A X B= C In 2+1d Tqft

We study the implications of the anyon fusion equation a x b = c on global properties of 2 + 1D topological quantum field theories (TQFTs). Here a and b are anyons that fuse together to give a unique anyon, c. As is well known, when at least one of a and b is abelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-forrn symmetries in a TQFT, of what we term "quasizero-form syrnrnetries" (as in the case of discrete gauge theories based on the largest Mathieu group, M-24), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chem-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.