Dimension as a quantum statistic and the classification of metaplectic categories

Abstract: We discuss several useful interpretations of the categorical dimension of objects in a braided fusion category, as well as some conjectures demonstrating the value of quantum dimension as a quantum statistic for detecting certain behaviors of anyons in topological phases of matter. From this discussion we find that objects in braided fusion categories with integral squared dimension have distinctive properties. A large and interesting class of non-integral modular categories such that every simple object has integral squared-dimensions are the metaplectic categories that have the same fusion rules as $SO(N)_2$ for some $N$. We describe and complete their classification and enumeration, by recognizing them as $\mathbb{Z}_2$-gaugings of cyclic modular categories (i.e. metric groups). We prove that any modular category of dimension $2^km$ with $m$ square-free and $k\leq 4$, satisfying some additional assumptions, is a metaplectic category. This illustrates anew that dimension can, in some circumstances, determine a surprising amount of the category's structure.