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

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.

• If I remember correctly, one difficulty in the (smooth or continuous) loop group case is that the central extension is topologically nontrivial. So one has to be careful by what is meant by the "cocycle for the central extension". – Victor Protsak May 16 '10 at 1:40
• You should look at the book by Pressley and Segal called Loop Groups. – user6119 May 16 '10 at 2:53
• The above cocycle is discontinuous because the bounding discs D_gamma cannot be made to depend continuously on gamma. The [BCSS] paper does it a little bit differently, and also defines \tilde{LG} as a topological group. – André Henriques May 16 '10 at 9:11
• Actually, the above cocycle is continuous around the origin. So one gets the topology of \tilde{LG} in a neighborhood of its identity element. By translating it around, this can be used to define the topology everywhere. – André Henriques May 16 '10 at 11:37
• Re new title: $C((t))$ is the field of formal Laurent series, which is the completion of the field of rational functions $C(t)$ at 0 or, alternatively, the field of fractions of the ring of formal power series $C[[t]].$ That is a right setting for a (completed) "algebraic loop group". But what do you mean by $C((t,t^{-1}))$? – Victor Protsak May 16 '10 at 23:13

## 2 Answers

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.

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.

• Hmmm... I thought that $K_2$ had made its appearance much earlier: according to Milnor in "Algebraic K-theory", work of Steinberg, Moore, and Matsumoto (on universal central extensions of Chevalley groups) was the main motivation for considering the new functor $K_2$. What did Bloch discover then? Deligne's "Symbole modere", on the other hand, only treats the case of commutative G. The difficulty for giving an explicit formula in all cases seems to be that the extension does not split topologically. – Victor Protsak May 16 '10 at 5:03
• $K_2$ is defined in terms of the Steinberg extension of $SL_n$ so perhaps the point is that you get a map from the kernel of the universal central extension to $K_2$ for all groups (if I remember correctly even for $SL_n$ and a field these kernels only stabilise from $n=3$ and onwards). In any case going to the cocycle question I think that from the Steinberg point of view it is clear that we get sections over the Bruhat cells and hence (in principle at least) a cocycle desciption for the algebraic loop group. ..... – Torsten Ekedahl May 16 '10 at 5:50
• ..... Note however that actual loops may wander across Bruhat cells so it doesn't seem to give a complete cocycle in the topological case. – Torsten Ekedahl May 16 '10 at 5:50
• @Torsten: 1. Comment about "wandering across Bruhat cells" is right on! 2. It appears that if G is split and simply-connected, the kernel of the universal central extension is Milnor $K_2(k)$ except when G of type C (which includes $SL_2$ in rank 1 case); for type C, it is somewhat larger - see Deligne and Brylinski, numdam.org/numdam-bin/item?id=PMIHES_2001__94__5_0, Introduction. – Victor Protsak May 16 '10 at 6:04
• My real point about the Steinberg extension was left out of my comment (because I forgot to make it...), I wanted to speculate that what Bloch did was relating the different kernels. – Torsten Ekedahl May 16 '10 at 8:05