### Universal Gerbes

#### Posted by Urs Schreiber

Jim Stasheff asked me to forward the following question to the $n$-Café audience:

There is a universal principal bundle for any group. Is there also a universal

gerbe?

I don’t know much about the answer. The only thing I am aware of is a little trick:

$U(1)$-gerbes on $X$ are classified (depending on your taste almost by definition) by $H^3(X,\mathbb{Z})$, i.e. by homotopy classes of maps from $X$ into an Eilenberg-MacLane space $K(\mathbb{Z},3)$.

But since the group $PU(H)$ of projective unitary operators on a separable Hilbert space $H$ is a $K(\mathbb{Z},2)$, this is also the classification of $PU(H)$-bundles.

For that reason, many people who find need of $U(1)$-gerbes in their daily work tend to resort to working with $PU(H)$-bundles.

Any $PU(H)$-bundle implcitily defines a $U(1)$-gerbe: its *lifting gerbe*.

Constructively, given a $PU(H)$-valued transition function describing a locally trivialized $PU(H)$-bundle, we may replace the function’s values $g_{ij}(x)$ everywhere by a chosen lift $\hat g_{ij}(x)$ in the projection

coming from the central extension

i.e. such that

In general, the resulting $\hat g_{ij}(x)$ will fail to satisfy the cocycle condition

but the failure is measured by a function $f$

Applying $p$ to this equation shows that $f$ takes values in the kernel of $p$, hence in $U(1)$.

While there may be no way to find a lift such that this $f$ vanishes, the $f$s will *always* satisfy a cocycle condition of their own, on quadruple overlaps.

This 2-cocycle charcterizes (the local trivialization of) a $U(1)$-gerbe. This is called the **lifting gerbe** associated to our original bundle. Its non-triviality measures the impossibility of lifting the structure group of our bundle from $PU(H)$ to $U(H)$.

For every $U(1)$-gerbe there is a $PU(H)$-bundle whose lifting to a $U(H)$-bundle is obstructed by that gerbe; $U(1)$-gerbes and $PU(H)$-bundles have the same classification.

So, given all that, we could pull a trick and declare that

*The universal $U(1)$-gerbe is the lifting gerbe of the universal $PU(H)$-bundle.*

That’s, at least, essentially the point of view adopted in

Alan L. Carey, Jouko Mickelsson
*The universal gerbe, Dixmier-Douady class, and gauge theory*

hep-th/0107207 .

For technical reasons, these authors in this article don’t quite use $U(H) \to PU(H)$, but something very similar.

## Re: Universal Gerbes

The answer is almost certainly “yes”. But what was the question? I forget what you mean by the word “gerbe” Urs. I tend to mean whatever geometric gadget is convenient for the job in hand and which is classified by the third integral cohomology group (or the appropriate Deligne cohomology group if I want a connection).