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Author Archives: Victoria Hain

Lectures: James Sparks

October 27, 2020
TITLE: GK geometry

Slides of Lecture

Lectures: Du Pei

October 27, 2020
TITLE: Branes in moduli spaces of Higgs bundles, II

Slides of Lecture

Lectures: Craig Lawrie

October 29, 2020
TITLE: G2, Spin(7) BPS Equations and T-branes

Slides of Lecture

Lectures: Laura Fredrickson

October 27, 2020
TITLE: Branes in the moduli spaces of Higgs bundles, I

Slides of Lecture

Lectures: Lara Anderson

October 26, 2020
TITLE: Higgs Bundles in String Compactifications

Slides of Lecture

Gianluca Zoccarato: Lectures

September 16, 2020 (jointly with Mirjam Cvetic)

TITLE: Unification of Higgs Bundles

ABSTRACT: We introduce a program to study compactifications of M-/F-theory on spaces with G2 and Spin(7) holonomy, which results in N=1 supersymmetry in four- and three-dimensions, respectively, by studying the world-volume gauge dynamics of Higgs bundles, which are associated with higher co-dimension singularities of special holonomy spaces.

Specifically, we will consider solutions of M-theory on local geometries with Spin(7) holonomy. The local geometry is an ADE fibration over a four manifold which gives a compactification of 7d supersymmetric Yang-Mills theory on Spin(7) manifold. The configurations of this system are controlled by a Higgs bundle on the four manifold. This Higgs bundle, which we will call the Spin(7) system, has the remarkable property of unifying other known Higgs bundles. We will show how solutions of other known Higgs bundles fit in the Spin(7) system and show how it is possible to use the Spin(7) system to create interpolating solutions between different Higgs bundles.

Slides of Lecture, Part I

Slides of Lecture, Part II

Justin Sawon: Lectures

Recorded June 2020
TITLE: Singular fibres of holomorphic Lagrangian fibrations

ABSTRACT: Fibrations on compact holomorphic symplectic manifolds are Lagrangian: their fibres must be Lagrangian with respect to the holomorphic symplectic structure. Moreover, the general fibre must be an abelian variety and singular fibres must occur in codimension one. In this talk I will survey results of Matsushita, Hwang-Oguiso, Christian Lehn, and myself that describe the structure of the singular fibres that can occur in these Lagrangian fibrations.
Slides of Lecture

Antoine Bourget: Lectures

Recorded June 2020
TITLE: Hasse diagrams for Symplectic Singularities via Magnetic Quivers

In this lecture I review the construction of a finite Hasse diagram encoding the structure of singularities and symplectic leaves in conic symplectic singularities. This construction uses the concept of magnetic quiver, which is a combinatorial description of certain conic symplectic singularities, and the algorithm known as “quiver subtraction”. I then show how this can be used to gain insight about these spaces, and how this is connected to various effects in classical and quantum physics (Higgs mechanism, structure of the chiral ring, moduli space of strongly coupled SCFTs in 5 and 6 dimensions).
Slides of Lecture

Roger Bielawski: Lectures

Recorded June 2020
TITLE: Twistor spaces and hyperkahler metrics

ABSTRACT: I discuss several questions about twistor spaces of hyperkaehler manifolds:
multiple components of the Kodaira moduli space of sections; differential
geometry of spaces of sections which include sections with “wrong” normal
bundle; and (pseudo-)hyperkaehler metrics arising on Hilbert schemes of higher
degrees curves in a twistor space.
On this last topic, in addition to older results, I discuss the current project
with Lorenzo Foscolo, where we construct (presumably QALF) hyperkaehler manifolds
from W-invariant hypertoric varieties. We suggest that a subclass of these
corresponds precisely to the Coulomb branches of 3-dimensional N=4 SUSY gauge theories.
Slides of Lecture

Claire Voisin: Lectures

Recorded June 2020
TITLE: Hodge theory and the topology of hyper-Kähler manifolds: an introduction

ABSTRACT: After a short discussion of hyper-Kähler manifolds in the Riemannian and complex geometry contexts,
I discuss the general topological properties of hyper-Kähler
manifolds obtained from the local study of the period map. At the end of the talk, I start discussing
hyper-Kähler manifolds in the algebraic geometry setting, which will be the subject of the second lecture.

The audience for this lecture is encouraged to download the slides and to follow
along in the slides while watching the video.

Slides of lecture

June 3, 2020
TITLE: Polarized variations of Hodge structures of hyper-Kähler type

ABSTRACT: Like for abelian varieties and complex tori, the theory of projective hyper-Kähler manifolds is very different from the theory of general Kähler ones. The former is related via the period map and Torelli theorem to the study of certain quotients of bounded Hermitian symmetric domains by arithmetic groups. I will discuss various results concerning these moduli spaces and constructions of hyper-Kähler manifolds via algebraic geometry.