Abstract
M-theory compactified on a ${G}_{2}$ manifold with resolved ${E}_{8}$ singularities realizes four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories coupled to gravity with three families of Standard Model fermions. Beginning with one ${E}_{8}$ singularity, three fermion families emerge when ${E}_{8}$ is broken by geometric engineering deformations to a smaller subgroup with equal rank. In this paper, we use the local geometry of the theory to explain the origin of the three families and their mass hierarchy. We linearize the blowing up of two-cycles associated with resolving ${E}_{8}$ singularities. After imposing explicit constraints on the effectively stabilized moduli, we arrive at Yukawa couplings for the quarks and leptons. We fit the high scale Yukawa couplings approximately which results in the quark masses agreeing reasonably well with the observations, implying that the experimental hierarchy of the masses is achievable within this framework. The hierarchy separation of the top quark from the charm and up is a stringy effect, while the spitting of the charm and up also depends on the Higgs sector. The Higgs sector cannot be reduced to having a single vacuum expectation value (VEV); all three VEVs must be nonzero. Three extra $U(1)\text{ }\text{ }\mathrm{s}$ survive to the low scale but are not massless, so $Z$ states are motivated to occur in the spectrum, but may be massive.
Highlights
M-theory has been met with considerable success [1,2,3,4]
We fit the high scale Yukawa couplings approximately which results in the quark masses agreeing reasonably well with the observations, implying that the experimental hierarchy of the masses is achievable within this framework
In this paper we focus on an M-theory calculation of the quark and charged lepton masses
Summary
M-theory has been met with considerable success [1,2,3,4]. One prediction of compactified M-theory is the existence of N 1⁄4 1 supersymmetry and its soft breaking via gluino condensation, while simultaneously stabilizing all moduli [4,5]. Suppose that the local model of X with ADE singularity is of the form C2=Γ × R3, where Γ is a finite subgroup of SUð2Þ (see Table 2) Under these circumstances, a super Yang-Mills N 1⁄4 1 multiplet with gauge group G 1⁄4 SUðkÞ; SOð2kÞ; E6; E7, and E8, respectively, 1ADE stands for A, D, and E Lie algebra. These singularities can be deformed to break the symmetry of the gauge group G to a subgroup of G with equal rank. As that would include an explicit method for computing matter content, in gauge symmetry breaking through deformation, and their coupling constants, the results would be applicable to a wider study of other matter interaction.
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