Abstract
In both quantum computing and black hole physics, it is natural to regard some deformations, infinitesimal unitaries, as \emph{easy} and others as \emph{hard}. This has lead to a renewed examination of right-invariant metrics on SU(2^N). It has been hypothesized that there is a critical such metric -- in the sense of phase transitions -- and a conjectural form suggested. In this note we explore a restriction that the ring structure on cohomology places on the global geometry of a critical metric.