Frederik Witt

Generalised G2-structures and type IIB superstrings

The recent mathematical literature introduces generalised geometries which are defined by a reduction from the structure group SO(d,d) of the vector bundle Td⊕Td* to a special subgroup. In this article we show that compactification of IIB superstring vacua on 7-manifolds with two covariantly constant spinors leads to a generalised G2-structure associated with a reduction from SO(7,7) to G2 × G2. We also consider compactifications on 6-manifolds where analogously we obtain a generalised SU(3)-structure associated with SU(3) × SU(3), and show how these relate to generalised G2-structures.

Calibrations and T–Duality

For a subclass of Hitchin’s generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold. Under a suitable integrability condition on the ambient geometry, these generalised calibrated submanifolds minimise a functional occurring as D–brane energy in type II string theories. Further, we investigate the behaviour of calibrated cycles under T–duality and construct non–trivial examples.

Generalised

We define new Riemannian structures on 7–manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type II supergravity theory with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skew–symmetric torsion. Finally, we construct explicit examples by introducing the device of T–duality.

G_2

We associate to each stable Higgs pair (A(0), Phi(0)) on a compact Riemann surface X a singular limiting configuration (A(infinity), Phi(infinity)), assuming that det Phi has only simple zeroes. We then prove. a desingularization theorem by constructing a family of solutions (A(t), t Phi(t) to Hitchins equations, which converge to this limiting configuration as t -> infinity. This provides a new proof via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

-manifolds

Abstract Let M be a topological G 2 -manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. For a connected boundary ∂Y of genus g, the index is given by ∫ ∂ Y c 1 ( ν X ) + 1 − g , where ν X denotes the orthogonal complement of T ∂ Y in T X | ∂ Y and c 1 ( ν X ) the first Chern class of ν X with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.

Ends of the moduli space of Higgs bundles

This is a companion paper to arXiv:1207.3529 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three. In this article we focus on the surface case. A salient feature here is the scale invariance of the functional which leads to a plenitude of critical points. Moreover, via the spinorial Weierstra{\ss} representation it relates to the Willmore energy of periodic immersions of surfaces into

Deformations of associative submanifolds with boundary

\mathbb{R}^3

A spinorial energy functional: critical points and gradient flow

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Metric bundles of split signature and type II supergravity

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2+1, in which there is a rich interplay with symplectic geometry.This is a survey about conformal mappings between pseudo-Riemannian manifolds and, in particular, conformal vector fields defined on such. Mathematics Subject Classification (2000). Primary 53C50; Secondary 53A30; 83C20.We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle

Limiting configurations for solutions of Hitchin’s equation

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