Norm, trace, and formal codegrees of fusion categories
aa r X i v : . [ m a t h . QA ] M a y Norm, trace, and formal codegreesof fusion categories
Andrew Schopieray ∗ May 29, 2020
Abstract
We prove several results in the theory of fusion categories using the product (norm)and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm.Furthermore, with two exceptions, all formal codegrees of spherical fusion categories withsquare-free norm are rational integers. This implies, with three exceptions, that everyspherical braided fusion category whose global dimension has prime norm is pointed.The reason exceptions occur is related to the classical Schur-Siegel-Smyth problem ofdescribing totally positive algebraic integers of small absolute trace.
Fusion categories and their many variants (tensor, spherical, braided, modular, etc.) are avast generalization of the representation theory of finite groups and finite-dimensional Hopfalgebras. The formal codegrees of a fusion category C , a finite collection of numerical invariantsassociated to representations of the underlying Grothendieck ring defined in Section 2.3, haveproven to be critical to this study and include the Frobenius-Perron dimension (FPdim( C )) andglobal dimension (dim( C )) for spherical fusion categories, as examples. Formal codegrees arerestrictive from a number-theoretic perspective because they are examples of totally positivecyclotomic integers, and the less-familiar algebraic d -numbers [27, Definition 1.1]. Frobenius-Schur indicators of semisimple quasi-Hopf algebras [22] and spherical fusion categories [24],and higher Gauss sums of modular tensor categories [25] are other examples of algebraic d -numbers. The main goal of this paper is to expand the general theory of formal codegrees offusion categories which is mainly contained in [27, 28, 29] thus far.For a formal codegree f ∈ C of a fusion category, let N ( f ) be the norm of f , or theproduct of its Galois conjugates, and Tr( f ) be the trace of f , or their sum. Theorem 3.1states that for each m ∈ Z ≥ there exist finitely-many fusion categories C up to equivalencewith N (dim( C )) = m . This norm finiteness is also true for FPdim( C ) but because the setof all Frobenius-Perron dimensions of fusion categories is a discrete subset of the positivereal numbers [6, Corollary 3.13]. In Section 4 we observe a unique feature of algebraic d -numbers: divisibility of algebraic d -numbers is equivalent to divisibility of their norms. As aresult (Theorem 4.4), if f is a formal codegree of a fusion category C which is not divisibleby any rational integer, then the formal codegrees of C are precisely the Galois orbit of f . ∗ This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley,California, during the Spring 2020 semester. C ( sl , κ − ad for prime κ ∈ Z ≥ coming from the representation theory of U q ( sl ) with q a root of unity [33]. The last resultof this section, Theorem 4.13, pertains to spherical fusion categories. Sphericality is a veryweak assumption as all known examples of fusion categories possess a spherical structure. Weprove that if C is a spherical fusion category with a formal codegree f of square-free norm,then f ∈ Z or f = (1 / ± √ / ± √ σ [26].We then apply our general theory to classification results for spherical fusion categories inSection 5. Recently in [38, Question 3.8] it was asked whether a spherical braided fusion cate-gory of prime global dimension is pointed (all simple objects have Frobenius-Perron dimension1) or equivalent to Fib ⊠ Fib σ . Theorem 5.12 answers this question in the affirmative andgeneralizes the result to include arbitrary number fields. That is to say any spherical braidedfusion category whose global dimension has prime norm is pointed, with the exception of Fib,Fib σ and Fib ⊠ Fib σ . The reason these exceptional (not pointed) spherical braided fusioncategories can occur is that the categorical dimensions of their simple objects are exceptionalcases of a classical result of Cassels [9, Lemma 3] on cyclotomic integers α such that the abso-lute trace (average of Galois conjugates) of | α | is less than 2. In Theorem 5.5, we remove theassumption of a braiding for spherical fusion categories with global dimension whose norm isa safe prime . Safe primes p ∈ Z ≥ are of the form p = 2 q + 1 where q ∈ Z ≥ is also prime. Itis expected, but currently not proven, that infinitely many safe primes exist. The reason theexceptional (not pointed) spherical fusion categories can occur in this case is that all of theirformal codegrees are of the form p · u where u is one of the three totally positive algebraicintegers of smallest absolute trace: 1 or (1 / ± √ λ <
2, the proof of finiteness,and classification of totally positive algebraic integers of absolute trace strictly less than λ isknown as the Schur-Siegel-Smyth trace problem. We encourage the reader to refer to [1] foran expository look at the long history of this problem which continues to this day. The firstresult in this direction was due to Schur [34, Satz VIII] who solved the finiteness problem for λ = √ e and the classification was subsequently solved by Siegel [35, Theorem III] for λ = 3 / p = 13 (Example 5.7) as a proof-of-concept that the methods we have developed for sphericalfusion categories can be pushed outside of the safe primes, but certainly more tools will beneeded to complete this classification for arbitrary fusion categories.This is not the first time the Schur-Siegel-Smyth trace problem, nor the results on absolutetrace of Cassels have appeared in the literature in relation to fusion categories. Gelaki, Naidu,and Nikshych used Siegel’s initial trace bound to study the number of zeroes in the S -matrixof a weakly integral modular tensor category [19, Proposition 6.2]. Their result can be viewedas an extension of similar results for zeroes of characters of finite groups (refer to [36] andreferences within). Later, Calegari, Morrison, and Snyder [8] improved upon Cassels result [9,Lemma 3] as a means to classify the smallest possible Frobenius-Perron dimensions of objectsin fusion categories; Calegari and Guo would make further improvements in [7]. The resultswe prove here on formal codegrees of fusion categories continue this tradition, and furtherilluminate the necessity for algebraic number theory in the study of fusion categories.2otation Meaning Notation Meaning α algebraic integer ζ n exp(2 πi/n ) K , L , . . . algebraic number fields Q ( ζ n ) + Q ( ζ n + ζ − n ) K × , L × , . . . unit groups [ n ] ∈ Q ( ζ m ) sin( nπ/m ) / sin( π/m )[ L : K ] dim K ( L ) N ( α ) Q σ ∈ Gal( Q ( α ) / Q ) σ ( α ) d α [ Q ( α ) : Q ] Tr( α ) P σ ∈ Gal( Q ( α ) / Q ) σ ( α )Figure 1: Recurring notation This exposition is as self-contained as one could hope; we include the following preliminarysections with references for readers less familiar with either elementary number theory orfusion categories. We also include a section about our main technical tool, formal codegrees,where the interplay of number theory and fusion categories is well illustrated. d -numbers What is needed here from algebraic number theory can be found in any undergraduate text-book on the subject (e.g. [2]). The following notation and language will be fixed throughout.The field of complex numbers, C , contains all other fields considered. Let Q ⊂ C be the fieldof rational numbers with algebraic closure Q . By a number field we will mean any extension K / Q with [ K : Q ] := dim Q ( K ) < ∞ . An algebraic integer is any α ∈ C which is a rootof a monic polynomial with coefficients in Z (the rational integers). For brevity, if α is analgebraic integer, we set d α := [ Q ( α ) : Q ]. The set of all algebraic integers has the structure ofa unital ring, A , and the group of invertible elements, or algebraic units , will be denoted A × .More generally, if K is any number field, then the ring of algebraic integers and the group ofalgebraic units contained in K will be denoted O K and O × K , respectively. Let α, β ∈ A , α = 0.Then α divides β if and only if there exists γ ∈ A such that αγ = β .All number fields of interest to us will be cyclotomic in light of [14, Theorem 8.51], whichis to say contained in Q ( ζ n ) where ζ n := exp(2 πi/n ) for some n ∈ Z ≥ . We refer the readerto [37] for further resources about cyclotomic number fields. One benefit of operating incyclotomic number fields is that they are all Galois extensions of Q , i.e. number fields K with K / Q a normal extension. In particular Gal( Q ( ζ n ) / Q )) ∼ = ( Z /n Z ) × . The maximal totally realsubfield of Q ( ζ n ) is Q ( ζ n ) + := Q ( ζ n + ζ − n ) with [ Q ( ζ n ) : Q ( ζ n ) + ] = 2 for n = 2. Algebraicunits of interest to us will be cyclotomic units : those products of a root of 1 and units ofthe form ( ζ n − / ( ζ −
1) where n ∈ Z ≥ and ζ is any root of 1. For example, let p be anodd prime. Lemma 8.1 of [37] describes multiplicative generators of the cyclotomic units interms of the real units [ a ] := sin( aπ/p ) / sin( π/p ) and ζ p . In particular, the cyclotomic unitsof Q ( ζ p ) + are multiplicatively generated by ± A := { [ a ] : 2 ≤ a ≤ ( p − / } .We will often refer to the product and sum of all Galois conjugates of algebraic integers.If K is a cyclotomic number field (more generally, if K / Q is Galois) we may safely define the norm function N K : K → Q by N K ( α ) := Y τ ∈ Gal( K / Q ) τ ( α ) . (1)3hen there is no risk of ambiguity, as will almost always be the case, we will reduce thisnotation to N ( α ) := N Q ( α ) ( α ). The norm function is multiplicative and when α ∈ O K , N K ( α ) ∈ Z . Therefore if Q ( α ) ⊂ K , N K ( α ) = N ( α ) [ K : Q ( α )] . Algebraic units are those u ∈ O K with N K ( u ) = ±
1. Likewise, we define the trace function Tr K : K → Q byTr K ( α ) := X τ ∈ Gal( K / Q ) τ ( α ) (2)and Tr( α ) := Tr Q ( α ) ( α ). The trace function is additive and when α ∈ O K , Tr K ( α ) ∈ Z .Therefore if Q ( α ) ⊂ K , then Tr K ( α ) = [ K : Q ( α )]Tr( α ). If x n + a x n − + · · · + a n − x + a n isthe minimal polynomial of a cyclotomic integer α , then Tr( α ) = − a and N ( α ) = ( − n a n .Lastly, a very specific type of algebraic integer appears abundantly in the study of fusioncategories, introduced by Ostrik in [27]. An algebraic d -number is a nonzero α ∈ A suchthat σ ( α ) /α ∈ A for all σ in the absolute Galois group Gal( Q / Q ). Lemma 2.7 of [27] givesmany alternative characterizations of a d -number. For our purposes the most importantcharacterization is that a nonzero α ∈ A is a d -number if and only if the minimal polynomialof α over Q (which lies in Z [ x ]), x n + a x n − + · · · + a n − x + a n , enjoys the property that( a n ) j divides ( a j ) n for all 1 ≤ j ≤ n . If α is a cyclotomic d -number, then by the proof of [27,Lemma 2.7], α d α = N ( α ) · u = ( − n a n · u (3)for some u ∈ O × Q ( α ) . This fact is crucial for future arguments. In what follows we will consider fusion categories over C in the sense of [13, 14]. These aresemisimple tensor categories [13, Definition 4.1.1] over C whose set of isomorphism classes ofsimple objects, O ( C ), is finite. One important property of tensor categories is the existenceof duality of objects: a permutation of O ( C ) which we denote by X X ∗ (without theassumption of semisimplicity, one would need to consider left and right duals separately [13,Proposition 4.8.1]). The decomposition of ⊗ -products into elements of O ( C ) (afforded bysemisimplicity) are known as the fusion rules of C . Example 2.1.
The canonical examples of fusion categories are Rep( G ) and Vec G , the cate-gories of finite-dimensional complex representations of a finite group G , and finite dimensionalcomplex G -graded vector spaces [13, Examples 2.3.4–2.3.6]. When G is the trivial group werecover the trivial fusion category Vec, the unique fusion category C (up to equivalence) with |O ( C ) | = 1.The fusion rules of a fusion category C are a property of the underlying Grothendieckring, K ( C ), hence a single Grothendieck ring can have many categorifications, correspondingto various associativity isomorphisms for these products. For example the group ring Z [ G ]is categorified by Vec G , but the associativity may also be twisted by a 3-cocycle ω to pro-duce potentially inequivalent categories Vec ωG [13, Example 2.3.8]. One can consider theseassociativity isomorphisms as solutions to large sets of algebraic equations gotten from therelations constraining associativity [13, Definition 2.1.1]. It is well-known that any solutionto this system of constraining equations corresponds to a categorification of the underlyingGrothendieck ring (see [3]), therefore for each fusion category C and Galois automorphism σ ∈ Gal( Q / Q ) one can construct a Galois conjugate fusion category C σ by applying σ to allstructure constants of C . A priori C σ could be equivalent to C , but many new examples offusion categories can be created by this Galois conjugation.4 .2.1 Dimensions There are many methods for differentiating between fusion categories via numerical invariants.For each X ∈ O ( C ) we define FPdim( X ) to be the maximal real eigenvalue of the nonnegativeinteger matrix of ⊗ -ing with X afforded by the Frobenius-Perron Theorem [17, 30]. Thesedimensions collect to a ring homomorphism FPdim : K ( C ) → C [13, Proposition 3.3.6 (1)].Then the Frobenius-Perron dimension of C FPdim( C ) := X X ∈O ( C ) FPdim( X ) (4)is one measurement of the size of a fusion category. Assumptions about Frobenius-Perrondimension are very limiting as this is a numerical invariant not only of the fusion categorybut of its underlying Grothendieck ring. More flexible numerical invariants are the squarednorms | X | for X ∈ O ( C ) introduced by M¨uger in [23] where one can find further details (wewill have no reason to refer to these in further sections). As with Frobenius-Perron dimension,we then define the global dimension of C :dim( C ) := X X ∈O ( C ) | X | . (5)As global dimension is defined in terms of morphisms of a fusion category C , for all σ ∈ Gal( Q / Q ) we have dim( C σ ) = σ (dim( C )) while FPdim( C σ ) = FPdim( C ) because the under-lying Grothendieck ring is unchanged. This makes the category C := ⊠ σ ∈ Gal( Q (dim( C )) / Q ) C σ (6)incredibly useful where ⊠ is the Deligne product of fusion categories [13, Section 4.6]. Inparticular, the global dimension is dim( C ) = N (dim( C )) while the Frobenius-Perron dimensionis FPdim( C ) = FPdim( C ) d dim( C ) , recalling that d dim( C ) = [ Q (dim( C )) : Q ]. A pivotal structure δ on a fusion category C is a collection of functorial tensor isomorphisms δ X : X → ( X ∗ ) ∗ for all X ∈ O ( C ). These may be represented by invertible scalars dim δ ( X ),the categorical dimensions of simple objects, which collect to form a ring homomorphismdim δ : K ( C ) → C [13, Proposition 4.7.12]. We say a pivotal structure δ is spherical ifdim δ ( X ) = dim δ ( X ∗ ) for all X ∈ O ( C ). It is a slight (common) abuse of language tosay δ is a spherical structure rather than δ is a pivotal structure which is spherical. A spherical fusion category is a fusion category C with a chosen spherical structure δ . But inwhat follows, the choice of spherical structure is irrelevant or uniquely determined so we willomit δ from all notation. If C is a spherical fusion category, one has dim( X ) = | X | for X ∈ O ( C ) [14, Corollary 2.10] so one may replace squared norms with categorical dimensionsin (5). Sphericality is potentially a very weak assumption on a fusion category as all currentlyknown examples of fusion categories posess a spherical structure. Whether this is true ingeneral has been an open problem of great interest. When dim( C ) = FPdim( C ) we saythat C is pseudounitary . Pseudounitary fusion categories have a unique spherical structuresuch that the categorical and Frobenius-Perron dimensions of all simple objects coincide [14,Proposition 8.23]. It will be assumed that a pseudounitary fusion category is equipped withthis distinguished spherical structure unless otherwise indicated.5 xample 2.2. If C is a fusion category such that FPdim( X ) = 1 for all X ∈ O ( C ) (such X are called invertible ), we say that C is pointed . Pointed categories are pseudounitary [14,Proposition 8.24] with FPdim( C ) = dim( C ) = |O ( C ) | and moreover are equivalent to Vec ωG forsome finite group G and 3-cocycle ω , as fusion categories. The correct notion of ⊗ -commutativity in fusion categories is that of a braiding. We say afusion category C is braided if there exists a family of functorial isomorphisms σ X,Y : X ⊗ Y → Y ⊗ X for all X, Y ∈ O ( C ) satisfying braid-like relations [13, Definition 8.1.1] which will beirrelevant for our arguments. The complexity of a braiding is measured by the symmetriccenter C ′ which has simple objects X such that σ Y,X σ X,Y = id X ⊗ Y for all Y ∈ O ( C ). Thetwo extremes are braided fusion categories for which C ′ = C or C ′ = Vec which we call symmetrically braided and nondegenerately braided , respectively. A spherical fusion categorywhich is nondegenerately braided is a modular tensor category . Modular tensor categories areso-named because they give two important |O ( C ) | -dimensional representations of the modulargroup SL ( Z ) generated by x, y such that x = 1 and ( xy ) = 1. First, there exists a projectiverepresentation x ˜ S , y ˜ T where ˜ T is a diagonal matrix consisting of roots of unity ˜ t X for x ∈ O ( C ) [13, Corollary 8.18.2] known as twists of simple objects. The collection of all scalars˜ s X,Y and ˜ t X for all X, Y ∈ O ( C ) is called the unnormalized modular data of C . We may thendefine the Gauss sums τ ± := X X ∈O ( C ) (˜ t X ) ± dim( X ) . (7)Choose any γ ∈ C which is a cube root of the root of unity τ + / p dim( C ) where p dim( C )is the positive square root of the global dimension. Then s := (1 / p dim( C ))˜ s and t := γ − ˜ t define a linear representation of SL ( Z ). The collection of all scalars s X,Y and t X for all X, Y ∈ O ( C ) is called the normalized modular data of C . A powerful characteristic of a modular tensor category is the existence of a Galois actionon its normalized modular data [10, 11]. Following [12, Theorem II], let n, ˜ n ∈ Z ≥ be theorders of t and ˜ t , respectively. Then the normalized and unnormalized modular data of C are contained in Q ( ζ n ) and Q ( ζ ˜ n ), respectively, with ˜ n | n , i.e. Q ( ζ ˜ n ) ⊆ Q ( ζ n ). And for each σ ∈ Gal( Q ( ζ n ) / Q ), there exists a unique permutation ˆ σ of O ( C ) such that σ (cid:18) s X,Y s ,Y (cid:19) = s X, ˆ σ ( Y ) s , ˆ σ ( Y ) (8)and σ ( t X ) = t ˆ σ ( X ) (9)for all X, Y ∈ O ( C ). Of particular interest is the square of Equation (8) when Y = whichdescribes the Galois action on squared categorical dimensions: σ (dim( X ) ) σ (dim( C )) = dim(ˆ σ ( X ) )dim( C ) . (10)Here we have used the fact that σ ( s X,Y ) = s X, ˆ σ ( Y ) = s σ ( X ) ,Y .6 .3 Formal codegrees In [27], Ostrik introduced the notion of the (multi-set of) formal codegrees of a fusion ring R , e.g. the Grothendieck ring of a fusion category (or more generally a based ring ). Herewe briefly recount the definition for completeness, although often in practice there are moreefficient methods for computing and studying these scalars which we subsequently summarize. The complexified ring S := R ⊗ Z C where R is a fusion ring possesses a nondegenerate pairing S × S ∗ → C and so for each irreducible representation ϕ of R , we define r ϕ ∈ S as that whichcorresponds to Tr( · , ϕ ) ∈ S ∗ (trace in the sense of linear algebra). The elements r ϕ ∈ S arecentral and thus act by a nonzero scalar f ϕ ∈ C on ϕ and by zero on any other irreduciblerepresentation of R . The scalars f ϕ over all irreducible representations of R (denoted Irr( R ))are called the formal codegrees of R and more specifically, if C is a fusion category, we saythe formal codegrees of K ( C ) are the formal codegrees of C itself. From this definition and[14, Theorem 8.51], the formal codegrees of a fusion category are cyclotomic integers whoseGalois conjugates are all greater than or equal to 1 [28, Remark 2.12]. It was shown in [27,Theorem 1.2] that the formal codegrees of a fusion category are algebraic d -numbers as well.Let B ⊂ R be a basis of a fusion ring with duality b b ∗ for all b ∈ B . If ϕ is the one-dimensional (hence, irreducible) representation induced by a ring homomorphism g ϕ : R → C ,then the corresponding formal codegree of R is f ϕ = X b ∈ B g ϕ ( b ) g ϕ ( b ∗ ) = X b ∈ B | g ϕ ( b ) | . (11)In what follows we will not notationally differentiate between a ring homomorphism and thecorresponding one-dimensional representation. Let C be a fusion category with Grothendieckring K ( C ). Then Frobenius-Perron theory provides one ring homomorphism: FPdim : K ( C ) → C , and when C is spherical, categorical dimension provides another: dim : K ( C ) → C . More-over, if σ ∈ Gal( Q / Q ), then one may conjugate any ring homomorphism ϕ : R → C of afusion ring R by σ . Specifically, σ ( ϕ ) : R → C is defined by σ ( ϕ )( x ) := σ ( ϕ ( x )) for all x ∈ R and evidently σ ( ϕ ) ∼ = ϕ if and only if σ ( ϕ ( x )) = ϕ ( x ) for all x ∈ R .One should often think of formal codegrees of a fusion category C as an abstraction orgeneralization of global/Frobenius-Perron dimension for multiple reasons. Firstly, FPdim( C )is the maximal real eigenvalue of the operator of ⊗ -ing with R := ⊕ X ∈O ( C ) X ⊗ X ∗ . But moregenerally [28, Remark 2.11], the remaining spectrum of this operator consists of dim( ϕ ) f ϕ forall ϕ ∈ Irr( K ( C )) appearing with multiplicity dim( ϕ ) . In concrete examples this can be anefficient method for computing formal codegrees. Secondly [28, Corollary 2.14], each formalcodegree f divides dim( C ) and by [28, Proposition 2.10] the following equalities hold: X ϕ ∈ Irr( K ( C )) dim( ϕ ) f ϕ = 1 ⇐⇒ X ϕ ∈ Irr( K ( C )) dim( ϕ ) dim( C ) f ϕ = dim( C ) . (12)For a modular tensor category, all irreducible representations of K ( C ) are one-dimensionalfrom commutativity, and Verlinde’s formula implies the formal codegrees are dim( C ) / dim( X ) over all X ∈ O ( C ), so (12) is the definition of global dimension in (5).7 .3.2 A series of examples We compute the formal codegrees of several fusion categories to illustrate techniques andconcepts used in further proofs.
Example 2.3.
Let n ∈ Z ≥ . A basis of K (Vec Z /n Z ) is indexed by elements of, and has fusionrules mimicking, the group operation of Z /n Z . As such, any irreducible representation is one-dimensional, corresponding to a ring homomorphism ϕ : K (Vec Z /n Z ) → C determined by ϕ ( x )where x ∈ Z /n Z is any generating element. Further, 1 = ϕ (1 R ) = ϕ ( x n ) = ϕ ( x ) n and ϕ ( x )must be an n th root of unity. Denote the ring homomorphisms ϕ j ( x ) := ζ jn for 0 ≤ j ≤ n − ϕ , . . . , ϕ n − are all nonisomorphic, andmany are Galois conjugate with one another (except ϕ ) depending on n . Using the formulain (11) we then have formal codegreesformal codegree n n · · · n n irr. rep. dim = FPdim = ϕ ϕ · · · ϕ n − ϕ n − (13)Observe that the formal codegrees are indistinguishable, but they correspond to distinctirreducible representations. In particular, the irreducible representations corresponding to theglobal and Frobenius-Perron dimensions are identical, and no other irreducible representationis Galois conjugate to them. Example 2.4.
Let Fib be the unique rank 2 spherical fusion category [26] with O (Fib) = { , X } , nontrivial fusion rule X ⊗ X ∼ = ⊕ X , and dim( X ) = [2] ∈ Q ( ζ ) + , often called theFibonacci category. We will compute the formal codegrees of Fib := Fib ⊠ Fib σ , where σ isthe nontrivial Galois automorphism of Q ( ζ ) + . One may easily compute FPdim(Fib) = 5[2] and dim(Fib) = 5. Therefore f FPdim = 5[2] , f σ (FPdim) = 5 σ ([2]) , and f dim = 5 are formalcodgrees of Fib. But dim has a “hidden” formal codegree: although σ fixes the formalcodegree 5, it does not fix the representation dim itself. In particular the values dim takes onthe nontrivial objects of Fib and Fib σ are swapped by σ by design. Therefore the cardinalityof the Galois orbit of dim is 2, while the cardinality of the Galois orbit of the correspondingformal codegrees is 1. Both orbits have cardinality 2 for the representation FPdim.formal codegree 5 5 5[2] σ ([2]) irr. rep. dim σ (dim) FPdim σ (FPdim) (14)This is a complete collection of formal codegrees of Fib using (12) since15 + 15 + 15[2] + 15 σ ([2]) = 1 . (15) Example 2.5.
Let C be the spherical fusion category constructed via the representation the-ory of U q ( so ) where q is a ninth root of unity [31, Section 5.2]. In particular, everythingwe consider below takes place in Q ( ζ ) + . This extension has degree 3 so let σ be the gener-ating Galois automorphism such that σ (cos( π/ π/ C ad [20, Section 3.1] is rank 6 and with a particular ordering of the basis, the Frobenius-Perronand categorical dimensions of simple objects are, respectively, 1 , σ ([2])[4] , [2][4] , [2] , [2] , [2] and1 , , − , [2] , σ ([2]) , σ ([2]). Hence all Galois conjugates (by σ , σ ) of dim and FPdim are noni-somorphic and using (11) one should verify the following collection of formal codegrees of C ad is complete using (12).formal codegree 9 9 9 9[4] σ ([4]) σ ([4]) irr. rep. dim σ (dim) σ (dim) FPdim σ (FPdim) σ (FPdim) (16)8n this example we see that rational integer global dimensions arise outside of weakly integralexamples and “constructed” examples by taking Deligne products over all Galois conjugates. Example 2.6.
As a final example we will compute the formal codegrees of the near-group fusion category N ( G, k ) [15, 21] where G is a finite abelian group and k ∈ Z ≥ using [27,Lemma 2.6]. We have |O ( N ( G, k )) | = | G | + 1: | G | of the simple objects are invertible andhave the fusion rules of the group G and for each g ∈ G , the remaining simple object ρ satisfies ρg = gρ = ρ . The only non-trivial fusion rule is ρ = k | G | ρ ⊕ L g ∈ G g . Thus FPdim( ρ ) is aroot of x − k | G | x − | G | . The matrix of ⊗ -ing with R = ρ ⊕ | G | is | G | + 1 1 · · · k | G | | G | + 1 · · · k | G | ... ... . . . ... ...1 · · · · · · | G | + 1 k | G | k | G | k | G | · · · k | G | | G | ( k | G | + 2) . (17)As an exercise one should compute that the eigenvalues of (17) are | G | with multiplicity | G | − x − | G | ( k | G | + 4) x + | G | ( k | G | + 4) which are FPdim( N ( G, k ))and its Galois conjugate. The fusion rules of N ( G, k ) are commutative, thus all irreduciblerepresentations of its Grothendieck ring are one-dimensional. Moreover these eigenvalues arethe formal codegrees of N ( G, k ) on the nose.
For any α ∈ A whose Galois conjugates are all greater than or equal to 1, it is clear that α ≤ N ( α ). Hence for any fusion category C , FPdim( C ) ≤ N (FPdim( C )) and moreover thereare finitely-many fusion categories C for any fixed N (FPdim( C )) ∈ Z ≥ by [6, Corollary 3.13].The similar result for global dimension is more subtle as it is not known if the set of all globaldimensions of fusion categories (or even spherical fusion categories) is discrete [29, Section1.2]. Recall the definition C := ⊠ σ ∈ Gal( Q (dim( C )) / Q ) C σ from Section 2.2.1 for the followingproof. Theorem 3.1.
Let m ∈ Z ≥ . There exist finitely-many fusion categories C up to equivalencewith N (dim( C )) = m . Proof.
Set g := dim( C ) for brevity. It is clear that m = N ( g ) = 1 if and only if C ≃
Vec.Ostrik [29, Theorem 4.1.1(i)] has shown that if C is a nontrivial spherical fusion category, then g > /
3. This implies that if m = N ( g ) >
1, then d g ≤ ⌊ log( m ) / log(4 / ⌋ := k . Theorem5.1.1 of [29] statesΣ := { eq. classes of D : D fusion category and dim( D ) = m } (18)is finite. Thus { FPdim( D ) : D ∈ Σ } is finite. But for each degree extension 1 ≤ d g ≤ k , wehave FPdim( C ) = (FPdim( C )) /d g which implies { FPdim( D ) : D spherical fusion category and D ∈ Σ } (19)is finite, and the spherical case is complete as the number of fusion categories of fixedFrobenius-Perron dimension (up to equivalence) is finite. This implies norm finiteness for arbi-trary fusion categories because each fusion category C posesses a sphericalization ˜ C [4, Section5.3] with dim( ˜ C ) = 2 g (hence Q (dim( C )) = Q (dim( ˜ C ))), and moreover N (dim( ˜ C )) = 2 d g · m where, as noted above, d g ≤ k . 9 roposition 3.2. Let C be a fusion category and g := dim( C ). Then d d g g ≤ |O ( C ) | d g ≤ N ( g ) . (20) Proof.
We have by [29, Lemma 4.2.2], |O ( C ) | d g = |O ( C ) | ≤ dim( C ) = N ( g ) . (21)The left-hand inequality in (20) is the fact that d g ≤ |O ( C ) | as the number of formal codegreesof C is less than or equal to the rank.If α ∈ A is totally positive with minimal polynomial x n − a x n − + · · · + ( − n − a n − x +( − n a n , set c j ( α ) := a j , a positive integer. Proposition 3.3.
Let j, m ∈ Z ≥ . There exist finitely-many fusion categories C up toequivalence with c j (dim( C )) = m . Proof.
Note that if C is nontrivial, each of the Galois conjugates of dim( C ) are greater than4 / m = c j (dim( C )) can be expressed as a symmetricfunction of its Galois conjugates, m > (cid:18) d dim( C ) j (cid:19) (4 / j > (3 / d dim( C ) , (22)where the second inequality follows from (4 / j > (3 / j and (cid:0) d dim( C ) j (cid:1) ≥ d dim( C ) /j for all j ∈ Z ≥ with j ≤ d dim( C ) . Hence k := (4 / m ≥ d dim( C ) . But by the d -number condition inEquation (3), N (dim( C )) j divides m d dim( C ) for all 1 ≤ j ≤ d dim( C ) . In particular N (dim( C )) ≤ j √ N k , a fixed value. Our result follows by Theorem 3.1. There has been much effort to study fusion categories whose Frobenius-Perron dimenson isa rational integer ( C is weakly integral ) with small prime factorizations. Some of the firstexamples of this are the classification of fusion categories whose Frobenius-Perron dimensionis a rational prime [14, Corollary 8.30] or the square of a rational prime [14, Proposition 8.32].When C is not weakly integral one can study the more extreme case when FPdim( C ) (hencethe remainder of the formal codegrees of C by Theorem 4.4) has no nontrivial rational integerdivisors. The most obvious case of this is if α , a formal codegree of a fusion category, were aunit. But any α ∈ A × which is totally real is either 1, or σ ( α ) < σ ∈ Gal( Q / Q )because N ( α ) = 1, hence α is not the formal codegree of any fusion category by [28, Remark2.12]. Thus the only fusion category with a unit for a formal codegree is Vec by [28, Proposition2.10]. Surprisingly, the rational integer divisors of a formal codegree of a fusion category canbe identified by only measuring its norm. This is far from true for arbitrary algebraic integers. Example 4.1.
Consider either root α of x + ax + b where a, b ∈ Z \ { } , gcd( a, b ) = 1, and d α = 2. Note that α is an algebraic integer, but α is not a d -number unless b = ±
1. Wehave N ( α ) = b which may have any nonzero rational integer coprime to a as a divisor. But α itself is not divisible by any m ∈ Z \ { , ± } . Indeed, x + ( a/m ) x + ( b/m ) is the minimalpolynomial of α/m over Q which lies in Z [ x ] if and only if m | a and m | b which cannothappen due to the coprime assumption. 10 roposition 4.2. Let α, β ∈ A be cyclotomic d -numbers (thus, nonzero) and L := Q ( α, β ).Then β divides α if and only if N L ( β ) divides N L ( α ). Proof.
By the d -number condition in Equation (3), there exist u α , u β ∈ A × such that α [ L : Q ] = ( α d α ) [ L : Q ( α )] = ( N ( α ) · u α ) [ L : Q ( α )] = N L ( α ) · u [ L : Q ( α )] α (23)and similary for β . Thus( α/β ) [ L : Q ] = ( N L ( α ) /N L ( β )) u [ L : Q ( α )] α u − [ L : Q ( β )] β . (24)Hence α/β ∈ A if and only if ( α/β ) [ L : Q ] ∈ A if and only if N L ( α ) /N L ( β ) ∈ A .Proposition 4.2 is true for arbitrary algebraic d -numbers but requires subtleties which arenot needed in what follows. Let J be an indexing set for the prime rational integers. Corollary 4.3.
Let α ∈ A be a cyclotomic d -number with N ( α ) = ± Q j ∈ J p k j j . The followingare equivalent:(a) 0 ≤ k j < d α for all j ∈ J , and(b) for all m ∈ Z ≥ , m does not divide α . Theorem 4.4.
Let C be a fusion category. If C has a formal codegree f such that for all m ∈ Z ≥ , m does not divide f , then(a) all formal codegrees of C are Galois conjugate to f ,(b) |O ( C ) | = d f , and(c) the Grothendieck ring of C is commutative. Proof.
Assume f has minimal polynomial x n − a x n − + · · · + ( − n − a n − x + ( − n a n , (25)where n, a j ∈ Z ≥ for all 1 ≤ j ≤ n as f is totally positive [28, Remark 2.12]. If f ∈ Z , then f = 1 and C ≃
Vec and the result is trivial, so we may assume n >
1. We have a n = N ( f )and as f is a d -number, a n − n divides a nn − [27, Lemma 2.7(v)]. Let p j be any rational primedividing a n . If p k j j is the largest power of p j dividing a n − and p ℓ j j is the largest power of p j dividing a n , then the largest power of p j dividing a nn − is p nk j j . Thus the above d -numbercondition implies nk j ≥ ( n − ℓ j , or k j ≥ ℓ j − ℓ j /n . Corollary 4.3 implies that ℓ j /n <
1, so k j ≥ ℓ j . Hence p ℓ j j divides a n − for all primes p j dividing a n , and therefore a n − ≥ a n . Thisimplies X σ ∈ Gal( Q ( f ) / Q ) σ ( f ) = a n − a n ≥ . (26)Moreover this sum is precisely 1 by [28, Proposition 2.10] and (a) follows. Claim (b) followsas the number of distinct formal codegrees must be less than or equal to |O ( C ) | , and claim(c) follows from [28, Example 2.18]. 11 xample 4.5. Consider the fusion categories T κ := C ( sl , κ − ad for prime κ ∈ Z ≥ (referto [32, Section 2.3] for a basic introduction). It is well-known that dim( T κ ) = κ/ (4 sin ( π/κ )).Hence Q (dim( C )) = Q (sin ( π/κ )). Let ϕ be the Euler totient function. For any n ∈ Z ≥ ,[ Q (sin(2 π/n )) : Q ] = (1 / ϕ ( n ) : n ≡ / ϕ ( n ) : n ≡ ϕ ( n ) : else (27)and [ Q (sin( π/κ )) : Q (sin ( π/κ ))] = 2 when κ is odd. Thus[ Q (dim( T κ )) : Q ] = (1 / ϕ ( κ ) = (1 / κ − . (28)We leave it as an exercise to verify that N (dim( T κ )) = κ ( κ − / . In particular, Theorem 4.4implies the set of all formal codegrees of T κ is the same as the set of Galois conjugates ofdim( T κ ). Moreover, if κ , . . . , κ n is any finite set of distinct odd primes and σ , . . . , σ n are any(not necessarily distinct) elements of Gal( Q / Q ), ⊠ nj =1 T σ j κ j satisfies the hypotheses of Theorem4.4 as well. Corollary 4.6.
Let C be a fusion category. If the Grothendieck ring of C is noncommutative,for every formal codegree f of C there exists an integer n f ∈ Z ≥ dividing f . Proof.
This is the contrapositive of Theorem 4.4 (c).
Example 4.7.
By some measures, the fusion category H corresponding to the extendedHaagerup subfactor [5] is the most interesting fusion category with respect to global dimensionthat has been constructed at this time. We know |O ( H ) | = 8 and it has noncommutative fusion rules. In particular, Corollary 4.6 implies the norm of every formal codegree of H mustcontain a prime power factor p k j j with k j ≥
3. For example, dim( H ) = FPdim( H ) is thelargest root of the polynomial x − x + 8450 x − . (29)Moreover N (dim( H )) = 21125 = 5 · as predicted.Recall that if K is any number field, each fusion category posesses a fusion subcategory C K whose objects are all X ∈ C such that FPdim( X ) ∈ K [18, Proposition 1.6]. Thus C Q is the largest integral fusion subcategory of C . Alternatively, one can consider the fusionsubcategory C ad , the trivial component of the universal grading of C [20, Section 3.2]. Corollary 4.8.
Let C be a fusion category. If C has a formal codegree satisfying the equivalentconditions of Corollary 4.3, then C Q = Vec and C = C ad . Proof.
As FPdim( C ) is a formal codegree, it cannot be divisible by any integer m ∈ Z ≥ by assumption. However, FPdim( C Q ) ∈ Z and divides FPdim( C ) [14, Proposition 8.15],hence it must be trivial. Similarly, if U ( C ) is the universal grading group of C , FPdim( C ) = | U ( C ) | FPdim( C ad ) [13, Theorem 3.5.2]. Thus | U ( C ) | = 1 and C = C ad . Lemma 4.9.
Let C be a fusion category. If σ ∈ Gal( Q (dim( C )) / Q ) is nontrivial, then C ⊠ C σ is not Galois conjugate to a pseudounitary fusion category.12 roof. We have FPdim( C ⊠ C σ ) = FPdim( C ) . Let s , . . . , s k be the distinct Galois conjugatesof dim( C ). If τ ∈ Gal( Q / Q ) is any Galois automorphism, then for some indices 1 ≤ i, j ≤ k , τ (dim( C ⊠ C σ )) = τ (dim( C )) τ ( σ (dim( C ))) = s i s j . (30)As dim( C ) = σ (dim( C )) by assumption, s i = s j . By [14, Proposition 8.22], we have s i ≤ FPdim( C ) for all 1 ≤ i ≤ k . Thus τ (dim( C ⊠ C σ )) = s i s j < FPdim( C ⊠ C σ ) which is to say C ⊠ C σ is not Galois conjugate to a pseudounitary fusion category. Lemma 4.10.
Let C be a spherical fusion category. If C has a formal codegree satisfying eitherof the equivalent conditions of Corollary 4.3, then C is Galois conjugate to a pseudounitaryfusion category. Proof.
When C is spherical, dim( C ) is a formal codegree of C and Theorem 4.4 implies it liesin the Galois orbit of FPdim( C ). Proposition 4.11.
Let C be a spherical fusion category, and f be a formal codegree of C whose norm has prime factorization N ( f ) = Q j ∈ J p k j j . There exists j ∈ J such that k j ≥ d f . Proof.
If not, k j < d f for all j ∈ J and thus all formal codegrees of C are Galois conjugate,and have the same norm by Theorem 4.4. Therefore, for any nontrivial σ ∈ Gal( Q (dim( C )) / Q ),FPdim( C ) is a formal codegree of C ⊠ C σ which satisfies the equivalent conditions of Corollary4.3. Moreover Lemma 4.10 implies C ⊠ C σ is pseudounitary, contradicting Lemma 4.9. Corollary 4.12.
Let C be a spherical fusion category. If C has a formal codegree f withsquare-free norm, then d f = [ Q ( f ) : Q ] is 1 or 2. Theorem 4.13.
Let C be a spherical fusion category with a formal codegree f with square-free norm. If f Z , then f = (1 / ± √
5) and C is equivalent to Fib or Fib σ as a sphericalfusion category. Proof.
Corollary 4.12 implies d f = 2, hence all formal codegrees are Galois conjugate, i.e. f is a Galois conjugate of dim( C ). Let N := N ( f ). Then f is a root of x − N x + N (seeEquation (26)), hence f = (1 / N ± √ N − N ) which both must be greater than 4 / / N − p N − N ) > / ⇒ ( N − / > N − N (32) ⇒ / > N. (33)Thus N = 5 because N − N < N = 1 , ,
3, and f ∈ Z when N = 4. All sphericalfusion categories of dim( C ) = (1 / ± √
5) were classified in [29, Example 5.1.2(iv)] andshown to be equivalent to Fib or Fib σ . Here we prove that aside from Fib, Fib σ , and Fib, all spherical braided fusion categorieswhose global dimension has prime norm p ∈ Z ≥ are pointed. By Theorem 4.13 this reducesto a classification of spherical braided fusion categories global dimension exactly p .13 .1 Results on spherical fusion categories Lemma 5.1.
Let C be a spherical fusion category. If dim( C ) = p ∈ Z ≥ is prime, then for allformal codegrees f of C , f = p · u f for some u f ∈ O × Q ( f ) . Proof.
We know f divides p [28, Corollary 2.14], thus N ( f ) = p k for some 0 ≤ k ≤ d f .Clearly k = 0 as C must be nontrivial. But if 0 < k < d f , then Lemma 4.10 implies C is(Galois conjugate to) a pseudounitary fusion category. Fusion categories of Frobenius-Perrondimension p are pointed [14, Corollary 8.30], hence our claim is proven with u f = 1. Otherwise N ( f ) = p d f , and moreover f d f = p d f · u for some u ∈ O × Q ( f ) by Equation (3) and our claimfollows. Proposition 5.2.
Let C be a spherical fusion category of prime global dimension p = 2. Forall formal codegrees f of C , d f divides ( p − / Proof.
The modular data of Z ( C ) is defined over Q ( ζ p n ) for some n ∈ Z ≥ by [6, Theorem3.9] as dim( Z ( C )) = p where Z ( C ) is the Drinfeld center of C [13, Section 7.13]. The fieldgenerated by the modular data of Z ( C ) includes the formal codegrees of C by [28, Theorem2.13]. But [ Q ( ζ p n ) : Q ] = p n − ( p − C are totallyreal [28, Remark 2.12]. Thus Q ( f ) ⊂ Q ( ζ p n ) + , and d f divides p n − ( p − /
2. But C has atmost |O ( C ) | ≤ p formal codegrees with equality if and only if C is pointed [29, Lemma 4.2.2],including dim( C ) = p , hence d f < p − Corollary 5.3.
Let C be a spherical fusion category of prime global dimension p ∈ Z ≥ .Then f ∈ Q ( ζ p ) + for all formal codegrees f of C . Proof. As Q ( ζ p n ) / Q is a cyclic extension, intermediate subfields are in one-to-one correspon-dence with rational integer divisors of p n − ( p − Q ( ζ p ) ⊂ Q ( ζ p n ) is the uniquesubfield K ⊂ Q ( ζ p n ) with [ K : Q ] = p −
1. Moreover [ Q ( f ) : Q ] divides p − Q ( f ) ⊂ Q ( ζ p ) + ⊂ Q ( ζ p ) as f is totally real. Proposition 5.4.
Let C be a spherical fusion category with prime global dimension p ∈ Z ≥ .Then X ϕ ∈ Irr( K ( C )) dim( ϕ ) u − f ϕ = p, (34)where u f ϕ ∈ O × Q ( ζ p ) + are the units from Lemma 5.1. Proof.
This follows from [28, Proposition 2.10] and Lemma 5.1.
Theorem 5.5.
Let p = 2 q + 1 where p, q ∈ Z ≥ are both prime. If C is a spherical fusioncategory with N (dim( C )) = p , then C is pointed or p = 5. Proof.
Proposition 5.2 implies d f = 1 , q for all formal codegrees f . This implies that eitherFPdim( C ) = p and C is pointed, or d FPdim( C ) = q . But [28, Corollary 2.15] states that allformal codegrees of C are contained in the field of categorical dimensions: Q (dim( X ) : X ∈ O ( C )) . (35)In particular the categorical dimension homomorphism dim : K ( C ) → C has at least q noni-somorphic Galois conjugates. Therefore there are precisely 2 q formal codegrees of C : p withmultiplicity q and the Galois conjugates of FPdim( C ). But Siegel’s trace bound for totally14ositive algebraic integers [35, Theorem III] implies Tr( u − C ) ) ≥ q/ ≥ q + 1 with equal-ity if and only if q = 2. Hence Equation (34) from Proposition 5.4 can only hold when p = 5or C is pointed. Note 5.6.
The primes p ∈ Z ≥ of the form p = 2 q + 1 where q ∈ Z ≥ is prime as well areknown as safe primes . The safe primes less than 500 are5 , , , , , , , , , , , , , , , , and 479 . (36) Example 5.7.
Theorem 5.5 and [29, Example 5.1.2] show that aside from Fib (up to equiva-lence), any spherical fusion category of prime global dimension p = 2 , , , ,
11 is pointed. Thenext smallest case is p = 13. Proposition 5.2 implies that d FPdim( C ) ∈ { , , , } . The case d FPdim( C ) = 1 is pointed and the case d FPdim( C ) = 6 = (13 − / d FPdim( C ) = 2, then FPdim( C ) ∈ Q ( √ ǫ := (1 / √ −
1, thus any totally positive unit in Q ( √
13) is ǫ n for some even n . But σ ( ǫ n ) = ǫ − n and moreover 13 ǫ − n < p / n ∈ Z ≥ thus no such number is the Frobenius-Perrondimension of a fusion category by [29, Theorem 4.2.1]. Lastly assume d FPdim( C ) = 3. Thenusing the absolute trace bound of Flammang [16], Tr( u − C ) ) > · . > Q ( ζ ) + . As the ring homomorphism dim must then have at leastthree Galois conjugates to satify [28, Corollary 2.15], we must have 6 ≤ Tr( u − C ) ) ≤ Q ( ζ ) + by evaluating √ ∆ where ∆ is the discriminant of x − ax + bx − ≤ a ≤
10 and b ∈ Z ≥ such that ∆ > u − C ) is totally real). One easily demonstrates this set isfinite and if u − C ) ∈ Q ( ζ ) + exists, then √ ∆ must be a power of 13. For example in thelargest case, for a = 10, 7 ≤ b ≤
25 and √ ∆ ∈ Z if and only if a = 17 when ∆ = 3 . Lemma 5.8.
Let C be a spherical fusion category of prime global dimension p ∈ Z ≥ . If C isbraided, then C is nondegenerately braided, hence modular, or symmetric. Proof.
By Lemma 5.1, FPdim( C ) = p · u FPdim( C ) for some u FPdim( C ) ∈ O × Q ( ζ p ) . HenceFPdim( C Q ) = 1 , p where C Q is the fusion subcategory consisting of objects of rational in-teger Frobenius-Perron dimension. In the latter case, C Q is pointed, hence rank( C ) ≥ p andthus C itself is pointed. Moreover FPdim( C ) = p and C is equivalent to Rep( Z /p Z ) as a fu-sion category [14, Corollary 8.30]. Braidings on Rep( Z /p Z ) are either symmetric or modularas Rep( Z /p Z ) has no proper nontrivial fusion subcategories. Otherwise C Q is trivial, whichcontains the symmetric center of C , i.e. C is modular.As spherical braided fusion categories of prime global dimension are nondegeneratelybraided, one can utilize the Galois action on the normalized modular data which will naturallyinvolve traces of real cyclotomic integers. Lemma 5.9.
Let p ∈ Z > be prime. If α ∈ Q ( ζ p ) + is a cyclotomic integer, then:(a) α = ± Q ( ζ p ) + ( α ) = ( p − / α is Galois conjugate to ± π/p ) with Tr Q ( ζ p ) + ( α ) = p −
2, or(c) Tr Q ( ζ p ) + ( α ) ≥ p . 15 roof. This is implied by [9, Lemma 3] which states that if α is a (real) cyclotomic integerwhich is not a root of unity or the sum of two roots of unity, then Tr( α ) ≥ · [ Q ( α ) : Q ].Assuming α ∈ Q ( ζ p ) + with p = 2, α being a root of unity or a sum of two roots of unity givescases (a) and (b) (refer to [8, Lemma 4.1.3], for example). Otherwise, [9, Lemma 3] impliesTr Q ( ζ p ) + ( α ) = [ Q ( ζ p ) + : Q ( α )] · Tr( α ) (37) ≥ [ Q ( ζ p ) + : Q ( α )] · · [ Q ( α ) : Q ] (38)= 2 · [ Q ( ζ p ) + : Q ] (39)= p. (40)Simple objects of categorical dimension of type (b) in Lemma 5.9 rarely occur under thecurrent assumptions. Lemma 5.10.
Let p ∈ Z ≥ be prime. If C is a spherical braided fusion category of globaldimension p , then dim( X ) = ± π/p ) for any X ∈ O ( C ). Proof.
If dim( X ) = ± π/p ) for some X ∈ O ( C ), then because C is modular, the Galoisconjugates of p (2 cos(2 π/p )) − are formal codegrees of C which are ( p − / K ( C ) → C has at least ( p − / C has formal codegree p withmultiplicity at least ( p − /
2. These formal codegrees f Y correspond to simple objects Y of squared categorical dimension 1 via f Y = dim( C ) / dim( Y ) , and together with the Galoisorbit of X , these p − O ( C ) as C cannot be pointed. Moreover p = dim( C ) = ( p − / X ) ) = ( p − / p − , (41)which implies p = 5. Proposition 5.11.
Let C be a spherical braided fusion category of prime global dimension p ∈ Z ≥ . If p = 5, then C is pointed. Proof.
We may assume p > p = 2 , , C is contained in Q ( ζ p n ) for some n ∈ Z ≥ [6, Theorem 3.9]. In particular the (multiplicative) central charge τ + / p dim( C ) is a p n th root of unity; let γ be any of its cube roots. The normalized twists t X for X ∈ O ( C ) are now ˜ t X /γ where ˜ t X is a p n th root of unity and σ ∈ Gal( Q / Q ) actson t X via σ ( t X ) = t ˆ σ ( X ) where ˆ σ : O ( C ) → O ( C ) is the corresponding Galois permutation[12, Theorem II (iii)]. As all normalized modular data lies in Q ( ζ p n ) we need only study theaction of G := Gal( Q ( ζ p n ) / Q ) ∼ = Gal( Q ( ζ ) / Q ) × Gal( Q ( ζ p n ) / Q ) (42) ∼ = Z / Z × Gal( Q ( ζ p n ) / Q ) . (43)Let H := { σ ∈ G : σ = τ for some τ ∈ G } . Because of the 2-torsion in (43), H can beidentified with a subgroup of Gal( Q ( ζ p n ) / Q ) (which is cyclic). As the subgroup of squares isindex 2 when the order of a finite cyclic group is even and the subgroups of each index areunique, then H ∼ = 1 × Gal( Q ( ζ p n ) + / Q ) which we will identify with the nontrivial factor.If ˜ t X = 1 for some X ∈ O ( C ), then ˜ t X is a primitive p k th root of unity for some 1 ≤ k ≤ n and ˜ t X has p k − ( p − / H -conjugates. Thus k = 1 (or else |O ( C ) | > p ) for all such X n = 1 above. This implies t ˆ σ ( X ) = t X for all σ ∈ H , hence the H -orbit (ˆ σ ( X )for all σ ∈ H ) of X has no fixed points. Moreoverdim( C ) ≥ X σ ∈ H dim(ˆ σ ( X )) = X σ ∈ H σ (dim( X ) ) = Tr Q ( ζ p ) + (dim( X ) ) (44)Furthermore, by Lemmas 5.9 and 5.10, Tr Q ( ζ p ) + (dim( X ) ) ≥ p unless dim( X ) = 1 because p was assumed to not be 5. Hence dim( X ) = 1 otherwise all simple objects are in the H -orbitof X , and hence all have nontrivial twist (but the unit must have trivial twist). An X ∈ O ( C )with ˜ t X = 1 does exist, otherwise C is integral, hence pointed.In summary, C posesses (at least) ( p − / C is pointed (c.f. [13, Exercise 9.6.2]).So finally we assume that the remainder of simple objects of C have trivial twists, and thesum of their squared categorical dimensions is necessarily p − ( p − / p + 1) /
2. Thisis sufficient information to compute (in principle) the Gauss sums τ ± of C , whose product is τ + · τ − = dim( C ) = p [13, Proposition 8.15.4]. If we define α := P σ ∈ H ˜ t ˆ σ ( X ) and τ to be a(cyclic) generator of Gal( Q ( ζ p ) / Q ). Then evidently α is either P odd a τ a ( ζ p ) or P even a τ a ( ζ p )for 0 < a < p as ˜ t X differs from ˜ t ˆ σ ( X ) by an even power of τ . These are complex conjugatesof one another, hence α + α = − p th roots of unitywhere here α is the complex conjugate of α . Moreover4 p = 4 τ + τ − = ( p + 1 + 2 α ) ( p + 1 + 2 α ) = ( p + 1) − p + 1) + 4 | α | . (45)Thus 4 p − ( p +1)( p −
3) = 4 | α | . But the left-hand side is negative for p > Theorem 5.12.
Let p ∈ Z ≥ be prime. If C is a spherical braided fusion category with N (dim( C )) = p , then C is pointed, or equivalent to Fib, Fib σ , or Fib as a spherical fusioncategory. Proof.
This follows from Proposition 5.11, [29, Example 5.1.2(v)], and Proposition 4.13.
Formal codegrees are necessary to the understanding of fusion categories, so here we discusspossible ways they may be inspected further.A classification of modular tensor categories C such that the Galois action on O ( C ) istransitive was recently announced by Ng, Wang, and Zhang and consists of the categories ⊠ nj =1 T σ j κ j from Example 4.5. We conjecture a result which extends to fusion categories whichare not necessarily modular. Conjecture 6.1.
Let C be a fusion category. If C has a formal codegree f such that for all m ∈ Z ≥ , m does not divide f , then C is equivalent to a fusion category of the form ⊠ nj =1 T σ j κ j .In Proposition 4.11, we were able to show that for each formal codegree f of a sphericalfusion category, there exists a prime dividing the norm of f with a large exponent relative tothe degree of the extension Q ( f ) / Q . But it is likely that more is true from an inspection ofknown examples. 17 uestion 6.2. Does there exist a spherical fusion category C with formal codegree f whosenorm has prime factorization N ( f ) = Q j ∈ J p k j j , and k j < d f for some j ∈ J ?One could also extend the results of Theorem 4.13 by studying formal codegrees f ofspherical fusion categories such that N ( f ) = Q j ∈ J p k j j with 0 ≤ k j ≤
2. Such a formalcodegree has [ Q ( f ) : Q ] ≤ N ( G, k ) (Example 2.6). It is expected that if G is a finitegroup, the near-group fusion categories N ( G, k ) exist for only finitely-many k ∈ Z ≥ . Onepeculiarity of existence for infinitely-many k ∈ Z ≥ is that this would be an infinite collectionof inequivalent fusion categories sharing the same formal codegree: | G | . This motivates thefollowing question. Question 6.3.
Let f ∈ C . Does there exist an infinite family of nonisomorphic categorifiablefusion rings each having f as a formal codegree?The proof of Theorem 5.12 was made possible by the theoretical framework of modulartensor categories. But Theorem 5.5 is significant evidence that the assumption of a braidingmay be superfluous. Regardless, the assumption of sphericality may be more difficult toremove. If dim( C ) = p on the nose, then one can classify fusion categories of global dimension2 p , which includes the sphericalization ˜ C , and work backwards. But when considering thenorm of global dimension the problem becomes significantly more difficult. In particularthe degree d dim( C ) = [ Q (dim( C )) : Q ] is bounded above by k := ⌊ log( p ) / log(4 / ⌋ (refer tothe proof of Theorem 3.1) and N (dim( ˜ C )) = 2 d dim( C ) N (dim( C )). Classifying spherical fusioncategories whose global dimension has large prime factors seems unlikely at this time, soanother method may be needed. Yet, we still present the following conjecture as a goal. Conjecture 6.4.
Let p ∈ Z ≥ be prime. If C is a fusion category whose global dimensionhas norm p , then C is pointed, or equivalent to Fib, Fib σ , or Fib. Acknowledgments.
We would like to thank Terry Gannon and Victor Ostrik for productivediscussions during the preparation of this manuscript, and we would like to thank ColleenDelaney and Julia Plavnik for reading an early draft.
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