- Organized by
Robert Bryant, Ph.D.Duke University
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, and those geometric spaces with constrained (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 surprising connections between these subjects that are yielding new insights in both mathematics and physics.
This annual meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics highlighted and explained the fundamental progress in the theory of spaces with special holonomy that has been made in the first three years of the Collaboration and described the goals of our continuing research program as well as the challenges that lie ahead.
An Overview of the Progress and Goals of the Special Holonomy Simons Collaboration
This talk will serve as an introduction to the meeting, including background on the area of special holonomy and an overview of the fundamental existence results, progress made by our collaboration (and others) and what we see as the major goals and challenges in current research in special holonomy.
Some recent progress in the Physics of Special Holonomy Spaces: Dark matter, Gauge theories and The String Landscape
Acharya will survey recent progress in understanding the physics of special holonomy spaces as models for the extra dimensions of space in string/M theory. He will begin by briefly reviewing how special holonomy spaces can aid our understanding of particle physics and cosmology and how recent progress may help shape our picture of the dark sector of the universe. The talk will then go on to apply recent progress on the construction of a class of G2-holonomy spaces as circle bundles over Calabi-Yau’s (Foscolo, Haskins, Nordstrom) to M theory/Type IIA duality and, in particular, to four-dimensional gauge theories. Finally Acharya surveys some recent progress in physics related to the conjecture “stable, compact Ricci flat manifolds have special holonomy” and how it can shape our understanding of the String Landscape.
\(G_2\) Geometry and Adiabatic Limits
The main focus of the talk will be \(G_2\) manifolds with co-associative fibrations and in particular the “adiabatic limit” when the fibers become very small and the structure can be described by solutions of a version of the maximal submanifold equation. Donaldson will discuss progress in the development of this theory and prospects for the future. In particular, he will explain the connection with boundary value problems for \(G_2\) structures and descriptions, in part conjectural, of calibrated submanifolds in the adiabatic limit.
Recent Developments in Special and Exceptional Holonomy Metrics
In this talk, Haskins will give an overview of some of the main recent developments in the construction of metrics with special and exceptional holonomy, concentrating on the progress made by members of the collaboration on resolving some long-standing open questions. He will indicate some of the symbiotic relationships that have developed between the two better understood cases of hyperKaehler 4-manifolds and Calabi-Yau 3-folds, and the more challenging case of 7-manifolds with holonomy \(G_2\).
One important unifying theme has been the more systematic study of special or exceptional holonomy metrics in either highly collapsed regimes or close to a suitable adiabatic limit: codimension one (circle) collapse of \(G_2\) holonomy metrics underpins the physical limit where M theory is modeled by Type IIA string theory; collapsed Calabi-Yau metrics with fibers close to flat tori appear in connection with degenerations of complex structure; K3 fibrations of Calabi-Yau 3-folds or coassociative K3 fibrations of \(G_2\) manifolds with small fiber size also both appear naturally. The latter adiabatic limit will be discussed in more detail in Simon Donaldson’s talk.
Topology in Special Holonomy
Nordstrom will discuss what scant information has been available about obstructions to the existence of \(G_2\)-holonomy metrics on closed \(7\)-manifolds and then go on to explain some recent progress on the topology of known and recently constructed examples. Many of these \(7\)-manifolds belong to classes where diffeomorphism classification results have been proved or are within reach, making it possible to map out a portion of the landscape of closed \(G_2\)-manifolds. In particular, we can exhibit phenomena such as homeomorphic, but not diffeomorphic, examples and the non-connectedness of the moduli space of \(G_2\)-metrics on specific \(7\)-manifolds.
Twistor Spaces and Special Holonomy
Adaptations of Penrose’s twistor theory to the Riemannian setting have played a role in the construction and study of metrics with special holonomy. The original idea was to encode conformal structures by objects from complex and algebraic geometry. After explaining this background, the talk will focus on a circle quotient of a well-known metric with holonomy \(G_2\) that fibers over the 4-sphere, investigated by Atiyah and Witten in the context of M-theory. The Gibbons-Hawking ansatz reduces the \(G_2\) space to R^6 endowed with a singular SU(3) structure invariant by a diagonal action of SO(3). The talk will describe some key features of this reduction (obtained jointly with Acharya and Bryant).
5d SCFTs: Geometry, Graphs and Gauge Theories
5d SCFT are intrinsically strongly coupled. We develop a systematic approach to study 5d SCFTs that descend from 6d SCFTs using resolutions of elliptic Calabi-Yau three-folds with non-minimal singularities. We show how to encode the network of descendant SCFTs using a simple graph-based approach, which encodes many of the salient features of the 5d SCFTs, including the strongly-coupled flavor symmetries, BPS states and gauge theory descriptions (including dualities).
From Gauge Theory to Calibrated Geometry and Back
Twenty years ago, Donaldson and Thomas proposed to define invariants of special holonomy manifolds using the equations of gauge theory. In the first part of Doan’s talk, he will discuss the basic ideas behind this fascinating proposal and survey some recent advances made by the Simons Collaboration. He will then focus on Calabi-Yau manifolds, in particular on a joint project with Thomas Walpuski, whose goal is to define new invariants of Calabi-Yau three-folds using gauge theory and pseudo-holomorphic curves.