Killing superalgebras for lorentzian five-manifolds

We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an $$ \mathfrak{sp} $$ sp (1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.

DIRAC OPERATORS ON LIE ALGEBROIDS

We compare the Dirac operator on transitive Riemannian Lie algebroid equipped by spin or complex spin structure with the one defined on to its base manifold‎. Consequently we derive upper eigenvalue bounds of Dirac operator on base manifold of spin Lie algebroid twisted with the spinor bundle of kernel bundle‎.

Gauge Field Theory in Natural Geometric Language

This monograph addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a not-too-short, integrated approach that exploits standard and non-standard notions in natural geometric language. The role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. Two-spinors yield a partly original ‘minimal geometric data’ approach to Einstein-Cartan-Maxwell-Dirac fields. The gravitational field is jointly represented by a spinor connection and by a soldering form (a ‘tetrad’) valued in a vector bundle naturally constructed from the assumed 2-spinor bundle. We give a presentation of electroweak theory that dispenses with group-related notions, and we introduce a non-standard, natural extension of it. Also within the 2-spinor approach we present: a non-standard view of gauge freedom; a first-order Lagrangian theory of fields with arbitrary spin; an original treatment of Lie derivatives of spinors and spinor connections. Furthermore we introduce an original formulation of Lagrangian field theories based on covariant differentials, which works in the classical and quantum field theories alike and simplifies calculations. We offer a precise mathematical approach to quantum bundles and quantum fields, including ghosts, BRST symmetry and anti-fields, treating the geometry of quantum bundles and their jet prolongations in terms Frölicher's notion of smoothness. We propose an approach to quantum particle physics based on the notion of detector, and illustrate the basic scattering computations in that context.

Harmonic spinors on the Davis hyperbolic 4-manifold

In this paper, we use the [Formula: see text]-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the [Formula: see text]-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.

A first-order Lagrangian theory of fields with arbitrary spin

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic “building block”. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann–Wigner equations and the [Formula: see text]-dimensional representation of the angular-momentum algebra needed for the Joos–Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric “main sector” — carrying an irreducible representation of the angular-momentum algebra — and an induced sequence of “ghost sectors”. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime, one gets plane-wave solutions that are characterized by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.

Klein and conformal superspaces, split algebras and spinor orbits

We discuss [Formula: see text] Klein and Klein-conformal superspaces in [Formula: see text] space-time dimensions, realizing them in terms of their functor of points over the split composition algebra [Formula: see text]. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions [Formula: see text] and [Formula: see text] as [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.

Seiberg-Witten Like Equations on Pseudo-RiemannianSpincManifolds withG2(2)∗Structure

We consider 7-dimensional pseudo-Riemannianspincmanifolds with structure groupG2(2)∗. On such manifolds, the space of 2-forms splits orthogonally into componentsΛ2M=Λ72⊕Λ142. We define self-duality of a 2-form by considering the partΛ72as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations onR4,3and give some solutions.

Invariant torsion and G2-metrics

We introduce and study a notion of invariant intrinsic torsion geometrywhich appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S3. This space is foliated by sixdimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G2 that arises from SO(3)-structures with invariant intrinsic torsion.