\section{(4 March 2009)}
\subsection*{Cyclic cohomology and higher traces.}
To describe topologically quantized transport coefficients we will 
use \( \underset{\text{algebra}}{\mathcal{A}} \), \( 
\underset{\text{trace}}{\tau} \), \( \underset{\text{comm. 
derivns on } \mathcal{A}}{\vec{\delta} = 
(\delta_{1} , \ldots , \delta_{n})} \) such that \( \tau \circ 
\delta_{\nu} = 0 \,\, \forall \nu \).  

This is the realm of \emph{higher 
traces} on Banach algebras.

Let \( \mathcal{B} \) be an associative algebra over \( \mathsf{k} \) 
(field).  Let \( C_{\lambda}^{n} (\mathcal{B}) = \{ \text{cyclic } n 
+ 1 \text{-forms}\} \).  So \( \eta \in C_{\lambda}^{n} (\mathcal{B}) 
\) this is a map 
\[ 
\eta : \underset{n + 1}{\underbrace{\mathcal{B} \times \dots \times 
\mathcal{B}} \rightarrow \mathsf{k}}
\]
which is linear in each argument and \emph{cyclic}:
\[ 
\eta ( A_{0} , \ldots , A_{n} ) = (-1)^{n} \eta ( A_{1} , \ldots , 
A_{n} , A_{0} ) \, , \quad A_{i} \in \mathcal{B}
\]
\( C_{\lambda}^{n} (\mathcal{B}) \) is a VS over \( \mathsf{k} \).

Define a differential operator
\begin{align*}
    \mathtt{b} & : C_{\lambda}^{n} (\mathcal{B}) \rightarrow 
    C_{\lambda}^{n + 1} (\mathcal{B}) \, , \text{where} \\
    \eta \in C_{\lambda}^{n} (\mathcal{B}) \longmapsto &  \\
    \quad \mathtt{b} \eta ( A_{0} 
    , \ldots , A_{n + 1} ) & = \sum_{i = 0}^{n} (-1)^{i} \eta ( A_{0} , 
    \ldots , A_{i} \cdot A_{i + 1} , \ldots , A_{n + 1} )   \\
    & +\, (-1)^{n + 1} \eta ( A_{n + 1} \cdot A_{0} , A_{1} , \ldots , 
    A_{n} )
\end{align*}

\begin{lemma*}[Exercise]
    \( \mathtt{b} \circ \mathtt{b} = 0 \).  The differential complex 
    \( (C_{\lambda}^{n} (\mathcal{B}), \mathtt{b}) \) is a 
    sub-complex (because of cyclic) of the complex for the Hochschild 
    cohomology \( H(\mathcal{B}, \mathcal{B}^{\ast}) \) (of \( 
    \mathcal{B} \) with coefficients in \( \mathcal{B}^{\ast} \)).
\end{lemma*}

\begin{defn*}
    \emph{Cyclic cohomology of} \( \mathcal{B} \) is the 
    cohomology of \( (C_{\lambda}^{n} (\mathcal{B}), \mathtt{b})  \), 
    denoted \( H^{n}C(\mathcal{B}) \).  \( \eta \in C_{\lambda}^{n} 
    (\mathcal{B}) \) is a \emph{cyclic cocycle} if \( \eta \in \krn 
    \mathtt{b} \).
\end{defn*}

Ex 1: \( \mathcal{B} \) a C*-algebra and \( \tau \) a bounded 
trace on \( \mathcal{B} \).
\[ 
\tau \in C_{\lambda}^{0} (\mathcal{B}) \Rightarrow \mathtt{b} 
\tau (A_{0} , A_{1}) = \tau (A_{0}A_{1}) - \tau 
(A_{1}A_{0}) = 0
\] since traces are cyclic.  So a bounded trace is a cyclic \( 0 \)-cocycle.

Ex 2:  \( \mathcal{M} \) an \( n \)-dim smooth, compact manifold 
without boundary, \( \mathcal{B} = C^{\infty} ( \mathcal{M} ) \) and 
for \( \tau \in C_{\lambda}^{n} (\mathcal{B}) \) (\( \diff \) is 
the exterior derivative on \( \mathcal{M} \)):
\[ 
\tau ( f_{0} , \ldots , f_{n} ) := \int_{\mathcal{M}} f_{0} 
\diff f_{1} \diff f_{2} \cdots \diff f_{n} \quad ((n + 
1) \text{-linear form})
\]

\begin{align*}
    f_{0} \diff f_{1} \cdots \diff f_{n} & = (-1)^{n-1}
    \underset{\diff(f_{n} f_{0}) - f_{n} \diff
    f_{0}}{\underbrace{(\diff f_{n}) f_{0}}} \diff f_{1}
    \cdots \diff f_{n-1} \text{ (sign from exterior product)} \\
     & = (-1)^{n - 1} \diff ( f_{n}f_{0} ) \diff f_{1} 
     \cdots \diff f_{n - 1} \, + \, (-1)^{n} f_{n} \diff f_{0} \cdots \diff f_{n - 1} \\
      \Longrightarrow &     \\
     \int_{\mathcal{M}} f_{0} \diff f_{1} \cdots \diff f_{n}  & = 0 + 
     (-1)^{n} \int_{\mathcal{M}} f_{n} \diff f_{0} \cdots \diff f_{n - 1}    \\
       & \hspace{1.5em} |    \\
       & \rule[0em]{0em}{1em}^{\diff ( f_{n}f_{0} ) \diff f_{1} 
     \cdots \diff f_{n - 1} \text{ is exact and there is no boundary}} 
\end{align*}
So \( \tau \) is cyclic.

Now 
\begin{gather*}
    \mathtt{b} \tau ( f_{0} , \ldots , f_{n + 1} ) = \sum_{i = 0}^{n} 
    (-1)^{i} \tau ( f_{0} , \ldots , f_{i} f_{i + 1} , \ldots , f_{n 
    + 1} )  \\
    + \, ( -1 )^{n + 1} \tau ( f_{n + 1}f_{0} , f_{1} , \ldots , f_{n} )  \\
    = \int_{\mathcal{M}} (f_{0} f_{1}) \diff f_{2} \diff 
    f_{3} \cdots \diff f_{n + 1} - \int_{\mathcal{M}} f_{0} \diff 
    (f_{1} f_{2}) \diff f_{3} \cdots \diff f_{n + 1} \\
    + \int_{\mathcal{M}} f_{0} \diff f_{1} \diff (f_{2} f_{3}) \cdots 
    \diff f_{n + 1} - \cdots  \\
    + (-1)^{n} \int_{\mathcal{M}} f_{0} \diff f_{1} \diff f_{2} 
    \cdots \diff (f_{n} f_{n + 1}) + (-1)^{n + 1} \int_{\mathcal{M}} 
    (f_{n + 1}f_{0}) \diff f_{1} \diff f_{2} \cdots \diff f_{n}  \\
    ¥  \\
    ¥
\end{gather*}
and from this expansion it follows that \( \mathtt{b} \tau ( f_{0} , 
\ldots , f_{n + 1} ) = 0 \) as follows:  in each integral except the 
first and last expand \( \diff (f_{i}f_{i + 1}) = f_{i} \diff f_{i+1} 
+ (\diff f_{i}) f_{i + 1} \); the resulting \( 2n + 2 \) integrals 
cancel in pairs (exercise).

So, \( \tau \) \emph{is a cyclic \( n \)-cocycle}

Result: almost, \( HC^{n}\left( C^{\infty} (\mathcal{M}) \right) = 
H^{n}_{de Rahm}(\mathcal{M}) \).

Little catch: \( C^{\infty}(\mathcal{M}) \) is not a C*-algebra.

Reassuring: de Rahm cohomology \( \cong \) \u{C}ech-cohomology and 
therefore is purely topological.

\subsection*{The pairing with \( K \)-theory}
A \underline{cyclic cocycle over a C*-algebra} \( \mathcal{A} \) 
(over \( \mathbb{C} \)) defines a homomorphism \[ K_{0}(\mathcal{A}) 
\rightarrow \mathbb{C} \]

Suppose \( \mathcal{A} \) is unital.  Recall that \( 
K_{0}(\mathcal{A}) \) is made from homotopy classes of projections in 
\( M_{k}(\mathcal{A}) \, \forall k \).  First extend a cyclic cocycle 
\( \eta \) of \( \mathcal{A} \) to one, \( \tilde{\eta} \), on \( 
M_{k}(\mathcal{A}) \).  Let \( \tr \) be the standard trace on \( 
M_{k}(\mathbb{C}) \).  Then, with \( M_{i} \in M_{k}(\mathbb{C}) \) 
and \( A_{i} \in \mathcal{A} \), define
\[ 
\tilde{\eta} ( M_{0} \otimes A_{0} , M_{1} \otimes A_{1} , \ldots , 
M_{n} \otimes A_{n} ) = \tr (M_{0}M_{1} \cdots M_{n}) \eta (A_{0} , 
\ldots , A_{n})
\]
with linear extension to the general element of \( M_{k} 
(\mathcal{A}) \).  Then  \( \tilde{\eta} \) is cyclic.

\begin{proposition*}
   Let \( p \) and \( q \) be homotopic projections in \( M_{k} 
   (\mathcal{A}) \), and let \( \eta \) be a cyclic \( n \)-cocycle.  
   Then
   \[ 
   \tilde{\eta} (\underset{n + 1}{\underbrace{p , p , \ldots , p}}) 
   =  \tilde{\eta} (\underset{n + 1}{\underbrace{q , q , \ldots , q}})
   \]
\end{proposition*}

\begin{remark*}[Marcy]
    If \( n \) is odd then, since \( \tilde{\eta} \) is cyclic, \( 
    \tilde{n} (p, \ldots ,p) = - \tilde{n} (p, \ldots ,p) \), and 
    therefore \( \tilde{n} (p, \ldots ,p) = 0 \).
\end{remark*}

\begin{proof}[Proof of the Proposition.]
    Suppose \( n = 2m \) and let \( m = 1 \) (other values of \( m \) 
    are left as an exercise).  Suppose there is a differentiable 
    homotopy \( p(t) \) between \( p \) and \( q \), so that \( p(0) 
    = p \) and \( p(1) = q \).
    \[ 
    \frac{\diff }{\diff t} \eta (p(t), \ldots , p(t)) = \eta (\dot{p} 
    , p , \ldots , p) + \eta (p , \dot{p} , \ldots , p) + \cdots = (n 
    + 1) \eta ( \dot{p} , p , \ldots , p)
    \]
    From the 27 Feb. lecture,
    \[ 
    \dot{p} = p \dot{p} p^{\perp} + p^{\perp} \dot{p} p
    \]
    
    Now for \( m = 1 \),
\begin{gather*}
     \eta (p \dot{p} p^{\perp} , p , p) = \underset{= 0 \text{ by
     cocycle hypoth.}}{\mathtt{b} \eta (p \dot{p} , p^{\perp} , p ,
     p)} + \underset{p^{\perp} p = 0}{\eta (p \dot{p} , p^{\perp} p ,
     p)} \\
    \underset{\text{cancel because } p^{2} = p}{- \eta (p \dot{p} ,
    p^{\perp} , p^{2}) + \eta (p^{2} \dot{p} , p^{\perp} , p)}  \\
    =0
\end{gather*}
    And a similar argument shows that \( \eta (p^{\perp} \dot{p} p , 
    p , p) = 0 \), so \( \eta (\dot{p} , p , p) = 0 \), and therefore 
    \( \eta (p(t), \ldots , p(t)) \) is constant.
\end{proof}

This allows us to define \( \eta \) and \( \tilde{\eta} \) on \( 
K_{0}(\mathcal{A}) \).