On Schrödinger superalgebras

Journal of High Energy Physics, 2008

We discuss super Schrödinger algebras with less supercharges from N =4 superconformal algebra psu(2,2|4). Firstly N =2 and N =1 superconformal algebras are constructed from the psu(2,2|4) via projection operators. Then a super Schrödinger subalgebra is found from each of them. The one obtained from N =2 has 12 supercharges with su(2) 2 ×u(1) and the other from N =1 has 6 supercharges with u(1) 3. By construction, those are still subalgebras of psu(2,2|4). Another super Schrödinger algebra, which preserves 6 supercharges with a single u(1) symmetry, is also obtained from N =1 superconformal algebra su(2,2|1). In particular, it coincides with the symmetry of N =2 non-relativistic Chern-Simons matter system in three dimensions.

Nuclear Physics B, 2006

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization of the Schrödinger algebra that may be extended into a representation of the conformal algebra in d+2 dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-1 2 Lévy-Leblond equation. An N = 2 supersymmetric extension of these equations leads, respectively, to a 'super-Schrödinger' model and to the (3|2)-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras osp(2|2) sh(2|2) and osp(2|4), respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schrödinger-Neveu-Schwarz algebra sns (N ) with N supercharges.

2013

Higher dimensional supersymmetric quantum mechanics is studied. General properties of the two dimensional case are presented. For three spatial dimesions or higher, a spin structure is shown to arise naturally from the nonrelativistic supersymmetry algebra. The study of supersymmetric quantum field theories in the low energy, non-relativistic limit is of special interest for various reasons- primarily because if supersymmetry is a symmetry of nature, what we see today must be the low energy remnant of it. In such a limit, the underlying field theory should approach a Galilean invariant supersymmetric field theory and, by the Bargmann superselection rule [1], such a field theory should be equivalent to a supersymmetric Schrödinger equation in each particle number sector of the theory. While one dimensional supersymmetric quantum mechanics has been studied exhaustively in the past [2, 3], there has only been a few attempts at generalizing this to higher dimensions [4, 5, 6]. More impo...

Proceedings of Fifth International Conference on Mathematical Methods in Physics — PoS(IC2006)

The complete classification of the irreducible representations of the N-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmetric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson's doublings of algebras in association with the N-extended supersymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N = 8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechanical systems to the dimensional reduction of higher-dimensional supersymmetric theories.

2004

We demonstrate that two-dimensional N = 8 supersymmetric quantum mechanics which inherits the most interesting properties of N = 2, d = 4 SYM can be constructed if the reduction to one dimension is performed in terms of the basic object, i.e. the N = 2, d = 4 vector multiplet. In such a reduction only complex scalar fields from the N = 2, d = 4 vector multiplet become physical bosons in d = 1, while the rest of the bosonic components are reduced to auxiliary fields, thus giving rise to the (2, 8, 6) supermultiplet in d = 1. We construct the most general action for this supermultiplet with all possible Fayet-Iliopoulos terms included and explicitly demonstrate that the action possesses duality symmetry extended to the fermionic sector of theory. In order to deal with the second-class constraints present in the system, we introduce the Dirac brackets for the canonical variables and find the supercharges and Hamiltonian which form a N = 8 super Poincarè algebra with central charges. Finally, we explicitly present the generalization of two-dimensional N = 8 supersymmetric quantum mechanics to the 2kdimensional case with a special Kähler geometry in the target space.

2010

The quantum nonrelativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N = 2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two self-adjoint extensions of the Hamiltonian that are compatible with the standard N = 2 supersymmetry. We show that only in these two cases one of the subsystems coincides with the original spinless Aharonov-Bohm model and comes accompanied by the super-partner Hamiltonian which allows a singular behavior of the wave functions. We find a family of additional, nonlocal integrals of motion and treat them together with local supercharges in the unifying framework of the trisupersymmetry. The inclusion of the dynamical conformal symmetries leads to an infinitely generated superalgebra, that contains several representations of the superconformal osp(2|2) symmetry. We present the application of the results in the framework of the two-body model of identical anyons. The nontrivial contact interaction and the emerging N = 2 linear and nonlinear supersymmetries of the anyons are discussed.

This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W_k, of the N=2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.

Annals of Physics, 1996

Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a − , a + ] = 1 + νK, involving the Klein operator K, {K, a ± } = 0, K 2 = 1. The connection of the minimal set of equations with the earlier proposed 'universal' vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken N = 2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2|2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. A possibility of 'superimposing' the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that osp(2|2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.

Communications in Mathematical Physics, 2007

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p + 2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the N = 1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the N = 2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.

We discuss two independent constructions to introduce an N-extended Supersymmetric Quantum Mechanics. The first one makes use of the Fierz identities while the second one (divided into two subcases) makes use of the Schur lemma. The N supercharges Q_I are square roots of a free Hamiltonian H given by the tensor product of a D-dimensional Laplacian and a 2d-dimensional identity matrix operator. We present the mutual relations among N, D and d. The mod 8 Bott's periodicity of Clifford algebras is encoded, in the Fierz case, in the Radon-Hurwitz function and, in the Schur case, in an extra independent function. Comment: 7 pages