arXiv research

Extends Chern character to non-abelian cohomology, linking to physics.

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in …

We develop a Chern character map for twisted equivariant non-abelian cohomology.

Trivial Massey product in specific cohomology groups.

Study shows non-abelian free groups' 4th cohomology is non-zero.

Researchers compute the cohomology ring of a foliation defined by a group action.

Explains quantum cohomology of Grassmannians using tt* equations.

New theory connects non-abelian bundle gerbes to abelian ones.

Non-Abelian actions are resolved using equivariant K-theory and delocalized cohomology.

M5-branes' flux quantization linked to non-abelian cohomology.

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bij…

New method recovers differential cohomology from diffeological spaces.

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…

The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.

New cohomology theory for diffeological spaces developed.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

We announce a solution to several enumeration problems in topology of surfaces. This includes an enumeration of homotopy classes of sections of locally trivial fiber bundles over surfaces and a computation of non-abelian 1-cohomology of surfaces.

In this paper we explain how non-abelian Hodge theory allows one to compute the $L^2$ cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new algebraic description for the fiberwise $L^2$ cohomology of a tame harmonic bundle o…

For a knot K in $S^3$ and a regular representation $ρ$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we con…

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the…

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi id…

The paper explores relationships between quandle cohomology, extensions, and automorphisms.

Study group extensions and bundles on manifolds.

We give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry. We prove the coincidence with the existing notions by comparing our construction with non-Abelian cohomology.

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified …

This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined wi…

This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternber…

We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological…

Extends T-duality to more general fibrations, proving isomorphic cohomology and algebroids.

New method for flux quantization on phase space stacks.

We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequ…

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups poss…

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references ad…

We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…

Study rigid Lie affine foliations on compact manifolds.

We give a complete and explicit description of the kinematical data of higher gauge theory on principal 2-bundles with the string 2-group model of Schommer-Pries as structure 2-group. We start with a self-contained review of the weak 2-category Bibun of Lie groupoids, bibundles and bibundle morphisms. We then construct…

New bounds on Euler characteristics for certain manifolds with finite groups.

Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links

Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.

Every symplectic Lie algebra with degenerate (including non-abelian nilpotent symplectic Lie algebras) has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism …

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=Σis a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over Σand coadjoint orbits in the dual of a central extension of the Lie algebra …

We show that for acylindrically hyperbolic groups $Γ$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $ρ$ of $Γ$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Γ;ρ)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(Γ)$ with …

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We…

We apply gauge theory to study the space $F_k(M)$ of smooth codimension-$k$ framed foliations on a smooth manifold $M$. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains $F_k(M)$ as a subspace. The notion of holonomy is natura…

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

The paper constructs toric vector bundles using spectral networks and non-abelianization.

Analyzes D-branes in string theory backgrounds using advanced cohomology.

Proves Hodge structures on modular functors, providing formulas for Hodge numbers.