Local Supersymmetry Transformations Research Articles

N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}=4 Supergravity with Local Scaling Symmetry in Four Dimensions

We construct the most general four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supergravity coupled to an arbitrary number n of vector multiplets in which the global scaling symmetry is gauged, in addition to a subgroup of SL(2, ℝ) × SO(6, n). The various gaugings are parametrized by an embedding tensor built out of 2n+63+4n+6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\left(\\begin{array}{c}n+6\\\\ {}3\\end{array}\\right)+4\\left(n+6\\right) $$\\end{document} parameters that satisfy a specific set of quadratic consistency constraints, to which we provide explicit solutions. We also derive the local supersymmetry transformation rules and the equations of motion for the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 matter-coupled supergravity with local scaling symmetry. Such supergravity theories do not possess an action, since the scaling symmetry is only an on-shell symmetry of the corresponding ungauged theories.

E7(7) exceptional field theory in superspace

We formulate the locally supersymmetric E7(7) exceptional field theory in a (4 + 56|32) dimensional superspace, corresponding to a 4D N = 8 “external” superspace augmented with an “internal” 56-dimensional space. This entails the unification of external diffeomorphisms and local supersymmetry transformations into superdiffeomorphisms. The solutions to the superspace Bianchi identities lead to on-shell duality equations for the p-form field strengths for p ≤ 4. The reduction to component fields provides a complete description of the on-shell supersymmetric theory. As an application of our results, we perform a generalized Scherk-Schwarz reduction and obtain the superspace formulation of maximal gauged supergravity in four dimensions parametrized by an embedding tensor.

Supersymmetric 3D model for gravity with SU(2) gauge symmetry, mass generation and effective cosmological constant

A Chern–Simons system in 2+1 dimensions invariant under local Lorentz rotations, SU(2) gauge transformations, and local supersymmetry (SUSY) transformations is proposed. The field content is that of (2+1)-gravity plus an SU(2) gauge field, a spin-1/2 fermion charged with respect to SU(2) and a trivial free abelian gauge field. A peculiarity of the model is the absence of gravitini, although it includes gravity and SUSY. Likewise, no gauginos are present. All the parameters involved in the system are either protected by gauge invariance or emerge as integration constants. An effective mass and effective cosmological constant emerge by spontaneous breaking of local scaling invariance. The vacuum sector is defined by configurations with locally flat Lorentz and SU(2) connections sporting nontrivial global charges. Three-dimensional Lorentz-flat geometries are spacetimes of locally constant negative—or zero—Riemann curvature, which include Minkowski space, AdS3, BTZ black holes, and point particles. These solutions admit different numbers of globally defined, covariantly constant spinors and are therefore good candidates for stable ground states. The fermionic sector in this system could describe the dynamics of electrons in graphene in the long wavelength limit near the Dirac points, with the spin degree of freedom of the electrons represented by the SU(2) label. If this is the case, the SU(2) gauge field would produce a spin–spin interaction giving rise to strong correlation of electron pairs.

Generators of local supersymmetry transformation from first-class constraints

We show how the generator of local supersymmetry transformations can be found from fermionic first-class constraints. This is done by adapting the approaches of Henneaux, Teitelboim and Zanelli, and of Castellani that have been used to find the generator of gauge transformations from bosonic first-class constraints. We illustrate how a supersymmetric gauge generator can be found by considering a spinning particle. The invariances that we find are not those presented in the original discussion of the spinning particle.

Supersymmetric completion of M-theory 4D-gauge algebra from twisted tori and fluxes

We present the supersymmetric completion of the M-theory free differential algebra resulting from a compactification to four dimensions on a twisted seven-torus with 4-form and 7-form fluxes turned on. The super--curvatures are given and the local supersymmetry transformations derived. Dual formulations of the theory are discussed in connection with classes of gaugings corresponding to diverse choices of vacua. This also includes seven dimensional compactifications on more general spaces not described by group manifolds.

Quantum inconsistency of Einstein supergravity

We show that N=1, D=4 Einstein-frame supergravity is inconsistent at one loop because of an anomaly in local supersymmetry transformations. A Jacobian must be added to the Einstein-frame Lagrangian to cancel this anomaly. We show how the Jacobian arises from the super-Weyl field redefinition that takes the superspace Lagrangian to the Einstein frame. We present an explicit example which demonstrates that the Jacobian is necessary for one-loop scattering amplitudes to be frame independent.

Noether superpotentials in supergravities

Straightforward application of the standard Noether method in supergravity theories yields an incorrect superpotential for local supersymmetry transformations, which gives only half of the correct supercharge. We show how to derive the correct superpotential through Lagrangian methods, by applying a criterion proposed recently by one of us. We verify the equivalence with the Hamiltonian formalism. It is also indicated why the first-order and second-order formalisms lead to the same superpotential. We rederive in particular the central extension by the magnetic charge of the N 4=2 algebra of SUGRA asymptotic charges.

Target-superspace in 2d dilatonic supergravity

The N=1 supersymmetric version of generalized 2d dilaton gravity can be cast into the form of a Poisson σ-model, where the target space and its Poisson bracket are graded. The target space consists of a 1+1 superspace and the dilaton, which is the generator of Lorentz boosts therein. The Poisson bracket on the target space induces the invariance of the worldsheet theory against both diffeomorphisms and local supersymmetry transformations (superdiffeomorphisms). The machinery of Poisson σ-models is then used to find the general local solution to the field equations. As a byproduct, classical equivalence between the bosonic theory and its supersymmetric extension is found.

N = 2 central charge superspaceand a minimal supergravity multiplet

We extend the notion of central charge superspace to the case of local supersymmetry. Gauged central charge transformations are identified as diffeomorphisms on the same footing as spacetime diffeomorphisms and local supersymmetry transformations. Given the general structure we then proceed to the description of a particular vector-tensor supergravity multiplet of 24 + 24 components, identified by means of rather radical constraints.

Generalized supergravity in two dimensions

Among the usual constraints of (1,1) supergravity in d = 2 the condition of vanishing bosonic torsion is dropped. Using the inverse supervierbein and the superconnection considerably simplifies the formidable computational problems. It allows us to solve the constraints for those fields before taking into account the (identically fulfilled) Bianchi identities. The relation of arbitrary functions in the seminal paper of Howe to supergravity multiplets is clarified. The local supersymmetry transformations remain the same, but, somewhat surprisingly, the transformations of zweibein and Rarita-Schwinger field decouple from those of the superconnection multiplet. A method emerges naturally of how to construct ‘non-Einsteinian’ supergravity theories with non-trivial curvature and torsion in d = 2 which, apart from their intrinsic interest, may be relevant for models of super black holes and for novel generalizations in superstring theories. Several explicit examples of such models are presented, some of which immediately allow for a dilatonic formulation for the bosonic part of the action.

(2, 2) supergravity in the light-cone gauge

Starting with the prepotential description of two-dimensional (2,2) supergravity we use local supersymmetry transformations to go to light-cone gauge. We discuss properties of the theory in this gauge and derive Ward identities for correlation functions defined with respect to the induced supergravity action.

The Ashtekar Formulation for CanonicalN= 2 Supergravity

We extend the Ashtekar formulation of canonical gravity to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt a chiral formulation, left- and right-handed supersym­ metries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersym­ metries are realized by constrained parameters.

THE ASHTEKAR FORMULATION FOR CANONICAL N=2 SUPERGRAVITY

Ashtekar’s formulation of canonical gravity is extended to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt the chiral formulation, left- and right-handed supersymmetries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersymmetries are realized by constrained parameters. By means of the canonical formalism, simple polynomial constraints are derived and we can show that the Poisson brackets of supersymmetry currents are actually closed. This guarantees the consistency of our supersymmetry transformations, which is unclear at the level of the algebra on the fields.

Superfield formulation of superparticles

A superfield action for the relativistic massless superparticle as a spinning particle is presented in the new gauge. The symmetries in the relativistic superparticle theory, that is to say, the invariances under, the reparametrization and the local supersymmetry-transformation in the parameter space, are manifest in this formalism. It is clear how these two kinds of transformations descend from a unified origin to be called “super-reparametrization”, which is a restricted form of the general coordinate transformation in the superspace. The action is manifestly invariant under these transformation by its constructions The minimal coupling with electromagnetic field is also constructed in the superfield formalism, and a manifestly invariant superfield action for the interacting superparticle is presented in our gauge. The formalism is extended to the construction of an action for theN=2 superparticle.

Functional representation of the superconformal group in two dimensions.

A functional representation for the local supersymmetry transformations of the superconformal group in two dimensions is explicitly constructed. The representation kernel acts on functionals of both a commuting field and a Grassmann field.

Path-integral formulation of closed strings.

We construct the covariant path integral for the Neveu-Schwarz-Ramond superstring in superspace, with manifest invariance under diffeomorphisms and local supersymmetry transformations. Spin structure is introduced, and the constraints imposed by modular invariance on fermionic string models examined. The critical exponents of the bosonic and fermionic string models in space-time dimensions less than the critical dimension are obtained for world surfaces of arbitrary topology.

Locally supersymmetric string Jacobian.

A representation of the superstring Jacobian is given for a general choice of the gauge slice e${^}_{a\ensuremath{\mu}}$(x), \ensuremath{\psi}${^}_{\ensuremath{\mu}}$(x), \^A(x). The expression involves an integral over the expected ghost fields ${c}_{a}$(x) and \ensuremath{\lambda}(x) and traceless antighost fields ${b}^{\mathrm{ab}}$(x) and \ensuremath{\beta}(x), and the action is invariant under local supersymmetry and super-Weyl transformations. Conserved currents of the ghost system are calculated directly from this action.

Quartic ghost field theories in two dimensions related to the string and superstring jacobians

The functional measure for the string and superstring contains jacobians which can be represented by integrals over ghost and antighosts with a bilinear action. The action is invariant under diffeomorphisms and Weyl transformations for the bosonic string and under local supersymmetry and super-Weyl transformations for the superstring. We extend these actions to include quartic ghost couplings while maintaining previous invariances. In the bosomic case, the new field theory is related to the massless Thirring model.

Fierz identities and invariance of 11-dimensional supergravity action.

A systematic method applicable to any dimension D is proposed and found to be useful for deriving various Fierz identities. With the help of this method we explicitly prove the invariance of D=11, N=1 supergravity action under local supersymmetry transformation.

Hamiltonian formulation of eleven-dimensional supergravity.

The Hamiltonian for eleven-dimensional supergravity is computed. Explicit expressions for the constraint generators of diffeohomorphisms, gauge transformations of the three-index form, local supersymmetry transformations, and boosts of the Cartan local frame are given.