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Is this a holomorphic one-form:

$$A=\frac{1}{2i}B\bar{z}\hspace{1mm}dz?$$

where $B$ is a constant. MY ANSWER:

I read that a one-form $\omega=f(z,\bar{z})dz$ is holomorphic if $f(z,\bar{z})$ is holomorphic. Therefore, since in my example $\frac{1}{2i}\bar{z}$ does not satisfy the Cauchy-Riemann equations it is not holomorphic and therefore $A$ is not a holomorphic one-form.

I also found that a complex one-form $$\omega=f(z,\bar{z})dz$$ is holomorphic if $$\bar{\partial}\omega=0$$ where $$\bar{\partial}\omega=\frac{\partial f(z,\bar{z})}{\partial \bar{z}}dz\wedge d\bar{z}.$$ Therefore for my $A$, $$\bar{\partial}A=\frac{B}{2i}\frac{\partial \bar{z}}{\partial \bar{z}}dz\wedge d\bar{z}=\frac{B}{2i}dz\wedge d\bar{z}$$ therefore not holomorphic.

Any suggestions?

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  • 2
    $\begingroup$ You are correct. $\endgroup$ – Kenny Wong Apr 5 '17 at 14:42
  • $\begingroup$ You checked your answer in two different ways and got the same result i.e. not holomorphic. What exactly is your question? $\endgroup$ – Semiclassical Apr 5 '17 at 14:54
  • $\begingroup$ I was reading a paper which claimed that the form was holomorphic. I want to see if it was a typo or if I didn't understand the definitions of the holomorphicity of forms. $\endgroup$ – Lewis Proctor Apr 5 '17 at 16:31
  • $\begingroup$ You're absolutely correct. The author of the paper is apparently confusing forms of type $(1,0)$ with holomorphic forms. Holomorphic $1$-forms, as you've pointed out, are $(1,0)$-forms $\alpha$ satisfying $\bar\partial\alpha = 0$. $\endgroup$ – Ted Shifrin Apr 5 '17 at 17:22
  • $\begingroup$ I talked to my supervisor about it and he says that the author probably meant that the form only has a $dz$ contribution. Which is obvious. In case people wanted background information to the problem the one-form $A$ is the gauge potential of a quantum system. Therefore computing the exterior derivative of $A$ would yield the magnetic field (if one uses $z=x+iy$ and $\bar{z}=x-iy$). $\endgroup$ – Lewis Proctor Apr 6 '17 at 15:56

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