Open Access Senior Thesis
Bachelor of Science
We determine several families of so-called associative 3-dimensional manifolds in R7. Such manifolds are of interest because associative 3-cycles in G2 holonomy manifolds such as R6 × S1, whose universal cover is R7, are candidates for representations of fundamental particles in String Theory. We apply the classic results of Harvey and Lawson to find 3-manifolds which are graphs of functions f : Im H → H and which are invariant under a particular 1-parameter subgroup of G2, the automorphism group of the Cayley numbers, O. Systems of PDEs are derived and solved, some special cases of a classic theorem of Harvey and Lawson are investigated, and theorems aiding in the classification of all such manifolds described here are proven. It is found that in most of the cases examined, the resulting manifold must be of the form of the graph of a holomorphic function crossed with R. However, some examples of other types of graphs are also found.
Weiner, Ian, "Super-Symmetric Three-Cycles in String Theory" (2001). HMC Senior Theses. 138.