Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
Study knot spaces and Atiyah duality in spectral categories.
problem Understanding the space of embeddings of a circle into higher-dimensional manifolds.
method Develop a cosimplicial model using Atiyah duality and prove a comodule version.
result Compute knot spaces in low degrees and establish isomorphisms on fundamental groups.
Paper establishes loop space T-duality formulae and refines earlier work.
problem Establishing T-duality on loop spaces with background flux.
method Developed loop Hori map and twisted Bismut-Chern character.
result Loop Hori map induces quasi-isomorphism on loop spaces.
We show that the graph TQFT for Heegaard Floer homology satisfies a strong version of Atiyah's duality axiom for a TQFT. As an application, we compute some Heegaard Floer mixed invariants of 4-dimensional mapping tori in terms of Lefschetz numbers on HF+.
Let Mn be a closed, connected n-manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the Spanier-Whitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M,S), whe…
Geometric framework for inverse problems using foliations and dual connections.
problem Reconstruction problems in inverse problems.
method Vaisman foliations and Atiyah--Molino sequences to induce transverse foliations and dual connections.
result Unique, path-independent reconstruction with vanishing torsion and curvature duality.
Detecting exotic spheres involves analyzing framed configuration spaces.
problem Detecting exotic spheres through homotopy type of truncated Disc-presheaves.
method Using gluing results for Disc-presheaves, Atiyah duality, and computations of mapping class groups.
result Conditions for detecting exotic spheres through framed configuration spaces.
Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.
problem Defining and linking novel Floer homologies for different manifold dimensions.
method Physics of a topologically-twisted 5d N=2 gauge theory, Vafa-Witten, Hitchin, and BF configurations.
result Derived novel gauge-theoretic and symplectic Floer homologies, and Atiyah-Floer correspondences.
Develops parametrised Poincaré duality for equivariant fixed points.
problem Understanding equivariant fixed points in non-presentable settings.
method Introduces parametrised Poincaré duality in parametrised higher category theory, proving basechange results.
result Generalises Cnossen's twisted ambidexterity to non-presentable settings and applies to isotropy separation methods.
The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P. Gilkey. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the…
Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…
A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with …
Study of mixed equation combining gauge theory and symplectic geometry.
problem Regularity and compactness of solutions to the mixed equation.
method Combining Uhlenbeck and Gormov compactness theorems.
result Moduli spaces of solutions to the mixed equation satisfy compactness properties.
A geometric model for twisted K-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of K-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric K-homology to the new g…
The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.
problem Characterizing Riemannian twistor spaces under vanishing curvature conditions.
method Moving frame method, classification, and nonexistence proofs.
result The only twistor space with parallel Bochner tensor is CP3. We study elliptic theory on manifolds with boundary represented as a covering space. Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. We show that elliptic elements define Fredholm operators and prove an index formula. …
Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.
problem Understanding the Atiyah and Todd classes of Lie algebroids.
method Analyzing the Atiyah sequence of Lie algebroids and proving class restrictions.
result Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence.
5D gauge theories are dual to 3D and 2D models via Floer homologies.
problem Exploring dualities in 5D gauge theories and their 3D and 2D counterparts.
method Using Landau-Ginzburg models and Floer homologies, the paper establishes dualities between different gauge theories and their associated homologies.
result Dual A∞-categories of Floer homologies are derived, proving mirror symmetry and Langlands duality. Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.
problem Atiyah-Floer conjecture for admissible SO(3)-bundles
method Adapted to admissible SO(3)-bundles
result A version of the Atiyah-Floer conjecture established
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a represent…
Atiyah reviewed holomorphic vector bundles and gauge theories.
problem Holomorphic vector bundles and gauge theories.
method Review of Atiyah's work from 1952-1990.
result Holomorphic vector bundles and gauge theories are interconnected.
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
Study connections on Lie groupoids and stacks using Atiyah sequences.
problem No specific problem stated; general connections on Lie groupoids and stacks.
method Construct connections using Atiyah sequences associated with transversal tangential distributions.
result Detailed study and construction of connections on Lie groupoids and stacks.
Paper develops a unified framework for Lie algebroid connections on various bundles.
problem Unified framework for Lie algebroid connections on vector and principal bundles.
method Generalized Atiyah algebroid structure and its short exact sequence.
result Explicit constructions of Atiyah classes for Lie algebroid connections.
The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra A when it is extended to L. In fact, given such an SH Lie pair (L,A), and any A-module E, there ass…
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Derives Atiyah sequence for noncommutative bundles.
problem Deciding when ∗-automorphisms lift to compatible ones. method Derivation-based Atiyah sequence derivation.
result Validates existence of compatible lifts.
The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.
problem Understanding the twistor spaces of Riemannian four-manifolds.
method Using the moving frame approach to analyze the twistor space Z of an oriented Riemannian four-manifold M. result Proves that first-order linear conditions on the almost complex structures of Z force the manifold M to be self-dual, and shows that the Atiyah-Hitchin-Singer twistor space bears a resemblance to a nearly Kähler manifold under first-order quadratic conditions. Revisits Vafa-Witten theory, deriving new invariants and homologies.
problem Understanding Vafa-Witten equations and their invariants.
method Physical derivation and mathematical analysis of Vafa-Witten equations.
result Novel invariants and Floer homologies derived from Vafa-Witten equations.
Established equivalence of Atiyah classes for generalized holomorphic vector bundles.
problem Defining and comparing Atiyah classes for generalized holomorphic vector bundles.
method Used three approaches: \(\check{C}\)ech cohomology, first jet short exact sequence, and Lie algebroid pairs.
result Equivalence of Atiyah classes defined by different methods.
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.
Constructs a triple on an Atiyah algebroid with connection.
problem Dynamics of systems on principal bundles and Atiyah algebroids.
method Constructs a Tulczyjew triple on a principal bundle with connection, then reduces to the Atiyah algebroid.
result Dynamics of systems on principal bundles and Atiyah algebroids are discussed and applied.
We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.
We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a K-theoretic formula for an index-type invariant. result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
For every Lie pair (L,A) of algebroids we construct a dg-manifold structure on the Z-graded manifold M=L[1]⊕L/A such that the inclusion ι:A[1]→M and the projection p:M→L[1] are morphisms of dg-manifolds. The vertical tangent bundle TpM then inherit…
Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.
problem Calculating the Atiyah-Patodi-Singer index without invertibility of boundary operator.
method Using an asymptotic gluing formula for eta invariants and a splitting principle.
result Formula expressing index in terms of eta invariants of domain-wall massive Dirac operators.
In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.
problem Constructing gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries.
method A gluing construction that captures the superposition of moduli spaces of centred SU(2) monopoles and Taub-NUT manifolds.
result Gravitational instantons are explicitly shown to be superpositions of Atiyah-Hitchin and Taub-NUT geometries.
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.
We present the details of our embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field Q admits a structure of L-infinity algebra with the Lie derivative LQ as unary …
The index theorem, discovered by Atiyah and Singer in 1963, is one of most important results in the twentieth century mathematics. It found numerous applications in analysis, geometry and physics. Since it was discovered numerous attempts to generalize it were made, see for example [5, 3, 4, 16, 12] to mention a few; s…
We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler's rescaling with Greiner's approach of the heat kernel asymptotics. As application we can easily compute the Connes-Moscovici cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index for…