# 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 trouble to understand a identity on the bottom of page 146 which is essential for the proof of the 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 some used notations (not all eg super trace and construction of spinor bundles are well explaned in the book):

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 right side?