I've noticed that there is some extremely intriguing self-similar/fractal branch cut structure on certain Eisenstein series. In particular, the following holomorphic, quasi-modular function on the upper-half complex plane:

$$\frac{1}{144}\big(E_{2}^{2}(\tau) - E_{4}(\tau)\big).$$

The plot I've included here is of the argument of this function. Clearly, it's invariant under $\tau \to \tau +1$, but most certainly not invariant under $\tau \to -1/\tau$.

enter image description here

I believe those black lines represent branch cuts. As you can see, this function exhibits what appears to be very rich branch cut structure, and it even appears to become fractal-like, at small imaginary parts of $\tau$.

Now I know that with functions on $\mathbb{C}$, branch cuts imply that the functions aren't really holomorphic on $\mathbb{C}$, but rather are holomorphic on some Riemann surface. Does anyone have any thoughts on an elegant geometrical way to think of the above function? As in, do we know really what space this thing is a function on? From this picture, it really doesn't look like it's natural to think of this as a function on the upper-half plane! Believe it or not, I actually have a physical reason why I don't think this is naturally considered as defined on the upper-half plane.

A couple of notes that might jog someone's thoughts:

(I) Those black "dots" at the top of the black lines are where the function vanishes identically,

(II) Those points that look like "saddle-points" are where the derivative of the function vanishes identically.

Thanks for any thoughts!

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    $\begingroup$ would you mind to share your physical reasoning? $\endgroup$ – tired Aug 8 '16 at 15:02
  • $\begingroup$ @tired Well, it requires a lot of details, and I'm not totally sure it's correct. But very roughly, in physics we often have quantum moduli spaces which are fibered over classical moduli spaces. In other words, the fibers are parameterized by "quantum deformations." The above function arises in relation to a theory which has an infinite, but discrete collection of classical vacua. Given the self-similarity, as well as the fact that $S=0$ at those "black holes", it's tempting think perhaps this encodes all of the classical vacua. $\endgroup$ – Benighted Aug 8 '16 at 18:34
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    $\begingroup$ as any modular form, it is natural to think to it as a meromorphic differential defined on the Riemann surface $\mathbb{C} \setminus \Gamma$ where $\Gamma$ is the modular group $\endgroup$ – reuns Sep 7 '16 at 16:04
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    $\begingroup$ this is a big word to say $f(\frac{a \tau+b}{c\tau+d})(c \tau+d)^{-4} = f(\tau)$ for every integer $a,b,c,d$ such that $ad-bc = 1$, i.e. $f(\tau) \partial^2\tau$ is modular (invariant under the action of $\Gamma$). this way the fractal you see is just the self-similarity under $\Gamma$ $\endgroup$ – reuns Sep 7 '16 at 16:05
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    $\begingroup$ @spietro I thought $E_2$ was $G_4$, yes it matters but not so much since $E_2(\tau) - Im(\tau)$ is modular (but not meromorphic), so it is some sort of harmonic differential defined on the Riemann surface $\endgroup$ – reuns Sep 8 '16 at 14:24

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