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.
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.
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.
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 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.
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.
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.
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.
Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.
problem Understanding spectral asymmetry for the massless Dirac operator.
method Constructing a negative order pseudodifferential asymmetry operator from spectral projections.
result Computed the principal symbol of the asymmetry operator, accounting for gauge invariance.
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
Study spectral functionals on manifolds with torsion.
problem Extending classical results to geometries with torsion.
method Using Wodzicki residue, investigate spectral functionals of Dirac and Laplace-type operators.
result Local densities recover fundamental geometric tensors.
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
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.
Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.
problem Finding spectral gaps for Dirac operators on hyperbolic surfaces.
method Explicit construction of spin hyperbolic surfaces with increasing genus.
result Uniform spectral gap for Dirac operator on constructed surfaces.
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.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
Wave operators and spectral stability for Dirac operators under Ricci flow.
problem Stability of the absolutely continuous spectrum of Dirac operators under Ricci flow.
method Proving existence and completeness of wave operators for Dirac operators and their squares under Ricci flow.
result Criterion for spectral stability of Dirac operators and their squares under Ricci flow without injectivity radius assumptions.
Study on spectral points of Inoue surfaces with Tricerri metric.
problem Identifying spectral points on Inoue surfaces.
method Analyzing chiral Dirac operators twisted by flat C∗-connections. result No spectral points inside the annulus α−1/4<∣z∣<α1/4, with spectral points on boundary. In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-fo…
The paper studies Dirac operators on large spectral three-manifolds.
problem Analyzing Dirac operators on spectrally large three-manifolds.
method Non-linear analysis of Seiberg-Witten equations and understanding transversality in monopole Floer homology.
result The locus of flat U(1)-connections on a three-torus where a twisted Dirac operator has kernel is a two-sphere.
Introduces new spectral triples for parabolic geometry.
problem Anisotropies and varying orders in parabolic geometry.
method Tangled spectral triples incorporating directional Dirac operators.
result Higher order spectral triples for hypoelliptic complexes and nilpotent group algebras.
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…
The aim of this paper is to study a possible "boundary phenomenon" for Spinc Dirac operators in a special case. If you parametrise Spinc Dirac operators by a family of connections on a Spinc 4-manifold with boundary, this boundary inherits also a family of Spinc Dirac operators which has a spectral section (in the sens…
Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.
problem Computing the residue cocycle for Dirac-type operators.
method Modified Getzler calculus for computation.
result Computed residue cocycle for a class of Dirac-type operators.
Researchers find spectral gaps in quantum flag manifolds using twisted operators.
problem Finding spectral gaps in quantum flag manifolds.
method Tensoring Laplace and Dolbeault-Dirac operators with negative Hermitian holomorphic modules.
result Twisting Dirac and Laplace operators by negative line bundles produces a spectral gap for q close to 1.
We formulate a noncommutative generalization of the Ricci flow theory in the framework of spectral action approach to noncommutative geometry. Grisha Perelman's functionals are generated as commutative versions of certain spectral functionals defined by nonholonomic Dirac operators and corresponding spectral triples. W…
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
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…
A formula is given in terms of secondary characteristic classes for the leading order contribution to the spectral flow for a path of twisted Dirac operators on an odd dimensional, Riemannian manifold when the twisting is done by a path of unitary connections with large curvature.
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
New lattice Dirac operator index method for curved boundaries.
problem Defining Dirac operator indices for curved boundaries and gravitational backgrounds.
method Employing K-theory and spectral flow to classify Wilson Dirac operator.
result Mod-2 index defined in both even and odd dimensions.
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
Lattice formulation captures Atiyah-Patodi-Singer index.
problem Capturing index of Dirac operators in lattice gauge theory.
method Generalized spectral flow for non-product structure near boundaries.
result Correctly captures continuum index for small lattice spacings.
We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corres…
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.
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. Study Dirac operators on finite warped cylinders with gauge fields.
problem Characterize spectral flow on finite warped cylinders with gauge fields.
method Identify endpoint operators, derive determinant characterization, introduce regularized APS conditions.
result Regularized APS conditions admit a spectral-flow framework, matching zero-mode sets.
The paper improves L2-estimates for Dirac-Dolbeault operators on complex manifolds.
problem Improving L2-estimates for Dirac-Dolbeault operators on complex manifolds. method Generalized classical method to handle mixed curvature cases and provided bounds on error terms.
result Full asymptotic expansion for Bergman kernel obtained.
Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the…
The paper shows how solutions of perturbed Dirac operators concentrate near singular sets.
problem Understanding concentration of solutions for perturbed Dirac operators.
method Analyzing the algebraic criterion on $(c, \A)$ and spectral properties of deformed Laplacians.
result Proves an index localization theorem based on spectral separation properties.
Study on Dirac operator spectrum on shrinking surfaces with cusps.
problem Behavior of Dirac operator spectrum on degenerating Riemannian surfaces.
method Adapted pseudodifferential calculus, including Dirac operators and their resolvents.
result Smoothness of spectral projectors and t2logt regularity for the cusp-surgery trace. 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…
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Given an open book decomposition (Σ,τ) of a three manifold Y, Thurston and Winkelnkemper [TW] construct a specific contact form a on Y. Given a spin-c Dirac operator D on Y, the contact form naturally associates a one parameter family of Dirac operators $D_r = D - \frac{ir}{2}\cl(a)$ for r≥0. When $r>>…