Journal of Geometry and Physics

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Journal of Geometry and Physics

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Modular invariance and anomaly cancellation formulas in odd dimension

Kefeng Liu

^a,b

, Yong Wang

^c,∗

aCenter of Mathematical Sciences, Zhejiang University, Hangzhou Zhejiang 310027, China

bDepartment of Mathematics, University of California at Los Angeles, Los Angeles CA 90095-1555, USA

cSchool of Mathematics and Statistics, Northeast Normal University, Changchun Jilin, 130024, China

a r t i c l e i n f o

Article history:

Received 2 May 2015 Accepted 14 October 2015 Available online 24 October 2015

MSC:

58C20 57R20 53C80 Keywords:

Modular invariance

Cancellation formulas in odd dimension Divisibilities

a b s t r a c t

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on(^4r −1)dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on(^4r −1) dimensional spin manifolds and some congruent formulas on characteristic number for (^4r−1)dimensional spin^cmanifolds.

©2015 Elsevier B.V. All rights reserved.

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



L-form and



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 spin^cmanifolds, then an analytic Ochanine congruence formula was derived. In [5], Qingtao Chen and Fei Han obtained more twisted cancellation formulas for 8kand 8k

+

4 dimensional manifolds and they also applied their cancellation formulas to study divisibilities on spin manifolds and congruences on spin^cmanifolds.

Another important application of modular invariance properties of characteristic forms is to prove the rigidity theorem on elliptic genera. For example, see [2,6–10]. For odd dimensional manifolds, we proved the similar rigidity theorem for elliptic genera under the condition that fixed point submanifolds are 1-dimensional in [11]. In [12], Han and Yu dropped off our condition and proved more general odd dimensional rigidity theorem for elliptic genera. In [12], in order to prove the rigidity theorem, they constructed some interesting modular forms under the condition that the 3th de-Rham cohomology of manifolds vanishes.

∗Corresponding author.

E-mail addresses:liu@ucla.edu.cn,liu@cms.zju.edu.cn(K. Liu),wangy581@nenu.edu.cn(Y. Wang).

http://dx.doi.org/10.1016/j.geomphys.2015.10.006 0393-0440/©2015 Elsevier B.V. All rights reserved.

In parallel, a natural question is whether we can get some interesting cancellation formulas in odd dimension by modular forms constructed in [12]. In this paper, we will give the confirmative answer of this question. That is, by studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on

(

^4r

−

1

)

dimensional manifolds. As an application, we derive some results on divisibilities on

(

^4r

−

1

)

dimensional spin manifolds and congruences on

(

^4r

−

1

)

dimensional spin^cmanifolds. In [13], a cancellation formula on 11-dimensional manifolds was derived. To the authors’ best knowledge, our cancellation formulas appear for the first time for general odd dimensional manifolds.

This paper is organized as follows: In Section2, we review some knowledge on characteristic forms and modular forms that we are going to use. In Section3, we prove some odd dimensional cancellation formulas and we apply them to get some results on divisibilities on the index of Toeplitz operators on spin manifolds. In Section4, we prove some odd cancellation formulas involving a complex line bundle. By these formulas, we get some congruent formulas on characteristic number for odd spin^cmanifolds.

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

LetM be a Riemannian manifold. Let

∇

^TM be the associated Levi-Civita connection onTMandR^TM

= ( ∇

^TM

)

^{2be the} curvature of

∇

^TM. Let



A

(

^TM

, ∇

^TM

)

^and



L

(

^TM

, ∇

^TM

)

be the Hirzebruch characteristic forms defined respectively by (cf. [14])



A

(

^TM

, ∇

^TM

) =

det¹²



^√₋₁

4π R^TM sinh

(

^√₄⁻_π^1RTM

)

 ,



L

(

^TM

, ∇

^TM

) =

det¹²



^√₋₁

2π R^TM tanh

(

^√₄⁻_π^1RTM

)



.

^(2.1)

LetF,F^′be two Hermitian vector bundles overMcarrying Hermitian connection

∇

^F

, ∇

^F′respectively. LetR^F

= ( ∇

^F

)

^2(resp.

R^F′

= ( ∇

^F′

)

²) be the curvature of

∇

^F (resp.

∇

^F′). If we set the formal differenceG

=

F

−

F^′, thenGcarries an induced Hermitian connection

∇

^Gin an obvious sense. We define the associated Chern character form as

ch

(

^G

, ∇

^G

) =

_tr



exp

 √

−

1 2

π

^R

F



−

_tr



exp

 √

−

1 2

π

^R

F′



.

^(2.2)

For any complex numbert, let

∧

_t

(

^F

) =

C

|

_M

+

tF

+

t²

∧

²

(

^F

) + · · · ,

^St

(

^F

) =

C

|

_M

+

tF

+

t²S²

(

^F

) + · · ·

denote respectively the total exterior and symmetric powers ofF, which live inK

(

^M

) [[

t

]]

. The following relations between these operations hold,

S_t

(

^F

) =

¹

∧

_−t

(

^F

) , ∧

_t

(

^F

−

F^′

) = ∧

_t

(

^F

)

∧

_t

(

^F′

) .

^(2.3)

Moreover, if

{ ω

i

} , { ω

^′j

}

are formal Chern roots for Hermitian vector bundlesF

,

^F′respectively, then ch

( ∧

_t

(

^F

)) = 

i

(

¹

+

e^ωit

).

^(2.4)

Then we have the following formulas for Chern character forms,

ch

(

^St

(

^F

)) = 

¹

i

(

¹

−

e^ωit

) ,

^ch

( ∧

_t

(

^F

−

F^′

)) =



i

(

¹

+

e^ωit

)



j

(

¹

+

e^ω′jt

) .

^(2.5)

IfWis a real Euclidean vector bundle overMcarrying a Euclidean connection

∇

^W, then its complexificationW_C

=

W

⊗

C is a complex vector bundle overMcarrying a canonical induced Hermitian metric from that ofW, as well as a Hermitian connection

∇

^WCinduced from

∇

^W. IfFis a vector bundle (complex or real) overM, set



F

=

F

−

dimFinK

(

^M

)

^orKO

(

^M

)

^. 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. [15]):

θ(v, τ) =

2q¹⁸sin

(πv)

∞



j=1

[ (

¹

−

q^j

)(

¹

−

e^2π

√

−1vq^j

)(

¹

−

e^−2π

√

−1vq^j

) ] ,

^(2.6)

θ

1

(v, τ) =

2q¹⁸ cos

(πv)

∞



j=1

[ (

¹

−

q^j

)(

¹

+

e^2π

√

−1vq^j

)(

¹

+

e^−2π

√

−1vq^j

) ] ,

^(2.7)

θ

2

(v, τ) =

∞



j=1

[ (

¹

−

q^j

)(

¹

−

e^2π

√

−1vq^j−12

)(

¹

−

e^−2π

√

−1vq^j−12

) ] ,

^(2.8)

θ

3

(v, τ) =

∞



j=1

[ (

¹

−

q^j

)(

¹

+

e^2π

√

−1vq^j−12

)(

¹

+

e^−2π

√

−1vq^j−12

) ] ,

^(2.9)

whereq

=

e^2π

√

−1τwith

τ ∈

H, the upper half complex plane. Let

θ

^′

(

⁰

, τ) = ∂θ(v, τ)

∂v |

_v=0

.

^(2.10)

Then the following Jacobi identity (cf. [15]) holds,

θ

^′

(

⁰

, τ) = πθ

1

(

⁰

, τ)θ

2

(

⁰

, τ)θ

3

(

⁰

, τ).

^(2.11)

DenoteSL₂

(

^Z

) =



_{a b}

c d



|

a

,

^b

,

^c

,

^d

∈

Z

,

^ad

−

bc

=

1



the modular group. LetS

=



_{0 −1}

1 0

 ,

^T

=



_{1 1}

0 1



be the two generators ofSL₂

(

^Z

)

. They act onHbyS

τ = −

_τ¹

,

^T

τ = τ +

1. One has the following transformation laws of theta functions under the actions ofSandT(cf. [15]):

θ(v, τ +

1

) =

e^π

√

−1

4

θ(v, τ), θ

 v, −

¹

τ



= √

¹

−

1

 √ τ

−

1



¹₂ e^π

√

−1τv²

θ(τv, τ) ;

(2.12)

θ

1

(v, τ +

1

) =

e^π

√

−1

4

θ

1

(v, τ), θ

1

 v, −

¹

τ



=

 √ τ

−

1



¹₂ e^π

√

−1τv²

θ

2

(τv, τ) ;

(2.13)

θ

2

(v, τ +

1

) = θ

3

(v, τ), θ

2

 v, −

¹

τ



=

 √ τ

−

1



¹₂ e^π

√

−1τv²

θ

1

(τv, τ) ;

(2.14)

θ

3

(v, τ +

1

) = θ

2

(v, τ), θ

3

 v, −

¹

τ



=

 τ

√

−

1



¹₂ e^π

√

−1τv²

θ

3

(τv, τ),

^(2.15)

θ

^′

(v, τ +

1

) =

e^π

√

−1

4

θ

^′

(v, τ), θ

^′



0

, −

¹

τ



= √

¹

−

₁

 √ τ

−

₁



¹₂

τθ

^′

(

⁰

, τ).

^(2.16)

Definition 2.1. A modular form overΓ, a subgroup ofSL₂

(

^Z

)

, is a holomorphic functionf

(τ)

^{onHsuch that} f

(

^g

τ) :=

f



a

τ +

b c

τ +

d



= χ(

^g

)(

^c

τ +

d

)

^kf

(τ), ∀

g

=



a b c d



∈

_Γ

,

^(2.17)

where

χ :

_Γ

→

C^⋆is a character ofΓ^.kis called the weight off. Let

Γ0

(

²

) =



a b c d



∈

SL₂

(

^Z

) |

c

≡

0

(

^{mod 2}

)

 ,

Γ⁰

(

²

) =



a b c d



∈

SL₂

(

^Z

) |

b

≡

0

(

^{mod 2}

)

 ,

be the two modular subgroups ofSL₂

(

^Z

)

. It is known that the generators ofΓ0

(

²

)

^areT

,

^ST2ST, the generators ofΓ⁰

(

²

)

^are STS

,

^T2STS(cf. [15]).

IfΓ is a modular subgroup, letMR

(

Γ

)

denote the ring of modular forms overΓ with real Fourier coefficients. Writing

θ

j

= θ

j

(

⁰

, τ),

¹

≤

j

≤

3, we introduce four explicit modular forms (cf. [2]),

δ

1

(τ) =

¹

8

(θ

2⁴

+ θ

3⁴

), ε

1

(τ) =

¹ 16

θ

2⁴

θ

3⁴

, δ

2

(τ) = −

¹

8

(θ

1⁴

+ θ

3⁴

), ε

2

(τ) =

¹

16

θ

1⁴

θ

3⁴

.

^(2.18)

They have the following Fourier expansions inq¹²:

δ

1

(τ) =

¹

4

+

6q

+

6q²

· · · , ε

1

(τ) =

¹

16

−

q

+

7q²

+ · · · , δ

2

(τ) = −

¹

8

−

3q¹²

−

3q

+ · · · , ε

2

(τ) =

q¹²

+

8q

+ · · · ,

^(2.19) where the ‘‘

· · ·

’’ terms are the higher degree terms, all of which have integral coefficients. They also satisfy the transformation laws,

δ

2



−

¹

τ



= τ

²

δ

1

(τ), ε

2



−

¹

τ



= τ

⁴

ε

1

(τ).

^(2.20)

Lemma 2.2 ([2]).

δ

1

(τ)

^(resp.

ε

1

(τ)

) is a modular form of weight2(resp.4) overΓ0

(

²

)

^,

δ

2

(τ)

^(resp.

ε

2

(τ)

) is a modular form of weight2(resp.4) overΓ⁰

(

²

)

and moreoverMR

(

Γ⁰

(

²

)) =

R

[ δ

2

(τ), ε

2

(τ) ]

.

3. Some cancellation formulas in odd dimension

LetMbe a

(

^4r

−

1

)

dimensional Riemannian manifold. Set Θ1

(

^TCM

) =

∞



n=1

S_qⁿ

(

^TCM

) ⊗

∞



m=1

∧

_qm

(

^TCM

),

Θ2

(

^TCM

) =

∞



n=1

S_qⁿ

(

^TCM

) ⊗

∞



m=1

∧

−q^m−¹₂

(

^TCM

),

Θ3

(

^TCM

) =

∞



n=1

S_qⁿ

(

^TCM

) ⊗

∞



m=1

∧

q^m−¹₂

(

^TCM

).

^(3.1)

We recall the odd Chern character of a smooth mapgfromM to the general linear groupGL

(

^N

,

^C

)

^{withNa positive} integer (see [14]). Letddenote a trivial connection onC^N

|

_M. We will denote byc_n

(

^M

, [

g

] )

the cohomology class associated to the closedn-form

c_n

(

^CN

|

_M

,

^g

,

^d

) =



1 2

π √

−

1



⁽ⁿ⁺₂¹⁾

Tr

[ (

^g−1dg

)

ⁿ

] .

^(3.2)

The odd Chern character form ch

(

^CN

|

_M

,

^g

,

^d

)

associated toganddby definition is ch

(

^CN

|

_M

,

^g

,

^d

) =

∞



n=1

n

!

(

²ⁿ

+

1

) !

^c2n+1

(

^CN

|

_M

,

^g

,

^d

).

^(3.3)

Let the connection

∇

_uon the trivial bundleC^N

|

_Mdefined by

∇

_u

= (

¹

−

u

)

^d

+

ug⁻¹

·

d

·

g

,

^u

∈ [

0

,

¹

] .

^(3.4)

Then we have

dch

(

^CN

|

_M

,

^g

,

^d

) =

ch

(

^CN

|

_M

,

^d

) −

ch

(

^CN

|

_M

,

^g−1

·

d

·

g

).

^(3.5) Now letg

:

M

→

SO

(

^N

)

and we assume thatNis even and large enough. LetEdenote the trivial real vector bundle of rankNoverM. We equipEwith the canonical trivial metric and trivial connectiond. Set

∇

_u

=

_d

+

_ug⁻¹_dg

,

^u

∈ [

₀

,

¹

] .

LetR_ube the curvature of

∇

_u, then

R_u

= (

^u2

−

u

)(

^g−1dg

)

²

.

^(3.6)

We also consider the complexification ofEandgextends to a unitary automorphism ofE_C. The connection

∇

_uextends to a Hermitian connection onE_Cwith curvature still given by(3.6). Let

△ (

^E

)

be the spinor bundle ofE, which is a trivial Hermitian bundle of rank 2^N2. We assume thatghas a lift to the Spin group Spin

(

^N

)

^:g△

:

M

→

Spin

(

^N

).

^Sog△can be viewed as an automorphism of

△ (

^E

)

preserving the Hermitian metric. We liftdonEto be a trivial Hermitian connectiond^△on

△ (

^E

)

^{, then}

∇

_u^△

= (

¹

−

u

)

^d△

+

u

(

^g△

)

⁻¹

·

d^△

·

g^△

,

^u

∈ [

0

,

¹

]

(3.7)

lifts

∇

^uonEto

△ (

^E

)

^{. LetQ}j

(

^E

)

^,j

=

1

,

²

,

3 be the virtual bundles defined as follows:

Q₁

(

^E

) = △ (

^E

) ⊗

∞



n=1

∧

_qn

(

E



_C

),

Q₂

(

^E

) =

∞



n=1

∧

−qⁿ−¹

2

(

E



_C

) ;

Q₃

(

^E

) =

∞



n=1

∧

qⁿ−¹

2

(

E



_C

).

^(3.8)

LetgonEhave a liftg^Qj(E)onQ_j

(

^E

)

^and

∇

_uhave a lift

∇

_u^Qj(E)onQ_j

(

^E

)

. Following [12], we defined ch

(

^Qj

(

^E

),

^gQj(E)

,

^d

, τ)

^for j

=

1

,

²

,

3 as follows:

ch

(

^Qj

(

^E

), ∇

₀^Qj(E)

, τ) −

ch

(

^Qj

(

^E

), ∇

₁^Qj(E)

, τ) =

dch

(

^Qj

(

^E

),

^gQj(E)

,

^d

, τ),

^(3.9) where

ch

(

^Q1

(

^E

),

^gQ1(E)

,

^d

, τ) = −

²

N/2

8

π

²



1

0

Tr



g⁻¹dg

θ

1^′

(

^Ru

/(

⁴

π

²

), τ) θ

1

(

^Ru

/(

⁴

π

²

), τ)



du

,

^(3.10)

and forj

=

2

,

³

ch

(

^Qj

(

^E

),

^gQj(E)

,

^d

, τ) = −

¹ 8

π

²



1

0

Tr



g⁻¹dg

θ

j^′

(

^Ru

/(

⁴

π

²

), τ) θ

j

(

^Ru

/(

⁴

π

²

), τ)



du

.

^(3.11)

By Proposition 2.2 in [12], we have ifc₃

(

^EC

,

^g

,

^d

) =

0, then for any integerl

≥

1 andj

=

1

,

²

,

^{3, ch}

(

^Qj

(

^E

),

^gQj(E)

,

^d

, τ)

^(4l−1) are modular forms of weight 2loverΓ0

(

²

)

^,Γ⁰

(

²

)

^andΓθrespectively. Let (see [12, Def. 2.3])

ΦL

( ∇

^TM

,

^g

,

^d

, τ) = 

L

(

^TM

, ∇

^TM

)

^ch

(

Θ1

(

^TM

), ∇

^Θ1(TM)

, τ)

^ch

(

^Q1

(

^E

),

^gQ1(E)

,

^d

, τ) ;

(3.12) ΦW

( ∇

^TM

,

^g

,

^d

, τ) = 

A

(

^TM

, ∇

^TM

)

^ch

(

Θ2

(

^TM

), ∇

^Θ2(TM)

, τ)

^ch

(

^Q2

(

^E

),

^gQ2(E)

,

^d

, τ) ;

(3.13) ΦW^′

( ∇

^TM

,

^g

,

^d

, τ) = 

A

(

^TM

, ∇

^TM

)

^ch

(

Θ3

(

^TM

), ∇

^Θ2(TM)

, τ)

^ch

(

^Q3

(

^E

),

^gQ3(E)

,

^d

, τ).

^(3.14) By Proposition 2.4 and Theorem 2.6 in [12], we have that ifc₃

(

^EC

,

^g

,

^d

) =

0, then for any integerl

≥

1 andj

=

1

,

²

,

^3,ΦL

( ∇

^TM

,

^g

,

^d

, τ)

^(4l−1)

,

ΦW

( ∇

^TM

,

^g

,

^d

, τ)

^(4l−1)andΦ_W^′

( ∇

^TM

,

^g

,

^d

, τ)

^(4l−1)are modular forms of weight 2loverΓ0

(

²

)

^, Γ⁰

(

²

)

^andΓθrespectively. We have



ΦL

( ∇

^TM

,

^g

,

^d

, τ), [

M

] 

= −

Ind

(

^T

⊗ △ (

^TM

) ⊗

_Θ1

(

^TM

) ⊗ (

^Q1

(

^E

),

^gQ1(E)

)) ;



ΦW

( ∇

^TM

,

^g

,

^d

, τ), [

M

] 

= −

Ind

(

^T

⊗

_Θ2

(

^TM

) ⊗ (

^Q2

(

^E

),

^gQ2(E)

)) ;



Φ_W^′

( ∇

^TM

,

^g

,

^d

, τ), [

M

] 

= −

Ind

(

^T

⊗

_Θ3

(

^TM

) ⊗ (

^Q3

(

^E

),

^gQ3(E)

)),

^(3.15) where Ind

(

^T

⊗ · · · )

denotes the index of the Toeplitz operator. Let

{±

2

π √

−

1x_j

}

for 1

≤

j

≤

2r

−

1 be the Chern roots of TM

⊗

C. Similar to the computations in [2], we have

ΦL

( ∇

^TM

,

^g

,

^d

, τ) =

2^2r−1



_2r−1



j=1

x_j

θ

^′

(

⁰

, τ) θ(

^xj

, τ)

θ

1

(

^xj

, τ) θ

1

(

⁰

, τ)



ch

(

^Q1

(

^E

),

^gQ1(E)

,

^d

, τ) ;

(3.16)

ΦW

( ∇

^TM

,

^g

,

^d

, τ) =



_2r−1



j=1

x_j

θ

^′

(

⁰

, τ) θ(

^xj

, τ)

θ

2

(

^xj

, τ) θ

2

(

⁰

, τ)



ch

(

^Q2

(

^E

),

^gQ2(E)

,

^d

, τ) ;

(3.17)

ΦW^′

( ∇

^TM

,

^g

,

^d

, τ) =



_2r−1



j=1

x_j

θ

^′

(

⁰

, τ) θ(

^xj

, τ)

θ

3

(

^xj

, τ) θ

3

(

⁰

, τ)



ch

(

^Q3

(

^E

),

^gQ3(E)

,

^d

, τ).

^(3.18)

Clearly,Θ1

(

^TCM

) ⊗

Q₁

(

^E

)

^andΘ2

(

^TCM

) ⊗

Q₂

(

^E

)

admit formal Fourier expansion inq¹² as Θ1

(

^TCM

) ⊗

Q₁

(

^E

) =

A₀

(

^TCM

,

^E

) +

A₁

(

^TCM

,

^E

)

^q

+ · · · ,

Θ2

(

^TCM

) ⊗

Q₂

(

^E

) =

B₀

(

^TCM

,

^E

) +

B₁

(

^TCM

,

^E

)

^q12

+ · · · ,

^(3.19) where theA_jandB_jare elements in the semi-group formally generated by Hermitian vector bundles overM. Moreover, they carry canonically induced Hermitian connections. If

B_j

(

^TCM

,

^E

) =

B_j,1

(

^TCM

) ⊗

B_j,2

(

^E

),

we let

ch

 (

^Bj

(

^TCM

,

^E

)) =

ch

(

^Bj,1

(

^TCM

))

^ch

(

^Bj,2

(

^E

),

^gBj,2(E)

,

^d

).

If

ω

is a differential form overM, we denote

ω

^(4r−1)its top degree component. Our main results include the following theorem.

Theorem 3.1. If c₃

(

^E

,

^g

,

^d

) =

0, then





L

(

^TM

, ∇

^TM

)

^ch

( △ (

^E

),

^g△(E)

,

^d

) 

(4r−1)

=

2^3r−1+N2 [^r₂]



l=1

2^−6lh_l

,

^(3.20)

where each h_l, 1

≤

l

≤ [

^r

2

]

, is a canonical integral linear combination of





A

(

^TM

, ∇

^TM

)

ch

 (

^Bj

(

^TCM

,

^E

)) 

(^4r−1)

,

1

≤

j

≤

l and h₁

,

^h2are given by(3.25)and(3.30).

Proof. Similarly to the computations in [2, P. 35] and by (2.26) in [12] and the conditionc₃

(

^E

,

^g

,

^d

) =

_{0, we have} ΦW



∇

^TM

,

^g

,

^d

, −

¹

τ



(4r−1)

= τ

^2r

2^2r−1+N2ΦL

( ∇

^TM

,

^g

,

^d

, τ)

^(4r−1)

.

^(3.21) ByΦW

( ∇

^TM

,

^g

,

^d

, τ)

^(4r−1)is a modular form of weight 2roverΓ⁰

(

²

)

^{. By}Lemma 2.2, we have

ΦW

( ∇

^TM

,

^g

,

^d

, τ)

^(4r−1)

=

h₀

(

⁸

δ

2

)

^r

+

h₁

(

⁸

δ

2

)

^r−2

ε

2

+ · · · +

h_[^r

2]

(

⁸

δ

2

)

^r−2[r2]

ε

2^[r2]

,

^(3.22) where eachh_l

,

⁰

≤

l

≤ [

^r

2

]

, is a real multiple of the volume form atx. By(2.20),(3.21)and(3.22), we get ΦL

( ∇

^TM

,

^g

,

^d

, τ)

^(4r−1)

=

2^2r−1+N2



h₀

(

⁸

δ

1

)

^r

+

h₁

(

⁸

δ

1

)

^r−2

ε

1

+ · · · +

h_[^r

2]

(

⁸

δ

1

)

^r−2[r2]

ε

1^[r2]

 .

^(3.23)

By comparing the constant term in(3.23), we get(3.20). By comparing the coefficients ofq^2j,j

≥

0 between the two sides of (3.22), we can use the induction method to prove thath₀

=

0 and eachh_l1

≤

l

≤ [

^r

2

]

, can be expressed through a canonical integral linear combination of





A

(

^TM

, ∇

^TM

)

ch

 (

^Bj

(

^TCM

,

^E

)) 

(4r−1)

,

1

≤

j

≤

l. Direct computations show that

Θ2

(

^TCM

) ⊗

Q₂

(

^E

) =

1

− (

^TCM

+

E



_C

)

^q12

+ (

^TCM

+ ∧

²T_CM

+ ∧

²E



_C

+

T_CM

⊗

E



_C

)

^q

+ · · · .

^(3.24) By(2.19)and comparing the coefficients ofq¹² of(3.22), we get

h₁

= ( −

1

)

^r−2





A

(

^TM

, ∇

^TM

)

ch

 (

^B1

(

^TCM

,

^E

)) 

(4r−1)

= ( −

1

)

^r−1





A

(

^TM

, ∇

^TM

)

^ch

(

^E

,

^g

,

^d

) 

(4r−1)

.

^(3.25)

By(2.19)and comparing the coefficients ofqof(3.22), we get h₂

= ( −

1

)

^r−4





A

(

^TM

, ∇

^TM

)

ch

 (

^B2

(

^TCM

,

^E

)) 

(4r−1)

+ [−

8

+

24

( −

1

)

^r

(

^r

−

2

) ]

h₁

.

^(3.26) By(2.3), then

ch

 (

^B2

(

^TCM

,

^E

)) =

ch

( ∧

²E



_C

,

^g

,

^d

) +

ch

(

^TCM

)

^ch

(

^E

,

^g

,

^d

) ;

(3.27)

∧

²E



_C

=

S²

(

^CN

) + ∧

²E_C

−

E_C

⊗

C^N

;

(3.28)

ch

( ∧

²E



_C

,

^g

,

^d

) =

ch

( ∧

²E_C

,

^g

,

^d

) −

Nch

(

^EC

,

^g

,

^d

).

^(3.29) By(3.25)–(3.29), we have

h₂

= ( −

1

)

^r−4





A

(

^TM

, ∇

^TM

)

^ch

(

^TCM

)

^ch

(

^E

,

^g

,

^d

) + 

A

(

^TM

, ∇

^TM

)

^ch

( ∧

²E

,

^g

,

^d

) 

(4r−1)

+ [ ( −

1

)

^r−4

(

⁷

−

N

−

4r

) −

24

(

^r

−

2

) +

8

( −

1

)

^r

] 



A

(

^TM

, ∇

^TM

)

^ch

(

^E

,

^g

,

^d

) 

(^4r−1)

.

(3.30)

Corollary 3.2. Let M be a

(

^4r

−

1

)

-dimensional spin manifold and c₃

(

^E

,

^g

,

^d

) =

0, then

Ind

(

^T

⊗ △

_TM

⊗ ( △ (

^E

),

^g△(E)

)) = −

₂^3r−1+N2

[^r₂]



l=1

2^−6lh_l

,

^(3.31)

where each h_l, 1

≤

l

≤ [

^r

2

]

, is a canonical integral linear combination ofInd

(

^T

⊗ (

^Bj

(

^TCM

,

^E

)))

^. Corollary 3.3. Let M be a

(

^4r

−

1

)

-dimensional spin manifold and c₃

(

^E

,

^g

,

^d

) =

0. If r is even, then

Ind

(

^T

⊗ △

TM

⊗ ( △ (

^E

),

^g△(E)

)) ≡

0

(

^{mod 2N2−1}

).

If r is odd, then

Ind

(

^T

⊗ △

TM

⊗ ( △ (

^E

),

^g△(E)

)) ≡

0

(

^{mod 2N2+2}

).

Corollary 3.4. Let M be a11-dimensional manifold and c₃

(

^E

,

^g

,

^d

) =

0. Then we have





L

(

^TM

, ∇

^TM

)

^ch

( △ (

^E

),

^g△(E)

,

^d

) 

(11)

=

2^N2+2





A

(

^TM

, ∇

^TM

)

^ch

(

^E

,

^g

,

^d

) 

(11)

.

^(3.32)

Corollary 3.5. Let M be a15-dimensional manifold and c₃

(

^E

,

^g

,

^d

) =

0. Then we have





L

(

^TM

, ∇

^TM

)

^ch

( △ (

^E

),

^g△(E)

,

^d

) 

(15)

=

2^N2−1



− (

¹¹³

+

N

) 

A

(

^TM

, ∇

^TM

)

^ch

(

^E

,

^g

,

^d

)

+

A

(

^TM

, ∇

^TM

)

^ch

(

^TCM

)

^ch

(

^E

,

^g

,

^d

) + 

A

(

^TM

, ∇

^TM

)

^ch

( ∧

²E

,

^g

,

^d

) 

(15)

.

^(3.33) By (5.21) in [5], we have

Θ1

(

^TM

) =

2

+

2

(

^TCM

−

4r

+

1

)

^q

+

2

[ ( −

8r

+

3

)

^TCM

+

T_CM

⊗

T_CM

+ (

^4r

−

1

)(

^4r

−

2

) ]

q²

+ · · · .

^(3.34) ch

(

^Q1

(

^E

),

^gQ1(E)

,

^d

) =

_ch

( △ (

^E

) ⊗ (

^ECq

+ ((

²

−

_4r

)

^EC

+ ∧

²_EC

)

^q2

+ · · · ),

^g

,

^d

).

^(3.35) Θ1

(

^TM

) ⊗

Q₁

(

^E

) =

2

△ (

^E

) ⊗

E_Cq

+ [

2

△ (

^E

) ⊗ ((

²

−

4r

)

^EC

+ ∧

²E_C

)

+

2

(

^TCM

−

4r

+

1

) ⊗ △ (

^E

) ⊗

E_C

]

q²

+

O

(

^q3

).

^(3.36) By (5.22) in [5], we have for 1

≤

l

≤ [

^r

2

] (

⁸

δ

1

)

^r−2l

ε

^l1

=

2^r−6l



1

+ (

^24r

−

64l

)

^q

+ (

^288r2

−

1536rl

+

2048l²

+

512l

−

264r

)

^q2

+

O

(

^q3

) 

.

^(3.37)

By comparing the coefficients ofqin(3.23), we get Theorem 3.6. If c₃

(

^E

,

^g

,

^d

) =

0, then





L

(

^TM

, ∇

^TM

)

^ch

( △ (

^E

) ⊗

E_C

,

^g

,

^d

) −

12r



L

(

^TM

, ∇

^TM

)

^ch

( △ (

^E

),

^g△(E)

,

^d

) 

(4r−1)

= −

2^3r+4+N2 [^r₂]



l=1

2^−6llh_l

.

^(3.38)

Corollary 3.7. Let M be a

(

^4r

−

1

)

-dimensional spin manifold and c₃

(

^E

,

^g

,

^d

) =

0.

If r is even, then

Ind

(

^T

⊗ △ (

^TM

) ⊗ ( △ (

^E

) ⊗

E_C

,

^g

)) −

12rInd

(

^T

⊗ △ (

^TM

) ⊗ ( △ (

^E

),

^g

)) ≡

0

(

^{mod 2N2+4}

).

^(3.39) If r is odd, then

Ind

(

^T

⊗ △ (

^TM

) ⊗ ( △ (

^E

) ⊗

E_C

,

^g

)) −

12rInd

(

^T

⊗ △ (

^TM

) ⊗ ( △ (

^E

),

^g

)) ≡

0

(

^{mod 2N2+7}

).

^(3.40) By comparing the coefficients ofq²in(3.23), we get