- Organized by
Robert Bryant, Ph.D.Duke University
The Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics held its second Annual Meeting at the Simons Foundation on September 13 & 14, 2018. Four of the collaboration’s principal investigators, together with postdoctoral fellows and guests, presented reports on the most recent developments in various aspects of the field of special holonomy, including the study of adiabatic limits, moduli problems, collapse, gluing constructions using methods from algebraic geometry, and connections with physics. They discussed research progress during the second year of the collaboration and the current directions of research.
Extra twisted connected sums and their 𝑣-invariants
Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy G₂ (i.e., G₂-manifolds). We would like to use this wealth of examples to guess further properties of G₂-manifolds and to find obstructions against holonomy G₂, taking into account the underlying topological G₂-structures.
The Crowley-Nordström v-invariant distinguishes topological G₂-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the v-invariant can assume all of its 48 possible values. This shows that G₂-bordism presents no obstruction against holonomy G₂. We also exhibit examples of 7-manifolds with disconnected G₂-moduli space. Our computation of the v-invariants involves integration of the Bismuth-Cheeger η-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind η-function.
A Ringel-Hall type construction of vertex algebras
‘Vertex algebras’ are complicated algebraic structures coming from physics, which arise in 2D conformal field theory and string theory, and also play an important role in mathematics, in areas such as monstrous moonshine and geometric Langlands. I will explain a new geometric construction of vertex algebras, which seems to be unknown. I discovered it by accident, while working on wall-crossing formulae for Donaldson-Thomas type invariants of Calabi-Yau 4-folds. The construction applies in many situations in algebraic geometry, differential geometry and representation theory, and produces vast numbers of new examples. It is also easy to generalize the construction in several ways to produce different types of vertex algebra, quantum vertex algebras and representations of vertex algebras. The construction seems to be closely related to, and is maybe the “correct” explanation for, a large body of work started by Grojnowski, Nakajima and others, which produces representations of interesting infinite-dimensional Lie algebras on the homology of moduli schemes, such as Hilbert schemes. Suppose A is a nice abelian category (such as coherent sheaves coh (X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as Dbcoh (X) or Dbmod − CQ ) over C . Let M be the moduli stack of objects in A or T, as an Artin stack or higher stack. Consider the homology H*(M) over some ring R. Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H* (Mpl) of a projective linear version Mpl of the moduli stack M. For example, if we take T = Dbmod − CQ, the vertex algebra H*(M) is the lattice vertex algebra attached to the dimension vector lattice Z Q₀ of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra. There is also a differential-geometric version: if X is a compact manifold equipped with an elliptic complex E (such as the de Rham complex or the Dirac operator), and M is the moduli stack (as a topological stack) of either all unitary connections on complex vector bundles on X , or all unitary connections on X satisfying a curvature condition depending on E (e.g., instantons on 4-manifolds, Hermitian-Einstein connections on Kahler manifolds, G₂-instantons or Spin(7)-instantons), then we can define a vertex algebra structure on H*(M). This should be part of the big picture into which other work in this collaboration on G₂-instantons and the Donaldson-Segal program fits. There must be a physical explanation for these vertex algebras, but so far, string theorists have not been able to give me one.
Supersymmetry, Special Holonomy, Ricci Flatness and the String Landscape
The main reason that physicists have been interested in Ricci flat spaces with special holonomy is supersymmetry. These spaces then lead to supersymmetric physical models defined on Minkowski spaces, primarily because the manifolds admit parallel spinors with respect to the Levi-Cevita connection. The rich relation between physics and mathematics has led to many significant results like mirror symmetry, Gromov-Witten theory, Donaldson-Thomas invariants and Gopakumar-Vafa, which leads one to suspect that special holonomy, Ricci flat manifolds are rather special objects. I will discuss the possibility that, perhaps all compact, simply connected, stable Ricci flat manifolds have special holonomy (i.e., the conjecture that could bear the slogan “supersymmetry equals Ricci flat equals special holonomy”).
I survey some of what is known about this question and discuss obstructions to the existence of Ricci flat metrics with generic holonomy, presenting some classes/constructions of manifolds which cannot admit a Ricci flat metric with generic holonomy. In dimension four, one of the main open questions is: Does S² ✕ S² admit a stable Ricci flat metric? I finish by discussing an instability in string theory that occurs for non-supersymmetric Ricci flat manifolds with finite fundamental group without parallel spinors (e.g. the Enriques surface) and suggest a connection to Witten’s ‘bubble of nothing.’
If the conjecture were true, it would strongly suggest that the only consistent theories of gravity in Minkowski spacetime are supersymmetric theories.
Collapsing of hyperkahler metrics on K3 surfaces
Hyperkahler metrics on K3 surfaces are prototypical examples of compact Ricci-flat metrics with special holonomy. I will explain some known results in this field and describe a new gluing construction, joint with Hans-Joachim Hein, Jeff Viaclovsky and Ruobing Zhang, of a family of hyperkahler metrics on K3 surfaces with multiscale collapsing phenomenon.
Gauge Theories and Associatives
I will discuss recent developments of gauge theories reduced on associative cycles in G₂-holonomy manifolds. The main focus will be on M5-branes or more generally 6d (2,0) superconformal theories with ADE gauge group on associative three-cycles M₃ in G₂ manifolds. The resulting theories preserve 3d N=1, i.e. minimal, supersymmetry. I will explain a generalization of the 3d-3d correspondence to such N=1 supersymmetric theories and determine the `dual’ topologial theories that compute the S³ partition function and Witten index of the 3d N=1 theory, respectively, to be real Chern-Simons theory and a generalized BF-model, respectively. The BPS equations for the latter are a set of generalized 3d Seiberg-Witten equations.
This is work in collaboration with Julius Eckhard (Oxford), Jin-Mann Wong (KIPMU), as well as work in progress with J.Eckhard and Heeyeon Kim (Oxford).
The 𝐺₂ Laplacian flow
The G₂ Laplacian flow was introduced by Bryant as a potential tool for studying the challenging problem of existence of holonomy G₂ metrics. I will introduce the flow, provide a brief survey of the general theory and describe some recent progress.
Infinitely many new families of complete cohomogeneity one 𝐺₂-manifolds
I will present joint work with Mark Haskins and Johannes Nordström on cohomogeneity one G₂ metrics, that is, G₂ 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 G₂-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 G₂ holonomy space with an isolated conical singularity in its interior and ALC geometry at infinity. Our infinitely many new simply connected AC G₂ manifolds are particularly noteworthy: only the three classic examples constructed by Bryant and Salamon in 1989 were previously known.
Asymptotic properties of toric 𝐺₂ manifolds
A toric G₂ manifold is a 7-manifold M equipped with a torsion-free G₂ structure, which is invariant under the action of a 3-torus T in such a way that there exist multi-moment maps associated to the G₂ 3-form and its Hodge dual. These are introduced and studied in a recent paper by Madsen and Swann, where they show that these multi-moment maps induce a local homeomorphism from the space of orbits M/T into R4. In other words, the multi-moment maps provide geometrically motivated local coordinates for M/T. In all of the known examples, this local homeomorphism is a global homeomorphism onto R4. I will describe some partial results toward showing that this is true in general.