# 2022 Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Annual Meeting

Date & Time

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 2022 annual meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics will highlight and explain the progress in our knowledge of spaces with special holonomy that has been made in recent years by the Collaboration, focusing on the construction of new examples and the definition of geometric invariants that distinguish such structures when they are defined on the same underlying manifold; the increasing role of solitons and limits of Ricci-flat spaces; moduli spaces and BPS invariants; and connections with physics, particularly the new areas of higher form symmetries. We will also describe goals of our continuing research program as well as the challenges that lie ahead.

• Agenda

#### Thursday

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Camillo De Lellis | Area-minimizing integral currents: singularities and structure 10:30 AM BREAK 11:00 AM Lorenzo Foscolo | Complete non-compact manifolds with holonomy G2 and ALC asymptotics 12:00 PM LUNCH 1:00 PM Dominic Joyce | Counting semistable coherent sheaves on surfaces 2:00 PM BREAK 2:30 PM Fabian Lehmann | An embedding problem for closed 3-forms on 5-manifolds 3:30 PM BREAK 4:00 PM Johannes Nordström | Invariants of twisted connected sum G_2-manifolds 5:00 PM DAY ONE CONCLUDES

#### Friday

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Simon Salamon | Spinors and instantons 10:30 AM BREAK 11:00 AM Max Hübner | Higher Symmetries via Cutting and Gluing of Orbifolds 12:00 PM LUNCH 1:00 PM Song Sun | Gravitational instantons and del Pezzo surfaces 2:00 PM MEETING CONCLUDES
• Abstracts

Camillo De Lellis
Area-minimizing integral currents: singularities and structure

Almgren’s famous Big Regularity Paper proves that the interior singular set of any $m$-dimensional area-minimizing integral current $T$ in any smooth Riemannian manifold $M$ has (Hausdorff) dimension at most $m{-}2$.  Except for the case $m=2$, when it was proved that interior singularities are isolated, little is known about the structure of the singular set.  Moreover, a recent theorem by Liu proves that we cannot expect it to be a $C^1$ $(m{-}2)$-dimensional submanifold (unless the ambient $M$ is real-analytic) as in fact, it can be a fractal set of any Hausdorff dimension $\alpha\le m{-}2$.  On the other hand, it seems likely that it is an $(m{-}2)$-rectifiable set, i.e., that it can be covered by countably many $C^1$ submanifolds.

In this talk, I will explain why the problem is challenging and how it can be broken down into  easier pieces following a recent joint work with Anna Skorobogatova.

Lorenzo Foscolo
Complete non-compact manifolds with holonomy G2 and ALC asymptotics

G2 manifolds are the Ricci-flat 7-manifolds with holonomy G2. Until recently there was
only a handful of known examples of complete non-compact G2 manifolds, all highly symmetric and arising from explicit solutions to ODE systems. In joint work with Haskins and Nordström, we produced infinitely many G2 manifolds on total spaces of principal circle bundles over asymptotically conical Calabi-Yau manifolds. The asymptotic geometry of the G2 metrics we produced is analogous to the geometry of 4-dimensional ALF (asymptotically locally flat) spaces and has been labelled ALC (asymptotically locally conical) in the physics literature. In this talk, I will discuss some further joint work on this class of manifolds, in particular consequences of the good deformation theory of ALC G2 manifolds and the construction of new examples with a slightly more complicated ALC asymptotic geometry analogous to the well-known Atiyah-Hitchin metric in 4-dimensional hyperkähler geometry.

Max Hübner
Higher Symmetries via Cutting and Gluing of Orbifolds

We study the higher symmetry structures of 4d, 5d, 7d quantum field theories (QFTs) which are geometrically engineered in M-theory by non-compact geometries with non-compact ADE loci. We characterize flavor and gauge Wilson lines of such QFTs via orbifold homology groups of the asymptotic boundary geometry and argue that the corresponding 0-form, 1-form and 2-group symmetries are mapped onto the Mayer-Vietoris sequence with respect to a covering derived from the asymptotic orbifold locus and its complement. Further, applying related cutting and gluing constructions to various compact geometries with ADE loci we discuss the global structure of the engineered supergravity gauge group.

Dominic Joyce
Counting semistable coherent sheaves on surfaces

In arXiv:2111.04694 I set out a program which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers. I will outline the general programme briefly, and go into detail on my current project, which uses the results to compute invariants counting semistable vector bundles and coherent sheaves on complex projective surfaces.

Invariants of this kind have a long history, notably in Donaldson theory of 4-manifolds, Vafa-Witten invariants, and so on, and are the subject of many conjectures in mathematics and String Theory. It is common to combine the invariants in formal power series — “generating functions” — which may have interesting number-theoretic properties such as rationality or modularity. My results explain a lot about the general structure of such generating functions, resolving some conjectures in the literature, and I hope in future to make progress on deeper questions such as Vafa-Witten modularity as well.

Fabian Lehmann
An embedding problem for closed 3-forms on 5-manifolds

The real part of a holomorphic volume form restricts to a closed 3-form on a 5-dimensional submanifold of a complex 3-fold with a Calabi-Yau structure. Vice versa, in analogy with the embedding problem for abstract CR-structures, this leads to the question which closed 3-forms on a given 5-manifold can be realised by an embedding into a Calabi-Yau 3-fold. We describe the structure induced on the 5-manifold by a closed 3-form and introduce a convexity notion. The main result is that in the “strongly pseudoconvex” case the embedding problem can be solved perturbatively if a finite dimensional vector space of obstructions vanishes. The proof uses the theory of sub-elliptic operators and the Nash-Moser inverse function theorem. The main example is the standard embedding of S^5 in C^3, for which the obstruction space vanishes. This is joint work with Simon Donaldson.

Johannes Nordström
Invariants of twisted connected sum G_2-manifolds

When trying to apply the h-cobordism theorem to prove that two manifolds are diffeomorphic, obstructions can often be captured in terms of invariants that measure the failure of a manifold with boundary to satisfy relations that hold for manifolds without boundary. I will discuss invariants of this kind for 7-manifolds with G_2-structures or stable complex structures, and recent progress with Crowley and Goette on applying these to holonomy G_2-manifolds obtained by variations of the twisted connected sum construction.

Simon Salamon
Spinors and instantons

Motivated in part by the existence of certain low-rank vector bundles on projective spaces, the talk will focus on the interplay between SU(3) as a global symmetry group, tensors satisfying twistor equations, and reduced holonomy in eight dimensions. It will relate work of Gray–Green 1970, Horrocks 1978, and Atiyah–Witten 2002, on Spin(7) structures, instantons and circle actions, to that of the Collaboration. The approach is based on a joint paper with Udhav Fowdar.

Song Sun
Gravitational instantons and del Pezzo surfaces

I will talk about the classification of gravitational instantons of type ALH* in terms of weak del Pezzo surfaces. This is based on joint work with Hein, Viaclovsky, and Zhang, arXiv 2111.09287.

• Ground Transportation

For those who are attending the satellite meeting at the Simons Center for Geometry and Physics, a shuttle will be available to take participants from New York City to Stony Brook on Saturday, September 10th.

• Accommodations

Group A – PIs & Speakers
Business-class airfare for flights over 5 hours
Hotel accommodations for up to 3 nights
Reimbursement of Local Expenses

Group B – Out-of-town Participants
Economy Airfare
Hotel Accommodations for up to 3 nights
Reimbursement of Local Expenses

Group C – Local Participants
No funding provided besides hosted conference meals.

Group D – Remote Participants
A Zoom link will be provided.

• Personal Car

For participants in Groups A & B driving to Manhattan, the James NoMad hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage; therefore, participants are encouraged to walk or take public transportation to the Simons Foundation.

• Hotel

Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad hotel. Any additional nights are at the attendee’s own expense.

22 E 29th St (corner of Madison Avenue)
New York, NY 10016

• COVID-19 Policy

ALL in-person meeting attendees must be vaccinated against the COVID-19 virus with a World Health Organization approved vaccine, be beyond the 14-day inoculation period of their final dose, and provide proof of vaccination upon arrival to the conference. Acceptable vaccines can be found at the bottom of this page on the WHO’s site.

• Reimbursement and Travel Policy

Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the foundation’s travel policy. An email will be sent within a week following the conclusion of the meeting with further instructions on submitting your expenses via the foundation’s web-based expense reimbursement platform.
Receipts are required for any expenses over \$50 USD and are due within 30 days of the conclusion of the meeting. Should you have any questions, please contact Emily Klein.

• Contacts

Registration and Travel Assistance
Claudio Proserpio
Ovation Travel Group
sfntravel@ovationtravel.com
Phone: +1 646-998-9865
Mobile: +1 646-320-2634
www.ovationtravel.com

Meeting Questions and Assistance
Emily Klein
Event Coordinator, MPS, Simons Foundation
eklein@simonsfoundation.org
(646) 751-1262

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