Unification of gauge couplings in Kaluza-Klein theory with two internal manifolds

Abstract

We consider a Kaluza-Klein theory whose ground state is ${\bf R}^4 \times {\bf M } \times {\bf K}$ where ${\bf M}$ and ${\bf K}$ are compact, irreducible, homogenous internal mani folds. This is the simplest ground state compatible with the existence of the graviton, gauge fields, massless scalar fields and the absence of the cosmological constan t. The requirement for these conditions to be satisfied are the odd dimensionality of ${\bf M}$ and ${\bf K}$, and the choice of a dimensionally continued Euler form action whose dimension is the same as the dimension of ${\bf M} \times {\bf K}$. We show that in such a theory, which is not simple due to presence of two internal manifolds, the gauge couplings $g^2_M$ and $g^2_K$ are actually unified provided that the internal space sizes are constant. For ${\bf M} \times {\bf K} = S^{2m+ 1} \times S^{2k+1}$ this gauge coupling unification relation reads $g^2_M / g^2_K = m / k$.