Explicit cocycle for the central extension of the algebraic loop group G(C((t)))

Question

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 $D _ {\gamma\delta}$.

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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.

Answer 1

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.

Answer 2

The Kac-Moody central extension can be described in terms of algebraic $K_2$. This was first discovered I think by Spencer Bloch in the early '80s. There is a scattered literature that spells this out in different contexts - the main published references I can think of are by Deligne-Brylinski (Central extensions of groups by $K_2$) and the papers it cites by Deligne (in particular Le Symbole Modéré), the papers by Brylinski-Mclaughlin on the Segal-Witten reciprocity law and symbols etc. (I learned of this from the famous unpublished manuscript of Beilinson-Kazhdan, I think it appears also in later published works of these two individually). Anyway this gives a formula for the Kac-Moody central extension in terms of the tame symbol. Actually one place where the whole story is spelled out beautifully is Kapranov's paper on Eisenstein series and S-duality.

To summarize briefly: $H^4(BG,Z)$ actually consists of algebraic cycle classes, i.e. it's equal to $Chow^2 (BG)$. Bloch showed (in the 70s) that this is the same as $H^2(BG,K_2)$ and used this to give a beautiful picture for second Chern classes. Anyway this can be interpreted as central extensions of $G$ by $K_2$. Now if you're over a local field (Laurent series say) the tame symbol is a kind of residue map, taking $K_2$ of the local field to $K_1$ (i.e. units) in the residue field. So you can push out the $K_2$ extension to get a $C^*$ extension of the loop group, as desired. While $K_2$ is an intimidating beast, this gives an explicit formula I think since the tame symbol is explicit... but I'm the wrong person to give you that formula.

BTW for the multiplicative group this ends up giving a POV on Weil reciprocity, that was spelled out by Witten in his gorgeous paper on Grassmannians, QFT and Algebraic Curves, and is explicated in a paper by Brylinski with a related title (Central Extensions and Reciprocity Laws) and most recently in a very pretty paper of Takhtajan.