Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group.
The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension
(see e.g.
Wikipedia) given by the cocycle
$c(f,g) = Res_0\langle f,dg\rangle$. Here, $\langle\ ,\ \rangle \colon \mathfrak{g}\otimes \mathfrak{g}\to\mathbb{C}$ denotes some invariant bilinear form on $\mathfrak{g}$, and $f dg$ is the $\mathfrak{g}\otimes \mathfrak{g}$-valued differential given by multiplying $f$ and dg.
Question: It there a similarly concrete cocycle for the central extension of $G(\mathbb{C}((t)))$ by $\mathbb{C}^\ast$?
To give you an idea of what I'm looking for, let me show
you a cocycle for central extension by $S^1$ of the smooth loop group $LG = \mathop{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.
Pick a bounding disc $D_\gamma$ : $D^2 \to G$ for each element $\gamma\in LG$. The cocycle is then given by
$$
c(\gamma,\delta) = \exp\left(i\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle
+i\int H^*\eta\right)
$$
where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.
References:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson,
and also on page 8 of Mickelsson's paper From Gauge anomalies to Gerbes and Gerbal actions.
Best Answer
For $SL_2$ a cocycle is given by $$ \sigma(g,h)=\left( \frac{x(gh)}{x(g)} , \frac{x(gh)}{x(h)} \right) $$ where for $g=\left(\begin{array}{ll} a & b \\ c & d\end{array}\right)\in SL_2(\mathbb{C}((t)))$, we define $x(g)=c$ unless $c=0$ in which case $x(g)=d$. $(\cdot,\cdot)$ is the tame symbol.
I'll see if I can come up with a good reference. I've seen this stuff over local fields, where this is attributed to Kubota, and Kazhdan-Patterson's paper on Metaplectic Forms has this formula in it (actually for $GL_2$). I would be suprised if there was a usable formula for higher rank groups.