Axions and the strong $\mathrm{CP}$ problem in $M$ theory

We examine the possibility that the strong $\mathrm{CP}$ problem is solved by string-theoretic axions in the strong-coupling limit of the ${\mathrm{E}}_8\times {\mathrm{E}}_8^{\prime }$ heterotic string theory $(M$ theory). We first discuss some generic features of gauge kinetic functions in compactified $M$ theory, and examine in detail the axion potential induced by the explicit breakings other than the QCD anomaly of the nonlinear $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{\mathrm{PQ}}$ symmetries of string-theoretic axions. It is argued based on supersymmetry and discrete gauge symmetries that if the compactification radius is large enough, there can be a $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{\mathrm{PQ}}$ symmetry whose breaking other than the QCD anomaly, whatever its microscopic origin is, is suppressed enough for the axion mechanism to work. Phenomenological viability of such a large radius crucially depends upon the quantized coefficients in gauge kinetic functions. We note that the large radius required for the axion mechanism is viable only in a limited class of models. For instance, for compactifications on a smooth Calabi-Yau manifold with a vanishing ${\mathrm{E}}_8^{\prime }$ field strength, it is viable only when the quantized flux of the antisymmetric tensor field in $M$ theory has a minimal nonzero value. It is also stressed that this large compactification radius allows the QCD axion in $M$ theory to be cosmologically viable in the presence of a late time entropy production.