Discreteness of global dimensions of spherical fusion categories

Determine whether the set of global dimensions of spherical fusion categories, X_s = {dim(C) : C is a spherical fusion category} contained in [1,∞), is discrete.

Background

The paper studies arithmetic constraints on the global dimension D of a spherical fusion category through the dimension block of the Drinfeld center's S-matrix. Before presenting its results, it recalls the broader question of whether the possible global dimensions form a discrete subset of the real line.

The paper proves substantial partial results: 1 is isolated, the interval below sqrt(5) contains only a specified finite set of possible values below any fixed bound, and consequently sqrt(2) is not a limit point. These results do not settle discreteness of X_s in its entirety.

References

It is an open question whether the set X_s:={\mathrm{dim}(\mathcal{C}):\mathcal{C}\text{ a spherical fusion category}}\subset[1,\infty) is discrete.

— The dimension block of a spherical fusion category  (2609.35287 - Schopieray, 28 Sep 2026) in Section 1, Introduction

Whether D_m\in X_s for m\geq4 remains open; the first open case is D_4\approx2.18847.

— The dimension block of a spherical fusion category  (2609.35287 - Schopieray, 28 Sep 2026) in Section 1, Introduction; Table 2