# Geometrizing the Third Cohomology of a Complex Lie Group

If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\mathcal{O}_{G_\mathbb{C}}^*$-banded gerbe on $G_\mathbb{C}$, essentially corresponding to a generator of $H^3(G_\mathbb{C}, \mathbb{Z})=\mathbb{Z}$; the gerbe comes with some extra "connective" structure. However, he cryptically says on p. 227 of my edition that

an explicit construction of [this gerbe] would involve, in one way or another, algebraic $K$-theory.

He goes on to give an explicit (ad hoc) construction in the case of $SL(2, \mathbb{C})$, but says no more about the algebraic $K$-theory construction. I can't figure out the relationship to algebraic $K$-theory at all, so my question is:

How does one construct this gerbe (with connective structure) via algebraic $K$-theory? I'd also be happy with other constructions, even if they are special to the case of $SL(n, \mathbb{C})$, say.

My interest in this question comes from the fact that it gives a (reasonably) concrete geometric interpretation of the $2$nd Chern class of a principal $G_\mathbb{C}$-bundle (say, $G_\mathbb{C}=SL(n, \mathbb{C})$) as follows: the second Chern class of $P$ may be viewed as the obstruction to building a gerbe with connective structure on the total space of $P$ which is fiberwise equivalent to the canonical gerbe on $G_\mathbb{C}$. Actually, this even gives a geometric interpretation of the "refined" $2$nd Chern class valued in the Deligne cohomology of the base.

• See the paper by Brylinski and Deligne available here math.ias.edu/people/faculty/deligne/preprints and the paper by Deligne on central extensions referred to therein. – SGP Jan 24 '13 at 0:35
• This gerbe is basically the string 2-group for $G_\mathbb{C}$, of which there are a number of constructions - but they may be specific to compact Lie groups. Note that $p_1/2$ (which I believe is $c_2$) is described elsewhere as the obstruction to the existence of a string structure for a principal Spin-bundle (and versions of this for other simply connected semisimple compact Lie groups). – David Roberts Jan 24 '13 at 1:21
• Is this answer by David Ben-Zvi useful? mathoverflow.net/questions/24845/… – André Henriques Jan 24 '13 at 2:09
• @SGP, Andre Henriques: Perhaps I'm being dull--but I don't see where the gerbe in question is coming from in these references. – Daniel Litt Jan 24 '13 at 2:30
• As the other comments mention, the gerbe comes from the second Chern class in H^4(BG,Z) which is in fact isomorphic to algebraic cycle classes on BG, which is H^2(BG,K_2). Analytically H^4(BG,Z) is the same as H^3(BG,U(1)), i.e. central extensions of G by BU(1), which are equivariant gerbes on G (the generator for G simple simply connected corresponds to the string group). However this class is algebraically not that of an O^* gerbe, which would live in H^3(BG, O^*), though maybe one can understand this difference by thinking about gerbes with connective structure. – David Ben-Zvi Jan 24 '13 at 16:16