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Structure of Collapsed Special Holonomy Spaces, Duke University, 9-13 April 2018

Arrival date: Sunday, April 8, 2018. Departure date: Friday, April 13, 2018.

The conference venue is the 21C Hotel in downtown Durham, North Carolina. 


Tentative Schedule

MON 9 APR

TUE 10 APR

WED 11 APR

THU 12 APR

FRI 13 APR

8:00 -9:00

Breakfast Breakfast Breakfast Breakfast Breakfast

9:00-10:00

R. Zhang J. Cheeger X. Rong R. Zhang D. Morrison

10:00-10:15

Q+A Q+A Q+A Q+A Q+A

10:15-11:00

Coffee Coffee Coffee Coffee Coffee

11:00-12:00

Y. Zhang R. Mazzeo R. Mazzeo Y. Li Y. Li

12:00-2:00

Lunch/Discussion Lunch/Discussion Lunch/Discussion Lunch/Discussion Lunch/Discussion

2:00-3:00

G. Wei G. Chen L. Foscolo M. Haskins S. Salamon

3:00-3:30

Coffee Coffee Coffee Coffee Coffee

3:30-4:30

R. Zhang Y. Zhang J. Viaclovsky Discussion

7:00-9:00

Conference banquet

Speakers:

 

Sir Simon Donaldson to deliver the Marston Morse Lecture Series

Sir Simon Donaldson, one of our Collaboration’s Principal Investigators, will deliver the Marston Morse Lectures at the Institute for Advanced Study during the first week of April, 2018, under the general title “Exceptional holonomy and related geometric structures.” The detailed schedule of lectures is here.

Robert Bryant delivers the McKnight-Zame Distinguished Lecture

Collaboration Director Dr. Robert Bryant delivered the 2018 McKnight-Zame Distinguished Lecture at the University of Miami on March 1, 2018. The lecture, entitled “From Mechanics, to Algebra, to Geometry: The Notion of Holonomy,” introduced a general mathematical audience to the main theme of this Collaboration. Further details about the lecture series and Dr. Bryant’s lecture can be found here.

Xenia de la Ossa: Lectures

January 8, 2018
TITLE: The geometry and moduli of heterotic G2 structures

ABSTRACT: A heterotic G_2 system is a quadrupole [(Y, \varphi), (V,A), (TY, \theta), H], where Y is a seven dimensional manifold with an integrable G_2 structure and \vaprhi is the corresponding associative three form, V is a bundle on Y with an instanton connection A, and \theta is an instanton connection on the tangent bundle TY. H is a three form given in terms of the B field and the Chern-Simons forms of A and \theta (the anomaly cancelation condition) which is further constrained so that it is equal to a natural three form uniquely determined by the G_2 structure on Y. This constraint mixes up the geometry of Y with that of the bundles.

I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative \cal Y acting on forms with values on the bundle {\cal Q} = T^*Y\oplus {\rm End}(V) \oplus {\rm End}(TY) which satisfies \check{\cal D}^2 = 0, for some appropriately defined projection of the operator \cal D. Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancellation condition. We show that the infinitesimal moduli space is given by the cohomology group H^1_{\check{\cal D}} and that it is finite dimensional. Our analysis leads to results that are of relevance to all orders in \alpha'. Time permitting, I will comment on work in progress about the finite deformations of heterotic G_2 systems and the relation to differential graded Lie algebras.

From the physics perspective these structures give rise to very interesting three dimensional gauge supergravity theories (on Minkowski or AdS3) with only N=1 supersymmetry. Very little is known about these theories, as opposed to those with N=2 or N=4, however we seem to have just enough supersymmetry to be able to deduce some interesting features of the effective field theories.

G2 Working Group at the Aspen Center for Physics

The Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics sponsored a G2 working group at the Aspen Center for Physics from August 7 through August 25, 2017.  The working group was attended by Andreas Braun (Oxford), Michele Del Zotto (SCGP), Jim Halverson (Northeastern), Magdalena Larfors (Uppsala), Dave Morrison (UC Santa Barbara), and Sakura Schafer-Nameki (Oxford).

The topics discussed included:

  1. gauge theory on G2 manifolds and G2 compactifications of the heterotic string
  2. singular fibers of SYZ fibrations of Calabi-Yau threefolds and what they might correspond to in K3-fibered G2 manifolds
  3. the structure of the gauge degrees of freedom in G2 compactifications, including BPS equations and Higgs bundle as function of the associative three-manifold base
  4. G2 compactifications of M-theory possesing both F-theory and heterotic duals, in particular the various moduli of such models including the role of the D3-branes from the F-theory realization.

Research papers which result from this activity will be included in the publication list on this website.  Stay tuned!

First Annual Meeting: September 14-15, 2017, Simons Foundation, New York City

Arrival date: Wednesday, September 13. Departure date : Friday afternoon, September 15, or Saturday, September 16. This conference will run from Thursday morning through Friday lunch.

Spaces with special holonomy are of intrinsic interest in both mathematics and mathematical physics; they appear in many contexts in Riemannian geometry, particularly Ricci-flat and Einstein geometry, minimal submanifold theory and the theory of calibrations, and gauge theory. The exceptional cases, which occur in dimensions 7 and 8, remain the most challenging and the least understood. Nevertheless, they share important features with the better-known case of SU(n) holonomy, where the three types of structures are known as Calabi-Yau spaces, Hermitian Yang–Mills connections, and special Lagrangian and complex submanifolds. The exceptional holonomy spaces play key roles in the study of fundamental physical theories such as M-theory and F-theory (generalizing the role that Calabi-Yau 3-folds play in string theory), and progress in these theories depends crucially on a better understanding of spaces (especially singular ones) with exceptional holonomy.

The Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics will hold its first Annual Meeting at the Simons Foundation on September 14 & 15, 2017. Four of its principal investigators and four of its postdoctoral fellows will present reports on the most recent developments in various aspects of the field of special holonomy, including the study of adiabatic limits, moduli problems, collapse, gluing constructions using methods from algebraic geometry, and connections with physics. They will discuss their research progress during the first year of the collaboration and the current directions of research

The speakers are:

  • Andreas Braun (Oxford)
  • Sir Simon Donaldson (Imperial College and SCGP)
  • Andriy Haydys (Freiberg)
  • Mark Haskins (Bath)
  • Dominic Joyce (Oxford)
  • Eirik Svanes (King’s College London)
  • David R. Morrison (UC Santa Barbara)
  • Yuguang Zhang (Imperial College London)

Special Holonomy: Progress and Open Problems 2017: September 10-13, 2017, SCGP, Stony Brook

Arrival date: Saturday, September 9.
Departure date: Wednesday afternoon, September 13, or Thursday, September 14.

Schedule

Sun. Sept. 10 Mon. Sept. 11 Tues. Sept. 12 Wed. Sept. 13
8:30 BREAKFAST
9:30 Simon Salamon Bobby Acharya Aleksander Doan Gao Chen
10:30 MORNING BREAK
11:00 Jason Lotay Daniel Butter Sebastian Goette Joel Fine
12:00 LUNCH
1:15 Gavin Ball Lorenzo Foscolo Sergei Gukov Thomas Walpuski
2:30 Robert Bryant Johannes Nordström Samson Shatashvili Song Sun
3:30 AFTERNOON TEA
4:00 Discussion Discussion Discussion

Speakers & Lecture titles:

This conference will be immediately followed by our First annual meeting held at the Simons Foundation in New York City.

Adam Jacob: Lectures

June 9, 2017
TITLE: Hermitian Yang Mills connections on reflexive sheaves

Slides of Lecture

Alexei Kovalev: Lectures

June 7, 2017
TITLE: Constructing compact, holonomy Spin(7) manifolds as generalised connected sums

Slides of lecture

Andreas Braun: Lectures

September 14, 2017
TITLE: Mirror Symmetry for G2 manifolds

ABSTRACT: String theories on different G_2 manifolds can lead to the same physics in a phenomenon called mirror symmetry. In this talk, I will review mirror symmetry for G_2 manifolds, focusing on recent progress. In particular, I will present constructions of mirror G_2 manifolds realized as twisted connected sums.

Slides of lecture

June 6, 2017
TITLE: Mirror Symmetry for G2 manifolds (see arXiv:1701.05202)

Slides of lecture