Critical Metrics and Covering Number

TL;DR

This paper investigates the geometry of right-invariant metrics on quantum unitary groups, exploring the existence of a critical metric related to phase transitions and the influence of cohomology ring structures.

Contribution

It examines the constraints imposed by cohomology ring structures on the global geometry of critical metrics in quantum and black hole physics contexts.

Findings

Hypothesized existence of a phase transition-like critical metric.

Cohomology ring structure restricts the geometry of critical metrics.

Provides a conjectural form for the critical metric.

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 $\operatorname{SU}(2^N)$. 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.