2021 Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Annual Meeting

Mark Haskins
Duke University

Solitons in Bryant’s Laplacian flow
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I will describe a geometric flow on 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group \(G_2\).

My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe a recent construction of non-compact shrinking, steady and expanding solitons in Laplacian flow all with asymptotically conical geometry. I will also describe an explicit complete steady soliton with one end, exponential volume growth and asymptotically constant negative scalar curvature.
 

Bobby Acharya
King’s College Londom & ICTP

The Physics of Some G₂ and SU(3) Holonomy Singularities
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After a brief survey of what is known about the physical interpretation of various kinds of singularities in special holonomy spaces, we describe some recent progress in understanding the physics of some classes of non-Abelian orbifold singularities and asymptotically conical \(G_2\) and \(SU(3)\) holonomy spaces. We expect that some of the \(G_2\) examples can be desingularised to give rise to new, complete \(G_2\)-holonomy manifolds.
 

Thomas Walpuski
Humboldt University of Berlin

The Gopakumar–Vafa finiteness conjecture
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In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

This talk is based on joint work with Eleny Ionel and Aleksander Doan.
 

Ruobing Zhang
Princeton University

Collapsing geometry of hyperkaehler manifolds in dimension four
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We will overview some recent developments on the degeneration theory of hyperkaehler 4-manifolds. As preliminaries, we will present some tools and conceptual ideas in understanding metric geometric aspect of degenerating hyperkaehler metrics. After explaining the background, this talk will focus on the following classification results in the volume collapsed setting.

First, we will classify the collapsed limits of the bounded-diameter hyperkaehler metrics on the K3 manifold. More generally, we will precisely characterize the limiting singularities of the hyperkaehler metrics with bounded quadratic curvature integral. We will also exhibit the classification of the complete non-compact hyperkaehler 4-manifolds with quadratically integrable curvature which are called gravitational instantons and appear as bubbling limits of degenerating hyperkaehler metrics.

The above ingredients constitute a relatively complete picture of the collapsing geometry of hyperkaehler metrics in dimension 4.
 

Mohammed Abouzaid
Columbia University

Lagrangian fibrations from the perspective of Floer theory
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While the construction of special Lagrangian fibrations remains largely open in higher dimensions, we have achieved significant progress in the last decade in constructing singular Lagrangian fibrations on large classes of symplectic manifolds, and understanding the associated Floer theoretic invariants. I discuss these developments and their mirror symmetry motivations, and touch upon questions related to the existence of special Lagrangians.
 

Mirjam Cvetič
University of Pennsylvania

Zero modes of Higgs bundles on local Spin(7) manifolds
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We employ Higgs bundles to study the 3D N=1 field theories obtained from M-theory compactified on a local Spin(7) space given as a four-manifold M4 of ADE singularities with further generic enhancements in the singularity type along one-dimensional subspaces. We emphasize topologically robust quantities such as “parity” anomalies of the 3D field theory which descend from topological data of geometric reflections of the compactification. Furthermore, we provide some explicit constructions of well-known 3D theories, including those which arise as edge modes of 4D topological insulators. The analysis also allows us to track the spectrum of extended objects and their transformations under higher-form symmetries.

In other better-understood geometric flows (e.g. Ricci flow and mean curvature flow) solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows. Time permitting I will make some comparisons with known solitons in Ricci flow and Lagrangian mean curvature flow. This is joint work with Johannes Nordström and also in part with Rowan Juneman (both at Bath).
 

Jason Lotay
Oxford University

Some remarks on contact Calabi-Yau 7-manifolds
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In geometry and physics it has proved useful to relate \(G_2\) and Calabi–Yau geometry via circle bundles. Contact Calabi–Yau 7-manifolds are, in the simplest cases, non-trivial circle bundles over Calabi–Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free \(G_2\)-structures. They also give useful backgrounds to examine the heterotic \(G_2\) system (also known as the \(G_2\)-Hull–Strominger system), which is a coupled set of equations arising from physics that involves the \(G_2\)-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi–Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.
 

David Morrison
UC-Santa Barbara

Non-Perturbative heterotic duals of M-Theory on G₂ orbifolds
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Joyce’s famous construction of a compact manifold with holonomy \(G_2\) started from an orbifold of a seven-torus; the hard part was to show that the singular points could be resolved while retaining the special holonomy. Ironically, for applications in physics one wants spaces of holonomy \(G_2\) with precisely this kind of orbifold singularity! In this talk, based on joint work with Bobby Acharya and Alex Kinsella, we examine the torus orbifolds underlying Joyce’s construction, and compare them with a “dual’’ formulation of the same physics, using the heterotic string compactified on an appropriate Calabi—Yau threefold (equipped with a bundle). Contrary to our expectations when embarking on this project, the connection between the dual models is quite subtle, involving non-perturbative effects on the heterotic side. We shall explain this in detail.