arXiv research

Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constru…

We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…

Constructs equivariant spectral flow for Dirac-type operators on manifolds.

We give a local formula for the index of a transverse Dirac-type operator on a compact manifold with a Riemannian foliation, under the assumption that the Molino sheaf is a sheaf of abelian Lie algebras.

We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We …

Classifies local boundary conditions for Dirac-type operators on manifolds.

The purpose of this note is to describe a unified approach to the fundamental results in the spectral theory of boundary value problems, restricted to the case of Dirac type operators. Even though many facts are known and well presented in the literature (cf. the monograph of Booss-Wojciechowski), we simplify and exten…

We prove the Fredholmness of Dirac operators of monopoles with Dirac-type singularities on oriented complete Riemannian $3$-folds, and we also calculate the $L^2$-indices of them.

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented t…

In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index t…

Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer bo…

Computes indices of mixed order Dirac-type operators and related tensor fields.

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize var…

In this paper we provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation …

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

In the previous papers, Furuta, Yoshida and the author gave a definition of analytic index theory of Dirac-type operator on open manifolds by making use of some geometric structure on an open covering of the end of the open manifold and a perturbation of the Dirac-type operator. In this paper we show the cobordism inva…

This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Dirac-type operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of th…

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple descripti…

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any $L^2$-section $φ$ contained in a closed A-invariant…

We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define t…

In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-typ…

Let $\mathcal{A}_0$ and $\mathcal{A}_1$ be two self-adjoint Fredholm Dirac-type operators defined on two non-compact manifolds. If they coincide at infinity so that the relative heat operator is trace-class, one can define their relative eta function as in the compact case. The regular value of this function at the zer…

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operat…

Local index formula for Lorentzian Dirac operators on spacetimes.

We give a superconnection proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid.

Novel heat flow estimates on ALE manifolds for Schrödinger operators.

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in $E$-theory. The other type is by module multiplications in $K$-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Hi…

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows to treat directly general elliptic operator which…

We clarify the relation between the Dixmier-Douady class on the space of self adjoint Fredholm operators (`universal B-field') and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple a…

The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies th…

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equa…

Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of diffe…

When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the $K-$theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreov…

We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type …

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …

We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\p M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a…

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert $C^\ast$-module bundles and prove an index theorem for such operators…

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant int…

We introduce and study new spectral invariant of two elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depends on both the eigenvalues and the eigensections of the operators, which is a equal to the regularized number of created particles from the vacuum…

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prov…