Unification Scale, Proton Decay, And Manifolds Of G_2 Holonomy

TL;DR

The work analyzes M-theory compactifications on $G_2$ manifolds and shows that gauge coupling threshold corrections for grand unification are governed by the Ray-Singer analytic torsion, enabling explicit results in simple topologies such as lens spaces. It derives precise relations between the four-dimensional unification data ($M_{GUT}$, $\alpha_{GUT}$) and the compactification data ($V_Q$, $L(Q)$), and connects these to Newton's constant via eleven-dimensional parameters, with $M_{GUT}$ parametrically below the 11D scale by ${\alpha_{GUT}^{1/3}}$. For $SU(5)$ models, the threshold corrections do not modify the prediction for $\sin^2\theta_W$ but shift the unification scale, and lens-space examples yield explicit $M_{GUT}-V_Q$ relations; the analysis also discusses how light matter multiplets and Higgs triplets modify beta functions and proton-decay channels. The proton decay amplitude receives a novel M-theory–driven contribution that can enhance certain modes (notably $p\to \pi^0 e^+_L$) relative to four-dimensional GUTs, though exact rates depend on uncertain constants and field localization. Overall, the paper links topological data of $G_2$ compactifications to phenomenological observables, highlighting the roles of membrane instantons, light multiplets, and SUSY breaking in realizing realistic physics.

Abstract

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 $α_{GUT}$, 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 $p\toπ^0e^+_L$ compared to other modes such as $p\to π^0e^+_R$ or $p\to π^+\barν_R$. Because of numerical uncertainties, we do not know whether to intepret this as an enhancement of the $p\to π^0e^+_L$ mode or a suppression of the others.