arXiv research

New cohomology ring for complex manifolds with group action.

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ri…

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycl…

We present a deRham model for Chen-Ruan cohomology ring of abelian orbifolds. We introduce the notion of \emph{twist factors} so that formally the stringy cohomology ring can be defined without going through pseudo-holomorphic orbifold curves. Thus our model can be viewed as the classical description of Chen-Ruan cohom…

Recently, there is an explosive growth of activities to understand stringy properties of orbifolds. In this article, we survey some of recent developments.

This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.

In this note, we reconcile two approaches that have been used to construct stringy multiplications. The pushing forward after pulling back that has been used to give a global stringy extension of the functors K_0,K^{top},A^*,H^* [CR, FG, AGV, JKK2], and the pulling back after having pushed forward, which we have previo…

In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related t…

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified grou…

We show global existence theorems for Gowdy symmetric spacetimes with type IIB stringy matter. The areal and constant mean curvature time coordinates are used. Before coming to that, it is shown that a wave map describes the evolution of this system.

We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these mo…

We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the definition of the Kontsevich integral to that of graphs in R^3 and study the beha…

The spaces of higher-order differential operators (in Dimension 1|2), which are modules over the stringy Lie superalgebra K(2), are isomorphic to the corresponding spaces of symbols as orthosymplectic modules in non resonant cases. Such an osp (2|2)-equivariant quantization, which has been given in second-order differe…

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kähler moduli space. A few basic examples are studied. In particular, we identify …

We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings, and is considerably more general than Lorentzian geometry. Our construction of geo…

Study examines corrections to heterotic geometry on SU(3) manifolds.

We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism …

We study the rigid limit of a class of hypermultiplet moduli spaces appearing in Calabi-Yau compactifications of type IIB string theory, which is induced by a local limit of the Calabi-Yau. We show that the resulting hyperkahler manifold is obtained by performing a hyperkahler quotient of the Swann bundle over the modu…

Extends cohomology theory for infinite volume transformation groups.

New cohomology theory for Lie 2-algebras extends classical theory.

Generics extended to new cohomologies.

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $θ$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[θ]$ of the odd cocycle $…

The article examines twisted cohomologies on algebraic and analytic varieties.

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

New cohomology theories for heaps and ternary operations linked to group cohomology.

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

Proves a vanishing property for symplectic manifold cohomology.

De Rham theorem extended to Orlicz cohomology.

In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…

New proof of blow-up formula for Morse-Novikov cohomology.

Survey on equivariant cohomology of Lie group actions.

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

Compute local cohomology of vector fields on manifolds.

The paper categorifies matroid characteristic polynomials using cohomology.

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been expl…

Unified theory of orbifolds and cohomology.

New cohomology theory for diffeological spaces developed.

This study introduces a unified cohomology theory for braided algebras.

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

The blow-down map is studied in Lie algebroid cohomology.

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…

New cohomological obstruction found for astheno-Kahler metrics.

Researchers redefine $\ell^\infty$-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.

Inequalities for symplectic cohomology groups are derived.

Extends Adams' theorem to periodic cohomology.

Study shows Dolbeault cohomology is unchanged by complex Lie group actions.