nLab RR field

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 – 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 IIA string theory

  $$F_{RR} = R_0 + R_2 + \cdots$$

• in type IIB string theory

  $$F_{RR} = R_1 + R_3 + \cdots$$

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 a suitable sense.

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

electric charge	magnetic charge
D0-brane	D6-brane
D1-brane	D5-brane
D2-brane	D4-brane
D3-brane	D3-brane

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.

Related concepts

• RR-field tadpole

• D-brane charge

• Hanany-Witten effect

• K-theory classification of D-brane charge

  • differential K-theory

  • KK-theory

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D = 2n)$	type IIA	$\,$	$\,$
D(-2)-brane	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
D8-brane	$\,$	$\,$
$(D = 2n+1)$	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-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G₂-manifold
topological M5-brane	$\,$	C6-field on G₂-manifold
S-brane
SM2-brane, membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

References

General

Early discussion in relation to D-branes:

• Joseph Polchinski, Dirichlet-Branes and Ramond-Ramond Charges, Phys. Rev. Lett. 75 (1995) 4724-4727 [arXiv:hep-th/9510017, doi:10.1103/PhysRevLett.75.4724]

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

• Daniel Freed, example 2.10 in: Dirac charge quantization and generalized differential cohomology_, Surveys in Differential Geometry, Int. Press, Somerville (2000) 129-194 [arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392]

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

• David Berenstein, Robert Leigh, Superstring Perturbation Theory and Ramond-Ramond Backgrounds, Phys. Rev. D 60 (1999) 106002 [arXiv:hep-th/9904104]

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

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032,2000 (arXiv:hep-th/9912279)

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, doi:10.1088/1126-6708/2000/05/032]

• Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 [arXiv:hep-th/0002027]

• Duiliu-Emanuel Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6 (2003) 1031-1134 [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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) [arXiv:0906.0795, slides]

See at orientifold for more on this. The relation to 11d Chern-Simons theory is made manifest in

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories [arXiv:hep-th/0611020]

Discussion of (Lagrangian densities for) D=10 type II supergravity with “duality-symmetric”/“democratic”/“pregeometric” for of the RR-fields:

• Gianguido Dall'Agata, Kurt Lechner, Mario Tonin, $D=10$, $N=IIB$ Supergravity: Lorentz-invariant actions and duality, JHEP 9807:017 (1998) $[$arXiv:hep-th/9806140, doi:10.1088/1126-6708/1998/07/017]

• Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Section 3 of: Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl. Phys. B 535(1998) 242-292 $[$arXiv:hep-th/9806106, doi:10.1016/S0550-3213(98)00552-5]

• Eric Bergshoeff, Renata Kallosh, Tomas Ortin, Diederik Roest, Antoine Van Proeyen, New Formulations of $D=10$ Supersymmetry and D8-O8 Domain Walls, Class. Quant. Grav. 18 (2001) 3359-3382 $[$arXiv:hep-th/0103233, doi:10.1088/0264-9381/18/17/303]

• Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, Various Faces of Type IIA Supergravity, Nucl. Phys. B 676 (2004) 189-228 $[$arXiv:hep-th/0307153, doi:10.1016/j.nuclphysb.2003.10.036]

• Karapet Mkrtchyan, Fridrich Valach: Democratic actions for type II supergravities, Phys. Rev. D 107 6 (2023) 066027 [doi:10.1103/PhysRevD.107.066027, arXiv:2207.00626]

• Jose Juan Fernandez-Melgarejo, Giacomo Giorgi, Carmen Gomez-Fayren, Tomás Ortín, Matteo Zatti: Democratic actions with scalar fields: symmetric sigma models, supergravity actions and the effective theory of the type IIB superstring, SciPost Phys. Core 7 4 (2024) 068 [arXiv:2401.00549, doi:10.21468/SciPostPhysCore.7.4.068, doi:10.21468/scipostphyscore.7.4.068]

Enhancement of the self-duality constraint on pregeometric RR-fields from (twisted) de Rham cohomology to (twisted) topological K-theory (under the hypothesized K-theory classification of D-brane charge) in terms of a quadratic form on differential K-theory:

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032 (2000) $[$arXiv:hep-th/9912279$]$

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031 (2000) $[$arXiv:hep-th/9912086, doi:10.1088/1126-6708/2000/05/032$]$

• Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 $[$arXiv:hep-th/0002027$]$

• Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry 7, Int. Press (2000) 129-194 $[$arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392$]$

An indication of a more refined discussion in twisted differential KR-theory:

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) $[$arXiv:0906.0795, slides$]$

See at orientifold for more on this.

Expressing the self-duality of pregeometric RR-fields in terms of 11d Chern-Simons theory:

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories $[$arXiv:hep-th/0611020$]$

Some review:

• Richard Szabo, section 3.6 and 4.6 of: Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology, ESI 2385 (2012) $[$arXiv:1209.2530, pdf$]$

Discussion in the context of flux quantization (here: D-brane charge quantization in K-theory):

• SS23, §3.3

Irrational RR-charge

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

• Constantin Bachas, Michael Douglas, Christoph Schweigert, around (2.8) of Flux Stabilization of D-branes, JHEP 0005:048, 2000 (arXiv:hep-th/0003037)

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

• Washington Taylor, D2-branes in B fields, JHEP 0007 (2000) 039 (arXiv:hep-th/0004141)

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 $C^{(1)}$ is constant. (from Zhou 01)

• Peter Rajan, D2-brane RR-charge on $SU(2)$, Phys.Lett. B533 (2002) 307-312 (arXiv:hep-th/0111245)

• Jian-Ge Zhou, D-branes in B Fields, Nucl.Phys. B607 (2001) 237-246 (arXiv:hep-th/0102178)

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

• Simon Burton, Hisham Sati, Urs Schreiber, Lift of fractional D-brane charge to equivariant Cohomotopy theory, J Geom Phys 161 (2021) 104034 (doi:10.1016/j.geomphys.2020.104034, arXiv:1812.09679)