Superspace unmasked: The physical interpretation of supergravity theory

Supergravity admits a geometric formulation in terms of an expanded algebra of functions on ordinary spacetime. This formulation, called graded manifold theory, is equivalent to the usual superspace supergravity constructions but avoids the anticommuting coordinates of superspace. The geometry suggests an operational interpretation of supergravity theory in terms of measurements made by local observers in spacetime.     

Graded non-commutative geometries

We present a short exposition of graded finite non-commutative geometries. The theory that serves as an example is based on the algebra of matrices M_n . This non-commutative algebra replaces the algebra of functions on a manifold. Consequently, vector fields (differentiations), forms and connections are constructed. The gauge theory can be introduced without the notion of internal manifold. We discuss some physical application, the similarities with the standard model, and the graded version of this geometry.     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Algebras from Multisymplectic Geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L _∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism.     

Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics

Our method of constructing representations of the spacetime diffeomorphism group DiffMin parametrized field theories is generalized to canonical geometrodynamics. The gravitational configuration space Riem Σ is extended by the space of embeddings of the spatial manifold Σ in the spacetimeM. Spacetime metrics are limited by Gaussian conditions with respect to an auxiliary foliation structure. As a result of these conditions, the super-Hamiltonian and supermomentum constraints are temporarily suspended. There are, however, new constraints in the theory associated with the canonical pair of the embedding variables and their conjugate momenta. By smearing the new constraint functions by vector fields V ∈ LDiffMrestricted to the embeddings, we construct a homomorphism from the Lie algebra of the spacetime diffeomorphism group into the Poisson bracket algebra of the dynamical variables on the extended geometrodynamical phase space. The dynamical evolution generated by such dynamical variables automatically preserves both the new and the old constraints and builds a Ricci-flat spacetime. The implications of the scheme for the canonical quantization of gravity are discussed.     

Heterotic superstrings in auxiliary field background and extended spacetime supersymmetry

We introduce the superstring vertices of the auxiliary connections of extended supergravity and elucidate the relation between the internal symmetry of the spacetime superconformal algebra and the Kac-Moody symmetry of the 2D superconformal algebra. The microscopic computation of effective lagrangians is outlined: it turns out that the low-energy effective lagrangians for 4D heterotic superstrings are of a standard supergravity form.     

Linear bosonic and fermionic quantum gauge theories on curved spacetimes

We develop a general setting for the quantization of linear bosonic and fermionic field theories subject to local gauge invariance and show how standard examples such as linearised Yang-Mills theory and linearised general relativity fit into this framework. Our construction always leads to a well-defined and gauge-invariant quantum field algebra, the centre and representations of this algebra, however, have to be analysed on a case-by-case basis. We discuss an example of a fermionic gauge field theory where the necessary conditions for the existence of Hilbert space representations are not met on any spacetime. On the other hand, we prove that these conditions are met for the Rarita-Schwinger gauge field in linearised pure $N=1$ supergravity on certain spacetimes, including asymptotically flat spacetimes and classes of spacetimes with compact Cauchy surfaces. We also present an explicit example of a supergravity background on which the Rarita-Schwinger gauge field can not be consistently quantized.     

A comment on superspace Bianchi identities and six-dimensional spacetime

We investigate the Bianchi identities for a superspace with a six-dimensional spacetime, under the assumption of certain superspace kinematic constraints. It is shown that these constraints lead to the supergravity theory recently discussed by Breitenlohner and Kabelschacht. However, we conclude that the resulting theory is consistent only if the curvature scalar of the six-dimensional spacetime vanishes.     

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov’s formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.     

On the determination of Cauchy surfaces from intrinsic properties

We consider the problem of determining from intrinsic properties whether or not a given spacelike surface is a Cauchy surface. We present three results relevant to this question. First, we derive necessary and sufficient conditions for a compact surface to be a Cauchy surface in a spacetime which admits one. Second, we show that for a non-compact surface it is impossible to determine whether or not it is a Cauchy surface without imposing some restriction on the entire spacetime. Third, we derive conditions for an asymptotically flat surface to be a Cauchy surface by imposing the global condition that it be imbedded in a weakly asymptotically simple and empty spacetime.This research was supported in part by the National Science Foundation grants PHY 70-022077 and PHY 76-20029 as well as the National Aeronautics and Space Administration grant NGR 21-002-010National Science Foundation Pre-doctoral Fellow     

Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras

The purpose of the Letter is to show how to use the cohomology of the Nijenhuis-Richardson graded Lie algebra of a vector space to construct formal deformations of each Lie algebra structure of that space. One then shows that the de Rham cohomology of a smooth manifold produces a family of cohomology classes of the graded Lie algebra of the space of smooth functions on the manifold. One uses these classes and the general construction above to provide one-differential formal deformations of the Poisson Lie algebra of the Poisson manifolds and to classify all these deformations in the symplectic case.     

Comment on “Classification of Cosmic Scale Factor via Noether Gauge Symmetries” [Int. J. Theor. Phys. 54, 2343 (2015)]

We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lemaître-Robertson-Walker spacetime is a well-known result in the literature.     

Deformed field theory on $\kappa$-spacetime

A general formalism is developed that allows the construction of a field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime (the Poincaré group) is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable *-product. Fields are elements of this function algebra. The Dirac and Klein-Gordon equation are defined and an action is found from which they can be derived.Received: 17 July 2003, Published online: 26 September 2003     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

Structure of supermanifolds and supersymmetry transformations

After giving a global, constraint-free Lagrangian formulation of theN=1 superspace supergravity in terms of super fibre bundles and differential forms over a supermanifold, we show that the concept of body manifold of a supermanifold provides a natural manner to reduce the theory to spacetime. This reduction, however, is not canonical, and the various ways in which it can be done give rise to transformations of the field variables which generalise the known invariances of theN=1 spacetime supergravity under supersymmetry transformations and spacetime diffeomorphisms.Research partly supported by the Gruppo Nazionale per la Fisica Matematica of the Italian Research Council and by the Italian Ministry of Public Education through the research project Geometria e Fisica     

Connections and geodesics in the spacetime tangent bundle

Recent interest in maximal proper acceleration as a possible principle generalizing the theory of relativity can draw on the differential geometry of tangent bundles, pioneered by K. Yano, E. T. Davies, and S. Ishihara. The differential equations of geodesics of the spacetime tangent bundle are reduced and investigated in the special case of a Riemannian spacetime base manifold. Simple relations are described between the natural lift of ordinary spacetime geodesics and geodesics in the spacetime tangent bundle.     

Second order Hamiltonian formalism and constraints algebra for non-polynomial supergravities

The second order Hamiltonian formalism for a non-polynomial N = 1D = 10 supergravity coupled to super Yang-Mills theory is developed. This is done by starting from the first order canoncial covariant formalism on group manifold. The Hamiltonian, generator of time evolution, is found as a functional of the first class constraints of this coupled system. These contraints close the constraint algebra and they are the generators of all the Hamiltonian gauge symmetries.     

Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.     

Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Quantum Geometry on Quantum Spacetime: Distance,Area and Volume Operators

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes.