An invariant operator due to F Klein quantizes H Poincaré's dodecahedral 3-manifold

Abstract

The eigenmodes of the Poincar\'e dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $\pi_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. ($\pi_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\cal H}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M}) \sim {\cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\cal H}^r_3$ as normal subgroup and from a second group ${\cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\cal H}_3$-invariant, we construct a novel hermitian ${\cal H}_3$-invariant polynomial (generalized Casimir) operator ${\cal K}$. Its eigenstates with eigenvalues $\kappa$ quantize a complete orthogonal basis on Poincar\'{e}'s dodecahedral 3-manifold. The eigenstates of lowest degree $\lambda=12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincar\'{e} 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${\cal K}$.