Gravitational couplings in \( \mathcal{N}=2 \) string compactifications and Mathieu Moonshine
HJE
2 string compactifications and Mathieu Moonshine
Aradhita Chattopadhyaya 0 1
Justin R. David 0 1
0 C.V. Raman Avenue , Bangalore 560012 , India
1 Centre for High Energy Physics, Indian Institute of Science
We evaluate the low energy gravitational couplings, Fg in the heterotic E8 ×E8 on K3 together with a 1/N shift along T 2. The orbifold g′ corresponds to the conjugacy classes of the Mathieu group M24. The holomorphic piece of Fg is given in terms of a polylogarithm with index 3−2g and predicts the GopakumarVafa invariants in the corresponding dual type II CalabiYau compactifications. We show that low lying GopakumarVafa invariants for each of these compactifications including the twisted sectors are integers. We observe that the conifold singularity for all these compactifications occurs only when states in the twisted sectors become massless and the strength of the singularity is determined by the genus zero GopakumarVafa invariant at this point in the moduli space.
Moonshine; Superstrings and Heterotic Strings; Discrete Symmetries; Topological Strings

1 Introduction
2 Heterotic string on orbifolds of K3 × T
2
3 The integral for gravitational thresholds
4 Evaluating the integral for gravitational thresholds
4.1
5 GopakumarVafa invariants and conifold singularities
5.1
5.2
5.3
Low lying coefficients of F¯ghol
Integrality of the GopakumarVafa invariants
Conifold singularities
6 Conclusions
A Modular transformations
B Details for g = 1 threshold integral
C Details for the g > 1, threshold integral
D Genus zero GV invariants for CHL orbifolds
E List of twisted elliptic genera
features is the R2 term in N = 4 theories in d = 4 which can be evaluated by a oneloop
computation in the type II frame, but yields predictions for an infinite sum of space time
instanton effects due to five branes in the heterotic frame [1, 2].
Let us focus on the gravitational coupling Fg of the low energy effective action of N = 2
string theories in d = 4. These couplings appear as the following terms in the effective
action
S =
Z
Fg(y, y¯)F 2g−2R2,
(1.1)
where F, R are the self dual part of the graviphoton and the Riemann curvature. The
coupling Fg depends on the vector multiplets y, y¯ of the theory. The canonical and well
studied example of such a theory is the E8 × E8 heterotic string theory compactified on
K3 × T 2 with the standard embedding of the spin connection in to a SU(2) of one of the
E8. Building on the earlier works [3–8], a detailed study of these couplings for this theory
has been carried out by [9] for g = 1 and in [10] for g > 1 by explicitly evaluating the
one loop threshold integrals. This compactification is also the standard and well studied
example of N = 2 string duality [11, 12]. The result from the heterotic side is one loop
exact. Moduli dependence of the one loop threshold corrections in this model have also
been studied earlier in [13–17]. On the type II A side, the theory is compactified on the
CalabiYau manifold X with Euler number χ(X) = −480. In fact the Euler number of the
dual CalabiYau is predicted from Fg.
Though the full expression of Fg is intricate, the holomorphic1 part of this coupling,
which can be obtained by sending y → ∞ is simple and predicts certain topological
invariants of the dual CalabiYau three fold X. Lets call this holomorphic part F¯g(y), the
genus g topological amplitude. The dual CalabiYau is generally a K3 fibration over a
base T 2. The heterotic side corresponds to the semiclassical limit when the volume of the
base is large. The topological amplitude F¯g(y) predicts the number of genus g holomorphic
curves in the K3 fibers of the dual CalabiYau X in the semiclassical limit. The most
convenient way to extract this information is using the observation of Gopakumar and
Vafa who showed that a generating function for F¯g(y) for any CalabiYau can be written
in terms of integer invariants which are called GopakumarVafa invariants [18, 19].
Therefore the explicit calculation of F¯g(y) from the heterotic side yields a prediction for the
GopakumarVafa invariants of the dual CalabiYau X in the semiclassical limit.
In this paper we generalize the evaluation of Fg to the heterotic E8 × E8 string theory
compactified on orbifolds of K3 × T 2. The orbifold g′ acts as a Z
K3 together with the 1/N shift on one of the circles of T 2. We consider the standard
N automorphism on
embedding in which the SU(2) spin connection of the K3 is embedded in one of the E8
of the theory. The orbifold g′ corresponds to the conjugacy classes of Mathieu group M24
listed in table 1 and they all preserve N = 2 supersymmetry. These compactifications
were introduced in [20, 21] with the motivation of exploring the role of M24 in string
compactifications which was first studied in the original K3 × T 2 model by [22]. We would
like to emphasise that these compactifications are distinct from that studied by [23]. In
this work the K3 was realized as a Z
N orbifold of T 4, with N = 2, 3, 4, 5 and the 1/N shifts
1It is actually the antiholomorphic part of Fg but we will take the complex conjugate and refer to it as
the holomorphic part.
– 2 –
where restricted to lie along the internal E8 × E8 lattice and not on the external T 2. Note
that the results of our paper does not hinge on the realization of K3 as an orbifold of T 4
and the shifts we consider are along the external T 2. One of our goals in evaluating the
Fg for the orbifolds under consideration in this paper is to determine the properties of the
dual CalabiYau geometry on the type II A side and to initiate a study of these geometries
and understand if at all M24 plays a role in these geometries. Recently the role of sporadic
symmetry groups in the elliptic genera of CalabiYau 5folds has been investigated [
24
].
We now summarize our main result for the holomorphic gravitational coupling Fg
for the orbifolds studied in this paper. Consider the twisted elliptic genus of K3 by an
automorphism g′ of order N , which is defined as
E2k are Eisenstein series of weight 2k and ζ refers to the zeta function. P2g is related to
the Schur polynomial S of order g by
P2g(x1, x2, · · · xg) = −S
1
1
x1, 2 x2, · · · g xg .
– 3 –
,
are Jacobi forms which transform under SL(2, Z) with index 1 and weight 0 and −2
respec(r,s) are numerical constants and βg(r′,s) is a weight 2 modular form under Γ0(N ).
After the discovery of the Mathieu moonshine symmetry in the elliptic genus of K3 [25],
the twining character F (0,1) for all the M24 conjugacy classes was first found in [26–28].
For g′ given in table 1, the corresponding to the full twisted elliptic genus M24 have been
explicitly determined in [29, 30]. We list them in appendix E for completeness. Now given
the twisted elliptic genus, consider the following weight 2g quasimodular form under Γ0(N )
f (r,s)(τ )P2g(G2, G4, G6, · · · , G2g) =
where
Then the topological amplitude F¯g is given by
The sum over lattice points m > 0 refers to the following lattice points (n1, n2), n1 ∈
n1T + n2U . The functions Li3−2g are polylogarithm functions of order 3 − 2g.
From comparing (1.5) and (1.7) we see that indeed it is the coefficients of the twisted
elliptic genus of K3 which forms the basic input data for the topological amplitude F¯g(y).
We wish to emphasise that this observation is the key result of this paper. This is a
generalization of the observation by [22] in which the elliptic genus of K3 which determines
the factor E4E6/η24, is the crucial input data for the topological amplitude for the
unorbifolded model. Our result is also a generalisation of the work of [10] which evaluated the
Fg for the unorbifolded heterotic compactification on K3 × T 2.
Now comparing the instanton contributions in F¯g(y) with the form of the topological
amplitude written in terms of the GopakumarVafa invariants we can extract out these
invariants. It is apriori not clear that the invariants will be integers since the coefficients
c(gr−,s1) in (1.5), themselves are not integers.
GopakumarVafa invariants are all integral. This forms a simple consistency check of our
result. In fact we will see that once the genus zero GopakumarVafa invariants are integers,
the higher genus invariants are assured to be integers. This is shown for genus g = 1, 2, 3.
The constant term in (1.7) contains the information of the Euler character of the dual
CalabiYau X. The Euler character of the CalabiYau dual to all the orbifolds considered
However to the level we have verified the
in this paper is listed in table 4.
We then study the conifold singularities of the F¯hol which correspond to points in
g
moduli space of enhanced gauge symmetry. We observe that there are no conifold
singularities from the untwisted sector and all conifold singularities are due to twisted sector
states becoming massless. The strength of this singularity is proportional to the genus zero
GopakumarVafa invariant corresponding to this state. The list of low lying genus zero
GopakumarVafa invariants for g′ which we refer to as CHL orbifolds2 are provided in
appendix D. For the rest of the g′ in table 1. we provide an ancillary Mathematica code from
which the genus zero GopakumarVafa invariants can be evaluated and seen to be integral.
We briefly mention the method we adopt to evaluate Fg. In the heterotic frame it is
given in terms of a one loop integral over the fundamental domain. We basically follow the
2These are geometric actions, which keep the Hodge number of K3; h1,1 ≥ 1.
– 4 –
method of orbits adopted in many works in this subject starting from [31]. See [32] for a
recent discussion of these methods. Therefore the methods used in this paper have been
developed and established by several previous works. However we do not directly apply the
lattice reduction theorem of Borcherds [33] as done by earlier works [10, 34, 35] for
evaluating Fg, g > 1. This is because the application of the reduction theorem when the integrand
has modular forms of Γ0(N ) is rather intricate and we find it is easier to proceed directly
and carry out each of the steps involved in the integrations. We would like to emphasise
that using the direct method we pursue, there are several manipulations which are
straightforward, but nevertheless important to arrive the standard form of Fg.3 For this reason we
find it instructive to provide the details of the way we implement the method of orbits in
the appendices B, C. Since this area of research has a rich history with several
contributions, it is important to mention that similar integrals involving modular forms of Γ0(N )
have occurred earlier for the specific case of N = 4 in [36] and for N = 2 in [37] and [34].
Modular forms transforming under Γ0(N ) for arbitrary N occurring in the theta lift of the
twisted elliptic genus of K3 were done in [38] and [39]. As far as we are aware the present
work is the first evaluation of gravitational couplings Fg for the orbifold compactifictions
introduced in [20] which generalizes the result of [10] to integrands involving Γ0(N ) forms.
The organization of the paper is as follows. In section 2, we review aspects of the Z
N
we will present the new supersymmetric index which forms the basic ingredient for the
gravitational threshold integral. Next in section 3 we write down the expressions for the
first evaluate the gravitational coupling F1. We do this because the integrand as well as
performing the integral is a straight forward generalization of the one done for the K3 × T 2
theory by [9] and provides checks for our calculation. Next we present the result for the
gravitational coupling Fg(y, y¯) for g > 1 and extract the holomorphic coupling F¯hol(y). In
g
section 5 we study the properties of the genus g topological amplitude and observe that
the GopakumarVafa invariants of the dual CalabiYau predicted by our calculation. We
then study the conifold singularities of F¯hol(y). In section 6 we present our conclusions.
g
The appendix A contains the modular properties of the integrand involved in evaluating
Fg(y, y¯). Appendices B, C contain the details of the integrations to obtain the gravitational
thresholds. Appendix D lists the GopakumarVafa invariants of for all g′ corresponding to
CHL orbifolds. Finally appendix E lists the elliptic genera of all the orbifolds considered
in the paper for completeness.
2
Heterotic string on orbifolds of K3 × T
2
In this section we briefly describe the general class of N = 2 compactifications we will
study. Consider the E8 × E8 heterotic string theory compactified on K3 × T 2 in which the
SU(2) spin connection of K3 is embedded in one of the E8’s which is called the standard
embedding. We then orbifold by a freely acting Z
N which acts as a g′ automorphism on
3For example we obtain the form given in (C.7), which requires the next steps to cast in the standard
form (C.11).
– 5 –
1A
2A
3A
5A
7A
11A
23A/B
4B
6A
8A
14A/B
15A/B
2B
3B
1
2
3
5
7
11
23
4
6
8
14
15
2
3
are such that the twining genera matches with the F (0,1) listed in [25]. These orbifolds are of order
4 and 9 respectively in our analysis. We refer to the classes 2A, 3A, 5A, 7A, 4B, 6A, 8A as the
CHL orbifolds.
K3 together with a 1/N shift along one of the circles of K3. The g′ action corresponds
to any of the 26 conjugacy classes of Mathieu group listed in the table. In the standard
embedding one of the E8 lattice breaks to a D6 ⊗ D2 lattice. The SU(2) spin connection of
K3 couples to the fermions of D2 lattice. These left moving fermions with the left moving
bosons of the K3 combines with the right moving supersymmetric K3 CFT to form a (6, 6)
conformal field theory. The internal conformal field theory is given by
H
(6,6) (6,0)
internal = HD2K3 ⊗ HD6 ⊗ HE8
(8,0)
⊗ HT 2
(2,3)
(2.1)
The g′ orbifold acts as a Z
N automorphism on the (6, 6) CFT which preserves its SU(2)
Rsymmetry together with a 1/N shift on one of the circles of the T 2 CFT. Thus these
compactifications preserves N
theories as compactifications of the E8 × E8 theory on the orbifold K3 × T 2/ZN .
= 2 supersymmetry in 4 dimensions. We refer to these
– 6 –
The starting point in evaluating both gauge and gravitational threshold corrections is
to obtain the new supersymmetric index of the internal CFT. This is defined by
Znew =
1
moving fermion number of the T 2 CFT together with the right moving fermion number of
β(0,s)(τ ) = − p + 1 Ep(τ ), for 1 ≤ s ≤ p − 1
β(r,rk)(τ ) =
π(p − 1) ∂τ [ln η(τ ) − ln η(N τ )]
– 7 –
The trace in (2.2) is taken over the Ramond sector of the right moving supersymmetric
HJEP05(218)
internal conformal field theory. Note that (c, c˜) = (22, 9).
In [21], the new supersymmetric index for compactifications on the orbifolds K3 ×
T 2/ZN was evaluated and it was shown that it can be written in terms of the twisted
elliptic genus of K3. Let us briefly review the result. First we define the twisted elliptic
genus of K3 which is given by
1
N
genus can be written in the general form
F (0,0)(τ, z) = αg(0′,0)A(τ, z),
are Jacobi forms which transform under SL(2, Z) with index 1 and weight 0 and −2
respec(r,s) are numerical constants and βg(r′,s) is a weight 2 modular form under Γ0(N ).
For example for g′ ∈ pA with p = 2, 3, 5, 7 we have
where
Here
2
2
12i
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
is a modular from which transforms with weight 2 under Γ0(N ). The twisted elliptic genus
transforms under SL(2, Z) as
F (r,s)
2T2U2  − m1U + m2 + n1T + n2T U 2,
12 p2R + m1n1 + m2n2
for
a, b, c, d ∈ Z,
ad − bc = 1
The indices in (2.9) cs + ar and ds + br are taken to be mod N .
Appendix E lists the twisted elliptic genus F (r,s)(τ, z) for corresponding to the
conjugacy classes g′ in table. Given the twisted elliptic genus, we an read out αg′
the general form given in (2.5). Then the new supersymmetric index for the standard
embedding compactifications of the E8 × E8 heterotic string on the orbifold K3 × T 2/ZN
(r,s), βg(r′,s) using
is given by
Znew(q, q¯) = −2 η24(τ ) Γ(2r,2,s)E4 4 αg′
the lattice sum on T 2 which is defined as
Here it is understood that the indices r, s are summed from 0 to N − 1. In (2.11), Γ(2r,2,s) is
1
1
=
X c(r,s)(l)ql.
l∈ NZ
– 8 –
and T, U are the K¨ahler and complex structure of the torus T 2. It is also understood
in (2.11), that there is a sum over the lattice momenta m1, m2 as well as the winding
numbers n1, n2. Note that in (2.11) its only the lattice sum that depends on both q and q¯,
the Eisenstein series E4, E6 as well as the Γ0(N ) weight 2 form βg(r′,s) depends only on the
holomorphic coordinate q. It is instructive to examine the situation in which there is no
orbifold performed. Then p = 1 and α1A = 8 and there are no twisted sectors. Thus the
new supersymmetric index reduces to
Znew(q, q¯) = −4 E4(τ )E6(τ )
Here Γ2,2 is the usual lattice sum without phases and twists. For later purpose we define
the following Γ0(N ) form, which occurs in the new supersymmetric index given in (2.11).
has the following transformation property under the SL(2, Z) generators
Using the fact that βg(r′,s)(τ ) is a Γ0(N ) form and (2.9) it is easy to see that the f (r,s)(τ )
f (r,s)(τ + 1) = f (r,s+r)(τ ),
f (r,s)
1
− τ
= (−iτ )−2f (N−r,s)(τ ) = (−iτ )−2f (r,N−s)(τ ).
Given the new supersymmetric index we can read out the difference between the
number of vectors and hypers in the spectrum [9]. This is given by
Nh − Nv = −
N−1
X c(0,s)(0).
s=0
(2.15)
(2.17)
(2.16)
Here, we have also included the four U(1)’s resulting from the metric and the
NeveuSchwarz Bfield with one index along the T 2 in the counting of the vectors. This data
also gives the Euler number of the corresponding type II dual CalabiYau compactification
which is given by
χ(X) = −2(Nh − Nv) = 2 X c(r,s)(0).
N−1
s=0
For the compactification on K3 × T 2 the value of Nh − Nv = 240 and therefore the
corresponding Euler number of the dual K3fibered CalabiYau manifold is χ(X) = −480.
These data for all the orbifolds considered in this paper is provided in table 4.
In the subsequent sections we will use the new supersymmetric index in (2.11) as a
starting point and evaluate the gravitational oneloop corrections for heterotic
compactifications on orbifolds K3 × T 2/ZN .
3
The integral for gravitational thresholds
In this section we obtain the integral for gravitational thresholds involving N = 2 heterotic
compactifications on orbifolds considered in this paper. Though these integrals were
obtained for the unorbifolded situation earlier in [8, 9] we find it instructive to go through
the analysis. This is because we wish to arrive at the key result that the integrals which
evaluate the thresholds for the orbifolds involve the twisted elliptic genus of K3.
It is first easy to discuss the one loop integral which captures the gravitational
correction with no graviphoton insertions. The result of this integral provides the moduli
dependence of the following higher derivative term in the N = 2 effective action for these
compactifications
I1(T, U ) = R+2F1(T, U ).
(3.1)
Here R+ refers to the antiselfdual Riemann tensor. From the earlier works of [3, 4], a
nice compact expression for the F1 has been obtained in [9]. This is given by4
1
F τ2
compactification for N = 1.
4Note that we have normalized F1 so that the result for the integral agrees with [9] for the K3 × T 2
– 9 –
The subtraction by bgrav is to ensure that the integral is well defined for τ2 → ∞. Note
that the trace over the internal conformal field theory is precisely what occurs in the new
supersymmetric index given in (2.2). Therefore to obtain the moduli dependence of the
gravitational correction for the orbifold compactifications in this paper, we substitute the
result for the new supersymmetric index (2.11) in (3.2) This results in
,
where we have ignored the constant necessary to regulate the integral. From the
properties (A.6) and (2.15) we see that the integrand is invariant under SL(2, Z) transformations.
Note that the key input that goes into the integral are the coefficients of the twisted
elliptic genus of K3 represented by the functions αg′
integral over the fundamental domain which is done in section 4.1.
(r,s), βg(r′,s). The next step is to perform the
Let us now discuss the integral which results in the moduli dependence of the following
term in the one loop effective action
Ig(T, U ) = R+2F+2g−2Fg(T, U ),
where F+ is the antiself dual field strength of the graviphoton and g > 1. From the world
sheet analysis of [8] we see that the one loop integral is given by
(3.3)
(3.4)
(3.5)
(3.6)
The super script (r, s) for pR indicates the sector over which the right moving momentum
belongs to. When r 6= 0 the winding number n1 on one of the circles of the T 2 is quantized
in units of Z + Nr since the right moving momentum belongs to the rth twisted sector of the
Fg =
1
2π2(g!)2
Z d2τ (
τ2
1
τ22η2(τ )
TrR h(i∂¯X)(2g−2)(−1)F F qL0− 2c4 q¯L˜0− 2c˜4 i
×h
Yg Z
i=1
the transverse noncompact space time bosons. The above result is obtained by carrying
out the Wick contractions of the vertex operators for the graviton and the graviphoton
insertions as mentioned in [8]. Note that the correlators of the space time bosons and
the internal conformal field theory factorize. The 2g − 2 insertions of ∂¯X arise from the
(2g − 3) graviphoton vertices in the (0)ghost picture and one ∂¯X that appears in the
picture changing operator. Since the correlator is evaluated on the torus, the insertions of
∂¯X can contribute only through the zero modes. Therefore this can be replaced as follows
(i∂¯X)(2g−2)
→
.
1/N shift action. Then trace over the internal conformal field theory can be written as5
This trace is evaluated using the input in (3.6) and repeating the steps which led to the
result in (2.11). The only difference is the insertions of the lattice momenta p(r,s). Note that
the sectors (r, s) are summed over, and it is always understood that the lattice momenta
and winding are summed.
Now we need to simplify the correlators over the free noncompact bosons. To this end
we follow [8] and use the generating function for these correlators
G(λ, τ, τ¯) = X
∞
1
g=1 (g!)2
λ
τ2
2g Yg Z
h
i=1
The correlation functions are normalized free field correlators of the spacetime bosons.
This generating function is given by To use this generating function for the correlators, we consider
G(λ, τ, τ¯) =
2πiλη3 2
θ1(λ, τ )
e− πτλ22 .
F (λ, T, U ) = X λ2gFg(T, U ).
∞
g=1
Then using the result (3.7) and (3.9) in (3.5), the one loop amplitude simplifies to
function admits the expansion
p(r,s)λ
λ˜ = √2T2U2
R
.
k=0
Here again note that the key input that goes into the integrand (3.11) for the gravitational
couplings for the orbifold compactifications considered in this paper is the twisted elliptic
To perform this integral we follow the approach in [10]. The reciprocal of the theta
2πλ˜η3 !
θ1(λ˜, τ )
2
∞
e− πτλ˜22 = X λ˜2kP2k(Gˆ2, . . . , G2k),
5In the analysis of [8], the trace over the internal conformal field theory was referred to as Cǫ(τ¯). Further
more, what we call as left movers is right movers in [8] and viceversa. We follow the notations of [10].
where P2k is a polynomial related to the Schur polynomial by
P2k(Gˆ2, . . . , , G2k) = −Sk Gˆ2, 2 G4, 3 G6, . . . , k G2k ,
1
1
1
where the Schur polynomials are in turn defined by the expansion
exp
" ∞
X xkzk
k=1
#
= X Sk(x1, · · · xk)zk,
∞
k=0
x
2
2
x
3
6
P2(Gˆ2) = −Gˆ2,
P4(Gˆ2, G4) = − 21 (Gˆ22 + G4),
P3(Gˆ2, G4, G6) = − 16 (Gˆ23 + Gˆ2Gˆ4) − 3 G6.
1
S1(x1) = x1,
S2(x1, x2) =
where the G’s are normalized Eisenstein series given by
Note that P is holomorphic, except for the occurrence of Gˆ2 and it transforms as a modular
form of weight 2k. Substituting the expansion (3.13) in (3.11), we obtain
F (λ, T, U ) = X
∞
k=0
λ2(k+1)
π2(2T2U2)k
1 (r,s)E6 − βg(r′,s)(τ )E4 (p(Rr,s))2kP2k+2.
Note that the integrand is an invariant under SL(2, Z). This can be seen using the properties
in (A.9) and (2.15)
4
4.1
g = 1
Evaluating the integral for gravitational thresholds
We will first perform the integral in (3.3). As it will be clear subsequently, it is simpler to
treat the case of g = 1 separately from the integral for g > 1. We will follow the unfolding
method developed in [9] for performing this integral. However we generalize the discussion
to integrands which contain modular forms transforming under Γ0(N ) rather than SL(2, Z).
Let us first define the Fourier expansions of the expressions in the integrand.
1
2η24 E4 4 αg′
Note that as expected, the twisted sectors admit fractional q expansions. Substituting this
expansion and the values of p2L and p2R from (2.12) into (3.3) we obtain
The first step to do the integral is to perform the Poisson resummation over the momenta
m1, m2 using the formula
X f (m)e2πism/N =
X
m∈Z
Z ∞
k∈Z+ Ns −∞
duf (u) exp(2πiku).
(4.3)
Using this identity and performing the integral over the corresponding variables u1, u2, we
Z d2τ
Therefore using (4.4) in (4.2) we can write the integral as
πT2
n1 k1 !
n2 k2
,
G(~n, ~k) = − U2τ2 A2 − 2πiT (det)A,
J (A, τ ) = T2 exp − U2 A2 − 2πiT detA f (r,s)(τ )Eˆ2(τ ),
Using (4.6), we can think of J as a function of the matrix A. Then the sum over r, s
and ~n, ~k in the right hand side of (4.7) can be thought of as the sum over matrices of the
I1(T, U ) =
F τ22 A
2
Z d τ X J (A, τ ).
Now using the modular transformation given in (2.15) and the definition of J in (4.8), it
This symmetry allows us to extend the integration over the fundamental domain to its
images under SL(2, Z) together with the restriction of the summation over A to its
inequivalent SL(2, Z) orbits. Lets denote the sum over inequivalent SL(2, Z) orbits as P′A,
then F1 becomes
I = X′ Z
A
d2τ
Now the label r, s is to interpreted as N n1 mod N and N k1 mod N respectively. The
region of integration FA depends on the orbit represented by A.
From the analysis of [9] we see that there are three inequivalent orbits. These are as
follows: the zero orbit
the nondegenerate orbit
and the degenerate orbit
ζ(3)
T2U2
3
1
2π
(4.10)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
The contribution from each of these orbits has been evaluated in the appendix B. Taking
the sum of these contributions the result for F1 is given by
F1 = T2 π6 E22(q)f (0,0)(q) q0
N−1
+ X
s=0
+4Re
X
k≥0,l≥−1
(k,l)6=(0,0)
k∈Z+r/N
where P˜ is given by
c˜(0,s)(0) − log(T2U2) +
U2 − κ
− c(0,s)(0)
π3
T2
6ζ(4, s/N ) U22 +
6 c(0,s)(0)
c(0,0)(0)
c(0,1)(0)
2A
3A
5A
7A
11A
23A
−120
−80
−48
−240/7
−240/11
−240/23
136
109
77
829/14
442/11
473/23
From the appendix it will be clear that the above result holds for T2 > U2 as in the analysis
of [9]. Let us examine the result for N = 1, the unorbifolded K3 × T 2 compactification.
We obtain For unorbifolded K3 this answer reduces to
F1 = −48πT2 − 264(− log(T2U2) − κ) − 88πU2
(4.18)
Let us recall that for each of the orbifolds we can read out the coefficients c˜(r,s) and c(r,s)
using their definition given in (4.1) and the explicit expressions for the twisted elliptic
genera given in the appendix E. Let us make a few observations from the result for F1.
Note that the coefficients which determine the moduli dependence of F1 in the first two
lines of (4.15) depends on low lying topological data of the new supersymmetric index.
This dependence does not involve exponentials in moduli T, U . The low lying coefficients
of the new supersymmetric index for the various orbifold compactifications corresponding
to the Mathieu moonshine conjugacy classes are listed in tables 2 and 3.
One also observes that for an order N orbifold c(0,s)(−1) = 1/N and c˜(0,s)(0) can be
written as
c˜(0,s)(0) = c(0,s)(0) − N
.
24
(4.19)
Using this topological data we can evaluate the linear and quadratic dependences of the
moduli T and U which result from the contribution of the integral along the zero orbit and
the degenerate orbit. These coefficients for the various orbifolds are listed in table 2 and 3.
Note that the coefficient of Tζ2(U3)2 is determined by the difference Nh − Nv or the Euler
character. An interesting observation is the occurrence of the low lying toplogical data
weighted with the Hurwitzzeta function ζ(2, s/N ) as well as ζ(4, s/N ) in the orbifolds.
This can be seen only for the orbifold compactifications. In fact if one is just given the
HJEP05(218)
c(0,0)(0) c(0,1)(0) s1
c(0,s1)(0) c(0,s2)(0)
X
πT2
three coefficients of U2, and U22/T2 and 1/T2U2 in the gravitational threshold F1 and
the properties of the low lying coefficients in, we can determine c(0,s) for all the orbifold
compactifications corresponding to the conjugacy classes of Mathieu Moonshine considered
here. A summary of these coefficients, including the Euler number for all the orbifolds
considered in this paper is given in table 4.
The higher level coefficients of the new supersymmetric index control the exponentially
suppressed terms in the last line of (4.15). Note that the exponential dependence of T
moduli carries the information of the twisted sectors.
4.2
g > 1
The gravitational threshold integral for g > 1 is given by
Fg =
1
where k = g − 1 > 0. Note that here the summation over (r, s) is implied. The steps to
perform the integral are similar to the case when g = 1, however one needs to keep track of
the extra insertions of the momentum p(r,s). First we perform the Poisson resummation
over the variables (m1, m2) to obtain
Fg(T, U ) =
N−1
X
r,s=0 n2,k2∈Z,n1,k1∈Z+ Nr
T2
2U2
k
2
ZF dτ 2τ J˜(A, τ ),
2
J˜(A, τ ) = T2A
− U2 A2 − 2πiT detA f (r,s)(τ )P2k+2(τ ),
6The Calabi Yau manifolds with χ values −480, 32, 276, 400, 520, 524, 644, 768 are known to exist and
they are listed in [40].
Now similar to the g = 1 case we can think of the sum over r, s and ~n, ~k as sum over the
matrices of the form in (4.6) with n2, k2 ∈
. Thus (4.21) can be written as
Ig(T, U ) =
Now using the modular transformation given in (2.15) and the definition of J˜ in (4.22), it
This symmetry allows us to extend the integration over the fundamental domain to its
images under SL(2, Z) together with the restriction of the summation over A to its
inequivalent SL(2, Z) orbits. Let us again denote the sum over inequivalent SL(2, Z) orbits
as P′A, then Fg becomes
Fg = X′ Z
A
FA
T2
2U2
k d2τ
2
τ 2 T2A
Now the label r, s is to interpreted as N n1 mod N and N k1 mod N respectively. The
region of integration FA depends on the orbit represented by A.
We can now look at the contribution of the three inequivalent orbits. First note that
the contribution of the zero orbit vanishes due to the presence of A
Thus we are left with the nondegenerate orbit which is characterized by the set of matrices
2k in the integrand.
A =
and the degenerate orbit
n1 j !
0 p
,
A =
p ∈ Z,
n1, j ∈ N
1 Z, n1 > j
0 j !
Note that here we have included the p = 0 case also in the nondegenerate orbit for
convenience. This is because the p = 0 can be treated uniformly together with the p 6= 0
situation in the nondegenerate orbit.
The detailed evaluation of the integral in the two orbits is carried out in the appendix C.
This integral was done in [10] for the case of K3 × T 2 compactification and in [34] for the
FHSV compactification. The latter situation involves only modular forms under Γ0(2)
in the integrand. Here we have generalized the evaluation of the integral for integrands
containing modular forms transforming under Γ0(N ). Further more as it will be clear in
the appendix C, we do not directly apply the reduction theorem of Borcherds [33] as done
in the earlier works to perform the integral. The application of the reduction theorem is
rather intricate and it easier to proceed straightforwardly from (4.25) and carry out the
steps involved in the integration.
To write the result in a convenient form, let us define the following two dimensional
vectors and the inner products.
m = (n1, n2),
y = (T, U ),
m · y = n1T1 + n2U2 + i(n1T1 + n2T2),
We also define the Fourier coefficients involved in the expansion of the modular forms as
Then, from the evaluation in appendix C, the result for the contribution from the
nondegenerate orbit is given by
Fgn>on1deg =
22(g−1)π2 r,s=0 m6=0 t=0 h=0
and m 6= 0 refers to any of the following cases,
sˆ + 1/2 = ν,
ν = 2k − h − j − t − 1/2,
k = g − 1
n1 > 0, n2 > 0;
n1 < 0, n2 < 0;
r
r
N
n1 = 0, n2 > 0 or n2 < 0;
n2 = 0, n1 > 0 or n1 < 0;
n1 =
, n2 < 0 or n1 = − N
, n2 > 0 with rn2 ≤ N, r > 0.
In (4.30), the index r in c(r,s) is related to n1 by r = N n1
result for the nondegenerate orbit is valid for T2, U2 > 0.7 Performing the integral for the
mod N and n2 ∈ Z. The
7Note that the result for the integral also contains the complex conjugate F¯g, which we have suppressed
for simplicity.
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
In the (0,0) sector one has and j < n1, j = 0, 1, . . . n1 −1 and p ∈
Z. So we can proceed
for the τ1 integral as done in [9] for the unorbifolded K3. We write τ1′ = j + pU1 + n1τ1,
then the relevant part for the τ1 integration in (B) becomes
2
Iˆ(τ1) = T2 Z d τ
Here we have focused on the τ1 dependent terms, and kept a generic term using the Fourier
expansion of f (0,0(q)Eˆ2(τ ).
Replacing τ1 by τ1′ we see that the only j dependence comes from
Now performing the sum over j forces n1n2 = n where n2 is an integer. Rest of the τ1
integration in the untwisted sector is Gaussian and the result is given by,
If we are in the untwisted sectors (0, s), we have j = j′ + s/N with j′ being an integer the
above result holds but with an additional factor of e−2πin2s/N and replacing c(0,0)(n1n2)
and c˜(0,0)(n1n2) by c(0,s)(n1n2) and c˜(0,s)(n1n2).
Let us now look at the τ2 integration. The τ2 integrand is given by,
I2 = pT2U2e−2πin2s/N Z ∞ dτ2
where F is given by,
F = −2πτ2n2n1 − U2τ2
πT2 (n1τ2 + pU2)2 − 2πiT n1p − 2πpn2U1 −
πn22U2τ2 .
T2
The τ2 integration is of the Bessel form and we use the following formulae to evaluate it,
Z dx
x3/2 e−ax+b/x = pπ/be−2√ab,
Z dx
x5/2 e−ax+b/x = √πe−2√ab 1 + 2√ab !
2b3/2
.
The result of this exercise yields for the untwisted sectors
I
untwisted = 4Re
X
n1>0,n2≥−1
n1n26=0
n1∈
Z
c˜(0,s)(m2/2)e−2πisn2/N Li1(e2πi(n1T +n2U))
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
3
c(0,s)(m2/2)e−2πisn2/N P˜(n1T + n2U ) ,
Let us go over to the twisted sectors. First consider an order N orbifold when N is
prime. In the twisted sectors, the n in the Fourier coefficients c(r,s)(n) and c˜(r,s)(n) is such
that n ∈ Z/N . While j = j′ + s/N, j′ ∈ Z and n1 ∈ r/N + Z. The modular forms f (r,s)
has the property that the transformations of τ → τ + 1 can be used to relate it to f (r,0).
After doing this, one effectively works in the (r, 0) twisted sector but now j runs from
0 to n1 − 1/N in steps of 1/N .13 Thus from (B.8) we see that the sum over all the values
of j for a prime N would be N n1 iff n/(n1N ) = n2 is an integer, else it is zero. Therefore
n = n1n2N , only integral values of n are picked up.
However one observes that the
coefficients of qn in f (r,s) are related to the coefficients of qn in f (r,0) under transformation
τ → τ + 1. If n is fractional the relation is through a multiplicative phase. Using this
property one can write the result in terms of the Fourier coefficients in the f (r,s) sector.
The final result for a prime N for the nondegenerate orbit can be given by
I
The above argument also holds when N is composite and the twisted sector r is such
that r, N are coprime. However if r divides N , then there are various suborbits for
the sectors (r, s) under transformations τ → τ + 1. For example if N = 4, then the
sectors (2, 0), (2, 2) and the sectors (2, 1), (2, 3) form distinct subsectors not related by
transformations of the kind τ → τ + 1. One then uses the argument developed for N prime
and carries it out for each of the suborbits. The end result works out to be the same as
that given in (B.14).
Degenerate orbit.
Here the determinant of A is 0. So we choose the matrix A as A =
0 j!
0 p
,
where j ∈
Z/N, p ∈
Z and (j, p) 6= (0, 0). Since n1 = 0 twisted sectors don’t contribute to
the degenerate orbit. There are logarithmic divergences due to the constant term in the
expansion of f (0,s)E2(τ ) which needs to be renormalized. We have
Iˆ =
Z
d2τ
X
A′,n,s FA τ22 T2 exp
πT2
− U2τ2 A
2
c˜(0,s)(n) − πτ2
3
c(0,s)(n) qn,
(B.15)
multiply the integrand for the c˜(0,s) with a factor of (1 − e−Λ/τ2 ).
where A2 = j + pU 2, p ∈ Z, j ∈ Z + s/N in (0, s) sector. To regularize the result we
13See [38] for a discussion of this step for integrands which involve a Γ0(2) from.
HJEP05(218)
The result of the integral from the second term is given by
Now for coefficient of c(0,s) one gets after integrating τ2 the result
− c˜(0,s)(0) log(Λ) + γE + 1 + log(2/3√3) .
−
6 U 2
X
j,p π3 T2
Z and use the following results [9]
Note that here j = j′ + s/N where j′ ∈ Z. This is the difference as compared to the
calculations of [9]. To evaluate these sums on j, p we need to divide the sum in j′ 6= 0, p = 0
X
j′∈Z (j′ + B)2 + C2 =
π "
C
1 +
e2πi(B+iC)
1 − e2πi(B+iC) +
e−2πi(B−iC) #
1 − e−2πi(B−iC)
Note that in applying these formula, the information of s is present in B as now B =
U1p + s/N .
When p = 0, the terms in the sum that contribute in the Λ → ∞ limit
from (B.17) and (B.19) result in
N−1
X
s=0
c˜(0,s)(0)
.
Now we carry out the sum over j′ for p 6= 0 in (B.19) using (B.21). When one does this,
there is a term that results from the action of the derivative on the first term of (B.20),
that is the term independent of B. This results in
The integration domain FA in this orbit is given by
The only dependence of τ1 comes from the qn and since it is integrated from −1/2 ≤ τ1 <
1/2 it picks up only the q0 term from f (0,s)Eˆ2. Performing the τ2 integration for the term
containing the coefficient c˜(0,s) we obtain
c˜(0,s)(0) π
X
j,p j + pU 2 − j + pU 2 + ΛU2/(πT2) −
1
Now, lets carry out the sum over j′ for p 6= 0 in the first two terms of (B.17).
Applying (B.20) from the B independent part we get
∞
X
p=1
2
p − pp2 + Λ/πT2U2
2
= − ln
πT2U2
Λ
− ln 4 + 2γE .
!
s=0
π2
N−1 6 c(0,s)(0)
X
ζ(3)
T2U2
Z
FA τ2
dτ2 (1 − e−Λ/τ2 ) .
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
(B.22)
(B.23)
(B.24)
This again comes in all the sectors of (0, s) and taking care of the moduli dependence and
combining (B.18) we tobain
N−1
s=1
X c˜(0,s)(− log T2U2 − κ),
(B.25)
4Re
X
l6=0,l≥−1
l Z
∈
Combining all these results
where κ = 1 − γE + log( 38√π3 ).
Finally keeping track of all the second and third terms in the r.h.s. of B.20 and its
derivative B.21, that is the B dependent part both in the sums (B.17) and (B.19) we obtain
c˜(0,s)(0)e2πisl/N Li1(e2πi(lU)) − πT2U2
3
c(0,s)(0)e2πisl/N P˜(lU ) .
(B.26)
HJEP05(218)
I
degen = (B.22) + (B.23) + (B.25) + (B.26).
(B.27)
C
Details for the g > 1, threshold integral
The integral we have to do is the one in (4.21). As discussed there is no zero orbit. We
have to perform the integral over the nondegenerate orbit and the degenerate orbit.
Nondegenerate orbit.
This orbit are characterized by matrices of the form For the nondegenerate orbit we apply the unfolding technique and choose a representative of the matrix A and integrate over two copies of the upper half plane. In this case A becomes A =
n1 j!
0 p
,
p ∈ Z, n1, j ∈ N
Z
, n1 ≥ j ≥ 0.
This gives det A = n1p and A = n1τ + j + pU . The domain for the integration now is two
copies of the upper half plane. In the (0, s) sectors and (r, s) sectors one can verify that
the limits on j, n1 are as in the last section for g = 1. The j dependence is in the phase
given in (B.8). Again after shifting τ1 to τ1′ = j + pU1 + n1τ1 which forces the n of the
Fourier coefficient to be integer as before.
The relevant part for the τ1 integration in (4.21) becomes
I(τ1) = T2
Z d2τ
τ 2+t
2
T2
2U2
k
(τ1′ + i(pU2 + n1τ2))2k
Note that the coefficients here would be written as c(r,s)(m2/2, t) in different sectors
and a factor of e−2πisn2/N will come as an extra phase to the integral in the corresponding
sectors along with e−2πiT n1p. Also a factor of 2 would be present due to the two copies
of the upper half plane. We shall not write these in the integral to avoid cumbersome
notation and will only reinstate them back in the final result. Let us denote the vector m~
with the components
m~ = (n1, n2),
m2/2 = n1n2,
(C.3)
where n2 ∈
Z and n1 ∈
Z + r/N .
This integral in (C.2) can be done by using binomial expansion for (τ1′ +i(pU2+n1τ2))2k
and shifting the variable τ1′ to τ˜1 = τ1′ − in2U2τ2/T2. Following through these replacements
in C.2, we obtain
2
I(τ1) = T2 Z d τ
T2
(τ˜1 + in2U2τ2/T2 + i(pU2 + n1τ2))2k
(C.4)
The τ˜1 can be integrated and we are left with
I(τ1) = T2 τ 2+t 2U2
2
1
T2
h ! 2k!
2j
h
×Γ(j + 1/2)
(πT2/U2τ2)j+1/2 e−πU2τ2n22/T2.
×exp − U2τ2
Here we have written down only the terms that involve τ1 dependence. Now putting back
all the terms which have τ2 dpendence we expand the complete integral as a polynomial
in τ2 which gives
×
Z ∞
0
dτ2 (τ2)l+h−j−3/2−t exp[−πT2U2p2/τ2 − πτ2/(T2U2)(n1T2 + n2U2)2]
Integrating term by term and replacing h by 2k − h, which still keeps the limits as 0 to 2k,
we obtain
I˜ = X
2k 2k−h [k−h/2] 2k − h! 2k!
X X
T2
T2
2U2
h=0 l=0 j=0
×2
pT2U2
2j
h
l+2k−h−j−t−1/2
×Kl+2k−h−j−t−1/2(2πip(n1T2 + n2U2)).
2c(r,s)(r2/2, t)e−2πisn2/N
The result of this integral (C.7) seems manifestly different from that obtained in [10]
at this step. To bring into in the from we perform changes in the order of summations and
limits. We outline these now. First we change the order of summation on l and h which gives
2k 2k−h [k−h/2]
X
X
X
h=0 l=0 j=0
→
X
X
X
.
k
e−2πiT1n1pΓ(j + 1/2)(−1)k−j
(in2U2τ2/T2)h−2j(ipU2 + in1τ2)2k−h
(C.5)
(C.6)
(C.7)
(C.8)
Now we call h − l as h+ and change the orders of the sums on h+ and l.
Now one needs to change the order of the sums j and l which can be done as
l=0
Ks+1/2(x) =
l
Lim(x) = X∞ x
.
r π
2x
l=1 lm
s
e−x X
(s + k)!
1
k=0 k!(s − k)! (2x)k
,
(C.9)
(C.10)
(C.11)
(C.12)
(C.13)
(C.14)
(C.15)
HJEP05(218)
One can shift to the variable j′ = k − j which also runs from 0 to k. We will relabel this
variable j. The resultant integral is a gamma function and since there is a sum over p it
The final sum over l is doable and this gives us
I˜ = 21−k X
X
e−2πisn2/N c(r,s)(m2/2, t) X
X
k+1
t=0 n1n26=0,p6=0
×ph+(T2U2)j+h++1/2−k(n2U2 + n1T2)2k−h+−2j(−1)k−j
×K2k−h+−j−t−1/2(2πp(n1T2 + n2U2))e−2πip(n1T2+n2U2).
(2k)!
j!h+!(2k − h+ − 2j)!(4π)j
pT2U2
2k−h+−j−t−1/2
The result (C.11 agrees with equation (4.29) of [10] when K3 × T 2 is not orbifolded.
In (C.11), the summation over the orbifold sectors is implied. To rewrite the result in terms of polylogarithms we use the relations
This leads to
k+1
I˜= 21−k X
X
X
X
X
2k [k−h+/2] s
t=0 n1,n26=0 h+=0 j=0
a=0 j!h+!(2k − h+ − 2j)! a!(s − a)! (4π)j+a
(2k)!
(s + a)! (−1)k−j+h+
×(T2U2)k−t(sgn Im(m · y))h+c(r,s)(m2/2, t)e−2πisn2/N Im(mˆ·y)t−j−aLi1+a+j+t−2k(e2πimˆ·y).
The case m = 0 needs to be treated separately. From (C.2) and (C.6) we see that we
can perform the integral also for the limit n1 = n2 = 0. After the τ1 integration, for the
m = 0 limit we obtain
I˜m=0 = X
2k!
j=0 2j
×
Z ∞
0
T2
T2
2U2
k
Γ(j + 1/2)(−1)k−j
dτ2 (τ2)j−3/2−t exp[−πT2U2p2/τ2](pU2)2k−2j(U2/(πT2))j+1/2.
forms a zeta function. A factor of 2 arises on taking two copies of the upper half plane.
However for values t = k the zeta function blows up so one can use a regularization by
adding −ǫ to the power of τ2. We have
So from C.15 the contribution from m = 0 to the nondegenerate part of the integral is
given by,
I
nondegenm=0 = 2 X
X(−1)j22j−4k+t
k k+1
j=0 t=0
(2k)!
(2j)!(k − j)!
e−2πisn2/N
c(0,s)(0, t) !
πt+1/2
×(πT2U2)−ǫΓ(1/2 + j + t − k + ǫ)ζ(1 + 2ǫ + 2t − 2k).
In the equation (C.17) we have used the fact
Γ(n + 1/2) =
(2n)!√π
n!4n .
We now use the equation (C.17) and consider ǫ → 0 limit to extract out the finite
contribution as follows,
(C.17)
(C.18)
(C.19)
(C.20)
HJEP05(218)
(πT2U2)−ǫ = 1 − ǫ log (πT2U2) ,
Γ(1/2 + s + ǫ) = Γ(1/2 + s)(1 + ǫ ψ(1/2 + s)),
ζ(1 + 2ǫ) =
+ γE,
1
2ǫ
where ψ(x) is the logarithmic derivative of the gamma function. The resulting sums in C.17
can be explicitly computed for some cases. We use the results Γ(1/2+n) = π1/22−n(2n+1)!!
and Γ(1/2 − n) = (−1)nπ1/22n/(2n − 1)!! with n ≥ 0 to arrive at
X(−1)j2j
j=0
Γ(1/2 + j) = π1/2δk,0.
Thus results for k = 0 that is genus one and k 6= 0 are different. For k > 0 we can write
the full nondegenerate orbit result as,
Ikn>on0d,meg=en0 = 2 X
N−1 Xk−1 c(0,s)(m2/2, t)
πt+1/2
ζ(1 + 2(t − k))(2T2U2)(k−t)
(C.21)
s˜=0
× X(−1)s˜22(s˜−2k)+t
N−1 c(0,s) k
+ X
s=0
23k πk s˜=0
X(−1)s
(2k)!
(2s˜)!(k − s˜)!
(2k)!
(2s˜)!(k − s˜)!
Γ(1/2 + s˜ + t − k)
ψ(1/2 + s˜).
Putting all the contributions together the contribution for Fgn>on1degen is given by
Fgn>on1degen =
X
X
X
X
r,s m6=0 t=0 h=0
X
j=0
a=0
X e−2πisn2/N c(r,s)(m2/2, t)
(C.22)
(sgn(Im(m · y)))h
(Im(mˆ·y))t−j−a
(T2U2)t
X(−1)s
s˜=0
Here m 6= 0 is defined in (4.32) and sˆ is defined in (4.31).
Degenerate orbit. The degenerate orbit is characterised by matrices of the form
A =
0 j !
0 0
,
Z
j ∈ N
, j 6= 0.
The evaluation of the integral in the degenerate orbit proceeds as follows. Let us first
assume T2 > U2 for this part of the calculation we have
Ideg =
T k+1 Z j2kτ2−t−2dτ2e−πj2T2/U2τ2−2πn′τ2 Z 1/2
2
dτ1e2πin′τ1 .
−1/2
Note that since we are in the untwisted sector for this orbit n′ ∈ Z. Furthermore to obtain
a nonzero result due to the integration we require n′ = 0. Then the result for Idegen is
given by
obtain [9, 33]14
(2U2)k
N−1
X
s=0 t
N−1
X
s=0 t
Ideg =
X π−t−1U2(T2/U2)k−tc(0,s)(0, t)t!ζ(2(t + 1 − k, s/N )).
If U2 > T2, we need to in fact we need start by performing at Tduality and then we
Idegen =
X π−t−1T2(U2/T2)k−tc(0,s)(0, t)t!ζ(2(t + 1 − k), s/N ).
Now reinstating all factors we obtain
Fgd>eg1en =
1
T22g−3
N−1 g
X
X c(0,s)(0, t) 22g−2πt+3
t!
T2
U2
t
ζ(2(2 + t − g), s/N ).
14The reason this has to be done has to do with the convergence of the integral for Fourier coefficient
when n′ = −1.
0
n(n1,n2)
0
n(n1,n2)
8606976768
115311621680
1242058447872
11292809553810
89550084115200
HJEP05(218)
(3/2,1)
(5/2,1)
(7/2,1)
16
8128
1212576
47890048
1055720304
Genus zero GV invariants for CHL orbifolds
In this appendix we list the genus zero GopakumarVafa invariants n0m corresponding to the
CHL orbifolds. We observe that they are integers. For the remaining orbifolds in table 1,
we provide an ancillary Mathematica code from which these invariants can be evaluated
and seen to be integers. One point worth mentioning, is that all the remaining orbifolds
especially the case of 2B and 3B do not have any geometric interpretation as actions on
the K3 surface. Therefore the fact that the GopakumarVafa invariants turn out to be
integers is interesting.
E
List of twisted elliptic genera
In this appendix we provide the list of the twisted elliptic genera used to determine the
f (r,s) defined in (2.14) in this paper. We call the classes 2A, 3A, 5A, 7A, 4B, 6A, 7A, CHL
(r/3, 0)
(1/3,3)
(1/3,9)
(1/3.12)
378
110646
6063822
164258118 2918573208
HJEP05(218)
orbifolds. When g′ corresponding to these classes act on K3, it can be seen that the values
of the Hodge number h1,1 ≥ 1. The twisted elliptic genus of classes 2B, 3B is such that their
twining character coincides with that constructed in [26–28]. But they do not correspond
to M24 symmetry. In fact in M24, the classes have order 2 and 3, however the orbifolds we
construct have order 4 and 9. The 2B twisted elliptic genus has been constructed in [52]
by considering K3 as 6, SU(2) WZW models at level 1. While 3B twisted elliptic genus
we give here was constructed in [29].
Conjugacy class pA, p ={1,2,3,5,7}.
F (0,0) = =
A(τ, z),
F (0,s) =
F (r,rs) =
8
p
8
8
p + 1 Ep(τ )
τ + s
p
(E.1)
1490560 103674750 1820321750
(r/5, 0)
(1/5,5)
(2/5,5)
(3/5,5)
(4/5,5)
250
72750
3892000 103674750 1820321750
HJEP05(218)
8
6
2
(n1, n2) (1/5, −1) (1/5,1)
(6/5,1)
(11/5,1)
(16/5,1)
14
1032
159648
7172574
175994068
(n1, n2) (2/5, −1) (2/5,1)
(7/5,1)
(12/5,1)
(17/5,1)
(n1, n2) (3/5, −1) (3/5,1)
(8/5,1)
(13/5,1)
(18/5,1)
3040
450704
17378724
375804020
4062
605260
11893715
260500896
(n1, n2) (4/5, −1) (4/5,1)
(9/5,1)
(14/5,1)
(19/5,1)
7082
1045848
39672656
840212212
Conjugacy class 11A. Going through these steps we obtain the following formula for
the twisted elliptic genus for 11A.
2
33
2
33
8
11
F (0,0) = =
A(τ, z),
F (0,s) =
F (r,rs) =
A(τ, z) − B(τ, z)
A(τ, z) + B(τ, z)
1
6 E11(τ ) − 52 η2(τ )η2(11τ ) ,
1
66 E11
τ + s
11
− 55
2 η2(τ + s)η2 τ + s
(E.2)
−642
187
54309
2881360 76296885 1333671600
(n1, n2) (1/7, −1) (1/7,1)
(8/7,1)
(15/7,1)
(22/7,1)
12
519
81348
3813632
96969840
(n1, n2) (2/7, −1) (2/7,1)
(9/7,1)
(16/7,1)
(23/7,1)
(17/7,1)
(24/7)
(18/7,1)
(25/7,1)
1525
227875
9059700
201601562
2034
304146
12117044
268891664
3540
522435
19770135
418523073
0
n(n1,n2)
−400
256
16
8
4
Conjugacy class 23A/B.
F (0,k)(τ, z) =
F (r,rk)(τ, z) =
where
1
1
23 3
1 1
23 3
A − B
A + B
23
1
12 E23
(n1, n2) (r/4, 0), r = 1, 3
(1/4,12)
(1/4,16)
(1/4, −2) (1/4,−3)
(1/2, −1)
(1/2,0)
(1/2,1)
(1/2,2)
(5/4,1)
(3/2,1)
(E.3)
f23,1(τ ) = 2
η3(τ )η3(23τ )
η(2τ )η(46τ )
+ 8η(τ )η(2τ )η(23τ )η(46τ ) + 8η2(2τ )η2(46τ ) + 5η2(τ )η2(23τ )
τ + k
1
− 22 f23,1
τ + k
0
n(n1,n2)
−524
132
246
12
8
8
(n1, n2) (r/6, 0), r = 1, 5
(1/6,24)
(1/6,30)
(n1, n2) (r/3, 0), r = 1, 2
(1/2,0)
17990607384
39912
2466900
71374608
1318893168
17990607384
HJEP05(218)
(1/6, −1)
(5/6, −1)
(1/6,1)
(5/6,1)
(1/6, −2)
(1/6,−4) (1/2, −1)
(1/3, −1)
(2/3,−1)
(1/3,1)
(2/3,1)
2032
303152
Conjugacy class 4B.
The twisted elliptic genus for this class is given by
1024
6072
4
6
4
(7/6,1)
155620
(1/2,1)
3054
(11/6,1)
(3/2,1)
(5/3,1)
F (0,1)(τ, z) = F (0,3)(τ, z) =
F (1,s)(τ, z) = F (3,3s) =
F (2,1)(τ, z) = F (2,3) =
F (0,2)(τ, z) =
F (2,2s)(τ, z) =
1
4
1
4
8A
8A
1 4A
3 + B
4A
3 − 3
4
1
4
4B
2B
3 − 3 E2(τ ) ,
3 + 3 E2
τ + s
2
− 3 E2(τ ) + 2E4(τ )
1
− 6 E2
τ + s
2
1
2 E4
B (5E2(τ ) − 6E4(τ ) ,
τ + s
4
0
n(n1,n2)
−644
6
8
8
16
60002304
(n1, n2) (r/8, 0), r = odd
(1/8,16) (r/4, 0), r = odd
(1/4,4)
Conjugacy class 6A. The twisted elliptic genus for 6A are given by
F (0,1) = F (0,5);
F (0,2) = F (0,4);
1
1
5
3 − B
− 6 E2(τ ) − 2 E3(τ ) + 2 E6(τ )
(1/8, −2)
(1/8,−3) (1/8,−4)
(1/8, −5)
(1/8,−6)
(1/8,−7)
F (1,k) = F (5,5k) =
1 2A
3 + B
1
− 12 E2
τ + k
2
1
− 6 E3
τ + k
3
τ + k
6
128
37888
2167808
(1/4, −2)
(1/4,−3) (1/2,−1)
(1/8, −1)
(1/4, −1)
16 2A − 2 BE3(τ ) ,
3
1 8A
4
3 − 3 BE2(τ ) .
517
2028
2
6
2
4
4
(1/2, 0)
(5,8,1)
4
(3/4,1)
8096
22976
(1/2,2)
22976
(7/8,1)
4056
2
(1/2,4)
(9/8,1)
1
(7/4,1)
(E.6)
, (E.7)
HJEP05(218)
F(2,2k+1) = A9 + 3B6 E3
F(4,4k+1) = A9 + 3B6 E3
τ +k+1 ,
3
3
F(3,1) = F(3,5) = A
B
9 − 12E3(τ)− 72E2
B
τ +1 + 8 E2
B
2
3τ +1 ,
2
F(3,2) = F(3,4) = A
F(2r,2rk) =
F(3,3k) =
16 2A+ 12BE3
τ +k ,
3
16 83A + 32BE2 2
τ +k .
Conjugacy class 8A.
where r = 1,3,5,7.
F(0,0)(τ,z) = A(τ,z),
F(0,1) = F(0,3) = F(0,5) = F(0,7),
81 23A − B −21E4(τ) + 73E8(τ) .
F(r,rk)(τ,z) = 81 23A + 8 −E4 4
B
τ + k + 3E8 8
7
τ + k
(E.11)
F(2,1) = F(6,3) = F(2,5) = F(6,7),
F(2,3) = F(6,5) = F(2,7) = F(6,1),
1 23A +
8
18 23A +
B
3
B
−E2(2τ) + 2E4 4
3
3
2τ + 1
2τ + 3
4
;
F(4,4s) = 81 83A + 23BE2
τ + s ,
2
F(4,2) = F(4,6) = 1 4A
8 3 − B3 (3E2(τ) − 4E2(2τ) ,
F(4,2k+1) = 81 23A + B 43E2(4τ) − 23E2(2τ) − 21E4(τ) .
(E.10)
(E.12)
(E.13)
F (0,1)(τ, z) = F (0,3) = F (0,5) = F (0,9) = F (0,11) = F (0,13);
A
14 3 − B
− 36 E2(τ ) − 12 E7(τ ) +
7
91
36 E14(τ )
1
1
− 72 E2
− 3 η(τ + k)η
η(τ )η(2τ )η(7τ )η(14τ )
τ + k
2
1
− 12 E7
τ + k
2
τ + k
7
τ + k
7
η
τ + k
τ + k
;
F (7,2k+1) =
F (7,2k) =
F (0,2k) =
F (2r,2rk) =
F (0,7) =
F (7,7k) =
A
A
+ B
+ B
1
1
14 3
14 3
14 3
1 8
14 3
1
2
7
3
7
7
3
− 3 η(τ + k)η(2τ + 2k)η
− 12 E7(τ ) +
+ 7 eiπ11/12η(τ )η(7τ )η
7τ + 1
2
− 12 E7(τ ) +
+ η(τ )η(7τ )η
τ
2
7τ
2
τ + k
7
2τ + 2k
7
2τ + 2k
7
1
.
τ
− 72 E2
τ + 1
2
1
− 72 E2 2
7τ
2
τ + 1
2
7τ + 1
2
7
1
4
2
A − 4 BE7(τ )
A +
τ + k
7
A − 3 BE2(τ ) ,
A +
τ + k
2
k runs from 1 to 6,
; k runs from 0 to 6.
k runs from 0 to 1.
where r=1,3,5,9,11,13 and rk is Mod 14.
F (2r,2rk+7) =
1
A
14 3
+ B
− 6 E2(τ ) − 12 E7
1
τ + k
7
where k runs from 0 to 6 and except 3 and r from 1 to 6.
(E.14)
(E.15)
(E.16)
(E.18)
(E.19)
(E.20)
F (0,1)(τ, z) = F (0,2) = F (0,4) = F (0,7) = F (0,8) = F (0,11) = F (0,13) = F (0,14);
1
1
15 3
A
A
+ B
− 48 E3
1
1
1
− 4 η(τ + k)η
τ + k
3
τ + k
3
τ + k
τ + k
+
7
48 E15
τ + k
τ + k
;
15 3 − B
− 16 E3(τ ) − 24 E5(τ ) +
16 E15(τ ) − 4
η(τ )η(3τ )η(5τ )η(15τ )
where r=1,2,4,7,8,11,13,14 and rk is mod 15. The sectors belonging to the 5A and 3A
suborbits are given by
F (0,0) =
F (0,3k) =
F (3r,3rk) =
F (0,5k) =
F (5r,5rk) =
4
4
15 3
15 3
A − 3 BE5(τ )
A +
1
τ + k
2A − 2 BE3(τ ) ;
2A +
τ + k
3
k runs from 1 to 4;
; k runs from 0 to 4.
k runs from 0 to 2.
Finally the remaining sectors are given by
F (3r,5+3rk) =
F (3r,10+3rk) =
1
A
3
A
3
+ B
+ B
1
1
1
3
1
− 4 E3(τ ) − 24 E5
− 4 η(τ + k)η(3τ + 3k)η
− 4 E3(τ ) − 24 E5
τ + k
τ + k
3
+ 8 E5
τ + k
3
+ 8 E5
where k runs from 0 to 4 and s=1 to 4.
F (5r,3s+5rk) =
1
+ B
τ + k
3
where k runs from 0 to 2 and s=1 to 2.
5
4
+ η(τ + k)η(5τ + 5k)η
τ + k
3τ + 3k
τ + k
3
τ + k
3
5τ + 5k
3
(E.21)
(E.22)
(E.23)
(E.24)
(E.25)
Conjugacy class 2B.
Conjugacy class 3B.
F (0,0)(τ, z) = 2A;
F (0,1)(τ, z) = F (0,3)(τ, z),
(E.28)
F (0,1)(τ, z) =
F (0,2)(τ, z) = −
B(τ, z)
F (1,s)(τ, z) = F (3,3s) = −
F (2,2s)(τ, z) = −
(E2(τ ) − E4(τ )),
3
B(τ, z)
4
B(τ, z)
B(τ, z)
3
E2
1
E2
3
3
E2(τ ),
τ + s
τ + s
− E4
τ + s
4
,
− 6 E2(τ ) +
3 E2(2τ ) ,
F (0,0)(τ, z) =
F (0,1)(τ, z) = F (0,2) = F (0,4) = F (0,5) = F (0,7) = F (0,8);
(E.29)
F (0,1)(τ, z) = −
F (0,3)(τ, z) = −
F (r,rs)(τ, z) =
F (3,1)(τ, z) = −
F (3,2)(τ, z) = −
3
A(τ, z)
,
η2(3τ )
B(τ, z)
4
η2( τ +3s ) ,
2B(τ, z) η6(τ + s)
2B(τ, z) e2πi/3 η6(τ )
2B(τ, z) e4πi/3 η6(τ )
η2(3τ )
η2(3τ )
,
,
E3(τ ),
= F (3,4) = F (3,7) = F (6,2) = F (6,8) = F (6,5);
r = 1, 2, 4, 5, 7, 8
= F (3,5) = F (3,8) = F (6,1) = F (6,7) = F (6,4);
F (3r,3rk)(τ, z) = −
A(τ, z)
B(τ, z)
τ + k
3
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any medium, provided the original author(s) and source are credited.
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