Let <i>G</i> be a simple Lie group and let <i>G</i>(&#8450;((t))) be its loop group.

The Lie algebra <i><b>g</b></i>[[t]][t<sup>-1</sup>] has a well known central extension
(see e.g. <a href="http://en.wikipedia.org/wiki/Affine_Lie_algebra">
Wikipedia</a>) given by the cocycle<br>
c(<i>f</i>,<i>g</i>) = Res<sub>0</sub> < <i>f dg</i> >. Here, < > : <i><b>g</b></i>&otimes;<i><b>g</b></i>&rarr;&#8450; denotes some invariant bilinear form on
<i><b>g</b></i>, and <i>f dg</i> is the (<i><b>g</b></i>&otimes;<i><b>g</b></i>)-valued differential given by multiplying <i>f</i> and <i>dg</i>.

> <b>Question:</b> It there a similarly concrete cocycle for the central extension of <i>G</i>(&#8450;((t))) by &#8450;*?

To give you an idea of what I'm looking for, let me show
you a cocycle for central extension by <i>S</i><sup>1</sup> of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc <i>D</i><sub>&gamma;</i></sub> : <i>D</i><sup>2</sup> &rarr; <i>G</i> for each element &gamma; &isin; <i>LG</i>. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle
+\int H^*\eta\quad \big)\big)$$

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,<br> and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.


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<b>Reference:</b><br> The cocycle for the smooth loop group can be found on page 19 of the paper<br> <a href="http://arxiv.org/pdf/math/0504123v2">From Loop groups to 2-groups</a>, by Baez, Crans, Schreiber, and Stevenson.