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Construction of new G2 manifolds using Calabi-Yau 3-folds

Final Activity Report Summary - G2 MANIFOLDS (Construction of new G2 manifolds using Calabi-Yau 3-folds)

Our research project was concerned with finding many new constructions of special geometric objects in seven dimensions, which are of great interest in current theories of physics, specifically in M-theory and supergravity. These objects are called compact G2 manifolds. For the first year of this Marie Curie fellowship the research fellow, Dr Spiros Karigiannis, learned the methods and techniques of geometric analysis, focussing on the analysis of elliptic partial differential equations on manifolds, in order to be able to apply them successfully for the main Marie Curie research project, which is described in the next paragraph. This initial training in analytic techniques resulted in two research papers by the fellow, on topics that were very closely related to the proposed research project, namely the desingularisation and the deformation of G2 manifolds with isolated conical singularities.

Up until now, exactly two construction procedures for compact G2 manifolds were known. The first, developed by Dominic Joyce from 1994 and extended in 2000, started from a seven-torus, and the second was constructed by Alexei Kovalev starting from two six-dimensional complex manifolds in 2000. It was important to find new examples of such objects, both for the physical theory and in order to better understand the set of all such possible objects in a purely mathematical context. We had a very specific method in mind to produce new examples of compact G2 manifolds, involving a ‘surgery’ procedure, where we ‘cut out’ the bad parts of a geometric object which was almost of the type we sought and ‘glued in’ new parts which had nice properties.

We then needed to argue, using methods of mathematical analysis, that the resulting object could be deformed to one of these G2 manifolds that we sought. The essential idea was to prove that on the ‘overlap region’, where we did the gluing, the deviation of our geometric object from being exactly of the type we wanted was very small. If we could show this, then a general theorem proved by Professor Joyce would ensure that our newly constructed geometric object could be perturbed to be exactly one of the special types that we sought. There were two separate subsets of the overlap region in which the above-mentioned deviation was measured in different ways. We had already established a way to perform our surgery construction to ensure that the deviation in one of those two subsets of the overlap region was indeed small enough. What remained was to verify that the same was true for the other subset of the overlap region. If it turned out not to be the case, then there was still hope for the project to succeed, through a slight modification of the manner in which we glued the two pieces together.

The fellow, Dr Karigiannis, resigned 10 months early from the Marie Curie fellowship, in order to accept a tenure-track assistant Professor position in his native country of Canada. Dr Karigiannis and Professor Joyce continued to work together on this difficult project and hoped to finish it in more or less the original timeframe of the fellowship, by later summer 2009.