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](http://en.wikipedia.org/wiki/Affine_Lie_algebra)) 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](http://arxiv.org/pdf/math/0504123v2), by Baez, Crans, Schreiber, and Stevenson,  
 and also on page 8 of Mickelsson's paper [From Gauge anomalies to Gerbes and Gerbal actions](http://arxiv.org/pdf/0812.1640v1).



Explicit cocycle for the central extension of the algebraic loop group G(C((t)))
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](https://eudml.org/doc/103999) has this formula in it (actually for $GL\_2$). I would be suprised if there was a usable formula for higher rank groups.


