Nonlinear Superalgebra Research Articles

Conformal bridge transformation, mathcal{PT} - and supersymmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcal{PT} -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcal{PT} -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

Particle in a self-dual dyon background: Hidden free nature and exotic superconformal symmetry

We show that a nonrelativistic particle in a combined field of a magnetic monopole and $1/{r}^{2}$ potential reveals a hidden, partially free dynamics when the strength of the central potential and the charge-monopole coupling constant are mutually fitted to each other. In this case the system admits both a conserved Laplace-Runge-Lenz vector and a dynamical conformal symmetry. The supersymmetrically extended system corresponds then to a background of a self-dual or anti-self-dual dyon. It is described by a quadratically extended Lie superalgebra $D(2,1;\ensuremath{\alpha})$ with $\ensuremath{\alpha}=1/2$, in which the bosonic set of generators is enlarged by a generalized Laplace-Runge-Lenz vector and its dynamical integral counterpart related to Galilei symmetry, as well as by the chiral ${\mathbb{Z}}_{2}$-grading operator. The odd part of the nonlinear superalgebra comprises a complete set of $24=2\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ fermionic generators. Here a usual duplication comes from the ${\mathbb{Z}}_{2}$-grading structure; the second factor can be associated with a triad of scalar integrals---the Hamiltonian, the generator of special conformal transformations, and the squared total angular momentum vector, while the quadruplication is generated by a chiral spin vector integral which exits due to the (anti-)self-dual nature of the electromagnetic background.

Symmetries of holographic super-minimal models

We compute the asymptotic symmetry of the higher-spin supergravity theory in AdS_3 and obtain an infinite-dimensional non-linear superalgebra, which we call the super-W_infinity[lambda] algebra. According to the recently proposed supersymmetric duality between higher-spin supergravity in an AdS_3 background and the 't Hooft limit of the N=2 CP^n Kazama-Suzuki model on the boundary, this symmetry algebra should agree with the 't Hooft limit of the chiral algebra of the CFT, SW_n. We provide two nontrivial checks of the duality. By comparing the algebras, we explicitly match the lowest-spin commutation relations in the super-W_infinity[lambda] with the corresponding commutation relations in the 't Hooft limit on the CFT side. We also consider the degenerate representations of the two algebras and find that the spectra of the chiral primary fields are identical.

Super-W∞ asymptotic symmetry of higher-spin AdS3 supergravity

We consider (2+1)-dimensional (N, M)-extended higher-spin anti-de Sitter supergravity and study its asymptotic symmetries. The theory is described by the Chern-Simons action based on a real, infinite-dimensional higher-spin superalgebra. We specify consistent boundary conditions on the higher-spin super-gauge connection corresponding to asymptotically anti-de Sitter spacetimes. We then determine the residual gauge transformations that preserve these asymptotic conditions and compute their Poisson bracket algebra. We find that the asymptotic symmetry is enhanced from the higher-spin superalgebra to some (N,M)-extended super-W(infinity) nonlinear superalgebra. The latter has the same classical central charge as pure Einstein gravity. Special attention is paid to the (1,1)-case. Truncation to the bosonic sector yields the previously found W(infinity) algebra, while truncation to the underlying finite-dimensional superalgebra reproduces the N-extended superconformal algebra (in its nonlinear version for N>2). We discuss string theory realization of these higher-spin anti-de Sitter supergravity theories as well as relations to previous treatments of super-W(infinity) in the literature.

Effective functional for the supercoherent state of spinless algebra in the Hubbard model

The supermatrix of a supercoherent state is calculated for deformed nonlinear superalgebra in the mode of spinless fermions for the Hubbard model with strong repulsion in sites. The effective functional is derived for spinless fermions. The superfield Lagrangian is calculated for spinless fermions in the Hubbard model with the help of the simple chiral model with N = 1 and self-action Ф3.

Supergroup approach to the Hubbard model

Based on the revealed hidden supergroup structure, we develop a new approach to the Hubbard model. We reveal a relation of even Hubbard operators to the spinor representation of the generators of the rotation group of four-dimensional spaces. We propose a procedure for constructing a matrix representation of translation generators, yielding a curved space on which dynamic superfields are defined. We construct a new deformed nonlinear superalgebra for the regime of spinless Hubbard model fermions in the case of large on-site repulsion and evaluate the effective functional for spinless fermions.

BRST STRUCTURE OF NONLINEAR SUPERALGEBRAS

In this paper we analyze the structure of the BRST charge of nonlinear superalgebras. We consider quadratic nonlinear superalgebras where a commutator (in terms of (super-)Poisson brackets) of the generators is a quadratic polynomial of the generators. We find the explicit form of the BRST charge up to cubic order in Faddeev–Popov ghost fields for arbitrary quadratic nonlinear superalgebras. We point out the existence of constraints on structure constants of the superalgebra when the nilpotent BRST charge is quadratic in Faddeev–Popov ghost fields. The general results are illustrated by simple examples of superalgebras.

Hidden nonlinear [formula omitted] superunitary symmetry of [formula omitted] superextended 1D Dirac delta potential problem

We show that the N=2 superextended 1D quantum Dirac delta potential problem is characterized by the hidden nonlinear su(2|2) superunitary symmetry. The unexpected feature of this simple supersymmetric system is that it admits three different Z2-gradings, which produce a separation of 16 integrals of motion into three different sets of 8 bosonic and 8 fermionic operators. These three different graded sets of integrals generate two different nonlinear, deformed forms of su(2|2), in which the Hamiltonian plays a role of a multiplicative central charge. On the ground state, the nonlinear superalgebra is reduced to the two distinct 2D Euclidean analogs of a superextended Poincaré algebra used earlier in the literature for investigation of spontaneous supersymmetry breaking. We indicate that the observed exotic supersymmetric structure with three different Z2-gradings can be useful for the search of hidden symmetries in some other quantum systems, in particular, related to the Lamé equation.

Nonlinear superconformal symmetry of a fermion in the field of a Dirac monopole

We study a longstanding problem of identification of the fermion–monopole symmetries. We show that the integrals of motion of the system generate a nonlinear classical Z2-graded Poisson, or quantum superalgebra, which may be treated as a nonlinear generalization of the osp(2|2)⊕su(2). In the nonlinear superalgebra, the shifted square of the full angular momentum plays the role of the central charge. Its square root is the even osp(2|2) spin generating the u(1) rotations of the supercharges. Classically, the central charge's square root has an odd counterpart whose quantum analog is, in fact, the same osp(2|2) spin operator. As an odd integral, the osp(2|2) spin generates a nonlinear supersymmetry of De Jonghe, Macfarlane, Peeters and van Holten, and may be identified as a grading operator of the nonlinear superconformal algebra.

Polynomial super-gl(n) algebras

We introduce a class of finite-dimensional nonlinear superalgebras L = L + L providing gradings of L = gl(n) sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree >1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Qa, b and even generators Eab, a, b = 1, ..., n, a three-parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed in order that the Jacobi identities are fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules and conditions for atypicality are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and spacetime supersymmetry, are discussed.

Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representation

We study the nonlinear (polynomial, N-fold,…) supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a nonlinear superalgebra which can be appropriately specified as N=2 SUSY. In this case we find a nontrivial hidden symmetry operator and rephrase it as a nonlinear function of the Hamiltonian on the physical state space. The full N=2 nonlinear SUSY algebra includes “central charges” both polynomial and nonpolynomial (due to a symmetry operator) in the Hamiltonian.

HIDDEN NONLINEAR SUPERSYMMETRIES IN PURE PARABOSONIC SYSTEMS

The existence of intimate relations between generalized statistics and supersymmetry is established by the observation of hidden supersymmetric structure in pure parabosonic systems. This structure is characterized generally by a nonlinear superalgebra. The nonlinear supersymmetry of parabosonic systems may be realized, in turn, by modifying appropriately the usual supersymmetric quantum mechanics. The relation of nonlinear parabosonic supersymmetry to the Calogero-like models with exchange interaction and to the spin chain models with inverse-square interaction is pointed out.

Hamiltonian Structure and Coset Construction of the Supersymmetric Extensions of N=2 KdV Hierarchy

A manifestly N=2 supersymmetric coset formalism is applied to analyze the "fermionic" extensions of N=2, a=4 and a=-2 KdV hierarchies. Both hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but nonlinear superalgebra by a Û (1) subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.