In the Euclidean setting, the Dirac operator was constructed so as to give the square of the Laplacian. Now for a K\"{a}hler manifold with a spin$^c$ structure we have the a corresponding Dirac operator $D$. Moreover, we have a Laplacian $(d+d^{\ast})$, where $d^{\ast}$ is the coadjoint $\ast d \ast $, for $\ast$ the Hodge $\ast$-mapping. Now in the case where the manifold is also symmetric we get a relationship between the square of the Dirac and the Laplacian that involves an extra curvature term. Does this extend to all K\"{a}hler manifolds, and if it does, what is the exact relationship?
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This is a very general statement valid for any first order differential operator such that its square has the same principal symbol as a Laplacian. One can then prove that the square of that first order operator differs from the covariant Laplacian $\nabla^*\nabla$ by a zeroth order term. For more details, see the monograph of Berligne, Getzler, Vergne. |
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