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Mark Haskins: Lectures

September 11, 2018
TITLE: Complete noncompact metrics of special and exceptional holonomy: the first 40 years

ABSTRACT: I will attempt to give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases G_2 and Spin_7 holonomy, and why the latter are much more difficult to construct.

In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G_2. Very recently mathematicians have constructed many, but by no means all, of these physically predicted G_2 metrics and also other G_2 metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry. If (by some miracle) time permits, I will describe some of the future prospects for the field.

April 12, 2018
TITLE: Uncollapsing highly collapsed G_2 holonomy metrics

ABSTRACT: In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant G_2 holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The asymptotic models for the geometry of these G_2 metrics are circle bundles with fibres of constant length l, so-called asymptotically local conical (ALC) geometry. These ALC G_2 metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length l goes to 0. A natural question is: what happens to these families of G_2 metrics as we try to make l large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In particular we suggest that a particular G_2 metric we construct should be regarded as a G_2 analogue of the Euclidean Taub—NUT metric on the complex plane.

Slides of lecture

September 15, 2017
TITLE: Codimension one collapse and special holonomy metrics

ABSTRACT:In this talk we describe recent developments and ongoing projects by members of the Collaboration related to codimension one collapse of exceptional holonomy metrics. Informally speaking, this is where a family of special holonomy metrics on a space of dimension n converges in some limit to a metric on a space of dimension n-1. Interesting examples occur for hyperkaehler 4-manifolds, G_2 holonomy manifolds and Spin_7 holonomy manifolds. The talk will focus on the G_2 holonomy case, but will also draw on the better understood hyperkaehler case for inspiration and for useful analogies.

These mathematical developments are closely related to important limits in physics, e.g. in the context of G_2 holonomy metrics it is related to the identification of the weak coupling limit of M theory compactified on a G_2 holonomy space being Type IIA String Theory on a 6-dimensional space. Inspiration for our work has already come from previous work of physicists studying M theory, including members of our Collaboration.

Slides of lecture

January 10, 2017 (jointly with Lorenzo Foscolo)
TITLE: Non-compact G2 manifolds collapsing to asymptotically conical Calabi-Yau 3-folds

ABSTRACT:We will present a new analytic construction of complete non-compact G_2 holonomy metrics, that yields infinitely many families of examples. The underlying 7-manifolds are all circle bundles over asymptotically conical (AC) Calabi-Yau 3-folds endowed with circle-invariant metrics. Their geometry at infinity is that of a circle bundle over a Calabi-Yau cone with circle fibres of fixed finite length. The G_2 manifolds we construct are therefore 7-dimensional analogues of 4-dimensional ALF hyperkähler metrics. Physicists have termed metrics with such asymptotics ALC (asymptotically locally conical). Within the Melrose school, ALC metrics form a special subclass of so-called fibred boundary metrics.

The dimensional reduction of the equations for G_2 holonomy in the presence of a Killing field was considered by Apostolov-Salamon and by several groups of physicists. We reinterpret the dimensionally-reduced equations in terms of a pair consisting of an SU(3) structure (with tightly constrained torsion) on the 6-dimensional orbit space coupled to an abelian Calabi-Yau monopole on this 6-manifold. We solve this nonlinear coupled system of PDEs by considering the geometric limit in which the circle fibres of the associated circle-invariant G_2 holonomy metrics collapse. In this collapsed limit, given the asymptotically conical Calabi-Yau 3-fold to which the 7-dimensional metrics collapse, the problem comes close to linearising and the crux of the matter is therefore to develop a sufficiently good understanding of solutions to these linearised equations in suitable spaces of functions. For this we need the well-known Fredholm theory for elliptic operators acting on weighted Sobolev spaces on asymptotically conical Calabi-Yau 3-folds and an extension of this theory to the setting of weighted Sobolev spaces on ALC manifolds.

The G_2 holonomy metrics we construct should be thought of as arising from (a suitable perturbation of) abelian Hermitian-Yang-Mills connections on AC Calabi-Yau 3-folds, especially those that arise as crepant resolutions of Calabi-Yau cones. From the physics viewpoint our solutions correspond to M theory uplifts of type IIA solutions in the absence of D6-branes. The existence of such a rich spectrum of “no brane” solutions does not seem to have been anticipated on the physics side, and is a new feature of G_2 holonomy metrics compared to the 4-dimensional hyperkähler setting. In particular all our examples provide instances of families of G_2 holonomy metrics that collapse with bounded curvature to Calabi-Yau 3-folds.

Time permitting we will outline extensions of the construction described here and how it fits into a broader programme.

The talk is closely related to the talks by Carron, Collins and Acharya and is based on joint work with Mark Haskins and Johannes Nordström.


September 9, 2016
TITLE: Collapse and special holonomy metrics

ABSTRACT:  The Gromov-Hausdorff topology provides a natural way to compactify the set of metrics with lower bounds on their Ricci curvature, in particular for Ricci-flat metrics (and therefore for special or exceptional holonomy metrics). Such Ricci-limit spaces need not be manifolds and a fundamental question that has attracted much attention is what one can say about the geometric structure of such limit spaces. In the non-collapsed case, i.e. when the limit space does not drop dimension, the theory is now very well developed, as will be described in Jeff Cheeger’s talk. By comparison the structure of collapsed Ricci-limit spaces is much less well understood.  In the Ricci-flat setting at least, relatively few families of collasping Ricci-flat metrics have been constructed.

In this talk we concentrate on the latter aspect and describe recent progress, work still in progress and future work on the construction of families of special holonomy metrics that collapse to a limit space of one dimension less. We focus on two cases: (i) the collapsing families of metrics on the K3 surface recently constructed by Foscolo and (ii) collapsing families of metrics on G_2 holonomy spaces (work in progress joint with Foscolo and Nordström). Understanding such 1-dimensional collapse in the context of G_2 holonomy metrics is intimately related to a rigorous mathematical understanding of an important phenomenon in physics: the weak coupling limit of M-theory compactified on a G_2 holonomy space being Type IIA String Theory on a 6-dimensional space. At various points inspiration for our work has already come from previous work of physicists, including members of our Collaboration.

The K3 case serves as a good warmup for the much more involved G_2 holonomy case and I will give an overview of Foscolo’s work in that setting. In the G_2 case I will outline what we have already achieved, what remains to be done and some prospects for the future. My talk is also closely related to the talks of Johannes Nordström and Song Sun.

Slides of lecture