Documentation

Cyclotomic Notation

Exact values are shown with radicals where concise and with the following cyclotomic abbreviations otherwise.

These ratios are always written with both superscripts, distinguishing them from the higher central charges $\xi_n$. Record pages can switch between the simplified display and the canonical cyclotomic expressions; downloadable exact data retain the canonical form.

Quadratic Forms

The Chern–Simons calculator writes finite metric groups as orthogonal sums of primary Jordan blocks, using $\perp$. A cyclic block $A_{p^r}^{a}$ has $q(x)=\exp(2\pi i a x^2/p^r)$ for odd $p$, and $q(x)=\exp(2\pi i a x^2/2^{r+1})$ for $p=2$. On $\mathbb Z_{2^r}^2$, the blocks $u_{2^r}$ and $v_{2^r}$ denote $q(x,y)=\exp(2\pi ixy/2^r)$ and $q(x,y)=\exp(2\pi i(x^2+xy+y^2)/2^r)$, respectively. $\mathrm{sVec}$ denotes the transparent-fermion block.

Finite Groups

Finite groups are listed for Tannakian subcategories and the symmetry groups of the modular data and fusion ring. In most cases, notation for these is standard. When possible, semidirect products are also written unambiguously as $\operatorname{SmallGroup}(n,k)$ using the identifier from the GAP package SmallGrp. For example, $\operatorname{SmallGroup}(20,3)\simeq\mathbb Z_5\rtimes\mathbb Z_4$. These can be accessed in GAP as:

G := SmallGroup(n, k);

Source Representations

The linear congruence representations of $\mathrm{SL}_2(\mathbb Z)$ from which the modular data derive are named according to their classification, as provided by the GAP package SL2Reps. These can be accessed in GAP as:

rho := First(SL2IrrepsOfDegree([irrep dimension]), x -> x.name = "[irrep name]");

Irrep dimensions are listed alongside them for convenience. Syntax is sensitive: for an irrep such as $R_2^0(1,3)_1$, the subscript comes before the exponent in the GAP name, R_2^0(1,3)_1. An irrep such as $R_1(3)^-$ is written R_1(3)-.

Representations of $\Gamma_\theta$ are not classified and are therefore left unnamed.

Realization Labels

Realizations indicate modular or supermodular data with known unitary constructions. Priority is given to Chern–Simons theories, as these account for the bulk of the realized categories.

We call a category Chern–Simons when it corresponds to a Chern–Simons theory with compact, connected gauge group $G$. All such groups are of the form

where $T\simeq\mathrm U(1)^r$ is a torus, $\widetilde G$ is a compact, simply connected, semisimple Lie group, and $\Gamma$ is a finite subgroup of the center of $T\times\widetilde G$. Therefore, the Chern–Simons modular and supermodular categories are of the form

$\mathcal P$ is any pointed modular category, or equivalently some $\mathrm U(1)^r_K$ Chern–Simons theory for a level matrix $K$. Similarly, the $\mathcal C(\mathfrak g,k)$ are affine Lie categories, each describing a Chern–Simons theory with simple, simply connected gauge group. Finally, the gauge-group quotient is realized categorically as condensation—as indicated by $(\mathcal C)^0_A$—by an algebra $A$ with support only on invertible objects; equivalently, it gauges an invertible one-form symmetry. Orientation reversals of these theories are denoted in the catalogue with an overline.

The methods by which these theories are identified are discussed in The Anyon Atlas. The labels are intended to identify which categories are realized by Chern–Simons theories, but they are in no way unique and may differ from standard nomenclature. To identify a particular Chern–Simons theory with compact connected gauge group, use the separate Chern–Simons condensation tool.

We also consider as Chern–Simons the gaugings of zero-form $H$ symmetries. These correspond to data with $\operatorname{Rep}(H)$ subcategories condensing to connected Chern–Simons theories. In the catalogue, the $H$-gauging of a Chern–Simons theory with connected gauge group $G$ is listed as $G\mathbin{//}H$. These gaugings include all compact disconnected gauge groups, including Dijkgraaf–Witten theories.

The catalogue uses the following definitions whenever the given name is of one of these forms:

Here $\mathrm{TC}=\mathrm U(1)^2_K$ denotes the toric code, with $K=\left(\begin{smallmatrix}0&2\\2&0\end{smallmatrix}\right)$, while $3\mathrm F=\mathrm U(1)^4_K$ denotes the 3-fermion theory, with $K=\left(\begin{smallmatrix}-2&1&0&0\\1&-2&1&1\\0&1&-2&0\\0&1&0&-2\end{smallmatrix}\right)$. When the symmetry is non-anomalous, it can be quotiented directly. When it is fermionic, the fermionic quotient is performed by first stacking with $\mathrm U(1)_1=\mathrm{sVec}$. In the remaining cases described as $\mathrm{PSp}$, $\mathrm{SO}$, $\mathrm{PSO}$, and $\mathrm{PSU}$, the theory is first stacked with a minimal abelian (pointed) theory that cancels the anomaly. In particular, $\mathcal C(\mathbb Z_n,q^{-1}_{n,k})$ denotes the metric-group category $\mathcal C(A,q)$ for an abelian group $A$ and quadratic form $q$. Being pointed, these categories have $\mathrm U(1)^r_K$ descriptions. Lastly, $\mathrm U(1)_{-2}=\mathrm E_{7,1}$ and $\mathrm U(1)_2=\mathrm{SU}(2)_1$.

Categories that are provably not obtained from Chern–Simons theories in this way are labeled Not CS.

Other realization labels are given for centers of exotic fusion categories and Dijkgraaf–Witten theories when these are not also realized by Chern–Simons theories.

Known Categorifiable

The catalogue search includes a Known Categorifiable selector. It begins with the realization labels above and closes them under Galois conjugation, products, Tannakian and fermion condensations, and factorizations.

We also include categorified half-integer-level affine Lie categories, negative-level $\mathrm U(1)$ categories, non-unitary Dijkgraaf–Witten theories, and odd metaplectic categories. Although these procedures do not exhaust all known categorifications, they identify the vast majority of them.

Derived Data from $(S,T)$

• Quantum dimensions. $d_i=S_{1i}$ in the normalization used here.
• Global dimension. $\mathcal D^2=\sum_i d_i^2$.
• Fusion rules. For modular categories, these are given by the Verlinde formula:
  $N_{ab}^{c}=\frac{1}{\mathcal D^2}\sum_{x\in\operatorname{Irr}(\mathcal C)}\frac{S_{ax}S_{bx}S_{\bar c x}}{S_{1x}}$

  In the supermodular case, the same formula using $\widehat S$, the $S$-matrix of the fermionic quotient, yields the sums $N_{ab}^{c}+N_{ab}^{c\otimes f}$. The full fusion rings compatible with balancing are determined from these sums. In general, there may be several such lifts.

• Invertible objects. A simple object $X$ is invertible when $X\otimes\bar X\cong\mathbf 1$. Abelian anyons are invertible objects.
• Chiral central charge. For modular categories,
  $\frac{1}{\mathcal D}\sum_{a\in\operatorname{Irr}(\mathcal C)}d_a^2\theta_a=e^{2\pi i c_-/8}$

  determines $c_-\bmod 8$. For supermodular categories, the linear representations from which our data derive satisfy

  $\rho(\mathfrak t)^2=e^{2\pi i c_-/12}\widehat T^{\,2}$

  yielding $c_-\bmod \tfrac12$, which can also be determined from minimal extensions when these are known. Here $\widehat T$ is the $T$-matrix of the fermionic quotient, as described in The Anyon Atlas.

• Second Frobenius–Schur indicator.
  $\nu_2(a)=\frac{1}{\mathcal D^2}\sum_{b,c\in\operatorname{Irr}(\mathcal C)}N_{bc}^{a}\,d_b d_c\left(\frac{\theta_b}{\theta_c}\right)^2$

  Duals are determined by the charge-conjugation matrix $C_{ab}=(S^2)_{ab}=\delta_{a\bar b}$. For self-dual objects, $\nu_2(a)=\pm1$, while for non-self-dual objects $\nu_2(a)=0$.

• Higher central charges. For modular categories, the higher central charges are
  $\xi_n(\mathcal C)=\frac{\displaystyle\sum_a d_a^2\theta_a^n}{\displaystyle\left|\sum_a d_a^2\theta_a^n\right|}$

  for $\gcd(n,\operatorname{ord}(T))=1$. These provide useful condensation invariants. For supermodular categories there is no corresponding formula, but the charges are inherited, up to sixteenth roots of unity, from a minimal extension when one is known.

Subcategories and Symmetry Groups

• Fusion subcategories. Collections of simple objects closed under fusion.
• Tannakian and super-Tannakian subcategories. A symmetric subcategory—one in which anyons braid trivially with each other—is either Tannakian, when $\theta=1$ for all of its anyons, or super-Tannakian. Tannakian subcategories are equivalent to categories $\operatorname{Rep}(G)$ of finite-dimensional representations of finite groups $G$, while super-Tannakian subcategories are equivalent to $\operatorname{sRep}(G)$. These subcategories are particularly interesting for condensation: a theory containing one has a description as a gauging of a zero-form $G$ symmetry.
• Modular-data symmetry group. The group $G_{\mathrm{mod}}$ of simple-object permutations preserving $S$ and $T$.
• Fusion-ring symmetry group. The group $G_{\mathrm{fus}}$ of simple-object permutations preserving the fusion rules.

Properties

• Prime. A modular category with no nontrivial factorization $\mathcal M\boxtimes\mathcal C$ into modular categories, or a supermodular category with no nontrivial factorization $\mathcal M\boxtimes_{\mathrm{sVec}}\mathcal C$ into supermodular categories. These products correspond to stacking.
• Pseudo-unitary. Having positive quantum dimensions, a necessary condition for unitarity.
• Weakly integral. $\mathcal D^2\in\mathbb Z$.
• Integral. $d_i\in\mathbb Z$ for every $i\in\operatorname{Irr}(\mathcal C)$.
• Pointed. Every simple object is invertible. All such categories are $\mathrm U(1)^r_K$ for some level matrix $K$.
• Multiplicity-free. The largest fusion coefficient is $1$.
• Simple. Having no nontrivial fusion subcategories.
• Spin. A modular category containing an invertible fermion $\psi$ with $\theta_\psi=-1$.
• Completely anisotropic. Having no nontrivial condensable algebras, corresponding to non-anomalous one-form symmetries.
• Split. A supermodular category admitting a factorization $\mathcal M\boxtimes\mathrm{sVec}$ with $\mathcal M$ modular.
• Primitive. A nonsplit supermodular category.

Condensation

• Condensable algebra. A non-anomalous, and hence gaugeable, one-form symmetry. $F$- and $R$-symbols are generally needed to guarantee existence, but strong necessary conditions can be tested on the modular data. Those conditions are listed in The Anyon Atlas.
• Condensation. Gauging a one-form symmetry. The potential condensations shown are those satisfying these conditions on the modular data.
• Minimal extension. The theory obtained by gauging $(-1)^F$ in a fermionic theory. Minimal extensions contain the full data needed to classify fermionic topological orders, up to invertible phases.
• Tannakian condensation. A Tannakian subcategory $\operatorname{Rep}(G)$ has an associated condensable algebra $A=\operatorname{Fun}(G)$. Its existence is guaranteed by the modular data, so these condensations are highlighted in the graphs.