The RR field or Ramond–Ramond field is a gauge field appearing in 10-dimensional type II supergravity.
Mathematically the RR field on a space $X$ is a cocycle in differential K-theory – or rather, in full generality, in twisted differential KR-theory subject to a self-dual higher gauge field constrained encoded by a quadratic form defining an 11-dimensional Chern-Simons theory on twisted differential KR cocycles.
Accordingly, the field strength of the RR field, i.e. the image of the differential K-cocycle in deRham cohomology, is an inhomogeneous even or odd differential form
in type IIB supergarvity
in type IIA supergarvity
The components of this are sometimes called the RR forms.
In the presence of a nontrivial Kalb–Ramond field the RR field is twisted: a cocycle in the corresponding twisted K-theory.
Moreover, the RR field is constrained to be a self-dual differential K-cocycle in some sense.
The RR field derives its name from the way it shows up when the supergravity theory in question is derived as an effective background theory in string theory. From the sigma-model perspective of the string the RR field is the condensate of fermionic 0-mode excitations of the type II superstring for a particular choice of boundary conditons called the Ramond boundary condititions. Since these boundary conditions have to be chosen for two spinor components, the name appears twice.
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
For a quick review see for instance
The self-dual higher gauge field nature (see there for more) in terms of a quadratic form on differential K-theory is discussed originally around
and (Freed 00) for type I superstring theory, and for type II superstring theory in
Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031,2000 (arXiv:hep-th/9912086)
Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 (arXiv:hep-th/0002027)
D. Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (arXiv:hep-th/0005090), summarised in A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)
with more refined discussion in twisted differential KR-theory in
See at orientifold for more on this. The relation to 11d Chern-Simons theory is made manifest in
Review is in