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I'm following the notes by Freed about the Dirac Operator. In section 5.4, equation (5.4.25-27), he makes the following claim about the Dirac operator. In different notation than he is using, he is considering the operator

$$Q = (\partial_\mu - \frac{1}{8}R_{\beta\mu\nu\sigma} \gamma^\nu \gamma^\sigma x^\beta)^2 = (\partial_\mu - \Omega_{\beta \mu} x^\beta)^2$$

so that $(\Omega_{\beta \mu}) = \frac{1}{8}R_{\beta\mu\nu\sigma} \gamma^\nu \gamma^\sigma$ is an antisymmetric matrix of Clifford algebra elements $\gamma^\mu$, in coordinates $x$ centered around the point of interest. He claims that one can write the matrix $\Omega_{\beta\mu}$ in a block diagonal form

$$\Omega_{\beta\mu} = \begin{pmatrix} 0 &\omega_1 &0 &0 &... &0 &0 \\ -\omega_1 &0 &0 &0 &... &0 &0 \\ 0 &0 &0 &\omega_2 &... &0 &0 \\ 0 &0 &-\omega_2 &0 &... &0 &0 \\ \vdots &\vdots &\vdots &\vdots & &\vdots &\vdots \\ 0 &0 &0 &0 &... &0 &\omega_{d/2} \\ 0 &0 &0 &0 &... &-\omega_{d/2} &0 \\ \end{pmatrix}$$

where the $\omega_i$ are Clifford algebra elements. If one does this, then we can more easily find the heat kernel, since the operator splits up into smaller chunks and use it to show the Dirac operator's index gives the $\hat{A}$ genus.

My questions is if such a decomposition is valid, and how one can make sense of these ideas? I've never seen the Riemann tensor decomposed like this and I couldn't find online any resources that indicate this is possible.

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    $\begingroup$ Is this possibly some other way of talking about using a basis made of bivectors? $\endgroup$
    – user4552
    Dec 11, 2019 at 6:10
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    $\begingroup$ Is he just using that any antisymmetric matrix can be written in block diagonal form? This is certainly true if the entries are real number. It is not so obvious to me that this remains true if the entries are elements of a Clifford algebra. (But it would not surprise me.) $\endgroup$
    – mmeent
    Dec 11, 2019 at 8:49
  • $\begingroup$ That's what I was thinking @mmeent. But I don't know how to justify that statement. $\endgroup$
    – Joe
    Dec 11, 2019 at 14:27
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    $\begingroup$ The Atiyah-Singer Theorem (or this specific case of Dirac Operator) says the index of the Dirac operator (which is manifestly an integer) is something called the $\hat{A}$ genus integrated over the manifold. The theorem shows that the index is related to how twisted a spin bundle over the manifold is. The non-trivial topology of this bundle is a topological invariant of the manifold because the spin bundles are intrinsic. A-S says that the index can be computed by an integral of a smooth local form. This is simliar to the 'Dirac quantization' $\int F \in 2\pi \mathbb{Z}$ for 2d gauge fields $\endgroup$
    – Joe
    Dec 11, 2019 at 18:51
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    $\begingroup$ If it means anything, the case of tensoring with a G-bundle (i.e. coupling to a G gauge field) can reproduce the chiral anomalies of chiral fermions coupled to gauge theories. This heat kernel method is essentially the method used by Fujikawa to compute the chiral anomaly, with the extra complication of coupling to gravity. This part I'm stuck on is the gravity part of the story. $\endgroup$
    – Joe
    Dec 11, 2019 at 18:59

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