Spinor Bundle Research Articles

Twistor representation of Yang–Mills fields on Minkowski space

A correspondence between (not necessarily self-dual nor anti-self-dual) Yang–Mills fields on Minkowski space, and pairs of cohomology classes γ, φ, on null twistor space PN■ is established; γ∈H1CR (PN■,g) defines a deformed Cauchy–Riemman (C–R)(γ) structure on a principal bundle over PN, and φ∈H1CR(γ) (PN■,O(−4)⊗g). The correspondence depends on the choice of a spacelike hyperplane in Minkowski space. Here, γ and φ provide initial values in the spin bundle over this hyperplane for a system of evolution equations, along the null geodesic spray congruence in the spin bundle over Minkowski space.

The Spectrum of the Dirac Operator on the Hyperbolic Space

AbstractWe represent the real hyperbolic space Hn as the rank one homogeneous space Spin (1, n)/Spin (n) and the spinor bundle S of H as the homogeneous bundle Spin (1, n) xSpin (n)VΔ where VΔ is the spinor representation space of Spin (n). The representation theoretic decomposition of L2(H, S) combined with the PARTHASARATHY formula for the DIRAC operator D yields the spectral representation of D2.

Spin characteristic classes and reduced 𝐾𝑆𝑝𝑖𝑛 group of a low-dimensional complex

This note studies relations between Spin bundles, over a CW-complex of dimension ≤ 9 \leq 9 , and their first two Spin characteristic classes. In particular by taking Spin characteristic classes, it is proved that the stable classes of Spin bundles over a manifold M M with dimension ≤ 8 \leq 8 are in one to one correspondence with the pairs of cohomology classes ( q 1 , q 2 ) ∈ H 4 ( M ; Z ) × H 8 ( M ; Z ) ({q_1},{q_2}) \in {H^4}(M;\mathbb {Z}) \times {H^8}(M;\mathbb {Z}) satisfying \[ ( q 1 ∪ q 2 + q 2 ) mod 3 + U 3 1 ∪ ( q 1 mod 3 ) ≡ 0 ({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0 \] , where U 3 1 ∈ H 4 ( M ; Z 3 ) U_3^1 \in {H^4}(M;{\mathbb {Z}_3}) is the indicated Wu-class of M M . As an application a computation is made for K Spin ~ ( M ) \widetilde {K\operatorname {Spin} }(M) , where M M is an eight-dimensional manifold with understood cohomology rings over Z , Z 2 , \mathbb {Z},{\mathbb {Z}_2}, , and Z 3 {\mathbb {Z}_3} .

Conformal nonlinear graviton

2. Let M be a four-dimensional complex manifold in which two holomorphic locally free spinor bundles S and S of rank 2 are given, together with the following identities: TM = S ® S, where TM is a tangent bundle and A2S = A2S. These identities define a holomorphic conformal structure in M, i.e., an invertible subbundle C ®2 in ~IM ® ~IM, where the symbol ® denotes the symmetric tensor product and C: = A=S *.

Some extensions of horrocks criterion to vector bundles on Grassmannians and quadrics

In this paper we prove that a vector bundle E on a grassmannian (resp. on a quadric) splits as a direct sum of line bundles if and only if certain cohomology groups involving E and the quotient bundle (resp. the spinor bundle) are zero. When rank E=2 a better criterion is obtained considering only finitely many suitably chosen cohomology groups.

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.

AN EXPLICIT COMPUTATION FOR THE BOSE-FERMI EQUIVALENCE ON RIEMANN SURFACES OF GENUS g

The instanton sum in the partition function for D bosons on a Riemann surface of genus g, with values in a general D-dimensional torus, TD = RD/ΛD is given explicitly. When the rational metric Q of the lattice, ΛD, is the identity we get the bosonization formula of Alvarez-Gaumé et al. for SO( 2D ). If Q is orthogonal, in the bosonization formula, we get the theta function associated with the quadratic form Q, if Q is generic we get rational Conformal Field Theory. Also we look for conditions on a twisted spin bundle LE, which may ensure that our partition functions arise from some generalized bosonization formulas.

Gauge anomalies on S2 and group extensions

A geometric connection is established between the group cocycle used by Mickelsson [Topological and Geometrical Methods in Field Theory (World Scientific, Singapore, 1986), pp. 117–131] to realize the Kac–Moody extension of the group of loops in SU(n) as the quotient of a topologically trivial extension of maps from the disk into SU(n) and the gauge anomaly for Quillen’s determinant bundle over the Cauchy–Riemann operators on the spin bundle over S2.

A pilot model in two dimensions for conformally invariant particle theory

The two-dimensional analog of the mathematical model for elementary particles treated in this journal vol 75, 1–57, is developed with greater explicitness than in the four-dimensional case, preliminary to quantization. In Part I the section space of the complex spannor bundle is studied. Invariant subspaces, invariant forms, and irreducible positive-energy factors under the connected conformal group are determined; transformation properties under discrete symmetries are also treated. Relations to the spinor bundle are treated; the Dirac operator is a canonical transform of the quadratic Casimir on the spannor bundle, and the section spaces have the same invariant factors; at the same time, the spinors constitute a deformation of the spannors. Two natural parallelizations of the spannor bundle and their intertwining operator are computed. The irreducible factors of the spannor section space under the conformal group are determined by their restrictions to Minkowski space, and this group acts continuously in the L 2-topology, in contrast to its action on the spinor section space.

On the twistor space of the six-sphere

The set of all complex lines of the right-handed Dirac spinor bundle of a standard six-sphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kähler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin(7), which acts transitively on the six-quadric, as a group of fiber-preserving isometries. We use a result due to Berard-Bérgery and Matsuzawa to show the existence of a non-Kähler, non symmetric, Hermitian-Einstein metric on the six-quadric, which is Spin(7)-invariant.

The primitive idempotents of the clifford algebras and the amorphic spinor fibre bundles

We intend to clarify the “primitive idempotent” method of defining spinor bundles as fields of minimal ideas in some Clifford algebra bundles. The fundamental notion of “geometric Cliffordian spinoriality groups” is introduced, a notion similar to that of “groupes de spinorialité” defined previously [1d] by the author, but not equivalent. Existence conditions depend on the choice of the standard left ideal and provide a “chaotic character” for physics. Existence conditions of primitive idempotent fields imply drastic conditions. We give several examples. The primitive idempotent method brings algebraic tools, but raises a lot of geometric obstacles and cannot replace, improve or generalize the “ r-isotropic” approach [1].

Spinor bundles on quadrics

We define some stable vector bundles on the complex quadric hypersurface Q n {Q_n} of dimension n n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q 4 ≃ Gr ⁡ ( 1 , 3 ) {Q_4} \simeq \operatorname {Gr} (1,\,3) . We call them spinor bundles. When n = 2 k − 1 n = 2k - 1 there is one spinor bundle of rank 2 k − 1 {2^{k - 1}} . When n = 2 k n = 2k there are two spinor bundles of rank 2 k − 1 {2^{k - 1}} . Their behavior is slightly different according as n ≡ 0 ( mod 4 ) n \equiv 0\;(\bmod 4) or n ≡ 2 ( mod 4 ) n \equiv 2\;(\bmod 4) . As an application, we describe some moduli spaces of rank 3 3 vector bundles on Q 5 {Q_5} and Q 6 {Q_6} .

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III ( S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983) , 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries. Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically. The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.

Real grassmannian ?-models and the extrinsic geometry of strings

We discuss the interplay between the extrinsic geometry of strings and the symmetric KahlerianG2,n σ-model coupled with two-dimensional gravity. It is shown that the recently proposed Polyakov's action classically corresponds to a particular topological sector of theG2,n model. The geometrical and physical meaning of the other topological sectors are also examined and shown to be related to the geometry of line bundles over Riemann surfaces (e.g. spinor bundles). The one-loop quantization of the model, around istanton-like solutions, provides scaling behaviour and asymptotic freedom for the adimensional coupling.

Some problems of spinor and algebraic spinor structures

A concrete realization of the Milnor–Lichnerowicz spinor bundle by algebraic spinors is considered in the case when the holonomy group of the Levi-Civita connection is equal to the Crumeyrolle group. Some relationships between the existence of parallel spinor fields on a space-time manifold ℳ and its topological invariants are given.

A notion of duality for twistor functions

A class of functions on the primed spin bundle of complexified Minkowski space, defined by ∇AA′∂f/∂πA′=0, is introduced and it is shown that these functions bear a close relationship to twistor functions, especially through their use in describing massless fields.

Gauge choice in Witten's energy expression

Witten's equation\(\not D\psi = 0\) can be interpreted as a gauge fixing condition for classical supergravity. We rigorously prove the existence of asymptotically constant solutions of the more general gauge condition\(\not D\psi = A\psi\) for almost all endomorphismsA of the spin bundle. Each gives an expression for the gravitational energy similar to Witten's. These include the choice\(A = \sqrt R\), which yields the particularly elegant energy expression first noticed by Deser.

A Note on the mod 2 Cohomology of BŜOn〈l6〉

Recall BSOn is the classifying space for the special orthogonal group of rank n. It is well known that the mod 2 cohomology ring of BSOn is given as follows:where wi is the i-th mod 2 Stiefel-Whitney class. For the 3-connective cover of BSOn, BSpinn, Quillen in [4] has determined H*(BSpinn) completely for all n. Let BŜOn〈8〉, BŜOn〈16〉 be the classifying spaces for n-plane spin bundle ξ satisfying w4(ξ) = 0 and w4(ξ) = w8(ξ) = 0 respectively. This note follows the method of A. Borel [2] and gives the mod 2 cohomology ring of BŜOn〈8〉 and BŜOn〈16〉 for small n. In particular we answer the question “when is H*(BŜOn〈8〉; Z2), or H*(BŜOn〈16〉; Z2) a polynomial algebra?”

Spinors as fundamental objects

We define, on the algebraic Dirac spinor space Ψ, some operators D± and T±. By means of them we show how the fundamental operations of Hermitian conjugation, complex conjugation, bar conjugation, and so on may be introduced in the Clifford algebra approach. These definitions depend on the geometrical property of the pure imaginary unit i of the Dirac algebra Ds,t and are the same only for mod 4 dimensions of vector space-times Rs,t. Furthermore, on the set Ψ×Ψ we introduce equivalence relations R± and define bijections χ± between Ψ×Ψ/R± and Ds,t. We investigate some properties of χ± and give the necessary and sufficient conditions for u∈Ds,t to belong to some minimal left ideal of Ds,t. Next we use the decomposition of the Dirac algebra Ds,t onto the Dirac spinor spaces to demonstrate two different ways of an action of any element s∈Spin(x,t). These considerations throw a new light onto the problem of the covariant derivative on the bundle of algebraic spinors over a space-time manifold.