2021 Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Annual Meeting
Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Home Page
Conference Organizer: Robert Bryant
The study of the geometry of higher dimensional spaces, important in many applications, both theoretical and applied, has led to an understanding of their properties in terms of *holonomy*, a way of describing global effects of curvature. Those geometric spaces with ‘special’ (i.e., reduced) holonomy have come to play a fundamental role in partial differential equations, algebraic geometry, calculus of variations, topology, and theoretical physics, often revealing connections between these subjects that are yielding new insights in both mathematics and physics.
The 2021 annual meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics will highlight and explain the progress in the theory of spaces with special holonomy that has been made in recent years by the Collaboration, focussing on the construction of new examples; the increasing role of limiting constructions, particularly solitons and limits of Ricciflat spaces; moduli spaces and BPS invariants; and connections with physics. We will also describe the goals of our continuing research program as well as the challenges that lie ahead.

Agenda
Thursday, September 9
9:30 AM Mark Haskins  Solitons in Bryant’s Laplacian flow 11:00 AM Bobby Acharya  The Physics of Some G_{2} and SU(3) Holonomy Singularities 1:00 PM Thomas Walpuski  The Gopakumar–Vafa finiteness conjecture 2:30 PM Ruobing Zhang  Collapsing geometry of hyperkaehler manifolds in dimension four 4:00 PM Mirjam Cvetič  Zero modes of Higgs bundles on local Spin(7) manifolds Friday, September 10
9:30 AM Mohammed Abouzaid  Lagrangian fibrations from the perspective of Floer theory 11:00 AM Jason Lotay  Some remarks on contact CalabiYau 7manifolds 1:00 PM David Morrison  NonPerturbative heterotic duals of MTheory on G_{2} orbifolds 
Abstracts & Slides
Mark Haskins
Duke UniversitySolitons in Bryant’s Laplacian flow
Download Slides (PDF)I will describe a geometric flow on 3forms, 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 noncompact 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 & ICTPThe Physics of Some G_{2} and SU(3) Holonomy Singularities
Download Slides (PDF)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 nonAbelian 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 BerlinThe Gopakumar–Vafa finiteness conjecture
Download Slides (PDF)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 sidestep questions regarding multiple covers and superrigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudoholomorphic 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 pseudoholomorphic 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 UniversityCollapsing geometry of hyperkaehler manifolds in dimension four
Download Slides (PDF)We will overview some recent developments on the degeneration theory of hyperkaehler 4manifolds. 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 boundeddiameter 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 noncompact hyperkaehler 4manifolds 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 UniversityLagrangian fibrations from the perspective of Floer theory
Download Slides (PDF)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 PennsylvaniaZero modes of Higgs bundles on local Spin(7) manifolds
Download Slides (PDF)We employ Higgs bundles to study the 3D N=1 field theories obtained from Mtheory compactified on a local Spin(7) space given as a fourmanifold M4 of ADE singularities with further generic enhancements in the singularity type along onedimensional 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 wellknown 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 higherform symmetries.
In other betterunderstood 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 longtime 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 UniversitySome remarks on contact CalabiYau 7manifolds
Download Slides (PDF)In geometry and physics it has proved useful to relate \(G_2\) and Calabi–Yau geometry via circle bundles. Contact Calabi–Yau 7manifolds are, in the simplest cases, nontrivial circle bundles over Calabi–Yau 3orbifolds. These 7manifolds provide testing grounds for the study of geometric flows which seek to find torsionfree \(G_2\)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 7manifold. I will report on recent progress on both of these directions in the study of contact Calabi–Yau 7manifolds, which is joint work with H. Sá Earp and J. Saavedra.
David Morrison
UCSanta BarbaraNonPerturbative heterotic duals of MTheory on G_{2} orbifolds
Download Slides (PDF)Joyce’s famous construction of a compact manifold with holonomy \(G_2\) started from an orbifold of a seventorus; 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 nonperturbative effects on the heterotic side. We shall explain this in detail.