# Central extensions of loop groups

Let $$LG=\operatorname{Maps}(S^1,G)$$ be the loop group of a compact Lie group $$G$$. I should add some adjectives to $$G$$, but for sake of simplicity let's just take $$G=SU(2)$$.

There is a central extension $$1\to S^1\to\widetilde{LG}\to LG\to 1$$ (these are classified by "level" in $$H^3(G)$$, but we may as well restrict attention to the "universal" such extension corresponding to a generator of this group). The constructions of this central extension that I have found so far (e.g. the one in Pressley--Segal) all go via first defining a closed $$2$$-form on $$LG$$, arguing it defines a unique $$S^1$$-bundle, and then putting a group structure on this bundle.

Is there a more intrinsic definition of $$\widetilde{LG}$$?

In other words, given a loop $$\gamma:S^1\to G$$, I would like to have an intrinsically defined principal $$S^1$$ homogeneous space (or, equivalently, a $$1$$-dimensional complex vector space).

For example, here is an answer "up to homotopy". Since $$\pi_1(G)=\pi_2(G)=0$$ and $$\pi_3(G)=\mathbb Z$$, given any loop $$\gamma:S^1\to G$$, the space of extensions $$\bar\gamma:D^2\to G$$ (i.e. $$\bar\gamma|_{\partial D^2=S^1}=\gamma$$) is homotopy equivalent to $$\Omega^2G$$ which is connected with fundamental group $$\mathbb Z$$. If we take the $$1$$-truncation of this space (add cells to kill all higher homotopy groups), we get $$S^1$$ (up to homotopy).

This gives an "intrinsically defined" space homotopy equivalent to $$S^1$$ defined in terms of a given loop $$\gamma:S^1\to G$$ (although it's not very explicit, and has questionable meaning/use). What about an honest one-dimensional complex vector space? (with a natural meaning). Even better, can we define intrinsically a holomorphic line bundle over the complexified loop group $$LG_{\mathbb C}$$?

• It sounds like you are looking for the determinant line bundle on the affine Grassmannian $LG/LG_+$ or its pullback to the loop group? A related question is mathoverflow.net/questions/24845/… , and the latest word (maybe) is arxiv.org/abs/1804.02567 – David Ben-Zvi Oct 9 '18 at 19:04
• or if you prefer a less algebraic version, the determinant bundle on the Grassmannian is the pullback from the line bundle defining the Plucker embedding of the Segal-Wilson Grassmannian, i.e. the construction of semi-infinite wedge powers – David Ben-Zvi Oct 9 '18 at 19:09

Fix a cocycle $$\omega\in C^3(G, \mathbb R)$$ such that $$\omega(H_3(G, \mathbb Z)) = \mathbb Z$$. (For $$G = SU(2)$$, we can take $$\omega$$ to be the standard volume form.) Fix a loop $$L$$ in $$G$$, and let $$\partial^{-1}(L) \subset C_2(G, \mathbb Z)$$ denote the set of 2-chains whose boundary is $$L$$. For $$x,y \in \partial^{-1}(L)$$, define an equivalence relation $$x \sim y$$ if $$\omega(\partial^{-1}(x - y)) \in \mathbb Z$$. Then $$\partial^{-1}(L)/\sim$$ is an $$S^1$$ torsor.
The above paragraph defines an $$S^1$$-bundle over $$LG$$, but it doesn't tell you how to multiply two elements of this bundle. For this, we use the fact that $$\pi_2(G)$$ is trivial and redefine $$\partial^{-1}(L)$$ to be maps of $$D^2$$ into $$G$$ which restrict to $$L$$ on the boundary. We can multiply the maps $$D^2 \to G$$ pointwise. This group structure will extend to the quotient if we choose $$\omega$$ to be invariant under left and right multiplication by elements of $$G$$. (I have not thought about this last claim as carefully as I should, so be skeptical here.)
Assuming I did not make a stupid error in the previous paragraph (time constraints!), we can summarize as follows: Elements of $$\widetilde{LG}$$ are represented by maps to $$D^2$$ to $$G$$, and two such maps are considered equivalent if the $$\omega$$-volume they cobound is an integer.