We survey a number of classification tools developed in recent years and employ them to classify pseudo-unitary rank 5 premodular categories up to Grothendieck equivalence.

## I. INTRODUCTION

Fusion categories axiomatize and generalize the theory of representation theory, and their study encompasses the representation theory of finite groups and certain Hopf algebras. In many of these situations, the fusion category associated with the underlying group/algebra enjoys extra structure, such as a notion of commutativity (*braiding*), duality (*rigidity*), or other conditions such as a *spherical structure* and non-degeneracy (*modularity*).^{1} The study of fusion categories has moved beyond its roots in groups and algebras and now has a more widespread use. For instance, non-degenerate ribbon braided fusion categories, i.e., modular categories, have broad uses in physics, where they describe topological phases of matter and topological quantum computers.^{2,3} Modular categories also have applications in pure mathematics providing knot, link, and 3-manifold invariants through Topological Quantum Field Theories (TQFT).^{4} More recently, ribbon braided fusion categories, i.e., premodular categories, have garnered increased attention. These categories are thought to describe higher dimensional TQFT and thus have relevance in manifold invariants and physics.^{5} Furthermore, the study of premodular and modular categories often follow a “leap-frogging” pattern whereby advances in the understanding of one type of category allow for advances in another. In recent years, researchers have found it useful to stratify fusion categories by a numeric parameter known as their *rank*. The low-rank classification of premodular categories has historically lagged the classification of modular categories. On the premodular side, categories are understood through rank 4,^{6–8} while on the modular side, they are completely characterized through rank 5.^{9,10} Additional conditions are often placed on modular categories to make further progress on classification. For instance, weakly integral modular categories have been classified through rank 7^{11,12} and maximally non-self dual modular categories have been classified through rank 11.^{11}

In this work, we aim to utilize a technique known as *(de-)equivariantization* to produce modular categories from premodular categories. This will allow us to leverage recent advances in arithmetic properties of modular categories^{13} and to classify pseudo-unitary premodular categories of rank 5 up to Grothendieck equivalence. Specifically, we will show the following:

*If* $C$ *is a pseudo-unitary rank 5 premodular category, then either*

$C$

*is Tannakian and is given by*Rep(*G*),*where G is*$Z5$,*D*_{8},*Q*_{8},*D*_{14}, $Z5\u22caZ4$, $Z7\u22caZ3$, $S4$*, or*$A5$*or*$C$

*is properly premodular and Grothendieck equivalent to one of the following:*Rep(

*D*_{14})*with*$C\u2032\u2245RepZ2$*and*$di=1,1,2,2,2$.$Rep(S4)$

*with*$C\u2032\u2245RepS3$, $di=1,1,2,3,3$*, and*$T=diag1,1,1,\u22121,\u22121$.Rep(

*D*_{8})*with*$C\u2032\u2245RepZ2$, $di=1,1,2,1,1$*, and*$T=diag1,1,\theta ,\u22121,\u22121$*, where θ is a root of unity satisfying a monic degree*4*polynomial over*$Z$.$SU(2)8/Z2$.

$C$

*is modular and it is Grothendieck equivalent to*$SU(2)4,SU(2)9/Z2,SU(5)1$*or*$SU(3)4/Z3$.^{13}

*Moreover, each case is realized.*

In Sec. II, we will review the basic theory of premodular categories. Having dispensed with the preliminaries, we will stratify premodular categories by the amount of degeneracy (rank of the Müger center). In Sec. III, we will analyze each case in turn to arrive at Theorem III.2.

## II. PRELIMINARIES

A premodular category $C$ is a balanced, braided fusion category. We will denote the isomorphism classes of simple objects by *X*_{a}, indexed such that $X0=I$ is the monoidal unit. We will denote the set of such isomorphism classes by $Irr(C)$. The fusion matrices, *N*_{a}, the *S*-matrix, *S* = (*s*_{x,y}), and the *T*-matrix, $T=(\delta x,y\theta x)$, are defined in the usual way.^{1} Here *θ*_{x} is the twist of the simple *x* and is known to have finite order.^{14} The triple ({*N*_{a}}, *S*, *T*) is known as **premodular datum**. Throughout we will assume $C$ is **pseudo-unitary**, and so we may take the categorical dimensions, *d*_{a}, to be the Frobenius-Perron eigenvalues of the *N*_{a}, i.e., the FP-dimensions. This simplifying assumption has physical relevance in that only pseudo-unitary categories can produce physical theories.^{5}

Let $C\u2032$ denote the Müger center of the category $C$.^{15} We say that $C$ is **modular** if $C\u2032=Vec$, **symmetric** if $C=C\u2032$, and **properly premodular** otherwise. Symmetric categories are completely characterized by the following result due to Deligne:^{16}

*If* $C$ *is a symmetric category, then* $C\u2245Rep(G,z)$, *where z is a central element of G of order at most* 2.

In the properly premodular setting, this result can be exploited to determine part of the premodular datum. For instance, the categorical dimensions of simples in $C\u2032\u2245Rep(G,z)$ follow from the representation theory of *G.* Of course, by definition of the *S*-matrix, for isomorphism classes of simples, *X* and *Y*, in $C\u2032$, one has *s*_{X,Y} = *d*_{X}*d*_{Y}. One can obtain similar information regarding the *T*-matrix, for instance:

*Lemma II.2.*

*Suppose* $C$ *is a symmetric ribbon category and* $X\u2208C$ *is an object such that* $X\u2297X\u2245I$*, then θ*_{X} = ±1*. Moreover, θ*_{X} = 1 *if X fixes an element of* $C$ *. In particular, this is true if* $C$ *is the Müger center of an odd rank premodular category.*

*Remark II.3.*

*There are non-Tannakian odd rank fusion categories; however these categories do not admit a ribbon structure*^{17}*[Proposition 2.6(i)]*. *In concrete cases, this can be seen directly by inspecting the balancing equation (1). For instance, consider the non-Tannakian symmetric fusion category* $RepQ8$*. This category has rank 5 with dimensions* 1, 1, 1, 1, 2*. Letting X denote the isomorphism class of the 2-dimensional simple object and f denote the isomorphism class of the generator of* sVec *in* $RepQ8$*. Then f* ⊗ *X* = *X. On the other hand, the category is symmetric and so* 2 = *d*_{X} = *d*_{X}*d*_{f} = *S*_{X,f}*. The right-hand side can be evaluated from the balancing equation to give* $SX,f\theta X\theta f=dX\u2297f\theta X\u2297f=dX\theta X=2\theta X$*. In particular, we have* 2 = *S*_{X,f} = 2/*θ*_{f} = −2*, an impossibility.*

In the case that $C$ has odd rank, Lemma II.2 tells us that the Müger center of $C$ is Tannakian. In particular, we can exploit Bruguières’ minimal modularization.^{18} This procedure is more generally known as de-equivariantization. Recent work by Natale and Burciu allows one to gain a great deal of insight into the structure of the minimal modularization of $C$.^{19} Moreover, it is often true that the structure enforced by the de-equivariantization procedure is at odds with the Galois theory of the modular category. By exploring the interplay between these two tools, one can make progress in the classification of premodular categories. A complete discussion of de-equivariantization and the Galois theory of modular categories is beyond the scope of this current work, but details can be found in many papers.^{12,18,19}

*T*-matrix,

*S*-matrix, and dimensions are related through the

**balancing equation**as follows:

^{1}

*d*

_{X}are further related through the

**dimension equation**as follows:

The structure of the fusion rules, knowledge of dimensions, and twists of a subcategory often allow one to produce a pair of polynomials from these equations from which the order of certain twists can be bounded. In particular, we have the following:

*Lemma II.4.*

*If*

**D**

*is a subset of*$da\u22231\u2264a\u2264r$, $f\u2208QD[x]$

*is a degree*2

*polynomial with leading coefficient a*

_{2}

*and constant coefficient a*

_{0}

*such that*$f\theta =0$

*for some θ, then*

*Proof.*

*α*be a primitive element of $QD$ Then the minimal polynomial of

*θ*over $QD$ divides

*f*in $QD$. In particular, if we let

*m*denote the degree of $QD,\theta $ over $QD$, then

*m*≤ 2. Note that the minimal polynomial of

*θ*over $QD$ must divide the minimal polynomial of

*θ*over $Q$. Since

*θ*is a root of unity, we can conclude that the minimal polynomial over $QD$ can only possibly have degree 2 if

*a*

_{2}≠ 0 and

*a*

_{0}/

*a*

_{2}is a unit. In particular,

Next note that for any *d* ∈ **D**, *d* satisfies a degree 2 monic polynomial over $QD\d$. In particular, $QD$ has degree at most $2D$ over $Q$. Thus $QD,\theta $ has degree at most $m2D$ over $Q$. Since $Q\theta $ is a subfield, the result follows.

*Lemma II.5.*

*Let* $C$ *be a rank r premodular category with rank r* − 1 *Müger center and order the simples so that the last object, X*_{r−1}*, is not in* $C\u2032$. *Then* $Sr\u22121,r\u22121=\u2212dimC\u2032$.

*Proof.*

*X*

_{r−1}is not in the Müger center, the (

*r*− 1)-th column of the

*S*-matrix must be orthogonal to the 0-th column. However,

*S*

_{ℓ,0}=

*d*

_{ℓ}for 0 ≤

*ℓ*≤

*r*− 1, and

*S*

_{k,r−1}=

*d*

_{k}

*d*

_{r−1}for 0 ≤

*k*≤

*r*− 2. Thus the orthogonality condition reads

*d*

_{r−1}≠ 0, the result follows.

## III. CLASSIFICATION OF RANK 5 PREMODULAR CATEGORIES

In this section, we will classify rank 5 pseudo-unitary premodular categories up to premodular datum, and in particular, Grothendieck equivalence. Our focus on pseudo-unitarity is physically motivated as pseudo-unitarity is a necessary condition for the category to lead to a physical theory.^{5} The classification of pseudo-unitary rank 5 modular categories was recently completed,^{13} so it suffices to consider symmetric and properly premodular categories.

We begin with symmetric premodular categories. By Theorem II.1 and Lemma II.2, it suffices to determine all groups which have exactly 5 irreducible representations. The number of irreducible representations can be related to the order of the group via a classical number theoretic argument^{11} [lemma 4.4 (ii)]. In particular, we have

Applying these bounds, one can perform an exhaustive search in GAP and Lemma II.2 to deduce the following:

*Proposition III.1.*

*If* $C$ *is a symmetric rank 5 category, then* $C$ *is Tannakian and is given by* Rep(*G*), *where G is* $Z5$, *D*_{8}, *Q*_{8}, *D*_{14}, $Z5\u22caZ4$, $Z7\u22caZ3$, $S4$*, or* $A5$.

Having completed the classification of rank 5 symmetric categories, we find it convenient to stratify our analysis of rank 5 properly premodular categories by the rank of the Müger center. From Propositions III.1, III.3, III.7, and III.11, we will be able to prove the following:

*If* $C$ *is a pseudo-unitary rank 5 premodular category, then one of the following holds:*

$C$

*is Tannakian and is given by*Rep(*G*),*where G is*$Z5$,*D*_{8},*Q*_{8},*D*_{14}, $Z5\u22caZ4$, $Z7\u22caZ3$, $S4$*, or*$A5$.$C$

*is properly premodular and Grothendieck equivalent to the following:*Rep(

*D*_{8})*with*$C\u2032\u2245RepZ2,di=1,1,2,1,1$*, and*$T=diag1,1,\theta ,\u22121,\u22121$*, where θ is a root of unity satisfying a monic degree 4 polynomial over*$Z$.Rep(

*D*_{14})*with*$C\u2032\u2245RepZ2$*and*$di=1,1,2,2,2$.$Rep(S4)$

*with*$C\u2032\u2245RepS3$, $di=1,1,2,3,3$*, and*$T=diag1,1,1,\u22121,\u22121$.$SU(2)8/Z2$.

$C$

*is modular and it is Grothendieck equivalent to*$SU(2)4,SU(2)9/Z2,SU(5)1$*or*$SU(3)4/Z3$*.*^{13}

*Moreover, each case is realized.*

### A. Rank 4 Müger center

*Proposition III.3.*

*There is no rank* 5 *properly premodular category with a rank* 4 *Müger center.*

In order to prove this result, we proceed through Propositions III.4, III.5, and III.6. These propositions consider the different structures of $C\u2032$.

By Theorem II.1, the structure of $C\u2032$ is Rep(*G*, *z*) as a symmetric category. Moreover, we have |*G*| ≤ 42^{11} [Lemma 4.4(ii)]. Using GAP, we can conclude that *G* is isomorphic to $Z4$, $Z2\xd7Z2$, *D*_{10}, or $A4$. In each of these cases, we may apply Lemma II.5 to deduce that *s*_{4,4} = −4. Furthermore, in each case, we can apply Theorem II.1 and Lemma II.2 along with (1) applied to *S*_{a,a} for 0 ≤ *a* ≤ 4 to deduce that *θ*_{a} = 1 for 0 ≤ *a* ≤ 3. In particular, the *S*- and *T*-matrices are determined by the dimension, *d*, and twist, *θ*, of *X*_{4}. Moreover, by Lemma II.4, we can conclude that the order, *n*, of *θ* satisfies $n\u22081,2,3,4,5,6,8,10,12$. The cases of $Z4$ and $Z2\xd7Z2$ are sufficiently similar that we consider them in tandem.

*Proposition III.4.*

*There is no rank 5 premodular category with Müger center Grothendieck equivalent to* $RepZ4$ *or* $RepZ22$.

*Proof.*

In both cases, the only $n\u22081,2,3,4,5,6,8,10,12$ consistent with Eqs. (1) and (2) is *n* = 10; consequently, $d=1+5$. Letting *G* denote the group such that $C\u2032\u2245RepG$, we can de-equivariantize to produce a modular category $CG$. Let $D^$ denote the set of orbits of isomorphism classes of the *G*-action, and let *D* denote a transversal of $D^$. Then the simples of $C$ can be understood via equivariantization as pairs $y,V$, where *y* runs over *D* and *V* ranges over the simple projective representations of $StabGy$.^{19} Since $I$ is stabilized by *G* and *G* is an abelian group, four of the simples in $C$ are of the form $I,\chi a$, where *χ*_{a} is a irreducible character of *G*. The remaining simple is $\u0177,V^$, where $V^$ is the sole simple projective representation of $StabG\u0177$. In the case of $G\u2245Z4$, we recall that $H2Z2k,C\xd7=0$ for all *k*, and so we may take $V^$ to be the sole irreducible linear representation. In particular, $dim\u0177=1+5/4\u2209Z\xaf$, an impossibility. So it suffices to consider $G\u2245Z22$. In this case, $H2Z22,C\xd7\u2245Z2$, and so *V* must be the 2-dimensional simple projective corresponding to the nontrivial 2-cocycle in $H2Z22,C\xd7$. In particular, $CZ22\u2245Fib$. This is not possible.^{20}

*Proposition III.5.*

*There is no rank* 5 *premodular category* $C$ *with* $C\u2032=Rep(D10)$.

*Proposition III.6.*

*There is no rank* 5 *premodular category* $C$ *with* $C\u2032=Rep(A4)$.

*Proof.*

The only choice for $n\u22081,2,3,4,5,6,8,10,12$ consistent with Eqs. (1) and (2) is *n* = 4 or 12. In both cases, we have $d=23$. Next we observe that $Z3\u2282A4$, and so $CA4$ is a premodular category with $Z3$ action. Letting *D* denote a transversal of the orbits of the simple object isomorphism classes under this action, we can conclude that the isomorphism classes of simple objects in $C$ are described by pairs $y,V$, where *y* ∈ *D* and *V* is a simple projective representation of $StabZ3y$. Since $y=I$ is fixed by the $Z3$-action, we can deduce that three of the simples are $I,\chi a$, where the *χ*_{a} are the irreducible characters of $Z3$. The remaining two objects we denote by $y,V$ and $z,W$. Since $Z3$ has no proper subgroups, it must be that *y* ≠ *z* and $StabZ3y=StabZ3z=1$. Applying the orbit-stabilizer theorem, we know that $Z3.y=Z3.z=3$. Thus *y* has dimension $2/3\u2209Z\xaf$, an impossibility.

### B. Rank 3 Müger center

*Proposition III.7.*

*If* $C$ *is a rank 5 properly premodular category with a rank* 3 *Müger center, then* $C$ *is Grothendieck equivalent to* $RepS4$, $C\u2032\u2245RepS3$*, and* $T=diag1,1,1,\u22121,\u22121$.

We will consider the different possibilities for $C\u2032$ in Propositions III.9 and III.10. However, before doing this, we find it useful to establish the following lemma.

*Lemma III.8.*

*Suppose* $C$ *is a modular category with d*_{a} = *d*_{b} ≠ 1 *for a* ≠ *b and there exists* $\sigma \u2208GalC$ *such that* $\sigma a=0$*, then* $d\sigma b=\xb11$*. In particular,* $Cpt\u2247Vec$.

*Proof.*

This follows from the fact that $d\sigma 0\sigma dk=d\sigma k\u03f5\sigma k,\sigma $ $\u2009\u21d2\u2009\sigma da=\sigma db$^{9} (Theorem 2.7).

Next, notice that groups with 3 irreducible representations must satisfy 3 ≤ |*G*| ≤ *A*_{3} ⇒3 ≤ |*G*| ≤ 6^{11} [Lemma 4.4(ii)]. In particular, $G\u2245S3$ or $Z3$.

*Proposition III.9.*

*There is no rank 5 properly premodular category whose Müger center is Grothendieck equivalent to* $RepZ3$.

*Proof.*

Suppose $C$ is such a properly premodular category. Then $C\u2032$ is Tannakian since $Z3$ has no order 2 central elements. Thus $CZ3$ is modular. Now let $D^$ be the set of orbits of isomorphism classes of simples in $CZ3$ under the $Z3$ action, and let *D* be a transversal of $D^$. Then the isomorphism classes of simples in $C$ are given by $I,\chi a$, $y,V$, and $z,W$, where *χ*_{a} are the irreducible linear characters of $Z3$ and *V* and *W* are simple projective representations of $StabZ3y$ and $StabZ3z$, respectively. Since $Z3$ has no proper subgroups and $H2Z3,C\xd7=0$, we may take *V* and *W* to be irreducible linear representations. Of course all irreducibles of the stabilizer must appear, and so *y* ≠ *z* and the stabilizers are trivial. Applying the orbit-stabilizer theorem, we can conclude that $Z3.y=Z3.z=3$ and $CZ3$ is a rank 7 modular category of global dimension 1 + 3*a*^{2} + 3*b*^{2}, where *a* = *d*_{3}/3 and *b* = *d*_{4}/3. So either *a* = *b* = 1^{12} or we may apply Lemma III.8 to conclude that *a* = 1. Applying the equidimensionality of the universal grading of $CZ3$,^{21,22} we can deduce that *b* = 1 or 2. The latter case cannot occur as every integral modular category of rank 7 is pointed^{12} (Theorem 5.8). Thus the global dimension of $C$ is 21. From here, we can conclude that $C$ is Grothendieck equivalent to $RepZ3\u22caZ7$^{23} (Theorem 6.3). Applying these fusion rules, balancing, and the symmetry *s*_{34} = *s*_{43} allows us to conclude that *θ*_{3} = *θ*_{4}. The last column of *S* is now determined in terms of *θ*_{3} by balancing and the fusion rules.

Applying orthogonality of the first and last columns of the *S*-matrix produces the equation $\theta 32+5\theta 3+1=0$. This is not possible as *θ*_{3} is a root of unity.

*Proposition III.10.*

*If* $C$ *is properly premodular and* $C\u2032$*has rank* 3*, then* $C$ *is Grothendieck equivalent to* $RepS4$, $C\u2032\u2245RepS3$*, and* $T=diag1,1,1,\u22121,\u22121$*. Such a category can be obtained via the* $S3$*-equivariantization of a 3-fermion theory.*

*Proof.*

Just as in the proof of Proposition III.9, we may de-equivariantize and understand the simples of $C$ in terms of equivariantization. Since $H2S3,C\xd7=0$, we have that the isomorphism classes of simples are $I,\rho a$, $y,V$, and $z,W$, where *ρ*_{a} are the irreducible representations of $S3$, while *V* and *W* are the irreducible linear representations of $StabS3y$ and $StabS3z$, respectively. There are now two cases to consider:

*Case 1:y* = *z*. In this case, $StabS3y$ can only have two irreducible linear representations and hence is isomorphic to $Z2$. By the orbit-stabilizer theorem, we can conclude that $CS3$ has rank 4 and thus is pointed.^{9}

^{19}and the balancing equation easily tell us that

*θ*

_{0}=

*θ*

_{1}=

*θ*

_{2}= 1. In addition, if we de-equivariantize by $Rep(Z2)\u2282Rep(S3)=C\u2032$, we get a premodular category with simples with dimensions 1, 1, 1, 3. From here, we can conclude that $x3=x3*$ and $x4=x4*$

^{19}(Proposition 3.21). In addition, we know that

*X*

_{1}⊗

*X*

_{3}=

*X*

_{4}⇒

*X*

_{3}⊗

*X*

_{3}does not contain

*X*

_{1}in its decomposition. If we know look at the dimension equation corresponding to

*X*

_{3}⊗

*X*

_{3}, we get the following Diophantine equation:

The latter case cannot happen since this would force 4 ≤ ||*N*_{3}||_{max} ≤ *d*_{3} = 3^{10} (Lemma 3.14). From here, we can finally conclude that $N233=N234=N243=N244=N134=1$ and $N333+N334=N443+N444=2$. If we now use the fact that all the *N*_{i} matrices commute, we can conclude that $N334=N444$ and $N443=N333$. If we combine the previous mentioned observations and the equation that you get by looking at *s*_{33}, we are able to conclude that *θ*_{3} = *θ*_{4} = −1.

Note that if we consider the rank 4 pointed modular categories, $D$, then the Grothendieck group is either $Z4$ or $Z22$. In the former case, there are two $S3$ equivariantizations of rank 9 and 12. In the latter case, if we take $D$ to be equivalent to the toric code, then there are again two $S3$ equivariantizations of rank 9 and 12. The final option is that $D$ is Grothendieck equivalent to the 3-fermion theory. In this case, $Aut\u2297brD\u2245S3$, and so the possible $S3$-actions are given by group homomorphisms, $HomS3,S3$. Once again, there are rank 9 and 12 equivariantizations. However, the identity automorphism yields an $S3$-action on $D$ with 2 orbits. The stabilizer of the nontrivial orbit is $Z3$, and so the $S3$-equivariantization under such an action would have rank 5 and have simples of dimension 1, 1, 2, 3, 3. Thus $C$ can be realized as an $S3$-equivariantization of the 3-fermion theory.

*Case 2:y* ≠ *z*. Since all irreducibles must appear, we have that the stabilizers of *y* and *z* in $S3$ are trivial. Applying the orbit-stabilizer theorem, we can conclude that $CS3$ is a rank 13 modular category of dimension 1 + 6*a*^{2} + 6*b*^{2}, where *a* = *d*_{3}/6 and *b* = *d*_{4}/6. In the case that $CS3$ is integral, by exhaustive search, we conclude that *a* = *b* = 1.^{11} On the other hand, if $CS3$ is not integral in which case we may apply Lemma III.8 to conclude that *a* = 1. Invoking the universal grading,^{21} we have *b* = 1 or $b=7$. In the latter case, $CS3$ has dimension 49, a contradiction^{24} (Proposition 8.32). Thus $CS3$ is pointed.

To see that this is not possible, let $D$ be a rank 13 pointed modular category. Of course, the orthogonal group on $Z13$, with quadratic form *q* coming from the twist on $D$, $OZ13,q$, is either trivial or isomorphic to $Z2$. So the actions of $S3$ on $D$ are given by elements of $HomS3,OZ13,q$, which are either trivial or the sign action. In the case of the trivial action, the isomorphism classes of simple objects in $D$ are fixed under the action of $S3$. In particular, the de-equivariantization has rank 39. In the case of the sign action, there are 7 orbits of simples under the $S3$ action, and hence, the de-equivariantization has rank 21. Thus, $C$ is not the $S3$-equivariantization of a rank 13 pointed modular category.

### C. Rank 2 Müger center

*Proposition III.11.*

*If* $C$ *is a rank 5 properly pseudo-unitary premodular category with a rank 2 Müger center, then one of the following is true:*

$C$

*is Grothendieck equivalent to*Rep(*D*_{8})*with*$C\u2032\u2245Rep(Z2)$*. Furthermore, the T-matrix is of the form*$T=diag1,1,\theta ,\u22121,\u22121$*for some root of unity θ, corresponding to that*2*-dimensional simple object, satisfying a monic quartic polynomial over*$Z$*or*$C$

*is Grothendieck equivalent to*$RepD14$.$C$

*is Grothendieck equivalent to*$SU(2)8/Z2$.

*Moreover, each is realized (i) by a*$Z2$

*-equivariantization of the toric code, (ii) by the adjoint subcategory of a*56

*-dimensional metaplectic category, e.g.,*$SO142ad$

*, and (iii) by*$SU28/Z2$.

*Proof.*

If a group *G* has only two irreducible representations, then $G\u2245Z2$^{11} [Lemma 4.4(ii)]. Thus $C\u2032$ is Grothendieck equivalent to $RepZ2$. Since $C$ has odd rank, we can further conclude that $C\u2032$ is Tannakian. Ordering the simples so that $X0=I$ and *X*_{1} generates $C\u2032$, we have two cases to consider.

*Case 1:X*_{1} ⊗ *X*_{3} = *X*_{4} and *X*_{1} ⊗ *X*_{2} = *X*_{2}. It follows immediately from these fusion rules that *X*_{2} is self-dual. Next note that the de-equivariantization $CZ2$ is a modular category of rank 4 and two of the objects have the same dimension. This forces $CZ2$ to be pointed. Since $C$ is pseudo-unitary, we either have Fib ⊠ Fib, Fib ⊠ Sem, or $Fib\u22a0Sem\xaf$.^{9}

*d*

_{2}= 2 and

*d*

_{3}=

*d*

_{4}= 1. Thus $X2\u2297X2=I\u2295X1\u2295X1\u2295X2$ or $I\u2295X1\u2295X3\u2295X4$. In the former case,

*X*

_{2}would generate a 6-dimensional fusion subcategory. This is not possible as 6 ∤ 8. Thus

*X*

_{2}generates $C$. Next note that this category is necessarily near group and hence can only be braided if $X3*=X4$

^{25,26}(Remark 4.4). It now follows that $C$ is Grothendieck equivalent to $RepD8$.

^{9}Examining the balancing equation for

*S*

_{1,3}, we see that

*θ*

_{3}=

*θ*

_{4}. Since

*X*

_{2}is self-dual, we can conclude that column 2 in

*S*is real. In particular,

*θ*

_{3}= ±1. Since

*X*

_{3}is not in the Müger center of $C$, we know that the 3-rd and 0-th column of

*S*must be orthogonal; hence,

*θ*

_{3}=

*θ*

_{4}= −1. From here, we can conclude that

This yields a quartic polynomial for *θ* over $Z$. Considering all possible *n* such that $\varphi n<4$ gives possible primitive roots. However, *θ* = *ζ*_{n} with $\varphi n<4$ satisfies (3). Such a category can be constructed by considering the equivariantization of the toric code under the nontrivial $Z2$-action.

In the case where $CZ2=Fib\u22a0Fib$, we get the following set of dimensions {1, 1, *φ*^{2}, *φ*^{2}, 2*φ*} for $C$, where *φ* is the golden mean. After computing the fusion rules using (1), we can conclude that $C$ is Grothendieck equivalent *SU*(2)_{8}.

If $CZ2=Fib\u22a0Sem$, we get a category $C$ with dimensions {1, 1, 1, 1, 2*φ*}. This category violates a condition given for near-group categories^{25} (Theorem 1.1); hence, it does not exist. The case of $Fib\u22a0Sem\xaf$ follows similarly.

*Case 2: X*_{1} ⊗ *X*_{a} = *X*_{a} for 2 ≤ *a* ≤ 4. In this case, the de-equivariantization is rank 7 modular with simples of dimension $1,d1^,d1^,d2^,d2^,d3^,d3^$. In particular, we know that $CZ2pt$ is either $RepZ3$ or $CZ2$ (Lemma III.8).^{12} In the former case, applying the universal grading and the pigeon-hole principle gives $d^22+d^32=3$. From here, notice that the global dimension of this category is 9, and we can conclude that $CZ2pt=CZ2\u2247RepZ3$^{24} (Proposition 8.32), a contradiction.

Thus $CZ2$ is pointed, and hence, *d*_{1} = *d*_{2} = *d*_{3} = 2. By the pigeon-hole principle and without loss of generality, we may assume *X*_{2} is self-dual. Just as in case 1, a dimension argument shows that *X*_{2} cannot be a sub-object of *X*_{2} ⊗ *X*_{2}. So without loss of generality, we have $X2\u2297X2=I\u2295X1\u2295X3$. Of course, the category generated by *X*_{2} cannot exclude *X*_{4} since its dimension must divide 14. Thus $C$ is cyclically generated by *X*_{2}. Furthermore, since *X*_{2} is self-dual, its tensor square must be as well. In particular, *X*_{3} is self-dual. The pigeon-hole principle reveals that *X*_{4} must also be self-dual. Thus $C$ is Grothendieck equivalent to $RepD14$^{26} (Theorem 4.2). One can arrive at the same conclusion by considering equivariantizations of the rank 4 pointed modular categories,^{9} as in the proof of Proposition III.10. Such categories may be realized as the adjoint categories of 56-dimensional metaplectic categories.^{27}

*Remark III.12.*

*It is interesting to note that none of the known algebraic conditions*^{8}*for premodular categories are sufficient to determine θ in Proposition III.11 beyond it being a root of a quartic monic polynomial over* $Z$.

## ACKNOWLEDGMENTS

The authors would like to thank César Galindo for enlightening discussions. The research described in this paper was, in part, conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy, Release No. PNNL-SA-120942.

## REFERENCES

^{3}m with m square-free