)
(cf. (3.9)). Also, using the transformation laws (3.5) and (4.1), it is easy to find

δQ ±a = τ 1 Q ±a + τ 2 c a b Q ±b . (4.7)
In what follows it will be convenient to choose the SU (2) frame (3.18) in which

q +1 = e (v 1 +iv 2 ) Q +1 , q +2 = e (v 1 -iv 2 ) Q +2 , (4.8
)

δQ 11 = (τ 1 + iτ 2 ) Q 11 , δQ 22 = (τ 1 -iτ 2 ) Q 22 , δQ 21 = (τ 1 + iτ 2 ) Q 21 , δQ 12 = (τ 1 -iτ 2 ) Q 12 . (4.9)
Like in the previous Section, the analyticity of q +a implies the "covariant analyticity" for Q +a :

D + q +1 = D+ q +1 = 0 ⇔ D + + D + (v 1 + iv 2 ) Q +1 = D+ + D+ (v 1 + iv 2 ) Q +1 = 0 , (4.10)
D + q +2 = D+ q +2 = 0 ⇔ D + + D + (v 1 -iv 2 ) Q +2 = D+ + D+ (v 1 -iv 2 ) Q +2 = 0 . (4.11)
From the analyticity of V ++ 1,2 it follows that the gauge connections in (4.10), (4.11) are linear in harmonics

D + (v 1 + iv 2 ) = A a (+) (t, θ, θ)u + a , D + (v 1 -iv 2 ) = A a (-) (t, θ, θ)u + a , D+ (v 1 -iv 2 ) = -Āa (+) (t, θ, θ)u + a , D+ (v 1 + iv 2 ) = -Āa (-) (t, θ, θ)u + a . (4.12)
Now it is time to properly fix the τ gauge freedom (4.9). Assuming that Q 12 = -(Q 21 ) possesses a non-zero constant background, a convenient gauge is

Q 12 = -Q 21 = 1 . (4.13)
Substituting this gauge into the covariant analyticity conditions for Q +a in (4.11), taking into account the relations (4.6), (4.12), and equating to zero the coefficients of three independent products of harmonics ((u + 1 ) 2 , (u + 2 ) 2 and u + 1 u + 2 ), we obtain

A 2 (+) = A 1 (-) = 0 , A 1 (+) = D 2 Φ , A 2 (-) = -D 1 Φ , (and c.c.) , D 1 Φ + ΦD 2 Φ = D1 Φ + Φ D2 Φ = 0 , D 2 Φ -ΦD 1 Φ = D2 Φ -Φ D1 Φ = 0 , (4.14)
where

Q 11 ≡ Φ , Q 22 = Φ .
Thus the only independent object that remains in the gauge (4.13) is a complex N =4 superfield Φ subjected to the constraints (4.14). These constraints are a nonlinear version of the twisted chirality constraints (3.23) and are easily recognized as a twisted version of the nonlinear N =4 chirality constraints [13]. It can be given the form of the ordinary nonlinear chirality constraints by relabelling the covariant derivatives just in the same way as in the case of the linear constraints (3.23).

It is worth mentioning a specific feature of the nonlinear chiral multiplet case as compared with the linear multiplet case. The original λ world constraints (2.1a), (4.2) preserve the whole automorphism SU (2) R group acting on the doublet indices of harmonics and Grassmann coordinates and break the Pauli-Gürsey SU (2) P G symmetry realized on the doublet index a of q +a down to a U (1) subgroup (due to the presence of the constant triplet c ab in (4.2)). The same symmetry structure is exhibited by the τ world constraints (4.5), (4.11) which are equivalent to (4.2) and the analyticity condition (2.1a). Before fixing the τ frame gauge as in (4.13), the superfields Q ia are transformed by SU (2) R linearly, in the same way as the spinor derivatives D i , Di , i.e. as

δ R Q ia ≃ Q ia′ (t, θ ′ , θ′ ) -Q ia (t, θ, θ) = λ i k Q ka , (λ ik ) = λ ik . λ i i = 0 , (4.15)
Here λ ik are constant SU (2) R parameters. After imposing the gauge (4.13), this transformation law becomes nonlinear, as it must be accompanied by the compensating τ gauge transformation needed for preserving (4.13)

(τ 1 -iτ 2 ) comp = -λ 12 + λ 11 Φ , (τ 1 + iτ 2 ) comp = λ 12 + λ 22 Φ , (4.16
)

whence δ R Φ = λ 11 + 2λ 12 Φ + λ 22 (Φ) 2 and c.c. , (4.17)
i.e. Φ and Φ are transformed as projective CP 1 coordinates of the 2-sphere S 2 ∼ SU (2) R /U (1) R . The constraints (4.14), where D i and Di are still transformed linearly with respect to their doublet indices and Φ , Φ are transformed according to the nonlinear transformation rule (4.17), are directly checked to be SU (2) R covariant. This interpretation of the nonlinear chiral N =4, d=1 superfields as parameters of S 2 was the starting point of the derivation of the constraints (4.14) in [13] (in an N =4 superspace parametrization twisted as compared to ours). Note that both Φ and Φ are inert under the U (1) remnant of the broken SU (2) P G symmetry. Let us now discuss how the actions of the nonlinear chiral multiplet are reproduced from the gauged q + actions.

The standard free action of q +a is obviously not invariant even under the rigid version of (4.1) due to the presence of the rescaling isometry in (4.1). This situation is quite similar to what we faced in [START_REF] Delduc | [END_REF] when deriving the action of the nonlinear (3, 4, 1) multiplet from the q + action with a gauged rescaling invariance. It was shown there that the simplest invariant action is a nonlinear action of the sigma-model type. In the case considered here we should start from the action which is simultaneously invariant under the rescalings and the U (1) transformations. The unique object invariant under both gauged isometries is constructed as

Y = q +a c ab q -b (q +a q - a ) , (4.18)
where q -a ≡ D --q +a -V --

1 q +a -V -- 2 c a b q +b (4.19)
and V -- 1,2 were defined in (3.12). It is the true nonlinear analog of the invariant X defined in (3.35). It is easy to check that Y admits an equivalent representation in terms of the τ world objects, such that it is manifestly invariant under the τ gauge transformations (4.7)

Y = Q +a c ab Q -b (Q +a Q - a ) , (4.20)
and satisfies the relations

(D ++ ) 2 Y = (D --) 2 Y = 0 , D ++ D --Y = 2Y , (D ++ Y D --Y ) -Y 2 = 1 . (4.21)
In the gauge (4.13) and frame (3.18)

Y = 2i 1 + Φ Φ Φ u + 2 u - 2 -Φ u + 1 u - 1 + 1 2 1 -Φ Φ u + 1 u - 2 + u + 2 u - 1 . (4.22)
A nonlinear analog of the relation (3.39) is

Φ 1 + Φ Φ = i Y u - 2 u + 2 - 1 2 D ++ Y u - 2 u - 2 - 1 2 D --Y u + 2 u + 2 . ( 4

.23)

From this relation and its conjugate it is easy to express Φ and Φ through Y and its harmonic derivatives.

The general sigma-model type action of the nonlinear chiral multiplet corresponds to the following gauged subclass of the general q +a actions

S chn = µ H L(u, Y, D ++ Y, D --Y ) . ( 4

.24)

After expressing Y in terms of the τ world objects, choosing the gauge (4.13), SU (2) frame (3.18) and preforming the integration over harmonics, (4.24) becomes the general sigma model action of the nonlinear chiral multiplet Φ, Φ as it was given in [START_REF] Bellucci | [END_REF]. Let us point out that this correspondence, as in other similar cases [START_REF] Delduc | [END_REF]5], allows one to equivalently deal with the action in the original λ frame q + representation by choosing the appropriate Wess-Zumino gauge for the "topological" gauge superfields V ±± 1,2 and using the residual two-parameter gauge freedom to trade two out of four original physical bosonic fields of q +a for two d=1 "gauge fields". The latter become just two auxiliary fields of the nonlinear chiral multiplet. To work in the q + representation is in some aspects easier than to use the τ frame where the same action looks as the general action of the superfields Φ and Φ .

It is interesting that in the case under consideration there also exists a unique SU (2) R invariant action of the WZW type which is gauge invariant up to a total derivative in the integral and in this sense is an analog of the free action of the linear chiral multiplet in the form (3.24), (3.25). The quantity

F = q +a q - a = e 2v 1 Q +a Q - a (4.25)
is manifestly SU (2) R invariant and invariant under the gauge Λ 2 transformations, while its Λ 1 transformation reads

δ Λ 1 F = 2Λ 1 F , δ Λ 1 log F = 2Λ 1 . (4.26)
Then the action

S su(2) = µ H log F = µ H 2v 1 + log(Q +a Q - a ) (4.27)
is gauge invariant since the full N =4 superspace integral of the analytic parameter Λ 1 vanishes. Using (4.4), it is easy to check that the bridge v 1 , modulo a constant and purely analytic term which vanish after integration, in the gauge (4.13) is reduced to

v 1 ⇒ log 1 -u + 1 u - 2 + 2Φu + 1 u - 1 + log 1 + u + 2 u - 1 + 2 Φu + 2 u - 2 .
(4.28)

It follows from the constraints (4.14) that the integral of any holomorphic or antiholomorphic function of Φ, Φ over the full N =4 superspace vanishes. So the bridge v 1 drops out from (4.27) in the gauge (4.13) and, taking into account that in this gauge It describes the N =4 superextension of the d=1 sigma model on S 2 ∼ SU (2) R /U (1) R and was derived in this form for the first time in [13]. The Lagrangian in (4.30) is just the corresponding Kähler potential3 . Actually, using eq. ( 4.23), we could write an equivalent representation for S su(2) through the universal tensorial invariant Y . However, the SU (2) R symmetry in such a representation is not manifest due to the presence of explicit harmonics over which one should integrate. Finally, we discuss the structure of two FI terms in the present case. They are originally given by the same V ++ 1,2 actions (3.41) as in the linear case. Further, passing to the integral over the full N =4 superspace, using the relation

Q +a Q - a = 1 + Φ Φ , ( 4
v 1 -iv 2 = log (q +a + ic a b q +b )u - a -log (Q +a + ic a b Q +b )u - a (4.31)
As a consequence of (5.4), the superfield Q +a is homogeneous in the harmonic variables

D ++ Q +a = 0 ⇒ Q +a (t, θ, θ, u) = Q ba (t, θ, θ)u + b , (Q ab ) = Q ab . (5.8)
The harmonic-independent superfields Q ba transform as

δQ ba = τ 1 Q ba -τ 2 ǫ ba .
(5.9)

The real parameter τ 2 may be used to gauge away the antisymmetric part of the tensor Q ba :

Q ab = Q (ab) . (5.10)
Then the target space scale invariance with the real parameter τ 1 may be used to fix the value of a component of this tensor. We choose the gauge

Q 21 = Q 12 = i ⇒ Q +1 = Q 11 u + 1 + iu + 2 , Q +2 = Q 22 u + 2 + iu + 1 . (5.11)
We now should take into account the analyticity constraints on the original superfields q +a . When expressed in terms of the new superfields, these constraints become

D + Q +a + D + v Q +a + (D + w + wD + v) u +a = 0 , D+ Q +a + D+ v Q +a + ( D+ w + w D+ v) u +a = 0 . (5.12)
Due to the analyticity of the gauge superfields V ++ , W ++ , the fermionic connections D + v, D + w + wD + v have a simple dependence on harmonic variables

D ++ D + v = 0 ⇒ D + v = A a (t, θ, θ)u + a , (5.13)
D ++ (D + w + wD + v) = 0 ⇒ D + w + w D + v = B a (t, θ, θ)u + a . (5.14)
Analogously, for the conjugate connections we have

D+ v = -Āa u + a , D+ w + w D+ v = -Ba u + a . (5.15)
Then, the constraints (5.12) imply, in ordinary superspace,

D 2 Q 11 + A 2 Q 11 + iA 1 -B 1 = 0 , B 2 = iA 2 , D 1 Q 22 + A 1 Q 22 + iA 2 + B 2 = 0 , B 1 = -iA 1 , (5.16)
D 1 Q 11 + A 1 Q 11 = 0 , D 2 Q 22 + A 2 Q 22 = 0 .
(5.17)

From eq. (5.16) one expresses the gauge connections

A 1 = iB 1 = 1 4 + Q 11 Q 22 2iD 2 Q 11 -Q 11 D 1 Q 22 , A 2 = -iB 2 = 1 4 + Q 11 Q 22 2iD 1 Q 22 -Q 22 D 2 Q 11 .
(

Substituting these expressions into eqs. (5.17) and then using complex conjugation yield the full set of the nonlinear constraints on the superfields

Q 11 ≡ Φ , Q 22 = Φ D 1 Φ + Φ 4 + Φ Φ 2iD 2 Φ -ΦD 1 Φ = 0 , D1 Φ + Φ 4 + Φ Φ 2i D2 Φ -Φ D1 Φ = 0 , (5.19) D 2 Φ + Φ 4 + Φ Φ 2iD 1 Φ -ΦD 2 Φ = 0 , D2 Φ + Φ 4 + Φ Φ 2i D1 Φ -Φ D2 Φ = 0 . (5.20)
Equations (5.19) and (5.20) may be interpreted as (twisted) non-linear chirality constraints on the superfields Φ and its complex conjugate Φ .

Let us now define the subclass of the general q + actions which respects the invariance under the rigid transformations (2.13) and, after gauging, under their local counterparts (5.1). In the τ frame it should yield the general sigma-model type action of the nonlinear chiral multiplet in question.

The invariance under the shift transformations in (2.13) just means that the corresponding superfield Lagrangian cannot depend on the trace part in q ia u + i , i.e. q ia → q (ib) . Next, the invariance under the target space scale transformations constrains the action to depend only on two independent ratios of three components of q (ab) , i.e. q (ab) → q 11 /q 12 , q 22 /q 12 . In other words, the appropriate general superfield q +a Lagrangian should be an arbitrary function of the superfields q 11 /q 12 , q 22 /q 12 which can be interpreted as projective coordinates of some twosphere S 2 . Being reformulated in terms of the superfields q ±a , this requirement amounts to the following particular choice of the q + lagrangian

L = L u ± , q±a |q| , (5.21)
where

q±a = q ±a - 1 2 u ±a q +b u - b -q -b u + b , |q| 2 = q+a q- a = (q +a u + a )(q -b u - b ) - 1 4 (q +a u - a + q -a u + a ) 2 = 1 2 q (ab) q (ab) . (5.22)
As in the previous case, the standard free action of q +a is not invariant under the target space rescalings. The Lagrangians from the subclass (5.21) are of the sigma-model type, with nonconstant bosonic target metrics. The gauging of these Lagrangians goes in the standard way, by subjecting q +a to the covariantized constraint (5.4) and defining q -a in a gauge-covariant way as q -a = D --q +a -V --q +a -W --u +a , δq -a = Λ 1 q -a + Λ 2 u -a .

(5.23)

The final gauge-covariantized action has the same form as the rigidly invariant one (5.21) but with the superfields q ±a defined in a gauge-covariant way. Just due to this covariance, the basic objects (5.22) admit the equivalent τ frame representation

q±a = e v Q ±a - 1 2 u ±a Q +b u - b -Q -b u + b ≡ e v Q±a , |q| 2 = e 2v (Q +a u + a )(Q -b u - b ) - 1 4 Q +a u - a + Q -a u + a 2 = 1 2 e 2v Q (ab) Q (ab) ≡ e 2v | Q| 2 .
(5.24)

The covariantized superfield argument in (5.21) does not depend on the bridge v , whence

L = L u ± , q±a |q| = L u ± , Q±a | Q| . (5.25)
As a result, in this gauge the superfields Φ and Φ are nonlinearly transformed under SU (2

) ′ R δ R ′ Φ = 2λ 12 Φ -iλ 11 2 + Φ Φ + iλ 22 (Φ) 2 , δ R ′ Φ = (δ R ′ Φ) , (5.34)
and so can be treated as coordinates of the coset S 2 ∼ SU (2) ′ R /U (1) ′ R in a particular parametrization. Obviously, there should exist an equivalence transformation to the stereographic projection parametrization in which the S 2 coordinates are transformed according to the holomorphic law (4.17). The precise form of this field redefinition is as follows

χ = i Φ 1 + √ 1 + Φ Φ , Φ = -2i χ 1 -χ χ , (5.35)
δ R ′ χ = λ 11 + 2λ 12 χ + λ 22 (χ) 2 and c.c. .

(5.36)

The transformation law (5.36) coincides with (4.17), which suggests that in this new holomorphic parametrization the constraints (5. 19), (5.20) take the form (4.14). Indeed, a simple calculation shows that after the field redefinition (5.35) the constraints (5. 19), (5.20) are equivalently rewritten as D 1 χ + χD 2 χ = 0 , D1 χ + χ D2 χ = 0 (and c.c.) .

(5.37)

Thus we see that the nonlinear chiral multiplet considered in this Section is in fact a disguised form of the nonlinear (twisted) chiral multiplet of ref. [13] rederived within the gauging procedure in the previous Section. This is rather surprising, because in the two cases we gauged two essentially different two-parameter groups, respectively, abelian and non-abelian ones (2.12b) and (2.13). The identity of these two multiplets amounts to the identity of their general actions, despite the fact that the classes of the appropriate q + actions one starts with in these two cases are essentially different. Here we again encounter the phenomenon of non-uniqueness of the inverse oxidation procedure as compared with the target space dimensional reduction [START_REF] Delduc | [END_REF]5]: the same off-shell multiplet can be recovered by gauging some non-equivalent isometries of the "root" multiplet. For instance, the N =4, d=1 multiplet [START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF][START_REF] Delduc | [END_REF][START_REF] Kuznetsova | [END_REF] and its most general action can be obtained from the q + multiplet and the appropriate set of the q + actions by gauging either the non-abelian SU (2) P G group or the abelian group of three independent shift isometries of q +a [5]. Basically, the difference between these two gauging procedures lies only in the fact that they start from different subclasses of the general set of q + actions. However, the final action of the reduced multiplet does not "remember" from which parent q + action it originated.

Taking for granted that all off-shell N =4, d=1 superfields can be recovered from the q +a superfield by gauging different symmetries realized on the latter and taking into account that only three independent two-parameter groups (defined in (2.12) and (2.13)) can be implemented on q +a , we conclude that only two essentially different off-shell N =4, d=1 multiplets with the content (2, 4, 2) exist: the standard linear chiral multiplet and the nonlinear chiral multiplet introduced in [13]. Any other version of the chiral multiplet should be reducible to one of these two via some field redefinition.

There is one more way to see that the constraints (5.19), (5.20) are equivalent to (4.14). After some algebra, using (5.19), (5.20) at the intermediate steps, the expressions for the spinor connections (5.18) can be cast in the following form

A 1 = D 2 iΦ 1 + √ 1 + Φ Φ -D 1 log 1 + 1 + Φ Φ , A 2 = D 1 i Φ 1 + √ 1 + Φ Φ -D 2 log 1 + 1 + Φ Φ .
(5.38)

Conclusions

In this article and two previous papers [START_REF] Delduc | [END_REF]5] we showed that all known off-shell N =4, d=1 multiplets with 4 physical fermions can be reproduced from the basic ("root") multiplet (4, 4, 0) by gauging some symmetries, abelian or non-abelian, realized on this multiplet. The corresponding general N =4 mechanics actions are recovered as the result of the proper gauge-fixing in the appropriate gauged subclasses of the general q + action, the subclasses which enjoy invariance under the symmetries just mentioned. Our gauging procedure uses the manifestly supersymmetric universal language of N =4 superspace and does not require to resort to component considerations at all. Another merit of our approach is that it reduces the whole set of non-equivalent superfield actions of the N =4 mechanics models to some particular cases of the generic q + action extended by non-propagating "topological" gauge superfields. Just the presence of the latter enables one to preserve the manifest supersymmetry at each step and to reveal the irreducible off-shell superfield contents of one or another model by choosing the appropriate superfield gauges and (in the cases considered in the present paper) by passing to the equivalent τ frame formulations. The alternative (and in many cases more technically feasible) way of doing suggested by the gauging approach is to always stay in the initial q + representation where the harmonic analyticity is manifest and to choose the WZ gauge for the relevant analytic non-propagating gauge superfields. Each "topological" gauge multiplet in the WZ gauge contributes just one scalar ("gauge") field which, after fully fixing the residual gauge freedom, becomes an auxiliary field of the new off-shell N =4 multiplet related to the q + multiplet via linear or nonlinear versions of the "automorphic duality" [START_REF] Gates | On Extended Supersymmetric Quantum Mechanics[END_REF]. Thus in the component formulation our approach automatically yields the explicit realization of this intrinsically onedimensional off-shell duality. The distinctions between various types of this duality are related to the differences between the global symmetry groups subjected to gauging. The basic peculiarity of the cases considered in this paper as compared to those treated in [START_REF] Delduc | [END_REF]5] is that the superfields describing the N =4 multiplets (2, 4, 2) do not "live" on the N =4 analytic harmonic subspace (as distinct from the multiplets (0, 4, 4), [START_REF] De Lima Rodrigues | The quantum mechanics SUSY algebra: an introductory review[END_REF][START_REF] Delduc | [END_REF][START_REF] Kuznetsova | [END_REF] and (3, 4, 2)). They are most naturally described after passing to the equivalent "τ frame" [16,[START_REF] Galperin | Harmonic Superspace[END_REF], with the ordinary N =4 superfield gauge parameters and the harmonic superfield bridges to the "λ frame" as the basic gauge objects. These bridges ensure the equivalence of the manifestly analytic λ frame picture one starts with and the picture in the τ frame. In the τ frame, the original gauge-covariantized analyticity-preserving harmonic constraints on the superfield q +a amount to the harmonic independence of the involved superfields. The harmonic Grassmann analyticity, which is manifest in the λ frame, in the τ frame amounts to the covariant analyticity conditions. After properly fixing N =4 supersymmetric τ gauges, these conditions become the linear or nonlinear N =4 chirality conditions, depending on which two-parameter symmetry group realized on q +a is subjected to gauging. There are only three such groups and they are listed in (2.12) and (2.13). We considered gauging of all these three groups and found that the gauging of the group (2.12a) leads to the linear chiral N =4 multiplet, while gaugings of (2.12b) and (2.13) lead to the same nonlinear chiral multiplet [13], despite the obvious non-equivalence of these two groups. This non-uniqueness is a manifestation of the general non-uniqueness of the oxidation procedure as inverse to the automorphic duality. Since only three two-parameter symmetries can be realized on q +a , from our results it follows, in particular, that no other non-equivalent nonlinear chiral N =4 multiplet can be defined.

Interesting venues for further applications of our gauge approach are provided by models of N =8 supersymmetric mechanics (see e.g. [19,[START_REF] Ivanov | Hierarchy of N=8 Mechanics Models[END_REF] and refs. therein). It was argued in [START_REF] Ivanov | Hierarchy of N=8 Mechanics Models[END_REF], by considering a wide set of examples, that the off-shell N =8 multiplet (8, 8, 0) is the true N =8 analog of the "root" N =4 multiplet (4, 4, 0) and that the whole set of the component actions of the N =8 mechanics models with 8 physical fermions (and finite numbers of auxiliary fields) follow from the general action of this basic N =8 multiplet via a linear version of the automorphic duality. It would be interesting to apply our techniques to these cases. Recall that our approach is bound by the requirement that the symmetries to be gauged commute with supersymmetry. In the N =8 case the target space scale and shift transformations still obey this criterion, so one can hope that the gauging would nicely work in this case too and could help to understand the relationships between the multiplet (8, 8, 0) and the rest of the N =8 multiplets in a manifestly N =4 supersymmetric superfield fashion. We can also hope to discover in this way new nonlinear N =8 multiplets and the corresponding new N =8 mechanics models, besides those already known [14,15]. The primary question to be answered is how to define an N =8 analog of the N =4 topological gauge multiplet which plays a crucial role in our approach. Another possible way of extending our study is to construct "topological" N =4, d=1 supergravity multiplets and to gauge, with their help, the R-symmetry SU (2) groups of N =4 supersymmetry, with new models of N =4 mechanics as an outcome. Finally, let us note that the nonlinear chiral multiplets exist also in dimensions d>1 [13], e.g. in d=3 [START_REF] Bellucci | [END_REF]. It would be interesting to inquire whether they can also be derived by gauging some symmetries realized on the appropriate analytic harmonic superfields, some analogs of q +a , i.e. whether they are also a disguised τ frame form of the harmonic analyticity conditions.

  .29) we obtain the simple final expression for the action (4.27) S su(2) = dtd 4 θ log(1 + Φ Φ).(4.30)

We use the same notation for the unrelated constant triplets in (2.12a) and (2.12b), hoping that this will not give rise to any confusion.

Both these R-symmetry SU (2) are manifest in the quartet notationD ii = (D i , Di ) = ( Di , Di ) , D ii = -D ii = (-Di , D i ) = (-Di , Di ) .

Formally, (4.30) looks also invariant under SU (2) P G , however the constraints (4.14) do not respect this second SU (2), so the latter is not a symmetry of (4.30).

Acknowledgements

The work of E.I. was supported in part by the RFBR grant 06-02-16684, the RFBR-DFG grant 06-02-04012-a, the grant DFG, project 436 RUS 113/669/0-3, the grant INTAS 05-7928 and a grant of Heisenberg-Landau program. He thanks Laboratoire de Physique, UMR5672 of CNRS and ENS Lyon, for the warm hospitality extended to him during the course of this work.

and its conjugate (they follow from (4.4)), as well as the analyticity of q +a , and, finally, performing the integration over harmonics in the gauge (4.13), the FI terms can be expressed through Φ, Φ as follows

Surprisingly, they are still linear in Φ, Φ, like their customary chiral superfield analogs (3.43), (3.44). Note that the SU (2) R invariance of (4.32) can be checked with the help of the basic constraints (4.14), taking into account that SU (2) R transforms the explicit θs in (4.32) in the standard way (rotates them in the doublet index), while Φ and Φ are transformed according to the law (4.17) and its conjugate.

Nonabelian gauge group

Let us now consider gauging of the last two-parameter group admitting a realization on the analytic superfield q +a , the non-abelian solvable group (2.13) consisting of a dilatation and a shift of q +a . The gauge transformation laws are

where as before Λ 1 and Λ 2 are charge-zero analytic superfields. Due to the nonabelian character of this group, its gauging is a little bit more tricky as compared to the previous two (abelian) cases. The commutation relations of the gauge transformations are given by

In order to covariantize the harmonic constraints, we need to introduce two analytic gauge superfields V ++ , W ++ with the transformation laws

3)

It is easy to check that the Lie bracket of two such transformations has the form (5.2). The covariant harmonic constraint on the superfield q +a now reads D ++ q +a -V ++ q +a -W ++ u +a = 0 .

(5.4)

The gauge superfields V ++ and W ++ are expressed through the corresponding non-analytic bridge superfields v, w as

As in other cases, the bridges v, w transform under two types of gauge transformations, the original ones with the analytic parameters Λ 1 (ζ, u) and Λ 2 (ζ, u), and new ones with the parameters τ 1 (t, θ, θ) and τ 2 (t, θ, θ) which are independent of harmonic variables:

We also define the new non-analytic "τ -world" superfield

(5.7)

In the gauges (5.10) and (5.11):

and the action corresponding to the Lagrangian (5.25), after performing the integration over harmonics, becomes the general off-shell action of the supermultiplet Φ, Φ. Note that the relations (5.26), like analogous relations of the previous cases, are invertible:

(5.27)

This ensures the possibility to express Φ, Φ through the basic gauge invariant object, q±a /|q| = Q±a /| Q| , and in fact proves the equivalence of the general N =4 action of superfields Φ, Φ and the particular class of the gauged q + actions defined above.

It is worth noting that the building blocks of the λ world gauge invariants can be successively reproduced from the simplest invariant of the shift λ gauge transformation (with the parameter Λ 2 )

q ++ = q +a u + a = q+a u + a , δq ++ = Λ 1 q ++ .

(5.28)

Acting on (5.28) by the covariant derivative D ---V --, we can produce new non-analytic superfields which are invariant under the Λ 2 transformations and covariant with respect to the Λ 1 transformations:

They are related to the superfields q ±a and q±a by

and can be used to form two independent gauge invariant ratios

q ++ q ---(q +-) 2 , X --= q -- q ++ q ---(q +-) 2 , (5.31) which are just independent harmonic projections of the superfield argument in (5.25).

Let us now dwell on the peculiarities of the realization of SU (2) R and SU (2) P G symmetries on the superfields Φ and Φ and the surprising relation to the nonlinear chiral multiplet discussed in the previous Section.

The basic gauge covariant constraint (5.4) clearly breaks the original SU (2) R × SU (2) P G symmetry realized on q +a , Grassmann and harmonic coordinates down to the diagonal Rsymmetry group SU (2) ′ R which uniformly rotates all doublet indices. The gauge (5.10) is

(5.32)

The gauge (5.11) is not preserved under (5.32), and in order to restore this gauge one should accompany the SU (2) ′ R transformations by a compensating τ 1 transformation with (τ 1 ) comp = i λ 22 Φλ 11 Φ .

(5.33) Substituting these expressions and their complex conjugates into (5.17) and complex conjugates of (5.17), we recover (5.37), with χ being related to Φ just by eqs. (5.35). Finally, as an instructive example, we present the SU (2) ′ R invariant action in terms of the original superfield variables, as well as the relevant FI term.

The SU (2) ′ R invariant action is given by an expression similar to (4.27)

It is manifestly invariant under the gauge shift Λ 2 transformation (since q+a is invariant), as well as under the scale and shift τ gauge transformations. It is also invariant under the scale gauge Λ 1 transformations since under the latter the Lagrangian in (5.39) is shifted by an analytic gauge parameter the integral of which over the full N =4 superspace vanishes:

To find the precise form of the action in terms of the nonlinear chiral superfields Φ, Φ, we should make use of eq. ( 5.26) and also compute the bridge part of the N =4 superspace integral in (5.39):

(5.41)

This integral can be evaluated by taking one spinor derivative, say D + , off the measure d 4 θ, throwing it on v, expressing D + v as in (5.13), doing the harmonic integral du, substituting the gauge-fixed expressions (5.38) for the spinor connections A 1 , A 2 and, finally, restoring the full Grassmann measure by taking the spinor derivatives D 1 , D 2 off these expressions. It turns out that only the second terms in the expressions (5.38) contribute, and we obtain Thus we obtain the expected result that the action (5.39), (5.43) is in fact identical to the previously considered SU (2) R invariant action (4.30).

As for the FI terms, in the present case only the gauge superfield V ++ possesses an abelian gauge transformation law, so one is able to construct only one FI term:

(5.44)

Inserting the identities 1 = D + θ -, 1 = -D+ θinto the round brackets in the r.h.s. of (5.44), integrating by parts with respect to spinor derivatives, using the relations (5.13) with (5.38) and their conjugates, doing harmonic integral and, at the end, integrating by parts once again, one finally finds

which coincides with one of the FI terms in (4.32).