2023 Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Annual Meeting

YEAR EIGHT (NCE) PRELIMINARY REPORT ON THE SIMONS COLLABORATION “SPECIAL HOLONOMY IN GEOMETRY, ANALYSIS, AND PHYSICS” ANNUAL MEETING AND FOLLOWING WORKSHOP (SEPTEMBER 7–13, 2023)

ROBERT L. BRYANT, DIRECTOR 

1. Introduction 

This is a brief update on the activities of our Simons Collaboration that have taken place since our seventh Annual Report was filed in June 2023. There have been two events, mainly in-person with some remote participants and speakers: An annual meeting at the Simons Foundation on September 7–8 and the following Year 8 (NCE) workshop of the Collaboration held at the Simons Center for Geometry and Physics in Stony Brook, NC during September 10–14. 

During the day-and-a-half annual meeting, we presented a series of eight talks, by Collaboration PIs, Postdocs, and one Invited Guest, about the important developments in the Collaboration’s research program in the past year. The format of the week-long workshop at the SCGP was our now-traditional “Progress and Open Problems”. See below for an expanded description. 

2. Seventh Annual Meeting 

This meeting was held September 7–8, 2023 at the Simons Foundation in NY, NY. 

• Organizer: Robert Bryant (with assistance from the other PIs of the Collaboration) 
• Themes: Singularity structure of minimizing cycles (in particular, calibrated cycles), categorial symmetries, Laplacian flow, homotopy associative cycles, calibrated fibrations, Calabi–Yau metrics, Lagrangian mean curvature flow. 
• Meeting Page for the 2023 Annual Meeting
• Speakers (the presentation slides have been submitted to the Foundation). 
  • Bobby Acharya, M-theory, particle physics and special singularities of special holonomy spaces: Past, Present and Future 
  • Camillo De Lellis, Area-minimizing integral currents: singularities and structure 
  • Simon Donaldson, Curves in K3 surfaces and fibered calibrated submanifolds (remote)
  • Sebastian Goette, Homotopy Associative Submanifolds in G₂-manifolds 
  • Mark Haskins, Recent developments in Laplacian flow 
  • Yang Li, Calabi–Yau metrics in the intermediate complex structure limit 
  • Jason Lotay, Joyce conjectures for the Lagrangian mean curvature flow of surfaces 
  • Sakura Schafer-Nameki, Categorical Symmetries and Geometric Engineering 
• Participants: There were 68 participants in person and 21 remote participants. (See Table 1 below.)

Table 1. Participants in Seventh Annual Meeting, September 7–8, 2023

Mohammed Abouzaid
Bobby Acharya
Izar Alonso
Daniel Baldwin
Gavin Ball
Gorapada Bera
Olivier Biquard
Robert Bryant
Nico Cavalleri
Gao Chen∗
Shih-Kai Chiu
Charles Cifarelli
Tristan Collins∗
Anuk Dayaprema
Camillo De Lellis
Simon Donaldson∗
Michael Douglas∗
Shubham Dwivedi
Gilles Englebert
Joel Fine
Anna Fino∗
Lorenzo Foscolo∗
Udhav Fowdar∗
Guido Franchetti∗
Mateo Galdeano
Dylan Galt
Sebastian Goette
Julius Grimminger
Dominik Gutwein
Mark Haskins
Andriy Haydys
Siqi He
Thorsten Hertl
Alfred Holmes
Max H ̈ubner
Thomas Jiang
Spiro Karigiannis∗
Aaron Kennon
Ilyas Khan
Alexei Kovalev∗
Thibault Langlais
Claude LeBrun
Fabian Lehmann
Jonas Lenthe
Yang Li
Mingyang Li mingyang
Jin Li
Hongyi Liu
Zhenhua Liu∗
Jason Lotay
Langte Ma∗
Jesse Madnick
Viktor Majewski
Agnese Mantione
Andres Moreno∗
Nikita Nekrasov∗
Tien Ngyuen
Johannes Nordström
Tristan Ozuch
Greg Parker
Alec Payne
Daniel Platt
Jacek Rzemieniecki
Henrique Sa Earp∗
Evyatar Sabag
Simon Salamon
Andrea Sangiovanni
Justin Sawon
Chris Scaduto∗
Sakura Schafer-Nameki
Enric Solė-Farrė
Jakob Stein
Song Sun
Eirik Svanes
Jiahua Tian
Ethan Torres
Valentino Tosatti
Federico Trinca
Thomas Walpuski
Mu-Tao Wang
Xiaowei Wang
Yuanqi Wang∗
Albert Wood
Jingxiang Wu
Dashen Yan
Ruobing Zhang
Junsheng Zhang
Wolfgang Ziller

∗ denotes remote attendee

3. Initial Workshop of Year 8 (NCE)

September 10–13, 2023 (Simons Center for Geometry and Physics)

This 3.5-day workshop was organized as a collection of 17 presentations on recent progress and challenges that the Collaboration would like to address. All the talks were recorded and are posted on the Collaboration website (see below). Most of the lectures were by present or former postdocs and graduate students of the Collaboration, but we did have a few speakers who were invited guests, such as Olivier Biquard, whose talk doubled as the Stony Brook geometry seminar for that week. The emphasis was on work by junior members of the Collaboration, since we wanted to highlight new directions that research in special holonomy is taking.

• Title: Progress and Open Problems
• Organizer: Mark Haskins and Simon Salamon
• Themes: Singularities of special holonomy manifolds and their resolutions, connections with physics (both applications of special holonomy to questions in physics and questions about special holonomy that physical theories suggest), holomorphic fibrations, mean curvature flow, asymptotically conical special holonomy manifolds, closed G₂-structures, new constructions from singularity resolutions, connections with knot and link invariants, instantons on various background metrics of interest.
• Speakers:
  • Mina Aganagic (UC Berkeley), Homological link invariants from Floer theory
  • Daniel Baldwin (King’s College London), Coulomb and Higgs phases of G₂ manifolds
  • Olivier Biquard (Sorbonne), Limits of K¨ahler-Einstein metrics with cone singularities, and Calabi-Yau metrics
  • Charles Cifarelli (Nantes), Steady gradient Kähler-Ricci solitons and Calabi–Yau metrics on ℂⁿ
  • Joel Fine (Universit´e Libre de Bruxelles), Knot invariants from hyperbolic SU(3) and G₂ geometries
  • Julius Grimminger (Oxford), Stratified hyper-K¨ahler moduli spaces and physics
  • Jonas Lente (Freiburg), Modular Mathai–Quillen currents
  • Jin Li (Freiburg), On the geometry of resolutions of G₂-manifolds with isolated conical singularities
  • Mingyang Li (UC Berkeley), Classification results for Hermitian non-K¨ahler gravitational instantons
  • Langte Ma (Shanghai Jiao Tong), Instantons on Joyce’s G₂-manifolds
  • Alec Payne (Duke), Closed G₂-Structures with Negatively Pinched Ricci Curvature
  • Daniel Platt (Imperial College London), Approximations of harmonic 1-forms on real loci of Calabi–Yau 3-folds
  • Evyatar Sabag (Oxford), G₂ Manifolds from 4d N = 1 Theories
  • Justin Sawon (UNC Chapel Hill), Lagrangian fibrations in four and six dimensions
  • Valentino Tosatti (NYU), Holomorphic Lagrangian fibrations and special K¨ahler geometry
  • Lu Wang (Yale), A mean curvature flow approach to density of minimal cones
  • Junsheng Zhang (UC Berkeley), On complete Calabi–Yau manifolds asymptotic to cones
• Meeting Page for September 2023 Workshop at SCGP2. The videos of the lectures are posted on the website and linked to this page.
• Participants: There were 59 participants in the meeting, 55 in-person and 4 remote. See Table 2 for the list. 4. Further Workshops of Year 8 (NCE)

The PIs of the Collaboration met during the SCGP conference (some attended by Zoom) to continue planning the remaining workshops of Year 8 (NCE). We plan to have two more workshops for the entire collaboration in Year 8 (NCE).

• IMPA, Rio de Janeiro, Brazil, March 11–15, 2024
• Durham, NC, May 13–17, 2024

The workshop at IMPA (Instituto de Matemática Pura e Aplicada in Rio de Janeiro, Brazil) will be a joint conference with IMPA entitled Special Holonomy and Complex Geometry. The workshop in Durham, NC will be the final official workshop of the Collaboration, and it is expected to be held at the 21c Museum Hotel in downtown Durham, NC, which was the site of an previous successful Collaboration conference.

Table 2. Participants in SCGP Workshop, September 10–13, 2023

Bobby Acharya
Izar Alonso Lorenzo
Daniel Baldwin
Gavin Ball
Gorapada Bera
Olivier Biquard
Robert Bryant
Nicolo Cavalleri
Shih-Kai Chiu
Charles Cifarelli
Mirjam Cvetic
Simon Donaldson∗
Gilles Englebert
Joel Fine
Mateo Galdeano
Dylan Galt
Sebastian Goette
Julius Grimminger
Dominik Gutwein
Mark Haskins
Andriy Haydys
Thorsten Hertl
Alfred Holmes
Max Hubner
Thomas Jiang
Dominic Joyce∗
Aaron Kennon
Ilyas Khan
Thibault Langlais
Fabian Lehmann
Jonas Lenthe
Jin Li
Mingyang Li mingyang
Yang Li
Hongyi Liu
Zhenhua Liu∗
Langte Ma∗
Jesse Madnick
Viktor Majewski
Agnese Mantione
Tien Nguyen
Johannes Nordström
Alec Payne
Daniel Platt
Evyatar Sabag
Simon Salamon
Andrea Sangiovanni
Justin Sawon
Enric Sole-Farrė
Jakob Stein
Jiahua Tian
Valentino Tosatti
Federico Trinca
Thomas Walpuski
Lu Wang
Albert Wood
Jingxiang Wu
Dashen Yan
Junsheng Zhang

∗ denotes remote attendee

Bobby Acharya
M-theory, particle physics and special singularities of special holonomy spaces: Past, Present and Future

Ricci flat spaces of special holonomy with special kinds of singularities provide models for the extra dimensions of physical space predicted by superstring/M-theory. After reviewing the intimate relationship between the singularities of these spaces and particle physics in four dimensions I will review some of the physics progress made by this Simons collaboration. Then I will review recent work (together with Daniel Baldwin) interpreting (a generalized version of) the Joyce-Karigiannis constructions of G2-manifolds in terms of Higgs and Coulomb phases of four dimensional gauge theories.

I will conclude with some perspectives on the future and some of the important open problems.
 

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

Let T be an area-minimizing integral current of dimension m in a smooth closed Riemannian manifold of dimension m + n. It is known since the work of De Giorgi, Fleming, Almgren, Simons, and Federer in the sixties and seventies that, when n = 1, the (interior) singular set of T has dimension at most m − 7. In higher codimension Almgren’s big regularity paper proved in 1980 that the singular set has dimension at most m − 2, laying the grounds for a theory which has been simplified and extended in the last 15 years. Both theorems are optimal, but at the qualitative level there is a quite important mismatch between the singular sets of the known examples and a general closed set of the same dimension. In a celebrated work in the nineties Simon proved, for n = 1, that the singular set is m − 7- rectifiable and that the tangent cone is unique Hm−7-a.e.. The counterpart of Simon’s theorem in higher codimension has been reached very recently by Paul Minter, Anna Skorobogatova and myself and, independently, by Krummel and Wickramasekera. Even though it would be natural to expect much stronger structural results, our theorem is indeed close to optimal, as a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension α ≤ m − 2.
 

Simon Donaldson
Curves in K3 surfaces and fibred calibrated submanifolds

We will discuss descriptions of calibrated submanifolds near the “adiabatic limit”. The situation we consider is where most of the ambient manifold is fibred by hyperkahler 4-manifolds and the calibrated submanifolds in question are modelled on fibrations with holomorphic curves as fibers. We will review first the case when the holomorphic curves are 2-spheres, which has been the scene of work by a number of authors (and the subject of previous talks in this Collaboration). We will then go on to consider the case of higher genus, which involves the geometry of moduli spaces of curves in K3 surfaces.
 

Sebastian Goette
Homotopy Associative Submanifolds in G₂-manifolds

Associative submanifolds are certain calibrated submanifolds in G₂-manifolds. There is the hope that counting them will reveal subtle information about the underlying G₂-structure. On the other hand, certain singular associatives can be resolved in exactly three different ways, so a naive count will be meaningless. In this talk, we will define homotopy associatives as cobordism classes of three-dimensional submanifolds that are adapted to the G₂-structure in a rather weak sense. We will see that a given cobordism class can be interpreted as a homotopy associative in exactly three different ways. This might help us to define a consistent counting scheme even when the naive number of associatives in a given cobordism class changes due to singularities.
 

Mark Haskins
Recent developments in Laplacian flow

Bryant’s Laplacian flow is a geometric flow of closed positive 3-forms on a 7-manifold that aims to give a parabolic PDE approach to constructing Riemannian manifolds with holonomy $G_2$. My talk will discuss some recent developments and open questions in this area. Along the way I will try to highlight ways in which Laplacian flow has both some similar and some distinctive features compared to more well-known and better-understood geometric flows, like Ricci flow, mean curvature flow and Lagrangian mean curvature flow.
 

Yang Li
Calabi-Yau metrics in the intermediate complex structure limit

Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture. We will discuss how to describe the Kahler potential at the C^0 level in the intermediate case for a large class of complete intersection examples.
 

Jason Lotay
Joyce conjectures for the Lagrangian mean curvature flow of surfaces

Building on the seminal work of Thomas-Yau, Joyce formulated an inspirational collection of conjectures concerning the Lagrangian mean curvature flow in Calabi-Yau manifolds which, in particular, relate the flow to notions of stability and the Fukaya category. I will present an overview of recent progress towards these conjectures in the case of Lagrangian surfaces. This is based on joint works with F. Schulze and G. Szekeleyhidi, and also with G. Oliveira.
 

Sakura Schafer-Nameki
Categorical Symmetries and Geometric Engineering

I will provide an overview of recent works on realizing generalized global symmetries (or categorical symmetries) in quantum field theories, that are constructed in string theory by geometric engineering.