Final Report Summary - G2 GEOMETRY (Aspects of G2 Geometry)
The "generic" oriented Riemannian manifold (M,g) has holonomy SO(n). Berger gave a complete classification of the possible "special" holonomy groups of (simply connected, irreducible, non-symmetric) compact Riemannian manifolds. This list includes the groups U(n), SU(n) and G2.
The U(n) case is classical: it corresponds to Kaehler manifolds. The SU(n) case has by now been extensively studied: it corresponds to Calabi-Yau (CY) manifolds. Roughly speaking, these are the Ricci-flat Kaehler manifolds and it follows from Aubin-Yau's solution to the Calabi conjecture that, up to a perturbation of the metric, they coincide with the class of Kaehler manifolds satisfying an additional topological condition: the first Chern class vanishes. Assuming M is simply connected, when n=2 these are the well-known K3 surfaces. When n is greater than 2, it is known that CY manifolds are projective, thus algebraic. The condition on the first Chern class can be readily studied using Algebraic Geometry, leading to many examples of compact CY manifolds. This gives a fair amount of information on the topology of CY manifolds, though not necessarily on the corresponding metrics which in general cannot be written down explicitly.
Much less is currently known about the G2 case, which is "exceptional" from several points of view.
The first non-compact examples of G2 manifolds are due to Bryant and Salamon in the 1980s; the first compact examples were constructed only in the 1990s by Joyce. How to obtain information on the possible topological and diffeomorphism types of G2 manifolds is an important open question: metrics with holonomy G2 are again Ricci-flat, but there is no analogue of the Calabi conjecture tying their existence to a topological condition.
The topic of special holonomy has close ties to the topic of minimal submanifolds. The link is made explicit via the notion of "calibrated submanifold", i.e. a submanifold whose behaviour is governed by a given "calibrating differential form" on (M,g): a simple argument shows that any calibrated submanifold is minimal, and actually volume-minimizing in its homology class. Again, calibrated submanifolds in the Kaehler case are classical: they coincide with the complex submanifolds.
Calibrated submanifolds in the CY case are known as "special Lagrangian" (SL).
They tend to have very nice geometric properties, e.g. smooth moduli spaces. In the G2 case the calibrated submanifolds are of two types: "associative" 3-folds and "coassociative" 4-folds.
CY geometry has played an important role in the mathematical foundation of String Theory since the 1980s. This accounts for a large part of the interest in CYs and has led to important advances both in Geometry and in Theoretical Physics. In particular, SL fibrations of CY manifolds are a key ingredient of Mirror Symmetry, one of the fundamental open problems in this field. More recently, with the advance of M-Theory, G2 geometry has also started to attract increasing amounts of attention among Physicists.
Even staying within the realm of Geometry, however, it is clear that the study of CY and G2 manifolds, more generally of calibrated geometry, has great potential for further development.
Recent work in this field includes:
(i)A new construction of compact G2 manifolds, due to Kovalev.
(ii) Several new explicit constructions of non-compact G2 metrics, partly prompted by Hitchin's work, which also details analogies between CY and G2 geometry. Also, new constructions of calibrated submanifolds and of various types of "weak" G2 structures, i.e. structures with torsion.
(iii) Links to higher-dimensional gauge theory and conjectural constructions of invariants of CY and G2 manifolds.
(iv) Convergence and collapsing results for Ricci-flat and CY metrics.
Some of the above work is still highly speculative but outlines promising directions of research for the long-term. To this end it is vital to continue building firm foundations and more concrete examples, as well as to continue investigating the general geometric properties of the above categories of manifolds and submanifolds.
Some of the main achievements of this project are:
(i)We construct huge numbers of new CY 3-folds, starting from weak and semi-Fano 3-folds. We then use these as ``building blocks" to consolidate and extend Kovalev's construction of G2 manifolds.
(ii) We obtain detailed topological and geometric information on the G2 manifolds built this way,sometimes determining the precise diffeomorphism type of the manifold.
(iii) We describe "geometric transitions" between G2 metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds.
(iv) We use rigid holomorphic curves in the 3-folds to build rigid compact associative submanifolds in the G2 manifolds. These are the first examples proven to be rigid. We also prove that many smooth 2-connected 7-manifolds can be realised in numerous ways; by varying the building blocks we can vary the number of rigid associative 3-folds constructed therein. We expect these examples will be very useful for constructing and testing G2 invariants based on an enumerative geometry of associative submanifolds.
(v)We develop a very general analytic and geometric framework for desingularizing certain categories of manifolds. We apply these techniques to SL submanifolds, but they are relevant to a much wider class of geometric problems.
(vi) We have started investigating the relationship between calibrations and certain geometric flows, related to Kaehler and G2 geometry, with particular focus on the properties of coupled systems of equations relating the ambient geometry to the geometry of submanifolds.