Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics First Annual Meeting

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Spaces with special holonomy are of intrinsic interest in both mathematics and mathematical physics; they appear in many contexts in Riemannian geometry, particularly Ricci-flat and Einstein geometry, minimal submanifold theory and the theory of calibrations, and gauge theory.  The exceptional cases, which occur in dimensions 7 and 8, remain the most challenging and the least understood. Nevertheless, they share important features with the better-known case of SU(n) holonomy, where the three types of structures are known as Calabi-Yau spaces, Hermitian Yang–Mills connections, and special Lagrangian and complex submanifolds. The exceptional holonomy spaces play key roles in the study of fundamental physical theories such as M-theory and F-theory (generalizing the role that Calabi-Yau 3-folds play in string theory), and progress in these theories depends crucially on a better understanding of spaces (especially singular ones) with exceptional holonomy.

The Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics held its first Annual Meeting at the Simons Foundation on September 14 & 15, 2017.  Four of its principal investigators and four of its postdoctoral fellows 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 their research progress during the first year of the collaboration and the current directions of research.
 

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We will begin by explaining how “maximal” sub manifolds in spaces of indefinite signature arise as formal collapsing (or adiabatic) limits of G₂ manifolds with co-associative fibrations.  Then we will discuss some analytical problems which arise in developing this idea, mostly having to do with the critical sets where the fibres become singular and the maximal submanifolds have branch points. In one direction we will discuss the deformation theory of these sets and in another we outline the relevance of recent constructions (by Yang Li and others) of new Calabi-Yau metrics on C³.
 

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Riemannian 7-manifolds with holonomy G₂ are a special class of Ricci-flat Riemannian manifolds, which are of interest to physicists working in M-theory. Associative 3-folds are calibrated 3-submanifolds in 7-manifolds with holonomy G₂, so they are a special kind of minimal submanifold.

There is a well-known analogy between G₂ manifolds X in dimension 7 and Calabi-Yau 3-folds Y in dimension 6. Under this analogy one should compare associative 3-folds in X with J-holomorphic curves in Y. Much of symplectic geometry — Gromov-Witten theory, Lagrangian Floer theory, and so on — is concerned with “counting” J-holomorphic curves, to get an answer which is independent of the (almost) complex structure J up to deformation. So we can ask: might there be interesting geometry of G₂ manifolds concerned with “counting” associative 3-folds, which gives an answer unchanged under deformations of the G₂ structure?

This talk, based on arXiv:1610.09836, presents a conjectural answer to this question. It is connected to conjectures of Donaldson and Segal on defining invariants by “counting” G₂ instantons on X with “compensation terms” counting pairs of a G₂ instanton and an associative 3-fold on X. At the end we will briefly discuss a proposed modification to the Donaldson-Segal conjecture, to correct for wall-crossing behaviour of associative 3-folds we discover during our investigation.
 

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I will talk about gauge theory on G₂ manifolds and its relation to the (generalized) Seiberg-Witten equations on three-manifolds. In particular, I will focus on the compactness properties for the corresponding moduli spaces and related problems.
 

In this talk, we study the collapsing of hyperkahler metrics on projective holomorphic symplectic manifolds along  holomorphic Lagrangian fibrations.  We prove that the Gromov-Hausdorff limits are compact metric spaces, which are half-dimensional special Kahler manifolds outside  singular sets of real Hausdorff codimension 2.
 

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Recent years have seen a renaissance in the construction and study of new examples of manifolds with exceptional holonomy, instanton bundles over these spaces and their applications in physics and string theory. Due to anomalies and alpha’ corrections, the bundle often has a non-trivial back-reaction on the base geometry, and it can be important to keep this in mind when studying aspects of the solutions such as the moduli problem. These corrections are of particular importance in the context of the heterotic string, and I will review some recent work that highlights this and discuss the heterotic moduli problem in particular.
 

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In this talk we describe recent developments and ongoing projects by members of the Collaboration related to codimension one collapse of exceptional holonomy metrics. Informally speaking, this is where a family of special holonomy metrics on a space of dimension n converges in some limit to a metric on a space of dimension n-1. Interesting examples occur for hyperkaehler 4-manifolds, G₂ holonomy manifolds and Spin_7 holonomy manifolds. The talk will focus on the G₂ holonomy case, but will also draw on the better understood hyperkaehler case for inspiration and for useful analogies.

These mathematical developments are closely related to important limits in physics, e.g. in the context of G₂ holonomy metrics it is related to the identification of the weak coupling limit of M theory compactified on a G₂ holonomy space being Type IIA String Theory on a 6-dimensional space. Inspiration for our work has already come from previous work of physicists studying M theory, including members of our Collaboration.
 

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String theories on different G₂ manifolds can lead to the same physics in a phenomenon called mirror symmetry. In this talk, I will review mirror symmetry for G₂ manifolds, focusing on recent progress. In particular, I will present constructions of mirror G₂ manifolds realized as twisted connected sums.
 

One route to describing a supersymmetric quantum theory of gravity in four-dimensional spacetime begins with the proposed eleven-dimensional quantum theory of gravity known as M-theory, which is then studied on the Cartesian product of a compact seven-dimensional space and a four-dimensional spacetime.  To obtain a supersymmetric theory of gravity, the compact space should be equipped with a Riemannian metric admitting a covariantly constant spinor; the latter leads to special holonomy. Geometric properties of the compact space determine physical properties of the four-dimensional theory.

Two key physical properties—non-abelian gauge fields and chiral matter—cannot be realized in this setup unless the compact space has singularities.  We will present some work in progress which modifies existing constructions of compact manifolds with holonomy G₂ to include (some of) the relevant singularities.