D-module representations of ${\cal N}=2,4,8$N=2,4,8 superconformal algebras and their superconformal mechanics

Abstract

The linear (homogeneous and inhomogeneous) (k, N , N − k) supermultiplets of the Nextended one-dimensional Supersymmetry Algebra induce D-module representations for the N = 2, 4, 8 superconformal algebras.For N = 2, the D-module representations of the A(1, 0) superalgebra are obtained. For N = 4 and scaling dimension λ = 0, the D-module representations of the A(1, 1) superalgebra are obtained. For λ = 0, the D-module representations of the D(2, 1; α) superalgebras are obtained, with α determined in terms of the scaling dimension λ according to: α = −2λ for k = 4, i.e. the (4, 4) supermultiplet, α = −λ for k = 3, i.e. (3, 4, 1), and α = λ for k = 1, i.e. (1, 4, 3). For λ = 0 the (2, 4, 2) supermultiplet induces a D-module representation for the centrally extended sl(2|2) superalgebra.For N = 8, the (8, 8) root supermultiplet induces a D-module representation of the D(4, 1) superalgebra at the fixed value λ = 1 4 . A Lagrangian framework to construct one-dimensional, off-shell, superconformal invariant actions from single-particle and multi-particles D-module representations is discussed. It is applied to explicitly construct invariant actions for the homogeneous and inhomogeneous N = 4 (1, 4, 3) D-module representations (in the last case for several interacting supermultiplets of different chirality).

From worldline to quantum superconformal mechanics with and without oscillatorial terms: D(2,1;α) and sl(Cunha1, Holanda2, Toppan3 2017Phys. Rev. D172100View full textAdd to dashboardCite

“…For N = 4 and k = 2, irreps of the exceptional superalgebras D(2, 1; α) are recovered, see [25,26,1], from the (k, 4, 4 − k) supermultiplets according to the relation…”

Chiral and Real N = 2 supersymmetric -conformal Galilei algebras

“…The first new feature is the criticality of λ. Unlike N = 2, λ enters the structure constants of N = 4, allowing to identify the one-dimensional superconformal algebra as D(2, 1; α) for α = (2 − k)λ [5]. For N = 8 four one-dimensional simple superconformal algebras (D(4, 1) for k = 0, 8, F (4) for k = 1, 7, A(3, 1) for k = 2, 6, D(2, 2) for k = 3, 5), are identified at the critical values λ = 1 k−4 (see [6] for details).…”

“…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]). …”

“…The non-vanishing (anti)commutators of G (1,2,1) are given in (2) and (4). Their respective D-module representations (realized by differential operators in d + 1 space-time dimensions) are given in (5) and (6).…”

“…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.…”