dg.differential geometry – Atiyah-Singer-Theorem in Heat Kernels and Dirac Operators


I’m reading “Heat Kernels and Dirac Operators” by Berline, Getzler and Vergne.
I have some troubles to understand a identity on the bottom of page 146 which is essential for the proof Atiyah-Singer-Theorem:

If $a in Gamma(M, C(M))$ and $b in Gamma(M, operatorname{End}_{C(M)}(mathcal{E}))$,
then the point-wise supertrace of the section
$a otimes b in Gamma(M, C(M) otimes operatorname{End}_{C(M)}(mathcal{E}))
cong Gamma(M, operatorname{End}(mathcal{E}))$
was
shown in (3.21) to equal the Berezin integral

$$operatorname{Str}_{mathcal{E}}(a(x) otimes b(x))=
(-2i)^{n/2} sigma_n(a(x)) operatorname{Str}_{mathcal{E}/S}(b(x))$$

I would like try to lose shortly some words on used notations:

Let $M$ be a compact oriented Riemannian manifold of even dimension $n$.
The Clifford bundle $C(M)$ of $M$ is the bundle of Clifford algebras over $M$
whose fibre at $x in M$ is the Clifford algebra $C(T^*_xM)$
of the Euclidean spaces $T^*_x M$. The Clifford bundle $C(M)$ is an associated
bundle to the orthonormal frame bundle,

$$C(M)=O(M) times_{O(n)} C(mathbb{R}^n).$$

Let $S$ be the spin group accociated to Clifford algebra $C(mathbb{R}^n)$. Assume,
that associated spinor bundle

$$mathcal{S}= operatorname{Spin}(M) times_{operatorname{Spin}(n)} S $$

on $M$ exists globally and let $mathcal{W}$ another
arbitrary finite dimensional vector bundle. Clearly $mathcal{S}$
is a Clifford module, since the action of $C(mathbb{R}^n)$ on $S$
leads to an action of the associated bundle $C(M)$.

Let $mathcal{W}$ another
arbitrary finite dimensional vector bundle on $M$ and define
$mathcal{E}= mathcal{W} otimes mathcal{S}$.
Observe that since $operatorname{End}(mathcal{S})= C(M)$ we obtain
$operatorname{End}_{C(M)}(mathcal{E})= operatorname{End}(mathcal{E})$.

If we now come back to indenity

$$operatorname{Str}_{mathcal{E}}(a(x) otimes b(x))=
(-2i)^{n/2} sigma_n(a(x)) operatorname{Str}_{mathcal{E}/S}(b(x))$$

I not understand, let take a look into proposition 3.21 which is proposed to
imply it:

Proposition 3.21. let $V$ is an even-dimensional oriented Euclidean vector and
$Q$ the induced bilinear form on $V otimes mathbb{C}$. Then if
the quadratic form $Q$ is non-degenerate, then there is,
up to a constant factor, a unique supertrace on $C(V)$, equal to $T circ sigma$. The
supertrace $operatorname{Str}(a)$ defined in (3.8) equals

$$operatorname{Str}(a)=(-2i)^{n/2} T circ sigma(a).$$

Here $sigma$ is the symbol map $sigma: C(V) otimes mathbb{C}
to wedge(V otimes mathbb{C})$
(page 104) ater complexification of $V$ and $T$ Berezin integral (page 42).

Now back to
$operatorname{Str}_{mathcal{E}}(a(x) otimes b(x))=
(-2i)^{n/2} sigma_n(a(x)) operatorname{Str}_{mathcal{E}/S}(b(x))$
. What
happens with Berezin on the left side?