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Lorenzo Foscolo: Lectures

March 15, 2023
TITLE: Hypertoric varieties, W-Hilbert schemes and Coulomb branches

ABSTRACT: In an influential paper motivated by IIA/M-theory duality, Sen discussed a construction of ALF spaces of dihedral type as a superposition of Z2-invariant multi-Taub-NUT metrics and the Atiyah-Hitchin metric. In this talk I will discuss joint work with R. Bielawski about a higher dimensional version of this story. We study transverse equivariant Hilbert schemes of hypertoric varieties invariant under the action of a Weyl group W. In particular, we show that Coulomb branches of 3-dimensional N=4 supersymmetric gauge theories, recently mathematically defined by Braverman, Finkelberg and Nakajima as holomorphic symplectic varieties, can be obtained as (symplectic quotients of) such Hilbert schemes. We also investigate the conjectural hyperkähler metric on these spaces in terms of twistor theory and Nahm’s equations.

September 8, 2022
TITLE: Complete non-compact manifolds with holonomy G2 and ALC asymptotics

ABSTRACT: G2 manifolds are the Ricci-flat 7-manifolds with holonomy G2. Until recently there was
only a handful of known examples of complete non-compact G2 manifolds, all highly symmetric and arising from explicit solutions to ODE systems. In joint work with Haskins and Nordström, we produced infinitely many G2 manifolds on total spaces of principal circle bundles over asymptotically conical Calabi-Yau manifolds. The asymptotic geometry of the G2 metrics we produced is analogous to the geometry of 4-dimensional ALF (asymptotically locally flat) spaces and has been labelled ALC (asymptotically locally conical) in the physics literature. In this talk, I will discuss some further joint work on this class of manifolds, in particular consequences of the good deformation theory of ALC G2 manifolds and the construction of new examples with a slightly more complicated ALC asymptotic geometry analogous to the well-known Atiyah-Hitchin metric in 4-dimensional hyperkähler geometry.

Slides of Lecture

January 14, 2022
TITLE: Examples of non-compact G2 manifolds

ABSTRACT: In past Collaboration meetings recent constructions of complete non-compact manifolds with G2 holonomy and AC (asymptotically conical) or ALC (asymptotically locally conical) asymptotic geometry, obtained in collaboration with Haskins and Nordström, have already been presented. The infinitely many new non-compact G2 manifolds (and related singular spaces) obtained from these constructions can be used in physics to produce 4-dimensional field theories with N=1 supersymmetries. In this talk I will discuss the properties of known examples that I understand to be most relevant to physical applications. These include the topology of the manifolds and their ends, the description of their L2 integrable harmonic forms, the existence of different AC G2-manifolds asymptotic to the same G2 cone, and similarities/differences between ALC and AC asymptotics. The examples I will discuss have been obtained in collaboration with Haskins and Nordström and with Acharya-Najjar-Svanes.

Slides of Lecture

October 28, 2020
TITLE: QALF hyperkähler spaces

ABSTRACT: I will report on work in progress with Roger Bielawski on the construction of non-compact hyperkähler metrics generalising to higher dimensions the geometry of ALF spaces. We reinterpret Sen’s construction of Dk ALF spaces as a superposition of Taub-NUT and Atiyah-Hitchin spaces in terms of twistor theory. Generalising this construction to higher dimensions, we define a holomorphic symplectic manifold and the twistor space of a conjectural QALF metric starting from a hypertoric manifold with an action of a Weyl group. In the simplest case where the hypertoric manifold is flat, I will justify the claim that the construction produces QALF metrics by relating the twistor space to Nahm’s equations. For special choices of data, the spaces we define are closely related to Coulomb branches of 3d N=4 SYM theories.
Slides of Lecture

September 14, 2018
TITLE: Infinitely many new families of complete cohomogeneity one G2-manifolds

ABSTRACT: I will present joint work with Mark Haskins and Johannes Nordström on cohomogeneity one G_2 metrics, that is, G_2 holonomy metrics acted upon by a group of isometries with generic orbits of codimension one. We construct infinitely many new one-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. At a special parameter value, the nature of the asymptotic geometry changes, and we obtain a unique member of each family with asymptotically conical (AC) geometry. On approach to a second special parameter value, the family of metrics collapses to an AC Calabi-Yau 3-fold. We also construct a closely related singular G_2 holonomy space with an isolated conical singularity in its interior and ALC geometry at infinity. Our infinitely many new simply connected AC G_2 manifolds are particularly noteworthy: only the three classic examples constructed by Bryant and Salamon in 1989 were previously known.

Slides of lecture

April 11, 2018
TITLE: Examples of collapsing 4-dimensional hyperkähler metrics

ABSTRACT: This talk is complementary to Ruobing Zhang’s lectures and Jeff Viaclovsky’s talk. I will survey older constructions of collapsing sequences of hyperkähler metrics on K3 surfaces, focusing in particular on Gross-Wilson’s study of Kähler Ricci-flat metrics on elliptic K3 surfaces and on my work on collapse to a 3-dimensional limit.

September 11, 2017
TITLE: ALC manifolds with special holonomy

ABSTRACT: At the meeting in January, I described a new analytic construction, obtained in joint work with Mark Haskins and Johannes Nordström, of complete non-compact G_2-manifolds. The construction starts with an asymptotically conical (AC) Calabi-Yau 3-fold B and produces a 1-parameter family of complete G_2-metrics on a suitable circle bundle over B. These G_2-metrics have asymptotically locally conical (ALC) geometry, the natural generalisation of the asymptotic geometry of ALF hyperkähler 4-manifolds to higher dimensions. In this talk I will describe some initial steps in two different directions related to that result. Firstly, I will show how in our construction of ALC G_2-manifolds rigid compact holomorphic curves in the AC Calabi-Yau 3-fold lift to rigid associative submanifolds in the ALC G_2-manifold. Secondly, I will describe the analogous construction of ALC Spin(7)-holonomy metrics on suitable circle bundles over AC G_2-manifolds.

January 10, 2017 (jointly with Mark Haskins)
TITLE: Non-compact G2 manifolds collapsing to asymptotically conical Calabi-Yau 3-folds

ABSTRACT:We will present a new analytic construction of complete non-compact G_2 holonomy metrics, that yields infinitely many families of examples. The underlying 7-manifolds are all circle bundles over asymptotically conical (AC) Calabi-Yau 3-folds endowed with circle-invariant metrics. Their geometry at infinity is that of a circle bundle over a Calabi-Yau cone with circle fibres of fixed finite length. The G_2 manifolds we construct are therefore 7-dimensional analogues of 4-dimensional ALF hyperkähler metrics. Physicists have termed metrics with such asymptotics ALC (asymptotically locally conical). Within the Melrose school, ALC metrics form a special subclass of so-called fibred boundary metrics.

The dimensional reduction of the equations for G_2 holonomy in the presence of a Killing field was considered by Apostolov-Salamon and by several groups of physicists. We reinterpret the dimensionally-reduced equations in terms of a pair consisting of an SU(3) structure (with tightly constrained torsion) on the 6-dimensional orbit space coupled to an abelian Calabi-Yau monopole on this 6-manifold. We solve this nonlinear coupled system of PDEs by considering the geometric limit in which the circle fibres of the associated circle-invariant G_2 holonomy metrics collapse. In this collapsed limit, given the asymptotically conical Calabi-Yau 3-fold to which the 7-dimensional metrics collapse, the problem comes close to linearising and the crux of the matter is therefore to develop a sufficiently good understanding of solutions to these linearised equations in suitable spaces of functions. For this we need the well-known Fredholm theory for elliptic operators acting on weighted Sobolev spaces on asymptotically conical Calabi-Yau 3-folds and an extension of this theory to the setting of weighted Sobolev spaces on ALC manifolds.

The G_2 holonomy metrics we construct should be thought of as arising from (a suitable perturbation of) abelian Hermitian-Yang-Mills connections on AC Calabi-Yau 3-folds, especially those that arise as crepant resolutions of Calabi-Yau cones. From the physics viewpoint our solutions correspond to M theory uplifts of type IIA solutions in the absence of D6-branes. The existence of such a rich spectrum of “no brane” solutions does not seem to have been anticipated on the physics side, and is a new feature of G_2 holonomy metrics compared to the 4-dimensional hyperkähler setting. In particular all our examples provide instances of families of G_2 holonomy metrics that collapse with bounded curvature to Calabi-Yau 3-folds.

Time permitting we will outline extensions of the construction described here and how it fits into a broader programme.

The talk is closely related to the talks by Carron, Collins and Acharya and is based on joint work with Mark Haskins and Johannes Nordström.


September 8, 2016
TITLE: ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ABSTRACT: The Kummer construction of Kähler Ricci-flat metrics on the (smooth 4-manifold underlying a complex) K3 surface provides the prototypical example of the formation of orbifold singularities in non-collapsing sequences of Einstein 4-manifolds. Much less is known about the structure of the singularities forming along sequences of collapsing Einstein metrics.

I will describe the construction of large families of Ricci-flat metrics on the K3 surface collapsing to the quotient of a flat 3-torus by an involution. The collapse occurs with bounded curvature away from finitely many points. The geometry around these points is modelled by ALF gravitational instantons (complete non-compact hyperkähler 4-manifolds with decaying curvature and cubic volume growth).

This 4-dimensional construction serves as a good warm-up for the much more involved case of collapsing families of G_2 holonomy metrics described in the talk of Mark Haskins.

Slides of lecture