arXiv research

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.

Review of non-abelian gerbes and their applications in string theory.

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.

The paper explores conditions for non-abelian tensor products to be linear.

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.

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…

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

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

Refines mixed Hodge structures using non-abelian Hodge theory.

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.

Survey on Higgs bundles via harmonic maps and their role in non-abelian Hodge correspondence.

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

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

Abstract: Proves non-abelian group of equivariant concordance.

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

The paper connects non-abelian zeta functions, Fokker-Planck equations, and projective flat connections.

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

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…

Defines non-Abelian gerbes with connections for physical applications.

Ancient solutions on bundles are found for non-abelian groups.

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

Researchers find non-abelian symmetric gravitating vortices on a sphere.

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.

We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

Study describes moduli spaces of flat bundles on Sasakian manifolds.

Quantum trace maps abelian character varieties to SL2 character varieties.

This paper is a rigorous study of two dual pairs of momentum maps arising in the context of fluid equations whose configuration Lie group is the group of automorphism of a trivial principal bundle, generically called here non-abelian fluids. It is shown that the actions involved are mutually completely orthogonal, whic…

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…

We present a new link invariant which depends on a representation of the link group in SO(3). The computer calculations indicate that an abelian version of this invariant is expressed in terms of the Alexander polynomial of the link. On the other hand, if we use non abelian representation, we get the squared non abelia…

Stability of non-abelian X-ray transform proven in higher dimensions.