The n-Category Café

Thanks! Fixed.

Since that was a rather boring comment,

Much of the process of finalizing an article is boring, unfortunately. But since it has to be done, I am grateful to you for bothering to look for typos! We fixed a bunch more now. Many probably still remain for the moment.

is this the point at which the hard-nosed string theorist takes note?

It’s a step in the right direction, I would think. Your question made me add a sentence to that “essay”.

More could be said here. Let me tell you what I am currently intrigued by:

there have been, roughly three major attempts for formalizing, mathematically, what higher supergravity is all about. I am thinking of

1) exceptional generalized geometry,

2) twisted generalized differential cohomology,

3) D’Auria-Fré theory.

We know now that all three find their natural home in nonabelian differential cohomology (section 4.3, section 4.4).

We currently understand some overlap between the three, but it seems clear that there is a natural unification of all three in a single natural cohesive structure.

There are also the Wu characteristic classes mod $p \gt 2$. Do they show up detecting higher torsion in your setting?

Let’s see:

The appearance of ordinary Wu classes originates here in the quadratic refinement of the intersection pairing on integral/differential cohomology, notably in higher dimensional Chern-Simons theory action functionals of the form

$$\exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma \hat G \hat \cup \hat G)$$

as they appear for instance in 7d supergravity dual to the M5-brane.

However, the full M-theory Chern-Simons term involves, in fact, the diagonal of a trilinear intersection pairing

$$\exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma \hat G \hat \cup \hat G \hat \cup \hat G)$$

(or rather “secondary” pairing, since here $dim \Sigma = 11$ with $deg \hat G = 4$).

and hence is asking for a cubic refinement. Generally one can consider higher dimensional cup product Chern-Simons functionals

$$\exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma (\hat G)^{\hat \cup n})$$

for all $n \in \mathbb{N}$. This will be asking for $n$-fold refinements.

Will that involve Wu classes with coefficients in $\mathbb{Z}_p$ for $p \gt 2$? I haven’t thought about this question yet. Will need to look up something on these Wu classes with other coefficients…