Preprints with Roger Bielawski, Hypertoric varieties, W-Hilbert schemes, and Coulomb branches, 2023. Abstract: We study transverse equivariant Hilbert schemes of W-invariant affine hypertoric varieties. In particular, we show that the Coulomb branches of Braverman, Finkelberg, and Nakajima can be obtained either as such Hilbert schemes or Hamiltonian reductions thereof. We also investigate the putative complete hyperkähler metrics on these objects. We show that they can be described as natural L2-metrics on moduli spaces of solutions to Nahm's equations on a star-shaped graph with poles at the outer ends of the edges and with the matching of solutions at the central vertex described by a hyperspherical variety canonically associated to a hypertoric variety. with Calum Ross, Calorons and constituent monopoles, 2022. Abstract: We study anti-self-dual Yang-Mills instantons on R^3 x S^1, also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge 1 monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group. with Roger Bielawski, Deformations of hyperkähler cones, 2020. Abstract: We use twistor methods to promote Namikawa's universal Poisson deformations of conic affine symplectic singularities to families of hyperkähler structures deforming hyperkähler cone metrics. The metrics we produce are generally incomplete, but for specific classes of hyperkähler cones, even these incomplete metrics have some interest and applications: we study in detail the case of the nilpotent cone of a simple complex Lie algebra, with applications to hyperkähler metrics with symmetries and hyperkähler quotients, and the case of Kleinian singularities, with applications to codimension-4 singularities of G2-holonomy metrics and their dual description in theoretical physics in terms of 3-dimensional gauge theory. Published paperswith Mark Haskins and Johannes Nordström, Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds, Duke Math. J. 170 (2021), no. 15, 3323-3416. Abstract: We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics, and are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperkähler geometry. We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known. with Bobby Acharya, Marwan Najjar and Eirik Svanes, New G2 conifolds in M-theory and their field theory interpretation, J. High Energ. Phys. 2021, 250 (2021). Abstract: A recent theorem of Foscolo–Haskins–Nordström which constructs complete G2-holonomy orbifolds from circle bundles over Calabi–Yau cones can be utilised to construct and investigate a large class of generalisations of the M-theory flop transition. We see that in many cases a UV perturbative gauge theory appears to have an infrared dual described by a smooth G2-holonomy background in M-theory. Various physical checks of this proposal are carried out affirmatively. with Mark Haskins and Johannes Nordström, Infinitely many new families of complete cohomogeneity one G2-manifolds: G2 analogues of the Taub-NUT and Eguchi-Hanson spaces, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 7, 2153-2220. Abstract: We construct infinitely many new 1-parameter families of simply connected complete non-compact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asyptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics sollapsed to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G2-manifolds. We also construct a closely related conically singular G2 holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G2-cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G2 analogue of the Taub-NUT metric in 4-dimensional hyperkähler geometry and that our new AC G2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperkähler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one. Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds, Geometry & Topology 25-1 (2021), 339-408. Abstract: We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold. ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface, Journal of Diff. Geom. 112 (2019), no 1, 79-120. Abstract: We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) hyperkähler metric arising from the Gibbons-Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperkähler metrics on the K3 surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric. Deformation theory of nearly Kähler manifolds, Journal of the London Math. Soc. 95 (2017), no 2, 586-612. Abstract: Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly Kähler structure on the flag manifold are all obstructed to second order. with Mark Haskins, New G2-holonomy cones and exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres, Annals of Mathematics 185 (2017), no 1, 59-130. Abstract: There is a rich theory of so-called (strict) nearly Kähler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kähler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6-manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions. A gluing construction for periodic monopoles, Int. Math. Res. Not. 2017, no. 24, 7504-7550. Abstract: Cherkis and Kapustin introduced the study of periodic monopoles (with singularities), that is, monopoles on R^2 x S^1 possibly singular at a finite collection of points. Four-dimensional moduli spaces of periodic monopoles with singularities are expected to provide examples of gravitational instantons, that is, complete hyperkähler four-manifolds with finite energy. In a previous paper we proved that the moduli space of charge k periodic monopoles with n singularities is either empty or generically a smooth hyperkähler manifold of dimension 4(k-1). In this paper we settle the existence question, constructing periodic monopoles (with singularities) by gluing methods. Deformation theory of periodic monopoles (with singularities), Comm. Math. Phys. 341 (2016), no. 1, 351–390. Abstract: We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on R^2 x S^1 possibly singular at a finite collection of points, are either empty or smooth hyperkähler manifolds. Furthermore, we prove an index theorem and therefore compute the dimension of the moduli spaces. Survey papersGravitational instantons and degenerations of Ricci-flat metrics on the K3 surface, in Lectures and Surveys on G2 manifolds and related topics, Fields Institute Communications, 84. Springer, New York, 2020. Abstract: The study of degenerations of metrics with special holonomy is an important theme unifying the study of convergence of Einstein metrics, the study of complete non-compact manifolds with special holonomy and the construction of spaces with special holonomy by singular perturbation methods. We survey three constructions of degenerating sequences of hyperkähler metrics on the (smooth 4-manifold underlying a complex) K3 surface — the classical Kummer construction, Gross–Wilson's work on collapse along the fibres of an elliptic fibration, and the author's construction of sequences collapsing to a 3-dimensional limit — describing how they fit into the general theory and highlighting the role played in each construction by gravitational instantons, i.e. complete non-compact hyperkähler 4-manifolds with decaying curvature at infinity. |