## Xenia de la Ossa: Lectures

### January 8, 2018

TITLE: The geometry and moduli of heterotic G2 structures

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

In this talk 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 acting on forms with values on the bundle which satisfies , for some appropriately defined projection of the operator

. 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

and that it is finite dimensional. Our analysis leads to results that are of relevance to all orders in . Time permitting, I will comment on work in progress about the finite deformations of heterotic 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 or , 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:

- gauge theory on G2 manifolds and G2 compactifications of the heterotic string
- singular fibers of SYZ fibrations of Calabi-Yau threefolds and what they might correspond to in K3-fibered G2 manifolds
- 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
- 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:

- Bobby Acharya (ICTP and King’s College London),
*M-theory/heterotic/type IIA duality* - Gavin Ball (Duke University),
*SO(4)-structures on 7-manifolds* - Robert Bryant (Duke University),
*Algebraically special associative submanifolds and special holonomy metrics* - Daniel Butter (Texas A&M),
*Eleven-Dimensional Supergravity in 4D, N=1 Superspace* - Gao Chen (Institute for Advanced Study),
*Rate of asymptotic convergence near isolated singularity of a G*_{2}manifold - Aleksander Doan (Stony Brook),
*Fueter sections and wall-crossing in Seiberg-Witten theory* - Joel Fine (Université Libre de Bruxelles),
*Hypersymplectic 4-manifolds and the G*_{2}Laplacian flow - Lorenzo Foscolo (Heriot-Watt University),
*ALC manifolds with special holonomy* - Sebastian Goette (Freiburg),
*The extended v-invariant — progress and problems* - Sergei Gukov (Caltech),
*Topological Phases and Special Holonomy* - Jason Lotay (University College London),
*Invariant coassociative 4-folds via gluing* - Johannes Nordström (Bath),
*New asymptotically conical G*_{2}-manifolds - Simon Salamon (King’s College London),
*Quotients and hypersurfaces of model metrics* - Samson Shatashvili (Trinity College Dublin and SCGP),
*G*_{2}superconformal theories and mirror symmetry revisited - Song Sun (Stony Brook),
*Singularities of Hermitian-Yang-Mills connections* - Thomas Walpulski (Michigan State),
*The (1,k)-ADHM Seiberg-Witten equation and k-fold covers of associatives*

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

## Alexei Kovalev: Lectures

### June 7, 2017

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

## Andreas Braun: Lectures

### September 14, 2017

TITLE: Mirror Symmetry for G_{2} manifolds

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

### June 6, 2017

TITLE: Mirror Symmetry for G_{2} manifolds (see arXiv:1701.05202)

## Sir Simon Donaldson receives honorary doctorate

Sir Simon Donaldson, one of our Collaboration’s Principal Investigators, has been awarded an honorary doctorate degree by the Universidad Complutense de Madrid. Professor Donaldson was presented with the degree at an award ceremony on January 20, 2017; he delivered a lecture entitled “Metrics on Algebraic Varieties” on the previous day. Details can be found here. Congratulations, Simon!

## Dominic Joyce awarded the Fröhlich Prize

Dr. Dominic Joyce, one of our Collaboration’s Principal Investigators, has been awarded the 2016 Frölich Prize by the London Mathematical Society “for his profound and wide-ranging contributions to differential and algebraic geometry.” Dr. Joyce was presented with a certificate honoring this achievement at the Annual General Meeting of the LMS on Friday, November 11, 2016. Congratulations, Dominic!

## Dominic Joyce: Lectures

01/09/2018: Lie brackets on the homology of moduli spaces, and wall-crossing formulae

09/14/2017: Conjectures on counting associative 3-folds in G₂ manifolds

06/06/2017: Constructing compact 8-manifolds with holonomy Spin(7)

06/05/2017: Constructing compact 7-manifolds with holonomy G_{2}

01/11/2017: Counting problems for G_{2} manifolds

09/07/2016: Derived differential geometry and moduli spaces in differential geometry

### January 9, 2018

TITLE: Lie brackets on the homology of moduli spaces, and wall-crossing formulae

ABSTRACT: Let be a field, and be the “projective linear” moduli stack of objects in a suitable -linear abelian category (such as the coherent sheaves on a smooth projective ) or triangulated category (such as the derived category ). I will explain how to define a Lie bracket [ , ] on the homology (with a nonstandard grading), making into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.

There is also a differential-geometric version of this: if is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, manifold, manifold) then we can define Lie brackets both on the homology of the moduli spaces of all or connections on for all , and on the homology of the moduli spaces of instanton or connections on for all .

All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.

Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with ), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear” moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs.

I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (, [ , ]).

### September 14, 2017

TITLE: Conjectures on counting associative 3-folds in G₂ manifolds

ABSTRACT: Riemannian 7-manifolds with holonomy G₂ are a special class of Ricci-flat Riemannian manifolds, which are of interest to physicists working in M-theory. Associative 3-folds are calibrated 3-submanifolds in 7-manifolds with holonomy G₂, so they are a special kind of minimal submanifold.

There is a well-known analogy between G₂ manifolds X in dimension 7 and Calabi-Yau 3-folds Y in dimension 6. Under this analogy one should compare associative 3-folds in X with J-holomorphic curves in Y. Much of symplectic geometry — Gromov-Witten theory, Lagrangian Floer theory, and so on — is concerned with “counting” J-holomorphic curves, to get an answer which is independent of the (almost) complex structure J up to deformation. So we can ask: might there be interesting geometry of G₂ manifolds concerned with “counting” associative 3-folds, which gives an answer unchanged under deformations of the G₂ structure?

This talk, based on arXiv:1610.09836, presents a conjectural answer to this question. It is connected to conjectures of Donaldson and Segal on defining invariants by “counting” G₂ instantons on X with “compensation terms” counting pairs of a G₂ instanton and an associative 3-fold on X. At the end we will briefly discuss a proposed modification to the Donaldson-Segal conjecture, to correct for wall-crossing behaviour of associative 3-folds we discover during our investigation.

### June 6, 2017

TITLE: Constructing compact 8-manifolds with holonomy Spin(7)

### June 5, 2017

TITLE: Constructing compact 7-manifolds with holonomy G_{2}

### January 11, 2017

TITLE: Counting problems for G_{2} manifolds

ABSTRACT: This talk(s) reviews my recent preprint arXiv:1610.0983.

Many important areas of geometry involve “counting” some kind of geometric object to define an “invariant” which is then shown to be unchanged under deformations of the base geometry. Examples include Donaldson invariants counting instantons on a compact oriented Riemannian 4-manifold (X,g) with b^2_+(X)>1 (which are unchanged under deformations of g), Gromov-Witten invariants of complex algebraic / symplectic manifolds (unchanged under deformations of the (almost) complex structure J), and so on.

Floer cohomology theories and Fukaya categories are similar: they involve “counting” objects such as J-holomorphic curves with boundary, and while the numbers are not deformation-invariant, one uses the numbers and some homological algebra to construct things which are deformation-invariant.

I will discuss whether there may be interesting “invariant” theories for compact -manifolds which are unchanged under deformations of the -structure (perhaps deformations preserving closure of 3- or 4-form), as considered by Donaldson and Segal in their 2009 paper “Gauge theory in higher dimensions II”.

The three obvious classes of objects in a -manifold one could try to “count” are associative 3-folds, coassociative 4-folds, and -instantons. I haven’t much to say about coassociatives.

For associative 3-folds, I argue that one cannot define deformation-invariant Gromov-Witten style invariants counting associatives, as there are singular behaviours of associatives which would change the numbers under deformation of the -structure. However, I suggest that there may still be interesting deformation-invariant information encoded in “numbers” of associatives, and I outline how one might define a supercommutative “ quantum cohomology algebra”, similar to quantum cohomology in symplectic geometry, but with features of a Floer theory.

On the way to this conclusion, I introduce some new ideas about associative 3-folds, including a way to orient moduli spaces of associatives, and make some conjectures on their singular behaviour.

For instantons, Donaldson and Segal proposed to define invariants, similar to Donaldson invariants of 4-manifolds, which “count” -instantons on the -manifold (X,\varphi). It is known that -instantons can “bubble” on an associative 3-fold under deformation of \varphi, and this would change the numbers of -instantons. So to make their invariants unchanged under deformation, they proposed to add “compensation terms” C(N,(P,A)) which count pairs of an associative N in X and a -instanton (P,A) on X with some weight, which would have the property that C(N,(P,A)) jumps by 1 when a -instanton bubbles on N leaving (P,A) after removal of singularities, so that the compensated sum remains unchanged. The precise definition of the weight C(N,(P,A)) has remained mysterious, though Walpuski and Haydys are working towards a definition.

I argue that for -instantons with group SU(2), it is not possible to define “compensation terms” C(N,(P,A)) with the required properties (for any conceivable definition, not just for current definitions), so that I do not believe that the Donaldson-Segal programme as currently formulated will succeed. The argument why not is actually a fairly simple thought-experiment using material from Donaldson and Segal’s paper, but it is based on the ideas about orienting associative moduli spaces mentioned earlier.

I won’t be able to say all this in an hour. So what I’ll probably actually do is give an 1-hour overview in three parts: 1. Introduction; 2. Counting associatives; 3. Counting -instantons; and then we can continue afterwards according to demand from those audience members who have not already run away screaming, if this is a nonempty set.

### September 7, 2016

TITLE: Derived differential geometry and moduli spaces in differential geometry

ABSTRACT: Derived Differential Geometry (DDG) is the study of “derived manifolds” and “derived orbifolds”, where “derived” is in the sense of the Derived Algebraic Geometry of Jacob Lurie and Toen-Vezzosi. They include ordinary manifolds and orbifolds, but also many spaces which are singular at the classical level. There are several approaches to DDG, due to Spivak, Borisov-Noel and myself, all more-or-less equivalent. The “Kuranishi spaces” studied in symplectic geometry by Fukaya-Oh-Ohta-Ono are a prototype notion of derived orbifold.

I claim that moduli spaces M of solutions of a nonlinear elliptic p.d.e. on a compact manifold should naturally have the structure of a derived manifold (if solutions have no symmetries) or a derived orbifold (if solutions have finite symmetry groups, e.g. Deligne-Mumford stable J-holomorphic curves). This includes many very interesting problems — instantons on 4-manifolds and other gauge-theoretic moduli problems, J-holomorphic curves in symplectic geometry and so on. In particular, it includes moduli spaces used to define enumerative invariants (Donaldson invariants, Gromov-Witten invariants, etc.). This is because compact, oriented derived manifolds and derived orbifolds have virtual cycles in homology, and these virtual cycles may be used to define the invariants.

I also claim that many natural (partial) compactifications of such moduli spaces M, e.g. by including J-holomorphic curves with nodes, should naturally have the structure of derived manifolds or derived orbifolds with corners, where the boundary represents the extra singular solutions.

I will outline a method to prove the existence of natural derived manifold and derived orbifold structures on differential geometric moduli spaces by a method of “universal families”, based on Grothendieck’s representable functors in algebraic geometry. That is, given a moduli problem, we define a notion of family F of solutions over a base derived manifold or orbifold S. A “universal family” is a family with a universal property w.r.t. all other families. If a universal family exists (I claim it should, under reasonable conditions) it is unique up to equivalence, and the base M of the family is the moduli space, with a derived manifold/orbifold structure.