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Evyatar Sabag: Lectures

September 10, 2023
TITLE: G2 Manifolds from 4d N=1 Theories

ABSTRACT: We propose new G2-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N = 1 duality domain walls of 5d N = 1 theories. These domain walls interpolate between different extended Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the geometric realization of such a 5d superconformal field theory and its extended Coulomb branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its Kahler cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the Calabi-Yau three-fold over a real line, whilst varying its K¨ahler parameters as prescribed by the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through a canonical singularity at the locus of the domain wall. Due to the 4d N = 1 supersymmetry that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy G2. Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are known to admit torsion-free G2-holonomy metrics. We develop several generalizations to new 7-manifolds, which realize domain walls in 5d SQCD theories and walls between 5d theories which are UV-dual.

Junsheng Zhang: Lectures

September 11, 2023
TITLE: On complete Calabi-Yau manifolds asymptotic to cones

ABSTRACT: We proved a “no semistability at infinity” result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds. Joint work with Song Sun.

Valentino Tosatti: Lectures

September 12, 2023
TITLE: Holomorphic Lagrangian fibrations and special Kähler geometry

ABSTRACT: Consider a compact hyperkähler manifold (aka irreducible holomorphic symplectic) with a nontrivial fiber space structure onto a lower-dimensional space. Classical work of Matsushita shows that the base must be half-dimensional, and the smooth fibers are holomorphic Lagrangian tori. A basic conjecture is that the base of such holomorphic Lagrangian fibrations should be projective space. I will discuss a new proof, joint with Yang Li, of a theorem of Hwang which shows that this conjecture is true when the base is smooth. Our arguments exploit crucially the differential geometry of a “special Kähler metric” that exists on the base away from the discriminant locus.

Langte Ma: Lectures

September 13, 2023
TITLE: Instantons on Joyce’s G2-manifolds

ABSTRACT: As 7-manifolds with special holonomy, examples of compact G2-manifolds were first constructed by Joyce as resolutions of flat G2-orbifolds. Later, Walpuski constructed non-trivial G2-instantons over Joyce’s manifolds via gluing techniques. In this talk, I will first explain how to define a deformation invariant of G2-orbifolds by counting flat connections, then describe the moduli space of instantons over certain non-compact G2-manifolds that appeared in Joyce’s construction, with the aim to give a complete description of moduli spaces over some examples in Joyce’s list.

Mingyang Li: Lectures

September 10, 2023
TITLE: Classification results for Hermitian non-Kahler gravitational instantons

ABSTRACT: We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications. These results are based on a previous work and an upcoming work by the speaker.

Jin Li: Lectures

September 12, 2023
TITLE: On the geometry of resolutions of G-2-manifolds with ICS

ABSTRACT: Given a compact G_2 manifold with isolated conical singularities (ICS), the process of resolutions of these singularities gives us a one-parameter family of torsion-free G_2 structures, which can be viewed as a curve in some moduli space. This talk reports the progress in estimating the length of the curve under the L^2 Riemannian metric on the moduli space.

Jonas Lente: Lectures

September 12, 2023
TITLE: Modular Mathai-Quillen currents

ABSTRACT: The Mathai-Quillen current is a correction term that appears in the Poincaré-Hopf theorem for manifolds with boundary, similar to the eta invariant in the Atiyah-Patodi-Singer theorem.
In this talk, I will explain how to extract a modular Mathai-Quillen current from modular cobordism invariants, such as the Witten genus. On the way, I will present an analogue of the Poincaré-Hopf theorem for supermanifolds. The hope is that this modular current provides a modular extension of the nu invariant for G_2-manifolds. This is part of my ongoing PhD project supervised by S. Goette and K. Wendland.

Julius Grimminger: Lectures

September 10, 2023
TITLE: Stratified hyper-Kähler moduli spaces and physics

ABSTRACT: Singular hyper-Kähler varieties are stratified into smooth subsets called symplectic leaves. In recent years the 3d Coulomb branch construction of hyper-Kähler varieties has been a powerful tool to study this stratification, going under the name of quiver subtraction. This algorithm is derived from intuition coming from physics, and in particular brane systems. We will review the concept of this stratification, as well as some computational methods and physical interpretations.

Charles Cifarelli: Lectures

September 11, 2023
TITLE: Steady gradient Kähler-Ricci solitons and Calabi-Yau metrics on Cn

ABSTRACT: I will present a new construction of complete steady gradient Kähler-Ricci solitons on C^n, using the theory of hamiltonian 2 forms, introduced by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, as an Ansatz. The metrics come in families of two types with distinct geometric behavior, which we call Cao type and Taub-NUT type. In particular, the Cao type and Taub-NUT type families have a volume growth rate of r^n and r^{2n-1}, respectively. Moreover, each Taub-NUT type family contains a codimension 1 subfamily of complete Ricci-flat metrics. This is joint work with V. Apostolov.

Olivier Biquard: Lectures

September 12, 2023
TITLE: Limits of Kähler-Einstein metrics with cone singularities, and Calabi-Yau metrics

ABSTRACT: There exist various constructions of open Calabi-Yau metrics (Kähler Ricci flat metrics on quasiprojective varieties). There are general questions about obtaining them as limits of Kähler-Einstein metrics with cone singularities on compactifications. I will discuss several cases, in particular the case of the Tian-Yau metric on the complement of an anticanonical divisor in a Fano manifold. Joint work with Henri Guenancia.