nLab
RR field

Context

String theory

Differential cohomology

Contents

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 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

  • in type IIB supergarvity

    F RR=C 0+C 2+ F_{RR} = C_0 + C_2 + \cdots
  • in type IIA supergarvity

    F RR=C 1+C 3+ 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.

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
D2-brane\,\,\,
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,
D8-brane\,\,
(D=2n+1)(D = 2n+1)type IIB\,\,
D(-1)-brane\,\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
D5-brane\,\,\,
D7-brane\,\,\,
D9-brane\,\,\,
(p,q)-string\,\,\,
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
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
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

References

General

For a quick review see for instance

  • Daniel Freed, example 2.10 Dirac charge quantization and generalized differential cohomology (arXiv)

Self-duality and quadratic form

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)

Revised on May 19, 2014 05:46:46 by Urs Schreiber (109.144.239.103)