The Classification of Extremal VertexOperator Algebras of Rank 2

J. Connor Grady

July 4, 2018

Abstract

In this paper we provide an overview of vector-valued modularforms, an algebraic structure that appears in number theory. Wedescribe how they can be used to study vertex operator algebras, whichare in turn algebraic representations of certain conformal field theories.Through this connection we prove that the number of character vectorsfor extremal vertex operator algebras with representation categoriesC satisfying rank(C) = 2 is finite. Moreover, we provide a list of allpossible candidates for these character vectors.

1 Introduction

One of the few classes of rigorously defined quantum field theories are confor-mal field theories (CFTs). A subset of these, unitary chiral conformal fieldtheories (χCFT), are associated to a mathematical structure called a vertexoperator algebra (VOA), which encodes the chiral algebra of the χCFT. Inaddition, nice enough χCFTs give rise to unitary modular tensor categorys(UMCs) which can be constructed from their associated VOAs. The questionof how to recreate a VOA from a UMC and how much additional informationin necessary to do so is still an active area of research [EG18].

Rather than tackle this problem directly, our goal is simply to use UMCsto classify the character vectors of VOAs, which were shown by [Zhu96] to bevector-valued modular forms. We restrict our attention to so-called extremal

1

VOAs, who’s well-behavedness allow us to use the results of [TW17, Thm.3.1]. By further restricting ourselves to UMCs C such that rank(C) = 2, thistheorem shows that the character vector of an associated VOA is uniquelydetermined by a single parameter, the central charge c. Our results show thatonly a finite number of pairs (C, c), called genera, produce viable candidatecharacter vectors. We further show that the list given in [TW17, Sec. 3.2] ofsome of these candidate character vectors is in fact complete.

We first give a brief outline of some relevant elementary algebra in section2. In section 3 we provide some background on modular froms. Then insection 4 we define vector-valued modular forms (VVMFs) and describe themethod given in [BG07] of constructing the fundamental matrix. In section5, we give an overview of extremal VOAs and how each induces a VVMF. Insection 6 we prove a theorem about VVMFs which allows one to completelyclassify the character vectors of extremal VOAs V with associated UMCRep(V) that satisfy rank(Rep(V)) = 2. In section 7 we prove the completelist of these character vectors.

2 Some Basic Algebra

A group is a pair (G, ·), where G is a set and · is a binary operation on theset, satisfying the following properties:

Closure: For all a, b ∈ G, a · b ∈ G.

Associativity: For all a, b, c ∈ G, a · (b · c) = (a · b) · c.

Identity: There exists an element 1 ∈ G that satisfies 1 · a = a · 1 = a forall a ∈ G.

Inverse: There exists an element a−1 ∈ G such that a−1 · a = a · a−1 = 1 forall a ∈ G.

An abelian group is a group which satisfies the following additional property:

Commutativity: For all a, b ∈ G, a · b = b · a.

By abuse of notation, from here on we will simply state that G is a groupwhen the binary operation · is clear from context. We will similarly use theunderlying set as a shorthand all of the algebraic structures mentioned in

2

this paper when they are clear from context. In addition, we will often dropthe ·, so a · b becomes ab.

A subgroup H of a group G is a subset of G that follows all of the groupaxioms when the binary operation is restricted to the set H. Given a sub-group H of G, and an element a ∈ G, a left coset is the set

aH = {ah|h ∈ H}.

The right coset Ha is defined analogously. A normal subgroup N of G is asubgroup such that for all a ∈ G, we have that aN = Na. Note that foran abelian group, all subgroups are normal. Given a group G and a normalsubgroup N , the quotient group G/N is the set of left cosets of N where thebinary operation is defined by

(aN)(bN) = (ab)N

for all a, b ∈ G.A group action · of a group G on a set X is a mapping · : G ×X → X

that satisfies the following properties:

Compatibility (ab) · x = a · (b · x) for all a, b ∈ G and x ∈ X.

Identity 1 · x = x for all x ∈ X.

The group G is said to act on X. Because of the compatibility condition andthe existence of inverse elements in G, the group action associates to eachelement of G a bijective map X → X.

A ring is a triple (R,+, ·), where (R,+) is an abelian group (with theidentity element denoted 0 instead of 1) and which satisfies the followingadditional properties:

Multiplicative Associativity: For all a, b, c ∈ R, (ab)c = a(bc).

Multiplicative Identity: There exists an element 1 ∈ R that satisfies 1 ·a = a · 1 = a for all a ∈ R.

Left Distributivity: a(b+ c) = ab+ ac for all a, b, c ∈ R.

Right Distributivity: (a+ b)c = ac+ bc for all a, b, c ∈ R.

3

The operations +, · are called addition and multiplication, respectively. Acommutative ring is a ring in which multiplication is also commutative. Afield is a commutative ring that contains multiplicative inverses for all non-zero elements.

A polynomial ring R[x] in the indeterminate x over the ring R is the setof polynomials in x with coefficients in R, i.e. elements of of the form

a(x) =n∑i=0

aixi,

where each ai ∈ R and n ∈ N. Note that, as usual, x0 = 1. The largesti ∈ {1, . . . , n} such that ai 6= 0 is called the degree of a(x). Let b(x) ∈ R[x],and without loss of generality assume deg(b(x)) = m ≥ n = deg(a(x)). Thenaddition in R[x] is defined by

a(x) + b(x) =m∑i=0

(ai + bi)xi

where if n < i ≤ m we set ai = 0. Multiplication in R[x] is defined by

a(x)b(x) =m+n∑i=0

( i∑j=0

ajbi−j

)xi

These operations make R[x] into a commutative ring.For any commutative ring R the general linear group GL(n,R) is the

group of invertible n× n matrices under matrix multiplication whose entriesare in R, where n ∈ N. The special linear group SL(n,R) is the subgroup ofGL(n,R) that contains only those matrices with determinant 1. The group ofscalar transformations Z(n,R) is the subgroup of GL(n,R) whose elementsare of the form λIn for some λ ∈ R − {0}, where In is the identity matrixof size n. We define SZ(n,R) to be the subgroup of Z(n,R) of elementswith unit determinant. The projective special linear group PSL(n,R) is thequotient group SL(n,R)/SZ(n,R).

A left module M over a ring R is an abelian group (M,+) along with amapping · : R×M →M that satisfies the following properites:

Left Distributivity: a · (x+ y) = a · x+ a · y for all a ∈ R and x, y ∈M .

Right Distributivity: (a+ b) · x = a · x+ b · x for all a, b ∈ R and x ∈M .

4

Compatibility: (ab) · x = a · (b · x) for all a, b ∈ R and x ∈M .

Identity: 1 · x = x for all x ∈M .

Again, it is customary to simply drop the mapping ·. Right modules overR are defined similarly. If R is a commutative ring, then left and right R-modules are equivalent and are simply called R-modules. Note that R is itselfand R-module.

A free module is a module that admits a basis. If F is a field, then anF -module is called a vector space over F . A submodule N of an R-moduleM is a subgroup of M such that for all a ∈ R and x ∈ N , ax ∈ N . Anirreducible module is a module that has no proper, non-trivial submodules.

For any algebraic structure, a homomorphism is a mapping that preservesthat structure. For example, if G,H are groups, a group homomorphismφ : G → H must satisfy φ(ab) = φ(a)φ(b) for all a, b ∈ G. An isomorphismis a homomorphism that is also a bijection. An isomorphism class is acollection of mathematical objects that are isomorphic. An automorphismis an isomorphism from a mathematical object to itself. In general, the setof automorphism for a given mathematical object create a group, where thegroup operation is composition of maps.

For a module M over a commutative ring R we define the dual moduleM∗ of M to be the R-module of module homomorphisms from M to R. IfM is actually a vector space V over a field F , then the dual module V ∗ iscalled the dual vector space of V .

For a vector space V over a field F we define the general linear groupGL(V ) to be the automorphism group of V . A representation of a group G onV is a group homomorphism φ : G→ GL(V ). If n is the dimension of V , thenGL(V ) and GL(n, F ) are isomorphic as groups, so we can equivalently thinkof a representation of G on V as a group homomorphism φ′ : G→ GL(n, F ).We define dim(φ) = dim(V ). A subrepresentation is the restriction of φ toa subspace W ⊂ V such that for all g ∈ G and for all w ∈ W , φ(g)w ∈ W .An irreducible representation is a representation with no proper, non-trivialsubrepresentations.

A matrix U ∈ GL(n,C) is unitary if U is invertible and U∗ = U−1, whereU∗ is the conjugate transpose of U . The unitary group U(n) is the subgroupof GL(n,C) consisting of all n×n unitary matrices. A unitary representationis a representation whose codomain is U(n).

5

3 Modular Forms

A linear fractional transformation is a mapping f : C→ C of the form

f(z) =az + b

cz + d

where a, b, c, d ∈ C and ad− bc 6= 0.We define the modular group to be PSL(2,Z). Note that for any λ ∈ Z

det(λI2) = λ2 det(I2) = λ2,

so SZ(2,Z) = {±I2}. The modular group is generated by the cosets

T =

[1 10 1

]{±I2},

S =

[0 −11 0

]{±I2}.

Proposition 3.1. For any coset γ ∈ PSL(2,Z) we choose a representative[a bc d

].

Then the action of γ on the upper half-plane H by linear fraction transfor-mation given by

γ · τ =aτ + b

cτ + d,

where τ ∈ H, is independent of the choice of representative of γ. Moreover,this is a group action.

Proof. First, note that

γ =

[a bc d

]{±In} =

{[a bc d

],−[a bc d

]}so there are only two elements in the coset γ. We immediately see that

(−a)τ + (−b)(−c)τ + (−d)

=aτ + b

cτ + d.

6

This then shows that the action of γ can be equally calculated from eitherof its representatives. Thus as a shorthand we will write elements of themodular group as single representatives in the future, rather than as wholecosets.

Next, we see that

T · τ =1τ + 1

0τ + 1= τ + 1

S · τ =0τ − 1

1τ + 0=−1

τ.

Since T and S generate PSL(2,Z) and these transformations then preserveH, H is closed under all γ ∈ PSL(2,Z).

Lastly, we calculate

γ′ · (γ · τ) = γ · aτ + b

cτ + d

=a′ aτ+bcτ+d

+ b′

c′ aτ+bcτ+d

+ d′

=(a′a+ b′c)τ + (a′b+ b′d)

(c′a+ d′c)τ + (c′b+ d′d)

= (γ′γ) · τ.

Thus the action is compatible with the group operation.

A function f : D ⊂ C→ C, where D is open, is called holomorphic on Dif it is differentiable in a neighborhood of every point in D. This conditionin fact implies that f is infinitely differentiable. We call f meromorphic if fis holomorphic except at at set of isolated points.

A modular form of weight k for the modular group is a function f : H→ Cthat is holomorphic in the upperhalf plane H and satisfies

f(γ · τ) = (cτ + d)kf(τ),

where τ ∈ H and

γ =

[a bc d

]∈ PSL(2,Z).

Modular functions are defined similarly to modular forms of weight 0, exceptthe holomorphicity requirement is relaxed to allow functions that are mero-morphic on H so long as their only pole is of finite order and is located at the

7

point i∞, called the cusp. That is to say, if we define q = e2πiτ for all τ ∈ Hand let f be a modular function, then the Laurent expansion of f takes theform

f(τ) =∞∑

n=−m

anqn

for some m ∈ N. This is the so-called q-expansion of f .Some specific modular forms that will be important to us are the Eisen-

stein series

Ek(τ) =∑

(m,n)∈Z2−(0,0)

1

(m+ nτ)k,

a modular form of weight k, and the modular discriminant

∆(τ) = q∞∏n=1

(1− qn)24,

a modular form of weight 12. Also of use to us will be the Klein J-invariant

J(τ) = q−1 +∞∑n=1

j(n)qn = q−1 + 196884q + · · · ,

which is the unique modular function of weight 0 that has only a simple poleat the cusp satisfying the relations J(i) = 984 and J(e2πi/3) = −744.

For the interested reader, further background on the modular group andmodular forms can be found in [Apo90].

4 Vector-Valued Modular Forms

Let ρ be an irreducible unitary representation of PSL(2,Z) of dimensiond. We define a vector-valued modular form (VVMF) to be a holomorphicfunction X : H→ Cd that satisfies

X(γ · τ) = ρ(γ)X(τ)

for all γ ∈ PSL(2,Z) and τ ∈ H and whose component functions Xi havefinite order poles at the cusp. We define M(ρ) to be the set of VVMFsassociated to ρ.

8

Fix some Λ ∈ U(d), called an exponent matrix of ρ, satisfying

e2πiΛ =∞∑n=0

(2πi)n

n!Λn = ρ(T ).

This is guaranteed to exist because ρ(T ) is unitary and thus invertible.

Lemma 4.1. e−2πiτΛX(τ) is invariant under integer translations in τ .

Proof.

e−2πi(τ+1)ΛX(τ + 1) = e−2πiτΛe−2πiΛX(T · τ)

= e−2πiτΛe−2πiτρ(T )X(τ)

= e−2πiτΛX(τ),

Again, we define q = e2πiτ . Then this lemma proves that the functionq−ΛX(log(q)/2πi) is a well-defined, single-valued function of q. In the restof this paper by abuse of notation we will simply write f(q) for the func-tion f(log(q)/2πi) and will switch freely between notations f(q) and f(τ)depending on what is most convenient.

We Fourier expand this expression as

q−ΛX(q) =∑n∈Z

X[n]qn

where X[n] are some coefficient vectors. We define the principle part map PΛ

to be the finite sumPΛX(q) =

∑n<0

X[n]qn.

We call Λ bijective if PΛ is an isomorphism of M(ρ). It can be shown [Gan14,Thm. 3.2] that bijective Λ exist for all ρ, that a bijective Λ must satisfy

Tr(Λ) =5d

12+

1

4Tr(ρ(S)) +

2

3√

3Re(e−iπ/6Tr(ρ(ST−1)), (1)

and that this condition is sufficient to ensure that Λ is bijective for d < 6[Gan14, Thm. 4.1].

9

Given bijective Λ, we define the canonical basis vector X(ξ;n) ∈ M(ρ) tobe the unique VVMF satisfying

[PΛX(ξ;n)]η = q−nδξη,

where δ is the Kronecker delta. Multiplication by powers of J takes M(ρ)to itself, implying that M(ρ) is a C[J ]-module. It was shown in [BG06] thatthe canonical basis vectors satisfy the recursion relation

X(ξ;n+1) = J(τ)X(ξ;n) −n−1∑m=1

j(n)X(ξ;n−m) −d∑η=1

χ(ξ;n)η X(η;1)

whereχ(ξ;n)η = X(ξ;η)[0]η

is the constant part of X(ξ;n). This allows us to express any canonical basisvector X(ξ;n) as a linear combination

X(ξ;n) =d∑η=1

aη(J)X(η;1),

where each aη(J) ∈ C[J ]. This implies that the set of canonical basis vectorswith simple poles {X(η;1)} span M(ρ). In fact, it was additionally shown in[BG07] that this set forms a basis for M(ρ) as a free C[J ]-module, implyingthat M(ρ) has rank d.

We define the fundamental matrix

Ξξη = [X(η;1)]ξ

and the characteristic matrix

χξη = [X(η;1)[0]]ξ

so χ is the constant part of Ξ. It was shown in [BG07] that one can solvefor χ uniquely, up to the conjugation by a matrix commuting with Λ. It wasfurther shown in [BG07, Sec. 2] that Ξ satisfies the recurrence relation

1

2πi

dΞ(τ)

dτ= Ξ(τ)D(τ) (2)

10

where

D(τ) =∆(τ)

E10(τ)

((J(τ)− 240)(Λ− Id) + χ+ [Λ, χ]

),

and [· , ·] is the commutator. This then allows one to find the rest of thecoefficients of Ξ, again up to the conjugation by a matrix commuting withΛ. By construction Ξ must satisfy the relation

Ξ(γ · τ) = ρ(γ)Ξ(τ),

so then solving this equation explicitly for fixed choices of γ and τ removesthe remaining ambiguity.

5 Vertex Operator Algebras

Vertex operator algebras (VOAs) are algebraic structures that correspondto the chiral algebras of χCFTs [Kac98]. In this section, rather than delveinto their structure, we will simply give a brief sketch of some importantproperties of VOAs and of how a class of VOAs can be associated to vector-valued modular forms. Throughout this paper, we assume certain nicenessproperties of VOAs that we will not define. Specifically, we assume that theyare of rational, C2-cofinite, and CFT type.

Let V be a nice VOA. Then V has finitely many isomorphism classesof irreducible modules, called V-modules. From each of these isomorphismclasses we choose representatives M0, · · · ,Md−1. Let L0 be an operator calledthe energy operator which acts on the Mi and the minimal energies hi bethe corresponding smallest eigenvalues. For all VOAs h0 = 0, and we willassume throughout that hi > 0 for all i ∈ {1, . . . , d−1}. We define Mi(n+hi)to be the eigenspace of L0 corresponding to the eigenvalue n + hi and thecharacters of the Mi as

(chMi)(τ) = q−c/24+hi∑n≥0

dim(Mi(n+ hi)) qn

where, as in the previous section, q = e2πiτ , and c is a number encoded in V

called the central charge. We will assume that c > 0.Define the representation category Rep(V) of V be the category of modules

of V. It was shown [Hua05] that Rep(V) is an algebraic structure called a

11

modular tensor category (MTC) [EGNO15]. Encoded in Rep(V) are numberscalled the twists of simple objects θi given by the relation

θi = e2πihi .

Thus from Rep(V) we can recover the hi mod 1. Also encoded in Rep(V)is another number called the multiplicative central charge χRep(V) which isrelated to c by

χRep(V) = eiπc/4.

This allows us to recover cmod 8 from Rep(V). We define the genus G(V) ofV to be the pair (Rep(V), c). Suppose C is an MTC. An admissible genus isa pair (C, c) such that χC = eiπc/4. If C is isomorphic to Rep(V), then we callV a realization of the admissible genus (C, c), and we call (C, c) realizable ifit admits a realization.

Rep(V) induces a representation ρ of SL(2,Z) that, in our examples, isirreducible and unitary. Assuming that the modules Mi are isomorphic totheir duals, ρ is then a representation of PSL(2,Z). It was proven in [Zhu96]that the character vector of V

ch(V) =

chM0...

chMd−1

is then a VVMF, i.e. an element of M(ρ). This means that we can use themethods of the previous section to study ch(V). If the Mi are not self-dual,we can use the trick in [BG07, Appendix A] to construct a representation %of PSL(2,Z) associated to ρ. Thus we can still use the theory of VVMFs tostudy cases where the Mi are not self-dual.

For a VOA V with central charge c and minimal energies h0, . . . , hd−1 itwas proven in [Mas07, Sec. 3] that

` =

(d2

)+dc

4− 6

d−1∑i=0

hi ∈ Z≥0. (3)

We call V extremal if ` < 6, and we call this condition the extremality con-dition. Since Rep(V) determines the hi mod 1, this implies that the extremalV have maximal

∑hi of all VOAs in the genus (Rep(V), c).

12

Suppose V is a VOA with associated representation ρ of PSL(2,Z). If theexponent matrix Λ given by

Λii = δi0 + hi −c

24, (4)

is bijective, then X(1;1) is the character vector of V. Since Λ must satisfyequation (1), this further implies that there are no other character vectors ofVOAs in the genus (Rep(V), c) with h′i ≥ hi for all i.

Contrapositively, suppose (C, c) is an admissible genus, ρ is the represen-tation of PSL(2,Z) associated to C, Λ is a bijective exponent matrix satisfyingΛ00 = 1− c/24, and Ξ is the fundamental matrix. Then if the coefficients ofthe q-expansion of the first column of Ξ are not positive integers, then thiscolumn cannot be a character vector of a VOA. This implies that there is noVOA realizing (C, c) with minimal energies hi ≥ Λii + c/24 for i > 0.

It was shown in [TW17, Thm. 3.1] the if C is a UMC, (C, c) is an ad-missible genus with 2 ≤ rank(C) ≤ 3, and V is an extremal VOA, thenthe exponent matrix given by equation (4) is bijective. This implies thatthe character vector of V is uniquely determined by the minimal energieshi. Then for an extremal choice of minimal energies hi, if the coefficients ofthe q-expansion of the first column of the fundamental matrix constructedfrom this choice of Λ are not all positive integers, there is no extremal VOAcorresponding to that choice of hi.

6 Main Result

The goal for this section is to completely enumerate all candidate charac-ter vectors of extremal VOAs V with rank(Rep(V)) = 2. To do this, firstnote that for any admissible genus (C, c) with rank(C) = 2, because C fixesthe hi mod 1 and h0 = 0, h1 is completely determined by the extremalitycondition. This allows us to state the following definition without ambiguity.

Definition 6.1. The extremal VOA bijective exponent matrix of an admis-sible genus (C, c) with rank(C) = 2 is given by

Λii = δi0 + hi −c

24

The strategy then will be to show that there exists a recursion relationbetween the diagonals of the characteristic matrices χ associated to the ex-tremal VOA bijective exponent matrix for the genera (C, c) and (C, c + 24).

13

We will then show that this recursion relation implies that, as c increases,eventually χ11 will no longer be a positive integer. Since cmod 8 is fixed by C,once the method explained in [BG07] is used to calculate the characteristicmatrices of 3 seed genera, only finitely many candidates remain for C. Sinceit was shown in [RSW09] that there are only 4 MTCs C with rank(C) = 2,we have thus reduced the problem to a finite set of computations which canbe done on a computer.

Lemma 6.2. Let (C, c) be an admissible genus with rank(C) = 2. Let Λ−and Λ+ be the extremal VOA bijective exponent matrices for the genera (C, c)and (C, c+ 24) respectively. Then

[Λ+]ii = [Λ−]ii + (−1)i+1.

Proof. Since h0 = 0, we have

[Λ+]00 = 1− c+ 24

24= [Λ−]00 − 1.

Since the representation ρ of PSL(2,Z) is the same for both genera, equa-tion (1) fixes Tr(Λ+) = Tr(Λ−), so

(Λ+)11 = Tr(Λ+)− (Λ+)00 = Tr(Λ−)− (Λ−)00 + 1 = (Λ−)11 + 1.

Corollary 6.3. The non-trivial minimal energies of (C, c) and (C, c + 24),denoted h− and h+, respectively, are related by h+ = h− + 2.

Lemma 6.4. Let everything be defined as in Lemma 6.2 and Corollary 6.3.Let χ− and χ+ be the characteristic matrices constructed from Λ− and Λ+,respectively, and ρ be the representation of PSL(2,Z) associated with C. Then

[χ+]11 =[χ−]22 + h−([χ−]11 − 240)

h− + 1

[χ+]22 =[χ−]11 + h−([χ−]22 + 240)

h− + 1.

Note that these statements are well-defined because we have been assumingthat h− > 0.

14

Proof. Let Ξ−(q) and Ξ+(q) be the q-expansions of the fundamental matricescorresponding to Λ− and Λ+ respectively. Then

q−Λ−Ξ−(q) = q−Λ− [X(1;1)− (q)|X(2;1)

− (q)]

=

[q−1 + a

(1)11 + a

(2)11 q + . . . a

(1)12 + a

(2)12 q + · · ·

a(1)21 + a

(2)21 q + . . . q−1 + a

(1)22 + a

(2)22 q + · · ·

]and similarly

q−Λ+Ξ+(q) = q−Λ+ [X(1;1)+ (q)|X(2;1)

+ (q)]

=

[q−1 + b

(1)11 + b

(2)11 q + . . . b

(1)12 + b

(2)12 q + · · ·

b(1)21 + b

(2)21 q + . . . q−1 + b

(1)22 + b

(2)22 q + · · ·

]where a

(k)ij , b

(k)ij ∈ C for all i, j ∈ {1, 2} and k ∈ N.

Then, by Lemma 6.2, we have

q−Λ−Ξ+(q) = q[−1 00 1 ]q−Λ+Ξ+(q)

=

[q−1 00 q

][q−1 + b

(1)11 + b

(2)11 q + . . . b

(1)12 + b

(2)12 q + · · ·

b(1)21 + b

(2)21 q + . . . q−1 + b

(1)22 + b

(2)22 q + · · ·

]

=

[q−2 + b

(1)11 + b

(2)11 q

−1 + . . . b(1)12 q

−1 + b(2)12 + · · ·

b(1)21 q + b

(2)21 q

2 + . . . 1 + b(1)22 + b

(2)22 q + · · ·

].

Since the sets {X(1;1)− ,X(2,1)

− } and {X(1;1)+ ,X(2,1)

+ } are both bases for M(ρ) as

a free C[J ]-module, this implies that a(1)21 X

(2;1)+ = X(1;1)

− . Since X(1;1)− 6= 0, we

see that a(1)21 6= 0, so

X(2;1)+ =

1

a(1)21

X(1;1)− .

In addition, the fact that M(ρ) is a free C[J ]-module implies that

X(1;1)+ = A(J)X(1;1)

− +B(J)X(2;1)−

for some A(J), B(J) ∈ C[J ]. Left multiplying both sides by q−Λ− yields[q−2 + b

(1)11 q

−1 + · · ·b

(1)21 q + b

(2)21 q

2 + · · ·

]=A(J)

[q−1 + a

(1)11 + · · ·

a(2)21 + a

(2)21 q + · · ·

]

+B(J)

[a

(1)12 + a

(2)12 q + · · ·

q−1 + a(1)22 + · · ·

].

15

Since J(q) has a simple pole, we quickly find

A(J) = J + a(1)22 −

a(2)21

a(1)21

, B(J) = −a(1)21 .

All together this gives

Ξ+ =

[(J + a

(1)22 −

a(2)21

a(1)21

)X(1;1)− − a(1)

21 X(2;1)−

∣∣∣∣ 1

a(1)21

X(1;1)−

]Now we calculate[

q−Λ+X(1;1)+ (q)

]1

=

[q[

1 00 −1 ]q−Λ−X(1;1)

+ (q)

]1

=q

(J(q) + a

(1)22 −

a(2)21

a(1)21

)[q−Λ−X(1;1)

− (q)

]1

− a(1)21 q

[q−Λ−X(2;1)

− (q)

]1

=q

(q−2 +

(a

(1)11 + a

(1)22 −

a(2)21

a(1)21

)q−1 +O(1)

)− a(1)

21 q

(a

(1)12 + a

(2)12 q +O(q2)

)=q−1 +

(a

(1)11 + a

(1)22 −

a(2)21

a(1)21

)+O(q),

and, similarly,[q−Λ+X(2;1)

+ (q)

]2

=

[q[

1 00 −1 ]q−Λ−X(2;1)

+ (q)

]2

=1

a(1)21

q−1

[q−Λ−X(1;1)

−

]2

=1

a(1)21

q−1

(a

(1)21 + a

(2)21 q +O(q2)

)=a

(1)21

a(1)21

q−1 +a

(2)21

a(1)21

+O(q).

16

By the chain rule1

2πi

dΞ(τ)

dτ= q

dΞ(q)

dq,

so, using the recursion relation given in equation (2), we find that

0 =

[q−Λ−

(qdΞ+(q)

dq− Ξ+(q)D(q)

)]21

=

(− a(1)

21

(a

(1)11 + h−(a

(1)22 + 240)

)+ a

(2)21 (h− + 1)

)q + . . . .

Rearranging this first term gives us

a(2)21

a(1)21

=a

(1)11 + h−(a

(1)22 + 240)

h− + 1

Thus we have that

[χ+]11 = a(1)11 + a

(1)22 −

a(2)21

a(1)21

=a

(1)22 + h−(a

(1)11 − 240)

h− + 1,

and

[χ+]22 =a

(2)21

a(1)21

=a

(1)11 + h−(a

(1)22 + 240)

h− + 1

Since [χ−]11 = a(1)11 and [χ−]22 = a

(1)22 , this completes the proof.

Thus the diagonals of χ+ can be completely solved for in terms of only thediagonals of χ−. This means that we can recursively solve for the diagonalsof the characteristic matrix corresponding to any central charge c+ 24n withn ∈ N. We can now think of the map f that takes the diagonals of χ− to thediagonals of χ+ as a discrete dynamical system and investigate its long termbehavior.

Lemma 6.5. Let

f(x, y, h) =(y+h(x−240)

h+1, x+h(y+240)

h+1, h+ 2

),

where h ∈ R+ and x, y ∈ C. Then for all n ∈ N

fn(x, y, h) =(ny+(h+n−1)(x−240n)

h+2n−1, nx+(h+n−1)(y+240n)

h+2n−1, h+ 2n

), (5)

where the superscript here denotes iteration, not differentiation.

17

Proof. The result clearly holds for n = 1. We proceed by induction on n.Suppose by way of induction that equation (5) holds for some value of n ∈ N.Then

[fn+1(x, y, h)]1 =[f(fn(x, y, h))]1

=

[f(ny+(h+n−1)(x−240n)

h+2n−1, nx+(h+n−1)(y+240n)

h+2n−1, h+ 2n

)]1

=1

h+ 2n+ 1

[nx+ (h+ n− 1)(y + 240n)

h+ 2n− 1

+ (h+ 2n)

(ny + (h+ n− 1)(x− 240n)

h+ 2n− 1− 240

)]=

1

(h+ 2n+ 1)(h+ 2n− 1)

[(h2 + 3hn+ 2n2 − h− n)x

+ (hn+ 2n2 + h+ n− 1)y

− 240(h2n+ 3hn2 + 2n3 + h2 + 2hn+ n2 + h+ n)]

=(h+ n)x+ (n+ 1)y − 240(h+ n)(n+ 1)

h+ 2n+ 1

=(n+ 1)y + [h+ (n+ 1)− 1][x− 240(n+ 1)]

h+ 2(n+ 1)− 1.

Switching the roles of x and y and replacing 240 with −240 in the abovecalculation shows that the hypothesis holds for [fn+1(h, x, y)]2. It is obviousthat [fn+1(h, x, y)]3 = h + 2(n + 1), so equation (5) has been proven for alln ∈ N.

Now we simply put all of these lemmas together to get the main theorem.

Theorem 6.6. Suppose (C, c) is an admissible genus with rank(C) = 2 andnon-trivial extremal minimal energy h. For n ∈ N, let χn denote the charac-ter matrix constructed from the extremal VOA bijective exponent associatedwith the genus (C, c+ 24n). Then for

n >1

480

([χ0]11 + [χ0]22 − 240(h− 1)

+

√([χ0]11 + [χ0]22 − 240(h− 1)

)2

+ 960(h− 1)[χ0]11

), (6)

[χn]11 is not a positive integer, so (C, c+ 24n) is not realizable.

18

Proof. First, it is easy to directly compute the characteristic matrices for allpossible seed genera, showing us that each [χ0]ij ∈ Q. We define f as inLemma 6.5. Then, by Lemma 6.4, we have

[χn]11 =[fn([χ0]11, [χ0]22, h

)]1

=n[χ0]22 + (h+ n− 1)([χ0]11 − 240n)

h+ 2n− 1

=−240n2 +

([χ0]11 + [χ0]22 − 240(h− 1)

)n+ (h− 1)[χ0]11

2n+ h− 1,

where the last line has just been rewritten in a more suggestive form. Weconsider the numerator and denominator of [χn]11 as functions of n. Sinceh > 0, the denominator is greater than 0 for all n ∈ N. The condition onn given in equation (6) is guarantees that n is larger than both roots of thenumerator. Since the numerator is concave down, this implies that for nsatisfying (6), [χn]11 < 0. Thus [χn]11 is not a positive integer.

This theorem gives an upper bound on the number of possible admissiblegenera constructed in this way from the seed genus (C, c). The fundamentalmatrices constructed from the extremal VOA bijective matrices of these finiteadmissible genera can then be computed individually.

Example 6.7. Let C = Rep(SU(2)1). For this choice of C, θ1 = i andχC = (1 + i)/

√2. Thus h1 mod 1 = 1/4 and cmod 8 = 1, so our seed genera

will be c = 1, 9, 17. Fixing c = 9, we see that

` =

(22

)+

2 · 94− 6

1∑i=0

hi =11

2− 6h1.

The extremality condition, 0 ≤ ` < 6, then shows that −1/2 < h1 ≤ 11/12,so h1 = 1/4. The extremal VOA bijective exponent matrix is then given by

Λ =

[1− 9

240

0 14− 9

24

]=

[58

00 −1

8

].

C also fixes the representation ρ of PSL(2Z) by

ρ(S) =1√2

[1 11 −1

]19

Let χn denote the characteristic matrix constructed from ρ and the extremalVOA bijective exponent associated with the genus (Rep(SU(2)1), 9 + 24n).Then the method laid out in [BG07] allows us to calculate

χ0 =

[251 267522 1

]Then equation (6) becomes

n >1

480

((251 + 1)− 240

(1

4− 1)

+

√(251 + 1− 240

(1

4− 1))2

+ 960(1

4− 1)

251

)

=432 + 12

√41

480.

This means that (Rep(SU(2)1), 9 + 24n) is not realizable for

n >

⌊432 + 12

√41

480

⌋= 1.

The only case to check then is n = 1, i.e. c = 33. We can again directlycalculate

χ1 =

[3 1

2

565760 249

],

which does in fact have all positive integers in the first column. As a sanitycheck we calculate

χ2 =1

13

[−1137 1

43520

1412731371520 4413

].

Thus (Rep(SU(2)1), 57) is not realizable, as we expected. The cases c = 1, 17proceed in the same way.

As mentioned at the beginning of this section, the choice of C fixes cmod 8,so there are only 3 unique seed genera for each C. In addition, there areonly 4 MTCs C of rank(C) = 2. Thus we have shown that the number ofadmissible genera that are possibly realizable to be finite. Having repeatedthe above calculations for all modular tensor categories of rank 2, we have

20

found no candidate character vectors that were not listed in [TW17, Sec.3.2]. In section 7 we recreate their list of candidate character vectors andrealizations (where a realization is known).

For future study, we hope to construct the realizations for all the candi-dates listed Section 7. In addition, we hope to extend our results to extremalVOAs V with rank(Rep(V)) = 3, as the results of [TW17] still apply to thisclass of VOAs, allowing us to use the theory of VVMFs to study them aswell.

7 Data

Below is a reproduction of the complete list of the candidate extremal char-acter vectors for MTCs C of rank(C) = 2 given in [TW17]. We include theextremal VOAs that realize the character vectors where they are known. Theone case in which there is no known extremal VOA realizing the candidateis marked ”unknown”.

7.1 C = Rep(SU(2)1)

θ1 = i, χC =1 + i√

2, ρ(S) =

1√2

[1 11 −1

]c h1 χ Character Vector Realization

1 1/4

[3 267522 −247

]q−1/24

[1 + 3q + 4q2 + · · ·

q1/4(2 + 2q + 6q2 + · · · )

]SU(2)1

9 1/4

[251 267522 1

]q−3/8

[1 + 251q + 4872q2 + · · ·

q1/4(2 + 498q + 8750q2 + · · · )

]SU(2)1 ⊗ E8,1

17 5/4

[323 881632 −319

]q−17/24

[1 + 323 + 60860q2 + · · ·q5/4(1632 + 162656q + · · ·

][GHM16]

33 9/4

[3 1

2

565760 249

]q−11/8

[1 + 3q + 86004q2 + · · ·

q9/4(565760 + 192053760q + · · · )

]UNKNOWN

7.2 C = Rep(E7,1)

θ1 = −i, χC =1− i√

2, ρ(S) =

1√2

[1 11 −1

]21

c h1 χ Character Vector Realization

7 3/4

[133 124856 −377

]q−7/24

[1 + 133q + 1673q2 + · · ·

q3/4(56 + 968q + 7506q2 + · · · )

]E7,1

15 3/4

[381 124856 −129

]q−5/8

[1 + 381q + 38781q2 + · · ·

q3/4(56 + 14856q + 478512q2 + · · · )

]E7,1 ⊗ E8,1

23 7/4

[69 10

32384 −65

]q−23/24

[1 + 69q + 131905q2 + · · ·

q7/4(32384 + 4418944q + · · · )

][GHM16]

7.3 C = Rep(G2,1)

θ1 = e4πi/5, χC = e−iπ/20, ρ(S) =1√

2 + φ

[1 φφ −1

], φ =

1 +√

5

2

c h1 χ Character Vector Realization

14/5 2/5

[14 128577 −259

]q−7/60

[1 + 14q + 42q2 + · · ·

q2/5(7 + 34q + 119q2 + · · · )

]G2,1

54/5 2/5

[262 128577 −10

]q−9/20

[1 + 262q + 7638q2 + · · ·

q2/5(7 + 1770q + 37419q2 + · · · )

]G2,1 ⊗ E8,1

94/5 7/5

[188 464794 −184

]q−47/60

[1 + 188q + 62087q2 + · · ·q7/5(4794 + 532134q + · · · )

][GHM16]

7.4 C = Rep(F4,1)

θ1 = e6πi/5, χC = eiπ/20, ρ(S) =1√

2 + φ

[1 φφ −1

], φ =

1 +√

5

2

c h1 χ Character Vector Realization

26/5 3/5

[52 377426 −296

]q−13/60

[1 + 52q + 377q2 + · · ·

q3/5(26 + 299q + 1702q2 + · · · )

]F4,1

66/5 3/5

[300 263774 −48

]q−11/20

[1 + 300q + 17397q2 + · · ·

q3/5(26 + 6747q + 183078q2 + · · · )

]F4,1 ⊗ E8,1

106/5 8/5

[106 1584717 −102

]q−53/60

[1 + 106q + 84429q2 + · · ·

q8/5(15847 + 1991846q + · · · )

][GHM16]

22

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