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 νZ2-gaugings of cyclic modular categories (i.e. metric groups). We prove that any modular category of dimension 2 k m with m square-free and k ≤ 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. 1

Symmetry defects and their application to topological quantum computing

“…These are the categories Ising (ν) with isomorphism classes of simple objects {1, σ , ψ}, where ν parametrizes the quantum twist of the anyon σ . These lead to 20 inequivalent MTCs of the form Ising (ν 1 ) ⊠ Ising (ν 2 ) [5]. Here we take ν 1 = ν 2 = 1, and by Ising we mean Ising (1) , the category with modular data given in [8].…”

“…Step 5: The diagram that remains contributes a scalar factor that can be calculated using equations (4) and (5) repeatedly until the empty diagram is obtained.…”

Integral metaplectic modular categories

“…There are three cases to consider depending on the value of N . The relevant fusion rules for each of these cases are listed below (note: fusion rules for √ N and N 2 dimensional objects are not included, but can be found here [1,3]).…”

Prime decomposition of modular tensor categories of local modules of Type D

“…5 , k) in this section by ordered pairs (s, t), with s, t ∈ Z ≥0 such that s + t ≤ k. The rank of C(so 5 , k) is therefore the triangular number (1/2)(k + 1)(k + 2). For even k, C(so 5 , k) pt ≃ Rep(Z/2Z) with unique nontrivial connected étale algebra R, otherwise C(so 5 , k) pt ≃ sVec.…”