In the previous section, we discussed the theory of even dimensional characteristic forms and classes. i.e.
\nLet $M$ be a smooth closed manifold and $E$ be a vector bundle with a connection $\\nabla^E$. We constructed a serial closed form
\n\\[
\n\\tr\\left[f\\left(\\frac{\\sqrt{-1}}{2\\pi}R^E \\right) \\right] \\in \\Omega^{even}(M),
\n\\]where $f$ is a power series in one variable. Then we obtain some even dimensional characteristic classes
\n\\[
\n\\Bigg[\\tr\\left[f\\left(\\frac{\\sqrt{-1}}{2\\pi}R^E \\right) \\right]\\Bigg] \\in H^{even}_{dR}(M,\\C).
\n\\]In this section, we will discuss an odd dimensional analogue of this result.<\/p>\n
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:
\n\\begin{gather*}
\ng:~M\\rightarrow GL(M,\\C),\\quad p\\rightarrow g(p)\\in GL(N,\\C),\\\\
\ni.e. \\quad g=(g_{ij})_{N\\times N} ~~where~~ g_{ij}\\in C^{\\infty}(M,\\C).
\n\\end{gather*}
\nLet $\\C^N|_M$ 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 $\\tr\\left[ \\left( g^{-1}dg \\right)^n\\right]$.
\n\\begin{enumerate}
\n \\item Assume $n$ is a positive even integer,
\n \\begin{align*}
\n \\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]\\\\
\n =&\\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]\\\\
\n =&\\tr\\left[ \\left(g^{-1}\\rd g \\right)^{n-1} ,\\left(g^{-1}\\rd g \\right)\\right]\\\\
\n =&0.
\n \\end{align*}
\n \\item Assume $n$ is a positive odd integer. Notice that $I=gg^{-1}$, then $\\rd g^{-1}=-g^{-1}(\\rd g)g^{-1}$. Hence
\n \\begin{align*}
\n \\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}\\\\
\n =&\\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}\\\\
\n =&\\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}\\\\
\n =&\\sum_{i=1}^n (-1)^{i} \\left(g^{-1}\\rd g\\right)^{n+1}\\\\
\n =&-\\left(g^{-1}\\rd g\\right)^{n+1}. \\quad [\\text{by $n$ is a odd positive integer}]
\n \\end{align*}
\n By the above case,
\n \\[
\n \\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.
\n \\]\\end{enumerate}
\nThis 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 $g$.
\n\\begin{lem}
\nIf $g_t:~M\\rightarrow GL(N,\\C)$ depends smoothly on $t\\in[0,1]$, then for any positive odd integer $n$, the following identity holds,
\n\\[
\n \\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].
\n\\]\\end{lem}
\nIf this lemma have been proved, one can integrate at both sides of above identity
\n\\begin{align*}
\n&\\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=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\\\\
\n=&\\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].
\n\\end{align*}
\nLet \\[
\n\\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),
\n\\] 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.
\n\\[
\n\\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].
\n\\]Now, we give the proof of lemma 1.
\n\\begin{proof} 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
\n\\begin{align*}
\n\\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]\\\\
\n=&\\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]\\\\
\n=&\\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=&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=&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=&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) \\\\
\n&- \\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)
\n\\Big]\\\\
\n=&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) \\\\
\n& + 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}
\n\\Big]\\\\
\n=&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) \\\\
\n& + g_t^{-1}\\left(\\frac{\\partial g_t}{\\partial t} \\right) \\left( g_t^{-1}\\rd g_t \\right)^{n}
\n\\Big]\\\\
\n=&n\\rd \\tr\\left[ g_t^{-1} \\frac{\\partial g_t}{\\partial t} \\left( g_t^{-1}\\rd g_t \\right)^{n-1} \\right].
\n\\end{align*}
\n\\end{proof}
\n\\begin{cor}
\nIf $f,g:~M\\rightarrow L(N,\\C)$ 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,
\n\\[
\n\\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.
\n\\]\\end{cor}
\n\\begin{proof}We consider the trivial vector bundle
\n\\[
\n\\C^{2N}|_M=\\C^{N}|_M\\oplus\\C^{N}|_M.
\n\\]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
\n\\[
\nh_u=\\left(\\begin{array}{cc}
\n f & 0 \\\\
\n 0 & 1
\n \\end{array}\\right)
\n \\left(\\begin{array}{cc}
\n \\cos(u) & \\sin(u) \\\\
\n -\\sin(u) & \\cos(u)
\n \\end{array}\\right)
\n \\left(\\begin{array}{cc}
\n 1 & 0 \\\\
\n 0 & g
\n \\end{array}\\right)
\n \\left(\\begin{array}{cc}
\n \\cos(u) & -\\sin(u) \\\\
\n \\sin(u) & \\cos(u)
\n \\end{array}\\right).
\n\\]Obviously,
\n\\[
\nh_0=\\left(\\begin{array}{cc}
\n f & 0 \\\\
\n 0 & g
\n \\end{array}
\n\\right),\\quad
\n\\left(\\begin{array}{cc}
\n fg & 0 \\\\
\n 0 & 1
\n\\end{array}\\right)
\n\\] 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
\n\\[
\n\\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
\n\\]
\ni.e.,
\n\\[
\n\\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.
\n\\]\\end{proof}
\nNow, 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
\n\\[
\n (e_1′,\\cdots,e_n’)=(e_1,\\cdots,e_n)A, \\quad A\\in \\Gamma\\left(Aut(\\C^N|M)\\right).
\n\\]
\nIf $\\rd’$ is the trivial connection associated $\\{e_1′,\\cdots,e_n’\\}$, then one can verify
\n\\[
\n d’=A^{-1}\\circ \\rd\\circ A.
\n\\]
\n\\begin{excs}
\nPlease verify $d’=A^{-1}\\circ \\rd\\circ A$.
\n\\end{excs}
\n\\begin{cor}\\label{cor:1-19}
\nLet $g\\in\\Gamma\\left(Aut(\\C^N|M)\\right)$. 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,
\n\\[
\n\\tr\\left[(g^{-1}\\rd g)^n \\right] =\\tr\\left[(g^{-1}\\rd’ g)^n \\right] +\\rd \\omega_n.
\n\\]\\end{cor}
\n\\begin{proof}
\nThere exists $A\\in\\Gamma\\left(Aut(\\C^N|M)\\right)$ such that
\n\\[
\n\\rd’=A^{-1} \\circ\\rd \\circ A=\\rd+A^{-1}\\circ(\\rd A).
\n\\]One deduces that
\n\\begin{align*}
\ng^{-1}\\rd’ g=&g^{-1}\\circ \\rd’\\circ g-\\rd’\\\\
\n=&g^{-1}\\circ A^{-1}\\circ \\rd’\\circ A\\circ g-A^{-1}\\circ \\rd\\circ A\\\\
\n=&A^{-1}\\left( A\\circ g^{-1}\\circ A^{-1}\\circ \\rd’\\circ A\\circ g\\circ A^{-1}-\\rd \\right)A\\\\
\n=&A^{-1}\\left( (AgA^{-1})\\rd (AgA^{-1}) \\right)A.
\n\\end{align*}
\nFrom above corollary, there exists $\\omega_n\\in\\Omega^{n-1}(M)$ for any positive odd integer $n$ such that
\n\\begin{align*}
\n\\tr\\left[(g^{-1}\\rd’ g )^n\\right]=& \\tr\\left[ A^{-1}\\left( (AgA^{-1})\\rd (AgA^{-1}) \\right)^n A \\right]\\\\
\n=&\\tr\\left[ \\left( (AgA^{-1})\\rd (AgA^{-1}) \\right)^n\\right]\\\\
\n=&\\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\\\\
\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)
\n\\end{align*}
\n\\end{proof}
\n\\begin{rem}The cohomology class determine by $\\tr\\left[(g^{-1}\\rd g)^n\\right]$ depends only on the homotopy class of $g:~M\\rightarrow GL(N,\\C)$.
\n\\end{rem}
\nWhen $n$ is a positive odd integer, we call the closed n-form
\n\\[
\n\\left(\\frac{1}{2\\pi\\sqrt{(-1)}}\\right)^{\\frac{n+1}{2}}\\tr\\Big[(g^{-1}\\rd g)^n\\Big]
\n\\]
\nthe n-th \\emph{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 $c_n([g])$.<\/p>\n
We define the \\emph{odd Chern character form} associated to $g,\\rd$ by
\n\\[
\n ch(g,\\rd)=\\sum_{n=0}^{\\infty}\\frac{n!}{(2n+1)!}c_{2n+1}(g,\\rd).
\n\\] Let $ch([g])$ denote the associated cohomology class which we call the \\emph{odd Chern character} associated to $[g]$.<\/p>\n","protected":false},"excerpt":{"rendered":"
In the previous section, we discussed th …<\/p>\n