The paper provides conditions for factoring equivariant spectral triples in unbounded KK-theory.
problem Factoring equivariant spectral triples in unbounded KK-theory.
method Sufficient conditions for factorization of equivariant spectral triples as a Kasparov product.
result Equivariant Dirac-type spectral triples on torus principal bundles always factorize.
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.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.
problem Equivariant invariants on manifolds with boundary.
method Analysis of Dirac operators, winding numbers, spectral flow, Maslov indices, and η-invariants.
result Established relation between equivariant η-invariants and Maslov triple indices.
We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) …
This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain for which was used in the first paper of the series. The computation has two main ste…
Constructs an analytic index for infinite dimensional manifolds with LT-action.
problem Analyzing infinite dimensional manifolds with LT-action. method Defines an analytic LT-equivariant index using a Hilbert space, Dirac operator, and crossed product. result Justifies the analytic index in terms of noncommutative geometry.
Constructs a new class for foliations to recover a secondary characteristic class.
problem Recovering the Godbillon-Vey invariant in equivariant KK-theory. method Groupoid equivariant Kasparov class for transversely oriented foliations.
result Chern character recovers the Connes-Moscovici cyclic cocycle for the Godbillon-Vey class.
We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.
This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal…
Extends Einstein-Hilbert action to higher-order spectral triples.
problem No specific problem stated; focuses on extending action.
method Introduced two second-order spectral triples and computed their Einstein-Hilbert actions.
result Demonstrated applicability of the theoretical framework.
New spectral triples for higher-rank graphs linked to wavelet decompositions.
problem Creating spectral triples for higher-rank graph C∗-algebras. method Generalizing spectral triples from Cuntz-Krieger algebras to higher-rank graph C∗-algebras and connecting them to wavelet decompositions. result Wavelet decompositions describe eigenspaces of Dirac operators in these spectral triples.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
Study noncommutative orbifolds using spectral triples and Morita equivalence.
problem Understanding noncommutative geometry of group actions on orbifolds.
method Developed Morita equivalence for crossed product spectral triples.
result Noncommutative orbifolds are Morita equivalence classes of spectral triples.
Construct spectral triples on C*-algebras with group actions.
problem Building spectral triples on C*-algebras with group actions.
method Systematic construction of spectral triples on A using geometry of AG and G. result Comparison with established examples of spectral triples.
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 spectral triples defined for SU(1,1) using harmonic analysis.
problem Defining new spectral triples for SU(1,1).
method Using harmonic analysis of SU(1,1) to construct pseudo-Riemannian and indefinite spectral triples.
result Triple (A,H,D) forms both pseudo-Riemannian and indefinite spectral triples. Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of σ-connections on finitely generated projective modules. This ma…
Defines curvature for spectral triples and applies to θ-deformations.
problem Defining curvature for noncommutative spectral triples.
method Using Levi-Civita connection, defines curvature tensors and derives Weitzenbock formula.
result Riemann and Ricci tensors transform naturally under θ-deformation, while scalar curvature is invariant.
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…
Researchers create spectral triples for twisted crossed products using Kasparov's external product.
problem Constructing spectral triples for twisted crossed products.
method Using Kasparov's external product, the construction of spectral triples for twisted crossed products is achieved.
result The construction of spectral triples for twisted crossed products is possible under suitable assumptions.
Develops spectral triples for quantum projective space.
problem Noncommutative geometry of quantum projective space.
method Introduces Kähler structures and spectral triples for quantum projective space.
result Produces an even spectral triple for quantum projective space.
Proves an index theorem for foliations using spectral triples.
problem Proving an Atiyah L2 covering index theorem for foliations. method Symbol calculus for foliations and spectral triples.
result Induces the same map on K-theory for two types of spectral triples.
We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the m…
We construct certain spectral triples in the sense of A. ~Connes and H. Moscovici (``The local index formula in noncommutative geometry'' {\it Geom. Funct. Anal.}, 5(2):174--243, 1995) that is transversally elliptic but not necessarily elliptic. We prove that these spectral triples satisfie the conditions which ensure …
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.
This thesis extends noncommutative geometry to semi-Riemannian manifolds and applies it to gauge theories.
problem Applying noncommutative geometry to Lorentzian manifolds for particle physics.
method Generalizing spectral triples to semi-Riemannian manifolds, constructing noncommutative gauge theories.
result Recovering the Standard Model using noncommutative geometry.
The paper derives spectral (0,4)-tensor functionals using the noncommutative residue.
problem Deriving spectral (0,4)-tensor functionals on compact spin manifolds.
method Using four one-forms and the Dirac operator, the noncommutative residue is applied to even-dimensional compact spin manifolds.
result Spectral (0,4)-tensor functionals are extended to a general spectral triple.
Defines an L2-signature for foliations using spectral triples.
problem Defining an L2-signature for foliations.
method Using Connes fibration and semi-finite spectral triples.
result Establishes compatibility of spectral triples for foliations.
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose ``Riemannian'' aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove …
To a finite, connected, unoriented graph of Betti-number g>=2 and valencies >=3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another example where non-commutative geometry provides a rigid framework for classification.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
Study spectral theory of non-Riemannian symmetric spaces.
problem Investigate spectral decomposition of invariant differential operators on compact quotients.
method Analyze geometry of properly transitive triples (G, H, L) and derive Casimir operator expressions.
result Discrete spectral decomposition for Type I triples, continuous for Type II.
The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…
Extends Bismut-Zhang's embedding formula to equivariant settings.
problem Mod Z embedding of eta invariants in equivariant families.
method Generalizes spectral flow to equivariant chern character.
result Equivariant embedding formula established.
Computing Chern-Simons action for perturbed Dirac triples
problem Computing Chern-Simons action for perturbed Dirac triples
method Computing Chern-Simons action for perturbed Dirac triples
result Computing Chern-Simons action for perturbed Dirac triples
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.
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.
We study the triple $(G,π,\prs)$ where G is a connected and simply connected Lie group, π and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on G such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the d…
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the sm…
Constructs equivariant cohomology models for differentiable stacks.
problem Developing cohomology theory for stacks with group actions.
method Extends classical results for smooth manifolds to differentiable stacks.
result Derives spectral sequences generalizing Bott's spectral sequence.
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.
Paper constructs new equivariant Floer cohomology and proves invariance properties.
problem Invariance properties of spectral sequences in symplectic Khovanov homology and Heegaard Floer homology.
method Flexible construction of equivariant Floer cohomology with finite group action.
result Proves invariance properties of spectral sequences and introduces new concordance homomorphism.
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the…
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.