## April 19, 2006

### Jurco on Gerbes and Stringy Applications

#### Posted by Urs Schreiber

In Vietri ($\to$) Branislav Jurčo gave a talk on

B. Jurčo
Nonabelian Gerbes, Differential Geometry and Stringy Applications.
$\mathrm{pdf}$

The slides for the talk have kindly been made available now.

Among other things, the talk recalls Killinback’s old result on how the Green-Schwarz anomaly can be understood as the (image in ${H}^{4}\left(M,ℝ\right)$ of the) obstruction to having a $\mathrm{String}\left(n\right)$-structure on target space $M$ ($\to$, $\to$) and how this can be understood ($\to$) as the obstruction to lifting a $\mathrm{Spin}\left(n\right)$ bundle on $M$ to a gerbe or 2-bundle on $M$, whose structure 2-group is $\left(\stackrel{̂}{\Omega \mathrm{Spin}\left(n\right)}\to P\mathrm{Spin}\left(n\right)\right)$ ($\to$, $\to$).

Closely related to that is the idea, promoted in

Paolo Aschieri, Branislav Jurco
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
hep-th/0409200,

on how the Diaconescu-Freed-Witten anomaly indicates that M5-branes support modules for abelian 2-gerbes - i.e. twisted 1-gerbes.

The reasoning here is precisely analogous, just one dimension higher, to how the Freed-Witten anomaly gives rise to D-branes supporting modules for abelian 1-gerbes - i.e. twisted bundles.

One day I should say more about modules for gerbes here. The concept was formally introduced in

P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, D. Stevenson
Twisted K-theory and K-theory of bundle gerbes
hep-th/0106194.

There is a nice description of the idea in terms of morphism of transport functors. Let $\mathrm{tra}$ be a transport $p$-functor that encodes parallel transport in a $\left(p-1\right)$-gerbe over a $p$-dimensional volume. There is an obvious notion for what it means for such a functor to be trivial (in the sense of how a bundle can be trivial). Let ${\mathrm{tra}}_{0}$ be a trivial $p$-transport. Now, a trivialization of $\mathrm{transport}$ (if it exists) is nothing but a choice of isomorphism

(1)$\mathrm{tra}\to {\mathrm{tra}}_{0}\phantom{\rule{thinmathspace}{0ex}}.$

In general, of course, no such isomorphism will exist. It may exist however if we enlarge the ambient category sufficiently. If that is the case, we call the triviaization a module for $\mathrm{tra}$.

For instance a module for an abelian bundle gerbe as defined in the above paper is essentially nothing but a trivialization of that bundle gerbe (a “stable isomorphism to the trivial bundle gerbe”), but in the (2-)category of nonabelian bundle gerbes instead of in the (2-)category of abelian bundle gerbes.

There is a general pattern at work here, which says that

Branes are modules ($\to$).

Posted at April 19, 2006 10:55 AM UTC

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