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.     

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.     

Analysis in space-time bundles. I. General considerations and the scalar bundle

Mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos (space-time) are presented. Bundles representative of physical fields are studied in the unique variant of Minkowski space M₀ enjoying similar properties of causality and symmetry. The “universal cosmos” M? is the universal cover of the causal compactification of M₀. The bundles studied are induced from representations of the scale-extended Poincaré group, which forms the isotropy group in M? of the universal cover $\text{G}\text{∽}\text{=}\text{S}\text{∽}\text{U(2,2)}$ of the connected component of the group of all causal transformations on M?. Discrete symmetries and higher-dimensional cases are also discussed.The primary focus is on the temporal evolution, especially stability (involving positivity of the energy), wave equations (implicative of finite propagation velocity), and the unitarity and/or composition series of associated actions of G?. General spin bundles on M? are treated, parallelized, and correlated with bundles on M₀. Associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case.     

Classifying Hilbert bundles. II

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, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

Discrete subgroups of GL(2, C) and spinor bundles on Riemann and Klein surfaces

Central Scientific Research Institute of Geodesy, Aero-Surveying, and Cartography. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 76–78, October–December, 1991.     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

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.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

Bordism rings with split normal bundles. II

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

Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Pic^d(C) such that h⁰(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in P^r as a projectively normal curve) if and only if (r+1)h¹≤g≤r(r−1)/2+2h¹, where h¹?g+r−d=h¹(C,L).     

String dynamics inD = 4 space-time. II. Quantum theory

The general scheme for covariant quantization of the string theory in the previously introduced nonstandard Hamiltonian formulation is proposed. The particular case of a bosonic string is studied in detail and the Regge spectrum is found.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 80–89, October, 1996.     

Threefolds with big and nef anticanonical bundles II

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K _X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X ⁺ are not both birational.     

Equations with nonnegative characteristic form. II

This monograph consists of two volumes and is devoted to second-order partial differential equations (mainly, equations with nonnegative characteristic form). A number of problems of qualitative theory (for example, local smoothness and hypoellipticity) are presented. To the memory of Ol’ga Arsen’evna Oleinik, my teacher Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 56, Partial Differential Equations, 2008.