New compact support differential cohomology theory with Pontryagin duality proof.
problem Developing a new mathematical framework for differential cohomology with compact support.
method Adapting Cheeger-Simons approach to introduce differential cohomology with compact support, proving functoriality, excision theorem, and using Pontryagin duality.
result Pontryagin duality for differential cohomology, showing isomorphism between ordinary differential cohomology and the smooth Pontryagin dual of compactly supported differential cohomology.
Characterizes primary operations in differential cohomology using stacks.
problem Understanding primary operations in differential cohomology.
method Characterization via stacks, explicit refinement of Steenrod squares and powers, interplay between different cohomology types.
result Developed techniques for differential cohomology, including Künneth decomposition.
We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilins…
Study spectral sequences in smooth generalized cohomology theories.
problem Systematic study of torsion in differential cohomology.
method Use Atiyah-Hirzebruch spectral sequences with filtrations by Cech resolutions.
result Explicit identification of differentials in spectral sequences for various theories.
Extends Massey products to differential cohomology using stacks.
problem Tackles Massey products in differential cohomology.
method Uses stacks to extend Massey products from cohomology to differential cohomology.
result Shows how properties of extended Massey products relate to classical ones.
Defines differential equivariant cohomology for Lie group actions.
problem Equivariant cohomology for quotient stacks.
method Principal bundles with connections, differential cohomology.
result Chern-Weil homomorphism factors through differential equivariant cohomology.
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.
We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…
The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by Hopkins-Singer and la…
Researchers twist Deligne cohomology for the first time.
problem No new problem introduced; existing Deligne cohomology is already versatile.
method Explicitly twisted Deligne cohomology by taking degree one twists of integral and de Rham cohomology.
result New properties of twisted Deligne cohomology are presented and illustrated.
Constructs AHSS for twisted differential generalized cohomology theories.
problem Generalizing AHSS for twisted settings and differential cohomology.
method Builds on previous work, uses bundles of spectra with flat connections.
result Establishes twisted differential spectra as bundles of spectra with flat connections.
The paper constructs cohomology classes on curve strata.
problem Understanding cohomology classes on curve strata.
method Using geometry of the boundary stratification of moduli space of multi-scale differentials.
result Construction of non-trivial and non-tautological cohomology classes.
Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.
problem Understanding cohomology classes on odd symplectic manifolds and their relation to Lagrangian submanifolds.
method Investigates complexes of differential, integral, and pseudo forms, introduces new operators, and proves cohomology isomorphisms.
result Proves isomorphism between de Rham cohomology and BV Laplacian cohomology on odd symplectic manifolds.
The paper introduces cohomology of coinvariant differential forms and studies its relations with other cohomologies.
problem Understanding the cohomology of coinvariant differential forms and its connections to other cohomologies.
method Defined cohomology of Γ-coinvariant forms and studied its relations with de Rham cohomology and invariant forms cohomology.
result Established relations between cohomology of coinvariant forms, de Rham cohomology, and cohomology of invariant forms.
Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence…
In this paper it is shown that multiplicative cohomology theories that are rationally even -- a technical condition that is often satisfied -- the Hopkins-Singer construction of generalized differential cohomology has a unital, graded commutative multiplicative structure. To this end, an explicit integration and a diff…
Develops axiomatic framework for differential cohomology and constructs generalized Cheeger-Simons characters.
problem Differential cohomology in the relative case.
method Axiomatic framework and construction of generalized Cheeger-Simons characters.
result Definition of integration map for fibre with boundary.
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of generalized cohomology theories including products and Umkehr maps.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
Study differential operators on specific manifolds and their harmonic forms.
problem Understanding harmonic forms on almost-Hermitian manifolds.
method Analysis of differential operators, Hodge Theory, and cohomologies.
result Comparison of harmonic forms and cohomologies with classical ones.
Computes cohomology classes of strata of meromorphic differentials.
problem Computing cohomology classes of differentials with prescribed poles.
method Defined a space of stable meromorphic differentials, stratified by zero multiplicities, computed Poincaré-dual cohomology classes, proved tautological, and provided an algorithm.
result All cohomology classes are tautological and can be computed.
The paper defines a new differential on manifolds with special holonomy and calculates their cohomology.
problem Defining and calculating cohomology on manifolds with special holonomy.
method Using the Frölicher-Nijenhuis bracket on parallel forms on G2- and mSpin(7)-manifolds. result Cohomology groups of differential forms and partial description of the cohomology of differential forms relative to the tangent bundle are calculated.
Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
problem Define and investigate differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
method Define Haefliger's differentiable cohomology for diffeomorphisms, investigate its structure, and generalize to flat Cartan groupoids.
result Define characteristic maps for geometric structures on manifolds associated to flat Cartan groupoids.
A new cohomology, induced by a vector field, is defined on pairs of differential forms (1--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an 1-differentable cohomology of Lichnerowicz type associated to an one form. Also, the case when the manifold is complex and …
A bicategory approach to differential cohomology is presented. Based on the axioms of Bunke-Schick, a symmetric monoidal groupoid is associated to differential refinements of cohomology theories. It is proven that such differential refinements are unique up to equivalence of the corresponding symmetric monoidal groupoi…
Modern differential cohomology explained with applications.
problem Understanding differential cohomology from a modern perspective.
method Sheaves on manifolds, Chern-Weil theory, differential characters, differential characteristic classes.
result Differential lift of the first Pontryagin class.
We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontry…
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
Survey explores cohomology's roles in applied math and sciences.
problem Understanding cohomology's role in solving differential equations.
method Examining differential complexes and structure-preserving discretizations.
result Various fundamental concepts in mechanics are formulated using differential complexes.
Higher gauge theory via differential nonabelian cohomology
problem Global infrared completion of higher gauge fields
method Maxwell-type higher gauge fields
result Electromagnetic flux quantization
We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M,g) and the Lq,p-cohomology of that manifold. The Lq,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in Lp(M) modulo the exact forms which are exterior diff…
We review and elaborate on some aspects of the quantization of certain classes of higher abelian gauge theories using techniques of generalized differential cohomology. Particular emphasis is placed on the examples of generalized Maxwell theory and Cheeger-Simons cohomology, and of Ramond-Ramond fields in Type II super…
In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…
In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…
Local study of foliation deformation cohomology.
problem Understanding deformations of singular foliations.
method Introducing and studying local deformation cohomology.
result Local deformation cohomology for singular foliations and related structures.
Explains model structures for higher orbifolds and applies them to quantum cohomology.
problem Understanding quantum cohomology of higher orbifolds.
method Develops model structures on higher orbifolds and applies them to quantum cohomology.
result Model structures provide insights into quantum cohomology of higher orbifolds.
Study cohomology of abelian differentials, find new stratifications.
problem Cohomology classes in abelian differentials strata.
method Unified topological approach to find explicit stratifications.
result Explicit stratification of spin moduli space for odd spin structures.
This paper proves cohomology invariants for differentiable stacks.
problem Understanding cohomology of differentiable stacks.
method Simplicial approach to representations up to homotopy.
result Cohomology with coefficients in a representation up to homotopy is a Morita invariant of the underlying stack.
Defines algebraic structures in Lagrangian Floer cohomology using differential forms.
problem Modeling algebraic structures in Lagrangian Floer cohomology.
method Defines two algebra structures using differential forms and a closed-open map, showing they coincide.
result Two algebra structures for the 2-dimensional Clifford torus coincide.
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ=0, and the algebraic o…
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invar…
The paper shows how to represent cohomology classes on Kähler manifolds using differential forms.
problem Representing cohomology classes on compact Kähler manifolds as differential forms.
method Representing Chern characters as Čech cocycles and analyzing their behavior under Hodge structure.
result Every rational cohomology class of type (p,p) on a compact Kähler manifold can be represented by a differential (p,p)-form.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…
We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
New method recovers differential cohomology from diffeological spaces.
problem Recovering differential cohomology from diffeological spaces.
method Introducing skeletal diffeologies and thin homotopies to recover differential cohomology.
result Ordinary differential cohomology can be recovered in terms of the homotopy theory of skeletal diffeological spaces.
Study cohomology spaces of sl(2) acting on n-ary differential operators.
problem Computing cohomology spaces for sl(2) action on n-ary differential operators.
method Analyzes polynomial μ-densities as sl(2) modules and computes cohomological spaces H^2.
result Computed cohomological spaces H^2 of sl(2) on n-ary differential operators.