Defines spectral sequences for fiberwise Dirac operators and proves adiabatic limit formula.
problem Calculating eta invariants for fibrations.
method Heat kernel method and analytic localization techniques.
result Extends remainder terms of eta invariants in fibrations.
Paper generalizes spectral flow formulas for compact Lie group actions.
problem Generalizing spectral flow formulas for compact Lie group actions.
method Equivariant version of Dai-Zhang higher spectral flow, embedding formula, adiabatic limit formula for Atiyah-Patodi-Singer eta invariants.
result Generalization of eta forms to equivariant Bismut-Cheeger eta forms.
It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a ''{lowest}'' action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for …
We consider a gauge invariant one parameter family of families of fiberwise twisted Dirac type operators on a fiberation with the typical fiber an even dimensional compact manifold with boundary, i.e., a family {Du},u∈[0,1] with D1=gD0g−1 for a suitable unitary automorphism g of the twisted bundle. Su…
We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise …
Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.
problem Understanding derivations and Lie algebras of vector bundles.
method Proving Lie algebras coincide through differential operators and Grothendieck constructions.
result Lie algebras coincide up to an isomorphism.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
The paper proves conditions for the triviality of L2-harmonic forms on Riemannian manifolds.
problem Conditions for the triviality of L2-harmonic forms on Riemannian manifolds. method Study of a covariant Schrödinger operator HX,V and its L2-kernel. result Sufficient conditions for the triviality of the L2-kernel of HX,V. Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Study on magnetic Dirac operators and their spectrum.
problem Understanding the spectrum of magnetic Dirac operators.
method Analysis of magnetic Dirac operators over complete Riemannian manifolds.
result Find sufficient conditions for maximal or discrete spectrum.
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…
Mathematical proof of index equality for lattice Dirac operators and continuum operators.
problem Equality of indices for lattice and continuum Dirac operators.
method Using K-theory, we prove equivalence of one-parameter families of continuum and lattice Dirac operators.
result Indices of continuum and lattice Dirac operators are equal.
Study perturbs APS boundary conditions for Lorentzian Dirac operators.
problem Maintaining Fredholmness of Dirac operators under perturbations of APS boundary conditions.
method Develop criteria for perturbing compact pairs of projections to remain Fredholm.
result Criteria for perturbing APS boundary conditions without losing Fredholmness.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
The paper calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
We use the spectra of Dirac type operators on the sphere Sn to produce sharp L2 inequalities on the sphere. These operators include the Dirac operator on Sn, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…
The paper proves formulas and theorems for Dirac-Witten operators on manifolds with or without boundaries.
problem Analyzing Dirac-Witten operators on manifolds with boundaries.
method Obtained Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
The paper calculates spectral torsion for rescaled Dirac operators on manifolds.
problem Computing spectral torsion for rescaled Dirac operators.
method Using trilinear Clifford multiplication and functional of differential one-forms.
result Computed spectral torsion for four types of rescaled Dirac operators.
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spinc Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing…
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.
problem Estimating eigenvalues of the Dirac-Witten operator on specific submanifolds.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Limiting-cases of eigenvalues studied and optimal bounds obtained.
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the Dirac operator with torsion on spin manifolds.
method Develops a Lichnerowicz type formula and proves a Kastler-Kalau-Walze type theorem.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator with torsion on 4D and 6D almost product Riemannian spin manifolds. The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the J-twist of the Dirac operator on spin manifolds.
method Lichnerowicz type formula and Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator on 3D and 4D almost product Riemannian spin manifolds with boundary.
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…
Optimizes eigenvalue bounds for submanifold Dirac operators.
problem Estimating eigenvalues of submanifold Dirac operators.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Optimal eigenvalue bounds established for submanifold Dirac operators.
This is the second part in a series of two papers. The k-Dirac complex is a complex of differential operators which are natural to a particular ∣2∣-graded parabolic geometry. In this paper we will consider the k-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …
The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.
problem Computing the noncommutative residue for a specific Dirac operator on 6D manifolds.
method Calculations and proofs for the rescaled Dirac operator fDh on 6D compact manifolds.
result Proof of the Kastler-Kalau-Walze type theorem for the rescaled Dirac operator on 6D compact manifolds with boundary.
The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.
problem Proving theorems for Dirac operators on manifolds with boundary.
method Establishing general Kastler-Kalau-Walze type theorems for conformal perturbations of Dirac operators.
result Proof of new theorems for Dirac operators on even-dimensional manifolds with boundary.
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
The article studies eigenvalues and spectrum of magnetic Dirac operators.
problem Fundamental mathematical properties of magnetic Dirac operators remain unexplored.
method Eigenvalue estimates and explicit spectrum computation for specific cases.
result Explicit computation of spectrum for magnetic fields on specific manifolds.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
New spectral functionals related to noncommutative residue and torsion Dirac operators.
problem Spectral functionals and Dirac operators with torsion.
method Noncommutative residue and Dirac operators with torsion.
result Extension of spectral functionals to noncommutative realm with torsion.
In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spe…
The paper broadens the class of manifolds where Dirac operator spectra are maximal.
problem Finding new manifolds with maximal Dirac operator spectra.
method Analyzing Lp-spectrum, using sufficient conditions, and applying the Weyl criterion. result New classes of manifolds with maximal Dirac operator spectra are identified.
In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boun…
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise (k,r)-j…
Classifies local boundary conditions for Dirac-type operators on manifolds.
problem Determining all local smooth boundary conditions for Dirac-type operators.
method Combining general theory of boundary value problems for Dirac operators and pointwise considerations.
result Classification of local self-adjoint regular boundary conditions for Dirac spinors in dimensions 3 and 4.