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I have been reading Jan Cnops' book, "An Introduction to Dirac Operators on Manifolds" (Birkhaeuser Boston, 2002) and various more standard texts on both Dirac operators and differential geometry in order to obtain a more rounded idea of the "Hodge-Dirac" operator and generalizations.

It is my understanding, along with Ari Stern in a joint paper arXiv:1401.1576 [math.NA] recently posted to the arXiv, that an abstract Hodge-Dirac operator on a Hilbert space can be defined using a suitable nilpotent operator $d$ and its adjoint $d^*$ as $d+d^*$, that it gives rise to a Hodge decomposition, and that the corresponding Hodge-Laplacian, with the same Hodge decomposition is $(d + d^*)^2 = d d^* + d^* d$. The operator $d$ can be thought of as an abstract exterior derivative, and this abstraction, as well as its relationship to Hilbert complexes is explained in the paper.

I am now on the lookout for more concrete definitions of the Hodge-Dirac operator. My trouble with Cnops is that his terminology does not correspond directly to standard terminology. So, on to the specific questions.

  1. In Section 4 of Chapter 2, Cnops defines what he calls the "exterior derivative $\nabla_X$". His Theorem 2.43 on pp. 41-42 indicates that his exterior derivative corresponds directly to a Levi-Civita connection, as does the elaboration in Section 2 of Appendix A. Does his "exterior derivative" correspond to a "covariant exterior derivative" in standard terminology?
  2. In Chapter 3, Section 3, Cnops defines what he calls a "Hodge Dirac operator $\nabla$" as $\nabla f = -\sum_{i=1}^m E_i^{-1} \nabla_{E_i} f.$ In Section 4, he defines what he calls a "Laplacian" or "Hodge Laplacian" $\Delta$ as $\Delta = -div \circ grad$. On p. 85 his Theorem 3.50 "Bochner-Weizenboeck" says "The Hodge Laplacian and Dirac operators are related by $\nabla^2 f = \Delta_m f + \frac{1}{4} R f,$ where $R$ is the curvature operator." So one or both of his definitions does not correspond to standard terminology. Which is non-standard, his "Hodge-Dirac" operator, his "Hodge-Laplacian", or both?
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  • $\begingroup$ Exterior covariant derivative is an operator on exterior algebra over the cotangent bundle, whereas Cnops considers Clifford algebra over tangent bundle instead. Regarding your second question -- why does it not correspond to standard terminology? $\endgroup$ – Vít Tuček Oct 12 '16 at 8:40
  • $\begingroup$ Regarding the second question, in my second paragraph above, I say that in my joint paper with Ari Stern, the abstract Hodge-Dirac operator is the exact square root of the corresponding Hodge-Laplacian. In Cnops, his "Hodge-Dirac" operator is not the exact square root of his "Hodge-Laplacian." $\endgroup$ – Penguian Oct 12 '16 at 12:47
  • $\begingroup$ In that case I think it is your definition that's not general enough. $\endgroup$ – Vít Tuček Oct 14 '16 at 22:28
  • $\begingroup$ Please elaborate. $\endgroup$ – Penguian Oct 15 '16 at 7:36

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