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Central extensions $$ 0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0 $$ in which $G$ is a principal $\mathbb{R}$-bundle over $\mathbb{R}^2$ (I suppose you mean that by "topological") are classified by …

The string group $String(n)$ is by definition a 3-connected cover of $Spin(n)$. This definition determines the homotopy type of the string group. [In a previous version of this question I screwed up …

I think that for $G$ a connected compact Lie group, we have $$ H^m_{cont}(G,M) = 0 $$ for $m>0$. This follows from the van Est-isomorphism $$ H^m_{cont}(G,M) \cong H^m(g,k; M), $$ which holds for $G$ …

Here is a solution for $G$ a compact, simple, connected, simply-connected Lie group and $\Gamma$ a subgroup of the center of $G$. The group $H^3(G,Z)$ classifies $U(1)$-gerbes over $G$. A gerbe $\math …

There is a nice geometrical treatment of the Chern-Simons action functional for a general Lie group $G$ - not necessarily simply connected - that relates it to "holonomy". It also clarifies why Chern- …

Iterated integrals define a map $$ \sigma: C(\Omega(M)) \to \Omega(LM) $$ where $C(\Omega(M))$ is the cyclic bar complex of $\Omega(M)$. It has various nice properties; for instance, it induces an iso …

Yes, it is exact, and there is in fact a canonical 2-form on each conjugacy class whose derivative is your $\Omega$. This was an important observation when studying D-branes in WZW models, see, e.g. h …

There should be an answer to Theo's question in terms of universal connections, but I don't know it. This universal connection is a connection on the universal principal $G$-bundle over $BG$, such th …

Gawedzki and I have investigated this question for all compact simple Lie groups using the descent of multiplicative bundle gerbes from simply connected covers to quotients by subgroups of the center: …