Mathematically the RR field on a space $X$ is a cocycle in differential K-theory. 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

$F_{RR} = C_0 + C_2 + \cdots$

in type IIA supergarvity

$F_{RR} = C_1 + C_3 + \cdots$

The components of this are sometimes called the RR forms.

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.