Schedule for: 23w5006 - Spinorial and Octonionic Aspects of G2 and Spin(7) Geometry

The octonions are an eight-dimensional analogue of the complex numbers, formed by adjoining seven square roots of -1 to the real numbers, instead of just one. They are nonassociative, and thus fall outside the scope of much of the usual theory of algebras and their modules that we learn in school. Nevertheless, this strange algebra turns up in surprising corners of mathematics, essentially whenever "exceptional" structures appear. This includes the G2 and Spin(7) manifolds that are our focus in this workshop. To get started with these geometric structures, I will introduce the octonions, and show how they naturally encode spinors in seven and eight dimensions.

When, in 1887, Wilhelm Killing unexpectedly found a new family of complex simple Lie groups [1], he gave the scientific community a precious gift: a group which can always be studied and continued to be amazing. Of course, we are talking about $\mathrm{G_2}$, the group of the thousand facets.

REFERENCES:
1. Killing, Wilhelm; Die Zusammensetzung der stetigen endlichen Transformations- gruppen. Math. Ann. 31 (1888), no. 2, 252–290.
2. Draper Fontanals, Cristina; Notes on $\mathrm{G_2}$: the Lie algebra and the Lie group. Differential Geom. Appl. 57 (2018), 23–74.
3. Cristina Draper, Francisco J. Palomo; Reductive homogeneous spaces of the compact Lie group $G_2$. Preprint arXiv:2211.06997 (proceedings NAART II to be published in Springer series PROMS)

I will begin with a brief explanation of what $\mathrm{G_2}$-instantons and $\mathrm{G_2}$-manifolds are. There is a general construction by Joyce-Karigiannis for $\mathrm{G_2}$-manifolds. Ignoring all analysis, I will explain one example of their construction. The example is the resolution of $K3 \times \mathbb{T}^3/\mathbb{Z}_2$ for a very explicit $K3$ surface. Furthermore, there is a construction method for $\mathrm{G_2}$-instantons on Joyce-Karigiannis manifolds. I will explain the ingredients needed for the construction, say nothing about the proof, and then explain one example of the ingredients.

(1) Alfred Holmes
Title: Spin(7) instantons and the ADHM Construction
Abstract: In this talk I'll give an overview of a potential way to construct Spin(7) instantons from solutions to the ADHM Seiberg Witten equations.

(2) Diego Artacho Obesso
Title: Generalised Spin$^r$ Structures on Homogeneous Spaces
Abstract: Spinorial methods have proven to be a powerful tool to study geometric properties of Spin manifolds. The idea is to make accessible the power of Spin geometry to manifolds which are not necessarily Spin. The concept of Spin$^c$ and Spin$^h$ structures provide examples of work in that direction. In this talk, I will present a generalisation of these structures and comment on what these structures look like on homogeneous spaces, particularly on spheres.

(3) Mario Garcia-Fernandez
Title: $\mathrm{SU(n)}$-instantons from the Hull-Strominger system
Abstract: I will explain how to construct an $\mathrm{SU(n)}$-instanton on a real orthogonal bundle, from a solution of the Hull-Strominger system on a (possibly non-K\"ahler) Calabi-Yau manifold. If time allows, I will comment on how this basic principle leads to obstructions to the existence of solutions and also on conjectural extensions to the $\mathrm{G_2}$ and Spin(7) heterotic systems. Based on joint work with Raúl Gonzalez Molina, in arXiv:2303.05274 and arXiv:2301.08236.

Abstract: In this talk I'll give an overview of a potential way to construct Spin(7) instantons from solutions to the ADHM Seiberg Witten equations.

Abstract: Spinorial methods have proven to be a powerful tool to study geometric properties of Spin manifolds. The idea is to make accessible the power of Spin geometry to manifolds which are not necessarily Spin. The concept of Spin$^c$ and Spin$^h$ structures provide examples of work in that direction. In this talk, I will present a generalisation of these structures and comment on what these structures look like on homogeneous spaces, particularly on spheres.

Abstract: I will explain how to construct an $\mathrm{SU(n)}$-instanton on a real orthogonal bundle, from a solution of the Hull-Strominger system on a (possibly non-K\"ahler) Calabi-Yau manifold. If time allows, I will comment on how this basic principle leads to obstructions to the existence of solutions and also on conjectural extensions to the $\mathrm{G_2}$ and Spin(7) heterotic systems. Based on joint work with Raúl Gonzalez Molina, in arXiv:2303.05274 and arXiv:2301.08236.

There are several relationships between (twisted) harmonic spinors and associative submanifolds. For example, the Dirac operator appears in the PDE for associative graphs, in the deformation theory for associative submanifolds, and in the second variation formula for volume. In the first part of this talk (joint work with Gavin Ball), we provide another relationship. If a 7-manifold $M$ has a closed or nearly-parallel $\mathrm{G_2}$-structure, we show that the second fundamental form of an associative can be viewed as a twisted spinor. Moreover, if $M$ has constant curvature (e.g., if $M = \mathbb{R}^7, \mathbb{S}^7$, or $\mathbb{T}^7$), then this twisted spinor is harmonic. Intuitively, this is a spin-geometric analogue of the classical Hopf differential for 2-dimensional surfaces. In the second part, we elaborate on this theme, highlighting the many analogies between harmonic spinors and holomorphic objects. In particular, we provide a “taxonomy” of Dirac equations that have arisen in the literature, which in turn suggests several avenues for further work.

In this talk, we generalize some results from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogues of Lie groups. The main components of this theory include a finite-dimensional smooth loop $\mathbb{L}$, together with its tangent algebra $\mathfrak{l}$ and pseudoautomorphism group $\Psi $, and a smooth manifold $M$ with a principal $\Psi $-bundle $\mathcal{P}$. A configuration in this theory is defined as a pair $\left( s,\omega \right) $, where $s$ is an $\mathbb{L}$-valued section and $\omega $ is a connection on $\mathcal{P}$. Each such pair determines the torsion $T^{\left( s,\omega \right) }$, which is a key object in theory. Given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm. We will also show how these results apply to $G_2$-geometry on $7$-dimensional manifolds.

$\mathrm{G_2}$-instantons are a special kind of connections on a Riemannian $7$-manifold, analogues of anti-self-dual connections in $4$ dimensions. I will start this talk by giving an overview of why we are interested in them and known examples. Then, I will explain how we construct $\mathrm{G_2}$-instantons in $\mathrm{SU(2)}^2$-invariant cohomogeneity one manifolds and give new examples of $\mathrm{G_2}$-instantons on $\mathbb{R}^4\times \mathbb{S}^3$ and $\mathbb{S}^4\times \mathbb{S}^3$. I will then discuss the bubbling behaviour of sequences of $\mathrm{G_2}$-instantons found.

The harmonic flow of an H-structure (aka the isometric flow) is the gradient flow of the energy functional for the intrinsic torsion. In recent years the cases when H= U(n), G2 and Spin(7) have been studied in great detail mainly due to their relation with special holonomy. In this talk I will discuss the case when H= Sp(2)Sp(1) (i.e. the quaternionic Kähler case) and shed some light into how the representation theory of H allows for a more unified approach. I will also discuss the cases when H= Sp(1) and Sp(2) to illustrate certain similarities and differences. Aside from analytical aspects of the flow, I will also describe explicit examples of non-trivial harmonic H-structures and as well as soliton solutions to the flow. This is a joint work with Henrique Sá Earp.

1) Henrik Naujoks
Title: Geometry and Spectral Properties of Aloff-Wallach Manifolds (Part I)
Abstract: The focus of our attention will be the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ . The family of manifolds depending on the embedding parameters $k, l$ will each be equipped with a metric depending on four additional parameters. These six parameters in total lead to various interesting structures (K-contact as well as Sasakian structures, Einstein metrics, etc.) on this set of Riemannian manifolds. The interplay of these structures will be discussed. Furthermore, we investigate the spectrum of the Laplace operator: The metrics on the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ are not normal, but for $k = l = 1$ some of them are isometric to a normal homogeneous space. For the latter, the spectrum of the Laplace operator can be explicitly computed using methods of representation theory.

2) Jonas Henkel
Title: Geometry and Spectral Properties of Aloff-Wallach Manifolds (Part II)
Abstract: The focus of our attention will be the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ . The family of manifolds depending on the embedding parameters $k, l$ will each be equipped with a metric depending on four additional parameters. These six parameters in total lead to various interesting structures (K-contact as well as Sasakian structures, Einstein metrics, etc.) on this set of Riemannian manifolds. The interplay of these structures will be discussed. Furthermore, we investigate the spectrum of the Laplace operator: The metrics on the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ are not normal, but for $k = l = 1$ some of them are isometric to a normal homogeneous space. For the latter, the spectrum of the Laplace operator can be explicitly computed using methods of representation theory.

3) Christina Tonnesen-Friedman
Title: Sasakian geometry on certain fiber joins
Abstract: This presentation will be based primarily on past and ongoing work with Charles P. Boyer. We will discuss the Sasakian geometry of certain $7$-manifolds constructed by the so-called fiber join construction for K-contact manifolds, introduced by T. Yamazaki around the turn of the century. This construction can be adapted to the Sasaki case and produces some interesting examples. We will talk about some of these examples and also discuss some limitations of the construction.

Abstract: The focus of our attention will be the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ . The family of manifolds depending on the embedding parameters $k, l$ will each be equipped with a metric depending on four additional parameters. These six parameters in total lead to various interesting structures (K-contact as well as Sasakian structures, Einstein metrics, etc.) on this set of Riemannian manifolds. The interplay of these structures will be discussed. Furthermore, we investigate the spectrum of the Laplace operator: The metrics on the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ are not normal, but for $k = l = 1$ some of them are isometric to a normal homogeneous space. For the latter, the spectrum of the Laplace operator can be explicitly computed using methods of representation theory.

2) Jonas Henkel
Title: Geometry and Spectral Properties of Aloff-Wallach Manifolds (Part II)
Abstract: The focus of our attention will be the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ . The family of manifolds depending on the embedding parameters $k, l$ will each be equipped with a metric depending on four additional parameters. These six parameters in total lead to various interesting structures (K-contact as well as Sasakian structures, Einstein metrics, etc.) on this set of Riemannian manifolds. The interplay of these structures will be discussed. Furthermore, we investigate the spectrum of the Laplace operator: The metrics on the Aloff-Wallach manifolds $\mathrm{SU(3)}/{S^1}_{k,l}$ are not normal, but for $k = l = 1$ some of them are isometric to a normal homogeneous space. For the latter, the spectrum of the Laplace operator can be explicitly computed using methods of representation theory.

Abstract: This presentation will be based primarily on past and ongoing work with Charles P. Boyer. We will discuss the Sasakian geometry of certain $7$-manifolds constructed by the so-called fiber join construction for K-contact manifolds, introduced by T. Yamazaki around the turn of the century. This construction can be adapted to the Sasaki case and produces some interesting examples. We will talk about some of these examples and also discuss some limitations of the construction.

In this talk, I will report on joint work with Jason Lotay on which we prove versions of the Thomas and Thomas-Yau conjectures regarding the existence of special Lagrangian submanifolds and the role of Lagrangian mean curvature flow as a way to find them. I will also report on some more recent work towards proving more recent conjectures due to Joyce.

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H \subset \mathrm{SO}(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen--Struwe for the harmonic map heat flow. This yields an $\epsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, with respect to the $L^\infty$-norm of initial torsion, in the spirit of Chen--Ding. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $\pi_n(\mathrm{SO}(n)/H)\neq\{1\}$; e.g. when $n=7$ and $H=\mathrm{G_2}$, or $n=8$ and $H=\mathrm{Spin}(7)$.

I will report on recent joint work with Spiro Karigiannis and Jesse Madnick where we discover, for a number of different calibrations, a characterization of calibrated submanifolds in terms of the first variation of the volume functional with respect to a special set of deformations of the ambient metric determined by the calibration form. Generalizing earlier such results due to Arezzo and Sun for complex submanifolds, we obtain variational characterizations for associative 3-folds and coassociative 4-folds in manifolds with $\mathrm{G_2}$-structures, as well as for Cayley 4-folds in manifolds with Spin(7)-structures.

The Laplacian flow and coflow are two of the most studied flows in $\mathrm{G_2}$ geometry. We will establish a correspondence between parabolic complex Monge--Ampère equations and these flows for initial data on a torus bundle over a complex Calabi--Yau $2$- or $3$-fold given from a Kähler metric. We will use estimates for these complex Monge--Ampère flows to show that both the Laplacian flow and coflow exist for all time and converge to a torsion-free $\mathrm{G_2}$ structure induced by a Ricci-flat Kähler metric. This is joint work with Sébastien Picard.

In this talk, we discuss two problems where compact quotients of Lie groups are useful for understanding topological properties of compact closed $\mathrm{G}_2$ manifolds that don´t admit any torsion-free $\mathrm{G}_2$ structure. These problems are related to the questions: Are simply connected compact closed $\mathrm{G}_2$ manifolds formal? Could a compact closed $\mathrm{G}_2$ manifold have third Betti number $b_3=0$? Using compact quotients of Lie groups, we first outline the construction of a manifold admitting a closed $\mathrm{G}_2$ structure that is not formal and has first Betti number $b_1=1$. Later, we show that compact quotients of Lie groups do not have any invariant $\mathrm{G}_2$ structure. The last result is joint work with Anna Fino and Alberto Raffero.

We discuss reducible $\mathrm{G_2}$-structures, more precisely $G$-structures with $G\subsetneq \mathrm{G_2}$ admitting a characteristic connection with parallel skew-torsion. We investigate how these structures can simplify the so-called heterotic $\mathrm{G_2}$ system. Our study focuses on a 1-parameter deformation of the characteristic connection. We find this family to contain two $\mathrm{G_2}$-instantons on $3$-$(\alpha, \delta)$-Sasaki manifolds and a new solution of the heterotic $\mathrm{G_2}$ system for arbitrary string parameter $\alpha'$ in the degenerate case. For further geometries we obtain approximate solutions. Joint work with Mateo Galdeano.

We use the Weitzenböck formulas to get information about the Betti numbers of nearly $\mathrm{G_2}$ and nearly Kähler manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.

(1) Shubham Dwivedi
Title : A gradient flow of Spin(7)-structures
Abstract: We will introduce a geometric flow of Spin(7)-structures which is the negative gradient flow of a natural energy functional on the space of Spin(7)-structures. We will evaluate the evolution of the Riemannian metric and show that the flow exists for a short time.

(2) Guilia Dileo
Title: Generalized Killing spinors on $3$-$(\alpha,\delta)$-Sasaki manifolds
Abstract: $3$-$(\alpha,\delta)$-Sasaki manifolds are a special class of Riemannian manifolds generalizing $3$-Sasaki manifolds, and admitting a canonical metric connection with totally skew-symmetric torsion. In the present talk I will show that every $7$-dimensional $3$-$(\alpha,\delta)$-Sasaki manifold admits a canonical $\mathrm{G_2}$-structure, which determines four generalized Killing spinors. The corresponding generalized Killing numbers are explicitly obtained, providing characterization of the cases where they coincide. This is part of a joint work with Ilka Agricola.

(3) Markus Upmeier
Title: Spinors, calibrated submanifolds, and instantons
Abstract: In the context of enumerative geometry for manifolds of special holonomy there is a deep connection between calibrated submanifolds and instantons through `bubbling'. During the talk, I will use spinors and Dirac operators to discuss an interesting link between the (linearized) deformation theories of calibrated submanifolds and instantons. The applications of the main result include a solution to the open problem of constructing orientation data in DT-theory for Calabi-Yau 4-folds.

Abstract: We will introduce a geometric flow of Spin(7)-structures which is the negative gradient flow of a natural energy functional on the space of Spin(7)-structures. We will evaluate the evolution of the Riemannian metric and show that the flow exists for a short time.

Abstract: $3$-$(\alpha,\delta)$-Sasaki manifolds are a special class of Riemannian manifolds generalizing $3$-Sasaki manifolds, and admitting a canonical metric connection with totally skew-symmetric torsion. In the present talk I will show that every $7$-dimensional $3$-$(\alpha,\delta)$-Sasaki manifold admits a canonical $\mathrm{G_2}$-structure, which determines four generalized Killing spinors. The corresponding generalized Killing numbers are explicitly obtained, providing characterization of the cases where they coincide. This is part of a joint work with Ilka Agricola.

Abstract: In the context of enumerative geometry for manifolds of special holonomy there is a deep connection between calibrated submanifolds and instantons through `bubbling'. During the talk, I will use spinors and Dirac operators to discuss an interesting link between the (linearized) deformation theories of calibrated submanifolds and instantons. The applications of the main result include a solution to the open problem of constructing orientation data in DT-theory for Calabi-Yau 4-folds.

Constructing associative and coassociative submanifolds of a $\mathrm{G_2}$-manifold is, in general, a difficult task. However, when the ambient manifold admits symmetries, finding cohomogeneity one calibrated submanifolds is more tractable. In this talk, I will discuss joint work with B. Aslan regarding the geometry of such calibrated submanifolds in $\mathrm{G_2}$-manifolds with a non-abelian cohomogeneity two symmetry. Afterwards, I will explain how to apply these results to describe new large families of complete associatives in the Bryant--Salamon manifold of topology $S^3\times \mathbb{R}^4$ and in the manifolds recently constructed by Foscolo--Haskins--Nördstrom.

The action of SO(3) by conjugation on the space of symmetric traceless matrices gives an embedding of SO(3) in SO(5). A 5-manifold whose structure group reduces to this copy of SO(3) is said to carry an SO(3)-structure. The integrable examples of these structures are the symmetric spaces R^5, SU(3)/SO(3) and SL(3)/SO(3), and general SO(3)-structures may be thought of as non-integrable analogues of these spaces. In my talk, I will describe work in progress on the local geometry of a subclass of SO(3)-structures called the nearly integrable SO(3)-structures. The nearly integrable condition was introduced by Bobienski and Nurowski as an analogue of the nearly Kahler condition in almost Hermitian geometry. However, despite the similarity of the definitions, it turns out that the local geometry of nearly integrable SO(3)-structures is significantly more restricted compared to the nearly Kahler case. The rigid nature of the local geometry suggests the possibility of giving a global classification of nearly integrable SO(3)-structures and I will sketch out such a program. If time permits, I will describe relations with G2-geometry.

There is a notion of non-degenerate $3$-form in six and seven dimensions which are the pointwise model for $\mathrm{G_2}$- and $\mathrm{SL}(3,\mathbb{C})$-structures, respectively. These are directly related, as the restriction of a $3$-form which defines a $\mathrm{G_2}$-structure on a $7$-manifold to a real hypersurface induces an $\mathrm{SL}(3,\mathbb{C})$-structure. I will describe the geometric structure induced on a real hypersurface inside a $6$-manifold with an $\mathrm{SL}(3,\mathbb{C})$-structure under a certain convexity condition. This is based on joint work with S. Donaldson.

The question of which manifolds are $spin$ or $spin^c$ has a simple and complete answer. In this talk we address the same question for the lesser known $spin^h$ manifolds which have appeared in geometry and physics in recent decades. We determine the first obstruction to being $spin^h$ and use this to provide an example of an orientable manifold which is not $spin^h$. The existence of such an example leads us to consider an infinite sequence of generalised spin structures. In doing so, we determine an answer to the following question: is there an integer $k$ such that every manifold embeds in a spin manifold with codimension at most $k$? This is joint work with Aleksandar Milivojevic.