arXiv research

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

Notes on harmonic maps between manifolds, existence and regularity covered.

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

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…

Paper extends Hodge correspondence to singular Kähler spaces.

Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.

We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…

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

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

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…

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

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abeli…

Study non-Abelian gauge theories using Poisson bracket structures.

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

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 …

In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of…

Holomorphic curves found in compact quotients of SL(2,C).

We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for the anomaly cancellation in supersymmetric sigma models, and in a geometric description of so-called non-geometric T-duals.

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…

This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape o…

This paper calculates the derivative of surface holonomy for non-abelian gerbes.

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…

Paper proves injectivity of non-abelian X-ray transform on certain spaces.

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…

A new diffeomorphism invariant of integral homology 3-spheres is defined using a non-abelian 'quaternionic' version of the Seiberg-Witten equations.

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …

New method for non-abelian parallel transport in higher gauge theories.

The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.

We study the twisted Alexander polynomial $Δ_{K,ρ}$ of a knot $K$ associated to a non-abelian representation $ρ$ of the knot group into $SL_2(\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every non-abelian representation, $Δ_{K,ρ}$ is monic and has degree $4g(K)-2$ where $g(K)$ is the genus of …

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

We introduce a new action $S_{standard}^{(ρ,h; Φ,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model --- based on maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ …

Abstract: Proves non-abelian group of equivariant concordance.

Develops a new non-abelian framework for Riemann surfaces and differential equations.

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

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

New proof of injectivity for broken non-abelian X-ray transform in Minkowski space.

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary map…

A new mathematical approach detects frequency-based alterations in brain networks.

The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we expl…

Study of harmonic maps with extreme Kerr-like singularities.

New algorithms learn multi-index models via harmonic analysis, achieving statistical and computational trade-offs.

Defines non-Abelian gerbes with connections for physical applications.

The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.