Twist deformations of the supersymmetric quantum

Central European Journal of Physics, 2010

The N -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even N one can identify the 1D N -extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators.

Nuclear Physics B, 2012

We prove that the invariance of the N = 2 superconformal quantum mechanics is controlled by subalgebras of a given twisted superalgebra made of 6 fermionic (nilpotent) generators and 6 bosonic generators (including a central charge). The superconformal quantum mechanics actions are invariant under this quite large twisted superalgebra. On the other hand, they are fully determined by a subalgebra with only 2 fermionic and 2 bosonic (the central charge and the ghost number) generators. The invariant actions are Q i -exact (i = 1, 2, . . . , 6), with a Q i ′ -exact (i ′ = i) antecedent for all 6 fermionic generators. It follows that the superconformal quantum mechanics actions with Calogero potentials are uniquely determined even if, in its bosonic sector, the twisted superalgebra does not contain the one-dimensional conformal algebra sl(2), but only its Borel subalgebra. The general coordinate covariance of the non-linear sigma-model for the N = 2 supersymmetric quantum mechanics in a curved target space is fully implied only by its worldline invariance under a pair of the 6 twisted supersymmetries. The transformation connecting the ordinary and twisted formulations of the N = 2 superconformal quantum mechanics is explicitly presented.

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.

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.

Physics Letters B, 2006

Non-anticommutative Grassmann coordinates in four-dimensional twistdeformed N = 1 Euclidean superspace are decomposed into geometrical ones and quantum shift operators. This decomposition leads to the mapping from the commutative to the non-anticommutative supersymmetric field theory. We apply this mapping to the Wess-Zumino model in commutative field theory and derive the corresponding non-anticommutative Lagrangian. Based on the theory of twist deformations of Hopf algebras, we comment the preservation of the (initial) N = 1 super-Poincaré algebra and on the consequent super-Poincaré invariant interpretation of the discussed model, but also provide a measure for the violation of the super-Poincaré symmetry. * We follow the convention in Ref. [19]. † It is also possible to use the Atiyah-Ward space time of the signature (+, +, −, −) [20]

2008

We show how some classical r-matrices for the D = 4 Poincaré algebra can be supersymmetrized by an addition of part depending on odd supercharges. These rmatrices for D = 4 super-Poincaré algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper [1] (arXiv:math/0712.3962v1). * The talk given by third author at the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'07) (Dubna

Journal of Physics A: Mathematical and General, 1998

The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra osp(1|2). The twist element is the same as for the sl(2) Lie algebra due to the embedding of the sl(2) into the superalgebra osp(1|2). The R-matrix has the direct sum structure in the irreducible representations of osp(1|2). The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group SL ξ (2) as subalgebra and Grassmann generators as well.

2011

We consider new quantum superspaces, obtained from the superextension of twist deformations of Minkowski spacetime providing Lie-algebraic noncommutativity of spacetime coordinates. The deformed superalgebraic relations describing new quantum superspaces are covariant under the twist-deformed Poincaré supersymmetries. New four classes of supertwist deformations of N = 1 Poincaré superalgebra are investigated and further their Euclidean counterpart presented. Because the proposed supertwists are in odd sector not unitary they are better adjusted to the description of deformed D = 4 Euclidean supersymmetries with independent left-chiral and right-chiral supercharges. Our supertwist deformations in the framework of Hopf-algebraic quantum deformations provide an alternative to the N = 1 2 SUSY Seiberg's star product deformation scheme.

Journal of Physics A: Mathematical and General, 2000

The minimal twist map introduced by Abdesselam et al for the nonstandard (Jordanian) quantum sl(2, R) algebra is used to construct the twist maps for two different non-standard quantum deformations of the (1+1) Schrödinger algebra. Such deformations are, respectively, the symmetry algebras of a space and a time uniform lattice discretization of the (1 + 1) free Schrödinger equation. It is shown that the corresponding twist maps connect the usual Lie symmetry approach to these discrete equations with non-standard quantum deformations. This relationship leads to a clear interpretation of the deformation parameter as the step of the uniform (space or time) lattice.