Let $M$is a closed oriented 2n-dimensional smooth manifold, $E$ is a 2n-dimensional oriented real vector bundle on $M$, with inner product on each fibers. Let $\tau=(\sqrt{-1})^{n}c(e_{1})c(e_{2})...c(e_{2n})$, use it we can define a $\mathbb{Z}_2$-grading on $\wedge(E^{*})\otimes\mathbb{C}=\wedge_{+}(E^{*})\otimes\mathbb{C}\oplus\wedge_{-}(E^{*})\otimes\mathbb{C}$, the Clifford action is defined by $c(v)\alpha=\varepsilon(v)\alpha-\iota(v)\alpha$.

I read a formula about Pfaffian like this:

$$Str[exp(-R^{\wedge(E^{*})\otimes\mathbb{C}})]=2^{n}(\sqrt{-1})^{-n}{\rm Pf}(-R^{E})$$

here $R^{E}$ and $R^{\wedge(E^{*})\otimes\mathbb{C}}$ are curvatures,

$E^{*}$ is the dual to $E$.

How to proof this formula? or any matrial about this? If anyone can tell me something I will be very thanks.

Superconnections, Thom classes and differential forms? They have similar formula in terms of "fermion Gaussian integrals". – José Figueroa-O'Farrill Aug 24 '10 at 11:44