Why RR cohomology is important in string theory?

Question

I want to know the RR cohomology in string theory or topological field theory in detail. (RR stands for Ramond Ramond). In following papers they compute the nilpotency of differential operator for RR fields. I know for example, from BRST symmetry, cohomology is important (in some sense?).

I want to know why some people compute RR cohomology [what is important in computing RR cohomology?, what is physics behind RR cohomology? ..]

Followings are some papers related with RR cohomology on the arxiv.

On the Picture Dependence of Ramond-Ramond Cohomology, K-Theory, D-Branes and Ramond-Ramond Fields

First of all i know the terminology cohomology, homology from my topology classes. Also i am somehow familiar with computing nilpotency of differential operators. But during the computation i came up with why RR cohomology is important in string theory, so i make a question here.

This post imported from StackExchange Physics at 2016-05-14 17:13 (UTC), posted by SE-user phy_math

Answer 1

RR fields in Type II string theory are background fields described by differential forms. The (locally defined) potentials, say C, are not uniquely defined. One can always shift by a gauge transformation, that is, by an exact form. C ~ C+dB.

The field strength of these potentials (that is, the curvature,dC, of the local potentials) can be patched together into a globally defined, closed differential form, F. One can ask if you can patch together the potentials into a globally defined differential form as well. If you can, then F = dC globally, and so F is an exact form, and is therefore a representative of the trivial de Rham cohomology class. 

Generally speaking, you can't patch together the potentials nicely, and so F is a closed form which is not (globally) exact. It represents a non-trivial class in the de Rham cohomology.

If this field has an (electric) source, then it satisfies

\(\begin{align} dF &= 0 \\ d \star F &= j \end{align}\)

where j is the current of the C field, and the star is the hodge star. The current j is Poincare dual to a compact oriented submanifold, which has the natural interpretation of the worldvolume of an object which acts as the source of C. The homology class of this submanifold (or equivalently, the counterpart of this homology class in de Rham COhomology) is identified with the charge of the extended object.

It follows that the RR charge associated to an object depends on the topology of the object via the cohomology/homology. 

(Note, it has been argued that cohomology is not an adequate description of RR charge, since it doesn't take into account the fact that a D-brane and an anti D-brane may annihilate to the vacuum. A more appropriate description is given by the K-theory, or in the presence of a non-trivial Neveu Schwarz background field, the twisted K-theory).