Ramond–Ramond field

As Joseph Polchinski argued in 1995, D-branes are the charged objects that act as sources for these fields, according to the rules of p-form electrodynamics.

It has been conjectured that quantum RR fields are not differential forms, but instead are classified by twisted K-theory.

The adjective "Ramond–Ramond" reflects the fact that in the RNS formalism, these fields appear in the Ramond–Ramond sector in which all vector fermions are periodic.

The field strength is, as usual defined to be the exterior derivative of the potential Gp+1 = dCp.

Unlike the case of electromagnetism, in the presence of a nontrivial Neveu–Schwarz 3-form field strength the field strength defined above is no longer gauge invariant and so also needs to be defined patchwise with the Dirac string off of a given patch interpreted itself as a D-brane.

This extra complication is responsible for some of the more interesting phenomena in string theory, such as the Hanany–Witten transition.

In the quantum theory Joseph Polchinski has shown that G0 is an integer, which jumps by one as one crosses a D8-brane.

It is often convenient to use the democratic formulation of type II string theories, which was introduced by Paul Townsend in p-Brane Democracy.

In D-brane Wess-Zumino Actions, T-duality and the Cosmological Constant Michael Green, Chris Hull and Paul Townsend constructed the field strengths and found the gauge transformations that leave them invariant.

Similarly to attempts to simultaneously include both electric and magnetic potentials in electromagnetism, the dual gauge potentials may not be added to the democratically formulated Lagrangian in a way that maintains the manifest locality of the theory.

Mathematically, in the case in which H vanishes, the resulting structure is the Deligne cohomology of the spacetime.

For nontrivial H, after including the Dirac quantization condition, it has been conjectured to correspond instead to differential K-theory.

They are also "twisted-quantized" in the sense that one can transform back to the original field strength whose integrals over compact cycles are quantized.

It is the original field strengths that are sourced by D-brane charge, in the sense that the integral of the original p-form field strength Gp over any contractible p-cycle is equal to the D(8-p)-brane charge linked by that cycle.

The usual convention in the string theory literature appears to be to not write this term explicitly in the action.

If instead one considers this Bianchi identity to be a field equation for Cp+1, then one says that the Dp-brane is electrically charged under the (p + 1)-form Cp+1.

The above equation of motion implies that there are two ways to derive the Dp-brane charge from the ambient fluxes.

The source free equations of motion for the improved field strengths F imply that the formal sum of all of the Fp's is an element of the H-twisted de Rham cohomology.

This is a version of De Rham cohomology in which the differential is not the exterior derivative d, but instead (d+H) where H is the Neveu-Schwarz 3-form.

The authors have shown that twisted Chern characters are always elements of the H-twisted de Rham cohomology.

These correspond to the cohomology classes in the Atiyah Hirzebruch Spectral Sequence construction of twisted K-theory, which are only defined up to terms which are closed under any of a series of differential operators.

The other equations of motion, such as those obtained by varying the NS B-field, do not have K-theory interpretations.