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Apparently, objects called half-forms exist in differential geometry. If for instance a one form could be written as $d\omega$, a half-form might be denoted $\sqrt{d\omega}$. These objects are very peculiar and I have not been able to find any real info online. Could someone give a short working-man's introduction to half-forms? How does one work with them? How are they related to one-forms? What pitfalls should one look out for? Thanks for any suggestion.

EDIT

I was asked in the comments to point to a source where the notion of half-forms is being used. Most recently I found them in this paper. See eq. (4.21) and the paragraph below it. The authors use these objects without making a fuss about it, which indicates that this should be standard knowledge.

Kagaratsch
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    Can you point to where you've seen such a thing referred to before? –  Sep 18 '15 at 22:38
  • Kostant's On the definition of quantization, and Blattner's Quantization and representation theory. Those are references where I believe this notions are used... – CvZ Sep 18 '15 at 22:52
  • I have seen "square roots" of forms in a particle physics paper once, I cannot remember the authors. – krvolok Sep 18 '15 at 23:01
  • See my edit of the question above. – Kagaratsch Sep 20 '15 at 03:59
  • Half-forms are a fancy way of defining spinors on Riemann surfaces. – Xenomorph Dec 27 '23 at 18:46
  • Given any line bundle $L$, you can define its absolute value $|L|$ by replacing the transition functions by their absolute value. You can then define $|L|^\alpha$ for any real $\alpha$. – Deane Dec 27 '23 at 21:54

2 Answers2

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Let $K$ denote the line bundle of top forms on a smooth manifold $X$. Sometimes you can equip such a manifold with extra structure which produces another line bundle $L$ such that $L^{\otimes 2} \cong K$; global sections of $L$ are then called half-forms. One reason these are important is that if $X$ is closed and oriented, you can integrate top forms, so the integration map on top forms induces a natural bilinear map on half-forms. This is important in geometric quantization; see, for example, Lectures on the Geometry of Quantization by Bates and Weinstein.

Qiaochu Yuan
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I just want to add some information about the physics related with it. Half forms are used by string theorists to define spinor fields on Riemann surfaces. Consider the square root of a canonical line bundle, aka

$$L\otimes L=K.$$

Notice that transition functions of $L$ are square roots of the transition function

$$\frac{\partial z_{\alpha}}{\partial z_{\beta}}$$

of $K$. But this has a sign ambiguity. A choice of transition function $S_{\alpha\beta}$ for $L$ on the intersection $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ will take values in $\mathbb{Z}_{2}$, representing the second Stieffel Witney class. i.e.

$$S_{\alpha\beta}\cdot S_{\beta\gamma}\cdot S_{\gamma\alpha}=\mathcal{w}_{\alpha\beta\gamma}.$$

A spinor bundle (or spinor structure) can be defined iff this class vanishes. So the second Stieffel Witney class plays a role here as a topological obstruction for the existence of spinors on the manifold.

To help you visualize what is going on, please remind yourself the spin-$1/2$ system in quantum mechanics. For a spinorial state $|\psi\rangle$, it requires a $720^{\circ}$ rotation to go back to its initial state. The spinor $|\psi\rangle$ behaves very similar to the situation of the Riemann surface $w=\sqrt{z}$, which is a double cover of the punctured complex plane. For this reason, people sometimes call the spinor bundle as the square root of a vector bundle, and so

$$L=\sqrt{K}.$$

You can find mathematically rigorous definitions from this lecture notes.

Xenomorph
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