M Theory Lesson 131

Speaking of the j invariant, recall that

$J(q) = j(q) - 744$

where $744 = 248 \times 3$, so in terms of the modular forms $\theta_0$ and $\theta_1$,

$J(q) = -3 [9 \frac{(\theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3})^{3}}{(\theta_{1}^{4} - \theta_{0}^{3} \theta_{1})^{3}} + 248]$

M theorists will recognise the number 248 as the dimension of the Lie group E8. Thus polynomials in $J(q)$ are a bit like polynomials in $\Theta$ and $\Pi$ (otherwise known as Eisenstein series $E_4$ and $E_6$). In fact, a classical result (mentioned in this paper) says that the modular discriminant of the denominator is given by

$\Delta = \frac{1}{1728} (\Theta^{3} - \Pi^{2})$

whereas the numerator is simply $\Theta^{3}$. The series $\Theta$ is sometimes written, for $q = e^{2 \pi i \tau}$ and $y = \textrm{Im} (\tau)$,

$\Theta (q) = 1 - \frac{3}{\pi y} - 24 \sum_{1}^{\infty} \frac{r q^{r}}{1 - q^{r}}$

For $y = 1$ (and ignoring the summation, which roughly equals 0.0018779 at $\tau = i$) this equals $1 - \frac{3}{\pi}$, which Louise Riofrio often likes to tell us is equal to 4.507034% (the observed fraction of baryonic matter). Is this another crackpot coincidence? Note that $0.0018779 \times 24$ is also equal to 4.507%, showing that $\Theta (q)$ is close to zero for $\tau = i$.