Atiyah-Singer index theorem 2

1. rough outline of heat kernel proof of Atiyah-singer index theorem

1.1. proof strategy

Theorem 1 (Mckean-Singer formula.)
$latex \displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)). &fg=000000$

from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory.
Main challenge:
1. in the expansion on heat kernel ,we need to proof when $latex { t \rightarrow 0}&fg=000000$, the limit exist and find a way to calculate it.
2. indentify the limit as $latex {t \rightarrow 0}&fg=000000$.
Proof: Our proof road锛� mckean-singer formula $latex {\rightarrow}&fg=000000$ local-index thm $latex {\rightarrow}&fg=000000$ A-S index thm $latex {\rightarrow}&fg=000000$ Riemann-roch-Hirzebunch theorem. $latex \Box&fg=000000$

1.2. preliminary work

superbundle: $latex {E=E^+ \oplus E^-}&fg=000000$.
on compact manifold $latex {M}&fg=000000$, $latex {D: \Gamma(M,E) \rightarrow \Gamma(M,E)}&fg=000000$ is a self-adjoint operator . $latex { D = }&fg=000000$. $latex {D^+=D|_E^+,D^-=D|_E^-}&fg=000000$.
observe that:
$latex {D}&fg=000000$ is symmetric $latex {\Longrightarrow}&fg=000000$ eigenvalue space of $latex {D^2}&fg=000000$ is finite dimention $latex {\Longrightarrow}&fg=000000$ in particular $latex {Ker D^2}&fg=000000$ is finite dimesion $latex {\Longrightarrow}&fg=000000$ $latex {Ker D}&fg=000000$ is finite dimension.
dimention of superspace $latex {E = E^+ \oplus E^-}&fg=000000$:

$latex \displaystyle dim E=dim E^+ – dim E^-.&fg=000000$

$latex \displaystyle kerD=kerD^+ \oplus ker D^-&fg=000000$


Def: $latex {ind D^+=dim ker D^+ -dim ker D^-}&fg=000000$.
1.Let $latex {D}&fg=000000$ be a self-adjoint Dirac operator on a clifford module $latex {E}&fg=000000$ over a compact manifold $latex {M}&fg=000000$,then

$latex \displaystyle \Gamma(M,E^{\pm})=ker D^{\pm} \oplus im D^{\mp} &fg=000000$

in particular,
$latex \displaystyle ind D^+ = dim ker D^+ -dim coker D^+&fg=000000$

where coker $latex {D^+ :=\Gamma(M,E^-)/im D^+}&fg=000000$.
2.Let $latex {D}&fg=000000$ be a differential operator acting on a $latex {Z_2}&fg=000000$-graded vector bundle $latex {E}&fg=000000$,then $latex {Str[D,K]=0}&fg=000000$.
the proof of this two lemma is easy,leave sas exercise.
1.3. Mckean-Singer formula

the formula is:

$latex \displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)). &fg=000000$


the expression of heat operator by spectral measure is:
$latex \displaystyle e^{-tD^2}=\int\limits_{0}^\infty d^{\lambda t}dE_{\lambda}.&fg=000000$


proof 1:
we have first eigenvalue estimate on compact manifold:
$latex \displaystyle |Str(e^{-tD^2}-P_0)| \leq Cvol(M)e^{-t\lambda}.&fg=000000$

$latex \displaystyle \Longrightarrow&fg=000000$

$latex \displaystyle \lim\limits_{t \rightarrow \infty}Str(e^{-tD^2}) = Str p_0 =dim kerD^+ -dim ker D^- =ind D^+.&fg=000000$


on the other hand ,we need to show $latex {Str e^{-tD^2}}&fg=000000$ is independent with $latex {t}&fg=000000$,in fact:
$latex \displaystyle \frac{d}{dt} Str (e^{-tD^2})=-Str(D^2 e^{-tD^2}).&fg=000000$

$latex {D}&fg=000000$ odd parity : $latex {\Longrightarrow \ D^2e^{-tD^2}=[D,D E^{-tD^2}]}&fg=000000$.
$latex {[\ \ ,\ \ ]}&fg=000000$ supercommunater $latex {\Longrightarrow \ \frac{d}{dt}Str (e^{-tD^2})= -Str[D,D e^{-tD^2}]=0.}&fg=000000$
by spectral decompositon of $latex {e^{-tD^2}}&fg=000000$:
$latex \displaystyle Str (e^{-tD^2})=\sum\limits_{\lambda \geq 0}(n_{\lambda}^+ – n_{\lambda}^-)e^{-t\lambda}&fg=000000$


observe that: $latex {n_{\lambda}^+=n_{\lambda}^-}&fg=000000$ for $latex {\lambda \not=0}&fg=000000$. $latex {\Longrightarrow ind D=n_0^+ – n_0^-}&fg=000000$. (detail in [BGV])
Corallary: the index of a smooth on-parameter family of Dirac operator is constant.
what we have proved is:
$latex \displaystyle ind D = Str (e^{-tD^2})=\int Str(k(x,y)).&fg=000000$

1.4. analytic formula of ind$latex {D^+}&fg=000000$

from the discuss of heat kernel in section 2,we know following result(section 2 only discuss the case of function but use the similar way we can get similar result on bundle):

$latex \displaystyle K_t(x,y) \sim (4 \pi t)^{\frac{n}{2}} \sum\limits_{i=0}^{+\infty} t^i K_i(x), K_i \in \Gamma(M,End(E)). &fg=000000$

on the other hand: $latex { D=\left[\begin{array}{ccc} 0 & D^- \\ D^+ & 0 \end{array}\right] \Longrightarrow D^2=\left[\begin{array}{ccc} D^-D^+ & 0\\ 0 & D^+D^- \end{array}\right] }&fg=000000$.
and use the Mckean-Singer formula,we get: ind (D^+)&=&Str(e^{-tD^2})
&=&\int\limits_M Tr(K_t(x,y,D^-D^+)) – \int\limits_M Tr(k_t(x,y,D^+D^-))
&=&\sum\limits_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum\limits_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-). where $latex {a_i}&fg=000000$ is the heat trace invariants.
take $latex { t \rightarrow 0}&fg=000000$,the only thing make sense is the series of order $latex {\frac{n}{2}}&fg=000000$,and we want to proof:
$latex \displaystyle ind(D^+) =a_{\frac{n}{2}}(D^-D^+) -a_{\frac{n}{2}}(D^+D^-).&fg=000000$

But the difficult thing is that the high order series is very hard ro calculate….
our strategy is following:

Step1: proof $latex {Str(K_t(x,y))}&fg=000000$ has a limit as $latex {t \rightarrow 0}&fg=000000$ i.e $latex {Str(K_t(x,y))\stackrel{t \rightarrow 0}{\longrightarrow}}&fg=000000$ index density.
step2:use a rescaling of space,time,clifford bundles ,to find a way that make us only need to calculate the leader coefficient.

1.5. From the McKean鈥揝inger formula to the index theorem

Let $latex {M}&fg=000000$ be a compact oriented Riemannian manifold of even dimension $latex {n}&fg=000000$. We will write $latex {k_t(x, y)}&fg=000000$ for the heat kernel associated to $latex {D^2}&fg=000000$. The diagonal $latex {k_t(x, x)}&fg=000000$ is a section of $latex {End(E )}&fg=000000$ which is iso- morphic to $latex {Cl(M) \otimes End_{Cl(M)}(E )}&fg=000000$. Using this isomorphism, we define a filtration on $latex {End(E )}&fg=000000$, induced by the filtration on $latex {Cl(M)}&fg=000000$. Elements of $latex {End_{Cl(M)}(E )}&fg=000000$ are given 0-degree. Denote by $latex {Cl_i(M)}&fg=000000$ the subbundle of $latex {Cl(M)}&fg=000000$ consisting of all elements of degree less or equal to $latex {i}&fg=000000$.
the following theorem hold:
Theorem 1. The following statements hold:
1. The coefficients $latex {k_i}&fg=000000$ have degree less or equal to $latex {2i}&fg=000000$. In other words, $latex {k_i \in \Gamma (M, Cl_{2i}(M) \otimes End_{Cl(M)}(E))}&fg=000000$. 2. If $latex { \sigma(k):= \sum \limits_{i=0}^{n/2}\sigma_{2i}(k_i) \in A(M,End_{Cl(M)}(E))}&fg=000000$,where $latex {\sigma_j :Cl_j(M)鈫扐_j(M)}&fg=000000$ denotes the $latex {i=0}&fg=000000$ restriction of the symbol map, then:

$latex \displaystyle \sigma (k) = A藛(M) exp(鈭扚E /S ).&fg=000000$

$latex \displaystyle ind (D^+)=Str(e^{-tD^2})=\sum\limits_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum\limits_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-). &fg=000000$

the important observation is that the $latex {Str}&fg=000000$ of order less than n all vanish,in the other word:

lemma2:for any quadratic space $latex {v}&fg=000000$ of dimension $latex {n}&fg=000000$,$latex {CL_{n-1}(V) =[CL(V),CL(V)]}&fg=000000$.

Proof: Let $latex {e_1,…,e_n}&fg=000000$ be a basis of $latex {V}&fg=000000$. For any multi-index $latex {I \subset \{1,…,n\}}&fg=000000$, denote by $latex {c_I}&fg=000000$ the Clifford product $latex {\prod_{i \in I} c}&fg=000000$.Then the set $latex {\{c_I\}}&fg=000000$ is a basis for $latex {Cl(V)}&fg=000000$.If$latex {|I|<n}&fg=000000$,there is at least one $latex {j}&fg=000000$ such that $latex {j \notin I}&fg=000000$, and we have
$latex \displaystyle c(e_I)=-\frac{1}{2}[c_j,c_j c_I]. &fg=000000$

so take $latex {t \rightarrow 0}&fg=000000$ , we get:
$latex \displaystyle ind (D) =(4 \pi)^{\frac{n}{2}} \sum\limits_{i = \frac {n}{2}}^{\infty} t^{i- \frac{n}{2}}Str(k_i(x)). &fg=000000$

now we want to identify the term $latex {Str(k_{\frac{n}{2}}(x))}&fg=000000$ as a characteristic form on $latex {M}&fg=000000$ :
Lemma 3. Let $latex {V}&fg=000000$ be a Euclidean space. There is, up to a constant factor, a unique supertrace on $latex {Cl(V)}&fg=000000$, equal to $latex {T \circ \sigma}&fg=000000$ where $latex {\sigma}&fg=000000$ denotes the symbol map and $latex {T}&fg=000000$ is the projection of $latex {\alpha \in \wedge V}&fg=000000$ onto the coefficient of $latex {e_1 \wedge, … ,\wedge e_n}&fg=000000$ if $latex {e_1,…,e_n}&fg=000000$ form an oriented orthonormal basis of $latex {V}&fg=000000$. Furthermore, the supertrace defined above equals:
$latex \displaystyle Str(a) = (鈭�2i)^{\frac{n}{2}} (T \circ \sigma(a)).&fg=000000$

rmk:(The map $latex {T}&fg=000000$ is also called the canonical Berezin integral.)
the dimension of $latex {Str}&fg=000000$ space is one because of $latex {CL_{n-1}(V)=[CL(V),CL(V)]}&fg=000000$ and it never be empty because there is a natural defined supertrace on $latex {CL(V)}&fg=000000$.so the only thing we need to do is to determine the constant,in face we only need to calculate the supertarce on any non-zero element .for instance the chirality operator $latex {\Gamma}&fg=000000$. We have that $latex { Str(\Gamma) = dim (\wedge P) = 2^{n/2}}&fg=000000$ and therefore $latex {Str(a) = (鈭�2i)^{n/2} T \circ \sigma (a)}&fg=000000$ for all $latex {a \in Cl(M)}&fg=000000$.
so we know:
$latex {\forall \ }&fg=000000$ section $latex {a \otimes b \in \Gamma(M \otimes CL(M) \otimes End_{CL(M)}(E)}&fg=000000$:
$latex \displaystyle Str_E( a \otimes b)(x) = (-2i)^{\frac{n}{2}} \sigma_n(a(x))Str _{E/S}(b(x)).&fg=000000$

$latex \displaystyle Str_E(k_{\frac{n}{2}})(x) = (-2i)^{\frac{n}{2}} Str _{E/S}(\sigma_n(k_{\frac{n}{2}})).&fg=000000$

theorem1 implies then the following theorem for the index of a Dirac operator associated to a Clifford connection which is known as the local index theorem.

Theorem 2. (Local index theorem) Let $latex {M}&fg=000000$ be a compact, oriented even-dimensional manifold and let $latex {E}&fg=000000$ be a Clifford module with Clifford connection $latex {\nabla E}&fg=000000$ . Let $latex {D}&fg=000000$ be the associated Dirac operator. Then $latex {\lim\limits_{t \rightarrow 0} Str(k_t(x, x))|dx| }&fg=000000$ exists and is obtained by taking the $latex {n}&fg=000000$ -th form piece of
$latex \displaystyle (2\pi i)^{鈭抧/2} \widehat A(M)ch(E /S ).&fg=000000$


rmk:This theorem only holds for Dirac operators associated to Clifford connections, which are those which are compatible with the Clifford action. However, since the index of a Dirac operator is independent of the Clifford superconnection used to define it, we get Atiyah鈥揝inger index formula for any Dirac operator.

Theorem 3. (Atiyah鈥揝inger Index Theorem) Let $latex {M}&fg=000000$ be a compact, oriented, even-dimensional manifold and let $latex {D}&fg=000000$ be a Dirac operator on a Clifford module $latex {E}&fg=000000$ . Then the index of $latex {D}&fg=000000$ is given by :
$latex \displaystyle ind D = (2 \pi i)^{-\frac{n}{2}}\int\limits_{M} \widehat A(M) ch(E/S) .&fg=000000$

hence theorem 3 is a consequence of theorem 1.
up to now,to prove index theorem ,we only need to prove theorem 1.
1.6. idea of the proof of theorem 1

we give the clear proof in appendix锛宼here we explain the idea:
To prove Theorem 4.11 we mainly follow Chapter 4 of [BGV] but rearrange the different steps in order to make the proof clearer, at least for us. Let us summarize the first part of the proof:

1. The idea of the proof is to work in normal coordinates $latex {x}&fg=000000$ around a point $latex {x_0 \in M}&fg=000000$. Near the diagonal, we use parallel transport to pull back the heat kernel $latex {k_t(x, x_0)}&fg=000000$ which is a section of the vector bundle $latex {E_x \otimes E_x^*}&fg=000000$ and define a new kernel $latex {k(t, x) := \iota(x_0, x)k_t(x, x_0)}&fg=000000$, a section of $latex {End(E_{x_0} ) \cong Cl(V^*) \otimes End(W)}&fg=000000$ for some twisting space $latex {W}&fg=000000$. Using the symbol isomorphism $latex {蟽}&fg=000000$, we can look at $latex {k(t, x)}&fg=000000$ as a section of $latex {\wedge(V^*) \otimes E (W)}&fg=000000$.

2. We use Lichnerowicz鈥� formula to get the explicit form of the operator $latex {L}&fg=000000$ such that the kernel $latex {k(t, x)}&fg=000000$ satisfies the heat equation $latex {(\partial t + L)k(t, x) = 0}&fg=000000$.

3. In a third step, we define a rescaling of space, time and the Clifford algebra, introduced by Getzler. This rescaling has the effect that the leading coefficient of the asymptotic expansion of the rescaled kernel is exactly the differential form $latex {\sigma(k)}&fg=000000$ of theorem 1 which leads to a reformulation of Theorem 1.




2. Chern-Weil theory

some text in Appendix A

3. Complete proof of theorem 1

some text in Appendix B