nLab RR field



String theory

Differential cohomology



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 XX 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

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 a suitable sense.

electric-magnetic duality of D-branes/RR-fields in type II string theory:

electric chargemagnetic charge

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).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d



Early discussion in relation to D-branes:

Review in the context of the K-theory classification of D-brane charge:

Discussion of the NSR string perturbation theory in RR-field background:

Self-duality and quadratic form

self-duality for pregeometric RR-fields – references

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

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

  • Richard Szabo, section 3.6 and 4.6 of: Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology [arXiv:1209.2530]

Discussion of Lagrangian densities for type II supergravity making the nature of the pregeometric RR-fields and their self-duality manifest:

Irrational RR-charge

An argument that RR-charge may occur in irrational ratios is due to

In a sequence of followup articles, authors found this problematic and tried to make sense of it:

In this article it was argued that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by the bulk contributions, but in this calculation it was implicitly assumed that the gauge field C (1)C^{(1)} is constant. (from Zhou 01)

Observation that this paradox is resolved under Hypothesis H, at least for fractional D-branes:

Last revised on June 24, 2023 at 09:57:09. See the history of this page for a list of all contributions to it.