Spinor Bundle Research Articles

Generically split octonion algebras and 𝔸1-homotopy theory

We study generically split octonion algebras over schemes using techniques of ${\mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another "mod $3$" invariant. We review Zorn's "vector matrix" construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille's analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.

Spinors and the tangent groupoid

The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a rescaled spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.

Spinors and the Tangent Groupoid

The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a "rescaled" spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.

Real spinor bundles and real Lipschitz structures

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in all dimensions and signatures. As a simple application, we show that the supersymmetry generator of eleven-dimensional supergravity in "mostly plus" signature can be interpreted as a global section of a bundle of irreducible Clifford modules if and $\textit{only if}$ the underlying eleven-manifold is orientable and spin.

Induced C*-Complexes in Metaplectic Geometry

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we obtain an elliptic $C^*$-complex in the sense of Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^*$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles

Chern-Dirac bundles on non-Kähler Hermitian manifolds

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with the Levi-Civita connection replaced by the Chern connection. We then show that the tensor product of the canonical and the anticanonical spinor bundles, called the $\mathcal{V} $-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic sections isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms among other types of harmonic sections of the $\mathcal{V} $-spinor bundle and twisted cohomology.

Additional fermionic fields onto parallelizable 7-spheres

The geometric Fierz identities are here employed to generate new emergent fermionic fields on the parallelizable (curvatureless, torsionfull) 7-sphere ($S^7$). Employing recently found new classes of spinor fields on the $S^7$ spin bundle, new classes of fermionic fields are obtained from their bilinear covariants by a generalized reconstruction theorem, on the parallelizable $S^7$. Using a generalized non-associative product on the octonionic bundle on the parallelizable $S^7$, these new classes of algebraic spinor fields, lifted onto the parallelizable $S^7$, are shown to correctly transform under the Moufang loop generators on $S^7$.

Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields

An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Cartan-Maxwell-Dirac fields based on “minimal geometric data”: the needed underlying structure is determined, via geometric constructions, from the unique assumption of a complex vector bundle $SM$ with 2-dimensional fibers, called a $2$-spinor bundle. Any further considered object is assumed to be a dynamical field; these include the gravitational field, which is jointly represented by a soldering form (the tetrad) relating the tangent space $M$ to the $2$-spinor bundle, and a connection of the latter (spinor connection). The Lie derivatives of objects of all considered types, with respect to a vector field ${\scriptstyle X}\colon M\rightarrow M$, turn out to be well-defined without making any special assumption about ${\scriptstyle X}$, and fulfill natural mutual relations.

A New $$\frac{1}{2}$$ 1 2 -Ricci Type Formula on the Spinor Bundle and Applications

Consider a Riemannian spin manifold $$(M^{n}, g)$$ $$(n\ge 3)$$ endowed with a non-trivial 3-form $$T\in \Lambda ^{3}T^{*}M$$ , such that $$\nabla ^{c}T=0$$ , where $$\nabla ^{c}:=\nabla ^{g}+\frac{1}{2}T$$ is the metric connection with skew-torsion T. In this note we introduce a generalized $$\frac{1}{2}$$ -Ricci type formula for the spinorial action of the Ricci endomorphism $${{\mathrm{Ric}}}^{s}(X)$$ , induced by the one-parameter family of metric connections $$\nabla ^{s}:=\nabla ^{g}+2sT$$ . This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrodinger–Lichnerowicz formula related to the square of the Dirac operator $$D^{s}$$ , induced by $$\nabla ^{s}$$ , under the condition $$\nabla ^{c}T=0$$ . In the same case, we provide integrability conditions for $$\nabla ^{s}$$ -parallel spinors, $$\nabla ^{c}$$ -parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kahler manifolds and nearly parallel $$\hbox {G}_2$$ -manifolds, in dimensions 5, 6 and 7, respectively.

Chiral splitting and mathcal{N}=4 Einstein-Yang-Mills tree amplitudes in 4d

We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional mathcal{N}le 4 supersymmetric Einstein-Yang-Mills theory, based on the refined scattering equations. This generalizes previously known formulas for all-trace purely bosonic, or supersymmetric single-trace amplitudes. We find this formula by applying a new chiral splitting formula for all CHY Pfaffians in 4d, into two determinants, of positive and negative helicity respectively. The splitting of CHY Pfaffians is shown to be a special case of the splitting of Tmathbb{M} valued fermion correlators on the sphere, which does not require the scattering equations to hold, and is a consequence of the isomorphism Tmathbb{M}simeq {mathbb{S}}^{+}otimes {mathbb{S}}^{-} between the tangent bundle of Minkowski space and the left- and right-handed spin bundles. We present and prove this general splitting formula.

Spectral triples from bimodule connections and Chern connections

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators \mathrlap{\,/}{D} starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of M_2(\mathbb C) , and also applies to the standard q -sphere and the q -disk with the right classical limit and all properties holding except for \mathcal J now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the q -sphere case provides the relevant bimodule connection.

Moduli Spaces of Higher Spin Klein Surfaces

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. The spaces of higher spin bundles on Klein surfaces are important because of their applications in singularity theory and real algebraic geometry, in particular for the study of real forms of Gorenstein quasi-homogeneous surface singularities. In this paper we describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of an Euclidean space by a discrete group. We also discuss applications to real forms of Brieskorn–Pham singularities.

G2-structures and octonion bundles

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonionic covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. We also interpret the choice of a particular G2-structure within the same metric class as a choice of gauge and show that under a change of this gauge, the torsion does transform as an octonion-valued connection 1-form. Finally, we also show an explicit relationship between the octonion bundle and the spinor bundle, define an octonionic Dirac operator and explore an energy functional for octonion sections. We then prove that critical points correspond to divergence-free torsion, which is shown to be an octonionic analog of the Coulomb gauge.

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.

Black Holes and Exotic Spinors

Exotic spin structures are non-trivial liftings, of the orthogonal bundle to the spin bundle, on orientable manifolds that admit spin structures according to the celebrated Geroch theorem. Exotic spin structures play a role of paramount importance in different areas of physics, from quantum field theory, in particular at Planck length scales, to gravity, and in cosmological scales. Here, we introduce an in-depth panorama in this field, providing black hole physics as the fount of spacetime exoticness. Black holes are then studied as the generators of a non-trivial topology that also can correspond to some inequivalent spin structure. Moreover, we investigate exotic spinor fields in this context and the way exotic spinor fields branch new physics. We also calculate the tunneling probability of exotic fermions across a Kerr-Sen black hole, showing that the exotic term does affect the tunneling probability, altering the black hole evaporation rate. Finally we show that it complies with the Hawking temperature universal law.

Ashtekar Variables: Structures in Bundles

Canonical gravity can be formulated by means of a densitized dreibein field together with an SU(2) connection. These so-called Ashtekar variables are the fundamental quantities, loop quantum gravity is resting on. In this paper, we review these variables from the perspective of fibre bundles. This is straightforward for the dreibein field as this is simply a frame field. The Ashtekar connection, however, is more complicated. It turns out that, at the level of the tangent bundle, it is given by the Levi-Civita connection plus a multiple of the Weingarten mapping, whose action on vector fields is induced from the vector product on \(\mathbb {R}^{3}\). Lifted to the spin bundle, one regains the well-known SU(2) Ashtekar connection. At the end, we apply our results to Friedmann–Robertson–Walker spacetimes.

Dirac operators on quasi-Hamiltonian G-spaces

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.

Manifolds with vectorial torsion

The present note deals with the properties of metric connections ∇ with vectorial torsion V on semi-Riemannian manifolds (Mn,g). We show that the ∇-curvature is symmetric if and only if V♭ is closed, and that V⊥ then defines an (n−1)-dimensional integrable distribution on Mn. If the vector field V is exact, we show that the V-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature—for example, a V-Ricci-flat connection with vectorial torsion in dimension 4, explaining some constructions occurring in general relativity. Finally, we investigate the Dirac operator D of a connection with vectorial torsion. We prove that for exact vector fields, the V-Dirac spectrum coincides with the spectrum of the Riemannian Dirac operator. We investigate in detail the existence of V-parallel spinor fields; several examples are constructed. It is known that the existence of a V-parallel spinor field implies dV♭=0 for n=3 or n≥5; for n=4, this is only true on compact manifolds. We prove an identity relating the V-Ricci curvature to the curvature in the spinor bundle. This result allows us to prove that if there exists a nontrivial V-parallel spinor, then RicV=0 for n≠4 and RicV(X)=X▪dV♭ for n=4. We conclude that the manifold is conformally equivalent either to a manifold with Riemannian parallel spinor or to a manifold whose universal cover is the product of R and an Einstein space of positive scalar curvature. We also prove that if dV♭=0, there are no non-trivial ∇-Killing spinor fields.

Spectral triples on proper étale groupoids

A proper étale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel–Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of the tangent bundle and the obstruction to lift the tangent bundle to a spinor bundle. In the case of an orientable and spin Lie groupoid, an invariant spinor bundle and an invariant Dirac operator will be constructed. This data gives rise to a spectral triple. The algebraic orientability axiom in noncommutative geometry is reformulated to make it compatible with the geometric model.

On seven dimensional 3-Sasakian manifolds

3-Sasakian manifolds in dimension seven have cocalibrated andnearly parallel G-structures. In this work, cocalibrated G-structure is deformed by one of the characteristic vector fields of the 3-Sasakian structure anda new G2structure is obtained whose metric has negative scalar curvature. Inaddition, the new G2structure has a nonzero Killing vector field. Then, by using this deformation, new covariant derivative on the spinor bundle is obtained and the new Dirac operator is written in terms of the Dirac operator before deformation