Superconformal mechanics inSU(2|1)superspace

Abstract

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).

Deformed supersymmetric quantum mechanics with spin variables

“…which are obtained from (4.3) just through the substitution m → −m. This feature is typical for the trigonometric type of superconformal symmetry [23]. As we will see below, the closure of these two types of transformations gives the superconformal group D(2, 1; −1/2) ∼ = OSp(4|2), where the superconformal Hamiltonian is defined as…”

“…[17,18,19,20] to the case of non-zero mass. On the other hand, in the papers [21,22,23,24] (see also [25,26]) there were found different realizations of N = 4, d = 1 superconformal groups and established some relations between the deformed supersymmetries and superconformal symmetries. One of the purposes of our paper is to clarify the role of the deformed SU(2|1) supersymmetry and its superconformal extension in the quantum spectrum of the systems with semi-dynamical variables.…”

Three-dimensional superconformal quantum mechanics with sl(2|1) dynamical symmetry

“…We mention that several different, both classical and quantum, supersymmetric models possessing (a real form of) sl(2|1) as dynamical symmetry have been investigated in the literature, see [31,25,26,32,33,29,34]. These models do not correspond to the three-dimensional Hamiltonians here presented.…”

A world-line framework for 1D topological conformal σ-models

“…In the λ → 0 limit, a representation of the superconformal algebra which only depends on ρ,ρ emerges. The corresponding supermultiplet which emerges in this limit has been called the inhomogeneous supermultiplet in [2] (see also [25]). The construction of the inhomogeneous superconformal actions has been discussed at length in that paper; since it can be straightforwardly applied to the present twisted case, it is sufficient to present here the final results.…”