Classification of Metaplectic Fusion Categories

Abstract

In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for $\mathfrak{so}{(2p+1)}_2$. 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.

1. Introduction

Landau [1] 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 [2], with the most famous example being the fractional quantum Hall systems [3,4].

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, [5].

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 [6,7]. 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 [8]. 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 [9], operator algebras [10], topological quantum field theories [11], and quantum invariants of 3-manifolds [12]. They also play an important role physically in the study of topological quantum computing [13] and topological phases of matter [14].

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 [15]. 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 [16,17,18], but there are some families [19,20] 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 [6,7] is a modular category Grothendieck equivalent to $\mathfrak{so}{(2p+1)}_2$, the category of affine $\mathfrak{so}(2p+1)$ representations with highest integer weight 2 [21]. Other common constructions for such categories come from $\mathcal{C}(B_p,2(2p+1),q)$[22] for q some $2(2p+1)$ root of unity, or from ${\mathbb{Z}}_2$ equivariantizations of $\mathcal{TY}({\mathbb{Z}}_N,\chi ,\tau )$[23] where $N=2p+1$. In ([24] [Theorem 3.2]), it was shown that all metaplectic modular categories arise from gauging the particle–hole symmetry in ${\mathbb{Z}}_{2p+1}$ modular categories.

Our main result is obtained by classifying solutions to the pentagon equations and hexagon equations coming from the $\mathfrak{so}{(2p+1)}_2$ 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 $\mathfrak{so}{(2p+1)}_2$ categories is known and the modular data computed for our solutions coincides. As such, we refine the classification of [24] one step further.

For fixed p, our categories are parameterized by pairs $(r,\kappa )$, where $\kappa =\pm 1$ and r a positive odd integer less than and co-prime to $2p+1$. Let R be the set of these r. Define ${\mathbb{G}}_{2p+1}^{\times }={\mathbb{Z}}_{2p+1}^{\times }/\langle 1,-1\rangle$ with $g:{\mathbb{Z}}_{2p+1}^{\times }\to {\mathbb{G}}_{2p+1}^{\times }$ the quotient map. Elements of ${\mathbb{Z}}_{2p+1}^{\times }$ can be represented as positive integers less than and co-prime to $2p+1$. Likewise, elements of ${\mathbb{G}}_{2p+1}^{\times }$ can be represented as positive integers less than or equal to p and co-prime to $2p+1$. Particularly, each $r\in R$ represents an element of ${\mathbb{Z}}_{2p+1}^{\times }$ and the action of ${\mathbb{G}}_{2p+1}^{\times }$ on R via $z·R=\{r{z}^{2}mod(2p+1)|r\in R\}$ makes sense. Similarly, the evaluation $g\left(r\right)|r\in R$ makes sense. Our main result can then be stated as follows

Theorem 1.

For fixed p, the following are true:

1. 

The monoidal classes of fusion categories constructed from $(r,\kappa )$, $\kappa =\pm 1$ and $r\in R$ are the fusion categories underlying metaplectic modular categories.

2. 

Let $(r,\kappa )$ and $(r^{\prime },{\kappa }^{\prime })$ parameterize two different solutions to the pentagon equations. Then the fusion categories constructed from these solutions are monoidally equivalent if and only if $\kappa ={\kappa }^{\prime }$ and there exists $z\in {\mathbb{G}}_{2p+1}^{\times }$ such that $g\left(r^{\prime }\right)=g\left(r{z}^2\right)$.

3. 

For $2p+1=p_1^{a_1}\dots p_l^{a_l}$, there are exactly $2^{l+1}$ monoidally inequivalent metaplectic modular categories if $\exists b\in Z_{2p+1}^{\times }|b^2=-1$, otherwise there are exactly $2^l$.

As a sanity check on this, following ([24] [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. We note that there could, in principle, be fusion categories which do not admit a modular structure.

The structure of this paper is as follows. In Section 2 we review the basic information for fusion and modular categories and their arithmetic descriptions, fusion and modular systems. In Section 3 we review the gauge invariants of [25] which allow us to extend deductions about our fusion systems to entire gauge and monoidal classes of fusion categories. In Section 4 we present our solutions to the pentagon and hexagon equations, and construct the modular data. In Section 5 we determine the monoidal equivalence classes of the fusion categories. In Section 6 we explicitly compute the equivalence classes and modular structures for several examples. Appendix A demonstrates that the explicit F- and R-symbols we present are indeed solutions to the pentagon and hexagon equations.

2. 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 [11,15,26,27], or now [28]. In the sequel we will always work over $k=\mathbb{C}$ for simplicity.

2.1. Fusion Categories and Modular Categories

Definition 1

([15] [Definitions 2.1 and 2.2]). A unital based ring $(R,B)$ is a ${\mathbb{Z}}^{+}$- ring R together with a set $B\subset R$ and identity $1_R\in B$ such that

• (Structure constants) There exist non-negative integers $N_{XY}^Z$ for $X,Y,Z\in B$ such that

  $$XY=\sum _{Z\in B}{N}_{XY}^{Z}Z.$$
• (Duality) A bijection $\ast :B\to B$ such that $1_R^{*}=1_R$ which extends to an anti-involution on $(R,B)$, i.e., ${\left(XY\right)}^{*}=X^{*}{Y}^{*},\forall X,Y\in B$.

By associativity, the structure constants must satisfy

$$\sum _{U\in B}{N}_{XY}^U{N}_{UZ}^W=\sum _{V\in B}{N}_{YZ}^V{N}_{XV}^W$$

(1)

for all $U,V,W,X,Y,Z\in B$.

The structure constants define a map $N:B^{\times 3}\to {\mathbb{Z}}^{+}$ which we can extend recursively to arbitrary $n+1>3$ via

$$N_{X_1\dots X_n}^Y:=\sum _{Z\in B}{N}_{X_1\dots X_{n-1}}^Z{N}_{Z{X}_n}^Y$$

for all $X_1,\dots ,X_n,Y\in B$. Those structure constants which are non-zero will play an important role and so we define

$$\Gamma (R,B)=\{(X,Y,Z)\in B^{\times 3}|N_{XY}^Z\ne 0\}$$

with ${\gamma }_{XY}^Z$ the notation for $(X,Y,Z)$. We will say that $(R,B)$ ismultiplicity freeif $N\left(\Gamma \right(R,B\left)\right)=\left\{1\right\}$.

For based rings $(R,B)$ and $(R,B^{\prime })$, every bijection $\rho :B\to B^{\prime }$ satisfying

$$N_{XY}^Z=N_{\varphi \left(X\right)\varphi \left(Y\right)}^{\varphi \left(Z\right)},\forall X,Y,Z\in B$$

(2)

defines a unique based ring isomorphism ${\rho }^{\prime }:(R,B)\to (R,B^{\prime })$ and vice versa. Let $Aut(R,B)$ to be the group of based ring automorphisms of $(R,B)$.

Remark 1.

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.

Let $(R,B)$ be a based ring and define $N_X$ be the matrix with entry ${\left(N_X\right)}_{YZ}=N_{XY}^Z$. The Frobenius–Perron dimension $FP\left(X\right)$ of X is the largest positive real eigenvalue of $N_X$. The Frobenius–Perron dimension of $(R,B)$ is $FP\left((R,B)\right)={\sum }_{X\in B}FP{\left(X\right)}^2$.

We now define fusion categories.

Definition 3.

Afusion category$\mathcal{C}$ over $\mathbb{C}$ is a monoidal semi-simple Abelian category with identity $\mathbf{1}$ such that (we will denote the monoidal bifunctor of a fusion category $\mathcal{C}$ by ⊗ and direct sum of objects by ⊕)

1. 

($\mathbb{C}$-linearity) $\mathcal{C}$ is enriched over $Ve{c}_{Fin}\left(\mathbb{C}\right)$. This is to say that $\mathcal{C}(a,b)$ is a finite dimensional vector space over k for all objects $a,b\in {\mathcal{C}}_0$.

2. 

(Finiteness) There are finitely many isomorphism classes of simple objects in ${\mathcal{C}}_0$ and $\mathcal{C}(a,a)\cong \mathbb{C}$ for all simple objects $a\in {\mathcal{C}}_0$.

3. 

(Rigidity) For every object $a\in {\mathcal{C}}_0$, there is an object $a^{*}\in {\mathcal{C}}_0$ and evaluation and co-evaluation maps

$$\begin{array}{cccc}\hfill e{v}_a:a\otimes a^{*}& \to \mathbf{1}\hfill & \hfill coe{v}_a:\mathbf{1}& \to a^{*}\otimes a\hfill \end{array}$$

such that

$$\begin{array}{cc}\hfill {\lambda }_a\circ (e{v}_a\otimes I{d}_a)\circ {\alpha }_{a,a^{*},a}\circ (I{d}_a\otimes coe{v}_a)\circ {\rho }_a^{-1}& =I{d}_a\mathrm{and}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\rho }_{a^{*}}\circ (I{d}_{a^{*}}\otimes e{v}_{a^{*}})\circ {\alpha }_{a^{*},a,a^{*}}^{-1}\circ (coe{v}_{a^{*}}\otimes I{d}_{a^{*}})\circ {\lambda }_{a^{*}}^{-1}& =I{d}_{a^{*}}.\hfill \end{array}$$

(4)

We denote by ${\mathcal{K}}_0\left(\mathcal{C}\right)$ the Grothendieck ring of $\mathcal{C}$. This is a based ring as defined in Definition 1 with basis elements corresponding to equivalence classes of simple objects $a,b,\dots \in {\mathcal{C}}_0$ and multiplication induced from ⊗ via $N_{a_1\dots a_n}^b=dim\left(\mathcal{C}(a_1\otimes \dots \otimes a_n,b)\right)$. We say that $\mathcal{C}$ is multiplicity free if ${\mathcal{K}}_0\left(\mathcal{C}\right)$ is multiplicity free. Given two fusion categories $\mathcal{C}$ and $\mathcal{D}$, we say that they are Grothendieck equivalent if and only if ${\mathcal{K}}_0\left(\mathcal{C}\right)\cong {\mathcal{K}}_0\left(\mathcal{D}\right)$. We say that a based ring $(R,B)$admits categorification if there exists a fusion category $\mathcal{C}$ such that $(R,B)\cong {\mathcal{K}}_0\left(\mathcal{C}\right)$.

Definition 4.

Let $\mathcal{C}$ be a fusion category and ϵ be a natural isomorphism $ϵ:**\to Id$ of the double dual and identity functors. For all equivalence classes of simple objects a of $\mathcal{C}$, this gives us a morphism ${ϵ}_a:a^{**}\to a$. From this, we can define the left and rightquantum traceof a morphism f on a,

$$\begin{array}{cc}\hfill t{r}_l\left(f\right)& =e{v}_a\circ (f\otimes I{d}_{a^{*}})\circ ({\left({ϵ}_a\right)}^{-1}\otimes I{d}_{a^{*}})\circ coe{v}_{a^{*}}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill t{r}_r\left(f\right)& =e{v}_{a^{*}}\circ (I{d}_{a^{*}}\otimes {ϵ}_a)\circ (I{d}_{a^{*}}\otimes f)\circ coe{v}_a\hfill \end{array}$$

(6)

where $e{v}_{\_}$ and $coe{v}_{\_}$ are the evaluation and coevaluation maps. A natural isomorphism ϵ such that ${ϵ}_{a\otimes b}\cong {ϵ}_a\otimes {ϵ}_b$ is called apivotal isomorphism. Apivotal fusion categoryis a fusion category equipped with a pivotal isomorphism.

For every isomorphism class of simple objects a in a pivotal fusion category $\mathcal{C}$, one defines the left quantum dimension $q_l\left(a\right)$ and the right quantum dimension $q_r\left(a\right)$ 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 $\mathcal{C}$, 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 $q\left(\mathcal{C}\right)={\sum }_a{\left(q_a\right)}^2$. A spherical fusion category is called pseudo-unitary iff $q\left(\mathcal{C}\right)=FP\left({\mathcal{K}}_0\left(\mathcal{C}\right)\right)$ and there exists a canonical choice of spherical structure such that $q_a=FP\left(a\right)$ for all a.

There is also the stronger notion of a unitary fusion category.

Definition 5.

Aconjugationon a fusion category is a family of conjugate linear maps $\mathcal{C}(x,y)\to \mathcal{C}(y,x)$, $f\to \overline{f}$ satisfying

$$\begin{array}{ccc}\stackrel{=}{f}=f,\hfill & \overline{f\otimes g}=\overline{f}\otimes \overline{g},& \hfill \mathrm{and}\overline{f\circ g}=\overline{g}\circ \overline{f}.\end{array}$$

A fusion category isunitaryif $f=0$ whenever $\overline{f}\circ f=0$ [29]. In this case then, all of the $\mathcal{C}(a\otimes b,c)$ spaces are Hilbert spaces[30]. Unitary implies $q\left(a\right)=FP\left(a\right)$ and so then pseudo-unitary and spherical.

Definition 6.

Let $\mathcal{C}$ be a fusion category. A braiding on $\mathcal{C}$ is a family of natural isomorphisms $c_{x,y}:y\otimes x\to x\otimes y$ satisfying the hexagon equations as in[11]. A braided fusion category is a fusion category equipped with a braiding.

For braided fusion categories, there is a canonical natural isomorphism

$${\psi }_a={\rho }_{a^{**}}\circ (I{d}_{a^{**}}\otimes e{v}_a)\circ {\alpha }_{a^{**},a,a^{*}}^{-1}\circ (c_{a,a^{**}}\otimes I{d}_a^{*})\circ {\alpha }_{a,a^{**},a^{*}}\circ (I{d}_a\otimes coe{v}_{a^{*}})\circ {\rho }_a^{-1}.$$

(7)

A balanced fusion category is a braided pivotal fusion category with balancing ${\theta }_a={ϵ}_a\circ {\psi }_a$. The trace of this morphism, by abuse of notation, will also be denoted ${\theta }_a$ and the diagonal matrix with entries ${\theta }_a$ 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 $\mathcal{C}$ is modular, then all ${\theta }_a$ are roots of unity. The matrices $\frac{1}{q\left(\mathcal{C}\right)}S$ and $T=diag\left({\theta }_a\right)$ give a representation of the modular group $SL(2,\mathbb{Z})$. It was believed that the pair $(S,T)$ uniquely determine the modular category and typically what one does in classifying modular categories (e.g., [5,31]) 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 [32].

2.2. Fusion and Modular Systems

An important question is how one goes about constructing fusion categories. One way to do so [26] is through a collection of numbers called a fusion system.

Definition 7.

A Fusion System $(L,N,F)\left(\mathcal{C}\right)$ over k consists of

1. 

A set of labels L containing an element called $\mathbf{1}$.

2. 

An involution $\ast :L\to L$ such that ${\mathbf{1}}^{*}=\mathbf{1}$.

3. 

A set map $N:L\times L\times L\to \{0,1\}$ (written $N_{ab}^c$ for $N(a,b,c)$) satisfying

$$\begin{array}{cc}\hfill {\delta }_a^b& =N_{a\mathbf{1}}^b=N_{\mathbf{1}a}^b=N_{a{b}^{*}}^{\mathbf{1}}=N_{b^{*}a}^{\mathbf{1}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill N_{abc}^d& :=\sum _e{N}_{ab}^e{N}_{ec}^d=\sum _f{N}_{af}^d{N}_{bc}^f\hfill \end{array}$$

(9)

We will define $\Gamma (L,N)=\left\{{\gamma }_{ab}^c\right|N(a,b,c)=1\}$.

4. 

For every quadruple $a,b,c,d\in L$, an invertible $N_{abc}^d\times N_{abc}^d$ matrix $F_{abc}^d$ with entries satisfying

$$\begin{array}{cc}\hfill F_{a\mathbf{1}b}^{c;ab}& =N_{ab}^c\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill F_{a{a}^{*}a}^{a;11}& \ne 0\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \sum _h{F}_{abc}^{h;fg}{F}_{agd}^{e;hi}{F}_{bcd}^{i;gj}& =F_{fcd}^{e;hj}{F}_{abj}^{e;fi}\hfill \end{array}$$

(12)

The entries of matrix $F_{abc}^{d;ef}$ are referred to as $6J$-symbols. Define $Aut\left(N\right)$ to be the group of permutations on L such that for all ν in $Aut\left(N\right)$

$$N_{ab}^c=N_{\nu \left(a\right)\nu \left(b\right)}^{\nu \left(c\right)}.$$

This then extends to an action on F via

$$\begin{array}{c}\hfill {\left(F_{abc}^{d;ef}\right)}^{\nu }:=F_{\nu \left(a\right)\nu \left(b\right)\nu \left(c\right)}^{\nu \left(d\right);\nu \left(e\right)\nu \left(f\right)}.\end{array}$$

(13)

Central to our analysis is the ability to interpolate between fusion categories over $\mathbb{C}$ and fusion systems over $\mathbb{C}$. Given a fusion system $(L,N,F)$ over $\mathcal{C}$, one can construct a fusion category $\mathcal{C}(L,N,F)$. Given a fusion category $\mathcal{C}$ over $\mathbb{C}$, a fusion system over $\mathbb{C}$, denoted $(L,N,F)\left(\mathcal{C}\right)$, can be extracted such that $\mathcal{C}\left(\right(L,N,F\left)\right(\mathcal{C}\left)\right)\cong \mathcal{C}$ as fusion categories. For our construction, this is proven in [26] as Proposition 3.7. A similar theorem also appears in [20].

It is by analyzing a family of fusion systems for $\mathfrak{so}{(2p+1)}_2$ Grothendieck rings that we develop our classification. To do this, we will utilize the following:

Proposition 1.

Two fusion categories $\mathcal{C}$ and $\mathcal{D}$ are Grothendieck equivalent if and only if given any two fusion systems $(L,N,F)$ and $(L^{\prime },N^{\prime },F^{\prime })$ extracted from them there exists a bijection $f:L\to L^{\prime }$ such that for all $\nu \in Aut\left(N\right)$ there exists ${\nu }^{\prime }\in Aut\left(N^{\prime }\right)$ such that $f\circ \nu ={\nu }^{\prime }\circ f$.

Proof. 

If $K_0\left(\mathcal{C}\right)\cong K_0\left(\mathcal{D}\right)$ there exists a based ring isomorphism in the sense of [15] 2.1.iv, and this is determined by its action on basis elements. □

This allows us to specify a common basis for $K_0\left(\mathcal{C}\right)$ from which to work.

There also exists arithmetic data for pivotal structures.

Proposition 2.

Let $(L,N,F)$ be a fusion system extracted from a fusion category $\mathcal{C}$ and define the set ${\left\{{ϵ}_a\right\}}_{a\in L}$. Pivotal structures on $\mathcal{C}$ are in 1-1 correspondence with solutions to

$$\begin{array}{cc}\hfill {ϵ}_c^{-1}{ϵ}_a{ϵ}_b& =F_{ab{c}^{*}}^{\mathbf{1};c{a}^{*}}{F}_{b{c}^{*}a}^{\mathbf{1};a^{*}{b}^{*}}{F}_{c^{*}ab}^{\mathbf{1};b^{*}c}.\hfill \end{array}$$

(14)

This is proven in ([26] [Proposition 3.12]).

Given a fusion category $\mathcal{C}$ with pivotal structure $ϵ$, if it is a spherical structure then there exists a fusion system $(L,N,F)\left(\mathcal{C}\right)$ such that all ${ϵ}_a=\pm 1$. By skeletalizing the rigidity conditions and using the choice of basis as in ([26] [Lemma 3.4]), the quantum dimensions can be computed from F and $ϵ$ as

$$\begin{array}{cccc}\hfill q_l\left(a\right)& ={ϵ}_a{\left(F_{a^{*}a{a}^{*}}^{a^{*};11}\right)}^{-1}\mathrm{and}\hfill & \hfill q_r\left(a\right)& ={\left({ϵ}_a{F}_{a{a}^{*}a}^{a;11}\right)}^{-1}\hfill \end{array}$$

(15)

If $\mathcal{C}$ admits the structure of a unitary fusion category, then there exists a fusion system $(L,N,F)\left(\mathcal{C}\right)$ such that all $F_{abc}^d$ matrices are unitary. Conversely (see [20], Section 4), if given $(L,N)$ there exists a solution to (10)–(12) such that all $F_{abc}^d$ matrices are unitary, then $\mathcal{C}(L,N,F)$ admits a unitary structure.

Definition 8.

A Modular System $(L,N,F,R,ϵ)$ is a fusion system $(L,N,F)$ such that $N_{ba}^c=N_{ab}^c,\forall {\gamma }_{ab}^c\in \Gamma (L,N)$, $ϵ={\left\{{ϵ}_a\right\}}_{a\in L}$ is a solution to(14), and $\left\{R_{ab}^c\right|{\gamma }_{ab}^c\in \Gamma (L,N)\}$ is a collection of numbers satisfying

$$\begin{array}{cc}\hfill R_{ac}^g{F}_{acb}^{d;gf}{R}_{bc}^f& =\sum _{e\in L}{F}_{cab}^{d;ge}{R}_{ec}^d{F}_{abc}^{d;ef}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\left(R_{ac}^g\right)}^{-1}{F}_{acb}^{d;gf}{\left(R_{bc}^f\right)}^{-1}& =\sum _{e\in L}{F}_{cab}^{d;ge}{\left(R_{ec}^d\right)}^{-1}{F}_{abc}^{d;ef}\hfill \end{array}$$

(17)

such that the matrix with entries

$${\widehat{S}}_{ab}=\sum _{c\in L}{F}_{a{b}^{*}b}^{a;c1}{R}_{a{b}^{*}}^c{R}_{b^{*}a}^c{F}_{a{b}^{*}b}^{a;1c}$$

(18)

is invertible.

Similar to (13), we can define the action of $\nu \in Aut\left(N\right)$ on R via

$$\begin{array}{c}\hfill {\left(R_{ab}^c\right)}^{\nu }:=R_{\nu \left(a\right)\nu \left(b\right)}^{\nu \left(c\right)}.\end{array}$$

(19)

The matrix $\widehat{S}$ is related to the matrix S via $S=D\widehat{S}D$. Additionally, we can compute the twists as

$${\theta }_a={\left(q_a\right)}^{-1}\sum _{c\in L}{q}_c{R}_{aa}^c$$

(20)

obtained by taking the trace of the morphism (7) above [30]. Finally, we give a formula for the topological central charge $c_{\mathrm{top}}$ (see [33]). We introduce

$$p_{\pm }=\sum _{a\in L}{\theta }_a^{\pm }{q}_a^2,$$

(21)

from which one obtains the topological central charge, defined modulo eight,

$$\begin{array}{cccc}\hfill e^{2\pi ıc/8}& =\frac{p_{+}}{\sqrt{{\sum }_a{q}_a^2}}\hfill & \hfill c_{\mathrm{top}}& =cmod8.\hfill \end{array}$$

(22)

Note that through teasing out definitions, all of the information above can be written in terms of F, R, and $ϵ$. Since, in our case, we have all of the F’s, R’s, and $ϵ$’s, so we can simply compute $(S,T)$ for the given the arithmetic data.

3. Monoidal Equivalence and Gauge Invariants

Given two fusion (modular) categories $\mathcal{C}$ and $\mathcal{D}$, a central question is one of whether or not they are equivalent in some suitable sense. The strongest of these is monoidal equivalence.

Definition 9.

$\mathcal{C}$ and $\mathcal{D}$ are said to bemonoidally equivalentiff there exists a pair of monoidal functors (see [9]) $\mathcal{F}:\mathcal{C}\to \mathcal{D}$ and $\mathcal{G}:\mathcal{C}\to \mathcal{D}$ such that $\mathcal{F}\circ \mathcal{G}$ and $\mathcal{G}\circ \mathcal{F}$ are naturally monoidal isomorphic to the identity functors on $\mathcal{C}$ and $\mathcal{D}$, respectively. If $\mathcal{C}$ and $\mathcal{D}$ are braided monoidal, then they arebraided monoidally equivalentif $\mathcal{F}$ and $\mathcal{G}$ are braided monoidal functors.

A family of categories ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_n$ 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 [27].

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 $Aut\left(N\right)$ on a fusion system $(L,N,F,R,ϵ)$ is given by (13) and it was shown in [26] that this gives an object permuting monoidal functor. We say that two fusion systems $F,F^{\prime }$ are permutation equivalent if there exists $\nu \in Aut\left(N\right)$ such that $F^{\prime }=F^{\nu }$. 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 $\mathcal{C}(a\otimes b,c)$ homspaces and can be represented arithmetically as follows:

Define G to be the set of functions $g:\Gamma (L,N)\to k^{\times }$ fixing the notation $g_{ab}^c:=g(a,b,c)$ and $g_c^{ab}:={\left(g(a,b,c)\right)}^{-1}$. Then given a solution F to (10)–(12) and $g\in G$, one obtains another solution $F^g$ via the equations

$$\begin{array}{cc}\hfill {\left(F^g\right)}_{abc}^{d;ef}& :=\left(g_{ab}^e\right)\left(g_{ec}^d\right)F_{abc}^{d;ef}\left(g_f^{bc}\right)\left(g_d^{af}\right)\hfill \end{array}$$

(23)

and two solutions F and $F^{\prime }$ are gauge equivalent if and only if there exists $g\in G$ such that $F^{\prime }=g$. (23) can be used to construct an object fixing monoidal functor. Given $\mathcal{C}$, it is clear from the definition that any two fusion systems extracted from $\mathcal{C}$ are gauge equivalent. Monoidal equivalence of fusion categories $\mathcal{C}$ and $\mathcal{D}$ is then determined by the existence of a permutation equivalence together with a gauge equivalence.

Remark 2.

The above extends readily to braided monoidal equivalence between modular categories. Given R and $R^{\prime }$, permutation equivalence requires the imposition of the extra condition $R=R^{\nu }$ as defined in(19). Gauge equivalence requires the imposition that $R^{\prime }=R^g$ as given by

$$\begin{array}{cc}\hfill \left(R^g\right)& :=\left(g_{ab}^c\right)R_{ab}^c\left(g_c^{ba}\right).\hfill \end{array}$$

(24)

Given a set of fusion(modular) systems, determining their monoidal equivalence is not an easy task in general. However, in [25], 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 $\Phi (L,N)$. F can then be interpreted as an algebra homomorphism $F:\mathbb{C}[\Phi (L,N\left)\right]\to \mathbb{C}$ with $F_{abc}^{d;ef}=F\left({\Phi }_{abc}^{d;ef}\right)$ for ${\Phi }_{abc}^{d;ef}\in \Phi$. From here, on an open subset U of $X(L,N)$, it is straightforward to extend F to ${\mathcal{O}}_X\left(U\right)$, 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 $Aut\left(N\right)⋉G$-orbits in $X(L,N)$. Given this, one can leverage the geometric invariant theory of [34] to obtain the following result:

Theorem 2.

Fix $(L,N)$. Let $\mathcal{N}$ be the number of gauge equivalence classes of fusion categories and $\mathcal{M}$ be the number monoidal equivalence classes of fusion categories. Then there exist $P\le \left(\genfrac{}{}{0pt}{}{\mathcal{N}}{2}\right)$G-invariant rational monomials with

$$\begin{array}{cccccccccc}\hfill m_i& ={\left({\varphi }_{i,1}\right)}^{k_{i,1}}\dots {\left({\varphi }_{i,j}\right)}^{k_{i,j}},\hfill & \hfill \mathrm{with}& \hfill & \hfill \varphi & \in \Phi (L,N)\hfill & \hfill \mathrm{and}& \hfill & \hfill k& \in \mathbb{Z}\hfill \end{array}$$

(25)

such that for fusion systems F and $F^{\prime }$, F and $F^{\prime }$ are gauge equivalent if and only if

$$\begin{array}{ccc}\hfill F\left(m_i\right)& =F^{\prime }\left(m_i\right),\hfill & \hfill i=1,\dots P.\end{array}$$

(26)

Similarly, there exist $Q\le \left(\genfrac{}{}{0pt}{}{\mathcal{M}}{2}\right)$$Aut\left(N\right)$-linear combinations $l_1,\dots ,l_q$ of G-invariant monomials as in(25)such that F and $F^{\prime }$ are monoidally equivalent if and only if

$$\begin{array}{ccc}\hfill F\left(l_j\right)& =F^{\prime }\left(l_j\right),\hfill & \hfill j=1,\dots Q.\end{array}$$

(27)

$Aut\left(N\right)$-linear combinations of G-invariants will be called $Aut\left(N\right)⋉G$-invariants for obvious reasons.

This is Theorem 1.2 of [25]. It is straightforward to extend this construction to classifying modular systems $(L,N,F,R,ϵ)$. 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 $Aut\left(N\right)⋉G$-invariants is not necessarily apparent a priori. However, as we will show in Section 5, there is an easy choice which works for our $\mathfrak{so}{(2p+1)}_2$ 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 [32], 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 $q^{\prime }s$, $S^{\prime }s$, and $T^{\prime }s$ can all be written using $F^{\prime }s$, $R^{\prime }s$, and ${ϵ}^{\prime }s$. Thus they all define G-invariant regular functions on X.

4. $\mathfrak{so}{(\mathbf{2}\mathit{p}+\mathbf{1})}_{\mathbf{2}}$ 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 $B_p^{\left(1\right)}$ (i.e., $\mathfrak{so}(2p+1)$) with highest integral weight 2. As determined by [35], these are equivalent (as modular categories) to the semi-simplification (à la [36]) of the representation category for $U_q\left(B_p\right)$ with $q={\mathbf{e}}^{\frac{\pi ı}{2p+1}}$. The affine Kac–Moody and quantum group constructions for the Grothendieck rings can be found in [21,22], respectively.

Another source for our fusion rules comes from Tambara–Yamagami categories as follows. Let p a positive integer, $\chi$ a symmetric bicharacter on ${\mathbb{Z}}_{2p+1}$ and $\nu =\pm$. In [23], it was shown that ${\mathbb{Z}}_2$-de-equivariantizations of the Tambara–Yamagami category $\mathcal{T}\mathcal{Y}({\mathbb{Z}}_{2p+1},\chi ,\nu )$ gives rise to an $\mathfrak{so}{(2p+1)}_2$ category.

4.1. Fusion Rules for $\mathfrak{so}{(2p+1)}_2$ Categories

For some $\mathfrak{so}{(2p+1)}_2$ category $\mathcal{C}$, we label the basis elements of $K_0\left(\mathcal{C}\right)$ (i.e., equivalence classes of simple objects) by elements of the set $L=\{\mathbf{1},ϵ,{\varphi }_i,{\psi }_{\pm }\}$ with $i=1,\dots p$. These have Frobenius–Perron dimensions $\{1,1,2,\sqrt{2p+1}\}$, respectively. $K_0\left(\mathcal{C}\right)$ is commutative with non-trivial products given by:

$$\begin{array}{cccccc}\hfill ϵ\otimes ϵ& \cong \mathbf{1}\hfill & \hfill {\varphi }_i\otimes {\varphi }_i& \cong \mathbf{1}\oplus ϵ\oplus {\varphi }_{g\left(2i\right)}\hfill & \hfill {\psi }_{\pm }\otimes {\psi }_{\pm }& \cong \mathbf{1}\underset{j=1}{\overset{p}{⨁}}{\varphi }_j\hfill \\ \hfill ϵ\otimes {\varphi }_i& \cong {\varphi }_i\hfill & \hfill {\varphi }_i\otimes {\varphi }_j& \cong {\varphi }_{g(i-j)}\oplus {\varphi }_{g(i+j)}\hfill & \hfill {\psi }_{\pm }\otimes {\psi }_{\mp }& \cong ϵ\underset{j=1}{\overset{p}{⨁}}{\varphi }_j\hfill \\ \hfill ϵ\otimes {\psi }_{\pm }& \cong {\psi }_{\mp }\hfill & \hfill {\varphi }_i\otimes {\psi }_{\pm }& \cong {\psi }_{\pm }\otimes {\psi }_{\mp }\hfill \end{array}$$

where $g:{\mathbb{Z}}_{2p+1}\to {\mathbb{G}}_{2p+1}$ with ${\mathbb{G}}_{2p+1}=\{0,\dots p\}$ is given by

$$g\left(a\right)=\left|a\right|.$$

Proposition 3.

Let $K_0\left(\mathcal{C}\right)$ be the Grothendieck ring of a $\mathfrak{so}{(2p+1)}_2$ category. Then

1. 

The automorphisms which permute the ${\varphi }_i$ are given by ${\mathbb{G}}_{2p+1}^{\times }:={\mathbb{Z}}_{2p+1}^{\times }/\langle 1,-1\rangle$,

2. 

The automorphisms which permute the ${\psi }_{\pm }$ are given by ${\mathbb{Z}}_2$, and

3. 

The automorphism group of $K_0\left(\mathcal{C}\right)$ is ${\mathbb{G}}_{2p+1}^{\times }\times {\mathbb{Z}}_2$.

Proof. 

That the fusion rules are invariant under exchange of ${\psi }_{\pm }$ is straight forward.

The automorphisms which permute the ${\varphi }_i$ follow restricting g to ${\mathbb{Z}}_{2p+1}^{\times }$ and promoting it to a group homomorphism. ${\mathbb{Z}}_{2p+1}^{\times }$ has even order and can be represented by integers in $\pm 1,\dots ,\pm p$. (Specifically the order has to be $\phi (2p+1)$ where $\phi$ is Euler’s totient function. For power k of prime r$\phi \left(r^k\right)=r^{(k-1)}(r-1)$, which is even. That the evaluation of $\phi$ is even for every $2p+1$ follows from prime factorization and the multiplicative property of $\phi$.)

If i is in ${\mathbb{Z}}_{2p+1}^{\times }$ so is $-i$, thus the quotient ${\mathbb{Z}}_{2p+1}^{\times }/\langle 1,-1\rangle$ is well defined. g is precisely this quotient map. (There is not an element of ${\mathbb{G}}_{2p+1}^{\times }$ for all $\pm i$. To see this consider ${\mathbb{Z}}_9^{\times }$, which has order six rather than eight, since three has no multiplicative inverse in ${\mathbb{Z}}_9$.) ${\mathbb{G}}_{2p+1}^{\times }$ acts on ${\varphi }_i$ 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 $K_0\left(\mathcal{C}\right)$ must be the direct product. This gives a separate proof from that in [21]. □

4.2. F-Matrices

Our solutions to (10)–(12) are indexed by pairs $(r,\kappa )$ where r is an odd integer between 1 and $2p+1$, such that $gcd(r,2p+1)=1$ and $\kappa =\pm 1$.

4.2.1. Notation

Fix $q={\mathbf{e}}^{\frac{\pi ı}{2p+1}}$. To write the general F-symbols, we first introduce the following matrices.

$$\begin{array}{cccccc}\hfill A(s_1,s_2,s_3,s_4)& =\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{s}_1& s_2\\ s_3& s_4\end{array}\right)\hfill & \hfill B& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\hfill & \hfill C& =\frac{1}{2}\left(\begin{array}{ccc}1& -1& \sqrt{2}\\ -1& 1& \sqrt{2}\\ \sqrt{2}& \sqrt{2}& 0\end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill D(r;t;s_1,s_2,s_3,s_4)& =\left(\begin{array}{cc}{s}_{1}Re\left(q^{rt}\right)& s_{2}Im\left(q^{rt}\right)\\ s_{3}Im\left(q^{rt}\right)& s_{4}Re\left(q^{rt}\right)\end{array}\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill E(r;t;s_1,s_2,s_3,s_4)& =\left(\begin{array}{cc}{s}_{1}Im\left(q^{rt}\right)& s_{2}Re\left(q^{rt}\right)\\ s_{3}Re\left(q^{rt}\right)& s_{4}Im\left(q^{rt}\right)\end{array}\right)\hfill \end{array}$$

(29)

where the $s_i$ take values $\pm 1$ and satisfy $s_1{s}_2{s}_3{s}_4=-1$ such that the associated F-matrices are orthogonal.

In addition, we define the matrices $G(r,\kappa )$, $H(r,\kappa )$ and $H^{\prime }(r,\kappa )$ from the following function:

$$J(i,j;r,\kappa )=\frac{2^{\zeta (i,j)}\kappa }{\sqrt{2p+1}}{\left(q^r\right)}^{ij}$$

(30)

where $\zeta \left(i,j\right)=\frac{2-{\delta }_{i0}-{\delta }_{j0}}{2}$.

$G(r,\kappa )$ is a $p\times p$ matrix with entries

$$G{(r,\kappa )}_{i,j}={(-1)}^{(i-1)(j-1)}Im\left(J(i,j;r,\kappa )\right)$$

(31)

whose indices run over $\{1,\dots ,p\}$. $H(r,\kappa )$ and $H^{\prime }(r,\kappa )$ are $(p+1)\times (p+1)$ matrices with entries

$$\begin{array}{cc}\hfill H{(r,\kappa )}_{i,j}& ={(-1)}^{ij}Re\left(J(i,j;r,\kappa )\right)\hfill \\ \hfill H^{\prime }{(r,\kappa )}_{i,j}& ={(-1)}^{{\delta }_{i0}+{\delta }_{j0}+1}H{(r,\kappa )}_{i,j}\hfill \end{array}$$

(32)

whose indices run over $\{0,\dots ,p\}$. We note that the matrices $G\left(r\right)$ and $H^{{(}^{\prime })}\left(r\right)$ are orthogonal provided that r is an odd integer relatively prime to $2p+1$.

4.2.2. 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: $(\mathbf{1},ϵ,{\varphi }_i,{\psi }_{\pm })$, where $i=1,\dots ,p$. Second, we will actually not give the values $F_{abc}^{d;ef}$ individually, but instead give the F-matrices $F_{abc}^d$.

In the basis we use to give the F-matrices, we have the following property

$$F_{bcd}^a={\left(F_{abc}^d\right)}^T.$$

(33)

In addition, it is implicitly assumed that the label d of $F_{abc}^d$ is in the tensor decomposition of $a\otimes b\otimes c$. 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 [37] that for fixed p the fusion rules for $\{1,ϵ,{\psi }_i\}$ give the tensor structure for $Rep\left(D_{2p+1}\right)$. $K_0\left(\mathcal{C}\right)$ is then a ${\mathbb{Z}}_2$-extension of $K_0\left(Rep\left(D_{2p+1}\right)\right)$. We will proceed by building subcategories Grothendieck equivalent to ${\mathbb{Z}}_2$ and $Rep\left(D_{2p+1}\right)$ before finally specifying F-matrices which correspond to the $\sqrt{2p+1}$ objects.

First, if one or more of the labels $a,b,c,d$ equals $\mathbf{1}$, $F_{abc}^d=\left(1\right)$. Next, we have $\{\mathbf{1},ϵ\}$ equivalent to ${\mathbb{Z}}_2$ and the action of $ϵ$ on the objects ${\varphi }_i$ is determined by

$$\begin{array}{cccccc}\hfill F_{ϵϵϵ}^{ϵ}& =\left(1\right)\hfill & \hfill F_{ϵ{\varphi }_iϵ}^{{\varphi }_i}& =\left(1\right)\hfill & \hfill F_{ϵϵ{\varphi }_i}^{{\varphi }_i}& =\left(-1\right)\hfill \end{array}$$

(34)

$$\begin{array}{cccc}\hfill F_{ϵ{\varphi }_i{\varphi }_j}^{{\varphi }_k}& =(-1^{jmod2})\hfill & \hfill & (j\le i)\hfill \end{array}$$

(35)

$$\begin{array}{cccc}\hfill F_{ϵ{\varphi }_i{\varphi }_j}^{{\varphi }_k}& =(-1^{jmod2})\hfill & \hfill & (j>i\wedge k=g(i+j\left)\right)\hfill \end{array}$$

(36)

$$\begin{array}{cccc}\hfill F_{ϵ{\varphi }_i{\varphi }_j}^{{\varphi }_k}& =(-1^{(j-1)mod2})\hfill & \hfill & (j>i\wedge k=g(i-j\left)\right)\hfill \end{array}$$

(37)

The rest of the F-matrices for the category $Rep\left(D_{2p+1}\right)$ are given by:

$$\begin{array}{cc}\hfill F_{{\varphi }_i{\varphi }_i{\varphi }_i}^{{\varphi }_i}& =C\hfill \end{array}$$

(38)

$$\begin{array}{cccc}\hfill F_{{\varphi }_i{\varphi }_j{\varphi }_i}^{{\varphi }_j}& =B\hfill & \hfill & (j\ne i)\hfill \end{array}$$

(39)

$$\begin{array}{cccc}\hfill F_{{\varphi }_i{\varphi }_i{\varphi }_j}^{{\varphi }_j}& =A(1,1,-1^{(j-i+1)mod2},-1^{(j-i)mod2})\hfill & \hfill & (j\ne i)\hfill \end{array}$$

(40)

$$\begin{array}{cccc}\hfill F_{{\varphi }_i{\varphi }_j{\varphi }_k}^{{\varphi }_l}& =\left(1\right)\hfill & \hfill & (\mathrm{O}.\mathrm{W}.)\hfill \end{array}$$

(41)

Proposition 4.

There are no other solutions to the pentagon equations for $Rep\left(D_{2p+1}\right)$ which extend to solutions for $\mathfrak{so}{(2p+1)}_2$.

Proof. 

By [38], all fusion categories $\mathcal{C}$ for $Rep\left(D_{2p+1}\right)$ are group theoretical and thus Morita equivalent to a pointed fusion category $\mathcal{D}$ of the form $(D_{2p+1},\omega )$ where $\omega \in H^3(D_{2p+1},{\mathbb{C}}^{\times })$, the group of $2p+1$ roots of unity. By [15] we know that (left) module categories over $D_{2p+1}$ are parameterized by $(H,\zeta )$ where $H\le G$ such that ${\omega |}_H=1$ and $\zeta \in H^2(H,{\mathbb{C}}^{\times })$.

In our case, we have that since the universal grading of $\mathfrak{so}{(2p+1)}_2$ is ${\mathbb{Z}}_2$, we have that $\{{\psi }_{+},{\psi }_{-}\}$ are indecomposable ${\mathbb{Z}}_2$ module categories over $Rep\left(D_{2p+1}\right)$. We can count which $Rep\left(D_{2p+1}\right)$ categories have ${\mathbb{Z}}_2$ module categories by looking at which $D_{2p+1}$ categories have ${\mathbb{Z}}_2$ module categories. Since $H^3(D_{2p+1},{\mathbb{C}}^{\times })\cong {\mathbb{Z}}_{2p+1}$ the only $\omega$ which fixes the ${\mathbb{Z}}_2$ subgroup of $D_{2p+1}$ is $\omega =1$ and its cohomology is ${\mathbb{Z}}_2$. Thus, only $Rep\left(D_{2p+1}\right)$ has indecomposable ${\mathbb{Z}}_2$ module categories and there are two of them. □

We now specify the F-matrices that are labeled by ${\psi }_{\pm }$ and start with the F-matrices whose labels consist of only ${\psi }_{\pm }$.

$$\begin{array}{cc}\hfill F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}& =H(r,\kappa )\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill F_{{\psi }_{\pm },{\psi }_{\mp },{\psi }_{\pm }}^{{\psi }_{\mp }}& =-H(r,\kappa )\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\mp }}^{{\psi }_{\mp }}& =H^{\prime }(r,\kappa )\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\mp }}& =\pm G(r,\kappa )\hfill \end{array}$$

(45)

There are three classes of F-matrices involving two labels that are ${\psi }_{\pm }$. The first class of this type is

$$\begin{array}{cc}\hfill F_{{\varphi }_i{\psi }_{\pm }{\varphi }_j}^{{\psi }_{\pm }}& =\mp {(-1)}^{ij}D(r;ij;-1,-1^{(i+j)},-1^{(i+j)},1)\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill F_{{\varphi }_i{\psi }_{\pm }{\varphi }_j}^{{\psi }_{\mp }}& =-{(-1)}^{ij}E(r;ij;-1^{(i+j)},1,1,-1^{(i+j+1)})\hfill \end{array}$$

(47)

where $i,j$ may or may not be equal. The second class of F-matrices with two labels equal to ${\psi }_{\pm }$ and $i\ne j$ is

$$\begin{array}{cc}\hfill F_{{\varphi }_i{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\pm }}& =A(1,1,1,-1)\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{\pm }}^{{\psi }_{\pm }}& =A(\pm 1,\pm 1,1,-1)\hfill & \hfill & (i-j)mod2=0\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{\pm }}^{{\psi }_{\pm }}& =A(1,-1,\pm 1,\pm 1)\hfill & \hfill & (i-j)mod2=1\hfill \\ \hfill F_{{\varphi }_i{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\mp }}& =A(-1,-1,{(-1)}^i,{(-1)}^{(i+1)})\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{+}}^{{\psi }_{-}}& =A({(-1)}^{(j+1)},{(-1)}^{(i+1)},{(-1)}^j,{(-1)}^{(i+1)})\hfill & \hfill & i<j\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{-}}^{{\psi }_{+}}& =A({(-1)}^i,{(-1)}^j,{(-1)}^i,{(-1)}^{(j+1)})\hfill & \hfill & i<j\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{+}}^{{\psi }_{-}}& =A({(-1)}^j,{(-1)}^i,{(-1)}^j,-1^{(i+1)})\hfill & \hfill & i>j\hfill \\ \hfill F_{{\varphi }_i{\varphi }_j{\psi }_{-}}^{{\psi }_{+}}& =A({(-1)}^{(i+1)},{(-1)}^{(j+1)},{(-1)}^i,-1^{(j+1)})\hfill & \hfill & i>j\hfill \end{array}$$

Finally, the third class of F-matrices with two labels equal to ${\psi }_{+}$ or ${\psi }_{-}$ is

$$\begin{array}{cc}\hfill F_{ϵ{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\pm }}& =F_{ϵ{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\mp }}=F_{ϵ{\psi }_{\pm }{\varphi }_i}^{{\psi }_{\pm }}=F_{ϵ{\psi }_{\pm }{\psi }_{\pm }}^{{\varphi }_i}=F_{ϵ{\psi }_{\pm }{\psi }_{\mp }}^{{\varphi }_i}=\left(1\right)\hfill \\ \\ \hfill F_{ϵ{\psi }_{\pm }{\varphi }_i}^{{\psi }_{\mp }}& =F_{ϵϵ{\psi }_{\pm }}^{{\psi }_{\pm }}=F_{ϵ{\psi }_{\pm }ϵ}^{{\psi }_{\pm }}=\left(-1\right)\hfill \end{array}$$

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 R-symbols $R_{ab}^c$, such that a and b are both one of ${\psi }_{\pm }$ by $-R_{ab}^c$. However, this is monoidally equivalent to our original solution via the automorphism ${\psi }_{\pm }\to {\psi }_{\mp }$. This gives us that for any p, solutions to the pentagon and hexagon equations can be uniquely identified by the tuple $(p,r,\kappa ,\lambda )$, where $\lambda =\pm 1$ 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

$$R_{ab}^c={\left({\sigma }_1\right)}_{ab}^c{\left({\sigma }_2\right)}_{ab}^c{(-1)}^{h_a+h_b-h_c},$$

where ${\left({\sigma }_1\right)}_{ab}^c=\pm 1$ and ${\left({\sigma }_2\right)}_{ab}^c=\pm 1$. Below, we specify the scaling dimensions $h_a$ for all the simple objects, as well as the signs ${\left({\sigma }_1\right)}_{ab}^c$ and ${\left({\sigma }_2\right)}_{ab}^c$.

The scaling dimensions are given by

$$\begin{array}{cc}\hfill h_{\mathbf{1}}& =0\hfill \\ \hfill h_{ϵ}& =1\hfill \\ \hfill h_{{\varphi }_i}& =\frac{ri(2p+1-i)}{2(2p+1)}\hfill \\ \hfill h_{{\psi }_{+}}& =\frac{r(p+\kappa s_p-(2p+1|r)+2)}{8}\hfill \\ \hfill h_{{\psi }_{-}}& =\frac{r(p+\kappa s_p-(2p+1|r)+6)}{8}\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill s_p& =-{(-1)}^{\frac{p(p+1)}{2}}=-\left(2\right|2p+1).\hfill \end{array}$$

The notation $\left(j\right|n)$ 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 ${\left({\sigma }_1\right)}_{ab}^c$ and ${\left({\sigma }_2\right)}_{ab}^c$.

We start with ${\left({\sigma }_1\right)}_{ab}^c$, and note that for the R-symbols presented here we have the property ${\left({\sigma }_1\right)}_{bc}^a={\left({\sigma }_1\right)}_{ab}^c$, 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

$${\left({\sigma }_1\right)}_{{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\pm }}={\left({\sigma }_1\right)}_{{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\mp }}=\left\{\begin{array}{cc}-1\hfill & \left(\right(imod4)=(1\vee 2\left)\right)\wedge (pmod2)=1\hfill \\ -1\hfill & \left(\right(imod4)=(2\vee 3\left)\right)\wedge (pmod2)=0\hfill \\ +1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

The second class of symbols is

$${\left({\sigma }_1\right)}_{{\varphi }_i{\varphi }_j}^{{\varphi }_{g(i+j)}}={(-1)}^{\left(ij\right)}.$$

(48)

The symbols ${\left({\sigma }_1\right)}_{ab}^c$ that are not specified by the rules above, are all equal to $+1$.

Finally, the signs ${\left({\sigma }_2\right)}_{ab}^c$ also satisfy the property ${\left({\sigma }_2\right)}_{bc}^a={\left({\sigma }_2\right)}_{ab}^c$ and there is only one class of non-trivial signs of type, namely

$${\left({\sigma }_2\right)}_{ϵ{\psi }_{\pm }}^{{\psi }_{\mp }}={\left({\sigma }_2\right)}_{{\varphi }_i{\psi }_{\pm }}^{{\psi }_{\mp }}={(-1)}^{\frac{r-1}{2}}.$$

The symbols ${\left({\sigma }_2\right)}_{ab}^c$ 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 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 $RFR=\sum FRF$. To derive the form of $h_{{\psi }_{\pm }}$ above, consider the hexagon equation associated to $F_{{\psi }_{+}{\psi }_{+}{\psi }_{+}}^{{\psi }_{+};\mathbf{11}}$. The sum on the right hand side gives rise to a quadratic Gauss sum (because one needs to sum over all ${\varphi }_i$ (as well as the identity)), which are closely related to the Jacobi symbols. This leads to their presence in the scaling dimensions $h_{{\psi }_{\pm }}$. See Appendix A for more details.

As stated above, the Jacobi symbols $\left(j\right|n)$ 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 $H(r,\kappa )$ and $G(r,\kappa )$ (whose sizes depend on p). In particular, one can show

$$\left(r\right|2p+1)=det(H(2r+2p+1))det(G(2r+2p+1)).$$

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, $\left(q\right|p)={\prod }_{n=1}^{(p-1)/2}\frac{sin(2\pi qn/p)}{sin(2\pi n/p)}$; see, for instance [39]. The difference with the function that we found, is that the values $\pm 1$ are the extrema of the function while this is certainly not the case for Eisenstein’s function.

Modular Data for $\mathfrak{so}{(2p+1)}_2$ 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 [23]. 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 ([31] [Lemma 2.4]). Since the group of invertible objects is ${\mathbb{Z}}_2$, there are exactly two pivotal structures for each category, both of which are spherical. It is straightforward to compute that for $\mathbf{1}$, $ϵ$, and ${\varphi }_i$, all pivotal coefficients must be 1. We also have ${ϵ}_{{\psi }_{+}}={ϵ}_{{\psi }_{-}}=\pm 1$, the choice of which switches the sign of $q_{{\psi }_{\pm }}$.

Below, we present the modular data for solutions with positive quantum dimensions. It is worth noting that for all p and r the pivotal coefficients which yield positive quantum dimensions for $\kappa =1$ are not those that yield positive quantum dimensions for $\kappa =-1$. This can be seen from noting the effect of $\kappa$ on (15). In particular, $q_{{\psi }_{\pm }}>0$ if we take ${ϵ}_{{\psi }_{\pm }}=\kappa$.

We then have that, for modular categories corresponding to the solution indexed by $(p,r,\kappa ,\lambda )$,

$$\begin{array}{cccccc}\hfill S_{\mathbf{11}}& =1\hfill & \hfill S_{ϵϵ}& =1\hfill & \hfill S_{{\varphi }_i{\varphi }_j}& =4cos\left(\frac{2\pi ijr}{2p+1}\right)\hfill \end{array}$$

(49)

$$\begin{array}{cccccc}\hfill S_{\mathbf{1}ϵ}& =1\hfill & \hfill S_{ϵ{\varphi }_i}& =2\hfill & \hfill S_{{\varphi }_i{\psi }_{\pm }}& =0\hfill \end{array}$$

(50)

$$\begin{array}{cccccc}\hfill S_{\mathbf{1}{\varphi }_i}& =2\hfill & \hfill S_{ϵ{\psi }_{\pm }}& =-q_{{\psi }_{\pm }}\hfill & \hfill S_{{\psi }_{\pm }{\psi }_{\pm }}& =-\kappa \left(2\right|2p+1)q_{{\psi }_{\pm }}\hfill \end{array}$$

(51)

$$\begin{array}{cccccc}\hfill S_{\mathbf{1}{\psi }_{\pm }}& =q_{{\psi }_{\pm }}\hfill & & & \hspace{1em}\hspace{1em}\hspace{1em}{S}_{{\psi }_{\pm }{\psi }_{\mp }}\hfill & =\kappa \left(2\right|2p+1)q_{{\psi }_{\mp }}\hfill \end{array}$$

(52)

For the twists we have

$$\begin{array}{cccc}\hfill {\theta }_{ϵ}& =1\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\theta }_{{\varphi }_i}\hfill & ={(-1)}^i{\mathbf{e}}^{ı\pi \frac{\lambda r{i}^2}{2p+1}}\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\theta }_{{\psi }_{\pm }}& =\mp {(-1)}^{\frac{\kappa \lambda r}{2}}{\mathbf{e}}^{\frac{ı\pi \lambda r}{4}((2p+1|r)+\kappa \left(2\right|2p+1)-p)}.\hfill \end{array}$$

(54)

For completeness, we also calculate the topological central charge from Equations (21) and (22). We first note that the contributions from ${\psi }_{\pm }$ 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 $p_{+}/\sqrt{{\sum }_a{q}_a^2}={ϵ}_{2p+1}\left(2\lambda r\right|2p+1)$, where ${ϵ}_{2p+1}=1$ for p even, and ${ϵ}_{2p+1}=ı$ for p odd. Using standard manipulations of the Jacobi symbol, one can finally write

$$c_{\mathrm{top}}=2p(\lambda +2p)-2+2\left(r\right|2p+1)mod8.$$

(55)

5. Monoidal Equivalence of $\mathfrak{so}{(\mathbf{2}\mathit{p}+\mathbf{1})}_{\mathbf{2}}$ 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 $(r,\kappa )$ and $(r^{\prime },{\kappa }^{\prime })$ are gauge equivalent if and only if $r=r^{\prime }$ and $\kappa ={\kappa }^{\prime }$. Our goal then is to find a set $\{l_1,\dots ,l_n\}$ of $Aut\left(N\right)$-invariant linear combination of G-invariant monomials as defined in Theorem 2 which determine the monoidal classes. We then determine the number of monoidal equivalence classes for a given p. For all pairs $(r,\kappa )$ as defined in Section 4.2, let $F_{(r,k)}$ be the evaluation map as defined for Theorem 2.

5.1. Determining Equivalence

We first note that the adjoint category is basically useless to us.

Proposition 5.

The monoidal classes of solutions specified in Section 4.2.2 cannot be distinguished by $Aut\left(N\right)⋉G$-invariants coming from $Rep\left(D_{2p+1}\right)$ subcategories.

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, $H^{{(}^{\prime })}$, and G matrices. These come from F-matrices involving at least one ${\psi }_{\pm }$ and so do not come from the $Rep\left(D_{2p+1}\right)$ subcategory. So if l is an $Aut\left(N\right)⋉G$-invariant which comes from $Rep\left(D_{2p+1}\right)$, i.e., only involves the objects 1, $ϵ$, and ${\varphi }_i$, $i=1,\dots p$ then

$$F_{(r,k)}\left(l\right)=F_{(r^{\prime },k^{\prime })}\left(l\right).$$

□

From here, we can narrow our search even further. From (42) to (47), the entries of

$$F_{{\varphi }_i{\psi }_{\pm }{\varphi }_j}^{{\psi }_{\pm }},F_{{\varphi }_i{\psi }_{\pm }{\varphi }_j}^{{\psi }_{\mp }},F_{{\psi }_{\pm }{\psi }_{\mp }{\psi }_{\mp }}^{{\psi }_{\mp }},\mathrm{and}{F}_{{\psi }_{\pm }{\psi }_{\mp }{\psi }_{\pm }}^{{\psi }_{\mp }}$$

can all be written in some fixed way as scalar multiples of entries in $F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}$ or $F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\mp }}$. These relationships are preserved under permutation. The entries of $F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}$ and $F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\mp }}$ are given by the matrices $H(r,\kappa )$ and $\pm G(r,\kappa )$, respectively.

Proposition 6.

Let $F_{(r,\kappa )},F_{(r^{\prime },{\kappa }^{\prime })}^{\prime }$ be solutions to(10)–(12)parameterized by $(r,\kappa )$ and $(r^{\prime },{\kappa }^{\prime })$. Then $(r,\kappa )=(r^{\prime },{\kappa }^{\prime })$ if and only if $H(r,\kappa )=H(r^{\prime },{\kappa }^{\prime })$.

Proof. 

The direction $(r,\kappa )=(r^{\prime },{\kappa }^{\prime })\Rightarrow H(r,\kappa )=H(r^{\prime },{\kappa }^{\prime })$ is obvious, so assume $H(r,k)=H(r^{\prime },k^{\prime })$. $H(r,\kappa )$ determines $G(r,\kappa )$ up to a sign and

$$\begin{array}{cccc}\hfill {\left(F_{(r,k)}\right)}_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}& ={\left(F_{(r^{\prime },k^{\prime })}\right)}_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}\hfill & \hfill & \mathrm{and}\hfill \\ \hfill {\left(F_{(r,k)}\right)}_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\mp }}& =\pm {\left(F_{(r^{\prime },k^{\prime })}\right)}_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\mp }}.\hfill \end{array}$$

This corresponds to a choice ${\psi }_{\pm }$, which is irrelevant to $(r,\kappa )$, so this implies $(r,\kappa )=(r^{\prime },{\kappa }^{\prime })$. □

Corollary 1.

$(r,\kappa )=(r^{\prime },{\kappa }^{\prime })$ if and only if

$$Diag\left(H(r,\kappa )\right)=Diag\left(H(r^{\prime },{\kappa }^{\prime })\right).$$

Proof. 

$H{(r,\kappa )}_{0,0}$ determines $\kappa$ and $H{(r,\kappa )}_{1,1}\propto Re\left(q^r\right)$ narrows r to one of two choices. Taken modulo $2p+1$ only one of r and $-r$ is odd, and so $H{(r,\kappa )}_{1,1}$ determines $H(r,\kappa )$. □

By inspection, we see that

$$\begin{array}{cc}\hfill {\left(F_{(r,k)}\right)}_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm ;{\varphi }_i{\varphi }_i}}{\left(F_{(r,k)}\right)}_{{\psi }_{\pm }{\varphi }_i{\psi }_{\pm }}^{{\varphi }_i;{\psi }_{\pm }{\psi }_{\pm }}& =H{(r,k)}_{i,i}\hfill \end{array}$$

so define

$$\begin{array}{cc}\hfill X_p(r,\kappa ,i)& =\frac{\sqrt{2p+1}}{2^{\zeta (i,i)}}\sum _{j\in \{+,-\}}\left({\left(F_{(r,k)}\right)}_{{\psi }_j{\psi }_j{\psi }_j}^{{\psi }_j;{\varphi }_i{\varphi }_i}{\left(F_{(r,k)}\right)}_{{\psi }_j{\varphi }_i{\psi }_j}^{{\varphi }_i;{\psi }_j{\psi }_j}\right)\hfill \\ \hfill & ={(-1)}^{i^2}cos\left(\frac{i^{2}r\pi }{2p+1}\right).\hfill \end{array}$$

and the $(p+1)$-tuple

$$\begin{array}{cc}\hfill X_p(r,\kappa )& =\left(X_p(r,\kappa ,i)\right|i=0,\dots ,p).\hfill \end{array}$$

Corollary 2.

$X_p(r,\kappa )$ uniquely determines the values for all G-invariant monomials and thus all $Aut\left(N\right)⋉G$-invariant monomials.

Corollary 3.

The permutation ${\psi }_{+}\to {\psi }_{-}$ acts as a gauge transformation.

There is a natural action of $z\in {\mathbb{G}}_{2p+1}^{\times }$ on $X(r,\kappa )$ given by

$$\begin{array}{cc}\hfill z·X_p(r,\kappa )& =(z·X_p(r,\kappa ,i)|i=0,\dots p)\hfill \\ \hfill & =\left(X_p(r,\kappa ,g(z\ast i))\right|i=0,\dots ,p)\hfill \\ \hfill & =\left(X_p(r{z}^2,\kappa ,i)\right|i=0,\dots p)\hfill \\ \hfill & =X_p(r{z}^2,\kappa ).\hfill \end{array}$$

Theorem 3.

Fix p and let $(r,\kappa )$ and $(r^{\prime },\kappa )$ be solutions as defined in Section 4.2. The following are equivalent:

1. 

$(r,\kappa )$ and $(r^{\prime },\kappa )$ are monoidally equivalent,

2. 

there exists $z\in {\mathbb{G}}_{2p+1}^{\times }$ such that $X_p(r^{\prime },\kappa )=X_p(r{z}^2,\kappa )$, and

3. 

there exists $z\in {\mathbb{G}}_{2p+1}^{\times }$ such that $g\left(r^{\prime }\right)=g\left(r{z}^2\right)$.

Proof. 

First assume that $(r,\kappa )$ and $(r^{\prime },\kappa )$ are monoidally equivalent via some $z\in {\mathbb{G}}_{2p+1}^{\times }$. z acts on $F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm }}$ by

$$\begin{array}{cc}\hfill z·F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm };{\varphi }_i{\varphi }_j}& =F_{{\psi }_{\pm }{\psi }_{\pm }{\psi }_{\pm }}^{{\psi }_{\pm };{\varphi }_{g\left(zi\right)}{\varphi }_{g\left(zj\right)}}=H(r{z}^2,\kappa )\hfill \end{array}$$

so

$$\begin{array}{cc}\hfill X_p(r^{\prime },\kappa )& =X_p(r{z}^2,\kappa ).\hfill \end{array}$$

This implies $r^{\prime }$ and $r{z}^2$ are congruent modulo $2p+1$ and so $g\left(r^{\prime }\right)=g\left(r{z}^2\right)$.

Assume $\exists z\in {\mathbb{G}}_{2p+1}^{\times }$ such that $g\left(r^{\prime }\right)=g\left(r{z}^2\right)$. Then, modulo $2p+1$, $r^{\prime }\cong r{z}^2$ or $r^{\prime }\cong -r{z}^2$, but regardless $X_p(r^{\prime },\kappa )=X_p(r{z}^2,\kappa )$. By Corollary 2, for any $Aut⋉G$-invariant l,

$$F_{(r^{\prime },\kappa )}\left(l\right)=F_{(r,\kappa )}\left(l\right).$$

Thus, $(r^{\prime },\kappa )$ and $(r,\kappa )$ are monoidally equivalent. □

5.2. 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 $(r,\kappa )$ and $(r^{\prime },{\kappa }^{\prime })$ are monoidally equivalent if and only if $\kappa ={\kappa }^{\prime }$ and there exists a $z\in {\mathbb{G}}_{2p+1}^{\times }$ such that $r^{\prime }=r{z}^2$. Thus, we define

$$\begin{array}{ccc}\hfill O_r& =& \left\{r{z}^2\right|z\in {\mathbb{G}}_{2p+1}^{\times }\}\subseteq {\mathbb{G}}_{2p+1}^{\times }.\hfill \end{array}$$

Each set contains elements such that the F-moves labeled by $(r,\kappa )$ and $(r^{\prime },\kappa )$ are monoidally equivalent. Furthermore, $(r,\kappa )$ and $(r^{\prime },\kappa )$ are monoidally equivalent if and only if $O_r=O_{r^{\prime }}$. We realize that $O_1$ is a normal subgroup and that $O_r=r{O}_1$ are conjugacy classes. Thus, the number of monoidally inequivalent F-moves must be twice (due to $\kappa$) $|{\mathbb{G}}_{2p+1}^{\times }|/|O_1|$.

It is possible to simplify this further by considering the map $f:{\mathbb{G}}_{2p+1}^{\times }\to O_1$ defined by $f\left(z\right)=z^2$. This map is clearly onto, i.e., the image of f is $O_1$. As we are dealing with the finite group,s we can use the first isomorphism theorem and deduce that $|{\mathbb{G}}_{2p+1}^{\times }|/|O_1|=|ker\left(f\right)|$ where $ker\left(f\right)$ is the kernel of f and is the subgroup

$$\begin{array}{ccc}\hfill ker\left(f\right)& =& \{r\in {\mathbb{G}}_{2p+1}^{\times }|r^2=1\}.\hfill \end{array}$$

Thus, the number of monoidally inequivalent F-moves is twice the number of elements of ${\mathbb{G}}_{2p+1}^{\times }$ that square to the identity (because of $\kappa$). We know that $\left|ker\left(f\right)\right|=2^m$ for some $m\in \mathbb{N}$.

If $r\in {\mathbb{Z}}_{2p+1}^{\times }$ satisfies $r^2=1$ or $-1$, then $r\in {\mathbb{G}}_{2p+1}^{\times }$ satisfies $r^2=1$. We can then deduce that

$$\begin{array}{ccc}\hfill \left|ker\right(f\left)\right|& =& \frac{1}{2}\left(|\{r\in {\mathbb{Z}}_{2p+1}^{\times }|r^2=1\}|+|\{r\in {\mathbb{Z}}_{2p+1}^{\times }|r^2=-1\}|\right).\hfill \end{array}$$

Firstly, consider the prime decomposition $2p+1=p_1^{a_1}...p_l^{a_l}$ where the $p_i$ are prime and $a_i>0$. We define sets

$$\begin{array}{ccc}\hfill B_1& =& \{r\in {\mathbb{Z}}_{2p+1}^{\times }|r^2=1\},\hfill \\ \hfill B_2& =& \{r\in {\mathbb{Z}}_{2p+1}^{\times }|r^2=-1\}.\hfill \end{array}$$

We note that as $B_1$ is also a group and that its order is precisely $2^l$. Next, suppose that $r,r^{\prime }\in B_2$ it follows that $r{r}^{\prime }\in B_1$, similarly if $r\in B_2$ and $r^{\prime }\in B_1$ then $r{r}^{\prime }\in B_2$. 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 $b_i^2\equiv -1mod{p}_i^{a_i}$ for all i.

From this, we can deduce the number of monoidal classes

$$\begin{array}{ccc}\hfill \#monoidalclasses& =& \left\{\begin{array}{cc}{2}^{l+1}& if\exists bs.t.b^2\equiv -1mod(2p+1),\hfill \\ 2^l& otherwise,\hfill \end{array}\right.\hfill \end{array}$$

(56)

where l is the number of primes that divide $(2p+1)$. Here we have used that ${\mathbb{Z}}_{2(2p+1)}^{\times }\cong {\mathbb{Z}}_{2p+1}^{\times }.$

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 ($\lambda =+1$) and their inverses ($\lambda =-1$), 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 ([24] [Theorem 3.2]).

6. Examples

In this section, we explicitly compute the classifying invariants for a number of “small” categories. The case of $\mathfrak{so}{\left(3\right)}_2$ is known to be Grothendieck equivalent to $\mathfrak{su}{\left(2\right)}_4$ and both the fusion and modular structures are fully classified (see [19]). Beyond this, there are several guideposts:

• The classification of weakly integral modular categories of dimension $4m$ is given in [40]. This contains those $\mathfrak{so}{(2p+1)}_2$ of our family for which $2p+1$ is square free.
• The classification of integral modular categories of dimension $4q^2$ is given in [41] where q is prime.
• Explicit formulae for the modular data of ${\mathbb{Z}}_2$-equivariantizations of Tambara–Yamagami categories is given in [23] to which our categories are Grothendieck equivalent.

6.1. $\mathfrak{so}{\left(3\right)}_2$

For $\mathfrak{so}{\left(3\right)}_2$, ${\mathbb{G}}_3^{\times }=\left\{1\right\}$. 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 (53):

$$\begin{array}{cc}\hfill diag(1,1,-{(-1)}^{1/3},-{(-1)}^{1/4},{(-1)}^{1/4})& \\ \hfill diag(1,1,{(-1)}^{2/3},{(-1)}^{3/4},-{(-1)}^{3/4})& \\ \hfill diag(1,1,-{(-1)}^{1/3},{(-1)}^{3/4},-{(-1)}^{3/4})& \\ \hfill diag(1,1,{(-1)}^{2/3},-{(-1)}^{1/4},{(-1)}^{1/4})\end{array}$$

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 [23]. In this case, the two braidings for each fusion category are inequivalent.

6.2. $\mathfrak{so}{\left(5\right)}_2$

There are two two-dimensional objects and ${\mathbb{G}}_5^{\times }\cong {\mathbb{Z}}_2$. Both elements square to the identity and so there are four fusion categories from (56). This can be seen from the sets

$$\begin{array}{cc}\hfill X_2(1,1)& =\left\{1,\frac{1}{4}\left(-1-\sqrt{5}\right),\frac{1}{4}\left(-1-\sqrt{5}\right)\right\},\mathrm{and}\hfill \\ \hfill X_2(3,1)& =\left\{1,\frac{-1}{4}\left(1-\sqrt{5}\right),\frac{-1}{4}\left(1-\sqrt{5}\right)\right\},\hfill \end{array}$$

From [23], there are also four modular categories, corresponding to solutions $(2,1,1,1)$, $(2,1,-1,1)$, $(2,3,1,1)$, and $(2,3,-1,1)$ with T-matrices

$$\begin{array}{cc}\hfill & diag(1,1,-{(-1)}^{1/5},{(-1)}^{4/5},-1,1),\hfill \\ \hfill & diag(1,1,-{(-1)}^{1/5},{(-1)}^{4/5},ı,-ı),\hfill \\ \hfill & diag(1,1,-{(-1)}^{3/5},{(-1)}^{2/5},-ı,ı),\mathrm{and}\hfill \\ \hfill & diag(1,1,-{(-1)}^{3/5},{(-1)}^{2/5},1,-1).\hfill \end{array}$$

In this case, the braidings corresponding to $\lambda =\pm 1$ are equivalent.

6.3. $\mathfrak{so}{\left(7\right)}_2$

There are three two-dimensional objects and $G_7^{\times }\cong {\mathbb{Z}}_3$. We have that

$$\begin{array}{cc}\hfill X_3(1,1)& =\left\{1,-cos\left(\frac{\pi }{7}\right),-sin\left(\frac{\pi }{14}\right),sin\left(\frac{3\pi }{14}\right)\right\},\hfill \\ \hfill X_3(3,1)& =\left\{1,-sin\left(\frac{\pi }{14}\right),sin\left(\frac{3\pi }{14}\right),-cos\left(\frac{\pi }{7}\right)\right\},\mathrm{and}\hfill \\ \hfill X_3(5,1)& =\left\{1,sin\left(\frac{3\pi }{14}\right),-cos\left(\frac{\pi }{7}\right),-sin\left(\frac{\pi }{14}\right)\right\}.\hfill \end{array}$$

These are all clearly related via permutation. Computing the modular data for our solutions, we find that there are four modular categories distinguished by their T-matrices

$$\begin{array}{cc}\hfill & diag(1,1,-{(-1)}^{1/7},{(-1)}^{4/7},{(-1)}^{2/7},-{(-1)}^{1/4},{(-1)}^{1/4})\hfill \\ \hfill & diag(1,1,{(-1)}^{6/7},-{(-1)}^{3/7},-{(-1)}^{5/7},{(-1)}^{3/4},-{(-1)}^{3/4})\hfill \\ \hfill & diag(1,1,-{(-1)}^{1/7},{(-1)}^{4/7},{(-1)}^{2/7},-{(-1)}^{3/4},{(-1)}^{3/4}),\mathrm{and}\hfill \\ \hfill & diag(1,1,{(-1)}^{6/7},-{(-1)}^{3/7},-{(-1)}^{5/7},{(-1)}^{1/4},-{(-1)}^{1/4}),\hfill \end{array}$$

corresponding to solutions $(3,1,1,1)$, $(3,1,1,-1)$, $(3,1,-1,1)$, and $(3,1,-1,-1)$, respectively.

6.4. $\mathfrak{so}{\left(9\right)}_2$

For $\mathfrak{so}{\left(9\right)}_2$, there are four two dimensional objects, but we have that ${\mathbb{G}}_9\cong {\mathbb{Z}}_3$ since there are three odd integers less than and coprime to 9, 1, 5, and 7. These correspond to the $S_4$ subgroup generated by the cycle $\langle 124\rangle$.

The sets $X_4(r,1)$ are then:

$$\begin{array}{cc}\hfill X_4(1,1)& =\left\{-cos\left(\frac{\pi }{9}\right),cos\left(\frac{4\pi }{9}\right),1,cos\left(\frac{2\pi }{9}\right)\right\},\hfill \\ \hfill X_4(5,1)& =\left\{cos\left(\frac{4\pi }{9}\right),cos\left(\frac{2\pi }{9}\right),1,-cos\left(\frac{\pi }{9}\right)\right\},\mathrm{and}\hfill \\ \hfill X_4(7,1)& =\left\{cos\left(\frac{2\pi }{9}\right),-cos\left(\frac{\pi }{9}\right),1,-cos\left(\frac{4\pi }{9}\right)\right\}.\hfill \end{array}$$

From this, it is easy to see that $\langle 124\rangle$ sends $X_4(5,1)$ to $X_4(1,1)$ and $\langle 142\rangle$ sends $X_4(7,1)$ to $X_4(1,1)$. Thus, there is only one equivalence class. Finally, as before, there are four modular structures distinguished by their T-matrices:

$$\begin{array}{cc}\hfill & diag(1,1,-{(-1)}^{1/9},{(-1)}^{4/9},1,-{(-1)}^{7/9},-1,1),\hfill \\ \hfill & diag(1,1,{(-1)}^{8/9},-{(-1)}^{5/9},1,{(-1)}^{2/9},-1,1),\hfill \\ \hfill & diag(1,1,-{(-1)}^{1/9},{(-1)}^{4/9},1,-{(-1)}^{7/9},-ı,ı),\mathrm{and}\hfill \\ \hfill & diag(1,1,{(-1)}^{8/9},-{(-1)}^{5/9},1,{(-1)}^{2/9},ı,-ı),\hfill \end{array}$$

corresponding to solutions $(4,1,1,1)$, $(4,1,1,-1)$, $(4,1,-1,1)$, and $(4,1,-1,-1)$, respectively.