Motivated by a recent interest in curved rigid supersymmetries, we construct a new type of N =4, d=1 supersymmetric systems by employing superfields defined on the cosets of the supergroup SU(2|1). The relevant worldline supersymmetry is a deformation of the standard N =4, d=1 supersymmetry by a mass parameter m. As instructive examples we consider, at the classical and quantum levels, the models associated with the supermultiplets (1,4,3) and (2,4,2) and find out interesting interrelations with some previous works on non-standard d=1 supersymmetry. In particular, the d=1 systems with "weak supersymmetry" are naturally reproduced within our SU(2|1) superfield approach as a subclass of the (1,4,3) models. A generalization to the N =8, d=1 case implies the supergroup SU(2|2) as the candidate deformed worldline supersymmetry.
Using the worldline SU (2|1) superfield approach, we construct N = 4 superconformally invariant actions for the d = 1 multiplets (1, 4, 3) and (2, 4, 2). The SU (2|1) superfield framework automatically implies the trigonometric realization of the superconformal symmetry and the harmonic oscillator term in the corresponding component actions. We deal with the general N = 4 superconformal algebra D(2, 1; α) and its central-extended α = 0 and α = −1 psu(1, 1|2) ⊕ su(2) descendants. We capitalize on the observation that D(2, 1; α) at α = 0 can be treated as a closure of its two su(2|1) subalgebras, one of which defines the superisometry of the SU (2|1) superspace, while the other is related to the first one through the reflection of µ, the parameter of contraction to the flat N = 4, d = 1 superspace. This closure property and its α = 0 analog suggest a simple criterion for the SU (2|1) invariant actions to be superconformal: they should be even functions of µ. We find that the superconformal actions of the multiplet (2, 4, 2) exist only at α = −1, 0 and are reduced to a sum of the free sigma-model type action and the conformal superpotential yielding, respectively, the oscillator potential ∼ µ 2 and the standard conformal inverse-square potential in the bosonic sector. The sigma-model action in this case can be constructed only on account of non-zero central charge in the superalgebra su(1, 1|2).
We construct a new version of the worldline SU(2|1) superspace as a deformation of the standard N = 4, d = 1 superspace and show that it naturally provides off-and on-shell description of general supersymmetric Kähler oscillator model considered earlier at the classical level within the Hamiltonian approach. The basic object is a generalized chiral SU(2|1), d = 1 superfield with the off-shell field content (2, 4, 2). The frequency of the oscillator and the strength of external magnetic field are defined by two parameters: the contraction parameter m and the new parameter λ which reflects the freedom in defining the chiral SU(2|1), d = 1 superspace. We treat both classical and quantum cases.1 This study was partly motivated by a recent interest in the theories possessing curved rigid analogs of the Poincaré supersymmetry (see, e.g., [2,3] and refs. therein).
We define the worldline harmonic SU(2|1) superspace and its analytic subspace as a deformation of the flat N = 4, d = 1 harmonic superspace. The harmonic superfield description of the two mutually mirror off-shell (4, 4, 0) SU(2|1) supermultiplets is developed and the corresponding invariant actions are presented, as well as the relevant classical and quantum supercharges. Whereas the σ-model actions exist for both types of the (4, 4, 0) multiplet, the invariant Wess-Zumino term can be defined only for one of them, thus demonstrating non-equivalence of these multiplets in the SU(2|1) case as opposed to the flat N = 4, d = 1 supersymmetry. A superconformal subclass of general SU(2|1) actions invariant under the trigonometric-type realizations of the supergroup D(2, 1; α) is singled out. The superconformal Wess-Zumino actions possess an infinite-dimensional supersymmetry forming the centerless N = 4 super Virasoro algebra. We solve a few simple instructive examples of the SU(2|1) supersymmetric quantum mechanics of the (4, 4, 0) multiplets and reveal the SU(2|1) representation contents of the corresponding sets of the quantum states.
SU(2|1) supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an N = 4 supersymmetrization of the quantum U(2) spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra su(2|1). The full system admits an SU(2|1) covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum SU(2|1) generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.2 It was shown in [17], that the centrally extended superalgebra su(2|1) can be represented as a semi-direct sum of su(2|1) and an extra R-symmetry generator: su(2|1) ≃ su(2|1)+ ⊃ u(1). The central charge is a combination of the R-symmetry generator and the internal U(1) generator of su(2|1). In the models under consideration the central charge operator is identified with the canonical Hamiltonian.3 The kinetic term of the variables Z k a in the action (1.5) is of the first-order in the time derivatives, in contrast to the dynamical variable X a b with the second-order kinetic term. Just for this reason we call Z k a ,Z a k semi-dynamical variables. In the Hamiltonian (see below), they appear only in the interaction terms and enter through the SU(2) current S (ik) .
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