# Transgression and twisted anomaly cancellation formulas on odd dimensional
manifolds

Yong Wang

###### 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.

Subj. Class.: Differential geometry; Algebraic topology
MSC: 58C20; 57R20; 53C80
Keywords: Transgression; elliptic genera; cancellation formulas

##  1 Introduction

In 1983, the physicists Alvarez-Gaumé and Witten [AW] discovered the
”miraculous cancellation” formula for gravitational anomaly which reveals a
beautiful relation between the top components of the Hirzebruch
$\widehat{L}$-form and $\widehat{A}$-form of a $12$-dimensional smooth
Riemannian manifold. Kefeng Liu [Li] 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 [HZ1],
[HZ2], 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 ${\rm spin}^{c}$ manifolds, then an
analytic Ochanine congruence formula was derived. For $(8k+2)$ and
$(8k+6)$-dimensional smooth Riemannian manifolds, F. Han and X. Huang [HH]
obtained some cancellation formulas.
On the other hand, motivated by the Chern-Simons theory, in [CH], 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 nature 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 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 $9$-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 $\nabla^{TM}$
be the associated Levi-Civita connection on $TM$ and
$R^{TM}=(\nabla^{TM})^{2}$ be the curvature of $\nabla^{TM}$. Let
$\widehat{A}(TM,\nabla^{TM})$ and $\widehat{L}(TM,\nabla^{TM})$ be the
Hirzebruch characteristic forms defined respectively by (cf. [Z])

$\widehat{A}(TM,\nabla^{TM})={\rm
det}^{\frac{1}{2}}\left(\frac{\frac{\sqrt{-1}}{4\pi}R^{TM}}{{\rm
sinh}(\frac{\sqrt{-1}}{4\pi}R^{TM})}\right),$
$\widehat{L}(TM,\nabla^{TM})={\rm
det}^{\frac{1}{2}}\left(\frac{\frac{\sqrt{-1}}{2\pi}R^{TM}}{{\rm
tanh}(\frac{\sqrt{-1}}{4\pi}R^{TM})}\right).$ $None$

Let $E$, $F$ be two Hermitian vector bundles over $M$ carrying Hermitian
connection $\nabla^{E},\nabla^{F}$ respectively. Let $R^{E}=(\nabla^{E})^{2}$
(resp. $R^{F}=(\nabla^{F})^{2}$) be the curvature of $\nabla^{E}$ (resp.
$\nabla^{F}$). If we set the formal difference $G=E-F$, then $G$ carries an
induced Hermitian connection $\nabla^{G}$ in an obvious sense. We define the
associated Chern character form as

${\rm ch}(G,\nabla^{G})={\rm tr}\left[{\rm
exp}(\frac{\sqrt{-1}}{2\pi}R^{E})\right]-{\rm tr}\left[{\rm
exp}(\frac{\sqrt{-1}}{2\pi}R^{F})\right].$ $None$

For any complex number $t$, let

$\wedge_{t}(E)={\bf C}|_{M}+tE+t^{2}\wedge^{2}(E)+\cdots,~{}S_{t}(E)={\bf
C}|_{M}+tE+t^{2}S^{2}(E)+\cdots$

denote respectively the total exterior and symmetric powers of $E$, which live
in $K(M)[[t]].$ The following relations between these operations hold,

$S_{t}(E)=\frac{1}{\wedge_{-t}(E)},~{}\wedge_{t}(E-F)=\frac{\wedge_{t}(E)}{\wedge_{t}(F)}.$
$None$

Moreover, if $\\{\omega_{i}\\},\\{\omega_{j}^{\prime}\\}$ are formal Chern
roots for Hermitian vector bundles $E,F$ respectively, then

${\rm ch}(\wedge_{t}(E))=\prod_{i}(1+e^{\omega_{i}}t).$ $None$

Then we have the following formulas for Chern character forms,

${\rm ch}(S_{t}(E))=\frac{1}{\prod_{i}(1-e^{\omega_{i}}t)},~{}{\rm
ch}(\wedge_{t}(E-F))=\frac{\prod_{i}(1+e^{\omega_{i}}t)}{\prod_{j}(1+e^{\omega^{\prime}_{j}}t)}.$
$None$

If $W$ is a real Euclidean vector bundle over $M$ carrying a Euclidean
connection $\nabla^{W}$, then its complexification $W_{\bf C}=W\otimes{\bf C}$
is a complex vector bundle over $M$ carrying a canonical induced Hermitian
metric from that of $W$, as well as a Hermitian connection $\nabla^{W_{\bf
C}}$ induced from $\nabla^{W}$. If $E$ is a vector bundle (complex or real)
over $M$, set $\widetilde{E}=E-{\rm dim}E$ in $K(M)$ or $KO(M)$.

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.
[Ch]):

$\theta(v,\tau)=2q^{\frac{1}{8}}{\rm sin}(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j})(1-e^{-2\pi\sqrt{-1}v}q^{j})],$
$None$ $\theta_{1}(v,\tau)=2q^{\frac{1}{8}}{\rm cos}(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1+e^{2\pi\sqrt{-1}v}q^{j})(1+e^{-2\pi\sqrt{-1}v}q^{j})],$
$None$
$\theta_{2}(v,\tau)=\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})(1-e^{-2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})],$
$None$
$\theta_{3}(v,\tau)=\prod_{j=1}^{\infty}[(1-q^{j})(1+e^{2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})(1+e^{-2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})],$
$None$

where $q=e^{2\pi\sqrt{-1}\tau}$ with $\tau\in\textbf{H}$, the upper half
complex plane. Let

$\theta^{\prime}(0,\tau)=\frac{\partial\theta(v,\tau)}{\partial v}|_{v=0}.$
$None$

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

$\theta^{\prime}(0,\tau)=\pi\theta_{1}(0,\tau)\theta_{2}(0,\tau)\theta_{3}(0,\tau).$
$None$

Denote $SL_{2}({\bf Z})=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\mid a,b,c,d\in{\bf Z},~{}ad-bc=1\right\\}$ the modular
group. Let $S=\left(\begin{array}[]{cc}\ 0&-1\\\
1&0\end{array}\right),~{}T=\left(\begin{array}[]{cc}\ 1&1\\\
0&1\end{array}\right)$ be the two generators of $SL_{2}(\bf{Z})$. They act on
H by $S\tau=-\frac{1}{\tau},~{}T\tau=\tau+1$. One has the following
transformation laws of theta functions under the actions of $S$ and $T$ (cf.
[Ch]):

$\theta(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta(v,\tau),~{}~{}\theta(v,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta(\tau v,\tau);$ $None$
$\theta_{1}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta_{1}(v,\tau),~{}~{}\theta_{1}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{2}(\tau v,\tau);$ $None$
$\theta_{2}(v,\tau+1)=\theta_{3}(v,\tau),~{}~{}\theta_{2}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{1}(\tau v,\tau);$ $None$
$\theta_{3}(v,\tau+1)=\theta_{2}(v,\tau),~{}~{}\theta_{3}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{3}(\tau v,\tau).$ $None$

Differentiating the above transformation formulas, we get that

$\theta^{\prime}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta^{\prime}(v,\tau),$
$\theta^{\prime}(v,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta(\tau v,\tau)+\tau\theta^{\prime}(\tau
v,\tau));$
$\theta^{\prime}_{1}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta_{1}^{\prime}(v,\tau),$
$\theta^{\prime}_{1}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{2}(\tau v,\tau)+\tau\theta^{\prime}_{2}(\tau
v,\tau));$ $\theta^{\prime}_{2}(v,\tau+1)=\theta_{3}^{\prime}(v,\tau),$
$\theta^{\prime}_{2}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{1}(\tau v,\tau)+\tau\theta^{\prime}_{1}(\tau
v,\tau));$ $\theta^{\prime}_{3}(v,\tau+1)=\theta_{2}^{\prime}(v,\tau),$
$\theta^{\prime}_{3}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{3}(\tau v,\tau)+\tau\theta^{\prime}_{3}(\tau
v,\tau))$ $None$

Therefore

$\theta^{\prime}(0,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}\tau\theta^{\prime}(0,\tau).$
$None$

Definition 2.1 A modular form over $\Gamma$, a subgroup of $SL_{2}({\bf Z})$,
is a holomorphic function $f(\tau)$ on H such that

$f(g\tau):=f\left(\frac{a\tau+b}{c\tau+d}\right)=\chi(g)(c\tau+d)^{k}f(\tau),~{}~{}\forall
g=\left(\begin{array}[]{cc}\ a&b\\\ c&d\end{array}\right)\in\Gamma,$ $None$

where $\chi:\Gamma\rightarrow{\bf C}^{\star}$ is a character of $\Gamma$. $k$
is called the weight of $f$.
Let

$\Gamma_{0}(2)=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\in SL_{2}({\bf Z})\mid c\equiv 0~{}({\rm
mod}~{}2)\right\\},$ $\Gamma^{0}(2)=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\in SL_{2}({\bf Z})\mid b\equiv 0~{}({\rm
mod}~{}2)\right\\},$ $\Gamma_{\theta}=\left\\{\left(\begin{array}[]{cc}\
a&b\\\ c&d\end{array}\right)\in SL_{2}({\bf Z})\mid\left(\begin{array}[]{cc}\
a&b\\\ c&d\end{array}\right)\equiv\left(\begin{array}[]{cc}\ 1&0\\\
0&1\end{array}\right){\rm or}\left(\begin{array}[]{cc}\ 0&1\\\
1&0\end{array}\right)~{}({\rm mod}~{}2)\right\\}$

be the three modular subgroups of $SL_{2}({\bf Z})$. It is known that the
generators of $\Gamma_{0}(2)$ are $T,~{}ST^{2}ST$, the generators of
$\Gamma^{0}(2)$ are $STS,~{}T^{2}STS$ and the generators of $\Gamma_{\theta}$
are $S,~{}T^{2}$ (cf.[Ch]).
If $\Gamma$ is a modular subgroup, let ${\mathcal{M}}_{{\bf R}}(\Gamma)$
denote the ring of modular forms over $\Gamma$ with real Fourier coefficients.
Writing $\theta_{j}=\theta_{j}(0,\tau),~{}1\leq j\leq 3,$ we introduce six
explicit modular forms (cf. [Li]),

$\delta_{1}(\tau)=\frac{1}{8}(\theta_{2}^{4}+\theta_{3}^{4}),~{}~{}\varepsilon_{1}(\tau)=\frac{1}{16}\theta_{2}^{4}\theta_{3}^{4},$
$\delta_{2}(\tau)=-\frac{1}{8}(\theta_{1}^{4}+\theta_{3}^{4}),~{}~{}\varepsilon_{2}(\tau)=\frac{1}{16}\theta_{1}^{4}\theta_{3}^{4},$
$\delta_{3}(\tau)=\frac{1}{8}(\theta_{1}^{4}-\theta_{2}^{4}),~{}~{}\varepsilon_{3}(\tau)=-\frac{1}{16}\theta_{1}^{4}\theta_{2}^{4}.$

They have the following Fourier expansions in $q^{\frac{1}{2}}$:

$\delta_{1}(\tau)=\frac{1}{4}+6q+\cdots,~{}~{}\varepsilon_{1}(\tau)=\frac{1}{16}-q+\cdots,$
$\delta_{2}(\tau)=-\frac{1}{8}-3q^{\frac{1}{2}}+\cdots,~{}~{}\varepsilon_{2}(\tau)=q^{\frac{1}{2}}+\cdots,$
$\delta_{3}(\tau)=-\frac{1}{8}+3q^{\frac{1}{2}}+\cdots,~{}~{}\varepsilon_{3}(\tau)=-q^{\frac{1}{2}}+\cdots,$

where the $"\cdots"$ terms are the higher degree terms, all of which have
integral coefficients. They also satisfy the transformation laws,

$\delta_{2}(-\frac{1}{\tau})=\tau^{2}\delta_{1}(\tau),~{}~{}~{}~{}~{}~{}\varepsilon_{2}(-\frac{1}{\tau})=\tau^{4}\varepsilon_{1}(\tau),$
$None$
$\delta_{2}(\tau+1)=\delta_{3}(\tau),~{}~{}~{}~{}~{}~{}\varepsilon_{2}(\tau+1)=\varepsilon_{3}(\tau).$
$None$

Lemma 2.2 ([Li]) $\delta_{1}(\tau)$ (resp. $\varepsilon_{1}(\tau)$) is a
modular form of weight $2$ (resp. $4$) over $\Gamma_{0}(2)$,
$\delta_{2}(\tau)$ (resp. $\varepsilon_{2}(\tau)$) is a modular form of weight
$2$ (resp. $4$) over $\Gamma^{0}(2)$, while $\delta_{3}(\tau)$ (resp.
$\varepsilon_{3}(\tau)$) is a modular form of weight $2$ (resp. $4$) over
$\Gamma_{\theta}(2)$ and moreover ${\mathcal{M}}_{{\bf R}}(\Gamma^{0}(2))={\bf
R}[\delta_{2}(\tau),\varepsilon_{2}(\tau)]$.

##  3 Transgressed forms and modularities on $4k-1$ dimensional manifolds

Let $M$ be a $4k-1$ dimensional Riemannian manifold and $\xi$ be a rank two
real oriented Euclidean vector bundle over $M$ carrying with a Euclidean
connection $\nabla^{\xi}$. Set

$\Theta_{1}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}),$
$\Theta_{2}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{q^{s}}(\widetilde{\xi_{C}}),$
$\Theta_{3}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m-\frac{1}{2}}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}).$
$None$

Let $c=e(\xi,\nabla^{\xi})$ be the Euler form of $\xi$ canonically associated
to $\nabla^{\xi}$. Set

$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\frac{\widehat{L}(TM,\nabla^{TM})}{{\rm
cosh}^{2}(\frac{c}{2})}{\rm
ch}(\Theta_{1}(T_{C}M,\xi_{C}),\nabla^{\Theta_{1}(T_{C}M,\xi_{C})}),$
$\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)={\widehat{A}(TM,\nabla^{TM})}{\rm
cosh}(\frac{c}{2}){\rm
ch}(\Theta_{2}(T_{C}M,\xi_{C}),\nabla^{\Theta_{2}(T_{C}M,\xi_{C})}),$
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)={\widehat{A}(TM,\nabla^{TM})}{\rm
cosh}(\frac{c}{2}){\rm
ch}(\Theta_{3}(T_{C}M,\xi_{C}),\nabla^{\Theta_{3}(T_{C}M,\xi_{C})}).$ $None$

Let $\\{\pm 2\pi\sqrt{-1}x_{j}|~{}1\leq j\leq 2k-1\\}$ and $\\{\pm
2\pi\sqrt{-1}u\\}$ be the Chern roots of $T_{C}M$ and $\xi_{C}$ respectively
and $c=2\pi\sqrt{-1}u.$ Through direct computations, we get (cf. [HZ2])

$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\sqrt{2}^{4k-1}\left\\{\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\frac{\theta_{1}^{2}(0,\tau)}{\theta_{1}^{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\right\\};$
$None$
$\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)=\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\frac{\theta_{2}^{2}(0,\tau)}{\theta_{2}^{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)};$
$None$
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)=\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{3}(x_{j},\tau)}{\theta_{3}(0,\tau)}\right)\frac{\theta_{3}^{2}(0,\tau)}{\theta_{3}^{2}(u,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}.$
$None$

Consider the following function defined on ${\bf C}\times{\bf H}$,

$f_{\Phi_{L}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{1}(z,\tau)}{\theta_{1}(0,\tau)},$
$f_{\Phi_{W}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{2}(z,\tau)}{\theta_{2}(0,\tau)},$
$f_{\Phi_{W}^{\prime}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{3}(z,\tau)}{\theta_{3}(0,\tau)}.$

Applying the Chern-Weil theory, we can express
$\Phi_{L},~{}\Phi_{W},\Phi_{W}^{\prime}$ as follows:

$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\sqrt{2}^{4k-1}{\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{L}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{1}^{2}(0,\tau)}{\theta_{1}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{3}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(0,\tau)}\right);$
$None$ $\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)={\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{W}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{2}^{2}(0,\tau)}{\theta_{2}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{3}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(0,\tau)}\right);$
$None$ $\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)={\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{W}^{\prime}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{3}^{2}(0,\tau)}{\theta_{3}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{1}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(0,\tau)}\frac{\theta_{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(0,\tau)}\right).$
$None$

Let $E$ be a vector bundle and $f$ be a power series with constant term $1$.
Let $\nabla_{t}^{E}$ be deformed connection given by
$\nabla_{t}^{E}=(1-t)\nabla_{0}^{E}+t\nabla_{1}^{E}$ and
$R^{E}_{t},~{}t\in[0,1],$ denote the curvature of $\nabla_{t}^{E}$.
$f^{\prime}(t)$ is the power series obtained from the derivative of $f(x)$
with respect to $x$. $\omega$ is a closed form. Recall the trivial
modification of Theorem 2.2 in [CH],

Lemma 3.1( [CH])

${\rm det}^{\frac{1}{2}}(f(R^{E}_{1}))\omega-{\rm
det}^{\frac{1}{2}}(f(R^{E}_{0}))\omega=d\int_{0}^{1}\frac{1}{2}{\rm
det}^{\frac{1}{2}}(f(R^{E}_{t}))\omega{\rm
tr}\left[\frac{d\nabla_{t}^{E}}{dt}\frac{f^{\prime}(R^{E}_{t})}{f(R^{E}_{t})}\right]dt.$
$None$

Now we let $E=TM$ and $A=\nabla_{1}^{TM}-\nabla_{0}^{TM}$, then by Lemma 3.1,
we have
$\Phi_{L}(\nabla_{1}^{TM},\nabla^{\xi},\tau)-\Phi_{L}(\nabla_{0}^{TM},\nabla^{\xi},\tau)$

$=\frac{1}{8\pi^{2}}d\int_{0}^{1}\Phi_{L}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

We define
$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k-1$ dimensional,
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
represents an element in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can compute the transgressed forms for $\Phi_{W},~{}\Phi_{W}^{\prime}$
respectively and define
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{2}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt;$
$None$

$CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}^{\prime}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$

which also lie in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]]$ and their
top components represent elements in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$.
As pointed in [CH], the equality (3.10) and the modular invariance properties
of $\Phi_{L}(\nabla_{0}^{TM},\nabla^{\xi},\tau)$ and
$\Phi_{L}(\nabla_{1}^{TM},\nabla^{\xi},\tau)$ are not enough to guarantee that
$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$ is a modular
form. However we have the following results.

Theorem 3.2 Let $M$ be a $4k-1$ dimensional manifold and
$\nabla_{0}^{TM},~{}\nabla_{1}^{TM}$ be two connections on $TM$ and $\xi$ be a
two dimensional oriented Euclidean real vector bundle with a Euclidean
connection $\nabla^{\xi}$, then we have
1)
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2)$;
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2);$
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,

$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau+1)=CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$

Proof. By (2.12)-(2.17), we have

$z\frac{\theta^{\prime}(0,-\frac{1}{\tau})}{\theta(z,-\frac{1}{\tau})}\frac{\theta_{1}(z,-\frac{1}{\tau})}{\theta_{1}(0,-\frac{1}{\tau})}=(\tau
z)\frac{\theta^{\prime}(0,{\tau})}{\theta(\tau z,{\tau})}\frac{\theta_{2}(\tau
z,{\tau})}{\theta_{2}(0,{\tau})};$ $None$
$\frac{1}{z}-\frac{\theta^{\prime}(z,-\frac{1}{\tau})}{\theta(z,-\frac{1}{\tau})}+\frac{\theta_{1}^{\prime}(z,-\frac{1}{\tau})}{\theta_{1}(z,-\frac{1}{\tau})}=\tau\left(\frac{1}{\tau
z}-\frac{\theta^{\prime}(\tau z,{\tau})}{\theta(\tau
z,{\tau})}+\frac{\theta_{2}^{\prime}(\tau z,{\tau})}{\theta_{2}(\tau
z,{\tau})}\right);$ $None$
$\frac{\theta_{1}^{2}(0,-\frac{1}{\tau})}{\theta_{1}^{2}(u,-\frac{1}{\tau})}\frac{\theta_{3}(u,-\frac{1}{\tau})}{\theta_{3}(0,-\frac{1}{\tau})}\frac{\theta_{2}(u,-\frac{1}{\tau})}{\theta_{2}(0,-\frac{1}{\tau})}=\frac{\theta_{2}^{2}(0,{\tau})}{\theta_{2}^{2}(\tau
u,{\tau})}\frac{\theta_{3}(\tau
u,{\tau})}{\theta_{3}(0,{\tau})}\frac{\theta_{1}(u\tau,{\tau})}{\theta_{1}(0,{\tau})}.$
$None$

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

$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$None$

Similarly we can show that

$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau),$
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\frac{\tau}{2})^{2k}\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau),$
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\tau)^{2k}\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$
$None$

From (3.17) and (3.18), we can get
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau})\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2).$ Similarly we can prove
that
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$ $\Box$
Let $M$ be a compact oriented smooth $3$-dimensional manifold, then our
transgressed forms are same as transgressed forms in the untwisted case which
have been computed in [CH]. 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
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and

$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(11)}=(2\tau)^{6}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}.$

By Lemma 2.2, we have

$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$

and by (2.19) and Theorem 3.2,

$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$

By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (3.19), we get

$z_{0}=-\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2}){\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right\\}^{(11)},$ $None$
$z_{1}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2})\left({\rm
ch}(T_{C}M,\nabla^{T_{C}M}_{t})-3(e^{c}+e^{-c}-2)\right)\right.$
$\left.\times{\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right.+$
$\left.\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm cosh}(\frac{c}{2}){\rm
tr}\left[A\left(-\frac{1}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{4\pi}+61\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right)\right]dt\right\\}^{(11)}.$ $None$

Plugging (3.21) and (3.22) into (3.20) and comparing the constant terms of
both sides, we obtain that

$\left\\{\int_{0}^{1}\frac{\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})}{{\rm
cosh}^{2}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(11)}=2^{3}(2^{6}z_{0}+z_{1}),$

so we have the following $11$-dimensional analogue of the twisted miraculous
cancellation formula.

Corollary 3.3 The following equality holds

$\left\\{\int_{0}^{1}\frac{\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})}{{\rm
cosh}^{2}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(11)}$
$=8\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2})\left({\rm
ch}(T_{C}M,\nabla^{T_{C}M}_{t})-3(e^{c}+e^{-c}-2)\right)\right.$
$\left.\times{\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right.+$
$\left.\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm cosh}(\frac{c}{2}){\rm
tr}\left[A\left(-\frac{1}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}-3\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right)\right]dt\right\\}^{(11)}.$ $None$

Next we consider the transgression of
$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau),~{}\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)$,
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)$ about $\nabla^{\xi}$. Let
$\nabla_{1}^{\xi},~{}\nabla_{0}^{\xi}$ be two Euclidean connections on $\xi$
and $B=\nabla_{1}^{\xi}-\nabla_{0}^{\xi}$. By (3.6)-(3.9), we have
$\Phi_{L}(\nabla^{TM},\nabla_{1}^{\xi},\tau)-\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\tau)$

$=\frac{1}{8\pi^{2}}d\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

We define
$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$

$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k-1$ dimensional,
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
represents an element in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can compute the transgressed forms for $\Phi_{W},~{}\Phi_{W}^{\prime}$
respectively and define
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{2}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$

$CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$

which also lie in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]]$ and their
top components represent elements in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$.
Similarly we have

Theorem 3.4 Let $M$ be a $4k-1$ dimensional manifold and $\nabla^{TM}$ be a
connection on $TM$ and $\xi$ be a two dimensional oriented Euclidean real
vector bundle with two Euclidean connections $\nabla_{1}^{\xi}$,
$\nabla_{0}^{\xi}$, then we have
1)
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2)$;
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2);$
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,

$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau).$

Proof. By (3.14),(3.16) and

$\frac{\theta_{2}^{\prime}(z,-\frac{1}{\tau})}{\theta_{2}(z,-\frac{1}{\tau})}+\frac{\theta_{3}^{\prime}(z,-\frac{1}{\tau})}{\theta_{3}(z,-\frac{1}{\tau})}-2\frac{\theta_{1}^{\prime}(z,-\frac{1}{\tau})}{\theta_{1}(z,-\frac{1}{\tau})}=\tau\left(\frac{\theta_{1}^{\prime}(\tau
z,{\tau})}{\theta_{1}(\tau z,{\tau})}+\frac{\theta_{3}^{\prime}(\tau
z,{\tau})}{\theta_{3}(\tau z,{\tau})}-2\frac{\theta_{2}^{\prime}(\tau
z,{\tau})}{\theta_{2}(\tau z,{\tau})}\right),$ $None$

we can get

$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{TM},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$None$

Similarly we can show that

$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau),$
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\frac{\tau}{2})^{2k}\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau),$
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\tau)^{2k}\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau).$
$None$

From (3.28) and (3.29), we can get
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau})\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2).$ Similarly we can prove
that
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$ $\Box$
Let $M$ be a compact oriented smooth $3$-dimensional manifold, we have

$\displaystyle CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi^{2}}\int_{0}^{1}{\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt$
$\displaystyle=$ $\displaystyle\frac{1}{8\pi^{4}}\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}\int_{0}^{1}{\rm
tr}[BR_{t}^{\xi}]dt.$

Since $\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}$
is a modular form of weight $2$ over $\Gamma_{0}(2),$ then it is a scalar
multiple of $\delta_{1}(\tau)$. Direct computations show

$\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}=2\pi^{2}+O(q^{\frac{1}{2}}),$

so

$\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}=8\pi^{2}\delta_{1}(\tau).$

By (4.15) in [CH], we have

$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{2\pi^{2}}\delta_{1}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right].$ $None$

Similarly, we obtain that

$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{8\pi^{2}}\delta_{2}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right],$ $None$
$CS\Phi^{\prime}_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{8\pi^{2}}\delta_{3}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right].$ $None$

Let $M$ be $11$ dimensional and $k=3$. We have that
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and

$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(11)}=(2\tau)^{6}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}.$

By Lemma 2.2, we have

$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$

and by (2.19) and Theorem 3.4,

$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$

By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (3.33), we get

$z_{0}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt\right\\}^{(11)},$ $None$

$z_{1}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi})\left(3{\rm
ch}(\xi_{C},\nabla^{\xi_{C}}_{t})-{\rm
ch}(T_{C}M,\nabla^{T_{C}M})+77\right)\right.$ $\left.\times{\rm
tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt+\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{3B}{2\pi}{\rm
sin}\frac{R_{t}^{\xi}}{2\pi}\right]dt\right\\}^{(11)}.$ $None$

Plugging (3.35) and (3.36) into (3.34) and comparing the constant terms of
both sides, we obtain that

Corollary 3.5 The following equality holds

$\left\\{\int_{0}^{1}\frac{\widehat{L}(TM,\nabla^{TM})}{{\rm
cos}^{2}(\frac{R_{t}^{\xi}}{4\pi})}{\rm tr}\left[B{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt\right\\}^{(11)}$
$=16\sqrt{2}\pi\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi})\left(3{\rm
ch}(\xi_{C},\nabla^{\xi_{C}}_{t})-{\rm
ch}(T_{C}M,\nabla^{T_{C}M})+13\right)\right.$ $\left.\times{\rm
tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt+\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{3B}{2\pi}{\rm
sin}\frac{R_{t}^{\xi}}{2\pi}\right]dt\right\\}^{(11)}.$ $None$

##  4 Transgressed forms and modularities on $4k+1$ dimensional manifolds

Let $M$ be a $4k+1$ dimensional Riemannian manifold. Set

$\Theta_{1}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}),$
$\Theta_{2}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{q^{s}}(\widetilde{\xi_{C}}),$
$\Theta_{3}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m-\frac{1}{2}}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}).$
$None$

Define

$\displaystyle\widetilde{\Phi_{L}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle\widehat{L}(TM,\nabla^{TM})\frac{{\rm
cosh}(\frac{c}{2})}{{\rm sinh}(\frac{c}{2})}$ $\displaystyle\cdot\left({\rm
ch}(\Theta_{1}(T_{C}M+\xi_{C},C^{2}))-\frac{{\rm
ch}(\Theta_{1}(T_{C}M+\xi_{C},\xi_{C}))}{{\rm cosh}^{2}(\frac{c}{2})}\right),$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle{\widehat{A}(TM,\nabla^{TM})}\frac{1}{2{\rm
sinh}(\frac{c}{2})}\left({\rm ch}(\Theta_{2}(T_{C}M+\xi_{C},C^{2}))\right.$
$\displaystyle\left.-{\rm cosh}(\frac{c}{2}){\rm
ch}(\Theta_{2}(T_{C}M+\xi_{C},\xi_{C}))\right)$
$\displaystyle\widetilde{\Phi^{\prime}_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle{\widehat{A}(TM,\nabla^{TM})}\frac{1}{2{\rm
sinh}(\frac{c}{2})}\left({\rm ch}(\Theta_{3}(T_{C}M+\xi_{C},C^{2}))\right.$
$\displaystyle\left.-{\rm cosh}(\frac{c}{2}){\rm
ch}(\Theta_{3}(T_{C}M+\xi_{C},\xi_{C}))\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.2)$

Through direct computations, we get (cf. [HH])

$\displaystyle\widetilde{\Phi_{L}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}^{4k+1}}{\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}-\frac{\theta_{1}(0,\tau)}{\theta_{1}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\right)$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}-\frac{\theta_{2}(0,\tau)}{\theta_{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\right)$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{3}(x_{j},\tau)}{\theta_{3}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}-\frac{\theta_{3}(0,\tau)}{\theta_{3}(u,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\right)~{}~{}~{}~{}~{}~{}(4.3)$

Applying the Chern-Weil theory and Lemma 3.1 again, we can transgress
$\widetilde{\Phi_{L}},~{}\widetilde{\Phi_{W}},\widetilde{\Phi_{W}^{\prime}}$
about $\nabla_{TM}$ and define transgressed forms as follows:
$CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{L}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k+1$ dimensional,
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
represents an element in $H^{4k+1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can define
$CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{W}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{2}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt;$
$None$

$CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$

$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{W}^{\prime}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$

Using the same discussions as Theorem 3.2, we obtain

Theorem 4.1 Let $M$ be a $4k+1$ dimensional manifold and
$\nabla_{0}^{TM},~{}\nabla_{1}^{TM}$ be two connections on $TM$ and $\xi$ be a
two dimensional oriented Euclidean real vector bundle with a Euclidean
connection $\nabla^{\xi}$, then we have
1)
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma_{0}(2)$;
$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma^{0}(2);$
$\\{CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,

$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k+1)}=(2\tau)^{2k+2}\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)},$
$CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau+1)=CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$

Let $M$ be $9$ dimensional and $k=2$. We have that
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and

$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(9)}=(2\tau)^{6}\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}.$

By Lemma 2.2, we have

$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$

and by (2.19) and Theorem 4.1,

$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$

By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (4.7), we get

$z_{0}=-\left\\{\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right\\}^{(9)},$ $None$
$z_{1}=\left\\{-\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm tr}\left[\frac{A}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}\right]dt\right.$
$+\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]$ $\left.\cdot\left((1-{\rm
cosh}\frac{c}{2})({\rm ch}(T_{C}M,\nabla_{t}^{T_{C}M})+61)+(1+2{\rm
cosh}\frac{c}{2})(e^{c}+e^{-c}-2)\right)dt\right\\}^{(9)}.$ $None$

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

$\left\\{\int_{0}^{1}\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})\frac{{\rm
sinh}\frac{c}{2}}{{\rm cosh}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(9)}$
$=8\left\\{-\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm tr}\left[\frac{A}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}\right]dt\right.$
$+\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]$ $\left.\cdot\left((1-{\rm
cosh}\frac{c}{2})({\rm ch}(T_{C}M,\nabla_{t}^{T_{C}M})-3)+(1+2{\rm
cosh}\frac{c}{2})(e^{c}+e^{-c}-2)\right)dt\right\\}^{(9)}.$ $None$

Acknowledgement This work was supported by NSFC No. 10801027\.

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

E-mail<EMAIL_ADDRESS>