The String Coffee Table

RR-Forms and Algebroids

Posted by Urs Schreiber

This morning we had a talk ($\to$) by Pietro Fré on issues of M-theory compactifications using the tool of “free differential algebras”. Before I say something about this talk, consider the following general question.

What on earth IS an RR-form, really ($\to$)?

It’s not quite something living in twisted K-theory ($\to$). Jarah Evslin today kindly tried to enlighten me a little about the latest ideas, but I guess I am being dense. His talk tomorrow ($\to$) might help.

Somehow, in the end, something like the following should be true, at least to my mind: over eleven dimensional spacetime there should be nothing but a plain 2-gerbe with connection, which is locally given by the supergravity 3-form. Upon compactifying down to IIA or IIB, that 2-gerbe with connection should decompose into the Kalb-Ramond 1-gerbe with connection and some other strucure in which the RR fields live.

It sounds like a straightforward task to examine the structure obtained by “compactifying a $p$-gerbe with connection” this way, which should completely resolve the issue of the nature of the RR-forms (or shouldn’t it?). But I don’t see that anyone has tried to do this.

(However the approach by Sati and Kriz, trying to identify elliptic cohomology as the correct refinement of K-theory should be closely related

Igor Kriz, Hisham Sati
Type IIB String Theory, S-Duality, and Generalized Cohomology
hep-th/0410293.

After all, it is expected on general grounds that ellitptic cohomology is to 2-gerbes as K-theory is to 1-gerbes ($\to$)).

But there are some tantalizing hints coming from the description of the field content of supergravity in terms of (free) (graded) differential algebras.

Leonardo Castellani, Alberto Perotto
Free Differential Algebras: Their Use in Field Theory and Dual Formulation
hep-th/9509031.

Recall ($\to$) that a graded differential algebra concentrated in the $p$ lowest degrees is nothing but a $p$-term $L_\infty$ algebra (-algebroid), which in turn is nothing but a semistrict Lie $p$-algebra. There is a way to encode connections on $p$-gerbes in terms of morphisms of such dg-algebras ($\to$, section 14), which is governed by the double complex obtained from deRham and the dg-algebra itself.

Now, precisely these structures have been found useful in organizing the field content of supergravity theories. This goes back to old seminal work by Fré and Auria on supergravity, and has recently again attracted increased attention, see for instance

A. Bandos, J.A. de Azcarraga, J.M. Izquierdo, M. Picon, O. Varela
On the underlying gauge group structure of D=11 supergravity
hep-th/0406020

Or see the recent article that Pietro Fré talked about today in Vietri sul Mare:

Pietro Fré
M-theory FDA, Twisted Tori and Chevalley Cohomology
hep-th/0510068.

The various $p$-form fields appearing in supergravity theories can nicely be regarded as components of a free graded differential algebra, with the differential being induced from the various generalized Maurer-Cartan equations. Field strengths and gauge transformations follow precisely the same logic as in the dg-algebra description of gerbes with connection.

For some reason I had not payed attention to this fact before. I am grateful to B. Jurco for addressing this point. To me it suggests that the 2-gerbe interpretation of RR-fields might be simply obtained by staring at this SUGRA dg-algebra long enough, comparing it with the algebroid description of gerbe connections. As soon as I find a working printer and some time (this may not be very soon), I should take a closer look at this.