The String Coffee Table

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.
$pdf$

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 $H^4(M,\mathbb{R})$ of the) obstruction to having a $\mathrm{String}(n)$-structure on target space $M$ ($\to$, $\to$) and how this can be understood ($\to$) as the obstruction to lifting a $\mathrm{Spin}(n)$ bundle on $M$ to a gerbe or 2-bundle on $M$, whose structure 2-group is $(\widehat{\Omega\mathrm{Spin}(n)}\to P\mathrm{Spin}(n))$ ($\to$, $\to$).

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 $(p-1)$-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

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 ($\to$).