Models of particle physics based on manifolds of G 2 holonomy are in most respects much more complicated than other string-derived models, but as we show here they do have one simplification: threshold corrections to grand unification are particularly simple. We compute these corrections, getting completely explicit results in some simple cases. We estimate the relation between Newton's constant, the GUT scale, and the value of α GU T , and explore the implications for proton decay. In the case of proton decay, there is an interesting mechanism which (relative to four-dimensional SUSY GUT's) enhances the gauge boson contribution to p → π 0 e + L compared to other modes such asBecause of numerical uncertainties, we do not know whether to intepret this as an enhancement of the p → π 0 e + L mode or a suppression of the others.e-print archive: http://lanl.arXiv.org/abs/hep-th/0211269

We study the moduli space of M -theories compactified on G 2 manifolds which are asymptotic to a cone over quotients of S 3 × S 3 . We show that the moduli space is composed of several components, each of which interpolates smoothly among various classical limits corresponding to low energy gauge theories with a given number of massless U (1) factors. Each component smoothly interpolates among supersymmetric gauge theories with different gauge groups.

A famous pre-Newtonian formula for π is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.The formula for π as the infinite product (1) that originates in physics, specifically in quantum mechanics. It is the purpose of this paper to show that this formula can in fact be derived from a variational computation of the spectrum of the hydrogen atom. The existence of such a derivation indicates that there are striking connections between well-established physics and pure mathematics [9] that are remarkably beautiful yet still to be discovered.The Schrödinger equation for the hydrogen atom is given by 1

We propose a new schematic model for mesons in which the building blocks are quarks and flavor-antisymmetric diquarks. The outcome is a new classification of the entire meson spectrum into quark-antiquark and diquark-antidiquark states which does not give rise to a radial quantum number: all mesons which have so far been believed to be radially excited are orbitally excited diquark-antidiquark states; similarly, there are no radially excited baryons. Further, mesons that were previously viewed as "exotic" are no longer exotic as they are now naturally integrated into the classification as diquark-antidiquark states. The classification also leads to the introduction of isorons (iso-hadrons), which are analogs of atomic isotopes, and their magic quantum numbers, which are analogs of the magic numbers of the nuclear shell model. The magic quantum numbers of isorons match the quantum numbers expected for low-lying glueballs in lattice QCD. We observe that interquark forces in mesons behave substantially differently from those in baryons: qualitatively, they are color-magnetic in mesons but color-electrostatic in baryons. We comment on potential models and the hydrogen atom. The implications of our results for confinement, asymptotic freedom, and a new set of relations between two fundamental properties of hadrons -their size and their energy -are discussed in our companion paper [1]