Classification of Metaplectic Fusion Categories

In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for $so(2p+1)_2$. These categories describe non-abelian anyons dubbed `metaplectic anyons'. We obtain explicit expressions for all the $F$- and $R$-symbols. Based on these, we conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes.


Introduction
Landau [Lan37] showed that most phases of matter can be classified by investigating in which way the underlying symmetry of the system can be broken. There are, however, also phases of matter that defy this classification, in particular, topologically ordered phases [Wen95], the most famous example being the fractional quantum Hall systems [TSG82,Lau83].
It is thus natural to try to classify topologically ordered phases, which can be described using the theory of modular tensor categories. Classifying all modular tensor categories is extremely difficult, with only partial results available, see for instance [RSW09].
In this paper, we study a family of fusion and modular systems that go under the name of 'metaplectic anyons'. Metaplectic anyon systems can be viewed as a generalization of the 'Ising anyon' system, and were studied, for instance, in [HNW14, HNW13]. Thus, we fix the fusion rules, and given these fusion rules, we try to find all possible, inequivalent physical solutions. In the process, we obtain explicit expressions for the F -and R-symbols, which is useful, for instance when one uses these metaplectic anyons to construct spin-chain like models as in [FFF18]. Because we use quite a bit of tensor category machinery, we are forced to take a mathematical point of view on the problem.
Fusion categories over a field k generalize categories of finite group representations; they are tensor categories with finitely many classes of simple objects, all having duals. They arise in representation theory [Kas95], operator algebras [EK98], topological quantum field theories [BK01], and quantum invariants of 3-manifolds [Tur10]. They also play an important role physically in the study of topological quantum computing [Wan10] and topological phases of matter [LW05].
As such, a classification of fusion categories up to various notions of equivalence is desirable. One such notion of equivalence is Grothendieck equivalence, wherein categories have isomorphic decategorifications, which is a based ring in the sense of [Ost03a]. Another notion of equivalence is monoidal equivalence, wherein categories are related via an invertible monoidal functor.
A natural line of inquiry is then: Given a based ring, does it admit categorification? If so, given its Grothendieck class of fusion categories, how many monoidal classes are there and how can they be distinguished? This is an incredibly difficult question to answer in general, especially at the level of fusion categories, with no further assumptions. Most results along these lines are either obtained for small examples which can be easily computed [Ost03b,Ost15,HH09], but there are some families [TY98,KW93] for which classifications are known.
In this paper we obtain an answer to these questions for a family of categories we will call metaplectic fusion categories. These are fusion categories underlying metaplectic modular categories. Recall that a metaplectic modular category [HNW14, HNW13] is a modular category Grothendieck equivalent to so(2p + 1) 2 , the category of affine so(2p + 1) representations with highest integer weight 2 [Gan02]. Other common constructions for such categories come from C(B p , 2(2p + 1), q) [Row06] for q some 2(2p + 1) root of unity, or from Z 2 equivariantizations of T Y(Z N , χ, τ ) [GNN09] where N = 2p + 1. In [ACRW16,Theorem 3.2] it was shown that all metaplectic modular categories arise from gauging the particle-hole symmetry in Z 2p+1 modular categories.
Our main result is obtained by classifying solutions to the pentagon equations and hexagon equations coming from the so(2p + 1) 2 based ring. We stop short of demonstrating that these are all of the categories as this is a much harder problem. However, the modular data for so(2p + 1) 2 categories is known and the modular data computed for our solutions coincides. As such, we refine the classification of [ACRW16] one step further.
For fixed p, our categories are parameterized by pairs (r, κ) where κ = ±1 and r a positive odd integer less than and co-prime to 2p + 1. Let R be the set of these r. Define G × 2p+1 = Z × 2p+1 / 1, −1 with g : Z × 2p+1 → G × 2p+1 the quotient map. Elements of Z × 2p+1 can be represented as positive integers less than and co-prime to 2p + 1. Likewise, elements of G × 2p+1 can be represented as positive integers less than or equal to p and co-prime to 2p + 1. Particularly, each r ∈ R represents an element of Z × 2p+1 and the action of G × 2p+1 on R via z · R = {rz 2 mod (2p + 1)|r ∈ R} makes sense. Similarly, the evaluation g(r)|r ∈ R makes sense. Our main result can then be stated as follows Theorem 1.1. For fixed p the following are true: (1) The monoidal classes of fusion categories constructed from (r, κ), κ = ±1 and r ∈ R are the fusion categories underlying metaplectic modular categories.
As a sanity check on this, following [ACRW16, Theorem 3.2], we note that the number of metaplectic fusion categories admitting the structure of a modular category is always less than or equal to the number of metaplectic modular categories 1 .
The structure of this paper is as follows. In 2 we review the basic information for fusion and modular categories and their arithmetic descriptions, fusion and modular systems. In 3 we review the gauge invariants of [HT15] which allow us to extend deductions about our fusion systems to entire gauge and monoidal classes of fusion categories. In 4 we present our solutions to the pentagon and hexagon equations, and construct the modular data. In 5 we determine the monoidal equivalence classes of the fusion categories.

Preliminaries
In this section we review relevant facts about fusion/modular categories and their arithmetic descriptions. We do not provide proofs, but most can be found in one of [ENO05,DHW13,Ost03a,BK01], or now [EGNO15]. In the sequel we will always work over k = C for simplicity.

Fusion Categories and Modular Categories.
1 There could, in principle, be fusion categories which do not admit a modular structure. • (Duality) A bijection * : B → B such that 1 * R = 1 R which extends to an anti-involution on (R, B), i.e. (XY ) * = X * Y * , ∀X, Y ∈ B. By associativity the structure constants must satisfy The structure constants define a map N : B ×3 → Z + which we can extend recursively to arbitrary n + 1 > 3 via Those structure constants which are non-zero will play an important role and so we define XY the notation for (X, Y, Z). We will say that (R, B) is multiplicity free if N (Γ(R, B)) = {1}.
For based rings (R, B) and (R, B ), every bijection ρ : B → B satisfying , ∀X, Y, Z ∈ B defines a unique based ring isomorphism ρ : (R, B) → (R, B ) and vice versa. Let Aut(R, B) to be the group of based ring automorphisms of (R, B).
Remark 2.2. Since all based rings in this paper are all multiplicity free, unless otherwise noted all future statements will be made under this assumption.
Definition 2.3. Let (R, B) be a based ring and define N X be the matrix with entry (N X ) Y Z = N Z XY . The Frobenius-Perron dimension F P (X) of X is the largest positive real eigenvalue of N X . The Frobenius-Perron dimension of (R, B) is F P ((R, B)) = X∈B F P (X) 2 .
We now define fusion categories.
Definition 2.4. A fusion category C over C is a monoidal semi-simple Abelian category 2 with identity 1 such that 2 We will denote the monoidal bifunctor of a fusion category C by ⊗ and direct sum of objects by ⊕ (1) (C-linearity) C is enriched over V ec F in (C). This is to say that C(a, b) is a finite dimensional vector space over k for all objects a, b ∈ C 0 . (2) (Finiteness) There are finitely many isomorphism classes of simple objects in C 0 and C(a, a) ∼ = C for all simple objects a ∈ C 0 . (3) (Rigidity) For every object a ∈ C 0 , there is an object a * ∈ C 0 and evaluation and co-evaluation maps ev a : a ⊗ a * → 1 coev a : 1 → a * ⊗ a such that We denote by K 0 (C) the Grothendieck ring of C. This is a based ring as defined in 2.1 with basis elements corresponding to equivalence classes of simple objects a, b, . . . ∈ C 0 and multiplication induced from ⊗ via N b a 1 ...an = dim(C(a 1 ⊗ . . . ⊗ a n , b)). We say that C is multiplicity free if K 0 (C) is multiplicity free. Given two fusion categories C and D we say that they are Grothendieck equivalent if and only if K 0 (C) ∼ = K 0 (D). We say that a based ring (R, B) admits categorification if there exists a fusion category C such that (R, B) ∼ = K 0 (C).
Definition 2.5. Let C be a fusion category and be a natural isomorphism : * * → Id of the double dual and identity functors. For all equivalence classes of simple objects a of C this gives us a morphism a : a * * → a. From this we can define the left and right quantum trace of a morphism f on a, where ev and coev are the evaluation and coevaluation maps. A natural isomorphism such that a⊗b ∼ = a ⊗ b is called a pivotal isomorphism. A pivotal fusion category is a fusion category equipped with a pivotal isomorphism.
For every isomorphism class of simple objects a in a pivotal fusion category C, one defines the left quantum dimension q l (a) and the right quantum dimension q r (a) as the quantum traces of the identity morphism on a. A spherical fusion category is a pivotal fusion category such that the left and right quantum dimensions of all objects coincide. In this case we refer simply to the quantum dimension of an object a as q a .
For a spherical fusion category C we define the matrix D to be the diagonal matrix with entries q a for all equivalence classes a. We also define the categorical quantum dimension q(C) = a (q a ) 2 . A spherical fusion category is called pseudo-unitary iff q(C) = F P (K 0 (C)) and there exists a canonical choice of spherical structure such that q a = F P (a) for all a.
There is also the stronger notion of a unitary fusion category.
Definition 2.6. A conjugation on a fusion category is a family of conjugate linear maps C(x, y) → C(y, x), f →f satisfyinḡ Müg03]. In this case then, all of the C(a ⊗ b, c) spaces are Hilbert spaces [Kit06]. Unitary implies q(a) = F P (a) and so then pseudo-unitary and spherical.
Definition 2.7. Let C be a fusion category. A braiding on C is a family of natural isomorphisms c x,y : y ⊗ x → x ⊗ y satisfying the hexagon equations as in [BK01]. A braided fusion category is a fusion category equipped with a braiding.
For braided fusion categories there is a canonical natural isomorphism (7) ψ a = ρ a * * •(Id a * * ⊗ev a )•α −1 a * * ,a,a * •(c a,a * * ⊗Id * a )•α a,a * * ,a * •(Id a ⊗coev a * )•ρ −1 a . A balanced fusion category is a braided pivotal fusion category with balancing θ a = a • ψ a . The trace of this morphism, by abuse of notation, will also be denoted θ a and the diagonal matrix with entries θ a will be called the T -matrix. A ribbon fusion category is a balanced fusion category that is also spherical.
There also exist a family of invariants S ab corresponding to the evaluations of Hopf links colored by each pair of equivalence classes (a, b). The matrix with entries S ab is called the S-Matrix and a modular category is a ribbon fusion category with non-degenerate S-matrix. In the case that C is modular than all θ a are roots of unity. The matrices 1 q(C) S and T = diag(θ a ) give a representation of the modular group SL(2, Z). It was believed that the pair (S, T ) uniquely determine the modular category and typically what one does in classifying modular categories (e.g. [RSW09], [BNRW16]) is enumerate the admissible pairs (S, T ). However, it has since then become clear that (S, T ) do not suffice to uniquely determine the modular category [MS21].

Fusion and Modular
Systems. An important question is how one goes about constructing fusion categories. One way to do so [DHW13] is through a collection of numbers called a fusion system.
(1) A set of labels L containing an element called 1.
(2) An involution * : L → L such that 1 * = 1. ( The entries of matrix F d;ef abc are referred to as 6J-symbols. Define Aut(N ) to be the group of permutations on L such that for all ν in Aut(N ) . This then extends to an action on F via Central to our analysis is the ability to interpolate between fusion categories over C and fusion systems over C. Given a fusion system (L, N, F ) over C, one can construct a fusion category C(L, N, F ). Given a fusion category C over C, a fusion system over C, denoted (L, N, F )(C), can be extracted such that C((L, N, F )(C)) ∼ = C as fusion categories. For our construction, this is proven in [DHW13] as Proposition 3.7. A similar theorem also appears in [TY98].
It is by analyzing a family of fusion systems for so(2p + 1) 2 Grothendieck rings that we develop our classification. To do this, we will utilize the following: Proposition 2.9. Two fusion categories C and D are Grothendieck equivalent if and only if given any two fusion systems (L, N, F ) and (L , N , F ) extracted from them there exists a bijection f : L → L such that for all there exists a based ring isomorphism in the sense of [Ost03a] 2.1.iv, and this is determined by its action on basis elements. This allows us to specify a common basis for K 0 (C) from which to work. There also exists arithmetic data for pivotal structures.
Proposition 2.10. Let (L, N, F ) be a fusion system extracted from a fusion category C and define the set { a } a∈L . Pivotal structures on C are in 1-1 correspondence with solutions to −1 c This is proven in [DHW13, Proposition 3.12]. Given a fusion category C with pivotal structure , if it is a spherical structure then there exists a fusion system (L, N, F )(C) such that all a = ±1. By skeletalizing the rigidity conditions and using the choice of basis as in [DHW13,Lemma 3.4], the quantum dimensions can be computed from F and as q l (a) = a (F a * ;11 a * aa * ) −1 and q r (a) = ( a F a;11 aa * a ) −1 (15) If C admits the structure of a unitary fusion category, then there exists a fusion system (L, N, F )(C) such that all F d abc matrices are unitary. Conversely (See [TY98], Section 4), if given (L, N ) there exists a solution to (10)-(12) such that all F d abc matrices are unitary, then C(L, N, F ) admits a unitary structure.
such that the matrix with entries Similar to 13, we can define the action of ν ∈ Aut(N ) on R via The matrixŜ is related to the matrix S via S = DŜD. Additionally, we can compute the twists as obtained by taking the trace of the morphism (7) above [Kit06]. Finally, we give a formula for the topological central charge c top (see [FG90]). We introduce from which one obtains the topological central charge, defined modulo eight, Note that through teasing out definitions, all of the information above can be written in terms of F , R, and . Since in our case we have all of the F 's, R's, and 's, so we can simply compute (S, T ) for the given the arithmetic data.

Monoidal Equivalence and Gauge Invariants
Given two fusion (modular) categories C and D, a central question is one of whether or not they are equivalent in some suitable sense. The strongest of these is monoidal equivalence.
Definition 3.1. C and D are said to be monoidally equivalent iff there exists a pair of monoidal functors(see [Kas95]) F : C → D and G : C → D such that F • G and G • F are naturally monoidal isomorphic to the identity functors on C and D respectively. If C and D are braided monoidal, then they are braided monoidally equivalent if F and G are braided monoidal functors.
A family of categories C 1 , . . . , C n which are monoidally equivalent are automatically Grothendieck equivalent. A difficult and important question goes the other direction: Given a Grothendieck equivalence class of categories, how many monoidal equivalence classes are there? That there are finitely many is known as Ocneanu rigidity [ENO05].
Given that we can describe fusion categories arithmetically, there is the natural question of whether or not an equivalence between them can be described arithmetically. The action Aut(N ) on a fusion system (L, N, F, R, ) is given by (13) and it was shown in [DHW13] that this gives an object permuting monoidal functor. We say that two fusion systems F, F are permutation equivalent if there exists ν ∈ Aut(N ) such that F = F ν . The other operation one can perform on (L, N, F ) to obtain equivalent categories is given by a gauge transformation. This corresponds to a change of basis on the C(a ⊗ b, c) homspaces and can be represented arithmetically as follows: Define G to be the set of functions g : Γ(L, N ) → k × fixing the notation g c ab := g(a, b, c) and g ab c := (g(a, b, c)) −1 . Then given a solution F to (10)-(12) and g ∈ G, one obtains another solution F g via the equations and two solutions F and F are gauge equivalent if and only if there exists g ∈ G such that F = g. (23) can be used to construct an object fixing monoidal functor. Given C, it is clear from the definition that any two fusion systems extracted from C are gauge equivalent. Monoidal equivalence of fusion categories C and D is then determined by the existence of a permutation equivalence together with a gauge equivalence.
Remark 3.2. The above extends readily to braided monoidal equivalence between modular categories. Given R and R , permutation equivalence requires the imposition of the extra condition R = R ν as defined in (19). Gauge equivalence requires the imposition that R = R g as given by Given a set of fusion(modular) systems, determining their monoidal equivalence is not an easy task in general. However, in [HT15] a construction is given for invariants which can distinguish gauge classes of fusion categories (with or without multiplicity). The key to this is noting that for a given (L, N ) solutions to the equations (10)-(12) define an algebraic scheme X(L, N ) in variables Φ(L, N ). F can then be interpreted as an algebra ho- , the ring of regular functions on X defined at U .
Gauge equivalence classes correspond to G-orbits in X(L, N ) and monoidal equivalence classes correspond to Aut(N ) G-orbits in X(L, N ). Given this, one can leverage the geometric invariant theory of [Mum99] to obtain the following result: such that for fusion systems F and F , F and F are gauge equivalent if and only if This is Theorem 1.2 of [HT15]. It is straightforward to extend this construction to classifying modular systems (L, N, F, R, ). One goes through the same arguments of using the scheme defined from (10)-(12),(14)-(17), and noting that they have the same essential properties (G is reductive, orbits are closed).
In the fusion case, the choice of Aut(N ) G-invariants is not necessarily apparent a priori. However, as we will show in section 5, there is an easy choice which works for our so(2p + 1) 2 categories.
For modular categories there are invariants which immediately present themselves. Even though we previously mentioned that the pair (S, T ) is not strong enough for classification [MS21], S and T prove to be useful. The quantum dimensions as defined in 15 are categorical invariants sufficient to distinguish between spherical structures, but the first row and column of the matrix S are essentially these numbers. It has also been previously mentioned that the q s, S s, and T s can all be written using F s, R s, and s. Thus they all define G-invariant regular functions on X.

so(2p + 1) 2 Fusion Systems
In the following we present arithmetic data for the F and R matrices which makes much of the structure for our categories apparent. We also provide explicit computations of the relevant categorical quantities.
There are several contexts in which the Grothendieck rings for our categories naturally arise, though two primary sources both come from the study of Lie algebras. As should be apparent from our notation, one of these is as Grothendieck rings for the categories of representations of the (untwisted) affine Kac-Moody algebras B (1) p (i.e. so(2p+1)) with highest integral weight 2. As determined by [Fin96], these are equivalent (as modular categories) to the semi-simplification (à la [And92]) of the representation category for U q (B p ) with q = e πı 2p+1 . The affine Kac-Moody and quantum group constructions for the Grothendieck rings can be found in [Gan02] and [Row06] respectively.
The automorphisms which permute the φ i follow restricting g to Z × 2p+1 and promoting it to a group homomorphism. Z × 2p+1 has even order 3 and can be represented by integers in ±1, . . . , ±p. If i is in Z × 2p+1 so is −i, thus the quotient Z × 2p+1 / 1, −1 is well defined. g is precisely this quotient map. 4 G × 2p+1 acts on φ i in the obvious way and since g is a group homomorphism, this preserves the fusion rules. There are no automorphisms on pointed objects or between objects of different Frobenius-perron dimension, and so the automorphism group of K 0 (C) must be the direct product. This gives a separate proof from that in [Gan02].
To write the general F -symbols, we first introduce the following matrices.

Arithmetic Data.
Before we give the specific 6J symbols, we make a few general remarks. First, the order of the entries in the F -matrices respects the order we specified above: (1, , φ i , ψ ± ), where i = 1, . . . , p. Second, we will actually not give the values F d;ef abc individually, but instead give the F -matrices F d abc . In the basis we use to give the F -matrices, we have the following property In addition, it is implicitly assumed that the label d of F d abc is in the tensor decomposition of a ⊗ b ⊗ c. This allows us to only specify a reduced set of F -matrices, while the others can be deduced by using this relation. We note that often, but not not always, the F -matrices are symmetric. In addition, all the F -symbols are real in our basis.
We first observe that from [NP14] that for fixed p the fusion rules for {1, , ψ i } give the tensor structure for Rep(D 2p+1 ). K 0 (C) is then a Z 2extension of K 0 (Rep(D 2p+1 )). We will proceed by building subcategories Grothendieck equivalent to Z 2 and Rep(D 2p+1 ) before finally specifying Fmatrices which correspond to the √ 2p + 1 objects. First, if one or more of the labels a, b, c, d equals 1, F d abc = (1). Next we have {1, } equivalent to Z 2 and the action of on the objects φ i is determined by The rest of the F -matrices for the category Rep(D 2p+1 ) are given by: Proposition 4.2. There are no other solutions to the pentagon equations for Rep(D 2p+1 ) which extend to solutions for so(2p + 1) 2 .
In our case we have that since the universal grading of so(2p + 1) 2 is Z 2 , we have that {ψ + , ψ − } are indecomposable Z 2 module categories over Rep(D 2p+1 ). We can count which Rep(D 2p+1 ) categories have Z 2 module categories by looking at which D 2p+1 categories have Z 2 module categories. Since H 3 (D 2p+1 , C × ) ∼ = Z 2p+1 the only ω which fixes the Z 2 subgroup of D 2p+1 is ω = 1 and its cohomology is Z 2 . Thus only Rep(D 2p+1 ) has indecomposable Z 2 module categories and there are two of them.
We now specify the F -matrices that are labeled by ψ ± and start with the F -matrices whose labels consist of only ψ ± .
There are three classes of F -matrices involving two labels that are ψ ± . The first class of this type is where i, j may or may not be equal. The second class of F -matrices with two labels equal to ψ ± and i = j is Finally, the third class of F -matrices with two labels equal to ψ + or ψ − is With these symbols, and the general rules described above, we have exhausted all the F -matrices.
4.3. R-Matrices. Given our solutions to the pentagon equations, one would anticipate it is not too complicated to construct solutions to the hexagon equations as well.
Given a solution to the hexagon equations one always has a second solution given by inverses. This simply corresponds to a choice of R and R −1 in (16) and (17). We may also obtain another solution by replacing all Rsymbols R c ab , such that a and b are both one of ψ ± by −R c ab . However, this is monoidally equivalent to our original solution via the automorphism ψ ± → ψ ∓ . This gives us that for any p, solutions to the pentagon and hexagon equations can be uniquely identified by the tuple (p, r, κ, λ), where λ = ±1 indicates whether or not one is referring to the R-symbols given below or their inverses. We then provide only one such solution.
We specify the R-symbols using a form inspired by conformal field theory. Namely, for each simple object type, we introduce the scaling dimensions h a which determine the R-symbols up to a sign R c ab = (σ 1 ) c ab (σ 2 ) c ab (−1) ha+h b −hc , where (σ 1 ) c ab = ±1 and (σ 2 ) c ab = ±1. Below we specify the scaling dimensions h a for all the simple objects, as well as the signs (σ 1 ) c ab and (σ 2 ) c ab . The scaling dimensions are given by The notation (j|n) denotes the Jacobi symbol, which is defined for j an integer, and n a positive, odd integer.
To completely specify the R-symbols we still need to specify the signs (σ 1 ) c ab and (σ 2 ) c ab . We start with (σ 1 ) c ab , and note that for the R-symbols presented here we have the property (σ 1 ) a bc = (σ 1 ) c ab , which implies that we only need to specify a limited set of these symbols. The non-trivial symbols come in two classes. The first class is The second class of symbols is The symbols (σ 1 ) c ab that are not specified by the rules above, are all equal to +1.
Finally, the signs (σ 2 ) c ab also satisfy the property (σ 2 ) a bc = (σ 2 ) c ab and there is only one class of non-trivial signs of type, namely The symbols (σ 2 ) c ab that are not specified by the rules above, are all equal to +1.
With this, we have completely specified one of the solutions of the hexagon equations, from which the other one follows by taking the inverse.
Remark 4.3. We close by commenting briefly on the appearance of the Jacobi symbols in the expression for the R-symbols. One can think of the list of hexagon equations as labeled by the different F -symbols, namely the one appearing on the left hand side of the symbolic form RF R = F RF . To derive the form of h ψ ± above, consider the hexagon equation associated to F ψ + ;11 ψ + ψ + ψ + . The sum on the right hand side gives rise to a quadratic Gauss sum (because one needs to sum over all φ i (as well as the identity)), which are closely related to the Jacobi symbols. This leads to their presence in the scaling dimensions h ψ ± . See Appendix A for more details.
As stated above, the Jacobi symbols (j|n) are defined for positive odd integers n and arbitrary integers j. Interestingly, we observed that the these Jacobi symbols can be written in terms of the matrices H(r, κ) and G(r, κ) (whose sizes depend on p). In particular, one can show (r|2p + 1) = det(H(2r + 2p + 1)) det(G(2r + 2p + 1)).
This gives an analytic function of r, that goes through all the Jacobi symbols, defined for r integer. We note that Eisenstein also constructed such a function, (q|p) = (p−1)/2 n=1 sin(2πqn/p) sin(2πn/p) , see for instance [Lem00]. The difference with the function that we found, is that the values ±1 are the extrema of the function while this is certainly not the case for Eisenstein's function.
4.3.1. Modular data for so(2p + 1) 2 Modular Systems. All of our F-Matrices are manifestly unitary, thus all of our categories admit at least one spherical structure. Since they are also braided, they must be ribbon and so we can use the formula (20). Our formula (18) is defined for any braided category, however one can obtain the more common form by taking DSD. We present the latter to more easily demonstrate that we obtain the same modular result as [GNN09]. However, we also provide an explicit classification at the fusion level.
Pivotal structures on modular categories are in bijective correspondence with the group of invertible objects and spherical structures with the maximal Abelian 2-subgroup [BNRW16, Lemma 2.4]. Since the group of invertible objects is Z 2 , there are exactly two pivotal structures for each category, both of which are spherical. It is straightforward to compute that for 1, , and φ i , all pivotal coefficients must be 1. We also have ψ + = ψ − = ±1, the choice of which switches the sign of q ψ ± .
Below, we present the modular data for solutions with positive quantum dimensions. It's worth noting that for all p and r the pivotal coefficients which yield positive quantum dimensions for κ = 1 are not those that yield positive quantum dimensions for κ = −1. This can be seen from noting the effect of κ on (15). In particular, q ψ ± > 0 if we take ψ ± = κ.
We then have that, for modular categories corresponding to the solution indexed by (p, r, κ, λ), S 11 = 1 S = 1 S φ i φ j = 4 cos 2πijr 2p + 1 (49) For the twists we have For completeness, we also calculate the topological central charge from eqs. (21),(22). We first note that the contributions from ψ ± in the sum to obtain p + cancel. Using similar arguments as given in Appendix A, one finds that the remaining terms constitute a quadratic Gauss sum, resulting in p + / a q 2 a = 2p+1 (2λr|2p + 1), where 2p+1 = 1 for p even, and 2p+1 = ı for p odd. Using standard manipulations of the Jacobi symbol, one can finally write (55) c top = 2p(λ + 2p) − 2 + 2(r|2p + 1) mod 8 .

Monoidal Equivalence of so(2p + 1) 2 Fusion Systems
In this section we introduce those gauge invariants which we will use to classify monoidally inequivalent solutions for a given p using the method of Section 3. At the level of gauge equivalence it is easy to see that two solutions (r, κ) and (r , κ ) are gauge equivalent if and only if r = r and κ = κ . Our goal then is to find a set {l 1 , . . . , l n } of Aut(N )-invariant linear combination of G-invariant monomials as defined in Theorem 3.3 which determine the monoidal classes. We then determine the number of monoidal equivalence classes for a given p. For all pairs (r, κ) as defined in Section 4.2 let F (r,k) be the evaluation map as defined for Theorem 3.3. 5.1. Determining Equivalence. We first note that the adjoint category is basically useless to us. Proof. For fixed p, the only F -matrix entries which differ are those in which r arises, i.e. those specified by the D, E, H ( ) , and G matrices. These come from F -matrices involving at least one ψ ± and so do not come from the Rep(D 2p+1 ) subcategory. So if l is an Aut(N ) G-invariant which comes from Rep(D 2p+1 ), i.e. only involves the objects 1, , and φ i , i = 1, . . . p then F (r,k) (l) = F (r ,k ) (l).
Corollary 5.4. X p (r, κ) uniquely determines the values for all G-invariant monomials and thus all Aut(N ) G-invariant monomials.

Calculating The Number of Monoidal Equivalence Classes.
It is not an easy task to calculate how many monoidally inequivalent sets of F-moves there are. Nevertheless we can construct a framework in which the question is seemingly simple. From the previous section we showed that the F-moves labeled by (r, κ) and (r , κ ) are monoidally equivalent if and only if κ = κ and there exists a z ∈ G × 2p+1 such that r = rz 2 . Thus we define . Each set contains elements such that the F-moves labeled by (r, κ) and (r , κ) are monoidally equivalent. Furthermore, (r, κ) and (r , κ) are monoidally equivalent if and only if O r = O r . We realize that O 1 is a normal subgroup and that O r = rO 1 are conjugacy classes. Thus the number of monoidally inequivalent F-moves must be twice (due to κ) |G × 2p+1 |/|O 1 |.
It is possible to simplify this further by considering the map f : G × 2p+1 → O 1 defined by f (z) = z 2 . This map is clearly onto, i.e. the image of f is O 1 . As we are dealing with the finite groups we can use the first isomorphism theorem and deduce that |G × 2p+1 |/|O 1 | = |ker(f )| where ker(f ) is the kernel of f and is the subgroup ker(f ) = {r ∈ G × 2p+1 |r 2 = 1}. Thus the number of monoidally inequivalent F-moves is twice the number of elements of G × 2p+1 that square to the identity (because of κ). We know that |ker(f )| = 2 m for some m ∈ N.
If r ∈ Z × 2p+1 satisfies r 2 = 1 or −1 then r ∈ G × 2p+1 satisfies r 2 = 1. We can then deduce that Firstly, consider the prime decomposition 2p + 1 = p a 1 1 ...p a l l where the p i are prime and a i > 0. We define sets We note that as B 1 is also a group and that its order is precisely 2 l . Next, suppose that r, r ∈ B 2 it follows that rr ∈ B 1 , similarly if r ∈ B 2 and r ∈ B 1 then rr ∈ B 2 . Thus we have that the cardinality of B 2 is zero or |B 2 | = |B 1 | = 2 l . For |B 2 | to be non-zero we must have that there exists b i such that b 2 i ≡ −1 mod p a i i for all i.
From this we can deduce the number of monoidal classes where l is the number of primes that divide (2p + 1). Here we have used that Z × 2(2p+1) . We note that for the cases in which there are 2 l monoidal classes, we obtain 2 l+1 modular categories, because of the R-symbols (λ = +1) and their inverses (λ = −1), which are not equivalent. However, in the cases in which there are 2 l+1 monoidal classes, the modular categories obtained by the R-symbols on the one hand and their inverses on the other, are pairwise equivalent. This means that also in this case, we obtain 2 l+1 modular categories, in accordance with [ACRW16, Theorem 3.2].

Examples
In this section we explicitly compute the classifying invariants for a number of "small" categories. The case of so(3) 2 is known to be Grothendieck equivalent to su(2) 4 and both the fusion and modular structures are fully classified (see [KW93]). Beyond this, there are several guideposts: • The classification of weakly integral modular categories of dimension 4m is given in [BGN + 16]. This contains those so(2p + 1) 2 of our family for which 2p + 1 is square free. • The classification of integral modular categories of dimension 4q 2 is given in [BGH + 14] where q is prime. • Explicit formulae for the modular data of Z 2 -equivariantizations of Tambara-Yamagami categories is given in [GNN09] to which our categories are Grothendieck equivalent.
6.1. so(3) 2 . For so(3) 2 , G × 3 = {1}. Thus we have only one two dimensional simple object for which the automorphism group is trivial. We have two solutions and two fusion categories. Up to permutation there are four modular categories, one for each of the four possible T-matrices coming from (54): Corresponding to solutions (1, 1, 1, 1), (1, 1, 1, −1), (1, 1, −1, 1) and (1, 1, −1, −1) respectively. This is in agreement with [GNN09]. In this case the two braidings for each fusion category are inequivalent. 6.2. so(5) 2 . There are two two dimensional objects and G × 5 ∼ = Z 2 . Both elements square to the identity and so there are four fusion categories from (56). This can be seen from the sets , and

Conclusions
In conclusion, we studied a family of fusion and modular systems with fusion rules that are equivalent to the fusion rules of so(2p + 1) 2 . The associated anyons are dubbed 'metaplectic anyons'. We obtained explicit expressions for all the F -and R-symbols (as well as the modular data, that is, the S-and T -matrices (i.e., the twists) and the central charge). Based on the explicit expressions for the F -symbols, we conjecture a classification of the monoidal equivalence classes via an analysis of suitably chosen gauge invariant combinations of F -symbols. We obtained an expression for the number of monoidal equivalence classes, and finished by giving some explicit examples. For future work, it would be interesting to see if the work in this paper can be extended to the fusion and modular systems equivalent to the fusion rules of so(2p) 2 .

A. Proof of Solution To Pentagon Equations
In this appendix, we briefly explain how we obtained the F -and Rsymbols that we present in the main text, and how we verified that they indeed are solutions to the pentagon and hexagon equations, for arbitrary tuples (p, r, κ, λ) with p a positive integer, r an odd integer such that 1 ≤ r < 2p + 1 and gcd(r, 2p + 1) = 1, κ = ±1 and λ = ±1.
We obtained the F -symbols, by numerically calculating them from the quantum group based on so(2p+1) at the appropriate roots of unity for small p. From this data, we were able to extract the general pattern, presented in section 4.2.
The main difficulty in verifying that the F -and R-symbols we obtained do indeed satisfy the pentagon and hexagon equations lies not so much in the actual verification of the equations, but rather in convincing oneself that all the pentagon and hexagon equations one needs to verify are indeed covered. The reason for this is the structure of the fusion rules of the dimension two objects, which are such that one has to consider many cases separately. This pertains both to all the possible labelings of the equations, as well as to the sum present in these equations.
In order to complete this task, we used the algebraic manipulation program Mathematica. We first carefully constructed all the pentagon equations that can occur algebraically, that is, for arbitrary tuples (p, r, κ). After having convinced ourselves that we indeed covered all the possible cases, we then verified this, for low, explicit values of p, by using the algebraic result to explicitly construct the labels of all the pentagon equations for these explicit values of p. For explicit p, one can also generate all possible cases directly. As expected, we indeed covered all the cases.
After we generated all the cases of the pentagon equations for the tuples (p, r, κ), we did the actual verification. In all but one class of equations, the equations are straightforwardly verified, and again, we used Mathematica for this. The most complicated cases occur when the sum over h in (10) runs over all dimension two objects (and possibly 1 or ). To check that the F -symbols satisfy these equations, one rewrites the (maximally triple) products of sines and cosines that occur as a sum of sines and cosines, and one uses Lagrange equations to perform the sum. One can actually use the same procedure to verify that the matrices G and H ( ) (31), (32) are orthogonal.
To verify that the solutions of the F -and R-symbols presented also satisfy the hexagon equations (16),(17) we used the same strategy as we used to verify the pentagon equations (10). That is, we explicitly constructed all different cases of the hexagon equations using Mathematica, which we also used to do the actual checking of the equations, except for one class of equations, that is non-trivial, and was checked by hand.
The class of equations that we explicitly verified has all the labels a, b, c, d of (16),(17) equal to ψ ± . The sum over e runs over all dimension two objects, and possibly 1 or . This class of equations falls apart into four distinct cases, of which we discuss the simplest case for ease of presentation. The other three cases can be shown to hold using similar arguments.
These equations hold if p is a positive integer, r is an odd integer 1 ≤ r < 2p + 1 with gcd(r, 2p + 1) = 1, κ = ±1 and i 1 , i 2 are integer. In the actual hexagon equations one has i 1 , i 2 = 1, 2, . . . , p. We note that (58) (and (57)) is obtained from (59) by setting i 2 = 0 (and i 1 = 0). We also note that in the right hand side of (60) is a sum of only p terms, in comparison to (57), (58) and (59), for which the right hand side is a sum of p + 1 terms.
We can now compare this result for the RHS of (57) with the LHS, by explicitly checking the LHS for the four different cases of p modulo 4. One finds that the LHS is indeed independent of κ = ±1, and that the LHS equals the RHS, as we wanted to show.