Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Connection colligations on Hilbert bundles

The theory of operator colligations in a Hilbert space is extended to connection colligations on Hilbert bundles. Any invariant subbundle of a Hilbert bundle leads to a decomposition of connection colligation into simpler components. The problem of finding all invariant subbundles of a Hilbert bundle is reduced to a search of subspaces of a fixed Hilbert space which are invariant with respect to the holonomy group of the linear connection.     

On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over ${{\mathbb{C}}}$ and Hilb ^d (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle ${{\mathcal{F}}_d(E)}$ over Hilb ^d (X). If E and V are semistable vector bundles on X such that ${{\mathcal{F}}_d(E)}$ and ${{\mathcal{F}}_d(V)}$ are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).     

Analysis in space-time bundles,II. The spinor and form bundles

The structures of the spin and form bundles over the universal cosmos M?, and their relations with corresponding bundles over the Minkowski space M₀ canonically imbedded in M?, are treated. Wave equations covariant with respect to the causal group G of M? are studied, their solution manifolds and other stable (essentially positive-energy) invariant subspaces of the section spaces of the bundles are determined, and the indecomposability of relevant invariant subspace chains is shown. Explicit parallelizations of the bundles are applied to the Dirac and Maxwell equations on M?. A basis for spinor fields that diagonalizes a complete set of K?-covariant quantum numbers (K? = maximal essentially compact subgroup of G?) is developed. Local multilinear invariants of bundles over M? are treated and specialized to convergent mathematical versions of the Fermi and Yukawa interaction Lagrangians that are G?-invariant for the appropriate conformal weights.     

Bordism rings with split normal bundles. II

Dedicated to Yuri Grigor'evich Reshetnyak on his sixtieth birthday.     

Hilbert symbol for Lubin-Tate formal groups. II

Explicit formulas are obtained for the generalized Hilbert symbol on the group of points of a formal Lubin-Tate group, simultaneously covering the cases of even and odd characteristics of the residue field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 85–96, 1983.     

Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers

Holomorphic vector bundles on the Riemann sphere and the 21 st Hilbert problem

This report was presented at the first French-Russian colloquium on Geometry in Luminy in May 1992.     

Hilbert space with non-associative scalars II

The theory of a Hilbert space over a finite associative algebra is formulated, and the spectral resolution theorem for bounded Hermitian operators on this space is obtained. The properties of series representations are discussed and are found to be analogous to the usual ones of the complex Hilbert space. It is then shown that the theory of the non-associative Hilbert space developed in our previous paper is contained in the more general theory for the special case in which the finite algebra is chosen to be the Cayley ring.     

Characteristic classes of flat bundles, II

On a smooth varietyX defined over a fieldK of characteristic zero, one defines characteristic classes of bundles with an integrableK-connection in a group lifting the Chow group, which map, whenK is the field of complex numbers andX is proper, to Cheeger-Simons' secondary analytic invariants, compatibly with the cycle map in the Deligne cohomology.     

Analysis in space-time bundles. III. Higher spin bundles

The universal cosmos M? is the unique four-dimensional globally causal space-time manifold to which the Dirac and Maxwell equations (among others) maximally and covariantly extend. A systematic treatment is presented of general fields over M?, of arbitrary spin; considered are fields induced from all irreducible representation of the isotropy group (scale-extended Poincaré group) to G?, the connected causal group of M?. Restricted to any species of such fields, the K?-invariant canonical Dirac operator (K? = maximal essentially compact subgroup of G?) is shown G?-covariant for a unique conformal weight. A normalized K?-finite basis for such fields is constructed. The basis actions thereon of the Dirac operator, infinitesimal generators of G?, discrete symmetries, second-order Casimir, and the essentially unique third-order noncentral quantum number (enveloping algebra element) invariant under K? are derived. Composition series under G? of a class of these field spaces—namely, the extension to M? of the relativistic fields considered by Bargman and Wigner, or arbitrary spin and conformal weight—are determined, distinguishing by invariance and causality features alone the essentially conventional positive-energy mass 0 subspaces and massive invariant sub-quotient spaces, whose unitarity under G? is given a new proof. The “completely positive” subclass (cf. below) of representations is determined. A more detailed treatment of spin one bundles (vector and two-form, of arbitrary conformal weight) is included; the exterior derivative transformations are diagonalized, and the conformally invariant massive spin one scalar product is identified with a mathematical version of the conventional electromagnetic field Lagrangian.     

On the geometry of orthonormal frame bundles II

We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications of the Sasaki-Mok metric depending on one real parameter c ≠ 0. The metrics are “strongly invariant” in some special sense. In particular, we consider the case when (M, g) is a space of constant sectional curvature K. Then, for dim M > 2, we find always, among the metrics , two strongly invariant Einstein metrics on OM which are Riemannian for K > 0 and pseudo-Riemannian for K < 0. At least one of them is not locally symmetric. We also find, for dim M ≥ 2, two invariant metrics with vanishing scalar curvature.