On the classification of N-extended supersymmetric quantum mechanical systems

Abstract

In this paper some properties of the irreducible multiplets of representation for the N = (p, q) -extended supersymmetry in one dimension are discussed. Essentially two results are here presented. At first a peculiar property of the one dimension is exhibited, namely that any multiplet containing 2d (d bosonic and d fermionic) particles in M different spin states, is equivalent to a {d, d} multiplet of just 2 spin states (all bosons and all fermions being grouped in the same spin). Later, it is shown that the classification of all multiplets of this kind carrying an irreducible representation of the N -extended supersymmetry is in one-to-one correspondence with the classification of real-valued Clifford Γ-matrices of Weyl type. In particular, p + q is mapped into D, the space-time dimensionality, while 2d is determined to be the dimensionality of the corresponding Γ-matrices. The implications of these results to the theory of spinning particles are analyzed. *

Chiral and Real N = 2 supersymmetric -conformal Galilei algebras

“…At first one considers the D-module reps of the supersymmetry in 0+1 dimensions (for N = 4, 8, see [3,4], they are given by the (k, N , N −k) field multiplets for k = 0, 1, . .…”

“…Our construction of the D-module reps for the cases at hand is based on the classified D-module reps for the relevant subalgebras, such as the N -extended supersymmetry in 0 + 1 dimensions (see [3] and [4]) and the D-module reps of the finite, simple, N -extended one-dimensional superconformal algebras (see [5] and [6]). …”

“…For a non-negative integer or half-integer value of the parameter they admit a consistent truncation as non semi-simple, finite, centerless, Lie superalgebras. The non-vanishing (anti)commutators of G (2,2) are presented in formulas (2) and (3). The non-vanishing (anti)commutators of G (1,2,1) are given in (2) and (4).…”

“…In this respect two main features are crucial for implementing the whole procedure. The first one requires the knowledge of the D-module reps, given in [3,4], of the algebra of the N -Extended Supersymmetric Mechanics in (0+1)-dimension (the algebra of the Supersymmetric Quantum Mechanics, see [15]). The second feature is based on the construction of the D-module reps for the supersymmetric extensions of the sl(2, R) algebra (see [5] and [6]), leading to the D-module reps of the finite, simple, one-dimensional Superconformal Lie algebras.…”

“…In the second case the scaling dimension assignment is the following: λ for the first boson, λ + 1 2 for the two fermions and λ + 1 for the remaining boson (known as the auxiliary field). In the superspace language the (2, 2) rep is known as the chiral representation, while the (1, 2, 1) rep is known as the real representation, see [3].…”