Dirac operator in harmonic coordinates

Question

It is well-known that the Laplace-Beltrami operator for a Riemannian manifold takes a relatively simple form with the harmonic coordinates. This operator should be obtained as a product of Dirac operators. So, my question is: What should the form be for this Dirac operator? Does some literature exist about?

Answer 1

Regardless of coordinates, the Laplace-DeRham operator may be written as the produce of Dirac operators $\Delta=(d+d^*)^2:\Omega^*(M) \to\Omega^*(M)$ where $d$ is the exterior derivative and $d^*$ is the formal adjoint of $d$ with respect to the L2 norm induced by the metric and an orientation. The restriction of $\Delta$ to $\Omega^0(M)=C^\infty(M)$ is the Laplace Beltran operator, and since $d^*$ restricted to $C^\infty(M)$ is zero, $\Delta_{Beltrami} =d^*d$.