• 3/14/2023: K3s at the intersection of Special Holonomy, Generalized Symmetries, and the Swampland
• 9/13/2022: Gravity, geometry and generalized symmetries

March 14, 2023
TITLE: K3s at the intersection of Special Holonomy, Generalized Symmetries, and the Swampland

ABSTRACT: K3 surfaces, a prime example of compact special holonomy manifolds, have been a fount of insights for physicists through string compactifications. In this talk, I will review some of the geometric properties that give rise to new (“Swampland”) constraints in effective field theories of quantum gravity, and connect these to generalized symmetries and their anomalies.

September 13, 2022
TITLE: Gravity, geometry and generalized symmetries

ABSTRACT: I will discuss how geometric constraints in string compactifications inspire consistency conditions of quantum theories of gravity, which can be formulated in the framework of generalized symmetries.

Slides of Lecture

• 5/14/2024: A Fredholm framework for singular deformation problems
• 9/13/2022: Gluing Z₂-Harmonic Spinors

May 14, 2024
TITLE: A Fredholm framework for singular deformation problems.

ABSTRACT: Recent work in several directions has led to singular elliptic operators with infinite-dimensional obstructions, for which the standard Fredholm theory does not apply. In particular, this is the case for singular Fueter sections. My previous work and the work of Donaldson and Takahashi has shown that for Fueter sections, the freedom to deform a singular set may be used to cancel the infinite-dimensional obstruction. The goal of this talk is to explain that this is not a trick for solving a particular problem, but rather the beginning of a unified approach where additional geometric freedom may be used to place these singular problems back within the standard Fredholm framework (almost). As a demonstration of this, I will talk about new gluing results for harmonic spinors and 1-forms based on joint work with Siqi He.

September 13, 2022
TITLE: Gluing Z₂-Harmonic Spinors

ABSTRACT: Z₂-harmonic spinors are singular generalizations of classical harmonic spinors and appear in two contexts in gauge-theory. First, they arise as limits of sequences of solutions to equations of Seiberg-Witten type on low-dimensional manifolds; second, they are the simplest type of (singular) Fueter section–objects which arise naturally on calibrated submanifolds in the study of gauge theory on manifolds with special holonomy. These two pictures are related by proposals for defining invariants of manifolds with special holonomy.

In this talk, after giving an introduction to these ideas, I will discuss a gluing result that shows a generic Z₂-harmonic spinor on a 3-manifold necessarily arises as the limit of a family of two-spinor Seiberg-Witten monopoles. Due to the singularities of the Z₂-harmonic spinor, the relevant operators in the gluing problem are only semi-Fredholm and possess an infinite-dimensional cokernel. To deal with this, the proof requires the analysis of families of elliptic operators degenerating to a singular limit, and the study of deformations of the singularities which are used to cancel the infinite-dimensional cokernel. At then end, I will discuss related problems in gauge theory and calibrated geometry.

September 13, 2022
TITLE: Deformations and gluing of asymptotically cylindrical associatives

ABSTRACT: Given a matching pair of asymptotically cylindrical (Acyl) G_2 manifolds the twisted connected sum construction produces a one parameter family of closed G_2 manifolds. We describe when we can construct closed rigid associatives in these closed G_2 manifolds by gluing suitable pairs of Acyl associatives in the matching pair of Acyl G_2 manifolds. The hypothesis and analysis in the gluing theorem requires some understanding of the deformation theory of Acyl associatives which will also be discussed. At the end we will describe examples of closed associatives coming from Acyl holomorphic curves or special Lagrangians.

September 12, 2022
TITLE: T^2-invariant associatives in G_2 manifolds with cohomogeneity-two symmetry

ABSTRACT: A classical way to construct calibrated submanifolds is via symmetry reduction. In this talk, we will consider G_2 manifolds with a T^2\times SU(2) structure-preserving action of cohomogeneity-two. For each of these manifolds, we describe the geometry of the T^2-invariant associative submanifolds using moment type maps for the group action. As an application, we describe an associative fibration, in the Karigiannis–Lotay sense, on the Bryant–Salamon manifolds S^3\times \mathbb{R}^4.
This is joint work in progress with B. Aslan.

September 12, 2022
TITLE: Heterotic systems, balanced SU(3)-structures and coclosed G_2-structures in cohomogeneity one manifolds

ABSTRACT: When considering compactifications of heterotic string theory down to 4D, the Hull–Strominger system arises over a six-dimensional manifold endowed with an invariant nowhere-vanishing holomorphic (3,0)-form. When compactifying down to 3D, we get the heterotic G_2 system over a manifold with a G_2-structure. In this talk, we describe these systems and then study the existence of some of the geometric structures required by them in the cohomogeneity one setting.

For the former one, we provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. For the later one, we find a family of coclosed G_2-structures on certain seven-dimensional cohomogeneity one manifolds. Part of this talk is based on a joint work with F. Salvatore.

Slides of Lecture

September 12, 2022
TITLE: Multiplicative vertex algebras and wall-crossing in equivariant K-theory

ABSTRACT: K-theory is an interesting multiplicative refinement of
cohomology, and many cohomological objects arising in enumerative
geometry have K-theoretic analogues — modular forms become Jacobi
forms, Yangians become quantum affine algebras, etc. I will explain
how this sort of refinement goes for vertex algebras. As an
application, Joyce’s recent “universal wall-crossing” machine, which
operates by making the homology of certain moduli stacks into vertex
algebras, can be lifted to equivariant K-theory, e.g. thereby proving
the main conjecture on semistable invariants in refined Vafa-Witten
theory. In a different direction, I expect there to be some hidden
multiplicative vertex algebra structure on the aforementioned quantum
affine algebras, which can be viewed as symmetry algebras controlling
various enumerative and physical theories.

September 11, 2022
TITLE: Geometric Flows of 3-Sasakian Structures

ABSTRACT: Geometric flows of G_2-Structures are expected to be valuable tools for determining when a G_2-Structure with torsion may be deformed to one which is torsion-free. Several flows of G_2-Structures have been proposed to provide insight into this question, including the Laplacian flow and the Laplacian coflow. Here we consider an alternative application of these geometric flows to the study of Nearly Parallel G_2-Structures, specifically those originating from 3-Sasakian geometry. We write down an ansatz for co-closed G_2-Structures given in terms of the 3-Sasakian data and consider how scaled versions of the Laplacian flow and coflow behave when we start the flows at one of these structures. These results provide us with insight into the stability/instability of the Nearly Parallel G_2-Structures which are special co-closed G_2-Structures in this ansatz. We then can compare these stability results with the analogous conclusions for the scaled Ricci flow starting at a G_2-metric corresponding to our ansatz for the G_2-Structure. This is joint work with Jason Lotay.

• 09/12/2023: On nearby special Lagrangians
• 09/11/2022: Coulomb and Higgs phases of G_2 manifolds

September 12, 2023
TITLE: Coulomb and Higgs phases of G_2 manifolds

ABSTRACT: We will discuss the physics of M-theory compactifications onto G_2-orbifolds of the type that can be resolved via the method of Joyce and Karigiannis i.e. orbifolds where one has a singular locus of A₁ singularities that admits a nowhere-vanishing Z_2-twisted) harmonic 1-form. We will see that topologically distinct desingularisations of such orbifolds can be physically interpreted as different branches of the 4d vacuum moduli space known as ‘Coulomb’ and ‘Higgs’ branches. The results suggest generalisations of the results of Joyce and Karigiannis to G_2-orbifolds with more diverse ADE singularities and higher order twists.

September 11, 2022
TITLE: On the physics of Joyce-Karigiannis manifolds

ABSTRACT: We will discuss the physics of M-theory compactifications onto G₂-orbifolds of the type that can be resolved via the method of Joyce and Karigiannis i.e. orbifolds where one has a singular locus of A₁ singularities that admits a nowhere-vanishing (Z₂-twisted) harmonic 1-form. We also discuss natural generalisations of these orbifolds (not currently resolvable by the method of Joyce-Karigiannis) and give proposals for the physics in these cases too.

Slides of Lecture

• 06/07/2022: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

June 7, 2020 (Jointly with Alec Payne)
TITLE: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

ABSTRACT: The Laplacian flow is a natural geometric flow which deforms closed G2-structures on 7-manifolds. This flow could be a major avenue to insight into manifolds with G2-holonomy. However as with many geometric flows, singularities are expected to form in finite time. Self-similar soliton solutions to the flow are expected to play a significant role in the analysis of these singularities. In this talk, we consider self-shrinking solitons (these are necessarily noncompact) with prescribed asymptotics on their ends. In particular, we consider the important class of asymptotically conical (AC) shrinkers, the first examples of which were recently constructed by Haskins-Nordstrom. We describe our proof of the uniqueness of AC gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. This is joint work of Mark Haskins, Ilyas Khan and Alec Payne.

Slides of Lecture

• 09/10/2023: Closed G2-Structures with Negatively Pinched Ricci Curvature
• 06/07/2022: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

September 10, 2023
TITLE: Closed G2-Structures with Negatively Pinched Ricci Curvature

ABSTRACT: The Goldberg conjecture is a well-known question about whether compact almost Kähler, Einstein manifolds must be Kähler. The analogue of the Goldberg conjecture for G_2-geometry asks whether an Einstein closed G_2-structure must be torsion-free, and this was proven for compact manifolds by Cleyton-Ivanov and Bryant. In this talk, we discuss how this result extends to noncompact manifolds. In general, we find that there is no closed G_2-structure on a noncompact manifold whose metric is complete and has sufficiently negatively pinched Ricci curvature.

June 7, 2022 (Jointly with Ilyas Khan)
TITLE: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

ABSTRACT: The Laplacian flow is a natural geometric flow which deforms closed G2-structures on 7-manifolds. This flow could be a major avenue to insight into manifolds with G2-holonomy. However as with many geometric flows, singularities are expected to form in finite time. Self-similar soliton solutions to the flow are expected to play a significant role in the analysis of these singularities. In this talk, we consider self-shrinking solitons (these are necessarily noncompact) with prescribed asymptotics on their ends. In particular, we consider the important class of asymptotically conical (AC) shrinkers, the first examples of which were recently constructed by Haskins-Nordstrom. We describe our proof of the uniqueness of AC gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. This is joint work of Mark Haskins, Ilyas Khan and Alec Payne.

Slides of Lecture