Idea

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. 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

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

• in type IIA supergarvity

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

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.

References

The concept can be found as example 2.10 in

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv)

In the remainder of the article various further details of the differential cohomology of the RR field are discussed.

The subtle self-duality conditions on differential cocycles found in these string theoretic supergravity backgrounds was what led Hopkins and Singer to their foundational work on differential cohomology:

• Hopkins, Singer, Quadratic functions in geometry, topology,and M-theory (arXiv)