a×b=c in 2+1D TQFT

Abstract

We study the implications of the anyon fusion equation a×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-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, M24), 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 Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.

Non-rational Narain CFTs from codes over $F_4$

“…This is likely to be true for c = 1, 2 and c = 8 as well: in the former cases the partition function is a sum of "characters" and in the latter case the lattice is the Barnes-Wall lattice, which can be constructed using a code over F 9 . More generally, any rational Narain CFT can be related to codes [23], and this likely extends to any finite CFT [24]. This observation prompts the question if the optimal theories for larger c could also be related to codes and if perhaps there are appropriate series of constructions which could describe optimal theories with an arbitrarily large central charge.…”

“…Now we clearly see this is not necessarily the case. It thus remains an open question to chart the space of theories with the code counterparts, see [23], and investigate if this relation can be extended beyond Narain theories.…”

Non-Abelian Anyons and Some Cousins of the Arad-Herzog Conjecture

“…Another interesting question is to understand to what degree fusion rules of the types we have been discussing constrain global properties of more general TQFTs. We will report on progress toward understanding this last question in [11].…”

Galois Orbits of TQFTs: Symmetries and Unitarity

“…Since Galois conjugation fixes rational numbers, it is clear that the space of integral MTCs (i.e., theories whose anyons all have integer quantum dimensions) is closed under it. An important class of integral MTCs are (twisted) discrete gauge theories (see [32] for a recent discussion of these theories, their subcategory structure, and their fusion rules). Since there are integral MTCs that are not (twisted) discrete gauge theories [33], we might naively imagine that these theories mix with discrete gauge theories under Galois conjugation.…”

“…We expect the equations (4.9) and (4.8) to give a unique solution up to symmetry gauge transformations. 32 Hence, any element p ∈ Gal(K U /K C ) acts on U g (a, b, c) to relate it to another set of solutions which is gauge equivalent to the one we started with.…”