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I am trying to learn a bit about loop group representation theory to understand its role in string geometry.

Let $G$ be a Lie group. I am thinking of $\text{Spin}(n)$, so you may assume $G$ to be simple and simply connected and compact and...

By the loop group $LG$ of $G$, I mean the group of smooth loops in $G$.

Now, in Pressley-Segal's book "Loop Groups", they give a definition of the level of a positive energy representation in terms of pure representation theory. [page 177].

However, in the paper "What is an elliptic object?", the authors talk about a level as an element in $H^4(BG)$.[the article can be found at: http://people.mpim-bonn.mpg.de/teichner/Math/Papers.html]

Now, my question is the following: In which way do these two notions connect, or even coincide?

Thanks for any answer; even a reference where this is worked out would be great!

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    $\begingroup$ The relationship should be made in explicit in some of the Freed-Hopkins-Teleman papers. journals.tubitak.gov.tr/math/issues/mat-01-25-1/… gives an introduction to the ideas $\endgroup$ – Daniel Pomerleano Sep 17 '14 at 16:16
  • $\begingroup$ Is the paper you mention this one: books.google.com/… $\endgroup$ – Jim Humphreys Sep 18 '14 at 13:39
  • $\begingroup$ Indeed, the mentioned article gives an introduction to the ideas underlying the work of Freed-Hopkins-Teleman. $\endgroup$ – Lars Borutzky Sep 18 '14 at 13:55
  • $\begingroup$ Indeed, the linked article by Freed gives an introduction to the ideas underlying the work of Freed-Hopkins-Teleman. However, the article does not answer the question. As far, as I can tell, my question seems to be too elementary, such that it does not get addressed in the Freed-Hopkins-Teleman paper: arXiv:0711.1906 $\endgroup$ – Lars Borutzky Sep 18 '14 at 14:10
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Let's first recall the case of finite groups $G$. A projective representation of $G$ is a homomorphism $\rho : G \to PGL_n(\mathbb{C})$. In trying to lift this to a genuine representation $G \to GL_n(\mathbb{C})$ we find an obstruction given by a class $c \in H^2(BG, \mathbb{C}^{\times})$. An invariant way to describe this obstruction is that the short exact sequence

$$1 \to \mathbb{C}^{\times} \to GL_n(\mathbb{C}) \to PGL_n(\mathbb{C}) \to 1$$

gives rise to a coefficient exact sequence ending

$$\cdots \to H^1(BG, GL_n(\mathbb{C})) \to H^1(BG, PGL_n(\mathbb{C})) \to H^2(BG, \mathbb{C}^{\times})$$

(where every group involved has the discrete topology). The class $c$ can be thought of as the level of the projective representation $\rho$.

If $G$ is perfect, meaning that $G/[G, G] \cong H_1(BG, \mathbb{Z})$ vanishes, then universal coefficients gives an identification

$$H^2(BG, \mathbb{C}^{\times}) \cong \text{Hom}(H_2(BG, \mathbb{Z}), \mathbb{C}^{\times})$$

and we can equivalently think about levels as follows: $G$ admits a universal central extension

$$1 \to H_2(BG, \mathbb{Z}) \to \widetilde{G} \to G \to 1$$

and we can identify projective representations of $G$ of level $c$ with ordinary representations of $\widetilde{G}$ where the central $H_2(BG, \mathbb{Z})$ acts by the character in $\text{Hom}(H_2(BG, \mathbb{Z}), \mathbb{C}^{\times})$ corresponding to $c$. In particular, by Schur's lemma, every irreducible representation of $\widetilde{G}$ has a well-defined level.

A similar but more complicated thing is happening in the case of loop groups, although I can't claim to know the details. Here is a guess at the details. With suitable hypotheses on a Lie group $G$, the loop group $LG$ should admit a universal central extension

$$1 \to S^1 \to \widetilde{LG} \to LG \to 1$$

and we should be able to think of projective representations of $LG$ in terms of representations of $\widetilde{LG}$ (I am ignoring the energy circle here) where the central $S^1$ acts by a fixed character; moreover, these characters should be identified with levels in a cohomological sense as above.

The identification should go something like this: given a level $k \in H^4(BG, \mathbb{Z})$ we can think of it as a class in $H^3(BG, \mathbb{C}^{\times})$ (where $\mathbb{C}^{\times}$ now has the usual topology) and then transgress it to a class in $H^2(LBG, \mathbb{C}^{\times})$. The group I actually wanted to land in is $H^2(BLG, \mathbb{C}^{\times})$, which at least looks like it has something to do with projective representations of $LG$, but in fact

$$\Omega LBG \cong L \Omega BG \cong LG$$

so $LBG \cong BLG$ as long as $LBG$ is connected, which should be ensured by $G$ being connected.

If $G$ is compact, simple, connected, and simply connected, then it's well-known that $\pi_2(G) \cong \pi_3(BG) \cong H_3(BG, \mathbb{Z})$ vanishes and that $\pi_3(G) \cong \pi_4(BG) \cong H_4(BG, \mathbb{Z}) \cong \mathbb{Z}$. Universal coefficients now gives an identification

$$H^4(BG, \mathbb{Z}) \cong \text{Hom}(H_4(BG, \mathbb{Z}), \mathbb{Z}) \cong \mathbb{Z}$$

which gets identified above with characters of the central $S^1$, although I'm not sure how; there should be some nice functorial description of this $S^1$ but I'm not sure what it is.

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