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Transgression and twisted anomaly cancellation formulas on odd dimensional manifolds
Journal of Geometry and Physics 60 (2010) 611–622
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Transgression and twisted anomaly cancellation formulas on odd dimensional manifolds Yong Wang School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
article
info
Article history: Received 27 November 2008 Received in revised form 29 July 2009 Accepted 29 December 2009 Available online 2 January 2010
abstract We compute the transgressed forms of some modularly invariant characteristic forms, which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifolds. © 2010 Elsevier B.V. All rights reserved.
MSC: 58C20 57R20 53C80 Subj. Class.: Differential geometry Algebraic topology Keywords: Transgression Elliptic genera Cancellation formulas
1. Introduction In 1983, the physicists Alvarez-Gaumé and Witten [1] discovered the ‘‘miraculous cancellation’’ formula for gravitational anomaly which reveals a beautiful relation between the top components of the Hirzebruch b L-form and b A-form of a 12 dimensional smooth Riemannian manifold. Kefeng Liu [2] established higher dimensional ‘‘miraculous cancellation’’ formulas for (8k + 4) dimensional Riemannian manifolds by developing modular invariance properties of characteristic forms. These formulas could be used to deduce some divisibility results. In [3,4], for each (8k + 4) dimensional smooth Riemannian manifold, a more general cancellation formula that involves a complex line bundle was established. This formula was applied to spinc manifolds, then an analytic Ochanine congruence formula was derived. For (8k + 2) and (8k + 6) dimensional smooth Riemannian manifolds, Han and Huang [5] obtained some cancellation formulas. On the other hand, motivated by the Chern–Simons theory, in [6], Qingtao Chen and Fei Han computed the transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. They studied the modularity properties of these secondary characteristic forms and relations among them. They also got an anomaly cancellation formula for 11 dimensional manifold. Thus a natural question is to get some twisted modular forms by transgression and some twisted anomaly cancellation formulas for odd dimensional manifolds. In this paper, we compute the transgressed forms of some modularly invariant characteristic forms, which are related to the ‘‘twisted’’ elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly
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Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
cancellation formulas on some odd dimensional manifolds. We hope that these new geometric invariants of connections with modularity properties obtained here could be applied somewhere. This paper is organized as follows: In Section 2, we review some knowledge on characteristic forms and modular forms that we are going to use. In Section 3, for (4k − 1) dimensional manifolds, we apply the Chern–Simons transgression to characteristic forms with modularity properties which are related to the ‘‘twisted’’ elliptic genera and obtain some interesting secondary characteristic forms with modularity properties. We also get two twisted cancellation formulas for 11 dimensional manifolds. In Section 4, for (4k + 1) dimensional manifolds, by transgression, we again obtain some interesting secondary characteristic forms with modularity properties. As a corollary, we get a twisted cancellation formula for nine dimensional manifolds. 2. Characteristic forms and modular forms The purpose of this section is to review the necessary knowledge on characteristic forms and modular forms that we are going to use. 2.1. Characteristic forms Let M be a Riemannian manifold. Let ∇ TM be the associated Levi-Civita connection on TM and RTM = (∇ TM )2 be the curvature of ∇ TM . Let b A(TM , ∇ TM ) and b L(TM , ∇ TM ) be the Hirzebruch characteristic forms defined respectively by (cf. [7]) 1 b A(TM , ∇ TM ) = det 2
b L(TM , ∇ TM ) = det
1 2
√
√
−1 TM R 4π√ , 1 TM sinh 4− R π
−1 TM R 2π √ . −1 TM tanh 4π R
(2.1)
Let E, F be two Hermitian vector bundles over M carrying Hermitian connection ∇ E , ∇ F respectively. Let RE = (∇ E )2 (resp. RF = (∇ F )2 ) be the curvature of ∇ E (resp. ∇ F ). If we set the formal difference G = E − F , then G carries an induced Hermitian connection ∇ G in an obvious sense. We define the associated Chern character form as
!# " !# √ √ −1 E −1 F R − tr exp R . ch(G, ∇ ) = tr exp 2π 2π "
G
(2.2)
For any complex number t, let
∧t (E ) = C|M + tE + t 2 ∧2 (E ) + · · · ,
St (E ) = C|M + tE + t 2 S 2 (E ) + · · ·
denote respectively the total exterior and symmetric powers of E, which live in K (M )[[t ]]. The following relations between these operations hold, St (E ) =
1
∧−t (E )
,
∧t (E − F ) =
∧t ( E ) . ∧t ( F )
(2.3)
Moreover, if {ωi }, {ωj0 } are formal Chern roots for Hermitian vector bundles E , F respectively, then ch(∧t (E )) =
Y
(1 + eωi t ).
(2.4)
i
Then we have the following formulas for Chern character forms,
ch(St (E )) = Q
1
(1 − eωi t )
i
Q (1 + eωi t ) ,
i
ch(∧t (E − F )) = Q
(1 + eωj t ) 0
.
(2.5)
j
If W is a real Euclidean vector bundle over M carrying a Euclidean connection ∇ W , then its complexification WC = W ⊗ C is a complex vector bundle over M carrying a canonical induced Hermitian metric from that of W , as well as a Hermitian connection ∇ WC induced from ∇ W . If E is a vector bundle (complex or real) over M, set e E = E − dim E in K (M ) or KO(M ).
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
613
2.2. Some properties about the Jacobi theta functions and modular forms We first recall the four Jacobi theta functions are defined as follows (cf. [8]): 1
θ (v, τ ) = 2q 8 sin(π v)
∞ Y
[(1 − qj )(1 − e2π
√
√
−1v j
q )(1 − e−2π −1v qj )],
(2.6)
j=1 1
θ1 (v, τ ) = 2q 8 cos(π v)
∞ Y
[(1 − qj )(1 + e2π
√
√
−1 v j
q )(1 + e−2π −1v qj )],
(2.7)
j =1
θ2 (v, τ ) =
∞ Y
[(1 − qj )(1 − e2π
√ −1v j− 21
q
)(1 − e−2π
√
−1v j− 12
q
)],
(2.8)
−1v j− 12
)],
(2.9)
j =1
θ3 (v, τ ) =
∞ Y
[(1 − qj )(1 + e2π
√ −1v j− 21
q
)(1 + e−2π
√
q
j =1
√
where q = e2π −1τ with τ ∈ H, the upper half complex plane. Let
∂θ (v, τ ) . θ (0, τ ) = ∂v v=0 0
(2.10)
Then the following Jacobi identity (cf. [8]) holds,
θ 0 (0, τ ) = π θ1 (0, τ )θ2 (0, τ )θ3 (0, τ ). n o a b Denote SL2 (Z) = | a, b, c , d ∈ Z, ad − bc = 1 the modular group. Let S = 01 c d
(2.11) −1 0
,T =
1 0
1 1
be the two
generators of SL2 (Z). They act on H by S τ = − τ , T τ = τ + 1. One has the following transformation laws of theta functions under the actions of S and T (cf. [8]): 1
21 √ 1 τ 1 2 eπ −1τ v θ (τ v, τ ); = √ θ v, − √ τ −1 −1 12 √ √ π −1 τ 1 2 4 eπ −1τ v θ2 (τ v, τ ); θ1 (v, τ + 1) = e = √ θ1 (v, τ ), θ1 v, − τ −1 12 √ τ 1 2 = √ eπ −1τ v θ1 (τ v, τ ); θ2 (v, τ + 1) = θ3 (v, τ ), θ2 v, − τ −1 12 √ 1 τ 2 eπ −1τ v θ3 (τ v, τ ). θ3 (v, τ + 1) = θ2 (v, τ ), θ3 v, − = √ τ −1 θ (v, τ + 1) = e
√ π −1 4
θ (v, τ ),
(2.12)
(2.13)
(2.14)
(2.15)
Differentiating the above transformation formulas, we get that √ π −1
θ 0 (v, τ + 1) = e 4 θ 0 (v, τ ), 21 √ √ 1 1 τ 2 0 θ v, − = √ eπ −1τ v (2π −1τ vθ (τ v, τ ) + τ θ 0 (τ v, τ )); √ τ −1 −1 √ π −1
θ10 (v, τ + 1) = e 4 θ10 (v, τ ), 21 √ √ 1 τ 2 0 θ1 v, − = √ eπ −1τ v (2π −1τ vθ2 (τ v, τ ) + τ θ20 (τ v, τ )); τ −1 θ20 (v, τ + 1) = θ30 (v, τ ), 21 √ √ 1 τ 2 θ20 v, − = √ eπ −1τ v (2π −1τ vθ1 (τ v, τ ) + τ θ10 (τ v, τ )); τ −1 θ30 (v, τ + 1) = θ20 (v, τ ), 21 √ √ 1 τ 2 θ30 v, − = √ eπ −1τ v (2π −1τ vθ3 (τ v, τ ) + τ θ30 (τ v, τ )). τ −1
(2.16)
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Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Therefore
θ
0
0, −
1
1
= √ −1
τ
τ √ −1
21
τ θ 0 (0, τ ).
(2.17)
Definition 2.1. A modular form over Γ , a subgroup of SL2 (Z), is a holomorphic function f (τ ) on H such that f (g τ ) := f
aτ + b
cτ + d
= χ (g )(c τ + d) f (τ ), k
∀g =
a c
b d
∈ Γ,
(2.18)
where χ : Γ → C? is a character of Γ . k is called the weight of f . Let
∈ SL2 (Z) | c ≡ 0 (mod 2) , a b ∈ SL2 (Z) | b ≡ 0 (mod 2) , Γ 0 (2) = c d a b a b 1 0 0 Γθ = ∈ SL2 (Z) | ≡ or Γ0 (2) =
a c
c
b d
d
c
d
0
1
1
1 (mod 2) 0
be the three modular subgroups of SL2 (Z). It is known that the generators of Γ0 (2) are T , ST 2 ST , the generators of Γ 0 (2) are STS , T 2 STS and the generators of Γθ are S , T 2 (cf. [8]). If Γ is a modular subgroup, let MR (Γ ) denote the ring of modular forms over Γ with real Fourier coefficients. Writing θj = θj (0, τ ), 1 ≤ j ≤ 3, we introduce six explicit modular forms (cf. [2,9]),
δ1 (τ ) =
1 8
(θ24 + θ34 ),
ε1 (τ ) =
1
δ2 (τ ) = − (θ14 + θ34 ),
ε2 (τ ) =
8
δ3 (τ ) =
1 8
(θ14 − θ24 ),
1 16
ε3 (τ ) = −
θ24 θ34 , 1
16 1 16
θ14 θ34 , θ14 θ24 .
They have the following Fourier expansions in q:
δ1 (τ ) =
1 4
+ 6q + · · · ,
ε1 (τ ) =
1 16
− q + ···,
1
1
8 1
ε2 (τ ) = q 2 + · · · ,
1
ε3 (τ ) = −q 2 + · · · ,
1
δ2 (τ ) = − − 3q 2 + · · · ,
1
δ3 (τ ) = − + 3q 2 + · · · , 8
where the ‘‘· · ·’’ terms are the higher degree terms, all of which have integral coefficients. They also satisfy the transformation laws,
δ2 −
1
τ
= τ δ1 (τ ), 2
δ2 (τ + 1) = δ3 (τ ),
ε2 −
1
τ
= τ 4 ε1 (τ ),
(2.19)
ε2 (τ + 1) = ε3 (τ ).
(2.20)
Lemma 2.2 ([2]). δ1 (τ ) (resp. ε1 (τ )) is a modular form of weight 2 (resp. 4) over Γ0 (2), δ2 (τ ) (resp. ε2 (τ )) is a modular form of weight 2 (resp. 4) over Γ 0 (2), while δ3 (τ ) (resp. ε3 (τ )) is a modular form of weight 2 (resp. 4) over Γθ (2) and moreover MR (Γ 0 (2)) = R[δ2 (τ ), ε2 (τ )]. 3. Transgressed forms and modularities on (4k − 1) dimensional manifolds Let M be a (4k − 1) dimensional Riemannian manifold and ξ be a rank two real oriented Euclidean vector bundle over M carrying with a Euclidean connection ∇ ξ . Set
Θ1 (TC M , ξC ) =
∞ O n =1
Sqn (T] C M) ⊗
∞ O m=1
e ∧qm (T] C M − 2ξC ) ⊗
∞ O r =1
∧
r− 1 q 2
(ξeC ) ⊗
∞ O s =1
∧ −q
s− 1 2
(ξeC ),
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Θ2 (TC M , ξC ) =
∞ O
Sqn (T] C M) ⊗
n=1
Θ3 (TC M , ξC ) =
∞ O
∞ O
∧
m=1
Sqn (T] C M) ⊗
n=1
∞ O
∧
m=1
m− 1 −q 2
m− 1 2 q
e (T] C M − 2ξC ) ⊗
∞ O
∧
r =1
e (T] C M − 2ξC ) ⊗
∞ O
r− 1 q 2
(ξeC ) ⊗
615
∞ O
∧qs (ξeC ),
s =1
∧qr (ξeC ) ⊗
∞ O
r =1
s=1
∧
s− 1 2
−q
(ξeC ).
(3.1)
Let c = e(ξ , ∇ ξ ) be the Euler form of ξ canonically associated to ∇ ξ . Set
b L(TM , ∇ TM ) ch(Θ1 (TC M , ξC ), ∇ Θ1 (TC M ,ξC ) ), cosh2 ( 2c ) c ch(Θ2 (TC M , ξC ), ∇ Θ2 (TC M ,ξC ) ), ΦW (∇ TM , ∇ ξ , τ ) = b A(TM , ∇ TM ) cosh 2 c 0 ch(Θ3 (TC M , ξC ), ∇ Θ3 (TC M ,ξC ) ). ΦW (∇ TM , ∇ ξ , τ ) = b A(TM , ∇ TM ) cosh ΦL (∇ TM , ∇ ξ , τ ) =
2
√
Let {±2π −1xj |1 ≤ j ≤ 2k − 1} and {±2π Through direct computations, we get (cf. [4])
√ √ −1u} be the Chern roots of TC M and ξC respectively and c = 2π −1u.
! ) θ 0 (0, τ ) θ1 (xj , τ ) θ12 (0, τ ) θ3 (u, τ ) θ2 (u, τ ) ΦL (∇ , ∇ , τ ) = 2 ; xj θ (xj , τ ) θ1 (0, τ ) θ12 (u, τ ) θ3 (0, τ ) θ2 (0, τ ) j =1 ! 2k −1 Y θ 0 (0, τ ) θ2 (xj , τ ) θ22 (0, τ ) θ3 (u, τ ) θ1 (u, τ ) TM ξ xj ; ΦW (∇ , ∇ , τ ) = θ (xj , τ ) θ2 (0, τ ) θ22 (u, τ ) θ3 (0, τ ) θ1 (0, τ ) j=1 ! 2k −1 Y θ 0 (0, τ ) θ3 (xj , τ ) θ32 (0, τ ) θ1 (u, τ ) θ2 (u, τ ) 0 TM ξ . ΦW (∇ , ∇ , τ ) = xj θ (xj , τ ) θ3 (0, τ ) θ32 (u, τ ) θ1 (0, τ ) θ2 (0, τ ) j=1 ξ
TM
√
4k−1
(
(3.2)
2k−1
Y
(3.3)
(3.4)
(3.5)
Consider the following function defined on C × H,
θ 0 (0, τ ) θ1 (z , τ ) , θ (z , τ ) θ1 (0, τ ) θ 0 (0, τ ) θ2 (z , τ ) fΦW (z , τ ) = z , θ (z , τ ) θ2 (0, τ ) θ 0 (0, τ ) θ3 (z , τ ) fΦ 0 (z , τ ) = z . W θ (z , τ ) θ3 (0, τ ) fΦL (z , τ ) = z
0 Applying the Chern–Weil theory, we can express ΦL , ΦW , ΦW as follows:
! Rξ Rξ θ12 (0, τ ) θ3 ( 4π 2 , τ ) θ2 ( 4π 2 , τ ) ΦL (∇ , ∇ , τ ) = 2 det fΦL ,τ det ; ξ 4π 2 θ12 ( 4Rπ 2 , τ ) θ3 (0, τ ) θ2 (0, τ ) ! TM Rξ Rξ R θ22 (0, τ ) θ3 ( 4π 2 , τ ) θ1 ( 4π 2 , τ ) 1 1 TM ξ ΦW (∇ , ∇ , τ ) = det 2 fΦW ,τ det 2 ; ξ 4π 2 θ22 ( 4Rπ 2 , τ ) θ3 (0, τ ) θ1 (0, τ ) ! TM Rξ Rξ R θ32 (0, τ ) θ1 ( 4π 2 , τ ) θ2 ( 4π 2 , τ ) 1 1 0 TM ξ 2 2 ΦW (∇ , ∇ , τ ) = det fΦ 0 ,τ det . ξ W 4π 2 θ32 ( 4Rπ 2 , τ ) θ1 (0, τ ) θ2 (0, τ ) ξ
TM
√
4k−1
1 2
RTM
1 2
(3.6)
(3.7)
(3.8)
Let E be a vector bundle and f be a power series with constant term 1. Let ∇tE be deformed connection given by ∇tE = (1 − t )∇0E + t ∇1E and REt , t ∈ [0, 1], denote the curvature of ∇tE . f 0 (t ) is the power series obtained from the derivative of f (x) with respect to x. ω is a closed form. Recall the trivial modification of Theorem 2.2 in [6], Lemma 3.1 ([6]). 1 2
det (f (
RE1
1 2
))ω − det (f ( ))ω = d RE0
1
Z 0
1 2
1 2
det (f (
REt
))ωtr
d∇tE f 0 (REt ) dt f (REt )
dt .
(3.9)
616
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Now we let E = TM and A = ∇1TM − ∇0TM , then by Lemma 3.1, we have
ΦL (∇1TM , ∇ ξ , τ ) − ΦL (∇0TM , ∇ ξ , τ ) =
1
Z
1 8π
d 2
ΦL (∇tTM , ∇ ξ , τ )tr
0
× A
1 RTM t
4π 2
RTM
−
θ 0 ( 4πt 2 , τ ) RTM
+
RTM
RTM
dt .
θ10 ( 4πt 2 , τ )
θ ( 4πt 2 , τ )
θ1 ( 4πt 2 , τ )
θ 0 ( 4πt 2 , τ )
(3.10)
We define CS ΦL (∇
TM 0
,∇
TM 1
ξ
, ∇ , τ ) :=
√ Z
1
2
8π 2
ΦL (∇
TM t
ξ
, ∇ , τ )tr A
0
1 RTM t
RTM
−
4π 2
RTM
+
θ ( 4πt 2 , τ )
RTM
RTM
dt
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(3.11)
1 2
(M , C)[[q ]]. Since M is (4k − 1) dimensional, {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) represents an element in 0 H 4k−1 (M , C)[[q ]]. Similarly, we can compute the transgressed forms for ΦW , ΦW respectively and define RTM RTM Z 1 θ 0 ( 4πt 2 , τ ) θ20 ( 4πt 2 , τ ) 1 1 TM TM ξ TM ξ dt ; + (3.12) CS ΦW (∇0 , ∇1 , ∇ , τ ) := ΦW (∇t , ∇ , τ )tr A TM − Rt RTM RTM 8π 2 0 t t , τ ) , τ ) θ ( θ ( 2 4π 2 4π 2 4π 2 TM TM R Z 1 0 Rt 0 t θ ( , τ ) θ ( , τ ) 1 1 2 2 3 4π 4π 0 0 dt CS ΦW ΦW + (3.13) (∇0TM , ∇1TM , ∇ ξ , τ ) := (∇tTM , ∇ ξ , τ )tr A TM − Rt RTM RTM 8π 2 0 t t θ ( , τ ) θ ( , τ ) 3 4π 2 4π 2 4π 2 which is in Ω
odd 1 2
1
1
which also lie in Ω odd (M , C)[[q 2 ]] and their top components represent elements in H 4k−1 (M , C)[[q 2 ]]. As pointed in [6], the equality (3.10) and the modular invariance properties of ΦL (∇0TM , ∇ ξ , τ ) and ΦL (∇1TM , ∇ ξ , τ ) are not enough to guarantee that CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ ) is a modular form. However we have the following results. Theorem 3.2. Let M be a (4k − 1) dimensional manifold and ∇0TM , ∇1TM be two connections on TM and ξ be a two dimensional oriented Euclidean real vector bundle with a Euclidean connection ∇ ξ , then we have (1) {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2); {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular 0 form of weight 2k over Γ 0 (2); {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular form of weight 2k over Γθ (2). (2) The following equalities hold,
CS ΦL
∇0TM , ∇1TM , ∇ ξ , −
1
(4k−1)
τ
= (2τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) ,
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ).
Proof. By (2.12)–(2.17), we have
θ 0 (0, − τ1 ) θ1 (z , − τ1 )
θ 0 (0, τ ) θ2 (τ z , τ ) ; θ (τ z , τ ) θ2 (0, τ ) θ (z , − τ1 ) θ1 (0, − τ1 ) θ 0 (z , − τ1 ) θ10 (z , − τ1 ) 1 1 θ 0 (τ z , τ ) θ20 (τ z , τ ) − + =τ − + ; z τz θ (τ z , τ ) θ2 (τ z , τ ) θ (z , − τ1 ) θ1 (z , − τ1 )
z
= (τ z )
θ12 (0, − τ1 ) θ3 (u, − τ1 ) θ2 (u, − τ1 ) θ12 (u, − τ1 ) θ3 (0, − τ1 ) θ2 (0, − τ1 )
=
θ22 (0, τ ) θ3 (τ u, τ ) θ1 (uτ , τ ) . θ22 (τ u, τ ) θ3 (0, τ ) θ1 (0, τ )
(3.14)
(3.15)
(3.16)
Note that we only take (4k − 1)-component, so by (3.6)–(3.8), (3.11), (3.12), (3.14)–(3.16), we can get
(4k−1) 1 TM TM ξ CS ΦL ∇0 , ∇1 , ∇ , − = (2τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) τ
Similarly we can show that CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ ),
CS ΦW
(4k−1) 1 τ 2k ∇0TM , ∇1TM , ∇ ξ , − = {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) , τ 2
(3.17)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
617
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ),
CS ΦW 0
TM 0
∇
,∇
TM 1
ξ
,∇ ,−
1
(4k−1)
0 = (τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) ,
τ
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ). ξ
(3.18)
(4k−1)
From (3.17) and (3.18), we can get {CS ΦL (∇ , ∇ , ∇ , τ )} is a modular form of weight 2k over Γ0 (2). Similarly 0 we can prove that {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )} is a modular form of weight 2k over Γ 0 (2) and {CS ΦW (∇0TM , ∇1TM , ∇ ξ , (4k−1) τ )} is a modular form of weight 2k over Γθ (2). TM TM 0 1 (4k−1)
Let M be a compact oriented smooth three dimensional manifold, then our transgressed forms are same as transgressed forms in the untwisted case which have been computed in [6]. From Theorem 3.2, we can imply some twisted cancellation formulas for odd dimensional manifolds. For example, let M be 11 dimensional and k = 3. We have that {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(11) is a modular form of weight 6 over Γ0 (2), {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) is a modular form of weight 6 over Γ 0 (2) and
CS ΦL
∇0TM , ∇1TM , ∇ ξ , −
1
(11)
τ
= (2τ )6 {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) .
By Lemma 2.2, we have
{CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) = z0 (8δ2 )3 + z1 (8δ2 )ε2 ,
(3.19)
and by (2.19) and Theorem 3.2,
{CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(11) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ].
(3.20)
1
By comparing the q 2 -expansion coefficients in (3.19), we get
z0 = −
z1 =
Z
b A(TM , ∇tTM ) cosh
0
Z
1
b A(TM , ∇tTM ) cosh 1
Z + 0
c 2
tr A
1 2RTM t
1
−
8π tan
RTM t 4π
dt
(11)
,
(3.21)
0
1
c
T M
ch(TC M , ∇t C ) − 3(ec + e−c − 2) tr A
2
1
2RTM t
c 1 RTM 1 1 b tr A − sin t + 61 TM − A(TM , ∇tTM ) cosh RTM 2 2π 4π 2Rt 8π tan 4tπ
1
−
RTM tan 4tπ (11)
8π dt
dt
.
(3.22)
Plugging (3.21) and (3.22) into (3.20) and comparing the constant terms of both sides, we obtain that
Z
1
√
2b L(TM , ∇tTM ) cosh2
0
c 2
tr A
1 2RTM t
1
−
RTM
4π sin 2tπ
(11) = 23 (26 z0 + z1 ),
so we have the following 11 dimensional analogue of the twisted miraculous cancellation formula. Corollary 3.3. The following equality holds
Z
1
√
2b L(TM , ∇tTM ) 1 tr A cosh2
0
=8
Z
c 2
1
b A(TM , ∇tTM ) cosh
0
Z
1 0
RTM
4π sin 2tπ
c 2
T M
ch(TC M , ∇t C ) − 3(ec + e−c
b A(TM , ∇tTM ) cosh
+
1
−
2RTM t
(11)
c 2
tr A −
1 2π
sin
RTM t 2π
1 − 2) tr A TM − 2Rt
− 3
1
1 2RTM t
−
1 RTM
8π tan 4tπ
RTM
8π tan 4tπ
dt
(11)
.
dt
(3.23)
618
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
ξ
0 Next we consider the transgression of ΦL (∇ TM , ∇ ξ , τ ), ΦW (∇ TM , ∇ ξ , τ ), ΦW (∇ TM , ∇ ξ , τ ) about ∇ ξ . Let ∇1 , ∇0 be two ξ
ξ
Euclidean connections on ξ and B = ∇1 − ∇0 . By (3.6)–(3.9), we have ξ
1
ξ
ΦL (∇ TM , ∇1 , τ ) − ΦL (∇ TM , ∇0 , τ ) =
8π
1
Z d 2
ξ
ΦL (∇ TM , ∇t , τ )tr 0
× B
R
ξ
R
ξ
θ20 ( 4πt 2 , τ )
+
θ2 ( 4πt 2 , τ )
R
ξ
R
ξ
θ30 ( 4πt 2 , τ )
−2
θ3 ( 4πt 2 , τ )
R
ξ
R
ξ
dt .
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(3.24)
We define ξ
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) :=
√ Z 8π 2
1
2
ξ
ΦL (∇ TM , ∇t , τ )tr B 0
R
ξ
θ20 ( 4πt 2 , τ ) ξ Rt
ξ R
+
θ2 ( 4π 2 , τ )
θ30 ( 4πt 2 , τ ) ξ Rt
R
−2
θ3 ( 4π 2 , τ ) ξ
1
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ1 ( 4π 2 , τ )
dt
(3.25)
ξ
which is in Ω odd (M , C)[[q 2 ]]. Since M is (4k − 1) dimensional, {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) represents an element in 1 2
0 H 4k−1 (M , C)[[q ]]. Similarly, we can compute the transgressed forms for ΦW , ΦW respectively and define
ξ
ξ
CS ΦW (∇ TM , ∇0 , ∇1 , τ ) :=
CS ΦW (∇ 0
TM
1
ξ
8π 2 1
ξ
, ∇0 , ∇1 , τ ) :=
8π 2
1
Z
ξ
ΦW (∇ TM , ∇t , τ )tr B 0
1
Z
ΦW (∇ 0
TM
ξ
, ∇t , τ )tr B
0
R
ξ
θ30 ( 4πt 2 , τ ) ξ Rt
R
+
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ3 ( 4π 2 , τ )
θ1 ( 4π 2 , τ )
R
ξ
R
ξ
R
ξ
R
ξ
θ20 ( 4πt 2 , τ )
+
θ2 ( 4πt 2 , τ )
θ10 ( 4πt 2 , τ )
ξ R
−2
θ20 ( 4πt 2 , τ ) ξ Rt
θ2 ( 4π 2 , τ ) ξ R
−2
θ30 ( 4πt 2 , τ ) R
θ1 ( 4πt 2 , τ )
ξ
θ3 ( 4πt 2 , τ )
1
dt , (3.26) dt , (3.27)
1
which also lie in Ω odd (M , C)[[q 2 ]] and their top components represent elements in H 4k−1 (M , C)[[q 2 ]]. Similarly we have Theorem 3.4. Let M be a (4k − 1) dimensional manifold and ∇ TM be a connection on TM and ξ be a two dimensional oriented ξ ξ Euclidean real vector bundle with two Euclidean connections ∇1 , ∇0 , then we have ξ
ξ
ξ
ξ
(1) {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2); {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ (2); {CS ΦW (∇ (2) The following equalities hold, 0
0
ξ
ξ
ξ
ξ
TM
ξ
ξ
, ∇0 , ∇1 , τ )}
(4k−1)
is a modular form of weight 2k over Γθ (2). ξ
ξ
{CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) = (2τ )2k {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ ) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ).
Proof. By (3.14), (3.16) and
θ20 (z , − τ1 ) θ2 (z , − τ1 )
+
θ30 (z , − τ1 ) θ3 (z , − τ1 )
−2
θ10 (z , − τ1 ) θ1 (z , − τ1 )
=τ
θ10 (τ z , τ ) θ30 (τ z , τ ) θ 0 (τ z , τ ) + −2 2 , θ1 (τ z , τ ) θ3 (τ z , τ ) θ2 (τ z , τ )
(3.28)
we can get
(4k−1) 1 ξ ξ ξ TM CS ΦL ∇ , ∇0 , ∇1 , − = (2τ )2k {CS ΦW (∇ TM , ∇0TM , ∇1 , τ )}(4k−1) . τ
(3.29)
Similarly we can show that ξ
ξ
ξ
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦL (∇ TM , ∇0 , ∇1 , τ ),
(4k−1) 1 τ 2k ξ ξ ξ ξ TM CS ΦW ∇ , ∇0 , ∇1 , − = {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , τ 2 ξ
ξ
ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ),
0 CS ΦW
(4k−1) 1 ξ ξ ξ ξ 0 = (τ )2k {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , ∇ TM , ∇0 , ∇1 , − τ ξ
ξ
ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ).
(3.30)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
619
ξ
From (3.29) and (3.30), we can get {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2). Similarly we can ξ
ξ
ξ
(4k−1)
ξ
0 prove that {CS ΦW (∇ , ∇0 , ∇1 , τ )} is a modular form of weight 2k over Γ 0 (2) and {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γθ (2). TM
Let M be a compact oriented smooth three dimensional manifold, we have ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) =
Since ∂∂z
θ20 (z ,τ ) θ2 (z ,τ )
+
√ Z
θ30 (z ,τ ) θ3 (z ,τ )
8π 2
1
2
ξ
ξ
ΦL (∇ TM , ∇t , τ )tr B
1 2π 2
=
8π 4
ξ
ξ Rt
ξ R
+
θ30 ( 4πt 2 , τ ) ξ Rt
1
θ2 ( 4π 2 , τ )
R
−2
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ3 ( 4π 2 , τ ) θ1 ( 4π 2 , τ ) ξ R 1 θ20 ( 4π 2 , τ ) θ30 ( 4π 2 , τ ) θ10 ( t 2 , τ ) dt tr B + − 2 4πξ ξ ξ Rt Rt Rt 0 θ2 ( 4π 2 , τ ) θ3 ( 4π 2 , τ ) θ 1 ( 4π 2 , τ ) 0 Z 1 0 0 ∂ θ2 (z , τ ) θ (z , τ ) θ (z , τ ) ξ + 3 −2 1 tr[BRt ]dt . ∂ z θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0 0 0
ξ Rt
Z
=
R
θ20 ( 4πt 2 , τ )
dt
ξ Rt
θ 0 (z ,τ ) − 2 θ11 (z ,τ ) |z =0 is a modular form of weight 2 over Γ0 (2), then it is a scalar multiple of δ1 (τ ). Direct
computations show
∂ ∂z
θ20 (z , τ ) θ30 (z , τ ) θ10 (z , τ ) = 2π 2 + O(q), + −2 θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0
∂ ∂z
θ20 (z , τ ) θ30 (z , τ ) θ 0 (z , τ ) + −2 1 = 8π 2 δ1 (τ ). θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0
so
By (4.15) in [6], we have ξ
1
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) =
2π
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B . 0 1 2 1
(3.31)
3
Similarly, we obtain that ξ
1
ξ
CS ΦW (∇ TM , ∇0 , ∇1 , τ ) = ξ
8π 1
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ ) =
8π
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B , 0 1 2 2
(3.32)
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B . 0 1 2 3
(3.33)
3
3
ξ
ξ
Let M be 11 dimensional and k = 3. We have that {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(11) is a modular form of weight 6 over ξ
ξ
Γ0 (2), {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) is a modular form of weight 6 over Γ 0 (2) and (11) 1 ξ ξ ξ ξ TM = (2τ )6 {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) . CS ΦL ∇ , ∇0 , ∇1 , − τ By Lemma 2.2, we have ξ
ξ
{CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) = z0 (8δ2 )3 + z1 (8δ2 )ε2 ,
(3.34)
and by (2.19) and Theorem 3.4, ξ
ξ
{CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(11) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ].
(3.35)
1 2
By comparing the q -expansion coefficients in (3.34), we get
(Z
ξ
1
b A(TM , ∇
z0 =
TM
) cos
0
(Z
ξ
b A(TM , ∇ TM ) cos 0
b A(TM , ∇ TM ) cos
+ 0
Rt
!
4π ξ
1
Z
" tr
4π
1
z1 =
!
Rt
Rt
4π
ξ
B 8π
tan
)(11)
#
Rt
,
dt
4π
(3.36)
ξC
3ch(ξC , ∇t ) − ch(TC M , ∇
!
" tr
3B 2π
ξ
sin
Rt
2π
TC M
) + 77 tr
"
B 8π
ξ
tan
Rt
4π
# dt
)(11)
# dt
.
Plugging (3.36) and (3.37) into (3.35) and comparing the constant terms of both sides, we obtain that
(3.37)
620
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Corollary 3.5. The following equality holds
" # (11) ξ Z 1 b L(TM , ∇ TM ) Rt tr B tan dt ξ 0 R 4π cos2 ( 4πt ) (Z ! " # ξ ξ 1 √ Rt Rt B ξ TM T M C C b A(TM , ∇ ) cos = 16 2π 3ch(ξC , ∇t ) − ch(TC M , ∇ ) + 13 tr tan dt 4π 8π 4π 0 ! " # )(11) Z 1 ξ ξ Rt Rt 3B TM b A(TM , ∇ ) cos + tr sin dt . 4π 2π 2π 0
(3.38)
4. Transgressed forms and modularities on (4k + 1) dimensional manifolds Let M be a (4k + 1) dimensional Riemannian manifold. Set
Θ1 (TC M + ξC , ξC ) =
∞ O
Sqn (TC ^ M + ξC ) ⊗
n=1
Θ2 (TC M + ξC , ξC ) =
∞ O
∞ O
∧qm (TC ^ M + ξC − 2ξeC ) ⊗
m=1
Sqn (TC ^ M + ξC ) ⊗
n=1
Θ3 (TC M + ξC , ξC ) =
∞ O
∞ O
n=1
∞ O m=1
∧
r =1
∧
m=1
Sqn (TC ^ M + ξC ) ⊗
∞ O
∧
m− 1 −q 2
m− 1 2 q
(TC ^ M + ξC − 2ξeC ) ⊗
∞ O
∧
r =1
(TC ^ M + ξC − 2ξeC ) ⊗
(ξeC ) ⊗
r− 1 q 2
∞ O
∞ O
∧ −q
s =1
r− 1 q 2
(ξeC ) ⊗
∞ O
s− 1 2
(ξeC ),
∧qs (ξeC ),
s=1
∧qr (ξeC ) ⊗
r =1
∞ O
∧ −q
s =1
s− 1 2
(ξeC ).
(4.1)
Define
eL (∇ Φ
TM
ξ
, ∇ , τ) =b L(TM , ∇ TM )
c 2 c 2
cosh
· ch(Θ1 (TC M + ξC , C )) − 2
sinh( )
TM g Φ , ∇ξ , τ ) = b A(TM , ∇ TM ) W (∇
TM 0 g Φ , ∇ξ , τ ) = b A(TM , ∇ TM ) W (∇
1
2 sinh( ) c 2
1
2 sinh( ) c 2
ch(Θ1 (TC M + ξC , ξC ))
! ,
cosh2 ( 2c )
ch(Θ2 (TC M + ξC , C 2 )) − cosh
c
ch(Θ3 (TC M + ξC , C 2 )) − cosh
c
2
2
ch(Θ2 (TC M + ξC , ξC ))
ch(Θ3 (TC M + ξC , ξC )) .
(4.2)
Through direct computations, we get (cf. [5])
√
4k+1
2k Y θ 0 (0, τ ) θ1 (xj , τ ) xj θ (xj , τ ) θ1 (0, τ ) j =1
!
θ 0 (0, τ ) · √ θ (u, τ ) π −1 ! 2k Y 1 θ 0 (0, τ ) θ2 (xj , τ ) θ 0 (0, τ ) TM ξ g Φ , ∇ , τ) = xj √ W (∇ θ (u, τ ) 2π −1 j=1 θ (xj , τ ) θ2 (0, τ ) ! 2k Y 1 θ 0 (0, τ ) θ3 (xj , τ ) θ 0 (0, τ ) TM g Φ , ∇ξ , τ ) = xj √ W (∇ θ (u, τ ) 2π −1 j=1 θ (xj , τ ) θ3 (0, τ ) 2
eL (∇ TM , ∇ ξ , τ ) = Φ
θ1 (u, τ ) θ1 (0, τ ) θ3 (u, τ ) θ2 (u, τ ) − θ1 (0, τ ) θ1 (u, τ ) θ3 (0, τ ) θ2 (0, τ ) θ2 (u, τ ) θ2 (0, τ ) θ3 (u, τ ) θ1 (u, τ ) · − θ2 (0, τ ) θ2 (u, τ ) θ3 (0, τ ) θ1 (0, τ )
·
θ3 (u, τ ) θ3 (0, τ ) θ2 (u, τ ) θ1 (u, τ ) − . θ3 (0, τ ) θ3 (u, τ ) θ2 (0, τ ) θ1 (0, τ )
(4.3)
0 g eL , Φ g Applying the Chern–Weil theory and Lemma 3.1 again, we can transgress Φ W , ΦW about ∇TM and define transgressed forms as follows:
eL (∇0TM , ∇1TM , ∇ ξ , τ ) := CS Φ
√ Z
1
2
8π 2
eL (∇tTM , ∇ ξ , τ )tr A Φ
0
RTM
1
−
RTM t
4π 2
θ 0 ( 4πt 2 , τ ) RTM
+
θ ( 4πt 2 , τ )
RTM
RTM
dt
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(4.4)
1
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k+1) represents an element in which is in Ω odd (M , C)[[q 2 ]]. Since M is (4k + 1) dimensional, {CS Φ 1
H 4k+1 (M , C)[[q 2 ]]. Similarly, we can define TM TM ξ g CS Φ W (∇0 , ∇1 , ∇ , τ ) :=
1 8π 2
1
Z 0
TM ξ g Φ W (∇t , ∇ , τ )tr A
1 RTM t
4π 2
RTM
−
θ 0 ( 4πt 2 , τ ) RTM
θ ( 4πt 2 , τ )
+
RTM
RTM
dt ;
θ20 ( 4πt 2 , τ ) θ2 ( 4πt 2 , τ )
(4.5)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
1 TM TM ξ 0 g CS Φ W (∇0 , ∇1 , ∇ , τ ) := 8π 2
1
Z 0
RTM
621 RTM
θ 0 ( 4πt 2 , τ ) θ30 ( 4πt 2 , τ ) 1 TM ξ 0 g dt . Φ (∇ , ∇ , τ ) tr A + − t W RTM RTM RTM t t t θ ( 4π 2 , τ ) θ 3 ( 4π 2 , τ ) 4π 2
(4.6)
Using the same discussions as Theorem 3.2, we obtain Theorem 4.1. Let M be a (4k + 1) dimensional manifold and ∇0TM , ∇1TM be two connections on TM and ξ be a two dimensional oriented Euclidean real vector bundle with a Euclidean connection ∇ ξ , then we have TM TM ξ (4k+1) eL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k+1) is a modular form of weight 2k + 2 over Γ0 (2); {CS Φ g (1) {CS Φ is a W (∇0 , ∇1 , ∇ , τ )}
0 TM TM ξ (4k+1) g modular form of weight 2k + 2 over Γ 0 (2); {CS Φ is a modular form of weight 2k + 2 over W (∇0 , ∇1 , ∇ , τ )} Γθ (2). (2) The following equalities hold,
(4k+1) 1 TM TM ξ (4k+1) g ∇0TM , ∇1TM , ∇ ξ , − = (2τ )2k+2 {CS Φ , W (∇0 , ∇1 , ∇ , τ )} τ TM 0 g g CS Φ , ∇ TM , ∇ ξ , τ + 1) = CS Φ (∇ TM , ∇ TM , ∇ ξ , τ ). W (∇
eL CS Φ
0
1
0
W
1
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(9) is a modular form of weight 6 over Let M be nine dimensional and k = 2. We have that {CS Φ TM TM ξ ( 9 ) g Γ0 (2), {CS Φ is a modular form of weight 6 over Γ 0 (2) and W (∇0 , ∇1 , ∇ , τ )}
(9) 1 TM TM TM TM ξ ξ (9) e g CS ΦL ∇0 , ∇1 , ∇ , − = (2τ )6 {CS Φ . W (∇0 , ∇1 , ∇ , τ )} τ
By Lemma 2.2, we have TM TM ξ (9) g {CS Φ = z0 (8δ2 )3 + z1 (8δ2 )ε2 , W (∇0 , ∇1 , ∇ , τ )}
(4.7)
and by (2.19) and Theorem 4.1,
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(9) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ]. {CS Φ
(4.8)
1
By comparing the q 2 -expansion coefficients in (4.7), we get
z0 = −
Z
1 0
(9) b A(TM , ∇tTM ) 1 1 c dt 1 − cosh tr A TM − , c TM R 2 sinh( 2 ) 2 2Rt 8π tan 4tπ
(4.9)
Z 1b TM b A(TM , ∇tTM ) A ( TM , ∇ ) c A RTM 1 1 t 1 − cosh tr sin t dt + tr A TM − RTM 2 sinh( 2c ) 2 2π 2π 2 sinh( 2c ) 2Rt 0 0 8π tan 4tπ )(9) c c c TC M −c · 1 − cosh (ch(TC M , ∇t ) + 61) + 1 + 2 cosh (e + e − 2) dt . (4.10)
( Z z1 = −
1
2
2
Plugging (4.9) and (4.10) into (4.8) and comparing the constant terms of both sides, we obtain that Corollary 4.2. The following equality holds
Z
1
√
2b L(TM , ∇tTM )
0
sinh cosh
c 2 c 2
tr A
1 2RTM t
−
1 RTM
4π sin 2tπ
(9)
Z 1b Z 1b A(TM , ∇tTM ) c A RTM A(TM , ∇tTM ) 1 1 t =8 − 1 − cosh tr sin dt + tr A TM − RTM 0 2 sinh( 2c ) 2 2π 2π 2 sinh( 2c ) 2Rt 0 8π tan 4tπ (9) c c c TC M · 1 − cosh (ch(TC M , ∇t ) − 3) + 1 + 2 cosh (e + e−c − 2) dt . (4.11) 2 2
622
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
ξ
Remark. As the referee points out, a natural question is to consider the case of two Euclidean connections ∇0 and ∇1 on ξ on a (4k + 1)-manifold M. Let g (z ) =
θ 0 (0, τ ) θ (z , τ )
θ1 (z , τ ) θ1 (0, τ ) θ3 (z , τ ) θ2 (z , τ ) − , θ1 (0, τ ) θ1 (z , τ ) θ3 (0, τ ) θ2 (0, τ )
(4.12)
then g 0 (z ) g (z )
=
2θ2 (0)θ3 (0)θ1 (z )θ10 − θ12 (0)(θ20 θ3 + θ30 θ2 )
θ2 (0)θ3 (0)θ12 − θ12 (0)θ2 θ3 ξ
ξ
−
eL (∇ TM , ∇t , τ ) about ∇t , the term When we transgress Φ ξ
holomorphic function, so
g 0 (Rt ) ξ g (Rt )
has no meaning like
1 ξ Rt
θ10 θ0 − . θ1 θ
ξ g 0 (Rt ) ξ g ( Rt )
will appear. A simple discussion shows that
(4.13) g 0 (z ) g (z )
is not a
and we can not get modular forms.
Acknowledgements The author is indebted to the referee for his careful reading and helpful comments. This work was supported by NSFC No. 10801027. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Alvarez-Gaumé, E. Witten, Gravitational anomalies, Nuclear Phys. B234 (1983) 269–330. K. Liu, Modular invariance and characteristic numbers, Comm. Math. Phys. 174 (1995) 29–42. F. Han, W. Zhang, Spinc -manifold and elliptic genera, C. R. Acad. Sci. Paris Serie I. 336 (2003) 1011–1014. F. Han, W. Zhang, Modular invariance, characteristic numbers and eta invariants, J. Differential Geom. 67 (2004) 257–288. F. Han, X. Huang, Even dimensional manifolds and generalized anomaly cancellation formulas, Trans. AMS 359 (11) (2007) 5365–5382. Q. Chen, F. Han, Elliptic genera, transgression and loop space Chern–Simons forms, Comm. Anal. Geom. 17 (1) (2009) 73–106. arXiv:0611104. W. Zhang, Lectures on Chern–Weil Theory and Witten Deformations, in: Nankai Tracks in Mathematics, vol. 4, World Scientific, Singapore, 2001. K. Chandrasekharan, Elliptic Functions, Springer-Verlag, 1985. K. Liu, Modular forms and topology, in: Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), in: Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 237–262.
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Transgression and twisted anomaly cancellation formulas on odd dimensional manifolds Yong Wang School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
article
info
Article history: Received 27 November 2008 Received in revised form 29 July 2009 Accepted 29 December 2009 Available online 2 January 2010
abstract We compute the transgressed forms of some modularly invariant characteristic forms, which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifolds. © 2010 Elsevier B.V. All rights reserved.
MSC: 58C20 57R20 53C80 Subj. Class.: Differential geometry Algebraic topology Keywords: Transgression Elliptic genera Cancellation formulas
1. Introduction In 1983, the physicists Alvarez-Gaumé and Witten [1] discovered the ‘‘miraculous cancellation’’ formula for gravitational anomaly which reveals a beautiful relation between the top components of the Hirzebruch b L-form and b A-form of a 12 dimensional smooth Riemannian manifold. Kefeng Liu [2] established higher dimensional ‘‘miraculous cancellation’’ formulas for (8k + 4) dimensional Riemannian manifolds by developing modular invariance properties of characteristic forms. These formulas could be used to deduce some divisibility results. In [3,4], for each (8k + 4) dimensional smooth Riemannian manifold, a more general cancellation formula that involves a complex line bundle was established. This formula was applied to spinc manifolds, then an analytic Ochanine congruence formula was derived. For (8k + 2) and (8k + 6) dimensional smooth Riemannian manifolds, Han and Huang [5] obtained some cancellation formulas. On the other hand, motivated by the Chern–Simons theory, in [6], Qingtao Chen and Fei Han computed the transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. They studied the modularity properties of these secondary characteristic forms and relations among them. They also got an anomaly cancellation formula for 11 dimensional manifold. Thus a natural question is to get some twisted modular forms by transgression and some twisted anomaly cancellation formulas for odd dimensional manifolds. In this paper, we compute the transgressed forms of some modularly invariant characteristic forms, which are related to the ‘‘twisted’’ elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly
E-mail address: [email protected]. 0393-0440/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2009.12.006
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Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
cancellation formulas on some odd dimensional manifolds. We hope that these new geometric invariants of connections with modularity properties obtained here could be applied somewhere. This paper is organized as follows: In Section 2, we review some knowledge on characteristic forms and modular forms that we are going to use. In Section 3, for (4k − 1) dimensional manifolds, we apply the Chern–Simons transgression to characteristic forms with modularity properties which are related to the ‘‘twisted’’ elliptic genera and obtain some interesting secondary characteristic forms with modularity properties. We also get two twisted cancellation formulas for 11 dimensional manifolds. In Section 4, for (4k + 1) dimensional manifolds, by transgression, we again obtain some interesting secondary characteristic forms with modularity properties. As a corollary, we get a twisted cancellation formula for nine dimensional manifolds. 2. Characteristic forms and modular forms The purpose of this section is to review the necessary knowledge on characteristic forms and modular forms that we are going to use. 2.1. Characteristic forms Let M be a Riemannian manifold. Let ∇ TM be the associated Levi-Civita connection on TM and RTM = (∇ TM )2 be the curvature of ∇ TM . Let b A(TM , ∇ TM ) and b L(TM , ∇ TM ) be the Hirzebruch characteristic forms defined respectively by (cf. [7]) 1 b A(TM , ∇ TM ) = det 2
b L(TM , ∇ TM ) = det
1 2
√
√
−1 TM R 4π√ , 1 TM sinh 4− R π
−1 TM R 2π √ . −1 TM tanh 4π R
(2.1)
Let E, F be two Hermitian vector bundles over M carrying Hermitian connection ∇ E , ∇ F respectively. Let RE = (∇ E )2 (resp. RF = (∇ F )2 ) be the curvature of ∇ E (resp. ∇ F ). If we set the formal difference G = E − F , then G carries an induced Hermitian connection ∇ G in an obvious sense. We define the associated Chern character form as
!# " !# √ √ −1 E −1 F R − tr exp R . ch(G, ∇ ) = tr exp 2π 2π "
G
(2.2)
For any complex number t, let
∧t (E ) = C|M + tE + t 2 ∧2 (E ) + · · · ,
St (E ) = C|M + tE + t 2 S 2 (E ) + · · ·
denote respectively the total exterior and symmetric powers of E, which live in K (M )[[t ]]. The following relations between these operations hold, St (E ) =
1
∧−t (E )
,
∧t (E − F ) =
∧t ( E ) . ∧t ( F )
(2.3)
Moreover, if {ωi }, {ωj0 } are formal Chern roots for Hermitian vector bundles E , F respectively, then ch(∧t (E )) =
Y
(1 + eωi t ).
(2.4)
i
Then we have the following formulas for Chern character forms,
ch(St (E )) = Q
1
(1 − eωi t )
i
Q (1 + eωi t ) ,
i
ch(∧t (E − F )) = Q
(1 + eωj t ) 0
.
(2.5)
j
If W is a real Euclidean vector bundle over M carrying a Euclidean connection ∇ W , then its complexification WC = W ⊗ C is a complex vector bundle over M carrying a canonical induced Hermitian metric from that of W , as well as a Hermitian connection ∇ WC induced from ∇ W . If E is a vector bundle (complex or real) over M, set e E = E − dim E in K (M ) or KO(M ).
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
613
2.2. Some properties about the Jacobi theta functions and modular forms We first recall the four Jacobi theta functions are defined as follows (cf. [8]): 1
θ (v, τ ) = 2q 8 sin(π v)
∞ Y
[(1 − qj )(1 − e2π
√
√
−1v j
q )(1 − e−2π −1v qj )],
(2.6)
j=1 1
θ1 (v, τ ) = 2q 8 cos(π v)
∞ Y
[(1 − qj )(1 + e2π
√
√
−1 v j
q )(1 + e−2π −1v qj )],
(2.7)
j =1
θ2 (v, τ ) =
∞ Y
[(1 − qj )(1 − e2π
√ −1v j− 21
q
)(1 − e−2π
√
−1v j− 12
q
)],
(2.8)
−1v j− 12
)],
(2.9)
j =1
θ3 (v, τ ) =
∞ Y
[(1 − qj )(1 + e2π
√ −1v j− 21
q
)(1 + e−2π
√
q
j =1
√
where q = e2π −1τ with τ ∈ H, the upper half complex plane. Let
∂θ (v, τ ) . θ (0, τ ) = ∂v v=0 0
(2.10)
Then the following Jacobi identity (cf. [8]) holds,
θ 0 (0, τ ) = π θ1 (0, τ )θ2 (0, τ )θ3 (0, τ ). n o a b Denote SL2 (Z) = | a, b, c , d ∈ Z, ad − bc = 1 the modular group. Let S = 01 c d
(2.11) −1 0
,T =
1 0
1 1
be the two
generators of SL2 (Z). They act on H by S τ = − τ , T τ = τ + 1. One has the following transformation laws of theta functions under the actions of S and T (cf. [8]): 1
21 √ 1 τ 1 2 eπ −1τ v θ (τ v, τ ); = √ θ v, − √ τ −1 −1 12 √ √ π −1 τ 1 2 4 eπ −1τ v θ2 (τ v, τ ); θ1 (v, τ + 1) = e = √ θ1 (v, τ ), θ1 v, − τ −1 12 √ τ 1 2 = √ eπ −1τ v θ1 (τ v, τ ); θ2 (v, τ + 1) = θ3 (v, τ ), θ2 v, − τ −1 12 √ 1 τ 2 eπ −1τ v θ3 (τ v, τ ). θ3 (v, τ + 1) = θ2 (v, τ ), θ3 v, − = √ τ −1 θ (v, τ + 1) = e
√ π −1 4
θ (v, τ ),
(2.12)
(2.13)
(2.14)
(2.15)
Differentiating the above transformation formulas, we get that √ π −1
θ 0 (v, τ + 1) = e 4 θ 0 (v, τ ), 21 √ √ 1 1 τ 2 0 θ v, − = √ eπ −1τ v (2π −1τ vθ (τ v, τ ) + τ θ 0 (τ v, τ )); √ τ −1 −1 √ π −1
θ10 (v, τ + 1) = e 4 θ10 (v, τ ), 21 √ √ 1 τ 2 0 θ1 v, − = √ eπ −1τ v (2π −1τ vθ2 (τ v, τ ) + τ θ20 (τ v, τ )); τ −1 θ20 (v, τ + 1) = θ30 (v, τ ), 21 √ √ 1 τ 2 θ20 v, − = √ eπ −1τ v (2π −1τ vθ1 (τ v, τ ) + τ θ10 (τ v, τ )); τ −1 θ30 (v, τ + 1) = θ20 (v, τ ), 21 √ √ 1 τ 2 θ30 v, − = √ eπ −1τ v (2π −1τ vθ3 (τ v, τ ) + τ θ30 (τ v, τ )). τ −1
(2.16)
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Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Therefore
θ
0
0, −
1
1
= √ −1
τ
τ √ −1
21
τ θ 0 (0, τ ).
(2.17)
Definition 2.1. A modular form over Γ , a subgroup of SL2 (Z), is a holomorphic function f (τ ) on H such that f (g τ ) := f
aτ + b
cτ + d
= χ (g )(c τ + d) f (τ ), k
∀g =
a c
b d
∈ Γ,
(2.18)
where χ : Γ → C? is a character of Γ . k is called the weight of f . Let
∈ SL2 (Z) | c ≡ 0 (mod 2) , a b ∈ SL2 (Z) | b ≡ 0 (mod 2) , Γ 0 (2) = c d a b a b 1 0 0 Γθ = ∈ SL2 (Z) | ≡ or Γ0 (2) =
a c
c
b d
d
c
d
0
1
1
1 (mod 2) 0
be the three modular subgroups of SL2 (Z). It is known that the generators of Γ0 (2) are T , ST 2 ST , the generators of Γ 0 (2) are STS , T 2 STS and the generators of Γθ are S , T 2 (cf. [8]). If Γ is a modular subgroup, let MR (Γ ) denote the ring of modular forms over Γ with real Fourier coefficients. Writing θj = θj (0, τ ), 1 ≤ j ≤ 3, we introduce six explicit modular forms (cf. [2,9]),
δ1 (τ ) =
1 8
(θ24 + θ34 ),
ε1 (τ ) =
1
δ2 (τ ) = − (θ14 + θ34 ),
ε2 (τ ) =
8
δ3 (τ ) =
1 8
(θ14 − θ24 ),
1 16
ε3 (τ ) = −
θ24 θ34 , 1
16 1 16
θ14 θ34 , θ14 θ24 .
They have the following Fourier expansions in q:
δ1 (τ ) =
1 4
+ 6q + · · · ,
ε1 (τ ) =
1 16
− q + ···,
1
1
8 1
ε2 (τ ) = q 2 + · · · ,
1
ε3 (τ ) = −q 2 + · · · ,
1
δ2 (τ ) = − − 3q 2 + · · · ,
1
δ3 (τ ) = − + 3q 2 + · · · , 8
where the ‘‘· · ·’’ terms are the higher degree terms, all of which have integral coefficients. They also satisfy the transformation laws,
δ2 −
1
τ
= τ δ1 (τ ), 2
δ2 (τ + 1) = δ3 (τ ),
ε2 −
1
τ
= τ 4 ε1 (τ ),
(2.19)
ε2 (τ + 1) = ε3 (τ ).
(2.20)
Lemma 2.2 ([2]). δ1 (τ ) (resp. ε1 (τ )) is a modular form of weight 2 (resp. 4) over Γ0 (2), δ2 (τ ) (resp. ε2 (τ )) is a modular form of weight 2 (resp. 4) over Γ 0 (2), while δ3 (τ ) (resp. ε3 (τ )) is a modular form of weight 2 (resp. 4) over Γθ (2) and moreover MR (Γ 0 (2)) = R[δ2 (τ ), ε2 (τ )]. 3. Transgressed forms and modularities on (4k − 1) dimensional manifolds Let M be a (4k − 1) dimensional Riemannian manifold and ξ be a rank two real oriented Euclidean vector bundle over M carrying with a Euclidean connection ∇ ξ . Set
Θ1 (TC M , ξC ) =
∞ O n =1
Sqn (T] C M) ⊗
∞ O m=1
e ∧qm (T] C M − 2ξC ) ⊗
∞ O r =1
∧
r− 1 q 2
(ξeC ) ⊗
∞ O s =1
∧ −q
s− 1 2
(ξeC ),
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Θ2 (TC M , ξC ) =
∞ O
Sqn (T] C M) ⊗
n=1
Θ3 (TC M , ξC ) =
∞ O
∞ O
∧
m=1
Sqn (T] C M) ⊗
n=1
∞ O
∧
m=1
m− 1 −q 2
m− 1 2 q
e (T] C M − 2ξC ) ⊗
∞ O
∧
r =1
e (T] C M − 2ξC ) ⊗
∞ O
r− 1 q 2
(ξeC ) ⊗
615
∞ O
∧qs (ξeC ),
s =1
∧qr (ξeC ) ⊗
∞ O
r =1
s=1
∧
s− 1 2
−q
(ξeC ).
(3.1)
Let c = e(ξ , ∇ ξ ) be the Euler form of ξ canonically associated to ∇ ξ . Set
b L(TM , ∇ TM ) ch(Θ1 (TC M , ξC ), ∇ Θ1 (TC M ,ξC ) ), cosh2 ( 2c ) c ch(Θ2 (TC M , ξC ), ∇ Θ2 (TC M ,ξC ) ), ΦW (∇ TM , ∇ ξ , τ ) = b A(TM , ∇ TM ) cosh 2 c 0 ch(Θ3 (TC M , ξC ), ∇ Θ3 (TC M ,ξC ) ). ΦW (∇ TM , ∇ ξ , τ ) = b A(TM , ∇ TM ) cosh ΦL (∇ TM , ∇ ξ , τ ) =
2
√
Let {±2π −1xj |1 ≤ j ≤ 2k − 1} and {±2π Through direct computations, we get (cf. [4])
√ √ −1u} be the Chern roots of TC M and ξC respectively and c = 2π −1u.
! ) θ 0 (0, τ ) θ1 (xj , τ ) θ12 (0, τ ) θ3 (u, τ ) θ2 (u, τ ) ΦL (∇ , ∇ , τ ) = 2 ; xj θ (xj , τ ) θ1 (0, τ ) θ12 (u, τ ) θ3 (0, τ ) θ2 (0, τ ) j =1 ! 2k −1 Y θ 0 (0, τ ) θ2 (xj , τ ) θ22 (0, τ ) θ3 (u, τ ) θ1 (u, τ ) TM ξ xj ; ΦW (∇ , ∇ , τ ) = θ (xj , τ ) θ2 (0, τ ) θ22 (u, τ ) θ3 (0, τ ) θ1 (0, τ ) j=1 ! 2k −1 Y θ 0 (0, τ ) θ3 (xj , τ ) θ32 (0, τ ) θ1 (u, τ ) θ2 (u, τ ) 0 TM ξ . ΦW (∇ , ∇ , τ ) = xj θ (xj , τ ) θ3 (0, τ ) θ32 (u, τ ) θ1 (0, τ ) θ2 (0, τ ) j=1 ξ
TM
√
4k−1
(
(3.2)
2k−1
Y
(3.3)
(3.4)
(3.5)
Consider the following function defined on C × H,
θ 0 (0, τ ) θ1 (z , τ ) , θ (z , τ ) θ1 (0, τ ) θ 0 (0, τ ) θ2 (z , τ ) fΦW (z , τ ) = z , θ (z , τ ) θ2 (0, τ ) θ 0 (0, τ ) θ3 (z , τ ) fΦ 0 (z , τ ) = z . W θ (z , τ ) θ3 (0, τ ) fΦL (z , τ ) = z
0 Applying the Chern–Weil theory, we can express ΦL , ΦW , ΦW as follows:
! Rξ Rξ θ12 (0, τ ) θ3 ( 4π 2 , τ ) θ2 ( 4π 2 , τ ) ΦL (∇ , ∇ , τ ) = 2 det fΦL ,τ det ; ξ 4π 2 θ12 ( 4Rπ 2 , τ ) θ3 (0, τ ) θ2 (0, τ ) ! TM Rξ Rξ R θ22 (0, τ ) θ3 ( 4π 2 , τ ) θ1 ( 4π 2 , τ ) 1 1 TM ξ ΦW (∇ , ∇ , τ ) = det 2 fΦW ,τ det 2 ; ξ 4π 2 θ22 ( 4Rπ 2 , τ ) θ3 (0, τ ) θ1 (0, τ ) ! TM Rξ Rξ R θ32 (0, τ ) θ1 ( 4π 2 , τ ) θ2 ( 4π 2 , τ ) 1 1 0 TM ξ 2 2 ΦW (∇ , ∇ , τ ) = det fΦ 0 ,τ det . ξ W 4π 2 θ32 ( 4Rπ 2 , τ ) θ1 (0, τ ) θ2 (0, τ ) ξ
TM
√
4k−1
1 2
RTM
1 2
(3.6)
(3.7)
(3.8)
Let E be a vector bundle and f be a power series with constant term 1. Let ∇tE be deformed connection given by ∇tE = (1 − t )∇0E + t ∇1E and REt , t ∈ [0, 1], denote the curvature of ∇tE . f 0 (t ) is the power series obtained from the derivative of f (x) with respect to x. ω is a closed form. Recall the trivial modification of Theorem 2.2 in [6], Lemma 3.1 ([6]). 1 2
det (f (
RE1
1 2
))ω − det (f ( ))ω = d RE0
1
Z 0
1 2
1 2
det (f (
REt
))ωtr
d∇tE f 0 (REt ) dt f (REt )
dt .
(3.9)
616
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Now we let E = TM and A = ∇1TM − ∇0TM , then by Lemma 3.1, we have
ΦL (∇1TM , ∇ ξ , τ ) − ΦL (∇0TM , ∇ ξ , τ ) =
1
Z
1 8π
d 2
ΦL (∇tTM , ∇ ξ , τ )tr
0
× A
1 RTM t
4π 2
RTM
−
θ 0 ( 4πt 2 , τ ) RTM
+
RTM
RTM
dt .
θ10 ( 4πt 2 , τ )
θ ( 4πt 2 , τ )
θ1 ( 4πt 2 , τ )
θ 0 ( 4πt 2 , τ )
(3.10)
We define CS ΦL (∇
TM 0
,∇
TM 1
ξ
, ∇ , τ ) :=
√ Z
1
2
8π 2
ΦL (∇
TM t
ξ
, ∇ , τ )tr A
0
1 RTM t
RTM
−
4π 2
RTM
+
θ ( 4πt 2 , τ )
RTM
RTM
dt
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(3.11)
1 2
(M , C)[[q ]]. Since M is (4k − 1) dimensional, {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) represents an element in 0 H 4k−1 (M , C)[[q ]]. Similarly, we can compute the transgressed forms for ΦW , ΦW respectively and define RTM RTM Z 1 θ 0 ( 4πt 2 , τ ) θ20 ( 4πt 2 , τ ) 1 1 TM TM ξ TM ξ dt ; + (3.12) CS ΦW (∇0 , ∇1 , ∇ , τ ) := ΦW (∇t , ∇ , τ )tr A TM − Rt RTM RTM 8π 2 0 t t , τ ) , τ ) θ ( θ ( 2 4π 2 4π 2 4π 2 TM TM R Z 1 0 Rt 0 t θ ( , τ ) θ ( , τ ) 1 1 2 2 3 4π 4π 0 0 dt CS ΦW ΦW + (3.13) (∇0TM , ∇1TM , ∇ ξ , τ ) := (∇tTM , ∇ ξ , τ )tr A TM − Rt RTM RTM 8π 2 0 t t θ ( , τ ) θ ( , τ ) 3 4π 2 4π 2 4π 2 which is in Ω
odd 1 2
1
1
which also lie in Ω odd (M , C)[[q 2 ]] and their top components represent elements in H 4k−1 (M , C)[[q 2 ]]. As pointed in [6], the equality (3.10) and the modular invariance properties of ΦL (∇0TM , ∇ ξ , τ ) and ΦL (∇1TM , ∇ ξ , τ ) are not enough to guarantee that CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ ) is a modular form. However we have the following results. Theorem 3.2. Let M be a (4k − 1) dimensional manifold and ∇0TM , ∇1TM be two connections on TM and ξ be a two dimensional oriented Euclidean real vector bundle with a Euclidean connection ∇ ξ , then we have (1) {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2); {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular 0 form of weight 2k over Γ 0 (2); {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) is a modular form of weight 2k over Γθ (2). (2) The following equalities hold,
CS ΦL
∇0TM , ∇1TM , ∇ ξ , −
1
(4k−1)
τ
= (2τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) ,
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ).
Proof. By (2.12)–(2.17), we have
θ 0 (0, − τ1 ) θ1 (z , − τ1 )
θ 0 (0, τ ) θ2 (τ z , τ ) ; θ (τ z , τ ) θ2 (0, τ ) θ (z , − τ1 ) θ1 (0, − τ1 ) θ 0 (z , − τ1 ) θ10 (z , − τ1 ) 1 1 θ 0 (τ z , τ ) θ20 (τ z , τ ) − + =τ − + ; z τz θ (τ z , τ ) θ2 (τ z , τ ) θ (z , − τ1 ) θ1 (z , − τ1 )
z
= (τ z )
θ12 (0, − τ1 ) θ3 (u, − τ1 ) θ2 (u, − τ1 ) θ12 (u, − τ1 ) θ3 (0, − τ1 ) θ2 (0, − τ1 )
=
θ22 (0, τ ) θ3 (τ u, τ ) θ1 (uτ , τ ) . θ22 (τ u, τ ) θ3 (0, τ ) θ1 (0, τ )
(3.14)
(3.15)
(3.16)
Note that we only take (4k − 1)-component, so by (3.6)–(3.8), (3.11), (3.12), (3.14)–(3.16), we can get
(4k−1) 1 TM TM ξ CS ΦL ∇0 , ∇1 , ∇ , − = (2τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) τ
Similarly we can show that CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ ),
CS ΦW
(4k−1) 1 τ 2k ∇0TM , ∇1TM , ∇ ξ , − = {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) , τ 2
(3.17)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
617
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ),
CS ΦW 0
TM 0
∇
,∇
TM 1
ξ
,∇ ,−
1
(4k−1)
0 = (τ )2k {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(4k−1) ,
τ
0 CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ + 1) = CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ ). ξ
(3.18)
(4k−1)
From (3.17) and (3.18), we can get {CS ΦL (∇ , ∇ , ∇ , τ )} is a modular form of weight 2k over Γ0 (2). Similarly 0 we can prove that {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )} is a modular form of weight 2k over Γ 0 (2) and {CS ΦW (∇0TM , ∇1TM , ∇ ξ , (4k−1) τ )} is a modular form of weight 2k over Γθ (2). TM TM 0 1 (4k−1)
Let M be a compact oriented smooth three dimensional manifold, then our transgressed forms are same as transgressed forms in the untwisted case which have been computed in [6]. From Theorem 3.2, we can imply some twisted cancellation formulas for odd dimensional manifolds. For example, let M be 11 dimensional and k = 3. We have that {CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(11) is a modular form of weight 6 over Γ0 (2), {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) is a modular form of weight 6 over Γ 0 (2) and
CS ΦL
∇0TM , ∇1TM , ∇ ξ , −
1
(11)
τ
= (2τ )6 {CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) .
By Lemma 2.2, we have
{CS ΦW (∇0TM , ∇1TM , ∇ ξ , τ )}(11) = z0 (8δ2 )3 + z1 (8δ2 )ε2 ,
(3.19)
and by (2.19) and Theorem 3.2,
{CS ΦL (∇0TM , ∇1TM , ∇ ξ , τ )}(11) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ].
(3.20)
1
By comparing the q 2 -expansion coefficients in (3.19), we get
z0 = −
z1 =
Z
b A(TM , ∇tTM ) cosh
0
Z
1
b A(TM , ∇tTM ) cosh 1
Z + 0
c 2
tr A
1 2RTM t
1
−
8π tan
RTM t 4π
dt
(11)
,
(3.21)
0
1
c
T M
ch(TC M , ∇t C ) − 3(ec + e−c − 2) tr A
2
1
2RTM t
c 1 RTM 1 1 b tr A − sin t + 61 TM − A(TM , ∇tTM ) cosh RTM 2 2π 4π 2Rt 8π tan 4tπ
1
−
RTM tan 4tπ (11)
8π dt
dt
.
(3.22)
Plugging (3.21) and (3.22) into (3.20) and comparing the constant terms of both sides, we obtain that
Z
1
√
2b L(TM , ∇tTM ) cosh2
0
c 2
tr A
1 2RTM t
1
−
RTM
4π sin 2tπ
(11) = 23 (26 z0 + z1 ),
so we have the following 11 dimensional analogue of the twisted miraculous cancellation formula. Corollary 3.3. The following equality holds
Z
1
√
2b L(TM , ∇tTM ) 1 tr A cosh2
0
=8
Z
c 2
1
b A(TM , ∇tTM ) cosh
0
Z
1 0
RTM
4π sin 2tπ
c 2
T M
ch(TC M , ∇t C ) − 3(ec + e−c
b A(TM , ∇tTM ) cosh
+
1
−
2RTM t
(11)
c 2
tr A −
1 2π
sin
RTM t 2π
1 − 2) tr A TM − 2Rt
− 3
1
1 2RTM t
−
1 RTM
8π tan 4tπ
RTM
8π tan 4tπ
dt
(11)
.
dt
(3.23)
618
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
ξ
0 Next we consider the transgression of ΦL (∇ TM , ∇ ξ , τ ), ΦW (∇ TM , ∇ ξ , τ ), ΦW (∇ TM , ∇ ξ , τ ) about ∇ ξ . Let ∇1 , ∇0 be two ξ
ξ
Euclidean connections on ξ and B = ∇1 − ∇0 . By (3.6)–(3.9), we have ξ
1
ξ
ΦL (∇ TM , ∇1 , τ ) − ΦL (∇ TM , ∇0 , τ ) =
8π
1
Z d 2
ξ
ΦL (∇ TM , ∇t , τ )tr 0
× B
R
ξ
R
ξ
θ20 ( 4πt 2 , τ )
+
θ2 ( 4πt 2 , τ )
R
ξ
R
ξ
θ30 ( 4πt 2 , τ )
−2
θ3 ( 4πt 2 , τ )
R
ξ
R
ξ
dt .
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(3.24)
We define ξ
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) :=
√ Z 8π 2
1
2
ξ
ΦL (∇ TM , ∇t , τ )tr B 0
R
ξ
θ20 ( 4πt 2 , τ ) ξ Rt
ξ R
+
θ2 ( 4π 2 , τ )
θ30 ( 4πt 2 , τ ) ξ Rt
R
−2
θ3 ( 4π 2 , τ ) ξ
1
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ1 ( 4π 2 , τ )
dt
(3.25)
ξ
which is in Ω odd (M , C)[[q 2 ]]. Since M is (4k − 1) dimensional, {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) represents an element in 1 2
0 H 4k−1 (M , C)[[q ]]. Similarly, we can compute the transgressed forms for ΦW , ΦW respectively and define
ξ
ξ
CS ΦW (∇ TM , ∇0 , ∇1 , τ ) :=
CS ΦW (∇ 0
TM
1
ξ
8π 2 1
ξ
, ∇0 , ∇1 , τ ) :=
8π 2
1
Z
ξ
ΦW (∇ TM , ∇t , τ )tr B 0
1
Z
ΦW (∇ 0
TM
ξ
, ∇t , τ )tr B
0
R
ξ
θ30 ( 4πt 2 , τ ) ξ Rt
R
+
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ3 ( 4π 2 , τ )
θ1 ( 4π 2 , τ )
R
ξ
R
ξ
R
ξ
R
ξ
θ20 ( 4πt 2 , τ )
+
θ2 ( 4πt 2 , τ )
θ10 ( 4πt 2 , τ )
ξ R
−2
θ20 ( 4πt 2 , τ ) ξ Rt
θ2 ( 4π 2 , τ ) ξ R
−2
θ30 ( 4πt 2 , τ ) R
θ1 ( 4πt 2 , τ )
ξ
θ3 ( 4πt 2 , τ )
1
dt , (3.26) dt , (3.27)
1
which also lie in Ω odd (M , C)[[q 2 ]] and their top components represent elements in H 4k−1 (M , C)[[q 2 ]]. Similarly we have Theorem 3.4. Let M be a (4k − 1) dimensional manifold and ∇ TM be a connection on TM and ξ be a two dimensional oriented ξ ξ Euclidean real vector bundle with two Euclidean connections ∇1 , ∇0 , then we have ξ
ξ
ξ
ξ
(1) {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2); {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ (2); {CS ΦW (∇ (2) The following equalities hold, 0
0
ξ
ξ
ξ
ξ
TM
ξ
ξ
, ∇0 , ∇1 , τ )}
(4k−1)
is a modular form of weight 2k over Γθ (2). ξ
ξ
{CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) = (2τ )2k {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ ) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ).
Proof. By (3.14), (3.16) and
θ20 (z , − τ1 ) θ2 (z , − τ1 )
+
θ30 (z , − τ1 ) θ3 (z , − τ1 )
−2
θ10 (z , − τ1 ) θ1 (z , − τ1 )
=τ
θ10 (τ z , τ ) θ30 (τ z , τ ) θ 0 (τ z , τ ) + −2 2 , θ1 (τ z , τ ) θ3 (τ z , τ ) θ2 (τ z , τ )
(3.28)
we can get
(4k−1) 1 ξ ξ ξ TM CS ΦL ∇ , ∇0 , ∇1 , − = (2τ )2k {CS ΦW (∇ TM , ∇0TM , ∇1 , τ )}(4k−1) . τ
(3.29)
Similarly we can show that ξ
ξ
ξ
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦL (∇ TM , ∇0 , ∇1 , τ ),
(4k−1) 1 τ 2k ξ ξ ξ ξ TM CS ΦW ∇ , ∇0 , ∇1 , − = {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , τ 2 ξ
ξ
ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ),
0 CS ΦW
(4k−1) 1 ξ ξ ξ ξ 0 = (τ )2k {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) , ∇ TM , ∇0 , ∇1 , − τ ξ
ξ
ξ
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ + 1) = CS ΦW (∇ TM , ∇0 , ∇1 , τ ).
(3.30)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
619
ξ
From (3.29) and (3.30), we can get {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γ0 (2). Similarly we can ξ
ξ
ξ
(4k−1)
ξ
0 prove that {CS ΦW (∇ , ∇0 , ∇1 , τ )} is a modular form of weight 2k over Γ 0 (2) and {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(4k−1) is a modular form of weight 2k over Γθ (2). TM
Let M be a compact oriented smooth three dimensional manifold, we have ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) =
Since ∂∂z
θ20 (z ,τ ) θ2 (z ,τ )
+
√ Z
θ30 (z ,τ ) θ3 (z ,τ )
8π 2
1
2
ξ
ξ
ΦL (∇ TM , ∇t , τ )tr B
1 2π 2
=
8π 4
ξ
ξ Rt
ξ R
+
θ30 ( 4πt 2 , τ ) ξ Rt
1
θ2 ( 4π 2 , τ )
R
−2
ξ
θ10 ( 4πt 2 , τ ) ξ Rt
θ3 ( 4π 2 , τ ) θ1 ( 4π 2 , τ ) ξ R 1 θ20 ( 4π 2 , τ ) θ30 ( 4π 2 , τ ) θ10 ( t 2 , τ ) dt tr B + − 2 4πξ ξ ξ Rt Rt Rt 0 θ2 ( 4π 2 , τ ) θ3 ( 4π 2 , τ ) θ 1 ( 4π 2 , τ ) 0 Z 1 0 0 ∂ θ2 (z , τ ) θ (z , τ ) θ (z , τ ) ξ + 3 −2 1 tr[BRt ]dt . ∂ z θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0 0 0
ξ Rt
Z
=
R
θ20 ( 4πt 2 , τ )
dt
ξ Rt
θ 0 (z ,τ ) − 2 θ11 (z ,τ ) |z =0 is a modular form of weight 2 over Γ0 (2), then it is a scalar multiple of δ1 (τ ). Direct
computations show
∂ ∂z
θ20 (z , τ ) θ30 (z , τ ) θ10 (z , τ ) = 2π 2 + O(q), + −2 θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0
∂ ∂z
θ20 (z , τ ) θ30 (z , τ ) θ 0 (z , τ ) + −2 1 = 8π 2 δ1 (τ ). θ2 (z , τ ) θ3 (z , τ ) θ1 (z , τ ) z =0
so
By (4.15) in [6], we have ξ
1
ξ
CS ΦL (∇ TM , ∇0 , ∇1 , τ ) =
2π
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B . 0 1 2 1
(3.31)
3
Similarly, we obtain that ξ
1
ξ
CS ΦW (∇ TM , ∇0 , ∇1 , τ ) = ξ
8π 1
ξ
0 CS ΦW (∇ TM , ∇0 , ∇1 , τ ) =
8π
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B , 0 1 2 2
(3.32)
2 ξ ξ δ (τ ) tr B [∇ , ∇ ] + B ∧ B ∧ B . 0 1 2 3
(3.33)
3
3
ξ
ξ
Let M be 11 dimensional and k = 3. We have that {CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(11) is a modular form of weight 6 over ξ
ξ
Γ0 (2), {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) is a modular form of weight 6 over Γ 0 (2) and (11) 1 ξ ξ ξ ξ TM = (2τ )6 {CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) . CS ΦL ∇ , ∇0 , ∇1 , − τ By Lemma 2.2, we have ξ
ξ
{CS ΦW (∇ TM , ∇0 , ∇1 , τ )}(11) = z0 (8δ2 )3 + z1 (8δ2 )ε2 ,
(3.34)
and by (2.19) and Theorem 3.4, ξ
ξ
{CS ΦL (∇ TM , ∇0 , ∇1 , τ )}(11) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ].
(3.35)
1 2
By comparing the q -expansion coefficients in (3.34), we get
(Z
ξ
1
b A(TM , ∇
z0 =
TM
) cos
0
(Z
ξ
b A(TM , ∇ TM ) cos 0
b A(TM , ∇ TM ) cos
+ 0
Rt
!
4π ξ
1
Z
" tr
4π
1
z1 =
!
Rt
Rt
4π
ξ
B 8π
tan
)(11)
#
Rt
,
dt
4π
(3.36)
ξC
3ch(ξC , ∇t ) − ch(TC M , ∇
!
" tr
3B 2π
ξ
sin
Rt
2π
TC M
) + 77 tr
"
B 8π
ξ
tan
Rt
4π
# dt
)(11)
# dt
.
Plugging (3.36) and (3.37) into (3.35) and comparing the constant terms of both sides, we obtain that
(3.37)
620
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
Corollary 3.5. The following equality holds
" # (11) ξ Z 1 b L(TM , ∇ TM ) Rt tr B tan dt ξ 0 R 4π cos2 ( 4πt ) (Z ! " # ξ ξ 1 √ Rt Rt B ξ TM T M C C b A(TM , ∇ ) cos = 16 2π 3ch(ξC , ∇t ) − ch(TC M , ∇ ) + 13 tr tan dt 4π 8π 4π 0 ! " # )(11) Z 1 ξ ξ Rt Rt 3B TM b A(TM , ∇ ) cos + tr sin dt . 4π 2π 2π 0
(3.38)
4. Transgressed forms and modularities on (4k + 1) dimensional manifolds Let M be a (4k + 1) dimensional Riemannian manifold. Set
Θ1 (TC M + ξC , ξC ) =
∞ O
Sqn (TC ^ M + ξC ) ⊗
n=1
Θ2 (TC M + ξC , ξC ) =
∞ O
∞ O
∧qm (TC ^ M + ξC − 2ξeC ) ⊗
m=1
Sqn (TC ^ M + ξC ) ⊗
n=1
Θ3 (TC M + ξC , ξC ) =
∞ O
∞ O
n=1
∞ O m=1
∧
r =1
∧
m=1
Sqn (TC ^ M + ξC ) ⊗
∞ O
∧
m− 1 −q 2
m− 1 2 q
(TC ^ M + ξC − 2ξeC ) ⊗
∞ O
∧
r =1
(TC ^ M + ξC − 2ξeC ) ⊗
(ξeC ) ⊗
r− 1 q 2
∞ O
∞ O
∧ −q
s =1
r− 1 q 2
(ξeC ) ⊗
∞ O
s− 1 2
(ξeC ),
∧qs (ξeC ),
s=1
∧qr (ξeC ) ⊗
r =1
∞ O
∧ −q
s =1
s− 1 2
(ξeC ).
(4.1)
Define
eL (∇ Φ
TM
ξ
, ∇ , τ) =b L(TM , ∇ TM )
c 2 c 2
cosh
· ch(Θ1 (TC M + ξC , C )) − 2
sinh( )
TM g Φ , ∇ξ , τ ) = b A(TM , ∇ TM ) W (∇
TM 0 g Φ , ∇ξ , τ ) = b A(TM , ∇ TM ) W (∇
1
2 sinh( ) c 2
1
2 sinh( ) c 2
ch(Θ1 (TC M + ξC , ξC ))
! ,
cosh2 ( 2c )
ch(Θ2 (TC M + ξC , C 2 )) − cosh
c
ch(Θ3 (TC M + ξC , C 2 )) − cosh
c
2
2
ch(Θ2 (TC M + ξC , ξC ))
ch(Θ3 (TC M + ξC , ξC )) .
(4.2)
Through direct computations, we get (cf. [5])
√
4k+1
2k Y θ 0 (0, τ ) θ1 (xj , τ ) xj θ (xj , τ ) θ1 (0, τ ) j =1
!
θ 0 (0, τ ) · √ θ (u, τ ) π −1 ! 2k Y 1 θ 0 (0, τ ) θ2 (xj , τ ) θ 0 (0, τ ) TM ξ g Φ , ∇ , τ) = xj √ W (∇ θ (u, τ ) 2π −1 j=1 θ (xj , τ ) θ2 (0, τ ) ! 2k Y 1 θ 0 (0, τ ) θ3 (xj , τ ) θ 0 (0, τ ) TM g Φ , ∇ξ , τ ) = xj √ W (∇ θ (u, τ ) 2π −1 j=1 θ (xj , τ ) θ3 (0, τ ) 2
eL (∇ TM , ∇ ξ , τ ) = Φ
θ1 (u, τ ) θ1 (0, τ ) θ3 (u, τ ) θ2 (u, τ ) − θ1 (0, τ ) θ1 (u, τ ) θ3 (0, τ ) θ2 (0, τ ) θ2 (u, τ ) θ2 (0, τ ) θ3 (u, τ ) θ1 (u, τ ) · − θ2 (0, τ ) θ2 (u, τ ) θ3 (0, τ ) θ1 (0, τ )
·
θ3 (u, τ ) θ3 (0, τ ) θ2 (u, τ ) θ1 (u, τ ) − . θ3 (0, τ ) θ3 (u, τ ) θ2 (0, τ ) θ1 (0, τ )
(4.3)
0 g eL , Φ g Applying the Chern–Weil theory and Lemma 3.1 again, we can transgress Φ W , ΦW about ∇TM and define transgressed forms as follows:
eL (∇0TM , ∇1TM , ∇ ξ , τ ) := CS Φ
√ Z
1
2
8π 2
eL (∇tTM , ∇ ξ , τ )tr A Φ
0
RTM
1
−
RTM t
4π 2
θ 0 ( 4πt 2 , τ ) RTM
+
θ ( 4πt 2 , τ )
RTM
RTM
dt
θ10 ( 4πt 2 , τ ) θ1 ( 4πt 2 , τ )
(4.4)
1
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k+1) represents an element in which is in Ω odd (M , C)[[q 2 ]]. Since M is (4k + 1) dimensional, {CS Φ 1
H 4k+1 (M , C)[[q 2 ]]. Similarly, we can define TM TM ξ g CS Φ W (∇0 , ∇1 , ∇ , τ ) :=
1 8π 2
1
Z 0
TM ξ g Φ W (∇t , ∇ , τ )tr A
1 RTM t
4π 2
RTM
−
θ 0 ( 4πt 2 , τ ) RTM
θ ( 4πt 2 , τ )
+
RTM
RTM
dt ;
θ20 ( 4πt 2 , τ ) θ2 ( 4πt 2 , τ )
(4.5)
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
1 TM TM ξ 0 g CS Φ W (∇0 , ∇1 , ∇ , τ ) := 8π 2
1
Z 0
RTM
621 RTM
θ 0 ( 4πt 2 , τ ) θ30 ( 4πt 2 , τ ) 1 TM ξ 0 g dt . Φ (∇ , ∇ , τ ) tr A + − t W RTM RTM RTM t t t θ ( 4π 2 , τ ) θ 3 ( 4π 2 , τ ) 4π 2
(4.6)
Using the same discussions as Theorem 3.2, we obtain Theorem 4.1. Let M be a (4k + 1) dimensional manifold and ∇0TM , ∇1TM be two connections on TM and ξ be a two dimensional oriented Euclidean real vector bundle with a Euclidean connection ∇ ξ , then we have TM TM ξ (4k+1) eL (∇0TM , ∇1TM , ∇ ξ , τ )}(4k+1) is a modular form of weight 2k + 2 over Γ0 (2); {CS Φ g (1) {CS Φ is a W (∇0 , ∇1 , ∇ , τ )}
0 TM TM ξ (4k+1) g modular form of weight 2k + 2 over Γ 0 (2); {CS Φ is a modular form of weight 2k + 2 over W (∇0 , ∇1 , ∇ , τ )} Γθ (2). (2) The following equalities hold,
(4k+1) 1 TM TM ξ (4k+1) g ∇0TM , ∇1TM , ∇ ξ , − = (2τ )2k+2 {CS Φ , W (∇0 , ∇1 , ∇ , τ )} τ TM 0 g g CS Φ , ∇ TM , ∇ ξ , τ + 1) = CS Φ (∇ TM , ∇ TM , ∇ ξ , τ ). W (∇
eL CS Φ
0
1
0
W
1
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(9) is a modular form of weight 6 over Let M be nine dimensional and k = 2. We have that {CS Φ TM TM ξ ( 9 ) g Γ0 (2), {CS Φ is a modular form of weight 6 over Γ 0 (2) and W (∇0 , ∇1 , ∇ , τ )}
(9) 1 TM TM TM TM ξ ξ (9) e g CS ΦL ∇0 , ∇1 , ∇ , − = (2τ )6 {CS Φ . W (∇0 , ∇1 , ∇ , τ )} τ
By Lemma 2.2, we have TM TM ξ (9) g {CS Φ = z0 (8δ2 )3 + z1 (8δ2 )ε2 , W (∇0 , ∇1 , ∇ , τ )}
(4.7)
and by (2.19) and Theorem 4.1,
eL (∇0TM , ∇1TM , ∇ ξ , τ )}(9) = 26 [z0 (8δ1 )3 + z1 (8δ1 )ε1 ]. {CS Φ
(4.8)
1
By comparing the q 2 -expansion coefficients in (4.7), we get
z0 = −
Z
1 0
(9) b A(TM , ∇tTM ) 1 1 c dt 1 − cosh tr A TM − , c TM R 2 sinh( 2 ) 2 2Rt 8π tan 4tπ
(4.9)
Z 1b TM b A(TM , ∇tTM ) A ( TM , ∇ ) c A RTM 1 1 t 1 − cosh tr sin t dt + tr A TM − RTM 2 sinh( 2c ) 2 2π 2π 2 sinh( 2c ) 2Rt 0 0 8π tan 4tπ )(9) c c c TC M −c · 1 − cosh (ch(TC M , ∇t ) + 61) + 1 + 2 cosh (e + e − 2) dt . (4.10)
( Z z1 = −
1
2
2
Plugging (4.9) and (4.10) into (4.8) and comparing the constant terms of both sides, we obtain that Corollary 4.2. The following equality holds
Z
1
√
2b L(TM , ∇tTM )
0
sinh cosh
c 2 c 2
tr A
1 2RTM t
−
1 RTM
4π sin 2tπ
(9)
Z 1b Z 1b A(TM , ∇tTM ) c A RTM A(TM , ∇tTM ) 1 1 t =8 − 1 − cosh tr sin dt + tr A TM − RTM 0 2 sinh( 2c ) 2 2π 2π 2 sinh( 2c ) 2Rt 0 8π tan 4tπ (9) c c c TC M · 1 − cosh (ch(TC M , ∇t ) − 3) + 1 + 2 cosh (e + e−c − 2) dt . (4.11) 2 2
622
Y. Wang / Journal of Geometry and Physics 60 (2010) 611–622
ξ
ξ
Remark. As the referee points out, a natural question is to consider the case of two Euclidean connections ∇0 and ∇1 on ξ on a (4k + 1)-manifold M. Let g (z ) =
θ 0 (0, τ ) θ (z , τ )
θ1 (z , τ ) θ1 (0, τ ) θ3 (z , τ ) θ2 (z , τ ) − , θ1 (0, τ ) θ1 (z , τ ) θ3 (0, τ ) θ2 (0, τ )
(4.12)
then g 0 (z ) g (z )
=
2θ2 (0)θ3 (0)θ1 (z )θ10 − θ12 (0)(θ20 θ3 + θ30 θ2 )
θ2 (0)θ3 (0)θ12 − θ12 (0)θ2 θ3 ξ
ξ
−
eL (∇ TM , ∇t , τ ) about ∇t , the term When we transgress Φ ξ
holomorphic function, so
g 0 (Rt ) ξ g (Rt )
has no meaning like
1 ξ Rt
θ10 θ0 − . θ1 θ
ξ g 0 (Rt ) ξ g ( Rt )
will appear. A simple discussion shows that
(4.13) g 0 (z ) g (z )
is not a
and we can not get modular forms.
Acknowledgements The author is indebted to the referee for his careful reading and helpful comments. This work was supported by NSFC No. 10801027. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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