UC Santa Barbara

M-theory and the heterotic string are two dual ways to obtain low-energy effective theories that may be engineered to reproduce the observed particles and forces of our universe. To achieve four-dimensional effective theories, one must compactify a number of extra spatial dimensions. In this dissertation, we study the particle spectra and dualities of the E8xE8 heterotic string and M-theory compactified on special holonomy orbifolds and local models built via torus fibrations.

After an introduction to string theory and special holonomy in Chapter 1, we investigate in Chapter 2 the way in which the heterotic gauge bundle mirrors effects from the M-theory geometry. By fibering the duality between the E8xE8 heterotic string on a three-torus and M-theory on K3, we study heterotic duals of M-theory compactified on toroidal G2 orbifolds. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the "gauge-locking" phenomenon seen in Horava-Witten compactifications. While the instanton configuration looks strange from the perspective of the E8xE8 heterotic string, it may be understood as T-dual SO(32) instantons along with winding shifts originating in a dual Type I compactification.

In Chapter 3, we consider three-dimensional local models resulting from a reduction of the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the torus directions, the Hermitian Yang-Mills conditions can be reinterpreted as a 3D complex flat connection satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the alpha prime-corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are new local solutions to the Hull-Strominger system. We also propose a method for computing the spectrum of certain non-abelian models, in close analogy with the Morse-Witten complex of the abelian models.