arXiv research

We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a s…

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary.

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's…

A generalization of Callias' index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index obtained depends on the choice of a family of Fredholm extensions, though as in the …

We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our result extends an index theorem of D. Freed to non-compact manifolds, thus provi…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callia…

Interprets coarse symbol and index classes for Callias type operators.

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…

Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.

We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete even dimensional Riemannian manifold $M$ (the odd dimensional case was considered in our previous paper arXiv:1706.06737). We use this index to define the relative $η$-invariant $η(A_1,A_0)$ of two strongly Calli…

The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The comp…

Study of spectral flow in symmetric Toeplitz operator families.

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection i…

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we t…

We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G…

The paper establishes distance estimates for manifolds with lower scalar curvature bounds.

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The …

The goal is to understand the index-theoretic aspects of the recent preprint of R. Nest and F. Radulescu, math.OA/9911042. The basic observation (due to E. Guenter/N. Higson) is that the index of the Toeplitz operator is equal to the index of an associated Callias type operator, i.e. a Dirac operator with potential, th…

We provide criteria for self-adjointness and τ-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of…

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 study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold $M$. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds $N_0$ and $N_1$. If the dimension of $M$ is odd we show that …

We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of…

Paper solves long neck problem on odd-dimensional spin manifolds.

Proves principles and estimates for initial data sets in Einstein equations.

Derives generalizations of the long neck principle and spectral width inequality.

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…

Formulates Index III lemma and Rauch III theorem with applications.

Proves a lattice version of the Atiyah-Singer index theorem.

Explain Arnold's proof of the Morse index theorem using Maslov index.

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

Researchers prove an equivariant index theorem on Euclidean space.

Extends a theorem for first-order elliptic operators on manifolds.

In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes the Zhang's index theorem for sub-signature operators.

The paper proves an index theorem for loop spaces of compact manifolds.

Proves an equivariant version of index theorem for geometric families.

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $η$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-o…

Extends index theorem to domain walls with discontinuous Riemannian connections.

In this note, we prove an index theorem on Galois covering for Heisenberg elliptic differential operators, which is not elliptic, analogous to Atiyah's $Γ$-index theorem. This note also contains an example of Heisenberg differential operators with non-trivial $Γ$-index.

Atiyah-Singer theorem links math fields, predicts topological insights.

Massive fermions help understand index theorems without chiral symmetry.

A recent anomaly computation of Horava and Witten is proved and generalized in the form of two index theorems in odd dimensions. Theorem A is a fixed point formula for orientation-reversing involutions. Theorem B is an index theorem for manifolds with boundary using local boundary conditions. Both hold for families of …

Proves Morse index theorem for geodesics in conic Finsler manifolds.

Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.

The paper proves index theorems for graph-based optimal control problems.