# 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? Posted on