II. Applications 141
5.6. Orbifolds of Lattice Vertex Operator Algebras
In this section we combine the results of Chapter 4 and this chapter to explicitly de-scribe orbifolds of holomorphic lattice vertex operator algebras where the vertex operator algebra automorphisms are obtained as lifts of lattice automorphisms.
Assumption L. Let L denote be an even, unimodular, positive-definite lattice. Let V = VL be the corresponding lattice vertex operator algebra, which is holomorphic, satisfies Assumption Nand has central charge c= rk(L).
Let ν∈Aut(L) be an automorphism of the lattice of order m= ord(ν)∈Z>0, which lifts to some automorphism ˆν of VL (described by the function u:L → {±1}) of order
ˆ
m= ord(ˆν)∈Z>0, some multiple ofm. Hence we are in the situation of AssumptionOP. We do not assume that ˆν is a standard lift. Moreover, assume that ˆν is of type ˆm{0}(see Proposition 4.9.5), i.e. the conformal weight of the unique ˆν-twistedVL-moduleVL(ˆν) is in (1/mˆ)Z.
For the rest of this section assume that Assumption L holds. Then the orbifold con-struction (4.6) forr = 0 yields a new holomorphic vertex operator algebra
Ve = M
i∈Zmˆ
W(i,0)
(assuming that representations φi are chosen as in Corollary 4.7.11) built from the ir-reducible modules of the fixed-point vertex operator subalgebra VLνˆ. The main goal of this section is to describe the computation of the character ch
Ve(τ), or more precisely its qτ-expansion. To this end we need to compute all the twisted trace functions
T(1, i, j, τ) = trVL(ˆνi)φi(ˆνj)qLτ0−c/24
for i, j ∈ Zmˆ. The twisted trace functions on VL =VL(ˆν0) are determined in Proposi-tion 5.5.3 to be
T(1,0, j, τ) = trVLνˆjqτL0−c/24= ϑLνj,w(τ) ηνj(τ) .
In general, the trace functions for i ∈ Zmˆ \ {0} cannot be obtained directly since we do not know all the representations φi explicitly in these cases (except if i and ˆm are coprime, see proof of Lemma 4.6.1).5
Modular Transformations
In order to obtain all the twisted trace functionsT(1, i, j, τ) we make use of the modular-transformation properties of the trace functions described in Section 4.8. For this we need to explicitly know the S- and T-transformations of the above trace functions.
The transformation behaviour of the eta function and of products thereof is explicitly known (see e.g. [Apo90], Theorem 3.4). To obtain the transformation behaviour of the generalised theta function in the numerator we have to write it in terms of ordinary theta functions.
Consider some lift ˆν ∈Aut(ˆL) of ν ∈Aut(L), not necessarily a standard lift, and fix j ∈ Zmˆ. We showed in Corollary 5.3.5 that the map α 7→ w(α) := u(α). . . u(νj−1α) defines a homomorphismLνj → {±1}. We consider the following decomposition of Lνj into inverse images with respect tow
Lνj =Lν0j⊕Lν1j :=w−1({1})⊕w−1({−1}).
Clearly,Lν0j = ker(w) is a sublattice ofLνj. The other inverse imageLν1j is a coset ofLν0j if it is non-empty. Namely, if β1 is some arbitrary element in Lν1j, thenLν1j =Lν0j+β1.
5Note that ifi∈Zmˆ\{0}andj= 0, we can in fact compute the trace functionsT(1, i,0, τ) directly since they are simply the characters of the irreducibleVL-modulesVL(ˆνi) but we omit this computation since we obtain them via the modular transformations below, which we have to use in any case.
Then
ϑLνj,w(τ) = X
α∈Lνj
u(α). . . u(νj−1α)qτhα,αi/2= X
α∈Lνj
w(α)qτhα,αi/2
=ϑLνj
0 (τ)−ϑLνj 1 (τ)
=
ϑLνj(τ) ifLν1j =∅, ϑLνj
0 (τ)−ϑβ
1+Lνj0 (τ) ifLν1j 6=∅.
The transformation behaviour of these theta functions under SL2(Z) is known:
Theorem 5.6.1 (Special case of [Bor98], Theorem 4.1). Let L be a positive-definite, even lattice with discriminant form D=L0/L. Define for γ+L∈D
ϑγ+L(τ) := X
α∈γ+L
qhα,αi/2τ . Then
ΘL(τ) := X
γ+L∈D
ϑγ+L(τ)eγ+L
is a modular form for the Weil representation ρD of Mp2(Z), the metaplectic cover of SL2(Z), on C[D]of weight rk(L)/2.
If Lν1j is empty, then ϑLνj,w(τ) = ϑLνj(τ) and we need to consider the modular-transformation behaviour of ΘLνj(τ).
Now assume thatLν1j is non-empty. Sinceβ1 ∈Lν1j ⊆Lνj ≤L≤L0 ≤(Lνj)0≤(Lν0j)0, we can view the componentsϑ
Lνj0 (τ) andϑ
Lνj0 +β1(τ) ofϑLνj,w(τ) as entries of the vector-valued modular form ΘLνj
0 (τ) and determine the transformation behaviour according to the above theorem.
While the transformation behaviour of vector-valued modular forms for the Weil rep-resentation is known in principle, it only becomes computationally feasible using the explicit formulæ developed [Sch09] for SL2(Z) and generalised in [Str13] to Mp2(Z).
Combining the modular properties of the eta function and the theta functions we can determine the qτ-expansion ofT(1,0, j, M.τ) for all M ∈SL2(Z).
Characters and Dimensions
It remains to determine theqτ-expansion ofT(1, i, j, τ) for any i, j∈Zmˆ. For this, let Mi,j := ∗ ∗
i gcd(i,j)
j gcd(i,j)
!
∈SL2(Z).
The matrix Mi,j and gcd(i, j) depend in fact not only oni, j ∈Zmˆ but on the choice of representatives modulo ˆm. Then
(0,gcd(i, j))Mi,j = (0,gcd(i, j)) ∗ ∗
i gcd(i,j)
j gcd(i,j)
!
= (i, j)
and hence
T(1, i, j, τ) =T(1,(0,gcd(i, j))Mi,j, τ) =T(1,0,gcd(i, j), Mi,j.τ)/Z(Mi,j)
according to (4.4), where the function Z takes values inU3. If the rank rk(L) ofL is a multiple of 24, the factorZ(M) = 1 for all M ∈SL2(Z).
Then we can determine the characters of the irreducibleVLˆν-modules W(i,j) via chW(i,j)(τ) = 1
n X
l∈Zmˆ
ξn−ljT(1, i, l, τ). and finally
chVe(τ) = X
i∈Zmˆ
chW(i,0)(τ) = 1 n
X
i,l∈Zmˆ
T(1, i, l, τ).
This allows us to read off the dimensions of the weight spacesVen forn∈Z≥0.
For all the examples considered in this text the above calculations are performed using Sage and Magma [Sag14, BCP97]. Moreover, the Weil representation is computed using the “modules” package of the PSAGE library by Nils-Peter Skoruppa, Fredrik Strömberg and Stephan Ehlen, with minor modifications and bugfixes by the author of this text [ESS14].
Example
To illustrate the above steps we study the following simple example, which we will revisit in the next chapter (see Section 6.3). LetL=N(A64) be the Niemeier lattice with root latticeA64, i.e.Lis an even, unimodular, positive-definite lattice of rank rk(L) =c= 24.
We consider a certain automorphismν ofL of order 5. It has cycle shape 1−155 and its fixed-point sublattice is isomorphic toA04(5), the dual lattice ofA4, with the quadratic form rescaled by 5. Let ˆν be a standard lift of ν. Since the order of ν is odd, ˆνk is a standard lift of νk for all k∈Z≥0 and ˆν also has order 5.
We can compute the conformal weight of VL(ˆν) and obtain ρ(VL(ˆν)) = 1 by Re-mark 5.4.1, i.e. ˆν is of type 5{0}.
We compute
T(1,0,0, τ) = ϑN(A6
4)(τ) η(τ)24 and
T(1,0, j, τ) = ϑA0
4(5)(τ) η(τ)−1η(5τ)5 forj∈Zn\ {0}. This yields
chW(0,0)(τ) = 1 5
X
l∈Z5
T(1,0, l, τ) = 1 5
ϑN(A6
4)(τ) η(τ)24 +4
5 ϑA0
4(5)(τ) η(τ)−1η(5τ)5
=qτ−1+ 28 + 39384qτ+ 4298760q2τ+ 172859970q3τ+. . . .
The S-transformation yields
T(1, i,0, τ) =T(1,0, i, S.τ) = 5 ϑA4(1/5)(τ) η(τ)−1η(τ /5)5
= 5 + 125qτ1/5+ 750q2/5τ + 3375qτ3/5+ 12250q4/5τ + 39375qτ + 114000q6/5τ + 307000qτ7/5+ 776250qτ8/5+ 1867125qτ9/5+ 4298750qτ2+. . .
fori∈Zn\ {0}. Then chW(i,0)(τ) = 1
5 X
l∈Z5
T(1, i, l, τ) = 1 5
X
l∈Z5
T(1, i,0, Ti−1l.τ)
= 1 5
X
l∈Z5
T(1, i,0, τ +l)
= 5 + 39375qτ+ 4298750qτ2+ 172860000qτ3+. . .
fori ∈Zn\ {0}. This is the T-invariant part ofT(1, i,0, τ), i.e. those monomials with integral qτ-exponents.
Finally we obtain chVe(τ) = X
i∈Z5
chW(i,0)(τ)
=q−1τ + 48 + 196884qτ+ 21493760qτ2+ 864299970qτ3+. . . .
We will see in Proposition 6.1.3 that for central chargec= 24 the character ofVe has to be exactly thej-invariant but with constant term given by dimC(Ve1). The only interesting term is hence the dimension of the weight-one space
dimC(Ve1) =hch
Ve(q)i(1−c/24) =hch
Ve(q)i(0) = 48.
As a test, we can also compute dimC(V1G) using the dimension formula in Proposi-tion 4.10.4. Since ρ(VL(ˆνi)) = 1 for i ∈ Z5\ {0}, the formula simplifies to (4.10). It is straightforward to compute dimC(V1) = 144 and dimC(V1G) = 28. This immediately yields dimC(Ve1) = 48.
Prime Order
From the above example it is clear that we can derive a simple formula for the dimension of Ve1 if the automorphism ˆν is of prime order, i.e. of type p{0} for some prime p. This means in particular that ˆν is a standard lift of ν, ord(ν) = ord(ˆν) = p and that all powers of ˆν are standard lifts of the corresponding powers of ν. The cycle shape of ν has to be of the form 1apb for somea, b∈Z witha+pb= rk(L) anda+b= rk(Lν).
Theorem 5.6.2. Let VL and νˆ be as in AssumptionL with ord(ˆν) =p for some prime p. Then
dimC((VeL)1) = N1(L) + (p−1)N1(Lν)
p + rk(Lν)
+ (p−1)√ pb p|(Lν)0/Lν|
"
ϑ(Lν)0(τ) η(τ)aη(τ /p)b
#
(1−c/24)
where N1(·) denotes the number of vectors α of normhα, αi/2 = 1 in a positive-definite lattice.
We conclude with a remark on the type p{r}of an arbitrary lifted lattice automorph-ism ˆν of prime order. Remark 5.4.1 implies that
ρ(VL(ˆν)) = (p−1)(p+ 1) 24p b.
For ˆν to be of type p{0} we have to demand that (p−1)(p+ 1)b/24∈Z. We observe:
Proposition 5.6.3. Let n ∈Z with 2 -n and 3 -n. Then n2−1 = (n+ 1)(n−1) is divisible by 24.
An immediate consequence is the following:
Corollary 5.6.4. Let p6= 2,3 be a prime. Then (p+ 1)(p−1) is divisible by 24.
In summary, if ˆν is of prime orderp, then it has typep{0}if and only if
• p= 2 and rk(L)−rk(Lν)∈8Z,
• p= 3 and rk(L)−rk(Lν)∈6Z or
• p≥5.
Forp= 3 this is the condition in Theorem B of [Miy13].