arXiv research

Numerical experiments support conjecture about opers and nonabelian Hodge.

The paper broadens a mathematical correspondence to include more balanced metrics.

We refine the Morgan's work on mixed Hodge structures on Sullivan's $1$--minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent variations of mixed Hodge structures over compact K"ahler manifolds.

Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.

Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…

The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singu…

Summary of main work 1999-2012

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotop…

Study on $GL(1|1)$ Higgs bundles over Riemann surfaces.

Extends Higgs fields theory to complex fiber bundles.

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

Paper generalizes Higgs bundle limits to parabolic setting.

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that…

Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.

We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kähler manifolds. We provide a si…

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-o…

New surface observables yield 2-knot invariants in nonabelian theories.

Sharp bounds found on nonabelian quotients of surface braid groups.

We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-f…

Higher gauge theory via differential nonabelian cohomology

The report presents the theory of harmonic maps from Kähler manifolds.

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details -- in particular the global moduli / instanton / magnetic charge structure -- have remained elusive. Here we deduce from…

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

Maps links in 3-manifolds to links in branched covers, relating quantum field theories.

Abelian gerbes and twisted bundles describe the topology of the NS-NS 3-form gauge field strength H. We review how they have been usefully applied to study and resolve global anomalies in open string theory. Abelian 2-gerbes and twisted nonabelian gerbes describe the topology of the 4-form field strength G of M-theory.…

This article studies the nonabelian localization results of Beasley and Witten, and considers the analogue of these results when the gauge group is U(1). It compares these results with results of Manoliu on abelian Chern-Simons theory, showing that the dependence on the coupling constant is the same.

The paper establishes a correspondence between Higgs torsors and connections on curves.

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structure…

We prove that the first order theory of nonabelian free groups eliminates the "there exists infinitely many" quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property. We also extend our results to torsion-free hyperbolic groups under some conditi…

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many…

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the con…

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than…

New space of currents defined for nonabelian free groups and malnormal subgroups.

We construct $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite subgroup of $\mathrm{SO}(4)$ and weakly equivariant $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite solvable group. In particular, this is the first example of $G$-corks for an infinite nonabelian group $G$ and answers a question…

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

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.

We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .

Study on Hodge theory for almost complex manifolds.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…

The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from …

Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.

Geometric model for Hodge filtered complex cobordism constructed.

Introduces a new Hodge theory using vector fields on manifolds.