Chern-weil Theory in Odd Dimension

In the previous section, we discussed the theory of even dimensional characteristic forms and classes. i.e.
Let $M$ be a smooth closed manifold and $E$ be a vector bundle with a connection $\nabla^E$. We constructed a serial closed form
\[
\tr\left[f\left(\frac{\sqrt{-1}}{2\pi}R^E \right) \right] \in \Omega^{even}(M),
\]where $f$ is a power series in one variable. Then we obtain some even dimensional characteristic classes
\[
\Bigg[\tr\left[f\left(\frac{\sqrt{-1}}{2\pi}R^E \right) \right]\Bigg] \in H^{even}_{dR}(M,\C).
\]In this section, we will discuss an odd dimensional analogue of this result.

Let $M$ be a smooth closed manifold. Let $g$ be a smooth map from $M$ to the general linear group $GL(N,\C)$ with $N>0$ a positive integer:
\begin{gather*}
g:~M\rightarrow GL(M,\C),\quad p\rightarrow g(p)\in GL(N,\C),\\
i.e. \quad g=(g_{ij})_{N\times N} ~~where~~ g_{ij}\in C^{\infty}(M,\C).
\end{gather*}
Let $\C^N|_M$ denote the trivial complex vector bundle of rank $N$ over $M$. Then the above $g$ can be viewed as a section of $Aut\left(\C^N|_M\right)$. Let $\rd$ denote a trivial connection on $\C^N|_M$. Then we consider n-form $\tr\left[ \left( g^{-1}dg \right)^n\right]$.

  1. Assume $n$ is a positive even integer,
    \begin{align*}
    \tr\left[\left(g^{-1}\rd g \right)^n \right]=&\frac{1}{2} \tr \left[\left( g^{-1}\rd g \right)^{n-1}\left(g^{-1}\rd g \right)+\left(g^{-1}\rd g \right)\left(g^{-1}\rd g \right)^{n-1} \right]\\
    =&\frac{1}{2} \tr \left[\left( g^{-1}\rd g \right)^{n-1}\left(g^{-1}\rd g \right) -(-1)^{(n-1)\cdot1}\left(g^{-1}\rd g \right)\left(g^{-1}\rd g \right)^{n-1} \right]\\
    =&\tr\left[ \left(g^{-1}\rd g \right)^{n-1} ,\left(g^{-1}\rd g \right)\right]\\
    =&0.
    \end{align*}
  2. Assume $n$ is a positive odd integer. Notice that $I=gg^{-1}$, then $\rd g^{-1}=-g^{-1}(\rd g)g^{-1}$. Hence
    \begin{align*}
    \rd \left[ \left(g^{-1}\rd g\right)^n\right]=&\sum_{i=1}^n (-1)^{i-1} \left(g^{-1}\rd g\right)^{i-1}\rd \left(g^{-1}\rd g\right) \left(g^{-1}\rd g\right)^{n-i}\\
    =&\sum_{i=1}^n (-1)^{i-1} \left(g^{-1}\rd g\right)^{i-1} \left( \rd g^{-1}\right)\left( \rd g\right) \left(g^{-1}\rd g\right)^{n-i}\\
    =&\sum_{i=1}^n (-1)^{i-1} \left(g^{-1}\rd g\right)^{i-1} \left( -g^{-1}(\rd g)g^{-1}\right)\left( \rd g\right) \left(g^{-1}\rd g\right)^{n-i}\\
    =&\sum_{i=1}^n (-1)^{i} \left(g^{-1}\rd g\right)^{n+1}\\
    =&-\left(g^{-1}\rd g\right)^{n+1}. \quad [\text{by $n$ is a odd positive integer}]
    \end{align*}
    By the above case,
    \[
    \rd \tr\Big[ \left(g^{-1}\rd g\right)^n \Big]=\tr\Big[\rd \left(g^{-1}\rd g\right)^n \Big]=- \tr\Big[ \left(g^{-1}\rd g\right)^{n+1} \Big]=0.
    \]
This show $\tr\left[ \left(g^{-1}\rd g\right)^n \right]$ is a closed form when $n$ is a positive odd integer. Roughly speaking, the cohomology class $\left[\tr\left[ \left(g^{-1}\rd g\right)^n \right]\right]\in H^{n}_{dR}(M,\C)$ depends on $g$ and trivial connection $\rd$. The following lemma shows that cohomology class does not depend on smooth deformations(homotopy) of $g$.
引理 1. If $g_t:~M\rightarrow GL(N,\C)$ depends smoothly on $t\in[0,1]$, then for any positive odd integer $n$, the following identity holds,
\[
\frac{\partial}{\partial t}\tr \Big[ \left( g_t^{-1}\rd g_t\right)^n \Big]=n\rd \left[ g_t^{-1}\frac{\partial g_t}{\partial t}\left( g_t^{-1}\rd g_t\right)^{n-1} \right].
\]

If this lemma have been proved, one can integrate at both sides of above identity
\begin{align*}
&\tr \Big[ \left( g_1^{-1}\rd g_1\right)^n \Big]-\tr \Big[ \left( g_0^{-1}\rd g_0\right)^n \Big]\\=&\int_{0}^1\frac{\partial}{\partial t}\tr \Big[ \left( g_t^{-1}\rd g_t\right)^n \Big]\rd t
=n\int_0^1\rd\Big[ g_t^{-1}\frac{\partial g_t}{\partial t}\left( g_t^{-1}\rd g_t\right)^{n-1}\Big]\rd t\\
=&\rd \left[n\int_0^1\Big[ g_t^{-1}\frac{\partial g_t}{\partial t}\left( g_t^{-1}\rd g_t\right)^{n-1}\Big]\rd t\right].
\end{align*}
Let \[
\eta=n\int_0^1\Big[ g_t^{-1}\frac{\partial g_t}{\partial t}\left( g_t^{-1}\rd g_t\right)^{n-1}\Big]\rd t\in \Omega^{n-1}(M),
\] then $\tr \Big[ \left( g_1^{-1}\rd g_1\right)^n \Big]-\tr \Big[ \left( g_0^{-1}\rd g_0\right)^n \Big]=\rd \eta$, i.e.
\[
\Big[\tr \left[ \left( g_1^{-1}\rd g_1\right)^n \right]\Big]=\Big[\tr \left[ \left( g_0^{-1}\rd g_0\right)^n \right]\Big].
\]Now, we give the proof of lemma 1.
证明 . By an analogue of $\rd g^{-1}=-g^{-1}(\rd g)g^{-1}$, one can obtain $\frac{\partial}{\partial t} g_t^{-1}=-g_t^{-1}\left(\frac{\partial g_t}{\partial t}\right)g_t^{-1}$. And, if $A\in\Omega^{odd}(M,End(\C^N))$, $B\in\Omega^{even}(M,End(\C^N))$, one easily verifies that $AB=BA$ by $\tr[A,B]=0$. Hence
\begin{align*}
\frac{\partial}{\partial t}\tr \left[\left( g_t^{-1}\rd g_t \right)^n\right]=&\tr \left[ \frac{\partial}{\partial t}\left( g_t^{-1}\rd g_t \right)^n\right]\\
=&\tr \left[ \sum_{i=1}^n \left( g_t^{-1}\rd g_t \right)^{i-1} \frac{\partial}{\partial t}\left( g_t^{-1}\rd g_t \right) \left( g_t^{-1}\rd g_t \right)^{n-i} \right]\\
=&\tr \left[ \sum_{i=1}^n \frac{\partial}{\partial t}\left( g_t^{-1}\rd g_t \right) \left( g_t^{-1}\rd g_t \right)^{n-1} \right] \qquad \text{by $n$ is an odd integer}\\
=&n\tr \left[\frac{\partial g_t^{-1}}{\partial t} \left(\rd g_t\right) \left( g_t^{-1}\rd g_t \right)^{n-1} + g_t^{-1} \left(\rd \frac{\partial g_t}{\partial t}\right) \left( g_t^{-1}\rd g_t \right)^{n-1}\right]\\
=&n\tr \left[-g_t^{-1}\frac{\partial g_t}{\partial t} g_t^{-1} \left(\rd g_t\right) \left( g_t^{-1}\rd g_t \right)^{n-1} + g_t^{-1} \left(\rd \frac{\partial g_t}{\partial t}\right) \left( g_t^{-1}\rd g_t \right)^{n-1}\right]\\
=&n\tr \Big[-g_t^{-1}\left(\frac{\partial g_t}{\partial t} \right) \left( g_t^{-1}\rd g_t \right)^{n} +\rd \left( g_t^{-1} \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1} \right) \\
&- \left(\rd g_t^{-1}\right) \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1} – g_t^{-1} \frac{\partial g_t}{\partial t}\rd \left(\left( g_t^{-1}\rd g_t \right)^{n-1}\right)
\Big]\\
=&n\tr \Big[-g_t^{-1}\left(\frac{\partial g_t}{\partial t} \right) \left( g_t^{-1}\rd g_t \right)^{n} +\rd \left( g_t^{-1} \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1} \right) \\
& + g_t^{-1}\left(\rd g_t\right)g_t^{-1} \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1}
\Big]\\
=&n\tr \Big[-g_t^{-1}\left(\frac{\partial g_t}{\partial t} \right) \left( g_t^{-1}\rd g_t \right)^{n} +\rd \left( g_t^{-1} \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1} \right) \\
& + g_t^{-1}\left(\frac{\partial g_t}{\partial t} \right) \left( g_t^{-1}\rd g_t \right)^{n}
\Big]\\
=&n\rd \tr\left[ g_t^{-1} \frac{\partial g_t}{\partial t} \left( g_t^{-1}\rd g_t \right)^{n-1} \right].
\end{align*}

推论 2. If $f,g:~M\rightarrow L(N,\C)$ are two smooth maps from $M$ to $GL(N,\C)$, then for any positive odd integer $n$, there exists $\omega_n\in\Omega^{n-1}(M)$ such that the following transgression formula holds,
\[
\tr\left[\left((fg)^{-1}\rd (fg)\right)^n\right]= \tr\left[\left(f^{-1}\rd f\right)^n\right]+\tr\left[\left(g^{-1}\rd g\right)^n\right]+\rd \omega_n.
\]

证明 . We consider the trivial vector bundle
\[
\C^{2N}|_M=\C^{N}|_M\oplus\C^{N}|_M.
\]We equip $\C^{2N}|_M$ with the trivial connection $\tilde{d}$ induced from $d$ on $\C^{N}|_M$. For any $u\in[0,\frac{\pi}{2}]$, let $h_u:~M\rightarrow GL(2N,\C)$ be defined by
\[
h_u=\left(\begin{array}{cc}
f & 0 \\
0 & 1
\end{array}\right)
\left(\begin{array}{cc}
\cos(u) & \sin(u) \\
-\sin(u) & \cos(u)
\end{array}\right)
\left(\begin{array}{cc}
1 & 0 \\
0 & g
\end{array}\right)
\left(\begin{array}{cc}
\cos(u) & -\sin(u) \\
\sin(u) & \cos(u)
\end{array}\right).
\]Obviously,
\[
h_0=\left(\begin{array}{cc}
f & 0 \\
0 & g
\end{array}
\right),\quad
\left(\begin{array}{cc}
fg & 0 \\
0 & 1
\end{array}\right)
\] Thus, $h_u$ provides a smooth homotopy between two sections $(fg,1)$ and $(f,g)$ in $\Gamma($ $Aut(\C^{2N}|_M))$. By the above lemma, for any positive odd inetger $n$, there exists $\omega_n\in\Omega^{n-1}(M)$ such that
\[
\tr\left[\left(h_0^{-1}\tilde{\rd}h_0\right)^n\right]=\tr\left[\left(h_{\frac{\pi}{2}}^{-1}\tilde{\rd} h_{\frac{\pi}{2}}^{-1}\right)^n\right]+\rd \omega_n
\]
i.e.,
\[
\tr\left[\left((fg)^{-1}\rd (fg)\right)^n\right]= \tr\left[\left(f^{-1}\rd f\right)^n\right]+\tr\left[\left(g^{-1}\rd g\right)^n\right]+\rd \omega_n.
\]

Now, we consider the change of $\tr\left[ \left(g^{-1}\rd g\right)^n \right]$ under different trivial connection. Let $E$ be a trivial complex vector bundle on $M$. As we take a global basis of $E$, a trivialization is determined, and a trivial connection is determined. Let $\rd$ be the trivial connection associated basis $\{e_1,\cdots,e_n\}$. Assume $\{e_1′,\cdots,e_n’\}$ is another basis of $E$, and
\[
(e_1′,\cdots,e_n’)=(e_1,\cdots,e_n)A, \quad A\in \Gamma\left(Aut(\C^N|M)\right).
\]
If $\rd’$ is the trivial connection associated $\{e_1′,\cdots,e_n’\}$, then one can verify
\[
d’=A^{-1}\circ \rd\circ A.
\]
练习 1. Please verify $d’=A^{-1}\circ \rd\circ A$.

推论 3. Let $g\in\Gamma\left(Aut(\C^N|M)\right)$. If $d’$ is another trivial connection on $\C^N|M)$, then for any positive odd integer $n$, there exists $\omega_n\in \Omega^{n-1}(M)$ such that the following transgression formula holds,
\[
\tr\left[(g^{-1}\rd g)^n \right] =\tr\left[(g^{-1}\rd’ g)^n \right] +\rd \omega_n.
\]

证明 . There exists $A\in\Gamma\left(Aut(\C^N|M)\right)$ such that
\[
\rd’=A^{-1} \circ\rd \circ A=\rd+A^{-1}\circ(\rd A).
\]One deduces that
\begin{align*}
g^{-1}\rd’ g=&g^{-1}\circ \rd’\circ g-\rd’\\
=&g^{-1}\circ A^{-1}\circ \rd’\circ A\circ g-A^{-1}\circ \rd\circ A\\
=&A^{-1}\left( A\circ g^{-1}\circ A^{-1}\circ \rd’\circ A\circ g\circ A^{-1}-\rd \right)A\\
=&A^{-1}\left( (AgA^{-1})\rd (AgA^{-1}) \right)A.
\end{align*}
From above corollary, there exists $\omega_n\in\Omega^{n-1}(M)$ for any positive odd integer $n$ such that
\begin{align*}
\tr\left[(g^{-1}\rd’ g )^n\right]=& \tr\left[ A^{-1}\left( (AgA^{-1})\rd (AgA^{-1}) \right)^n A \right]\\
=&\tr\left[ \left( (AgA^{-1})\rd (AgA^{-1}) \right)^n\right]\\
=&\tr\left[ (A^{-1}\rd A )^n \right] + \tr\left[ (A\rd A^{-1} )^n\right] + \tr\left[(g^{-1}\rd g )^n\right] – \rd \omega_n\\
=&\tr\left[(g^{-1}\rd g )^n \right]- \rd \omega_n. \quad \left(\text{It is from $\rd A=-A(\rd A^{-1})A$}\right)
\end{align*}

注记 1. The cohomology class determine by $\tr\left[(g^{-1}\rd g)^n\right]$ depends only on the homotopy class of $g:~M\rightarrow GL(N,\C)$.

When $n$ is a positive odd integer, we call the closed n-form
\[
\left(\frac{1}{2\pi\sqrt{(-1)}}\right)^{\frac{n+1}{2}}\tr\Big[(g^{-1}\rd g)^n\Big]
\]
the n-th Chern form associated to $g,\rd$ and denote it by $c_n(g,\rd)$. The associated cohomology class will be called the n-th Chern class associated to the homotopy class $[g]$, denote it by $c_n([g])$.

We define the odd Chern character form associated to $g,\rd$ by
\[
ch(g,\rd)=\sum_{n=0}^{\infty}\frac{n!}{(2n+1)!}c_{2n+1}(g,\rd).
\] Let $ch([g])$ denote the associated cohomology class which we call the odd Chern character associated to $[g]$.

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