Deformations of nearly Kähler instantons

We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly Kähler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3)

Deformation theory of nearly K\"ahler manifolds

Nearly K\"ahler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly K\"ahler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly K\"ahler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly K\"ahler structure on the flag manifold are all obstructed to second order

Deformations of Asymptotically Conical G2-Instantons

This thesis develops the deformation theory of instantons on asymptotically conical G2-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted Dirac operator on the G2-manifold and to the eigenvalues of a twisted Dirac operator on the nearly Kähler link. As an application, we use this framework to study the moduli spaces of known examples of G2-instantons living on the Bryant-Salamon manifolds and on R7. We develop two methods for determining eigenvalues of twisted Dirac operators on nearly Kähler 6- manifolds and apply this to calculate the virtual dimension of the moduli spaces that we study. In the case of the instanton of Günaydin-Nicolai, which lives on R7; we show how knowledge of the virtual dimension of the moduli space can be used to study uniqueness properties of this instanton

Gauge theory in dimension 77

We first review the notion of a $G_2$-manifold, defined in terms of a principal $G_2$ ("gauge") bundle over a $7$-dimensional manifold, before discussing their relation to supergravity. In a second thread, we focus on associative submanifolds and present their deformation theory. In particular, we elaborate on a deformation problem with coassociative boundary condition. Its space of infinitesimal deformations can be identified with the solution space of an elliptic equation whose index is given by a topological formula.Comment: 15 page

SU(3)-instantons and G2,Spin(7)G_2, Spin(7)-heterotic string solitons

Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimensions 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in $G_2$ arises from our considerations and Hitchin's flow equations, which seems to be new. Compact examples of $SU(3), G_2$ and $Spin(7)$ instanton satisfying the anomaly cancellation conditions are presented.Comment: LaTex, 22 pages, Corrected anomaly cancellation, final version to appear in Commun. Math. Phy

A method of deforming G-structures

We consider deformations of G-structures via the right action on the frame bundle in a base-point-dependent manner. We investigate which of these deformations again lead to G-structures and in which cases the original and the deformed G-structures define the same instantons. Further, we construct a bijection from connections compatible with the original G-structure to those compatible with the deformed G-structure and investigate the change of intrinsic torsion under the aforementioned deformations. Finally, we consider several examples.Comment: 14 pages; v3: references added, published in Journal of Geometry and Physic

Special submanifolds in nearly Kähler 6-manifolds

This thesis is on J-holomorphic curves and special Lagrangians of nearly Kähler manifolds, with a focus on nearly Kähler CP^3. We consider the following four problems. Firstly, we relate classical geometric properties of surfaces in four-manifolds to properties of their twistor lifts, based on the work of Eells, Salamon and Friedrich. This leads to the construction of deformation invariant quantities for J-holomorphic curves in certain twistor spaces, such as CP^3 or the manifold of complete flags in C^3. We give an example of how the twistor lift of the discriminant locus of a family of quadrics in CP^3 performs a desingularisation. Secondly, we introduce the class of transverse J-holomorphic curves in CP^3, for which we define angle functions. It turns out that the angle functions essentially encode the geometry of the curve, which results in classification results for J-holomorphic curves with special geometric properties. We derive a system of PDEs for the angle functions which enables us to establish a Bonnet-type theorem for transverse J-holomorphic curves. By constructing toric moment-type maps we relate them to the theory of U(1) invariant minimal surfaces in S^4. Thirdly, we consider the deformation problem for J-holomorphic curves in general nearly Kähler manifolds. We turn to infinitesimal deformations and show that they are eigensections of a twisted Dirac operator on the normal bundle of the curve. By solving this equation explicitly we show that homogeneous tori in CP^3 and S^6 are rigid and compute the spectrum of the Dirac operator in these cases. Lastly, we derive the structure equations for special Lagrangians in CP^3. This yields a classification of totally geodesic special Lagrangians. By introducing moment maps we also classify all SU(2) invariant special Lagrangians in CP^3 and provide new homogeneous examples

Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds

A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of the metric is below - 3 , then the G2 4-form is exact if and only if the manifold is Euclidean R7. A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below - 3 , then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R6