Let *G* be a simple Lie group and let *G*(ℂ((t))) be its loop group.

The Lie algebra * g*[[t]][t

^{-1}] has a well known central extension (see e.g. Wikipedia) given by the cocycle

c(

*f*,

*g*) = Res

_{0}<

*f dg*>. Here, < > :

*⊗*

**g***→ℂ denotes some invariant bilinear form on*

**g***, and*

**g***f dg*is the (

*⊗*

**g***)-valued differential given by multiplying*

**g***f*and

*dg*.

Question:It there a similarly concrete cocycle for the central extension ofG(ℂ((t))) by ℂ*?

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 $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc *D*_{γ} : *D*^{2} → *G* for each element γ ∈ *LG*. 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,

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.