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We are familiar with the expansion of the j-function,

$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$

and maybe with the approximation,

$$e^{\pi\sqrt{652}} = (640320^3+744)^2-2\cdot196883.999999999918\dots$$

but can somebody give a short, non-specialist explanation (if it is even possible), why the relationship $a\approx b$ between,

$$\begin{aligned} \log(196883)\; &\approx 12.19 = a\\ 4\pi\; &\approx 12.56 = b \end{aligned}$$

is suddenly mentioned, of all places, in quantum gravity? (Witten's paper is here.)


$\color{brown}{Edit:}$ (To address possible comments)

Witten defines a certain function $Z_k(q)$ in page 30, and for the first few $k$,

$$\begin{aligned} J(q) = Z_1(q) &= q^{-1}+196884q+\dots\\ Z_2(q) &= q^{-2}+1+42987520q+\dots\\ Z_3(q) &= q^{-3}+q^{-1}+1+2593096794q+\dots\\ Z_4(q) &= q^{-4}+q^{-2}+q^{-1}+2+81026609428q+\dots \end{aligned}$$

On a hunch, I used Mathematica's Integer Relations and checked these coefficients with the coefficients $c_n>1$ of $J(q)$,

$$c_n =196884, 21493760, 864299970, 20245856256,\dots$$

(OEIS A014708) and, sure enough, they were just simple linear combinations,

$$\begin{aligned} 196884\; &=c_1\\ 42987520\; &= 2c_2\\ 2593096794\; &= c_1+3c_3\\ 81026609428\; &= c_1+2c_2+4c_4 \end{aligned}$$

Using the general formula at the bottom of p.34,

$$\begin{aligned} \log(c_1)\;&\approx 12.19\\ 4\pi\sqrt{1}\; &\approx 12.56\\[2.5mm] \log(2c_2)\;&\approx 17.57\\ 4\pi\sqrt{2}\; &\approx 17.77\\[2.5mm] \log(c_1+3c_3)\;&\approx 21.67\\ 4\pi\sqrt{3}\; &\approx 21.76\\[2.5mm] \log(c_1+2c_2+4c_4)\;&\approx 25.12\quad\quad\quad\quad\\ 4\pi\sqrt{4}\; &\approx 25.13\\ \end{aligned}$$

and the paper states that "...agreement improves rapidly if one increases k..." for the Bekenstein-Hawking entropy. (Whatever that is.)

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    $\begingroup$ 12.19 is approximately 12.56? I'm not sure why such a crummy approximation would be mentioned anywhere. $\endgroup$ – Gerry Myerson Jan 6 '15 at 15:05
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    $\begingroup$ @GerryMyerson: They define a function they call $Z_k(q)$. That was just $k=1$. To quote Witten's paper in page 32, "...agreement improves rapidly if one increases $k$." For $k=4$, they used the first coefficient of the q-expansion as, $$\log(81026609428)\approx 25.12,\quad 8\pi \approx. 25.13$$ $\endgroup$ – Tito Piezas III Jan 6 '15 at 16:45
  • $\begingroup$ for Bekenstein-Hawking entropy see en.wikipedia.org/wiki/Black_hole_thermodynamics $\endgroup$ – jjcale Jan 6 '15 at 19:09
  • $\begingroup$ @GerryMyerson: You are right: it's not a good approximation and, in fact, the original paper does not say it is. Quoting from the paper (p. 32): "It is illuminating to compare the number $196883$ to the Bekenstein-Hawking formula. An exact quantum degeneracy of $196883$ corresponds to an entropy of $\log 196883 \approx 12.19$. By contrast, the Bekenstein- Hawking entropy at $k = 1$ and $L_0 = 1$ is $4π \approx 12.57$. We should not expect perfect agreement, because the Bekenstein-Hawking formula is derived in a semiclassical approximation which is valid for large $k$." $\endgroup$ – José Figueroa-O'Farrill Jan 6 '15 at 20:47
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    $\begingroup$ Can you clarify what is the question that you want answered? Is it the mathematical derivation of the approximation? Do you want something beyond what is on the subsequent pages of Witten's paper? Or do you want to understand the physics behind the relation? If so, do you have a specific question beyond or about what's in the paper? $\endgroup$ – Aaron Bergman Jan 7 '15 at 13:43
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I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional universe behave like thermodynamic objects. They have temperature, and entropy (reflecting some hidden microstates), and the entropy is proportional to the surface area of the event horizon. Since the derivation of their formula did not use quantum gravity, one expects that quantum corrections become relevant when one considers very small black holes.

The black holes that appear in this question live in $AdS_3$ space, which is a 2+1 dimensional spacetime that has $SL_2(\mathbb{R})$-geometry (which is kind of negatively curved). While this universe is quite different from our own, one obtains an entropy versus surface area relationship for black holes by similar reasoning, and again one expects some quantum corrections to show up in the small entropy regime. Explicit black hole solutions to Einstein's equations were found by Bañados, Teitelboim, and Zanelli, and they were found to have event horizons with positive surface area (which is really circumference when we consider 2+1 dimensions).

When quantum gravity is brought into the picture, the sizes of possible black holes become quantized. Following AdS/CFT, Witten conjectures that size corresponds to conformal weight of a primary field, and this is why you see the formula $4\pi\sqrt{k}$.

The near-integer behavior of $e^{2\pi \sqrt{163}}$ does not seem particularly connected to any of this. It is basic class field theory, with some Hecke operators. See Chapter 3 (I think) of Silverman's Advanced Topics and this MathOverflow question.

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  • $\begingroup$ Thanks. This question was motivated by this MSE post about $e^{\pi\sqrt{d}}$. The OP wanted to know other interesting properties. $\endgroup$ – Tito Piezas III Jan 14 '15 at 4:39
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It's not clear to me what kind of an answer you want. However, at a very shallow level, I think even the Wikipedia article you link to shows that this mention in quantum gravity is not that "sudden".

Is there something dissatisfying about what's in the article?

As will become more clear below, I'm certainly no specialist, but from that article, I gather that:

(a) There is a mathematical object called the "moonshine module" which connects the j-function, the Monster group, etc, and which is at the heart of the study of "moonshine".

(b) This "moonshine module" also has a physical interpretation in terms of a conformal field theory / string theory, and that interpretation goes back (at least?) to Frenkel, Lepowsky and Meurman, 1988.

(c) String theory is a theory for quantum gravity, and hence it's quite plausible to me that someone like Witten, studying models for quantum gravity and looking at certain kinds of black holes might conjecture in 2007 that some flavor of moonshine is relevant to 3D quantum gravity.

(d) Hence I also find it plausible that Witten could find the number $a=\log(196883)$ (after some suitable string theory / physics calculations) as the entropy of some kind of black hole.

(e) I have no reason to doubt the article when it says that the Beckenstein–Hawking estimate $b=4\pi$ is another estimate for this quantity, coming from another approach to the physics of black holes.

(f) Lo-and-behold, the two numbers $a$ and $b$ (really, $a_k$ and $b_k$, where $k$ is some parameter related to the black hole) are "close" (as $k$ becomes large), providing evidence (along with 80ish pages of other arguments) for the conjecture.

For more details to the above, I imagine that following up on the many references cited in the article (plus probably several years of study) one could begin to make this chain of word associations more precise.

There's also this recent review titled "Moonshine" by Duncan, Griffin and Ono, which devotes quite a bit of space to Witten's conjecture (see e.g. Section 6).

The above in some sense just gives a very crude rendering of "what" $a\approx b$ is. Now if you want an answer as to why moonshine appears in string theory or quantum gravity in the first place, that's more a question about the philosophy of mathematics and physics, which I am even less well-qualified to talk about here.

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    $\begingroup$ Not dissatisfying, but the connection to black holes and why $e^{\pi\sqrt{163}}$ is a near-integer doesn't seem to be immediately obvious, don't you think? In fact, as pointed out by some comments, the relation $a \approx b$ or $\log(196883) \approx 4\pi$ is beneath notice, let alone somehow related to quantum gravity. $\endgroup$ – Tito Piezas III Jan 6 '15 at 23:22
  • $\begingroup$ I didn't mean to say that any of the above is obvious, but actually rather hoped to prod you to sharpen your question a bit. $\endgroup$ – j.c. Jan 7 '15 at 7:09
  • $\begingroup$ Thanks. I've elaborated on my question in an answer/comment: mathoverflow.net/a/193394/12905 $\endgroup$ – Tito Piezas III Jan 7 '15 at 19:11
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I now realize my question had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

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