Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Second Annual Meeting
 Organized by

Robert Bryant, Ph.D.Duke University
Simons Collaboration on Special Holonomy
in Geometry, Analysis and Physics Home Page
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.
Download the Progress Report (PDF)
Talks
Sebastian Götte
Extra twisted connected sums and their 𝑣invariants
Joyce’s orbifold construction and the twisted connected sums by Kovalev and CortiHaskinsNordströmPacini provide many examples of compact Riemannian 7manifolds with holonomy G₂ (i.e., G₂manifolds). We would like to use this wealth of examples to guess further properties of G₂manifolds and to find obstructions against holonomy G₂, taking into account the underlying topological G₂structures.
The CrowleyNordström vinvariant distinguishes topological G₂structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the vinvariant can assume all of its 48 possible values. This shows that G₂bordism presents no obstruction against holonomy G₂. We also exhibit examples of 7manifolds with disconnected G₂moduli space. Our computation of the vinvariants involves integration of the BismuthCheeger ηforms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind ηfunction.
Dominic Joyce
A RingelHall 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 wallcrossing formulae for DonaldsonThomas type invariants of CalabiYau 4folds. 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 infinitedimensional 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 modCQ of a quiver Q) or T is a nice triangulated category (such as D^{b}coh (X) or D^{b}mod − 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 = D^{b}mod − 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 KacMoody algebra. There is also a differentialgeometric 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 4manifolds, HermitianEinstein connections on Kahler manifolds, G₂instantons 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 G₂instantons and the DonaldsonSegal 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.
Bobby Acharya
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 LeviCevita connection. The rich relation between physics and mathematics has led to many significant results like mirror symmetry, GromovWitten theory, DonaldsonThomas invariants and GopakumarVafa, 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 nonsupersymmetric 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.
Song Sun
Collapsing of hyperkahler metrics on K3 surfaces
Hyperkahler metrics on K3 surfaces are prototypical examples of compact Ricciflat metrics with special holonomy. I will explain some known results in this field and describe a new gluing construction, joint with HansJoachim Hein, Jeff Viaclovsky and Ruobing Zhang, of a family of hyperkahler metrics on K3 surfaces with multiscale collapsing phenomenon.
Sakura SchaferNameki
Gauge Theories and Associatives
I will discuss recent developments of gauge theories reduced on associative cycles in G₂holonomy manifolds. The main focus will be on M5branes or more generally 6d (2,0) superconformal theories with ADE gauge group on associative threecycles M₃ in G₂ manifolds. The resulting theories preserve 3d N=1, i.e. minimal, supersymmetry. I will explain a generalization of the 3d3d 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 ChernSimons theory and a generalized BFmodel, respectively. The BPS equations for the latter are a set of generalized 3d SeibergWitten equations.
This is work in collaboration with Julius Eckhard (Oxford), JinMann Wong (KIPMU), as well as work in progress with J.Eckhard and Heeyeon Kim (Oxford).
Jason Lotay
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.
Lorenzo Foscolo
Infinitely many new families of complete cohomogeneity one 𝐺₂manifolds
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 oneparameter families of simply connected complete noncompact G₂manifolds with controlled geometry at infinity. The generic member of each family has socalled 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 CalabiYau 3fold. 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.
Kael Dixon
Asymptotic properties of toric 𝐺₂ manifolds
A toric G₂ manifold is a 7manifold M equipped with a torsionfree G₂ structure, which is invariant under the action of a 3torus T in such a way that there exist multimoment maps associated to the G₂ 3form and its Hodge dual. These are introduced and studied in a recent paper by Madsen and Swann, where they show that these multimoment maps induce a local homeomorphism from the space of orbits M/T into R^{4}. In other words, the multimoment maps provide geometrically motivated local coordinates for M/T. In all of the known examples, this local homeomorphism is a global homeomorphism onto R^{4}. I will describe some partial results toward showing that this is true in general.