# Classification of Metaplectic Fusion Categories

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- 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 ${2}^{l}$.

## 2. Preliminaries

#### 2.1. Fusion Categories and Modular Categories

**Definition**

**1**

- (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., ${\left(XY\right)}^{*}={X}^{*}{Y}^{*},\forall X,Y\in B$.

**Remark**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- 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$$\begin{array}{cc}\hfill {\lambda}_{a}\circ (e{v}_{a}\otimes I{d}_{a})\circ {\alpha}_{a,{a}^{*},a}\circ (I{d}_{a}\otimes coe{v}_{a})\circ {\rho}_{a}^{-1}& =I{d}_{a}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\rho}_{{a}^{*}}\circ (I{d}_{{a}^{*}}\otimes e{v}_{{a}^{*}})\circ {\alpha}_{{a}^{*},a,{a}^{*}}^{-1}\circ (coe{v}_{{a}^{*}}\otimes I{d}_{{a}^{*}})\circ {\lambda}_{{a}^{*}}^{-1}& =I{d}_{{a}^{*}}.\hfill \end{array}$$

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

#### 2.2. Fusion and Modular Systems

**Definition**

**7.**

- 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$$\begin{array}{cc}\hfill {\delta}_{a}^{b}& ={N}_{a\mathbf{1}}^{b}={N}_{\mathbf{1}a}^{b}={N}_{a{b}^{*}}^{\mathbf{1}}={N}_{{b}^{*}a}^{\mathbf{1}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {N}_{abc}^{d}& :=\sum _{e}{N}_{ab}^{e}{N}_{ec}^{d}=\sum _{f}{N}_{af}^{d}{N}_{bc}^{f}\hfill \end{array}$$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$$\begin{array}{cc}\hfill {F}_{a\mathbf{1}b}^{c;ab}& ={N}_{ab}^{c}\hfill \end{array}$$$$\begin{array}{cc}\hfill {F}_{a{a}^{*}a}^{a;11}& \ne 0\hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{h}{F}_{abc}^{h;fg}{F}_{agd}^{e;hi}{F}_{bcd}^{i;gj}& ={F}_{fcd}^{e;hj}{F}_{abj}^{e;fi}\hfill \end{array}$$

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Definition**

**8.**

## 3. Monoidal Equivalence and Gauge Invariants

**Definition**

**9.**

**Remark**

**2.**

**Theorem**

**2.**

## 4. $\mathfrak{so}{(\mathbf{2}\mathit{p}+\mathbf{1})}_{\mathbf{2}}$ Fusion Systems

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

**Proposition**

**3.**

- 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.**

#### 4.2. F-Matrices

#### 4.2.1. Notation

#### 4.2.2. Arithmetic Data

**Proposition**

**4.**

**Proof.**

#### 4.3. R-Matrices

**Remark**

**3.**

#### Modular Data for $\mathfrak{so}{(2p+1)}_{2}$ Modular Systems

## 5. Monoidal Equivalence of $\mathfrak{so}{(\mathbf{2}\mathit{p}+\mathbf{1})}_{\mathbf{2}}$ Fusion Systems

#### 5.1. Determining Equivalence

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**3.**

- 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.**

#### 5.2. Calculating the Number of Monoidal Equivalence Classes

## 6. Examples

- 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 $4{q}^{2}$ 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. $\mathfrak{so}{\left(3\right)}_{2}$

#### 6.2. $\mathfrak{so}{\left(5\right)}_{2}$

#### 6.3. $\mathfrak{so}{\left(7\right)}_{2}$

#### 6.4. $\mathfrak{so}{\left(9\right)}_{2}$

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Solution to Pentagon Equations

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Ardonne, E.; Finch, P.E.; Titsworth, M.
Classification of Metaplectic Fusion Categories. *Symmetry* **2021**, *13*, 2102.
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Ardonne E, Finch PE, Titsworth M.
Classification of Metaplectic Fusion Categories. *Symmetry*. 2021; 13(11):2102.
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Ardonne, Eddy, Peter E. Finch, and Matthew Titsworth.
2021. "Classification of Metaplectic Fusion Categories" *Symmetry* 13, no. 11: 2102.
https://doi.org/10.3390/sym13112102