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.)*

anywhere. $\endgroup$...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$3more comments