Superspace Research Articles

Superspace Lagrangian Model of Supersymmetric Quantum Mechanics in Three-Dimensional Space

En employant le formalisme de superespace, on construit un modele de mecanique quantique supersymetrique dans l'espace a 3 dimensions. On peut considerer que ce modele decrit le mouvement d'une particule a spin dans des potentiels central et tensoriel a un corps, dont l'ensemble est determine par un seul superpotentiel. On etudie le modele le plus simple obtenu en prenant le superpotentiel harmonique. On montre que la supersymetrie est une symetrie exacte de ce modele et que le modele possede le supergroupe SU(3/3) comme groupe de symetrie maximale

Super lattices and gauge theory.

There has been a fundamental flaw in all previous attempts to put supersymmetry on the lattice. Because supersymmetry closes on the Poincare group, and since the Wilson lattice breaks Poincare invariance, the standard lattice must necessarily break supersymmetry. We solve this difficulty by putting supersymmetry on a random super lattice, where each random site is a point in super space (x/sub i/,xi/sub i/). We construct the action out of unitary SU(N) superfields connecting two such super lattice sites and sum the traces over simplexes. The theory is manifestly supersymmetric because there is no preferred direction in either real or Grassmann space. This supersymmetric lattice gauge theory on a random lattice reduces to the usual supersymmetric gauge theory in the continuum limit. We discuss applications, such as calculating nonperturbative corrections to the vacuum energy, which may yield a large enough supersymmetry breaking to explain the hierarchy problem. We discuss using random lattices to describe a lattice version of gravity and supergravity.

The Berezin integral as a contour integral

In superspace formulations of supersymmetric field theories, use is made of the Berezin “integral”, a linear map which assigns a real number to each polynomial function of the Grasmann coordinates θ α. However, this map has not been shown to be an integral in the mathematical sense of a limit of sums or an inverse to differentiation. Consequently, its mathematical basis is unclear and its generalization to arbituary curved supermanifolds is problematic. I will show that certain normalized contour integrals in the space of θ α possess the properties of Berezin “integrals”, including translation invariance and correct transformation properties under changes of coordinates. These contour integrals are mathematically well defined and may generalize to arbitrary supermanifolds. Alice Rogers' definition of super space is used.

Superspace Formulation of the Classical Spectral Transform Method for the Massive Thirring Model

On presente une formulation super-espace de la methode de transformee spectrale classique pour le modele de Thirring massif dont des variables de champ ψ et ψ* ont des proprietes d'anticommutation

Unitary representations of non-compact supergroups

We give a general theory for the construction of oscillator-like unitary irreducible representations (UIRs) of non-compact supergroups in a super Fock space. This construction applies to all non-compact supergroupsG whose coset spaceG/K with respect to their maximal compact subsupergroupK is “Hermitean supersymmetric”. We illustrate our method with the example of SU(m, p/n+q) by giving its oscillator-like UIRs in a “particle state” basis as well as “supercoherent state basis”. The same class of UIRs can also be realized over the “super Hilbert spaces” of holomorphic functions of aZ variable labelling the coherent states.

Superspaces of (s) with basis

Superspaces of (s) with basis

Integration over surfaces in superspace

Integration over surfaces in superspace

Reduction of the super phase space for a massless Dirac particle

We show how to get the reduced super phase space for a classical spinning particle and how to quantize this theory. This technique introduces a Grassmannian ’’Hamiltonian’’ and a corresponding Grassmannian ’’time’’ parameter.

An introduction to superspace symmetry: the three-dimensional coloured crystals as four-dimensional semi-crystals

An introduction to superspace symmetry: the three-dimensional coloured crystals as four-dimensional semi-crystals

SUPER KILLING EQUATION AND SUPERSYM-METRY TRANSFORMATION

In super space (x, θ), as the metric tensor field GAB(x, θ) is given, we calculated the Riemann curvature tensor RDABC of fourth rank and its generalized cyclicity.The equation that must be satisfied by isometry in the super space, i.e. super Killing equation: ξA:B+ηabξB:A=0, is deduced.In flat super space with zero curvature tensor, we have obtained the general solu-tions of the super Killing equation and the commutation relations of the corresponding generators. In the case of constant curvature, we have obtained a special solution of the super Killing equation.

Algorithms, anyone?

Many a reader of the Journal must in his youth have played one of those games in which some figurine — a race horse or windjammer in my day, but now surely a space capsule appropriate for a game perhaps called "Triple-S" (for "Super Space Spy") — is moved along a prescribed track in response to a throw of the dice. At frequent points in the track, however, the young player may encounter instructions such as "forfeit two turns," "advance five steps" or, perhaps to prepare him for life's calamities, "go back to the launching pad." Those of us thus . . .

MATHEMATICA-aided Study Of Lie AlgebrasAnd Their Cohomology - From Supergravity ToBallbearings And Magnetic Hydrodynamics

We describe applications of a MATHEMATICA-based package for the study of Lie algebras and their cohomology such as (1) the possibility to write Supergravity Equations for any TV-extended Minkowski superspace and to find out the possible models for these super spaces; (2) the possibility of studying stability of nonholonomic systems (ballbearings, gyroscopes, electromechanical devices like a rotor collector with a gliding contact; waves in plasma, etc.); (3) description of the analogue of the curvature tensor for nonlinear nonholonomic constraints and the fields of solids or their surfaces, e.g., cones, as in optimal control; (4) a new method for the study of integrability of dynamical systems. The above problems are particular instances of the general problem to compute cohomology or homology of the given Lie algebra or superalgebra with various coefficients. The package SuperLie makes it possible to determine (1) Lie algebras via defining relations, from the Cartan matrix, realized via vector fields, as polynomials with Poisson or contact (Legendre) bracket, etc., (2) various modules over these Lie algebras (tensors, with vacuum vector, etc.), (3) list central extensions and deformations and even (4) back up the Leites conjecture (an analog of Kostrikin-Shafarevich conjecture) classifying simple Lie algebras over the algebraically closed field of characteristic 2 with new examples. For the details see references.

Super Hilbert Space and the Quantum-Mechanical Time Operators

The basic idea of super Hilbert space is to represent physical states by continuous linear functionals on a space of good functions, rather than by functions in a Hilbert space. Since L2 is in one-to-one correspondence with a subset of super Hilbert space, everything that can be done in L2 can be done in super Hilbert space. In addition, however, it is possible to have a time operator and thus to base relativistic quantum mechanics on covariant four-dimensional commutation relations.

Super Atiyah Classes and Obstructions to Splitting of Supermoduli Space

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S. We evaluate these classes explicitly for the moduli space of super Riemann surfaces (super moduli space) and its reduced space, the moduli space of spin curves. These classes are interpreted in terms of certain extensions arising from line bundles on the square of the varying (super) Riemann surface. These results are used to give a new proof of the non- projectedness of Mg,1, the moduli space of super Riemann surfaces with one puncture.

The quantum chiral Minkowski and conformal superspaces

We give a quantum deformation of the chiral super Minkowski space in four dimensions as the big cell inside a quantum super Grassmannian. The quantization is performed in such way that the actions of the Poincar\'e and conformal quantum supergroups on the quantum Minkowski and quantum conformal superspaces are presented.