Sati: M-theory and characteristic classes

[Urs Schreiber]

Recently we had some discussion about if nonabelian gerbes are the right
language to talk about M5-branes:

http://groups.google.de/groups?selm=34517tF461o32U1-100000%40individual.net

The argument was that the coupling of the M-theory 3-form C_3 to the M2
cannot be anything but the holonomy of an abelian 2-gerbe. By an argument
completely analogous to how in one dimension lower the existence of
nonabelian bundles (0-gerbes) on D-branes follows from the gerbe description
of the coupling of the KR 2-form B to the string, it follows that the
boundary of an M2 inside an M5 should couple to nonabelian 1-gerbe holonomy.

In a recent paper

Hisham Sati:
M-theory and characteristic classes,
hep-th/0501245

the author argues for the 2-gerbe description of C_3 and embeds the
discussion into a larger framework of "M-theory characteristic classes".

He ends by saying:

>>>
For K-theory, the objects corresponding to the two-form curvature F_2 are
vector bundles. What are the corresponding M-objects, i.e. the ones related
to G_4? Not surprisingly, we propose that they would be "2-objects", e.g.
2-gerbes, 2-vector bundles, ect... This will be discussed seperately.

What is the theory that we are looking for? From the general structure of
the character, from the mod 24 congruence of the string class, and from
previous work [...], we expect such a theory to be some form/refinement of
elliptic cohomology (e.g. related to the theory of topological modular
forms).
<<<

Regarding the part

> "2-objects", e.g. 2-gerbes, 2-vector bundles

of course I cannot resist to mention that 1-gerbes and 2-bundles are to a
good degree the same, as demonstrated in hep-th/0412325 in terms of cocycle
data. (There are however quite some subtleties in making this "to a good
degree" into a true equivalence of categories.) The rough idea is that a
2-bundle with certain base 2-space has a 1-gerbe of 2-sections.

The relation to elliptic cohomology and the entire stringy/categorified
setup leads me to expect that in order to get characteristic classes here in
the general case we will need to consider 2-maps from a loop/path space X
into a classifying 2-space BG. But that's just speculation so far.