I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an action of the braid group $B_3$. Here is what he says about this action:

First, we lift modular forms $f : \mathbb{H} \to \mathbb{C}$ to functions $\phi_f : SL_2(\mathbb{R}) \to \mathbb{C}$ as follows: let

$$\phi_f \left( \left[ \begin{array}{cc} a & b \\\ c & d \end{array} \right] \right) = f \left( \frac{ai + b}{ci + d} \right) (ci + d)^{-k}.$$

Thinking of $f$ as a function on $SL_2(\mathbb{R})$ invariant under $SO_2(\mathbb{R})$, we have now exchanged invariance under $SO_2(\mathbb{R})$ for invariance under $SL_2(\mathbb{Z})$. ($SO_2(\mathbb{R})$ now acts by the character corresponding to $k$.) In moduli space terms, an element $g \in SL_2(\mathbb{R})$ can be identified with the elliptic curve $\mathbb{C}/\Lambda$ where $\Lambda$ has basis the first and second columns (say) of $g$, and $\phi_f$ is a function on this space invariant under change of basis but covariant under rotation.

Second, $SL_2(\mathbb{R})$ admits a universal cover $\widetilde{SL_2(\mathbb{R})}$ in which the universal central extension $B_3$ of $SL_2(\mathbb{Z})$ sits as a discrete subgroup. Unfortunately, Gannon doesn't give an explicit description of this universal cover (presumably because it's somewhat complicated).

**Question:** What is a good explicit description of this universal cover and of how $B_3$ sits in it (hence of how it acts on modular forms)? In particular, does it have a moduli-theoretic interpretation related to the description of $B_3$ as the fundamental group of the space $C_3$ of unordered triplets of distinct points in $\mathbb{C}$? (These triplets $(a, b, c$) can, of course, be identified with elliptic curves $y^2 = 4(x - a)(x - b)(x - c)$.)