AMS/IP Studies in Advanced Mathematics

Decomposition of Witten–Reshetikhin–Turaev Invariant: Linking Pairing and Modular Forms Kazuhiro Hikami Abstract. We study the SU (2) Witten–Reshetikhin–Turaev invariants for Seifert manifolds associated with the Arnold 14 unimodal singularities. We show that the invariants are decomposed based on a value of linking pairings. Discussed also is a relationship with modular forms with weight-3/2.

1. Introduction There has been a great progress in quantum topology during 20 years since Witten constructed quantum Chern–Simons invariant for 3-manifolds M [37], Z Zk (M ) = eiS(A) DA, (1.1) where A is a G-gauge connection on the trivial bundle over M , and S(A) is the Chern–Simons action   Z k 2 S(A) = Tr A ∧ dA + A ∧ A ∧ A , (1.2) 4π M 3 with k ∈ Z. Throughout this article, we study SU (2) gauge group. Due to that the critical points of S(A) are the flat connection on M , it is expected [37, 8, 7] that in the large k limit we have 2πi 1 3 Xp Zk (M ) ∼ e− 4 πi Tα e2πi(k+2) CS(Aα ) e− 4 Iα . (1.3) 2 α Here a sum of α denotes a gauge equivalent class of flat connections, and CS(Aα ), Tα , and Iα denote the classical Chern–Simons invariant, the Reidemeister torsion, and the spectral flow, respectively. Asymptotic behavior (1.3) has been computed exactly by use of explicit form of the Witten–Reshetikhin–Turaev (WRT) invariants [32] for Seifert manifolds (see, e.g., [34, 25]). Based on these results, Lawrence and Zagier [26] found a remarkable structure of quantum invariants. It was shown that the WRT invariant for the Poincar´e homology sphere is a limiting value of the Eichler integral of vectorvalued modular form with weight-3/2. By use of nearly modular properties of the 2000 Mathematics Subject Classification. 57M27, 58J28, 11F23 . c

2010 American Mathematical Society and International Press

1

2

K. HIKAMI

Eichler integral, not only contributions from critical points (1.3) but also perturbative invariants arising from an asymptotic expansion of the trivial connection contribution can be computed. From the viewpoint of modular forms, perturbative invariants are from period function while the Chern–Simons invariant correspond to the Eichler integral at integers. This result was generalized to Seifert homology spheres [13, 14, 17, 18], the Seifert manifolds associated with the ADE singularities [15], and torus knots [20, 11, 38]. It was also realized that the WRT invariant for Seifert manifolds is closely related to the Ramanujan mock theta functions [26, 12]. Results of [15] indicate that the WRT invariant for rational homology sphere M is decomposed as X (λ (`,`)) τN (M ) = e2πiλM (`,`)N τN M (M ), (1.4) `∈Tors H1 (M ;Z) (λ

(`,`))

where λM is the linking pairing, and the decomposed WRT invariant τN M (M ) is a limiting value of certain q-series in τ & N1 . In case that M is the spherical Seifert manifold, each decomposed WRT invariant was identified with the Eichler integral, and the perturbative invariants can also be decomposed for each value of the linking pairings. Purpose of this article is to study the WRT invariants for Seifert manifolds associated with the Arnold unimodal singularities [2]. We shall show that the WRT invariants can be written in terms of the Eichler integrals, and that the decomposition (1.4) is fulfilled. One sees that the Arnold strange duality can be seen as λM = −λM ∗ from the viewpoint of (1.4) as studied in [19]. This article is organized as follows. In Section 2 we introduce the Seifert manifolds as a spherical neighborhood of isolated singularities, and study their linking pairings. Here we pay attention to ADE and unimodal singularities. In Section 3 we define the Eichler integral of certain weight-3/2 vector modular form, and recall that the WRT invariants for E7 and E8 can be written as a linear combination of these Eichler integrals. We briefly study their unified WRT invariants [10]. In Section 4 we show results on the WRT invariants for Seifert manifolds associated with the unimodal singularities. We study asymptotic expansions by use of the Eichler integrals. Given are the Chern–Simons invariants and generating functions of perturbative invariants. 2. Singularities and Seifert Manifolds 2.1. Seifert Manifolds Associated with Singularities. We set f (x, y, z) as the weighted homogeneous polynomials of weight (d1 , d2 , d3 ) and degree d. We have a natural C∗ -action on V as
f td1 x, td2 y, td3 z = td f (x, y, z), (2.1) for t ∈ C∗ . We assume that the variety V = {f (x, y, z) = 0} ,

(2.2)

has an isolated singularity at the origin, and we introduce a closed oriented 3manifold M as a spherical neighborhood of the singularity, M = V ∩ S5.

(2.3)

DECOMPOSITION OF WRT INVARIANT

3

Here S 5 is a sufficiently small sphere centered at the origin. With respect to the S 1 -action induced from the C∗ -action (2.1) on V , M is the Seifert manifold. The Seifert invariant (g; b; (p1 , q1 ), (p2 , q2 ), (p3 , q3 )) of M , where g is a genus of orientable base, and (pj , qj ) are coprime integers, is fixed explicitly from the weighted polynomial (2.1) [31]. In the following, we study the case of g = 0. The Seifert manifold M = M (0; b; (p1 , q1 ), (p2 , q2 ), (p3 , q3 )) has a surgery description depicted in Fig. 1. The fundamental group of M has a presentation [36] + * h is center, pi −qi for i = 1, 2, 3, . (2.4) π1 (M ) = x1 , x2 , x3 , h xi = h x1 x2 x3 = hb −b

p1 /q1

p3 /q3

p2 /q2

Figure 1. Depicted is a surgery description of the Seifert manifold M (0; b; (p1 , q1 ), (p2 , q2 ), (p3 , q3 )). 2.2. Classifications of Singularities. Classifications of the isolated singularities have been widely studied (see [2] and references therein). Amongst others we pay attention to two classes of the singularities hereafter; the ADE singularities (Table 1) and the Arnold 14 unimodal singularities (Table 2). • the ADE singularities When the weighted homogeneous polynomial f (x, y, z) is of the ADE singularities, the Seifert invariant, b and (pj , qj ), is given in Table 1 [31]. We have p11 + p12 + p13 > 1, and the ADE singularities are the quotient singularities associated with spherical triangle ∆(p1 , p2 , p3 ) with angles π/p1 , π/p2 , and π/p3 . Correspondingly the Seifert manifold (2.3) is SU (2)/Γ where Γ is a discrete subgroup of SU(2). • the Arnold 14 unimodal singularities The weighted homogeneous polynomial f (x, y, z) given in Table 2 has the exceptional unimodal singularity at the origin. In that Table, the Seifert invariant is also given. We have the hyperbolic triangle ∆(p1 , p2 , p3 ) satisfying p11 + p12 + p13 < 1. All Seifert manifolds associated with these singularities are of M (g = 0; b = −1; (p1 , 1), (p2 , 1), (p3 , 1)). It is known that, among the Arnold 14 unimodal singularities, there is a strange duality between the Dolgachev number and the Gabrielov number of singularity X; there exists a unimodal singularity X ∗ whose Gabrielov number (b∗1 , b∗2 , b∗3 ) coincides with the Dolgachev number (p1 , p2 , p3 ) of X. This duality was interpreted from various viewpoints, such as the weight system [35], the polar duality [24], and the mirror symmetry [33].

D4K+3

8K

8K + 2

(3, 4, 6) (4, 6, 9) (6, 10, 15)

x3 y + y 3 + z 2

x5 + y 3 + z 2

E7

E8

(2, 3, 5)

(2, 3, 4)

(2, 3, 3)

(2, 2, 4K + 1)

(2, 2, 4K − 1)

(2, 2, 4K, 1)

(2, 2, 4K − 2)

(p1 , p2 , p3 )

Table 1. ADE singularities

30

18

12

(2, 4 K + 1, 4 K + 2) 8 K + 4

x +y +z

2

(2, 4 K − 1, 4 K)

(2, 4 K, 4 K + 1)

E6

3

2

2

d

(2, 4 K − 2, 4 K − 1) 8 K − 2

(d1 , d2 , d3 )

2

x

4

+ xy + z

4K+2

+ xy + z

x

D4K+1

2

2

4K

x

+ xy + z

D4K+2

2

x4K−1 + x y 2 + z 2

4K+1

D4K

f (x, y, z)

0

Z2

Z3

Z4

Z4

Z2 ⊕ Z2

Z2 ⊕ Z2

H1 (M ; Z) 0

1 2

2






1 4 2 3 1 2

∅




3 4

⊕

1 2

0


1



λM

1 2




CS

1 − 48 , − 25 48

 1 49 − 120 , − 120



 1 − 24

for 0 ≤ m ≤ 2K − 1

for  0 ≤ m ≤ 2Kff− 2 (2m+1)2 − 16K+4

for  0 ≤ m ≤ 2Kff− 1 (2m+1)2 − 16K−4

for  0 ≤ m ≤ 2Kff− 2 (2m+1)2 − 16K

 ff (2m+1)2 − 16K−8

4 K. HIKAMI

(d1 , d2 , d3 ) (6, 14, 21) (4, 10, 15) (3, 8, 12) (6, 8, 15) (4, 6, 11) (3, 5, 9) (4, 5, 10) (3, 4, 8) (6, 8, 9) (4, 6, 7) (3, 5, 6) (4, 5, 6) (3, 4, 5) (3, 4, 4)

f (x, y, z)

x7 + y 3 + z 2

x5 y + y 3 + z 2

x8 + y 3 + z 2

x5 + x y 3 + z 2

x4 y + x y 3 + z 2

x6 + x y 3 + z 2

x5 + y 4 + z 2

x4 y + y 4 + z 2

x4 + y 3 + x z 2

x3 y + y 3 + x z 2

x5 + y 3 + x z 2

x4 + y 2 z + x z 2

x3 y + y 2 z + x z 2

x4 + y 3 + z 3

E12

E13

E14

Z11

Z12

Z13

W12

W13

Q10

Q11

Q12

S11

S12

U12

(4, 4, 4)

(3, 4, 5)

(2, 5, 6)

(3, 3, 6)

(2, 4, 7)

(2, 3, 9)

(3, 4, 4)

(2, 5, 5)

(3, 3, 5)

(2, 4, 6)

(2, 3, 8)

(3, 3, 4)

(2, 4, 5)

(2, 3, 7)

(p1 , p2 , p3 )

Dolgachev

Z4 ⊕ Z4

Z13

Z8

Z3 ⊕ Z3

Z2 ⊕ Z3

Z3

Z8

Z5

Z2 ⊕ Z3

Z2 ⊕ Z2

Z2

Z3

Z2

0

H1 (M ; Z)

(4, 4, 4)

(3, 4, 5)

(3, 4, 4)

(3, 3, 6)

(3, 3, 5)

(3, 3, 4)

(2, 5, 6)

(2, 5, 5)

(2, 4, 7)

(2, 4, 6)

(2, 4, 5)

(2, 3, 9)

(2, 3, 8)

(2, 3, 7)

(b1 , b2 , b3 )

Gabrielov

Table 2. Unimodal singularities

12

13

16

15

18

24

16

20

18

22

30

24

30

42

d




1 3




1 2




1 3

2 1 4

2 3

1 3

2 3

1 2




1 4 1 2

5 13

8

⊕
3

⊕




5 8

5

⊕
3

⊕

1







1 2

1 2




1 2




2 3

2

∅
1

λM















73 96 , 96 3 11 16 , 16 2 19 15 , 30









9 1 16 , 4





7 19 3 , , n10 133037 10o 60 , 60 , 133 157 240 , 240

, 25 , − 5 n72 1972 75 24o 112 , 112 , 29 27 , 112 n− 112 o 5 17 24 , 24 , 1 2 6,3

13 6 , 24

1

9 8 , 40

5











1 25 12 , 48

9 49 80 , 80

 25





25 47 168 , − 168

 49



CS

DECOMPOSITION OF WRT INVARIANT 5

6

K. HIKAMI

2.3. Linking Pairing. For closed oriented 3-manifolds M , we have the linking pairing λM , λM : Tors H1 (M ; Z) ⊗ Tors H1 (M ; Z) → Q/Z, where Tors H1 (M ; Z) denotes the torsion part of H1 (M ; Z). For a, a0 ∈ Tors H1 (M ; Z), we define λM as follows. We choose s ∈ Z6=0 such that s a = 0 ∈ H1 (M ; Z), and set a 2-chain B which is bounded as ∂B = s a. The linking pairing is given from the intersection number as # (B · a0 ) mod Z. (2.5) s See [23] for the classification of linking pairing on 3-manifolds. For M = M (0; b; (p1 , q1 ), (p2 , q2 ), (p3 , q3 )), abelianization of π1 (M ) (2.4) gives   pi xi + qi h for i = 1, 2, 3, ∼ H1 (M ; Z) = spanZ {x1 , x2 , x3 , h} / x1 + x2 + x3 + h 4 4 ∼ = Z /AM Z , λM (a, a0 ) =

where AM is the linking matrix defined by  p1  p2 AM =   p3 1 1 1

 q1 q2  . q3  −b

(2.6)

We have |H1 (M ; Z)| = |det AM | = |p1 p2 p3 EM |, where EM is the Euler number EM = −b −

3 X qj . p j=1 j

(2.7)

We assume det AM 6= 0, and the torsion part can be identified with Z4 /AM Z4 . Thus for ` = (`1 , `2 , `3 , `4 )T ∈ Z4 representing a generator `1 x1 +`2 x2 +`3 x3 +`4 h, we may set m = (m1 , m2 , m3 , m4 )T ∈ Z4 such that s ` = AM m. As mi is the number of meridian disks of the i-th solid torus, the intersection number in (2.5) gives 1 −1 λM (`, `0 ) = mT diag (q1 , q2 , q3 , 1) `0 = `T (A0M ) `0 , s where A0M is given by  p1  1 q1 p 2  1 q2 . A0M =  p3  1 q3 1 1 1 −b −1

By use of an integral unimodular matrix P, the matrix (A0M ) diagonalized as −1

PT (A0M )

P = ΛM ⊕ (±1) ⊕ · · · ⊕ (±1)

is block-

mod Z.

0

As a result, with a, a ∈ Tors H1 (M ; Z) the linking pairing is written as λM (a, a0 ) = aT ΛM a0 .

(2.8)

See Tables 1 and 2 for explicit forms of λM for Seifert manifolds related to the ADE and unimodal singularities.

DECOMPOSITION OF WRT INVARIANT

7

3. WRT Invariant for Seifert Manifolds 3.1. SU(2) WRT Invariant. Mathematically rigorous definition of the Witten invariant (1.1) was given by Reshetikhin and Turaev [32]. The SU(2) Reshetikhin–Turaev invariant τN (M ) for 3-manifold M is related to the Witten invariant (1.1) with SU(2) gauge group by Zk (M ) =

τk+2 (M ) , τk+2 (S 2 × S 1 )

(3.1)


q 1 where we have τN S 2 × S 1 = N2 sin(π/N ) . Explicitly when M is constructed by rational surgery pj /qj on the j-th component of the n-component link L, we have πi N −2 N

τN (M ) = e 4

Pn

(

j=1

Φ(U(pj ,qj ) )−3 sign(L)) N −1 X

×

JL (k1 , . . . , kn )

n h Y j=1

k1 ,...,kn =1

i ρ(U(pj ,qj ) )

. (3.2) kj ,1

Here JL (k1 , . . . , kn ) is the colored Jones polynomial with color-kj on the j-th component of L, and L is the linking matrix of L. The Rademacher function Φ(U(p,q) ) for U(p,q) = ( pq rs ) ∈ SL(2; Z) is   ( p+s p r q − 12 s(q, p), for q 6= 0, Φ = q s 0, for q = 0. The function s(b, a) is the Dedekind sum s(b, a) = sign(a)

|a|−1

X

k=1

( ((x)) =

k   k b  , a a

x − bxc − 21 , if x 6∈ Z, 0, if x ∈ Z.

ρ(U(p,q) ) is given by h i ρ(U(p,q) ) a,b

2 πi sign(q) − πi4 Φ(U (p,q) )+ 2N q sb = −i p e 2 N |q|  πi  X 2 πi πi × e 2N q pγ e N q γb − e− N q γb . (3.3)

γ mod 2N q γ=a mod 2N

The WRT invariant for Seifert manifolds M (0; b; (p1 , q1 ), (p2 , q2 ), (p3 , q3 )) is computable from (3.2) via Fig. 1. 3.2. Unified WRT Invariants. For integral homology sphere M , Habiro constructed the unified WRT invariant Iq (M ) [10]. The invariant Iq (M ) takes values in lim Z[q]/(q)n , and is written as an infinite sum ←− n

Iq (M ) =

∞ X n=0

fn (q) · (q)n ,

(3.4)

8

K. HIKAMI

with fn (q) ∈ Z[q]. Here and hereafter we use a standard notation of q-analysis, (x)n = (x; q)n = (1 − x)(1 − x q) · · · (1 − x q n−1 ). The WRT invariant is given by evaluation at the root of unity, τN (M ) = evq=e2πi/N [Iq (M )] ,

(3.5)

in which the infinite series (3.4) terminates at a finite sum. The expression (3.4) follows from a fact that the colored Jones polynomial for knot K, which is normalized to be Junknot (N ) = 1 and JK (1) = 1, has a cyclotomic expansion [9] ∞ X JK (N ) = CK (n) · (q 1+N )n (q 1−N )n , (3.6) n=0

where CK (n) ∈ Z[q, q

−1

]. For example, the trefoil has ∞ X Jtrefoil (N ) = q n (q 1+N )n (q 1−N )n .

(3.7)

n=0

We note that an inverse of (3.6) is n X 1 (1 − q k+1 ) (1 − q 2k+2 ) JK (k + 1). CK (n) = q n (−1)k q 2 k(k−1) (q)n−k (q)n+k+2 k=0

A case of the rational homology sphere was studied in [3, 4] by use of the surgery formula [5]. For instance, when Ms is s-surgery of knot K whose colored Jones polynomial is (3.6), we have ∞ X n(n+3) (1 − q) Iq (M+1 ) = (−1)n q − 2 CK (n) (q n+1 )n+1 , (3.8) n=0

(1 − q) Iq (M−1 ) =

∞ X

CK (n) (q n+1 )n+1 .

(3.9)

n=0

We have τN (M±2 ) = 0 for odd N , and τN (M±2 ) for even N follows from √ 1X 1 1 n (1 − q) Iq (M2 ) = 2 q 4 (−1)n q − 2 CK (n) (q 2 ; −q 2 )2n+1 ,

(3.10)

n≥0

(1 − q) Iq (M−2 ) =

√

1

2q4

∞ X

1

1

CK (n) (q 2 ; −q 2 )2n+1 .

(3.11)

n=0

3.3. WRT Invariants and Modular Form. We use q = e2πiτ where τ is in the upper half-plane, τ ∈ H. For P ∈ Z>0 and a ∈ Z, we define n2 1X (a) (a) n ψ2P (n) q 4P , (3.12) ΨP (τ ) = 2 n∈Z

where

(a) ψ2P (n)

is an odd function of period 2 P , ( ±1, for n ≡ ±a (a) ψ2P (n) = 0, otherwise.

mod 2 P ,

(a)

(3.13)

The q-series ΨP (τ ) is a vector-valued modular form with weight 3/2 satisfying (a)

ΨN

3

[η]

(γ(τ )) =

N −1 X b=1

(b)

[ρ(γ)]a,b

ΨN

3

[η]

(τ ),

DECOMPOSITION OF WRT INVARIANT

9

where γ ∈ SL(2; Z), and ρ is given in (3.3). η denotes the Dedekind η-function, 1 η(τ ) = q 24 (q)∞ . The Eichler integral [26, 38] is defined by e (a) (τ ) = Ψ P

∞ X

n2

(a)

ψ2P (n) q 4P ,

(3.14)

n=0 (a)

which can be regarded as a half-integration of ΨP (τ ) with respect to τ . A limiting value of the Eichler integral in τ & Q can be computed, e.g.,   2P N X k2 k (a) e (a) (1/N ) = − 2P N πi B Ψ , (3.15) ψ (k) e 1 P 2P 2P N k=0   a2 πiN e (a) (N ) = 1 − a e 2P Ψ , (3.16) P P xt

where N ∈ Z, and Bk (x) denotes the k-th Bernoulli polynomial defined by ette−1 = P∞ Bk (x) k k=0 k! t . These two limiting values of the Eichler integral (3.14) are related to each other by nearly modular property as [11] r P −1 r   N X 2 ab (a) e (b) (−N ) e ΨP (1/N ) + sin π Ψ P i P P b=1    k ∞ L −2 k, ψ (a) X 2P πi ' . (3.17) k! 2P N k=0   (a) Here L k, ψ2P denotes the Dirichlet L-function, and at negative integers it is written in terms of the Bernoulli polynomial as k 2P  n    (2 P ) X (a) (a) ψ2P (n) Bk+1 . L −k, ψ2P = − k + 1 n=1 2P

The generating function of the L-function in (3.17) is   ∞ L −2 k, ψ (a) X 2P sinh((P − a) z) = z 2k , sinh(P z) (2 k)!

(3.18)

(3.19)

k=0

where 0 < a < P . Note that an infinite series in (3.17) follows from a period integral (a) of ΨP (τ ), and that such transformation formula is reminiscent of the Ramanujan mock theta function. Since the work of Lawrence–Zagier [26], the WRT invariants for certain Seifert manifolds are known to be related to limiting values of the Eichler integral (3.15). We show an example. We take M to be the Poincar´e homology sphere, E8 in Table 1. The WRT invariant is computed to be [26]   2πi πi 121 1 e (1)+(11)+(19)+(29) e 60N πi e N − 1 · τN (E8 ) = e 60N − Ψ (1/N ). (3.20) 2 30 e ka (a)+kb (b)+··· (•) = ka Ψ e (a) (•) + kb Ψ e (b) (•) + Here and hereafter we use a notation, Ψ P P P · · · , for brevity. By use of (3.17), we obtain an asymptotic expansion r        2πi  121 1 49 N 2 π 2π πi − 60 πiN − 60 πiN 60N N √ e e − 1 ·τN (E8 ) ' sin e + sin e i 5 5 5

10

K. HIKAMI

+e

πi 60N

  (1)+(11)+(19)+(29)  k ∞ πi 1 X L −2 k, ψ60 . (3.21) − 2 k! 60 N k=0

πi

One finds that an infinite series with the term e 60N generates perturbative invariants [28, 27]. Note that the L-function is given by   (1)+(11)+(19)+(29) ∞ cosh(5 z) cosh(9 z) X L −2 k, ψ60 2 = z 2k . cosh(15 z) (2 k)! k=0

The unified WRT invariant Iq (E8 ) is as follows. Recalling that the Poincar´e homology sphere is (−1)-surgery of trefoil, we get from (3.7) and (3.9) ∞ X

1 + q (1 − q) Iq (E8 ) =

q n (q n )n .

(3.22)

n=0

Iq (E8 ) coincides with τN (E8 ) (3.20) at the N -th root of unity. We see that the right hand side in (3.22) is 2 − χ0 (1/q) [12], where χ0 (1/q) means an extended value outside the unit circle of the fifth order mock theta function χ0 (q) defined by χ0 (q) =

∞ X

qn

n=0

(q n+1 )n

.

Note that the unified WRT invariant generates the L-function in (3.21) when we set q = e−t and take a limit t & 0,    k ∞ L −2 k, ψ (1)+(11)+(19)+(29) ∞ X X 60 1 −t 1 . q n (q n )n |q=e−t = e 120 t 2 k! 120 n=0 k=0

This type of identities has been studied in [38, 16, 21, 1, 29]. 3.4. Decomposition of the WRT Invariants. It is conjectured [15] that the WRT invariant for rational homology sphere M is decomposed as (1.4). In (0) case that M is a homology sphere, we have λM = ∅, and τN (M ) is an evaluation of the unified WRT invariant Iq (M ) at the N -th root of unity [9]. Explicit (0) forms of τN (M ) were given in [13,
14, 17] for some Seifert homology spheres. When H1 (M ; Z) = Z2 and λM = 21 , we have τN (M ) = 0 for odd N , and the (1)

(0)

decomposition (1.4) is read as τN (M ) = τN2 (M ). Corresponding to (1.4) the perturbative invariants arising from an asymptotic expansion of the trivial connection contribution can also be decomposed in terms of the linking pairing. The first verification of (1.4) based on the explicit form of the SU (2) WRT invariant is for the lens space L(p, q), which is constructed by a p/q-surgery on trivial knot. The WRT invariant (3.2) is computed as [22]  2πi  6s(q,p)+1 e− N πi e N − 1 · τN (L(p, q)) p X

n ε 2πi q n+ε 1+2N πi Np . (3.23) √ e p p ε=±1 n=1 q
As we have H1 (L(p, q); Z) = Zp , and a linking pairing is λL(p,q) = p , the conjecture (1.4) is true for the lens space.

=

q

e2πi p n

2

N

X

DECOMPOSITION OF WRT INVARIANT

11

Our previous results [15] on the Seifert manifolds associated with the ADE singularities in Table 1 also support (1.4). Therein the decomposed WRT invariants were explicitly constructed as a linear combination of the
Eichler integrals (3.15). For instance, in case of E7 in Table 1 we have λE7 = 12 , and  2πi  37 e 24N πi e N − 1 τN (E7 ) √ 
2  πi e (1)+(5)+(7)+(11) (1/N ) , (3.24) = 1 + (−1)N 2 e 24N − Ψ 12 4 (0)

(1)

which is a sum of τN (E7 ) and τN2 (E7 ). Nearly modularity (3.17) gives an asymptotic expansion as r  2πi   1 + (−1)N πi 23 37 N 1  − 1 πiN πi √ √ e N − 1 · τN (E7 ) ' e 24 + e 24 πiN + e 24N e 24N i 2 2   √  k ∞ L −2 k, ψ (1)+(3)+(7)+(11) X
24 2 πi − 1 + (−1)N . (3.25) 4 k! 24 N k=0

The first term is a contribution (1.3) from the critical points in the Chern–Simons path integral. The second term and an infinite series are the perturbative invariants, and a generating function of the L-function is   (1)+(5)+(7)+(11) ∞ cosh(3 z) cosh(2 z) X L −2 k, ψ24 = z 2k . 2 cosh(6 z) (2 k)! k=0

In case that N is even, τN (E7 ) is an evaluation of the unified invariant at the root of unity, which can be computed from (3.7) and (3.11) as [6] (1 − q) Iq (E7 ) =

√

2q

1 4

∞ X

1

1

q n (q 2 ; −q 2 )2n+1 .

(3.26)

n=0

We note that Iq (M ) generates the L-function in (3.25) as    k ∞ L −2 k, ψ (1)+(5)+(7)+(11) ∞ X X 24 1 1 1 49 t −t t n 2 2 48 e + q (q ; −q )2n+1 −t = e . 2 k! 48 q=e n=0 k=0

4. WRT Invariants Associated with the Unimodal Singularities We shall give an explicit form of the WRT invariant for the Seifert manifolds associated with the Arnold unimodal singularities, and show that they are written as a linear combination of the Eichler integrals. We omit details of computations, as they are tedious but analogous to [26, 15]. These results support the decomposition conjecture (1.4), and we list in Table 2 the classical Chern–Simons invariant given from an asymptotic expansion (1.3). It should be noted that the Arnold strange duality can also be interpreted from the linking pairing [19], λM = −λM ∗ , ∗

(4.1)

where M and M are dual in Arnold’s sense. In the following, we show results separately for self-dual cases (E12 , Z12 , W12 , Q12 , S12 , U12 ) and other cases (E13 ↔ Z11 , Q10 ↔ E14 , Q11 ↔ Z13 , S11 ↔ W13 ). Also given are generating functions for linking-pairing decomposed perturbative invariants in terms of hyperbolic functions.

12

K. HIKAMI

4.1. Self-Dual Cases. 4.1.1. E12 . This is the Brieskorn homology sphere, whose WRT invariant was computed in [13]. We have (orientation is opposite to [13])  2πi  1 1 e (1)−(13)−(29)+(41) e− 84N πi e N − 1 · τN (E12 ) = Ψ (−1/N ). (4.2) 2 42 By use of (3.17), we can compute the asymptotic expansion in N → ∞ as  2πi  1 e− 84N πi e N − 1 τN (E12 )       √ 25 2π 3π 2 − 47 πiN πiN 84 84 − sin e − sin e ' − iN √ 7 7 7   ∞ L −2 k, ψ (1)−(13)−(29)+(41)  π k X 84 1 , − 2 k! 84 N i k=0

where the perturbative invariants are generated from   ∞ L −2 k, ψ (1)−(13)−(29)+(41) X 84 2 sinh(6 z) sinh(14 z) = z 2k . cosh(21 z) (2 k)! k=0

The unified invariant is given from (3.7) and (3.8) as [18] (q − 1) Iq (E12 ) =

∞ X

(−1)n q −

n(n+1) 2

(q n+1 )n+1 ,

(4.3)

n=0

which also generates the L-function ∞ X

1 (−1)n q − 2 n(n+1) (q n+1 )n+1

q=e−t

n=0

=

1 − 1 t e 168 2

   k ∞ L −2 k, ψ (1)−(13)−(29)+(41) X 84 t . k! 168 n=0

4.1.2. Z12 . The WRT invariant is given by  2πi 
(1)−(5)−(7)+(11) 1 1 e e− 24N πi e N − 1 · τN (Z12 ) = 1 + (−1)N Ψ (−1/N ). 12 2

(4.4)

An asymptotic expansion follows by use of (3.17) as  2πi   √ 1 11 1  3 πiN e− 24N πi e N − 1 · τN (Z12 ) ' − i N · √ e8 + e 8 πiN 3   ∞ L −2 k, ψ (1)−(5)−(7)+(11)  π k X
24 1 + 1 + (−1)N , 2 k! 24 N i k=0

where the perturbative invariants are generated by   ∞ L −2 k, ψ (1)−(5)−(7)+(11) X 24 2 sinh(3 z) sinh(2 z) = z 2k . cosh(6 z) (2 k)! k=0

DECOMPOSITION OF WRT INVARIANT

13

4.1.3. W12 . We have  2πi  1 1 n e (1)−2(5)+(9) e− 20N πi e N − 1 · τN (W12 ) = √ Ψ (−1/N ) 10 2 5 o 2π 4π 4 e (1)−2 cos 5 (5)+(9) (−1/N ) + 2 e− 54 πiN Ψ e (1)−2 cos 5 (5)+(9) (−1/N ) . + 2 e 5 πiN Ψ 10

10

(4.5) (± 2 )

(0)

Here we have three decomposed WRT invariants, τN (W12 ) and τN 5 (W12 ). An asymptotic expansion in N → ∞ is written as  2πi  1 e− 20N πi e N − 1 · τN (W12 )       π √ √ 5 9 2 2π 4π πiN πiN 4 20 ' − iN · 4 sin sin e e − 5 cos 5 5 5 5 ∞  1 X 1  π k n  (1)−2(5)+(9) L −2 k, ψ20 + √ 2 5 k=0 k! 20 i N    o 4 4 (1)−2 cos 2π (1)−2 cos 4π 5 (5)+(9) 5 (5)+(9) +2 e 5 πiN L −2 k, ψ20 + 2 e− 5 πiN L −2 k, ψ20 . (0)

(± 2 )

The perturbative invariants from decomposed invariants, τN and τN 5 , are generated from (a = 0, 1, 2)   (1)−2 cos 2aπ

5 (5)+(9) ∞ L −2 k, ψ a a X 20 sinh 2z + 5 πi sinh 2z − 5 πi 2 = z 2k . cosh(5 z) (2 k)! k=0

4.1.4. Q12 . We get  2πi  1 1 n e 5(1)+(3)−4(5) e− 24N πi e N − 1 · τN (Q12 ) = Ψ6 (−1/N ) 6 o 2 e (1)−(3)+(5) (−1/N ) + e− 32 πiN Ψ e 2(1)+(3)+2(5) (−1/N ) , (4.6) +2 e 3 πiN Ψ 6 6 which induces an asymptotic expansion as  2πi  1 e− 24N πi e N − 1 · τN (Q12 )  √ √ 1 √ 4 17 1  1 πiN ' − iN · √ 2 e 12 + 4 e 12 πiN + 2 3 e 3 πiN + 3 e 3 πiN 4 3 ∞  1 X 1  π k n  5(1)+(3)−4(5) + L −2 k, ψ12 6 k! 12 i N k=0    o 2 2 (1)−(3)+(5) 2(1)+(3)+2(5) +2 e 3 πiN L −2 k, ψ12 + e− 3 πiN L −2 k, ψ12 . Generating functions of the decomposed perturbative invariants are     X ∞ L −2 k, ψ 5(1)+(3)−4(5) 12 sinh(4 z) sinh(2 z) 2 cosh(z) 5 −4 = z 2k , sinh(6 z) sinh(6 z) (2 k)! k=0   ∞ L −2 k, ψ (1)−(3)+(5) X 12 1 = z 2k , 2 cosh(z) (2 k)! k=0

14

K. HIKAMI

  2(1)+(3)+2(5) ∞ 4 cosh(2 z) + 1 X L −2 k, ψ12 = z 2k . 2 cosh(3 z) (2 k)! k=0

4.1.5. U12 . We have  2πi  1 e− 16N πi e N − 1 · τN (U12 ) o 1 n e (1)−3(3) e (1)+(3) (−1/N ) , (4.7) = Ψ4 (−1/N ) + 3 (−1)N Ψ 4 2 and an asymptotic expansion in N → ∞ is given by  2πi   √ 1 1 1 √ 9 πiN e− 16N πi e N − 1 · τN (U12 ) ' − i N · √ 2 e8 + e 2 πiN 2 ∞   o 1 X 1  π k n  (1)−3(3) (1)+(3) + L −2 k, ψ8 + 3 (−1)N L −2 k, ψ8 . 2 k! 8 N i k=0

Generating functions of the decomposed perturbative invariants are   ∞ L −2 k, ψ (1)−3(3) X 8 sinh3 (z) = z 2k , 4 sinh(4 z) (2 k)! k=0   ∞ L −2 k, ψ (1)+(3) X 8 cosh(z) = z 2k . cosh(3 z) (2 k)! k=0

4.1.6. S12 . The WRT invariant is written as e

2πi 287 N 240



e

×

780  1 X e (c) Ψ780 (−1/N ) − 1 τN (S12 ) = √ 2 13 c=1  ! 6 X 10 2ac (13)−(37)−(43)−(53) (2a−1)2 πiN 13 1+2 e cos π ψ120 (c), (4.8) 13 a=1

2πi N

which we do not find a simplified expression. An asymptotic expansion is given by e

2πi 287 N 240

   √ 13 1 √ 2π − 1 τN (S12 ) ' − i N · √ 2 sin e 30 πiN 5 5 !    π  37  π  133 √ 157 2π + sin e 120 πiN + 2 sin e 30 πiN + sin e 120 πiN 5 5 5



e

2πi N

∞

780

k X 1 X π (13)−(37)−(43)−(53) + √ ψ120 (c) 2 13 k=0 1560 i N c=1  ! 6 (c) X 2 10 2ac L(−2 k, ψ1560 ) (2a−1) πiN × 1+2 e 13 cos π . 13 k! a=1 4.2. Non Self-Dual Cases.

DECOMPOSITION OF WRT INVARIANT

15

4.2.1. E13 . We have  2πi 
(1)−(9)−(11)+(19) 21 1 e e− 40N πi e N − 1 ·τN (E13 ) = √ 1 + (−1)N Ψ (−1/N ), (4.9) 20 2 2 which gives an asymptotic expansion in N → ∞ as r    2πi   √ 49 21 9 2 2π πi − 40N e N − 1 · τN (E13 ) ' − i N · e sin e 40 πiN + e 40 πiN 5 5   (1)−(9)−(11)+(19) ∞ L −2 k, ψ  π k X
40 1 + √ 1 + (−1)N . k! 40 N i 2 2 k=0

The perturbative invariants follow from 2 sinh(5 z) sinh(4 z) = cosh(10 z)

  ∞ L −2 k, ψ (1)−(9)−(11)+(19) X 40 (2 k)!

k=0

z 2k .

4.2.2. Z11 . We get  2πi 
(1)−(7)−(17)+(23) 1 πi e (−1/N ), (4.10) e e N − 1 ·τN (Z11 ) = √ 1 + (−1)N Ψ 24 2 2 which in large-N limit behaves like    2πi   √ 23 73 3 π  25 πiN 1 e 48N πi e N − 1 · τN (Z11 ) ' − i N · √ sin e 48 + e 48 πiN 8 2   (1)−(7)−(17)+(23) ∞ L −2 k, ψ  π k X
48 1 + √ 1 + (−1)N . k! 48 N i 2 2 23 48N

k=0

Here a generating function of L-functions is   ∞ L −2 k, ψ (1)−(7)−(17)+(23) X 48 2 sinh(3 z) sinh(8 z) = z 2k . cosh(12 z) (2 k)! k=0

The unified WRT invariant is computed from (3.7) and (3.11) [6] as ∞ √ 1 X n 1 1 (−1)n q 2 (q 2 ; −q 2 )2n+1 , (q − 1) Iq (Z11 ) = 2 q 4

(4.11)

n=0

which also generates the L-function in t & 0 ∞ X n=0

n 1 1 (−1)n q 2 (q 2 ; −q 2 )2n+1

q=e−t

=

1 47 t e 96 2

   k ∞ L −2 k, ψ (1)−(7)−(17)+(23) X 48 t . k! 96

k=0

4.2.3. Q10 . We have  2πi  35 e 36N πi e N − 1 · τN (Q10 ) o 2 1 n e (1)−(5)−(13)+(17) e 2(1)+(5)+(13)+2(17) (−1/N ) . (−1/N ) + e− 3 πiN Ψ = √ Ψ18 18 2 3 (4.12) An asymptotic expansion is then given by  2πi  35 e 36N πi e N − 1 · τN (Q10 )

16

K. HIKAMI

√ 2 ' − iN · 3

 sin

2π 9

 e

49 36 πiN

 + sin

4π 9

√

 e

25 36 πiN

3 − 5 πiN + e 12 2

!

∞ 1 X 1  π k + √ 2 3 k=0 k! 36 N i n    o 2 (1)−(5)−(13)+(17) 2(1)+(5)+(13)+2(17) × L −2 k, ψ36 + e− 3 πiN L −2 k, ψ36 .

The decomposed perturbative invariants are generated as   ∞ L −2 k, ψ (1)−(5)−(13)+(17) X 36 2 sinh(6 z) sinh(2 z) = z 2k , cosh (9 z) (2 k)! k=0   ∞ L −2 k, ψ 2(1)+(5)+(13)+2(17) X 36 2 cosh(8 z) + cosh(4 z) = z 2k . cosh(9 z) (2 k)! k=0

(0)

Based on the decomposed WRT invariant τN (Q10 ) constructed from surgery on trefoil, we have a hypergeometric generating function for perturbative invariants as ∞ n X X (q)2n+1 n k+1 61 k(k+3) q (−1) q (q)n−k (q)n+k+1 n=0 k=−n−1 q=e−t    k ∞ L −2 k, ψ (1)−(5)−(13)+(17) X 36 1 71 t t 72 = e . (4.13) 2 k! 72 k=0

4.2.4. E14 . We obtain  2πi  25 e− 24N πi e N − 1 · τN (E14 ) o 2 1 n e (1)−(7)−2(9) e (1)−(7)+(9) (−1/N ) . (4.14) = √ Ψ12 (−1/N ) + 2 e 3 πiN Ψ 12 2 3 An asymptotic expansion is given by  2πi   √ 25 25 1 √ 1 πiN e− 24N πi e N − 1 · τN (E14 ) ' − i N · 2 e6 + e 24 πiN 2 ∞   o 2 1 X 1  π k n  (1)−(7)−2(9) (1)−(7)+(9) + √ L −2 k, ψ24 + 2 e 3 πiN L −2 k, ψ24 , k! 24 N i 2 3 k=0

where generating functions of the decomposed perturbative invariants are   (1)−(7)−2(9) ∞ 4 sinh (3 z) sinh2 (4 z) X L −2 k, ψ24 = z 2k , sinh (12 z) (2 k)! k=0   ∞ L −2 k, ψ (1)−(7)+(9) X 24 2 cosh(8 z) + 1 sinh (3 z) = z 2k . sinh (12 z) (2 k)! k=0

DECOMPOSITION OF WRT INVARIANT

17

4.2.5. Q11 . We have  2πi 
25 1 e 56N πi e N − 1 · τN (Q11 ) = √ 1 + (−1)N 2 6 n (3)−(11)−(17)+(25)−(31)+(39)+(45)−(53)+(59)−(67)−(73)+(81) e × Ψ84 (−1/N ) o 2 e 2(3)+(11)+(17)−(25)+(31)+2(39)+2(45)+(53)−(59)+(67)+(73)+2(81) (−1/N ) , + e− 3 πiN Ψ 84 (4.15) which induces an asymptotic expansion as  2πi  25 e 56N πi e N − 1 · τN (Q11 ) r        √ 75 27 2 2 π  19 πiN 3 π  − 29 πiN πiN πiN 56 56 56 56 ' − iN · sin e + sin e +e +e 7 7 7 ∞ 1 + (−1)N X 1  π k √ + k! 168 N i 2 6 k=0 n   (3)−(11)−(17)+(25)−(31)+(39)+(45)−(53)+(59)−(67)−(73)+(81) × L −2 k, ψ168  o 2 2(3)+(11)+(17)−(25)+(31)+2(39)+2(45)+(53)−(59)+(67)+(73)+2(81) + e− 3 πiN L −2 k, ψ168 . Here we have − sinh(7 z) − sinh(21 z) + sinh(35 z) 2 sinh(4 z) cosh(42 z)   (3)−(11)−(17)+(25)−(31)+(39)+(45)−(53)+(59)−(67)−(73)+(81) ∞ L −2 k, ψ X 168 = z 2k , (2 k)! k=0

2 cosh(18 z) + cosh(10 z) 2 sinh(21 z) sinh(4 z) 2 cosh(21 z) + cosh(42 z) cosh(42 z)   ∞ L −2 k, ψ 2(3)+(11)+(17)−(25)+(31)+2(39)+2(45)+(53)−(59)+(67)+(73)+2(81) X 168 = z 2k . (2 k)! k=0

4.2.6. Z13 . We get  2πi 
17 1 e− 30N πi e N − 1 · τN (Z13 ) = √ 1 + (−1)N 2 6 n (2)−(8)−2(12)+2(18)+(22)−(28) e × Ψ30 (−1/N ) 1

(2)−(8)+(12)−(18)+(22)−(28)

e +2 e− 3 πiN Ψ 30

o (−1/N ) . (4.16)

An asymptotic expansion in N → ∞ is computed to be r    2πi   √ 17 19 2 2 π  4 πiN − 30N πi N e e − 1 · τN (Z13 ) ' − i N · sin e 15 + e 15 πiN 5 5 ∞   n   k
X 1 1 π (1)−(4)−2(6)+2(9)+(11)−(14) + √ 1 + (−1)N L −2 k, ψ30 k! 15 N i 2 6 k=0  o 1 (1)−(4)+(6)−(9)+(11)−(14) +2 e− 3 πiN L −2 k, ψ30 ,

18

K. HIKAMI

where the decomposed perturbative invariants are generated from  

∞ L −2 k, ψ (1)−(4)−2(6)+2(9)+(11)−(14) 2 5 3 X 30 sinh 2 z sinh 2 z
= 4 z 2k . 15 (2 k)! sinh 2 z k=0   (1)−(4)+(6)−(9)+(11)−(14)
∞ 3 L −2 k, ψ X 30 sinh 2 z
= z 2k . (2 k)! sinh 25 z k=0

4.2.7. S11 . We have  2πi  13 1 n e (2)−(7)−(8)+(13)−(17)+(22)+(23)−(28) e 30N πi e N − 1 · τN (S11 ) = √ Ψ30 (−1/N ) 2 2 e (2)+(7)−(8)−(13)+(17)+(22)−(23)−(28) (−1/N ) + (−1)N Ψ 30 o 3 e (2)+(8)+(22)+(28) (−1/N ) . (4.17) +2 e− 4 πiN Ψ 30 An asymptotic expansion is given by  2πi  13 e 30N πi e N − 1 · τN (S11 )       √ 7 3 4 π 2 π  19 πiN πiN πiN 5 15 5 ' − iN · √ sin e e + sin +e 5 5 30 ∞  1 X 1  π k n  (2)−(7)−(8)+(13) L −2 k, ψ30 + √ 2 2 k=0 k! 60 N i    o 3 (2)+(7)−(8)−(13) (2)+(8)+(22)+(28) +(−1)N L −2 k, ψ30 + 2 e− 4 πiN L −2 k, ψ60 , where generating functions of each L-function are   ∞ L −2 k, ψ (2)±(7)−(8)∓(13) X 30 cosh(10 z) ± cosh(5 z) 2 sinh(3 z) = z 2k , sinh(15 z) (2 k)! k=0   (2)+(8)+(22)+(28) ∞ 2 cosh(10 z) cosh(3 z) X L −2 k, ψ60 = z 2k . cosh(15 z) (2 k)! k=0

4.2.8. W13 . The WRT invariant is computed as  2πi  7 1 n e (1)−2(4)−(5) e− 12N πi e N − 1 · τN (W13 ) = √ Ψ6 (−1/N ) 2 2 o e (1)+2(4)−(5) (−1/N ) + 2 e 34 πiN Ψ e (1)−(5) (−1/N ) , (4.18) +(−1)N Ψ 6 6 which gives an asymptotic expansion in N → ∞ as  2πi   √ 7 13 1  1 πiN e− 12N πi e N − 1 · τN (W13 ) ' − i N · √ e3 + e 12 πiN 2 ∞   n   X k 1 1 π (1)−2(4)−(5) + √ L −2 k, ψ12 2 2 k=0 k! 12 i N   o  3 (1)+2(4)−(5) (1)−(5) +(−1)N L −2 k, ψ12 + 2 e 4 πiN L −2 k, ψ12 .

DECOMPOSITION OF WRT INVARIANT

19

Generating functions of the decomposed perturbative invariants are   ∞ L −2 k, ψ (1)−2(4)−(5) 2 3 X 12 sinh(2 z) sinh ( 2 z) 4 = z 2k , sinh(6 z) (2 k)! k=0   (1)+2(4)−(5) ∞ 2 3 sinh(2 z) cosh ( 2 z) X L −2 k, ψ12 4 = z 2k , sinh(6 z) (2 k)! k=0   (1)−(5) ∞ sinh(2 z) sinh(3 z) X L −2 k, ψ12 = z 2k . 2 sinh(6 z) (2 k)! k=0

5. Concluding Remarks We have shown that the WRT invariant for M can be decomposed as (1.4) by use of the linking pairing when M is the Seifert manifold associated with the Arnold unimodal singularities. The classical Chern–Simons invariants can be given from a nearly modular property of the Eichler integral (3.17). Although the linking pairing has a duality corresponding to the Arnold strange duality [19], it is not clear at this stage whether there exists duality for the classical Chern–Simons invariant and the decomposed perturbative invariants. Variants of unified WRT invariants are constructed for three of the Seifert manifolds, E12 , Z11 , and Q10 , based on an s-surgery formula on trefoil. They have an interesting property from the viewpoint of modular form [6]. By computer experiments as a generalization of (4.13), hypergeometric generating function for s ∈ Z6=0,−6 seems to have an expansion in t & 0 ∞ n X X s−2 2 1 (q)2n+1 n k+1 k + k 2 q (−1) q 2s (q)n−k (q)n+k+1 n=0 k=−n−1 q=e−t
∞ X L −2 n, ψ2`(s)  t n s 1 (1− 24(6+s) t ) = e , (5.1) 2 n! 4 `(s) n=0 where `(s) = lcm(2, 3, s, 6 + s), and the L-function follows from    
`(s) ∞ sinh `(s) z sinh z X L −2 k, ψ2`(s) 2k 3 6+s   2 = z . (2 k)! cosh `(s) k=0 2 z To conclude, one sees that the WRT invariants for rational homology spheres could be treated by use of the Eichler integral, which is intimately related to mock modular form (see [39, 30] for recent developments). It is not clear whether the decomposition (1.4) holds when M is a hyperbolic rational homology sphere. Analytic properties of the WRT invariant for hyperbolic manifolds shall provide an interesting research objects [40]. Acknowledgments The author would like to thank organizers of “Chern–Simons Theory: 20 years after”. This work is supported in part by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

20

K. HIKAMI

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DECOMPOSITION OF WRT INVARIANT

21

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