The study of the geometry of higher dimensional spaces, important in many applications, both theoretical and applied, has led to an understanding of their properties in terms of *holonomy*, a way of describing global effects of curvature. Those geometric spaces with ‘special’ (i.e., reduced) holonomy have come to play a fundamental role in partial differential equations, algebraic geometry, calculus of variations, topology, and theoretical physics, often revealing connections between these subjects that are yielding new insights in both mathematics and physics.
The 2022 annual meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics will highlight and explain the progress in our knowledge of spaces with special holonomy that has been made in recent years by the Collaboration, focusing on the construction of new examples and the definition of geometric invariants that distinguish such structures when they are defined on the same underlying manifold; the increasing role of solitons and limits of Ricci-flat spaces; moduli spaces and BPS invariants; and connections with physics, particularly the new areas of higher form symmetries. We will also describe goals of our continuing research program as well as the challenges that lie ahead.
Lorenzo Foscolo Complete non-compact manifolds with holonomy G2 and ALC asymptotics
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.
Max Hübner Higher Symmetries via Cutting and Gluing of Orbifolds
We study the higher symmetry structures of 4d, 5d, 7d quantum field theories (QFTs) which are geometrically engineered in M-theory by non-compact geometries with non-compact ADE loci. We characterize flavor and gauge Wilson lines of such QFTs via orbifold homology groups of the asymptotic boundary geometry and argue that the corresponding 0-form, 1-form and 2-group symmetries are mapped onto the Mayer-Vietoris sequence with respect to a covering derived from the asymptotic orbifold locus and its complement. Further, applying related cutting and gluing constructions to various compact geometries with ADE loci we discuss the global structure of the engineered supergravity gauge group.
Dominic Joyce Counting semistable coherent sheaves on surfaces
In arXiv:2111.04694 I set out a program which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers. I will outline the general programme briefly, and go into detail on my current project, which uses the results to compute invariants counting semistable vector bundles and coherent sheaves on complex projective surfaces.
Invariants of this kind have a long history, notably in Donaldson theory of 4-manifolds, Vafa-Witten invariants, and so on, and are the subject of many conjectures in mathematics and String Theory. It is common to combine the invariants in formal power series — “generating functions” — which may have interesting number-theoretic properties such as rationality or modularity. My results explain a lot about the general structure of such generating functions, resolving some conjectures in the literature, and I hope in future to make progress on deeper questions such as Vafa-Witten modularity as well.
Fabian Lehmann An embedding problem for closed 3-forms on 5-manifolds
The real part of a holomorphic volume form restricts to a closed 3-form on a 5-dimensional submanifold of a complex 3-fold with a Calabi-Yau structure. Vice versa, in analogy with the embedding problem for abstract CR-structures, this leads to the question which closed 3-forms on a given 5-manifold can be realised by an embedding into a Calabi-Yau 3-fold. We describe the structure induced on the 5-manifold by a closed 3-form and introduce a convexity notion. The main result is that in the “strongly pseudoconvex” case the embedding problem can be solved perturbatively if a finite dimensional vector space of obstructions vanishes. The proof uses the theory of sub-elliptic operators and the Nash-Moser inverse function theorem. The main example is the standard embedding of S^5 in C^3, for which the obstruction space vanishes. This is joint work with Simon Donaldson.
Johannes Nordström Invariants of twisted connected sum G₂-manifolds
When trying to apply the h-cobordism theorem to prove that two manifolds are diffeomorphic, obstructions can often be captured in terms of invariants that measure the failure of a manifold with boundary to satisfy relations that hold for manifolds without boundary. I will discuss invariants of this kind for 7-manifolds with G₂-structures or stable complex structures, and recent progress with Crowley and Goette on applying these to holonomy G₂-manifolds obtained by variations of the twisted connected sum construction.
Song Sun Gravitational instantons and del Pezzo surfaces
I will talk about the classification of gravitational instantons of type ALH* in terms of weak del Pezzo surfaces. This is based on joint work with Hein, Viaclovsky, and Zhang, arXiv 2111.09287.
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