Table Of Content= 4 SUPERCONFORMAL ALGEBRA AND THE ENTROPY OF
N HYPERKA¨HLER MANIFOLDS
TOHRU EGUCHI AND KAZUHIRO HIKAMI
Abstract. We study the elliptic genera of hyperK¨ahler manifolds using the representa-
tiontheoryof =4superconformalalgebra. Weconsiderthedecompositionoftheelliptic
0 N
generain terms of =4 irreducible characters,andderive the rateof increaseofthe mul-
1 N
tiplicities of half-BPS representations making use of Rademacher expansion. Exponential
0
2 increase of the multiplicity suggests that we can associate the notion of an entropy to the
geometry of hyperK¨ahler manifolds. In the case of symmetric products of K3 surfaces our
n
a entropy agrees with the black hole entropy of D5-D1 system.
J
4
1
]
h 1. Introduction
t
-
p
e It has been known for some time [18] that characters of the BPS representations of
h
the extended superconformal algebra do not in general have a good modular property.
[
This is because of the existence of special singular vectors coming from the BPS condition
3
v (Gi h = 0). Thus BPS characters are not regular theta functions but are mock (pseudo)
0 | i
theta functions of the kind first introduced by Ramanujan [1, 9]. Systematic understanding
1
4 of mock theta functions, however, was not available until very recently. Intrinsic structure
0
. behind them was first revealed by Zwegers several years ago [48], and they are identified as
9
0 the holomorphic part of the harmonic Maass forms (see Appendix for definition). Since the
9
work of Zwegers, the theory of the mock theta function has been applied to the theory of
0
: partitions [2], and its relationship with the quantum invariant for links and 3-manifolds has
v
i been clarified [23, 24, 25, 26, 27, 31, 46] (see Ref. 37 for a review on recent developments).
X
r
a Thispaperisasequeltoourpreviouspapers[10,11],wherewehavestudiedrepresentation
theory of the = 4 superconformal algebras using the method of Zwegers and in particular
N
the decomposition of the elliptic genus of the K3 surface in terms of irreducible characters
of = 4 algebra.
N
In general the elliptic genus of hyperKa¨hler manifold of complex-dimension 2k has an
expansion
k
elliptic genus = c BPS representation : h = ,ℓ
ℓ
4
ℓ∈ 0X,1,...,k (cid:20) (cid:21)
{ 2 2}
Date: January 14, 2010.
2000 Mathematics Subject Classification.
Key words and phrases.
1
2 T. EGUCHI ANDK. HIKAMI
∞
k
(ℓ)
+ p (n) non-BPS representation : h = n+ ,ℓ . (1.1)
k 4
Xn=1ℓ∈ 1X,1,...,k (cid:20) (cid:21)
{2 2}
Here h and ℓ respectively denote the conformal dimension and isospin of highest weight
states. In this paper we introduce the Rademacher expansion and determine the asymptotic
(ℓ)
behavior of the multiplicity factors p (n) as n becomes large. We shall show that they have
k
an exponential growth and at large k behave as
p(ℓ)(n) exp 2π√kn ℓ2 . (1.2)
k ∼ −
(cid:16) (cid:17)
Such an exponential behavior of the degeneracy is reminiscent of the entropy of black holes.
In the elliptic genus the right-moving sector is held fixed at the Ramond ground state
and hence the non-BPS states in (1.1) are actually the half-BPS states (BPS (non-BPS) in
the right-(left-)moving sector). Counting the asymptotic degeneracy of states protected by
supersymmetry amounts to computing the entropy of systems. Actually as we see below,
when one considers the case of symmetric product of K3 surfaces K3[k] it in fact agrees
with the entropy of the standard D5-D1 black holes in AdS3 S3 K3 [6, 42]. Positivity
× ×
inside the square root of (1.2) corresponds to the cosmic censorship in classical general
relativity [6, 7].
We propose in this paper that arbitrary hyperKa¨hler manifolds carry entropy as defined
above. In (1.1) a bad modular property of BPS characters is exactly compensated by the
equally bad modular property of the infinite series p(ℓ)(n)qn. Thus the lack of modular
n k
behavior of BPS characters is the origin of entropy in hyperKa¨hler manifolds.
P
This paper is organized as follows. In Section 2 we briefly review our previous results in
Refs. 10, 11. In Section 3 we study the Rademacher expansion of the Fourier coefficients of
the vector-valued harmonic Maass form by use of the Poincar´e–Maass series. In Section 4
we study the decomposition of the elliptic genera of the hyperKa¨hler manifolds in terms
of irreducible characters. By use of the Rademacher expansion, we derive the asymptotic
behavior of the multiplicity of the non-BPS representations. We present the cases of level-2
and -3 in some detail. The last section contains concluding remarks.
2. Superconformal Algebras and Mock Theta Functions
2.1. Characters of Superconformal Algebras. The = 4 superconformal algebra at
N
level k has a central charge c = 6k, and contains an affine SU(2) algebra. Its highest
k
weight state Ω is labeled by the conformal weight h and the isospin ℓ,
| i
L Ω = h Ω ,
0
| i | i
T3 Ω = ℓ Ω .
0 | i | i
The character of a representation is defined by
chk,h,ℓ(z;τ) = TrH e4πizT03 qL0−2c4 , (2.1)
(cid:16) (cid:17)
3
where q = e2πiτ with τ H, and denotes the Hilbert space of the representation. In
∈ H
the following we often use ζ = e2πiz with z C. In = 4 theory we have two types of
∈ N
representations [16, 17, 18]; massless (BPS) and massive (non-BPS) representations. In the
Ramond sector, their character formulas are given as follows;
massless representations (h = k, and ℓ = 0, 1,..., k),
• 4 2 2
i [θ (z;τ)]2 e4πiε((k+1)m+ℓ)z
chR (z;τ) = 10 ε q(k+1)m2+2ℓm, (2.2)
k,k4,ℓ θ11(2z;τ) · [η(τ)]3 ε=±1m∈Z (1+e−2πiεzq−m)2
X X
massive representations (h > k and ℓ = 1,1,..., k),
• 4 2 2
chRk,h,ℓ(z;τ) = qh−kℓ+21−k4 [θ1[0η((zτ;)τ]3)]2 χk−1,ℓ−21(z;τ), (2.3)
where χ (z;τ) denotes the affine SU(2) character
k,ℓ
ϑ ϑ
k+2,2ℓ+1 k+2,−2ℓ−1
χ (z;τ) = − (z;τ), (2.4)
k,ℓ
ϑ ϑ
2,1 2,−1
−
with the theta series defined by
ϑP,a(z;τ) = q(2P4nP+a)2 e2πiz(2Pn+a). (2.5)
n∈Z
X
Note that the denominator of the affine character equals
(ϑ ϑ )(z;τ) = iθ (2z;τ). (2.6)
2,1 2,−1 11
− −
Characters in other sectors are obtained by spectral flow: z z + 1 (R), z z + τ (NS),
→ 2 → 2
z z + 1+τ (NS).
→ 2
e
At the unitarity boundary h = k, the non-BPS representation decomposes into a sum of
g 4
the BPS representations. For instance, in the R sector (R sector with ( 1)F insertion) we
−
have
hlցimk chRk,h,ℓ(z;τ) = (−1)2ℓ+1q−kℓ+21 [eθ1[1η((zτ;)τ]3)]2 χk−1,ℓ−21(z;τ)
4 e
= chR (z;τ)+2 chR (z;τ)+chR (z;τ). (2.7)
k,k,ℓ k,k,ℓ−1 k,k,ℓ−1
4 4 2 4
e e e
2.2. Conformal Characters and Mock Theta Functions. For notational convenience
we set the holomorphic function C (z;τ) to be the massless superconformal character with
k
isospin-0 in R sector; 1
C (z;τ) = chR (z;τ)
ek k,h=k,ℓ=0
4
[θ e(z;τ)]2 i 1+qne2πiz
= 11 q(k+1)n2 e4πi(k+1)nz . (2.8)
[η(τ)]3 θ (2z;τ) 1 qne2πiz
11 n∈Z −
X
1In this article we slightly modify the notations from our previous papers [10, 11].
4 T. EGUCHI ANDK. HIKAMI
Note that massless representation carries the Witten index
C (z = 0;τ) = 1. (2.9)
k
It is known that the function C (z;τ) does not have a good behavior under modular trans-
k
formation: one has to find its suitable “completion” which has a good modular behavior.
The following completion of C (z;τ) has been obtained in our previous work [10]
k
k
1
(a) (a)
C (z;τ) = C (z;τ) R (τ)B (z;τ). (2.10)
k k − i 2(k+1) k k
a=1
X
Here the basis funbctions B(a)(z;τ) are pproportional to the massive characters of = 4
k N
algebra (2.3)
[θ (z;τ)]2
(a) 11
Bk (z;τ) = [η(τ)]3 χk−1,a−21(z;τ)
[θ (z;τ)]2 ϑ ϑ
11 k+1,a k+1,−a
= − (z;τ). (2.11)
[η(τ)]3 · ϑ ϑ
2,1 2,−1
−
(a)
The non-holomorphic function R (τ) is defined as
k
R(a)(τ) = i 2(k+1) sgn n+ 1 E n ℑτ q−2(kn+21), (2.12)
k 2 − k +1
n∈Z " (cid:18) (cid:19) r !#
p X
n=a mod2(k+1)
where E(x) is the error function
x
E(x) = 2 e−πt2 dt = 1 erfc √πx .
−
Z0
(cid:0) (cid:1)
(a)
The function R (τ) can be rewritten as a period integral,
k
i∞ Ψ(a)(z)
R(a)(τ) = k dz, (2.13)
k
Z−τ z+τ
i
q
(a)
where Ψ (τ) denotes a vector-valued modular form with weight-3/2 proportional to the
k
affine SU(2) character;
ϑ ϑ
Ψk(a)(τ) = [η(τ)]3χk−1,a−21(0;τ) = [η(τ)]3 kϑ+12,,a1 −ϑk2,+−11,−a(0;τ). (2.14)
−
In the sense of Zagier [47], the massless superconformal character C (z;τ) is a mock theta
k
(a)
function whose shadow is Ψ (τ). The completion C (z;τ) is a real analytic Jacobi form
k k
with weight-0 and index-k. Its modular properties are summarized as follows;
b
Ck(z;τ) = e−2πikzτ2 Ck z; 1 ,
τ −τ
(cid:18) (cid:19)
b b (2.15)
C (z;τ +1) = C (z +1;τ) = C (z;τ),
k k k
bCk(z +τ;τ) =bq−ke−4πikzCkb(z;τ).
b b
5
(a)
We notice that the basis function B (z;τ) is a vector-valued Jacobi form with weight-
k
( 1/2) and index-k;
−
k
B(a)(z;τ) = τ e−2πikzτ2 2 sin ab π B(b) z; 1 ,
k − i k +1 k +1 k τ −τ
r b=1 r (cid:18) (cid:19) (cid:18) (cid:19)
X
B(a)(z;τ +1) = e2(ka+21)πiB(a)(z;τ), (2.16)
k k
(a) (a)
B (z +1;τ) = B (z;τ),
k k
B(a)(z +τ;τ) = q−ke−4πikzB(a)(z;τ).
k k
2.3. Harmonic Maass Form. Next we define the elements of a k k matrix B (z;τ) as
k
×
(B (z;τ)) = B(b)(z ;τ), (2.17)
k ab k a
for 1 a,b k, and z C. We introduce [10]
a
≤ ≤ ∈
k
H(a)(z ,...,z ;τ) = B (z;τ)−1 C (z ;τ), (2.18)
k 1 k k ab k b
b=1
X(cid:0) (cid:1)
whose completion is
k
H(a)(z ,...,z ;τ) = B (z;τ)−1 C (z ;τ). (2.19)
k 1 k k ab k b
b=1
X(cid:0) (cid:1)
We then have the modbular transformation laws, b
k
i 2 ab z z 1
(a) (b) 1 k
H (z ,...,z ;τ) = sin π H ,..., ; ,
k 1 k − τ k +1 k +1 k τ τ −τ
r b=1 r (cid:18) (cid:19) (cid:18) (cid:19)
X
b b
H(a)(z ,...,z ;τ +1) = e−2(ka+21)πiH(a)(z ,...,z ;τ), (2.20)
k 1 k k 1 k
(a) (a)
Hk b(z1,...,zb +1,...,zk;τ) = Hk (z1b,...,zb +τ,...,zk;τ)
(a)
= H (z ,...,z ;τ).
b bk 1 k
(a) (a)
We note that H (z ;τ) and H (z ;τ) are related as
k i k i b
(a)
R (τ)
b H(a)(z , ,z ;τ) = H(a)(z , ,z ;τ) k . (2.21)
k 1 ··· k k 1 ··· k − i 2(k +1)
b p
From (2.13) it follows that
(a)
∂ Ψ ( τ)
R(a)(τ) = k − , (2.22)
∂τ k √2 τ
ℑ
and we obtain
∂ i 1
(a) (a)
H (z ,...,z ;τ) = Ψ ( τ). (2.23)
∂τ k 1 k 2(k+1) √2 τ k −
ℑ
b p
6 T. EGUCHI ANDK. HIKAMI
(a)
As a result, the completion H (z ,...,z ;τ) is a harmonic Maass form, and is an eigen-
k 1 k
function of the differential operator
b
(a)
∆ H (z ,...,z ;τ) = 0. (2.24)
1 k 1 k
2
Here ∆ denotes the hyperbolic Laplacian (τ = u+iv)
ℓ b
∂2 ∂2 ∂ ∂
∆ = v2 + +iℓv +i
ℓ − ∂u2 ∂v2 ∂u ∂v
(cid:18) (cid:19) (cid:18) (cid:19)
∂ ∂
= 4 ( τ)2−ℓ ( τ)ℓ . (2.25)
− ℑ ∂τ ℑ ∂τ
(a)
Correspondingly, the function H (z ,...,z ;τ) defined in (2.18) is regarded as a holomor-
k 1 k
phic part of the harmonic Maass form.
2.4. Jacobi Form. The Jacobi form f(z;τ) with weight-k and index-m obeys the following
transformation laws [19];
f z; 1 = τke2πimzτ2 f(z;τ),
τ −τ
(cid:18) (cid:19)
(2.26)
f(z;τ +1) = f(z +1;τ) = f(z;τ),
f(z +τ;τ) = q−me−4πimzf(z;τ).
It is known [19] that the space of the Jacobi form with even weight is spanned by
E (τ),E (τ),φ (z;τ),φ (z;τ) ,
4 6 −2,1 0,1
{ }
and a basis of Jacobi forms with weight-k and index-m is given by
[E (τ)]a [E (τ)]b [φ (z;τ)]c [φ (z;τ)]d, (2.27)
4 6 −2,1 0,1
with non-negative integers a, b, c, d satisfying
4a+6b 2c = k, c+d = m.
−
Here E (τ) and E (τ) are the Eisenstein series,
4 6
∞
E (τ) = 1+240 σ (n)qn
4 3
n=1
X
= 1+240q+2160q2+6720q3+17520q4+30240q5+ ,
···
∞
E (τ) = 1 504 σ (n)qn
6 5
−
n=1
X
= 1 504q 16632q2 122976q3 532728q4 ,
− − − − −···
where
σ (n) = rk.
k
r|n
X
7
The remaining two functions with index-1 are defined by
[θ (z;τ)]2
11
φ (z;τ) = (2.28)
−2,1 − [η(τ)]6
= ζ 2+ζ−1 + 2ζ2 +8ζ 12+8ζ−1 2ζ−2 q
− − − −
+ ζ3 12ζ2+39ζ 56+39ζ−1 12ζ−2+ζ−3 q2 + ,
(cid:0) (cid:1) (cid:0) (cid:1)
− − − ···
(cid:0) 2 2 2 (cid:1)
θ (z;τ) θ (z;τ) θ (z;τ)
10 00 01
φ (z;τ) = 4 + + (2.29)
0,1
θ (0;τ) θ (0;τ) θ (0;τ)
"(cid:18) 10 (cid:19) (cid:18) 00 (cid:19) (cid:18) 01 (cid:19) #
= ζ +10+ζ−1 + 10ζ2 64ζ +108 64ζ−1+10ζ−2 q
− −
+ ζ3 +108ζ2 513ζ +808 513ζ−1+108ζ−2+ζ−3 q2 + .
(cid:0) (cid:1) (cid:0) (cid:1)
− − ···
It is noted that (cid:0) (cid:1)
φ (0;τ) = 0,
−2,1
(2.30)
φ (0;τ) = 12,
0,1
and that φ (z;τ) is just one-half of the elliptic genus of the K3 surface [13, 30].
0,1
2.5. Character Decomposition of Elliptic Genera. Interms ofthecompletion C (z;τ)
k
(a)
of the massless character and the harmonic Maass form H (z ,...,z ;τ), we introduce the
k 1 k
function J (z;w ,...,w ;τ) as [10] b
k 1 k
b
k
(a) (a)
J (z;w ,...,w ;τ) = C (z;τ) H (w ,...,w ;τ) B (z;τ) (2.31)
k 1 k k − k 1 k k
a=1
X
b k b
(a) (a)
= C (z;τ) H (w ,...,w ;τ) B (z;τ). (2.32)
k − k 1 k k
a=1
X
Non-holomorphic dependence in (2.31) cancels each other, and the function
J (z;w ,...,w ;τ) is holomorphic as is seen in (2.32). J (z;w ,...,w ;τ) transforms like
k 1 k k 1 k
a Jacobi form with weight-0 and index-k [19];
Jk(z;w1,...,wk;τ) = e−2πikzτ2 Jk z; w1,..., wk; 1 ,
τ τ τ −τ
(cid:18) (cid:19)
J (z +1;w ,...,w ;τ) = J (z;w ,...,w +1,...,w ;τ)
k 1 k k 1 a k
= J (z;w ,...,w +τ,...,w ;τ)
k 1 a k (2.33)
= J (z;w ,...,w ;τ +1)
k 1 k
= J (z;w ,...,w ;τ),
k 1 k
J (z +τ;w ,...,w ;τ) = q−ke−4πikzJ (z;w ,...,w ;τ).
k 1 k k 1 k
By construction, the function J (z;w ,...,w ;τ) vanishes at z = w for a = 1,...,k,
k 1 k a
J (w ;w ,...,w ;τ) = 0, (2.34)
k a 1 k
and we also have
J (0;w ,...,w ;τ) = 1, (2.35)
k 1 k
8 T. EGUCHI ANDK. HIKAMI
because of (2.9). In the following we choose w to be half-periods, w 1, 1+τ, τ , and
a a ∈ 2 2 2
use the notation w
(k2,k3,k4) (cid:8) (cid:9)
w =
(k2,k3,k4)
1 1+τ τ
w ,...,w k = # w = ,k = # w = ,k = # w = .
1 k 2 a 3 a 4 a
2 2 2
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
(cid:12) (cid:16) (cid:17)
Then it is possible to sh(cid:12)ow that
(cid:12)
θ (z;τ) 2k2 θ (z;τ) 2k3 θ (z;τ) 2k4
J (z;w ;τ) = 10 00 01 , k = k +k +k
k (k2,k3,k4) θ (0;τ) θ (0;τ) θ (0;τ) 2 3 4
(cid:18) 10 (cid:19) (cid:18) 00 (cid:19) (cid:18) 01 (cid:19)
(2.36)
This is a vector-valued Jacobi form, and is a building block of the elliptic genera
for hyperKa¨hler manifold with complex dimensions 2k [14, 15]. Symmetrization of
J (z;w ;τ) in k , k , k gives a Jacobi form with weight-0 and index-k.
k (k2,k3,k4) 2 3 4
For our convenience we introduce the following notation,
Σ(a) (τ) = H(a) w ;τ . (2.37)
(k2,k3,k4) k (k2,k3,k4)
symmetrizatiXonof(k2,k3,k4) (cid:0) (cid:1)
Here k + k + k = k and without loss of generality we set k k k . Completion
2 3 4 2 3 4
≥ ≥
(a)
Σ (τ) is defined as
(k2,k3,k4)
Σ(a) (τ) = H(a) w ;τ . (2.38)
b (k2,k3,k4) k (k2,k3,k4)
symmetrizatiXonof(k2,k3,k4) (cid:0) (cid:1)
b b
(a)
As C (z;τ) and B (z;τ) are the massless and massive characters (2.8) and (2.11)
k k
respectively, the formula (2.32) is used to give the decomposition of elliptic genera in
terms of = 4 irreducible representations. In particular, the Fourier coefficients of
N
H(a)(w ;τ) counts the number of massive representations in elliptic genera. Since
k (k2,k3,k4)
in elliptic genera the right-moving sectors are always fixed to the ground state, massive
representations in the left-moving sector correspond to the overall half-BPS. Then the as-
ymptotic behavior of the growth of the multiplicity of non-BPS states in elliptic genera is
relatedtotheblackholeentropyinstringcompactificationonhyperKa¨hler manifolds. Aswe
shall see in the standard case of D5-D1 black hole in string compactification on K3 surface,
we will reproduce the black hole entropy from the growth of massive representations.
3. Harmonic Maass Form and Poincar´e–Maass Series
3.1. Jacobi Form and Theta Series. In the formula (2.32), the Fourier coefficients of
(a)
H (w ,...,w ;τ) count the multiplicity of non-BPS representations. Our purpose is to
k 1 k
compute these Fourier coefficients. As the parameters w are specialized to half-period, our
a
problem is to construct a vector-valued harmonic Maass form (k 1 and 1 a k),
≥ ≤ ≤
(a)
∆ Σ (τ) = 0, (3.1)
1 k
2
b
9
which transforms as (2.20);
k
i 2 ab 1
(a) (b)
Σ (τ) = sin π Σ ,
k − τ k +1 k +1 k −τ
r b=1 r (cid:18) (cid:19) (cid:18) (cid:19) (3.2)
X
b b
(a) − a2 πi (a)
Σ (τ +1) = e 2(k+1) Σ (τ).
k k
Once we are given such a mbodular form, we can cbonstruct a real analytic Jacobi form
(z;τ) of weight-0 and index-k by
k
J
k
b (z;τ) = Σ(a)(τ)B(a)(z;τ). (3.3)
Jk k k
a=1
X
(a) b b
Note that, when Σ (τ) is holomorphic and trivially satisfies (3.1), the function (z;τ)
k Jk
becomes a holomorphic Jacobi form.
b b
(a)
On the contrary, we can invert the above relation and determine the function Σ (τ) in
k
terms of a real analytic Jacobi form (z;τ). If we introduce a function
k
J
b
θ (2z;τ)
(z;τ) = i [η(τ)]3 11 (z;τ)
Jk −b [θ (z;τ)]2 Jk
11
for convenience, which is ae real analytic Jacobi form withbweight-1 and index-(k + 1), we
(a)
can in fact express the function Σ (τ) as a Fourier integral
k
Σ(a)(τ) =bq−4(ka+21) z0+1 (z;τ)e−2πiazdz, (3.4)
k Jk
Zz0
where z0 C is arbitrarby. Proof of (3.4) is ratheer standard [19]; due to the periodicity of
∈
(z;τ) in z z +1, we can expand
k
J →
n2
(z;τ) = q4(k+1) Σn(τ)e2πinz,
e Jk k
n∈Z
X
n2 e b
where q4(k+1) is inserted for convenience. Coefficients Σn(τ) are given by
k
Σnk(τ) = q−4(kn+21) z0+1Jk(z;bτ)e−2πinzdz.
Zz0
Quasi-periodicity of (zb;τ) in z z +τ implieseΣn(τ) = Σn+2(k+1)(τ). We thus obtain
Jk → k k
(2(k+1)m+a)2
Jk(z;eτ) = Σak(τ)e2πbi(2(k+1)m+ba)z q 4(k+1)
m∈Z amod2(k+1)
X X
e b
= Σa(τ)ϑ (z;τ).
k k+1,a
amod2(k+1)
X
b
Since (z;τ) is odd with respect to z and ϑ ( z;τ) = ϑ (z;τ), we recover (3.3).
k k+1,a k+1,−a
J −
In the case when (z;τ) is real analytic, for example J (z;w ,...,w ;τ) C (z;τ) as
e Jk k 1 k − k
in (2.31), formula (3.4) is valid when we replace (z;τ) with J (z;w ,...,w ;τ) C (z;τ).
k k 1 k k
J −
It is possible to see tbhat also in the holomorphic case the relation (3.4) is validbwhen we
b b
10 T. EGUCHI ANDK. HIKAMI
(a) (a)
replace Σ (τ) by Σ (τ) and (z;τ) by J (z;w ,...,w ;τ) C (z;τ). This is due to the
k k Jk k 1 k − k
relationship (2.10) and (2.21).
b b
Using the fact that
[η(τ)]3
iθ (2z;τ)
− 11 [θ (z;τ)]2
11
1+ζ
= + ζ2 ζ−2 q +2 ζ3 ζ−3 q2 +2 ζ4 ζ−4 q3
1 ζ − − −
−
(cid:0) +(cid:1)2ζ5+ζ(cid:0)4 ζ−4 (cid:1)2ζ−5 q(cid:0)4 +2 ζ6(cid:1) ζ−6 q5 + ,
− − − ···
integrality of the Fourier coefficients(cid:0)of Σ(a)(τ) in (3.4) follo(cid:1)ws straig(cid:0)htforwar(cid:1)dly once one
k
has integrality of the Fourier coefficients of the Jacobi form (z;τ).
k
J
b
In the case of k = 1 we take the Jacobi form to be the elliptic genus of the K3 surface
b
2φ (z;τ). Then we find
0,1
θ (2z;τ)
i [η(τ)]3 11 [2φ (z;τ) 24C (z;τ)]
− [θ (z;τ)]2 0,1 − 1
11
= 2ζ +2ζ−1 + 2ζ3+90ζ 90ζ−1 2ζ−3 q
− − −
+ 90ζ3 +462ζ 462ζ−1+90ζ−3 q2
(cid:0) (cid:1) (cid:0) (cid:1)
− −
+ 2ζ5 462ζ3+1540ζ 1540ζ−1+462ζ−3+2ζ−5 q3
(cid:0) (cid:1)
− − −
+ 90ζ5 1540ζ3+4554ζ 4554ζ−1+1540ζ−3 90ζ−5 q4 + .
(cid:0) (cid:1)
− − − ···
One finds that the Fou(cid:0)rier coefficients of ζ or ζ−1 are nothing but the multiplicit(cid:1)y of massive
representations in the K3 surface discussed in [11].
3.2. Multiplier System. We shall construct a solution of (3.1) and (3.2) in the form of
the Poincar´e–Maass series. It is a generalization of the discussion in our previous paper [11]
where a case of k = 1 was studied as an application of the Rademacher expansion for the
mock theta function. See Refs. 2, 3 for recent studies on the Poincar´e–Maass series.
We utilize the following multiplier system for the SU(2) affine character (2.4). For γ =
a b
Γ(1) = SL(2;Z), we set
c d ∈
(cid:18) (cid:19)
k
χk−1,a12−1 (0;γ(τ)) = [ρ(γ)]a1,a2 χk−1,a22−1(0;τ). (3.5)
aX2=1
Here we have
0 1 2 a a
1 2
ρ − = sin π ,
1 0 k +1 k +1
(cid:20) (cid:18)(cid:18) (cid:19)(cid:19)(cid:21)a1,a2 r (cid:18) (cid:19)
(3.6)
1 1 a12 −1 πi
ρ = e 2(k+1) 4 δ ,
0 1 (cid:18) (cid:19) a1,a2
(cid:20) (cid:18)(cid:18) (cid:19)(cid:19)(cid:21)a1,a2
