## May 23, 2006

### Bunke on H, Part I

#### Posted by Urs Schreiber

Regrettably, I did not find the time, yet, to write more about a couple of interesting talks we heard in Vienna ($\to$).

Among the unwritten reports is one on Simon Willerton’s talk ($\to$) as well as one on U. Bunke’s talk.

Fortunately, though, Ulrich Bunke visits Hamburg today in order to give the same (I gather) talk again. So I get a second chance.

Since in Vienna Ulrich Bunke filled the board with plenty of high-powered notation, I have used this second chance for taking a look at his (unpublished) notes

U. Bunke, Th. Schick & M. Spitzweck
Sheaf theory for stacks in manifolds and twisted cohomology for ${S}^{1}$-gerbes
(dvi),

which are available on his website ($\to$).

This work, as well as several other papers by U. Bunke on T-duality, revolves around the clean mathematical understanding of string theoretic phenomena in the presence of Kalb-Ramond flux, i.e. in the presence of a (nontrivial) $U\left(1\right)$-gerbe that the string couples to.

With such a nontrivial background structure present, many familiar objects living on target space $X$ receive a “twist” encoded by the class in

(1)${H}^{3}\left(X,ℤ\right)$

that classifies the gerbe, or rather by the 3-form

(2)$H\in {H}^{3}\left(X,ℝ\right)$

which represents this class in deRham cohomology and which is nothing but the 3-form field strength of the Kalb-Ramond field - known otherwise as the $H$-flux.

In the presence of nontrivial $H$-flux, D-branes, which usually support $\mathrm{SU}\left(N\right)$-bundles over them, now instead support twisted $\mathrm{SU}\left(N\right)$-bundles. That’s because these D-branes really constitute gerbe modules ($\to$) for the gerbe given by $H$.

This twisting manifests itself in different incarnations in different places.

Naturally, like ordinary bundles and hence ordinary D-branes (in a background with vanishing $H$-flux) are classified by K-theory, these twisted bundles are classified by some twisted K-theory.

Similarly, where one usually expects to encounter ordinary deRham cohomology $\left({\Omega }^{•}\left(X\right),d\right)$, one now finds $H$-twisted deRham cohomology

(3)$\left({\Omega }^{•}\left(X\right),\phantom{\rule{thinmathspace}{0ex}}{d}_{H}:=d+H\wedge \right)\phantom{\rule{thinmathspace}{0ex}},$

where the ordinary deRham differential is accompanied by the operator of exterior multiplication with $H$.

While this may sound obvious, things are really a little more subtle. There are a couple of dualities in the game, most prominently T-duality and S-duality, which we know all this twisted business should be compatible with. This forces one to find not the most naive, but the most natural way to describe all these things - a way that most cleanly expresses the true nature of the objects under considerations which then, usually, is also the one most easily seen to be compatible with the symmetries (dualities, here) of the setup.

There are a bunch of people thinking about twisted K-theory and its invariance (or rather non-invariance) under S-duality. I had recently mentioned Jarah Evlins’s ideas on this ($\to$).

The work by Ulrich Bunke, that I want to say a little about here, on the other hand, concentrates on T-duality and, particularly, twisted deRham cohomology.

More precisely, the topic of his latest talk is about the following:

The definition of twisted deRham cohomology given above is nice and simple, but fails to satisfy some desireable naturality conditions. More precisely this means that it does not transform functorially under transformations of the gerbe that defines the twist.

This indicates that looking at ${d}_{H}=d+H\wedge$ fails to realize the crucial nature of twisted deRham cohomology.

The main point of the above cited work is a proposal for what should be the “true” definition of twisted deRham cohomology.

The idea is actually quite elegant and simple.

We can realize ordinary deRham cohomology ${H}^{•}\left(X,ℝ\right)$ using sheaves on $X$. Now, a gerbe (a stack) $G$ on $X$ may be regarded as nothing but a certain generalized space ($\to$). If we knew how to define sheaves on these generalized spaces, then we might be able to similarly define a cohomology for these generalized spaces. With a little luck, the cohomology of $G$ is then essentially the same thing as the twisted cohomology on $X$, with the twist given by $G$.

Apparently, nobody had considered sheaf theory for stacks this way before. Bunke, Schick & Spitzweg work it out and then demonstrate that the above guess is actually correct.

The point is, that the cohomology of a gerbe $G$ obtained this way is functorial (hence “natural”) and is isomorphic to the simple-minded twisted cohomology - but not in a canonical way.

Ordinary twisted deRham cohomology is thus shown to be a somewhat unnatural point of view on the true entity we are interested in.

To be continued…

Posted at May 23, 2006 12:50 PM UTC

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