On near-group centers and super-modular categories

Besides their interest in algebraic category theory and topology, modular (resp. super-modular) categories are important in condensed matter physics as they describe bosonic (resp. fermionic) topological phases of matter in two spacial dimensions (see eg. [29, 6]). This connection goes back to the mathematical study of conformal field theory in [25, 1]. It is of substantial interest to classify these categories both for these applications and their intrinsic beauty.

It is known [9, 22] that for any $r$ there are finitely many modular (resp. super-modular) categories with precisely $r$ isomorphism classes of simple objects, i.e. rank $r$. Although a complete classification of modular (resp. super-modular) categories is probably out of reach without some general structure theorems, for small $r$, classifications are known [29, 8, 27, 7, 10], at least up to modular data.

From any modular category $\mathcal{C}$ one obtains a (projective) representation of the modular group $\operatorname{SL}(2,\mathbb{Z})$ as the mapping class group of the torus. This representation is determined by a pair of matrices $(S,T)$ called the modular data of $\mathcal{C}$. The matrices $S$ and $T$ satisfy a number of remarkable constraints, including the key result of [28] that says that the $\operatorname{SL}(2,\mathbb{Z})$ representation factors through $\operatorname{SL}(2,\mathbb{Z}/N)$ for some minimal $N$, called the level of the representation, and that $N$ is the (finite) order of the $T$-matrix. The paper [27] introduced a computational method for classifying (low) rank $r$ modular categories roughly as follows:

1. Step 1

   Construct all irreducible prime-power-level representations of $\operatorname{SL}(2,\mathbb{Z})$ of dimension at most $r$, with the property that the image of $\mathfrak{s}:=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ is symmetric and the image of $\mathfrak{t}:=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ is diagonal. That is, all such representations factoring over $\operatorname{SL}(2,\mathbb{Z}/p^{k})$ for primes $p$ and $k\geq 1$.

2. Step 2

   Construct all finite-level representations $\rho$ of dimension $r$ by considering direct sums of tensor products of representations from Step 1.

3. Step 3

   Construct possible modular data $(S,T)$ of rank $r$ by studying matrices of the form $S=U^{-1}\rho(\mathfrak{s})U$ where $U$ is orthogonal [27, Theorem 3.4] and commutes with $\rho(\mathfrak{t})$ for each $\rho$ from Step 2.

4. Step 4

   Use the numerous constraints on modular data to eliminate as many pairs $(S,T)$ as possible.

5. Step 5

   For each remaining pair $(S,T)$ find a modular category $\mathcal{C}$ with this modular data.

In [27] this approach was successfully applied to classify rank $6$ modular categories up to modular data.¹¹1In principle there could be inequivalent categories with the same modular data, but they would of course have the same fusion rules so the ambiguity is modest. The number of cases to be considered proliferates rapidly, which requires significant assistance from computational software.

Super modular categories are slight generalizations of modular categories. Modular categories $\mathcal{C}$ are non-degenerate in the sense that the symmetric center $\operatorname{Sym}(\mathcal{C})$ is the trivial category $\operatorname{Vec}$, while super-modular categories $\mathcal{C}$ are slightly degenerate [14]: $\operatorname{Sym}(\mathcal{C})$ is equivalent to the braided fusion category $\operatorname{sVec}$ of super-vector spaces. A technical, but inessential, additional assumption is that a super-modular category should also be unitary. While super-modular categories have $S$ and $T$ matrices, they do not immediately yield a representation of a group, since $S$ will be degenerate. However, after an appropriate reduction one obtains a representation of the index 3 subgroup $\Gamma_{\theta}\subset\operatorname{SL}(2,\mathbb{Z})$ generated by $\mathfrak{s}$ and $\mathfrak{t}^{2}$. The idea is that both $S$ and $T^{2}$ have a well-defined tensor-decomposition into $S=\hat{S}\otimes\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$ and $T^{2}=\hat{T}^{2}\otimes\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ where $\hat{S}$ and $\hat{T}^{2}$ give a projective representation of $\Gamma_{\theta}$. The pair $(\hat{S},\hat{T}^{2})$ is called the super-modular data of $\mathcal{C}$ – note that $\hat{T}$ is only defined up to sign choices.

A natural problem is to extend the above-described approach for classifying low rank modular categories to super-modular categories. The crucial Step 1 is justified by the recently proved Minimal Modular Extension (MME) Theorem of [21] and [4] which together show that the representations of $\Gamma_{\theta}$ coming from super-modular categories factor over the finite group $\Gamma_{\theta}\cap\operatorname{SL}(2,\mathbb{Z}/N)$ for some $N$. Then one can prove appropriate modifications of the steps above for the super-modular setting. As a more modest goal one might hope to produce new super-modular data to aid in the classification.

Under certain assumptions, Steps 1-4 were used in [12] to carry out a partial classification of super-modular data of rank $8$ and rank $10$. The expectation is that the rank $8$ classification is complete, as it coincides with the partial classification in [10]. The authors of [12] produced 2 families of super-modular data $(\hat{S},\hat{T}^{2})$ of rank 10 that were not known to have a realization, see (3.1) and (3.2).

The main motivation for this article is to realize the super-modular data (3.1) and (3.2) by finding super-modular categories with these super-modular data. To do this we must find modular categories containing a fermion $f$ (i.e. spin modular categories [6]) so that the centralizer $C_{\mathcal{C}}(\langle f\rangle)$ of $f$ (also denoted by $\langle f\rangle^{\prime}$) is a super-modular category with the given data. Here by a fermion we mean an object $f$ with $f\otimes f\cong\mathbf{1}$ that has self-braiding $c_{f,f}=-\operatorname{Id}_{f\otimes f}$. If such a spin modular category exists there is a 16-fold ambiguity, since there are precisely 16 minimal modular extensions of any given super-modular category [6, 24]. Our main insight is that the super-modular data in [12] bears some similarity with modular data found in [19], which is conjecturally the modular data associated with the Drinfeld centers of near-group fusion categories [31] (see Section 2.2). In particular, we find that near-group categories associated with the groups $\mathbb{Z}/6$ and $\mathbb{Z}/4\times\mathbb{Z}/2$ yield such categories, after taking their Drinfeld centers, condensing a boson in the second case, and discarding pointed modular factors. The work of [18] and [20] lay the groundwork for our approach, with the main difficulty being producing the modular data of these Drinfeld centers – this involves solving a large system of non-linear equations. We have the following:

We remark that for case $(b)$ we must first condense a boson, i.e. take the $\mathbb{Z}/2$-de-equivariantization with respect to the symmetric Tannakian category $\langle b\rangle$ where $b$ is a boson: an object $b$ so that $b\otimes b\cong\mathbf{1}$ and the self-braiding satisfies $c_{b,b}=\operatorname{Id}_{b\otimes b}$.

Along the way we noticed that our approach is quite general: we can often construct spin and hence super-modular categories from Drinfeld centers of near-group categories (for groups of even order). We illustrate this with some examples.

Acknowledgments E.R. and H.S. were partially supported by NSF grants DMS-2000331 and DMS-2205962. The authors thank T. Gannon, A. Schopieray, Y. Wang, A. Bagheri and P. Gustafson for enlightening conversations.

We shall need a few standard, but technical notions from the theory of braided fusion categories. Throughout this paper, we use the notation $\zeta_{n}:=e^{2\pi i/n}$, $\mathbb{T}=\{x\in\mathbb{C};|x|=1\}$, and $\chi_{n}^{m}=m+\sqrt{n}$. We occasionally employ the shorthand $XY:=X\otimes Y$ for notational convenience, and also write $X\oplus Y$ as $X+Y$. We denote by $e(r)=exp(2\pi ir)$ for any rational number $r$.

A braiding on a fusion category $\mathcal{C}$ is a natural isomorphism $c_{X,Y}:X\otimes Y\rightarrow Y\otimes X$ satisfying the hexagon equations [17]. A braided fusion category is a fusion category equipped with a braiding. We call a braided fusion category $\mathcal{C}$ symmetric if $c_{Y,X}c_{X,Y}=\operatorname{Id}_{X\otimes Y}$ for all $X,Y\in\mathcal{C}$. Let $\mathcal{C}$ be a braided fusion category and $\mathcal{D}\subset\mathcal{C}$ a fusion subcategory. The Müger centralizer $C_{\mathcal{C}}(\mathcal{D})$ of $\mathcal{D}$ in $\mathcal{C}$ is the symmetric fusion subcategory of $\mathcal{C}$ generalized by $X\in\mathcal{C}$ such that $c_{Y,X}c_{X,Y}=\operatorname{Id}_{X\otimes Y}$ for all $Y\in\mathcal{D}$. We will often use the shorthand notation $\mathcal{D}^{\prime}=C_{\mathcal{C}}(\mathcal{D})$ when no confusion can arise. The Müger center of $\mathcal{C}$ is the fusion subcategory $C_{\mathcal{C}}(\mathcal{C})=\mathcal{C}^{\prime}$, which is also called the symmetric center and sometimes denoted $\operatorname{Sym}(\mathcal{C})$. A premodular category is a spherical braided fusion category (in other notation, a ribbon fusion category). A premodular category $\mathcal{C}$ is called modular if $\operatorname{Sym}(\mathcal{C})\cong\operatorname{Vec}$, that is, $\operatorname{Sym}(\mathcal{C})$ is equivalent to the category of finite-dimensional vector spaces over $\mathbb{C}$. For a premodular category $\mathcal{C}$, the $\tilde{S}$ matrix is defined by $\tilde{S}_{X,Y}=\operatorname{Tr}(c_{Y,X^{*}}c_{X^{*},Y})$ and the $T$ matrix by $T_{X,Y}=\delta_{X,Y}\theta_{X}$, where $\theta_{X}$ is the (scalar form of the) ribbon twist. The $S$-matrix is a normalized version of $\tilde{S}$, namely $S:=\frac{\tilde{S}}{\sqrt{\dim(\mathcal{C})}}$. An alternative definition of a modular category is a premodular category whose $S$ matrix is non-degenerate. A super-modular category $\mathcal{B}$ is a premodular category with $\operatorname{Sym}(\mathcal{B})$ equivalent to the category $\operatorname{sVec}$ of super-vector spaces. In this case, the $S$ matrix is degenerate. For any super-modular category it is more convenient to consider $(\hat{S},\hat{T}^{2})$, where $S=\hat{S}\otimes\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$ and $T^{2}=\hat{T}^{2}\otimes\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$. Then, $\hat{S}$ is non-degenerate. A super-modular category $\mathcal{C}$ is called split if there is a modular category $\mathcal{D}$ such that $\mathcal{C}\cong\mathcal{D}\boxtimes\operatorname{sVec}$ as braided fusion categories. Clearly non-split super-modular categories are of greatest interest. The following recent result is crucial:

As the minimal dimension of a (pseudo-unitary) modular category containing a super-modular category $\mathcal{C}$ is $2\dim(\mathcal{C})$, such a category is called a minimal modular extension of $\mathcal{C}$. A modular category $\mathcal{D}$ with a fermion $f$ is called a spin modular category. In this case $\mathcal{D}=\mathcal{D}_{0}\oplus\mathcal{D}_{1}$ is $\mathbb{Z}/2$-graded, where the trivial component $\mathcal{D}_{0}=C_{\mathcal{D}}(\langle f\rangle)$. By results of [6, 24] there are exactly $16$ minimal modular extensions of any super-modular category. Clearly if $\mathcal{C}=\mathcal{D}\boxtimes\operatorname{sVec}$ is a split super-modular category then $\mathcal{D}\boxtimes\mathcal{K}$ is a minimal modular extension where $\mathcal{K}$ is one of the $16$ minimal modular extensions of $\operatorname{sVec}$ [23]. Thus a super-modular category $\mathcal{C}$ is split if and only if any (hence, every) minimal modular extension factors in this way.

We recall some results and notation from [17, Section 8.4]. A braided fusion category is pointed if every simple object is invertible. In a pointed braided fusion category, the isomorphism classes of simple objects form a finite abelian group. For a finite abelian group $G$, a quadratic form on $G$ is a map $q:G\rightarrow\mathbb{F}^{\times}$ such that $q(g)=q(-g)$ and the map $\langle\;,\;\rangle:G\times G\rightarrow\mathbb{F}^{\times}$ with $\langle g,h\rangle=\frac{q(g+h)}{q(g)q(h)}$ is a symmetric bicharacter. Let $G$ be a finite abelian group and $q:G\rightarrow\mathbb{F}^{\times}$ a quadratic form on $G$. The pair $(G,q)$ is called a pre-metric group. Consider the pointed braided fusion category with (isomorphism classes of) simple objects labeled by $g\in G$ and braiding $c_{g,h}$. We can define a quadratic form $q:G\rightarrow\mathbb{F}^{\times}$ on $G$ by sending $g$ to $c_{g,g}\in\mathbb{F}^{\times}$. Then, $(G,q)$ is a pre-metric group. In fact, up to braided equivalence, for each pre-metric group $(G,q)$, there is a unique pointed braided fusion category, denoted by $\mathcal{C}(G,q)$. By equipping $\mathcal{C}(G,q)$ with the spherical structure $\theta_{g}=q(g)$, we get a premodular category with a symmetric $S$ matrix defined by $S_{g,h}=\frac{1}{\sqrt{|G|}}\overline{\langle g,h\rangle}$ and a diagonal $T$ matrix defined by $T_{g,g}=q(g)$ (we use the conventions of [19]). If $\langle\;,\;\rangle$ is a non-degenerate bicharacter, then $q$ is a non-degenerate quadratic form. In this case, the pair $(G,q)$ is called a metric group. The corresponding pointed category for a metric group, $\mathcal{C}(G,q)$, is then a modular category with $S$ and $T$ as its modular data.