# Exceptional algebra

Calabi–Yau manifolds have arisen as non-trivial solutions to Einstein's theory of gravity. As a detailed description of these bizarre multi-dimensional spaces is not available, EU-funded scientists have constructed a vast number of distinct manifolds.

Over the three-year lifetime of the 'Aspects of G2 geometry' (G2 GEOMETRY) project, numerous algebraic techniques have been successfully used in the construction of Calabi–Yau three-folds. In three complex dimensions, six dimensions of the 10D universe that string theory says we live in were curled up. To our senses, only four are accessible — just like Earth looks flat on the small scales that we see. In the Calabi–Yau three-folds, six dimensions were hidden so that the resulting 4D theory would keep some amount of supersymmetry. Supersymmetry is one of the best candidates for physics beyond the Standard Model, the theory describing electromagnetic, weak and strong nuclear interactions. Starting with these Calabi–Yau three-folds as building blocks, the G2 GEOMETRY scientists went on to construct many new G2 manifolds. G2 manifolds are models of the extra dimensions of M-theory, going beyond supersymmetry. Specifically, the M-theory contains gravity and is supersymmetric as well as quantum mechanically consistent. G2 GEOMETRY mathematicians constructed compact G2 manifolds from twisted connected sums of Calabi–Yau manifolds. Having generalised Kovalev's methodology, they were able to compute different metrics on the G2 manifolds built this way. In these G2 manifolds, compact sub-manifolds were constructed with the use of rigid curves. Although G2 manifolds can be interconnected in many ways, these are the first examples of rigid sub-manifolds connecting different 7D spaces. While many properties of the G2 manifolds have been discovered within the G2 GEOMETRY project, more questions on how to differentiate between categories of manifolds have been raised and probed. Much of the research in the G2 GEOMETRY project was inspired by questions arising in theoretical physics. The progress in differential geometry achieved should, on the other hand, further our understanding of the fundamental theory behind superstring and M-theories.

## Keywords

Algebra, Calabi–Yau, manifolds, multi-dimensional spaces, G2 geometry, geometry, supersymmetry, Standard Model, G2 manifolds, M-theory