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Recently I've been looking into the ``half-differential'' as a definition of spinor fields, as talked about in one of the answers to this post:

Is there Geometric Interpretation of Spinors?

or in this post:

What are half-forms?

I've also found some articles talking about it in the context of the Schwarzian derivative.

Is there a local definition (e.g. as the pointwise-dual to some half-derivative) of half-differentials or half-forms on a Riemannian manifold? I'm aware of the definition as a line bundle, but I'm not sure where to begin with a local construction. Are they the same as the fractional derivatives used in fractional calculus?

Usual differential forms are constructed as the dual to derivatives (which can be defined in terms of limits in an arbitrarily small neighborhood), is the same true of half-forms?

Thank you!

iglizworks
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  • Please clarify what yoy mean by a "local definition." For instance, how would you define a differential form in terms of "derivatives/limits"? Differential forms are dual to vector fields, and the latter can be defined using the language of limits. – Moishe Kohan Aug 19 '24 at 16:03
  • @MoisheKohan Tried to clarify a little. By local I mean a point wise dual to a local half-derivative (e.g.). Then the question becomes how to define a half-derivative locally in terms of limits. – iglizworks Aug 19 '24 at 19:26

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