Sections Of Vector Bundle Research Articles

Algebraic and analytic Dirac induction for graded affine Hecke algebras

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

A Class of Generalized Laplacians on Vector Bundles Devoted to Multi-Channel Image Processing

In the context of fibre bundles theory, there exist some differential operators of order 2, called generalized Laplacians, acting on sections of vector bundles over Riemannian manifolds, and generalizing the Laplace-Beltrami operator. Such operators are determined by covariant derivatives on vector bundles. In this paper, we construct a class of generalized Laplacians, devoted to multi-channel image processing, from the construction of optimal covariant derivatives. The key idea is to consider an image as a section of an associate bundle, that is a vector bundle related to a principal bundle through a group representation. In this context, covariant derivatives are determined by connection 1-forms on principal bundles. We construct optimal connection 1-forms by the minimization of a variational problem on principal bundles. From the heat equations of the generalized Laplacians induced by the corresponding optimal covariant derivatives, we obtain diffusions whose behaviors depend of the choice of the group representation. We provide experiments on color images.

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

We consider Schrodinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on \(\mathsf{C }^{\infty }_0\), and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian \(3\)-manifolds.

A NOTE ON THE CONFORMAL INVARIANCE OF G-GENERALIZED GRADIENTS

We consider generalized gradients in the general context of G-structures. They are natural first-order differential operators acting on sections of vector bundles associated to irreducible G-representations. We study their geometric properties and show in particular their conformal invariance.

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL ( n , C ) is an equivariant fibration X → X S of the complete flag variety X of C n . To each such fibration we associate an algebra J S of operators on L 2 ( X ) , or more generally on L 2 -sections of vector bundles over X . This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X → X S and X → X T , is a compact operator if S ∪ T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU ( n ) , which may be described as ‘essential orthogonality of subrepresentations’.

The chiral ring of AdS3/CFT2and the attractor mechanism

We study the moduli dependence of the chiral ring in N = (4,4) superconformal field theories, with special emphasis on those CFTs that are dual to type IIB string theory on AdS3xS3xX4. The chiral primary operators are sections of vector bundles, whose connection describes the operator mixing under motion on the moduli space. This connection can be exactly computed using the constraints from N = (4,4) supersymmetry. Its curvature can be determined using the tt* equations, for which we give a derivation in the physical theory which does not rely on the topological twisting. We show that for N = (4,4) theories the chiral ring is covariantly constant over the moduli space, a fact which can be seen as a non-renormalization theorem for the three-point functions of chiral primaries in AdS3/CFT2. From the spacetime point of view our analysis has the following applications. First, in the case of a D1/D5 black string, we can see the matching of the attractor flow in supergravity to RG-flow in the boundary field theory perturbed by irrelevant operators, to first order away from the fixed point. Second, under spectral flow the chiral primaries become the Ramond ground states of the CFT. These ground states represent the microstates of a small black hole in five dimensions consisting of a D1/D5 bound state. The connection that we compute can be considered as an example of Berry's phase for the internal microstates of a supersymmetric black hole.

Two-loop superstrings VII. Cohomology of chiral amplitudes

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.

Discontinuous non-linear mappings on locally convex direct limits

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the space of functions supported in K is smooth (and thus continuous), for each compact subset K of the line. More generally, we construct mappings with analogous pathological properties on spaces of compactly supported smooth sections in vector bundles over non-compact bases. The results are useful in infinite-dimensional Lie theory, where they can be used to analyze the precise direct limit properties of test function groups and groups of compactly supported diffeomorphisms.

Foundations of a nonlinear distributional geometry

Colombeau's construction of generalized functions (in its special variant) is extended to a theory of generalized sections of vector bundles. As particular cases, generalized tensor analysis and exterior algebra are studied. A point value characterization for generalized functions on manifolds is derived, several algebraic characterizations of spaces of generalized sections are established and consistency properties with respect to linear distributional geometry are derived. An application to nonsmooth mechanics indicates the additional flexibility offered by this approach compared to the purely distributional picture.

Heat kernel approach in quantum field theory

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.

HEAT KERNEL ASYMPTOTICS OF OPERATORS WITH NON-LAPLACE PRINCIPAL PART

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part -∇μ∇μ. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

The transverse geometry of $G$-manifolds and Riemannian foliations

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension of the orbit may be different from the dimension of the leaf closure. Conversely, given a Riemannian foliation, there is a metric on the basic manifold (an $O(q)$-manifold associated to the foliation) such that the leaf closure space is isometric to the $O(q)$-orbit space of the basic manifold via an isometry that preserves the volume of the leaf closures of maximal dimension. Thus, the orbit space of any Riemannian G-manifold is isometric to the orbit space of a Riemannian $O(q)$-manifold via an isometry that preserves the volumes of orbits of maximal dimension. Consequently, the spectrum of the Laplacian restricted to invariant functions on any $G$-manifold may be identified with the spectrum of the Laplacian restricted to invariant functions on a Riemannian $O(q)$-manifold. Other similar results concerning the spectrum of differential operators on sections of vector bundles over Riemannian foliations and $G$-manifolds are discussed.

INDEX THEORY OF PERTURBED DOLBEAULT OPERATORS: SMOOTH POLAR DIVISORS

This paper generalizes the index-theoretic content of the physical models studied in [3, 7]. The paper calculates the homology Chern character, and thus the index formula, for a broad class of perturbed Dolbeault operators on complete noncompact complex manifolds. The manifolds studied are complements of smooth polar divisors of meromorphic sections of vector bundles over closed complex manifolds. These sections define complexes of Koszul type that are used to construct the perturbations of the Dolbeault operators.

Nonstandard solvability for linear operators between sections of vector bundles

Given a certain kind of linear operator A (possibly a differential operator or a properly supported pseudodifferential operator) between sections of Hermitian vector bundles over a Riemannian manifold, a necessary and sufficient condition is obtained for the operator A to be solvable in a class of nonstandard sections in a generalized sense of weak solutions. The existence of a fundamental-solution-like internal section is established in the solvable case.

Dixmier's trace for boundary value problems

Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal \(\) for the Hilbert space \(\) of L2 sections in vector bundles E over X, F over ∂X.

On the multiplicative structure of sections of vector bundles on an algebraic curve

AbstractHere we study (in a more general setting) the following problem. Let C be a smooth projective curve, E and F vector bundles on C and V ⊆ H0 (C, E) (resp. W ⊆ H0 (C, F)) vector spaces generically spanning E (resp. F); find lower bounds for the dimension of the image of the multiplication map V ⊗ W → H0 (C, E ⊗ F) generalizing the case rank(E) = rank(F) = 1.

Nonlinear parabolic equations in sections of vector bundles

We study the Cauchy problem for a nonlinear parabolic equation relative to a section of a vector bundle. The special procedure of covariant differential extension makes it possible to reduce this equation to a system of quasilinear parabolic equations, which can be studied using the theory of stochastic equations. We obtain a probabilistic representation of the solution of both the auxiliary quasilinear parabolic system and the original nonlinear equation. Bibliography: 5 titles.

Path reparametrization in functional integrals on vector bundles

The procedure of path reparametrization (“substitution of a new time”) in functional integrals describing the evolution of sections of vector bundles is considered. The reparametrization Jacobian is found by means of a transformation of the random process that generates the measure in a functional integral of Wiener type.

Bounded $H_\infty$-calculus for elliptic operators

It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. 0. Introduction. It is the main purpose of this paper to prove under mild regularity assumptionsthat Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers. In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8&0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0 , n), we can define a bounded linear operator j(A) on Lp, and an estimate of the form llf(A)II.ccLp) ::::; c llflloo is valid. This means that elliptic operators possess a bounded R 00-calculus in the sense of Mcintosh [16]. Choosing, in particular, f(z) :=zit fortE JR, it follows that A possesses bounded imaginary powers ( cf. Section 2 below for more precise statements). There are two main reasons for our interest in this problem. First, it is known (cf. [22], [24]) that the complex interpolation spaces [E, D(A)]& coincide with the domains of the fractional powers A & for 0 < e < 1, provided A is a densely defined linear operator on the Banach space E possessing bounded imaginary powers. Second, by a result of Dore and Venni [1 OJ, the fact that A possesses bounded imaginary powers is intimately connected with 'maximal regularity results' for abstract evolution equations of the form u +Au = f (t). Both these results are of great use in the Received for publication August 1993. AMS Subject Classifications: 35J45, 47F05

A comparison theorem for eigenvalues of the covariant Laplacian

We present comparison theorems for the eigenvalues of the covariant Laplacian acting on the sections of vector bundle with the structure group U( N).